System, method, and computer program product for analyzing multivariate time series using convolutional Fourier networks

Abstract

A system for analyzing multivariate time series is provided, the system including at least one processor programmed or configured to receive a time series of historical data points, determine a historical time period, determine a contemporary time period, determine a first time series of data points associated with a historical transaction metric from the historical time period, determine a second time series of data points associated with a historical target transaction metric from the historical time period, determine a third time series of data points associated with a contemporary transaction metric from the contemporary time period, and generate a machine learning model, wherein the machine learning model is configured to provide an output including a predicted time series of data points associated with a contemporary target transaction metric. Methods and computer program products are also provided.

Description

[0001] Cross - Reference to Related Applications

[0002] This application claims the benefit of priority to U.S. Provisional Patent Application No. 63 / 192,259, filed on May 24, 2021, which is incorporated herein by reference in its entirety. Technical Field

[0003] The present disclosure generally relates to systems, methods, and computer program products for predicting metrics using machine - learning models, and in one particular embodiment, to systems, methods, and computer program products for predicting metrics using a machine - learning model including a convolutional Fourier network. Background Art

[0004] Machine learning can be a field of computer science that uses statistical techniques to provide a computer system with the ability to learn (e.g., gradually improve performance) tasks using data, without explicitly programming the computer system to perform the tasks. In some cases, a machine - learning model can be developed for a dataset such that the machine - learning model can perform tasks (e.g., tasks associated with prediction) regarding the dataset.

[0005] In some cases, machine - learning models, such as predictive machine - learning models, can be used to make predictions associated with risk or opportunity based on data. A predictive machine - learning model can be used to analyze the relationship between the performance of a unit and one or more known characteristics of the unit based on data associated with the unit. The purpose of a predictive machine - learning model can be to evaluate the likelihood that a similar unit will exhibit the performance of the unit. A predictive machine - learning model can be used as a fraud - detection model. For example, a predictive machine - learning model can perform calculations based on data associated with a payment transaction to evaluate the risk or opportunity of a payment transaction involving a customer in order to guide the decision of whether to authorize the payment transaction.

[0006] A multivariate time series can refer to a time series having more than one time - dependent variable. In some cases, in a multivariate time series, each time - dependent variable can depend not only on the past values of the time - dependent variable but also have some dependence on other time - dependent variables. This dependence can be used to predict future values of the time - dependent variables.

[0007] A payment processor can be an entity (e.g., a business entity such as an association, a company, etc.) that enables payment transactions to occur between different parties (e.g., a consumer and a merchant) with reliability, convenience, and security. The payment processor can process (e.g., clear and / or settle) payment transactions using a payment processing network. The payment processing network can include a network of communication devices configured to allow communication between parties (e.g., service providers, financial institutions, and / or account holders) such that payment transactions can occur.

[0008] Since an unexpected interruption of the payment processor's payment processing network can impair payment transactions enabled by the payment processor, the payment processor can use a predictive machine learning model to proactively monitor payment transactions (e.g., transaction data associated with the payment transactions) of a predictive system, and the predictive machine learning model can take a multivariate time series as an input. In some cases, if certain aspects of the transaction data (e.g., transaction volume and / or transaction amount) deviate from the normal situation (e.g., a predetermined level) in different geographical regions, an unexpected network interruption may be detected in a timely manner. In addition, suspicious activities, such as an automated teller machine (ATM) cash-out attack, can be detected based on determining that an abnormal transaction decline rate of the issuer system is occurring.

[0009] However, latency in the payment processing network can cause latency in the availability of transaction data used with the predictive machine learning model. For example, some transaction data can be provided hourly (e.g., via a process of aggregating transaction data), while other aspects of the transaction data may not be available for a longer period, such as for several days. If the predictive system may need to wait for all transactions to be ready, the predictive system may not keep up with the dynamics in real-time payment transactions. In addition, external factors such as economic, geopolitical, and / or pandemic factors can suddenly change the behavior of payment transactions and subsequently cause a sudden conceptual shift, which may reduce the quality of predictions made by the predictive system.

[0010] In addition, various transaction metrics may be used within the predictive system, and many of the applications mentioned above require the system to estimate time-varying patterns within the transaction metrics. However, only multivariate time series transaction data may not be sufficient for accurate predictions. It can be challenging for a predictive machine learning model to learn patterns from multiple incoming transaction metrics received simultaneously.

[0011] In addition, when analyzing a multivariate time series, such as a multivariate time series including aspects of data related to seasonality, due to the nature of the multivariate time series, predictive techniques based on the true values of time-dependent variables in the multivariate time series may be ineffective. SUMMARY OF THE INVENTION

[0012] Accordingly, systems, apparatuses, products, devices, and / or methods for predicting metrics using convolutional Fourier networks are disclosed, which overcome some or all of the deficiencies of the prior art.

[0013] Other embodiments or aspects are set forth in the numbered clauses below.

[0014] Clause 1: A system, comprising: at least one processor programmed or configured to: receive a time series of historical data points, wherein the historical data points include values of a plurality of transaction characteristics for a plurality of time intervals during an initial time period; determine a historical time period, wherein the historical time period includes a first time period of the initial time period; determine a contemporary time period, wherein the contemporary time period includes a second time period of the initial time period that is after the historical time period, wherein the historical time period is longer than the contemporary time period; determine a first time series of data points associated with a historical transaction metric from the historical time period; determine a second time series of data points associated with a historical target transaction metric from the historical time period; determine a third time series of data points associated with a contemporary transaction metric from the contemporary time period; generate a machine learning model, wherein the machine learning model is configured to provide an output including a predicted time series of data points associated with a contemporary target transaction metric, wherein when generating the machine learning model, the at least one processor is programmed or configured to: train the machine learning model based on the first time series of data points associated with the historical transaction metric, the second time series of data points associated with the historical target transaction metric, and the third time series of data points associated with the contemporary transaction metric; and generate the output of the machine learning model, wherein the output of the machine learning model includes the contemporary target transaction metric, and wherein the contemporary target transaction metric includes values of the target transaction metric during a target prediction period.

[0015] Clause 2: The system according to Clause 1, wherein when training the machine learning model, the at least one processor is programmed or configured to: provide a first time series of the data points associated with the historical transaction metric, a second time series of the data points associated with the historical target transaction metric, and a third time series of the data points associated with the contemporary transaction metric as inputs to a processing layer of the machine learning model, wherein the processing layer includes a fast Fourier transform (FFT) layer; provide an output of the processing layer as an input to a feature extraction component of the machine learning model; provide an output of the feature extraction component of the machine learning model as an input to a dual attention component of the machine learning model; and provide an output of the dual attention component of the machine learning model as an input to a learning and prediction component of the machine learning model.

[0016] Clause 3: The system according to Clause 1 or 2, wherein when providing the first time series of the data points associated with the historical transaction metric, the second time series of the data points associated with the historical target transaction metric, and the third time series of the data points associated with the contemporary transaction metric as inputs to the processing layer of the machine learning model, the at least one processor is programmed or configured to: provide each feature of a first plurality of features of the first time series of the data points associated with the historical transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the first plurality of features; provide each feature of a second plurality of features of the second time series of the data points associated with the historical target transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the second plurality of features; and provide each feature of a third plurality of features of the third time series of the data points associated with the contemporary transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the third plurality of features.

[0017] Clause 4: The system according to any one of Clauses 1 to 3, wherein when training the machine learning model, the at least one processor is programmed or configured to: combine the real parts of each of the first plurality of features to generate a combined real part of the first plurality of features; combine the imaginary parts of each of the first plurality of features to generate a combined imaginary part of the first plurality of features; combine the real parts of each of the second plurality of features to generate a combined real part of the second plurality of features; combine the imaginary parts of each of the second plurality of features to generate a combined imaginary part of the second plurality of features; combine the real parts of each of the third plurality of features to generate a combined real part of the third plurality of features; and combine the imaginary parts of each of the third plurality of features to generate a combined imaginary part of the third plurality of features; and wherein the output of the processing layer of the machine learning model includes the combined real part of the first plurality of features, the combined imaginary part of the first plurality of features, the combined real part of the second plurality of features, the combined imaginary part of the second plurality of features, the combined real part of the third plurality of features, and the combined imaginary part of the third plurality of features.

[0018] Clause 5: The system according to any one of Clauses 1 to 4, wherein when training the machine learning model, the at least one processor is programmed or configured to: provide the combined real part of the first plurality of features as an input to a one-dimensional feature extraction convolutional layer; provide the combined imaginary part of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the combined real part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the combined imaginary part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the combined real part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the combined imaginary part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; and generate the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer.

[0019] Clause 6: The system according to any one of Clauses 1 to 5, wherein the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part, where the real input part includes: a first real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features; a second real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features; a third real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features; and wherein the imaginary input part includes: a first imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features; a second imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features; a third imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features; and wherein when training the machine learning model, the at least one processor is programmed or configured to: generate a real component attention matrix based on the real input part, where the real component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features; and generate an imaginary component attention matrix based on the imaginary input part, where the imaginary component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features.

[0020] Clause 7: The system according to any one of clauses 1 to 6, wherein the at least one processor is further programmed or configured to: generate the real input portion of the input to the dual attention component of the machine learning model; and generate the imaginary input portion of the input to the dual attention component of the machine learning model; wherein when generating the real input portion of the input to the dual attention component of the machine learning model, the at least one processor is programmed or configured to: provide the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the first plurality of features as an input to the dropout layer to generate a real contemporary transaction metric feature vector as the first real input; provide the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input; and wherein when generating the imaginary input portion of the input to the dual attention component of the machine learning model, the at least one processor is programmed or configured to: provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the first plurality of features as an input to the dropout layer to generate an imaginary contemporary transaction metric feature vector as the first imaginary input; provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the second plurality of features as an input to the dropout layer to generate an imaginary historical transaction metric feature vector as the second imaginary input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the third plurality of features as an input to the dropout layer to generate an imaginary historical target transaction metric feature vector as the third imaginary input.

[0021] Clause 8: The system according to any one of Clauses 1 to 7, wherein when generating the real component attention matrix, the at least one processor is programmed or configured to: generate the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; provide the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a softmax layer to generate a real attention feature vector; and generate the dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein when generating the imaginary component attention matrix, the at least one processor is programmed or configured to: generate the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; provide the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a softmax layer to generate an imaginary attention feature vector; and generate the dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

[0022] Clause 9: The system according to any one of Clauses 1 to 8, wherein when generating the real component attention matrix, the at least one processor is programmed or configured to: perform an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a tanh activation function; provide the output of the tanh activation function as an input to a softmax layer to generate a real attention feature vector; and generate the dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein when generating the imaginary component attention matrix, the at least one processor is programmed or configured to: perform an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; provide the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a tanh activation function; provide the output of the tanh activation function as an input to a softmax layer to generate an imaginary attention feature vector; and generate the dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

[0023] Clause 10: The system according to any one of Clauses 1 to 9, wherein when providing the output of the dual attention component of the machine learning model as the input to the learning and prediction component of the machine learning model, the at least one processor is programmed or configured to: concatenate the real component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation; provide the first concatenation as the input to a feature learning one-dimensional convolutional layer; concatenate the imaginary component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation; provide the second concatenation as the input to the feature learning one-dimensional convolutional layer; provide the output of the feature learning one-dimensional convolutional layer for the first concatenation as the input to a reshape layer to generate a real component feature vector; and provide the output of the feature learning one-dimensional convolutional layer for the second concatenation as the input to the reshape layer to generate an imaginary component feature vector.

[0024] Clause 11: The system according to any one of Clauses 1 to 10, wherein when generating the output of the machine learning model, the at least one processor is programmed or configured to: generate a complex number feature vector based on the real component feature vector and the imaginary component feature vector; and provide the complex number feature vector as the input to an inverse FFT layer to generate the contemporary target transaction metric.

[0025] Clause 12: A method includes: receiving, by at least one processor, a time series of historical data points, where the historical data points include values of multiple transaction characteristics for multiple time intervals during an initial time period; determining, by at least one processor, a historical time period, where the historical time period includes a first time period of the initial time period; determining, by at least one processor, a contemporary time period, where the contemporary time period includes a second time period of the initial time period that is after the historical time period, and where the historical time period is longer than the contemporary time period; determining, by at least one processor, from the historical time period a first time series of data points associated with a historical transaction metric; determining, by at least one processor, from the historical time period a second time series of data points associated with a historical target transaction metric; determining, by at least one processor, from the contemporary time period a third time series of data points associated with a contemporary transaction metric; generating, by at least one processor, a machine learning model, where the machine learning model is configured to provide an output that includes a predicted time series of data points associated with a contemporary target transaction metric, and where generating the machine learning model includes: training the machine learning model based on the first time series of data points associated with the historical transaction metric, the second time series of data points associated with the historical target transaction metric, and the third time series of data points associated with the contemporary transaction metric; and generating an output of the machine learning model, where the output of the machine learning model includes the contemporary target transaction metric, and where the contemporary target transaction metric includes values of the target transaction metric during a target prediction period.

[0026] Clause 13: The method according to Clause 12, where training the machine learning model includes: providing the first time series of data points associated with the historical transaction metric, the second time series of data points associated with the historical target transaction metric, and the third time series of data points associated with the contemporary transaction metric as inputs to a processing layer of the machine learning model, where the processing layer includes a fast Fourier transform (FFT) layer; providing an output of the processing layer as an input to a feature extraction component of the machine learning model; providing an output of the feature extraction component of the machine learning model as an input to a dual attention component of the machine learning model; and providing an output of the dual attention component of the machine learning model as an input to a learning and prediction component of the machine learning model.

[0027] Clause 14: The method according to Clause 12 or 13, wherein providing the first time series of data points associated with the historical transaction metrics, the second time series of data points associated with the historical target transaction metrics, and the third time series of data points associated with the contemporary transaction metrics as inputs to the processing layer of the machine learning model includes: providing each feature of the first plurality of features of the first time series of data points associated with the historical transaction metrics as an input to the FFT layer to generate the real part and the imaginary part of each feature of the first plurality of features; providing each feature of the second plurality of features of the second time series of data points associated with the historical target transaction metrics as an input to the FFT layer to generate the real part and the imaginary part of each feature of the second plurality of features; and providing each feature of the third plurality of features of the third time series of data points associated with the contemporary transaction metrics as an input to the FFT layer to generate the real part and the imaginary part of each feature of the third plurality of features.

[0028] Clause 15: The method according to any one of Clauses 12 to 14, wherein training the machine learning model includes: combining the real parts of each feature of the first plurality of features to generate a combined real part of the first plurality of features; combining the imaginary parts of each feature of the first plurality of features to generate a combined imaginary part of the first plurality of features; combining the real parts of each feature of the second plurality of features to generate a combined real part of the second plurality of features; combining the imaginary parts of each feature of the second plurality of features to generate a combined imaginary part of the second plurality of features; combining the real parts of each feature of the third plurality of features to generate a combined real part of the third plurality of features; and combining the imaginary parts of each feature of the third plurality of features to generate a combined imaginary part of the third plurality of features; and wherein the output of the processing layer of the machine learning model includes the combined real part of the first plurality of features, the combined imaginary part of the first plurality of features, the combined real part of the second plurality of features, the combined imaginary part of the second plurality of features, the combined real part of the third plurality of features, and the combined imaginary part of the third plurality of features.

[0029] Clause 16: The method according to any one of Clauses 12 to 15, wherein training the machine learning model includes: providing the combined real part of the first plurality of features as an input to a one-dimensional feature extraction convolutional layer; providing the combined imaginary part of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; providing the combined real part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; providing the combined imaginary part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; providing the combined real part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; providing the combined imaginary part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; and generating the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer.

[0030] Clause 17: The method according to any one of Clauses 12 to 16, wherein the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part, wherein the real input part includes: a first real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features; a second real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features; a third real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features; and wherein the imaginary input part includes: a first imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features; a second imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features; a third imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features; and wherein training the machine learning model includes: generating a real component attention matrix based on the real input part, wherein the real component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features; and generating an imaginary component attention matrix based on the imaginary input part, wherein the imaginary component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features.

[0031] Clause 18: The method according to any one of Clauses 12 to 17, wherein the method further comprises: generating a real input portion of the input to the dual attention component of the machine learning model; and generating an imaginary input portion of the input to the dual attention component of the machine learning model; wherein generating the real input portion of the input to the dual attention component of the machine learning model comprises: providing the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the first plurality of features as an input to a dropout layer to generate a real contemporary transaction metric feature vector as the first real input; providing the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and providing the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input; and wherein generating the imaginary input portion of the input to the dual attention component of the machine learning model comprises: providing the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the first plurality of features as an input to the dropout layer to generate an imaginary contemporary transaction metric feature vector as the first imaginary input; providing the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the second plurality of features as an input to the dropout layer to generate an imaginary historical transaction metric feature vector as the second imaginary input; and providing the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the third plurality of features as an input to the dropout layer to generate an imaginary historical target transaction metric feature vector as the third imaginary input.

[0032] Clause 19: The method according to any one of Clauses 12 to 18, wherein generating the real component attention matrix includes: generating a dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; providing the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a softmax layer to generate a real attention feature vector; and generating a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein generating the imaginary component attention matrix includes: generating a dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; providing the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a softmax layer to generate an imaginary attention feature vector; and generating a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

[0033] Clause 20: The method according to any one of Clauses 12 to 19, wherein generating the real component attention matrix includes: performing an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; providing the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a tanh activation function; providing the output of the tanh activation function as an input to a softmax layer to generate a real attention feature vector; and generating a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein generating the imaginary component attention matrix includes: performing an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; providing the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a tanh activation function; providing the output of the tanh activation function as an input to a softmax layer to generate an imaginary attention feature vector; and generating a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

[0034] Clause 21: The method according to any one of Clauses 12 to 20, wherein providing the output of the dual attention component of the machine learning model as the input to the learning and prediction components of the machine learning model includes: concatenating the real component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation; providing the first concatenation as the input to a feature learning one-dimensional convolutional layer; concatenating the imaginary component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation; providing the second concatenation as the input to the feature learning one-dimensional convolutional layer; providing the output of the feature learning one-dimensional convolutional layer for the first concatenation as the input to a reconstruction layer to generate a real component feature vector; and providing the output of the feature learning one-dimensional convolutional layer for the second concatenation as the input to the reconstruction layer to generate an imaginary component feature vector.

[0035] Clause 22: The method according to any one of Clauses 12 to 21, wherein generating the output of the machine learning model includes: generating a complex feature vector based on the real component feature vector and the imaginary component feature vector; and providing the complex feature vector as the input to an inverse FFT layer to generate the contemporary target transaction metric.

[0036] Clause 23: A computer program product comprising at least one non-transitory computer-readable medium, the at least one non-transitory computer-readable medium including one or more instructions that, when executed by at least one processor, cause the at least one processor to: receive a time series of historical data points, where the historical data points include values of a plurality of transaction characteristics for a plurality of time intervals during an initial time period; determine a historical time period, where the historical time period includes a first time period of the initial time period; determine a contemporary time period, where the contemporary time period includes a second time period of the initial time period that is after the historical time period, where the historical time period is longer than the contemporary time period; determine a first time series of data points associated with a historical transaction metric from the historical time period; determine a second time series of data points associated with a historical target transaction metric from the historical time period; determine a third time series of data points associated with a contemporary transaction metric from the contemporary time period; generate a machine learning model, where the machine learning model is configured to provide an output including a predicted time series of data points associated with a contemporary target transaction metric, where the one or more instructions that cause the at least one processor to generate the machine learning model cause the at least one processor to: train the machine learning model based on the first time series of data points associated with the historical transaction metric, the second time series of data points associated with the historical target transaction metric, and the third time series of data points associated with the contemporary transaction metric; and generate an output of the machine learning model, where the output of the machine learning model includes the contemporary target transaction metric, and where the contemporary target transaction metric includes values of the target transaction metric during a target prediction period.

[0037] Clause 24: The computer program product according to Clause 23, where the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: provide the first time series of data points associated with the historical transaction metric, the second time series of data points associated with the historical target transaction metric, and the third time series of data points associated with the contemporary transaction metric, as inputs to a processing layer of the machine learning model, where the processing layer includes a fast Fourier transform (FFT) layer; provide an output of the processing layer, as an input to a feature extraction component of the machine learning model; provide an output of the feature extraction component of the machine learning model, as an input to a dual attention component of the machine learning model; and provide an output of the dual attention component of the machine learning model, as an input to a learning and prediction component of the machine learning model.

[0038] Clause 25: The computer program product according to clause 23 or 24, wherein the one or more instructions that cause the at least one processor to provide the first time series of data points associated with the historical transaction metric, the second time series of data points associated with the historical target transaction metric, and the third time series of data points associated with the contemporary transaction metric as inputs to the processing layer of the machine learning model cause the at least one processor to: provide each feature of the first plurality of features of the first time series of data points associated with the historical transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the first plurality of features; provide each feature of the second plurality of features of the second time series of data points associated with the historical target transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the second plurality of features; and provide each feature of the third plurality of features of the third time series of data points associated with the contemporary transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the third plurality of features.

[0039] Clause 26: The computer program product according to any one of clauses 23 to 25, wherein the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: combine the real parts of each feature of the first plurality of features to generate a combined real part of the first plurality of features; combine the imaginary parts of each feature of the first plurality of features to generate a combined imaginary part of the first plurality of features; combine the real parts of each feature of the second plurality of features to generate a combined real part of the second plurality of features; combine the imaginary parts of each feature of the second plurality of features to generate a combined imaginary part of the second plurality of features; combine the real parts of each feature of the third plurality of features to generate a combined real part of the third plurality of features; and combine the imaginary parts of each feature of the third plurality of features to generate a combined imaginary part of the third plurality of features; and wherein the output of the processing layer of the machine learning model includes the combined real part of the first plurality of features, the combined imaginary part of the first plurality of features, the combined real part of the second plurality of features, the combined imaginary part of the second plurality of features, the combined real part of the third plurality of features, and the combined imaginary part of the third plurality of features.

[0040] Clause 27: The computer program product according to any one of Clauses 23 to 26, wherein the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: provide the real part of the combination of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the imaginary part of the combination of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the real part of the combination of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the imaginary part of the combination of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the real part of the combination of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; provide the imaginary part of the combination of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; and generate the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer.

[0041] Clause 28: The computer program product according to any one of Clauses 23 to 27, wherein the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part, wherein the real input part includes: a first real input based on the output of the one-dimensional feature extraction convolutional layer for the real part of the combination of the first plurality of features; a second real input based on the output of the one-dimensional feature extraction convolutional layer for the real part of the combination of the second plurality of features; a third real input based on the output of the one-dimensional feature extraction convolutional layer for the real part of the combination of the third plurality of features; and wherein the imaginary input part includes: a first imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the imaginary part of the combination of the first plurality of features; a second imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the imaginary part of the combination of the second plurality of features; a third imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the imaginary part of the combination of the third plurality of features; and wherein the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: generate a real component attention matrix based on the real input part, wherein the real component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features; and generate an imaginary component attention matrix based on the imaginary input part, wherein the imaginary component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features.

[0042] Clause 29: A computer program product according to any one of Clauses 23 to 28, wherein the one or more instructions further cause the at least one processor to: generate the real input portion of the input to the dual attention component of the machine learning model; and generate the imaginary input portion of the input to the dual attention component of the machine learning model; wherein the one or more instructions that cause the at least one processor to generate the real input portion of the input to the dual attention component of the machine learning model cause the at least one processor to: provide the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the first plurality of features as an input to the dropout layer to generate a real contemporary transaction metric feature vector as the first real input; provide the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined real portion of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input; and wherein the one or more instructions that cause the at least one processor to generate the imaginary input portion of the input to the dual attention component of the machine learning model cause the at least one processor to: provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the first plurality of features as an input to the dropout layer to generate an imaginary contemporary transaction metric feature vector as the first imaginary input; provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the second plurality of features as an input to the dropout layer to generate an imaginary historical transaction metric feature vector as the second imaginary input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary portion of the third plurality of features as an input to the dropout layer to generate an imaginary historical target transaction metric feature vector as the third imaginary input.

[0043] Clause 30: The computer program product according to any one of Clauses 23 to 29, wherein the one or more instructions that cause the at least one processor to generate the real component attention matrix cause the at least one processor to: generate the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; provide the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a softmax layer to generate a real attention feature vector; and generate the dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein the one or more instructions that cause the at least one processor to generate the imaginary component attention matrix cause the at least one processor to: generate the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; provide the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a softmax layer to generate an imaginary attention feature vector; and generate the dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

[0044] Clause 31: The computer program product according to any one of Clauses 23 to 30, wherein the one or more instructions that cause the at least one processor to generate the real component attention matrix cause the at least one processor to: perform an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a tanh activation function; provide the output of the tanh activation function as an input to a softmax layer to generate a real attention feature vector; and generate the dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein the one or more instructions that cause the at least one processor to generate the imaginary component attention matrix cause the at least one processor to: perform an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; provide the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a tanh activation function; provide the output of the tanh activation function as an input to a softmax layer to generate an imaginary attention feature vector; and generate the dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

[0045] Clause 32: The computer program product according to any one of Clauses 23 to 31, wherein the one or more instructions that cause the at least one processor to provide the output of the dual attention component of the machine learning model as the input to the learning and prediction components of the machine learning model cause the at least one processor to: concatenate the real component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation; provide the first concatenation as the input to a feature learning one-dimensional convolutional layer; concatenate the imaginary component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation; provide the second concatenation as the input to the feature learning one-dimensional convolutional layer; provide the output of the feature learning one-dimensional convolutional layer for the first concatenation as the input to a reconstruction layer to generate a real component feature vector; and provide the output of the feature learning one-dimensional convolutional layer for the second concatenation as the input to the reconstruction layer to generate an imaginary component feature vector.

[0046] Clause 33: The computer program product according to any one of Clauses 23 to 32, wherein the one or more instructions that cause the at least one processor to generate the output of the machine learning model cause the at least one processor to be programmed or configured to: generate a complex feature vector based on the real component feature vector and the imaginary component feature vector; and provide the complex feature vector as the input to an inverse FFT layer to generate the contemporary target transaction metric.

[0047] After considering the following description and the appended claims in light of the accompanying drawings, these and other features and characteristics of the present disclosure, as well as the methods of operation and functions of the associated structural elements and combinations of the various parts, and the manufacturing economy, will become more apparent, all of the drawings forming a part of this specification, wherein like reference numerals designate corresponding parts in the various figures. However, it should be clearly understood that the drawings are for illustrative and descriptive purposes only and are not intended to define a limitation of the present disclosure. Unless the context clearly dictates otherwise, the singular forms "a" and "the" as used in this specification and the claims include plural referents. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] The additional advantages and details of the present disclosure are explained in more detail below with reference to the exemplary embodiments illustrated in the accompanying drawings, in which:

[0049] Figure 1 is a diagram of a non-limiting embodiment or aspect of an environment in which the systems, devices, products, apparatuses, and / or methods described herein can be implemented in accordance with the principles of the present disclosure;

[0050] Figure 2 is Figure 1 a diagram of a non-limiting embodiment or aspect of components of one or more apparatuses;

[0051] Figure 3 is a flowchart of a non - limiting example or aspect of a process for analyzing multivariate time series using a machine learning model including a convolutional Fourier network;

[0052] Figure 4 is a flowchart of a non - limiting example or aspect of a process for generating a machine learning model including a convolutional Fourier network;

[0053] Figures 5A - 5E is a diagram of a non - limiting example or aspect of a machine learning model architecture for analyzing multivariate time series;

[0054] Figure 6 is a diagram of a non - limiting example or aspect of an implementation of a process for training a machine learning model in a distributed environment; and

[0055] Figure 7 is a diagram of a non - limiting example or aspect of an implementation of a process for training a machine learning model in a distributed environment. DETAILED DESCRIPTION

[0056] For purposes of description below, the terms "end", "upper", "lower", "right", "left", "vertical", "horizontal", "top", "bottom", "lateral", "longitudinal", and derivatives thereof shall refer to the orientation of the present disclosure as it appears in the figures. However, it should be understood that, unless explicitly specified to the contrary, the present disclosure may assume various alternative variations and sequences of steps. It should also be understood that the specific apparatus and processes illustrated in the figures and described in the following description are merely exemplary examples or aspects of the present disclosure. Accordingly, unless otherwise indicated, the specific dimensions and other physical characteristics associated with the embodiments or aspects of the embodiments disclosed herein should not be considered limiting.

[0057] As used herein, aspects, components, elements, structures, acts, steps, functions, instructions, etc. should not be understood as critical or necessary unless explicitly so described. Additionally, as used herein, the article "a" is intended to include one or more items and may be interchangeable with "one or more" and "at least one". Further, as used herein, the term "set" is intended to include one or more items (e.g., related items, unrelated items, combinations of related and unrelated items, etc.) and may be interchangeable with "one or more" or "at least one". Where only one item is intended, the term "one" or similar language is used. Also, as used herein, the term "having" and the like are intended to be open - ended terms. Additionally, unless otherwise explicitly stated, the phrase "based on" is intended to mean "at least partially based on". In appropriate cases, the phrase "based on" may also mean "responsive to".

[0058] As used herein, the terms "communicate" and "convey" can refer to the receipt, reception, sending, transmission, provisioning, etc. of information (e.g., data, signals, messages, instructions, commands, etc.). A unit (e.g., a device, a system, a component of a device or system, a combination thereof, etc.) communicating with another unit means that the one unit is capable of receiving information from and / or conveying (e.g., sending) information to the other unit, either directly or indirectly. This can refer to a direct or indirect connection that is wired and / or wireless in nature. Additionally, although the information being sent may be modified, processed, relayed, and / or routed between a first unit and a second unit, the two units can still communicate with each other. For example, even if a first unit receives information passively and does not actively send information to a second unit, the first unit can still communicate with the second unit. As another example, if at least one intermediate unit (e.g., a third unit located between the first unit and the second unit) processes the information received from the first unit and sends the processed information to the second unit, the first unit can communicate with the second unit. In some non-limiting embodiments or aspects, a message can refer to a network packet (e.g., a data packet, etc.) that includes data.

[0059] As used herein, the terms "issuer", "issuer institution", "issuer bank", or "payment device issuer" can refer to one or more entities that provide an account to an individual (e.g., a user, a customer, etc.) for conducting payment transactions such as credit payment transactions and / or debit payment transactions, etc. For example, an issuer institution can provide an account identifier, such as a primary account number (PAN), that uniquely identifies one or more accounts associated with the customer. In some non-limiting embodiments or aspects, an issuer can be associated with a bank identification number (BIN) that uniquely identifies the issuer institution. As used herein, an "issuer system" can refer to one or more computer systems operated by or on behalf of an issuer, such as a server that executes one or more software applications. For example, an issuer system can include one or more authorization servers for authorizing transactions.

[0060] As used herein, the term "transaction service provider" can refer to an entity that receives a transaction authorization request from a merchant or other entity and, in some cases, provides payment assurance through an agreement between the transaction service provider and the issuer institution. For example, a transaction service provider can include a payment network, such as American or any other entity that processes transactions. As used herein, the term "transaction service provider system" may refer to one or more computer systems operated by or on behalf of a transaction service provider, such as a transaction service provider system that executes one or more software applications. The transaction service provider system may include one or more processors and, in some non-limiting embodiments or aspects, may be operated by or on behalf of a transaction service provider.

[0061] As used herein, the term "merchant" may refer to one or more entities (e.g., an operator of a retail business) that provide goods and / or services and / or access to goods and / or services to users (e.g., customers, consumers, etc.) based on transactions such as payment transactions. As used herein, "merchant system" may refer to one or more computer systems operated by or on behalf of a merchant, such as a server that executes one or more software applications. As used herein, the term "product" may refer to one or more goods and / or services provided by a merchant.

[0062] As used herein, the term "acquirer" may refer to an entity that is licensed by and approved by a transaction service provider to initiate transactions (e.g., payment transactions) involving a payment device associated with the transaction service provider. As used herein, the term "acquirer system" may also refer to one or more computer systems, computer devices, etc. operated by or on behalf of an acquirer. Transactions that an acquirer may initiate may include payment transactions (e.g., purchases, original credit transactions (OCTs), account funding transactions (AFTs), etc.). In some non-limiting embodiments, an acquirer may be authorized by a transaction service provider to contract with a merchant or service provider and initiate transactions involving a payment device associated with the transaction service provider. An acquirer may contract with a payment service provider to enable the payment service provider to provide sponsorship to a merchant. An acquirer may monitor the compliance of a payment service provider in accordance with transaction service provider regulations. An acquirer may conduct due diligence on a payment service provider and ensure that appropriate due diligence occurs prior to contracting with a sponsored merchant. An acquirer may be responsible for all transaction service provider programs operated or sponsored by the acquirer. An acquirer may be responsible for the actions of an acquirer payment service provider, a merchant sponsored by an acquirer payment service provider, etc. In some non-limiting embodiments, an acquirer may be a financial institution, such as a bank.

[0063] As used herein, the term "payment gateway" may refer to an entity and / or a payment processing system operated by or on behalf of such entity, where the entity (such as a merchant service provider, payment service provider, payment service provider, payment service provider contracted with an acquirer, payment aggregator, etc.) provides payment services (such as transaction service provider payment services, payment processing services, etc.) to one or more merchants. The payment services may be associated with the use of a portable financial device managed by a transaction service provider. As used herein, the term "payment gateway system" may refer to one or more computer systems, computer devices, servers, server groups, etc. operated by or on behalf of a payment gateway.

[0064] As used herein, the terms "client" and "client device" may refer to one or more computing devices, such as a processor, a storage device, and / or similar computer components that access services that may be provided by a server. In some non-limiting embodiments or aspects, the client device may include an electronic device configured to communicate with one or more networks and / or facilitate payment transactions, such as but not limited to one or more desktop computers, one or more portable computers (e.g., tablet computers), one or more mobile devices (e.g., cellular phones, smartphones, personal digital assistants, wearable devices such as watches, glasses, lenses, and / or clothing, etc.), and / or other similar devices. Additionally, the term "client" may also refer to an entity that owns, uses, and / or operates a client device to facilitate a transaction with another entity.

[0065] As used herein, the term "server" may refer to one or more computing devices, such as a processor, a storage device, and / or similar computer components, that communicate with client devices and / or other computing devices over a network such as the Internet or a private network, and in some instances, facilitate communication between other servers and / or client devices.

[0066] As used herein, the term "system" may refer to one or more computing devices or a combination of computing devices, such as but not limited to a processor, a server, a client device, a software application, and / or other similar components. Additionally, as used herein, a reference to a "server" or a "processor" may refer to the previously described server and / or processor stated to perform a previous step or function, a different server and / or processor, and / or a combination of servers and / or processors. For example, as used in the specification and claims, a first server and / or a first processor stated to perform a first step or function may refer to the same or a different server and / or processor stated to perform a second step or function.

[0067] Non-limiting embodiments or aspects of the present disclosure relate to systems, methods, and computer program products for predicting metrics using convolutional Fourier networks. In some non-limiting embodiments or aspects, a system for estimating transaction metrics during a time interval may include at least one processor programmed or configured to: receive a time series of historical data points, where the historical data points include values of multiple transaction characteristics for multiple time intervals during an initial time period; determine a historical time period, where the historical time period includes a first time period of the initial time period; determine a contemporary time period, where the contemporary time period includes a second time period of the initial time period that is after the historical time period, and where the historical time period is longer than the contemporary time period. In some non-limiting embodiments or aspects, the at least one processor may be programmed or configured to determine a first time series of data points associated with historical transaction metrics from the historical time period; determine a second time series of data points associated with historical target transaction metrics from the historical time period; determine a third time series of data points associated with contemporary transaction metrics from the contemporary time period, and generate a machine learning model.

[0068] In some non-limiting embodiments or aspects, the machine learning model is configured to provide an output including a predicted time series of data points associated with contemporary target transaction metrics. In some non-limiting embodiments or aspects, when generating the machine learning model, the at least one processor is programmed or configured to: train the machine learning model based on the first time series of data points associated with the historical transaction metrics, the second time series of data points associated with the historical target transaction metrics, and the third time series of data points associated with the contemporary transaction metrics. In some non-limiting embodiments or aspects, when training the machine learning model, the at least one processor is programmed or configured to: provide the first time series of data points associated with the historical transaction metrics, the second time series of data points associated with the historical target transaction metrics, and the third time series of data points associated with the contemporary transaction metrics as inputs to a feature extraction component of the machine learning model; provide an output of the feature extraction component of the machine learning model as an input to a dual attention component of the machine learning model; provide an output of the dual attention component of the machine learning model as an input to a learning and prediction component of the machine learning model; and generate an output of the learning and prediction component of the machine learning model, where an output of the prediction component of the machine learning model includes the contemporary target transaction metric, and where the contemporary target transaction metric includes a value of the target transaction metric during a target prediction period.

[0069] In this manner, and by using Fourier transforms and / or inverse Fourier transforms as convolutional Fourier networks, the system can provide the ability to process data based on the latency involved in the payment processing network, such as the latency in the availability of transaction data for a machine learning model. Additionally, the system may be robust and able to provide accurate results when analyzing factors that may change payment transaction behavior and lead to concept drift. Further, the system may be able to accurately analyze a multivariate time series based on the ability to analyze data-specific aspects associated with the data in the multivariate time series, including seasonal aspects.

[0070] Now refer to Figure 1 , Figure 1 which is a diagram of an example environment 100 in which the apparatuses, systems, and / or methods described herein may be implemented. As Figure 1 shown, environment 100 includes a time series analysis system 102, a transaction service provider system 104, a user device 106, and a communication network 108. The time series analysis system 102, the transaction service provider system 104, and / or the user device 106 may be interconnected (e.g., establish connections to communicate) via a wired connection, a wireless connection, or a combination of wired and wireless connections.

[0071] The time series analysis system 102 may include one or more apparatuses configured to communicate with the transaction service provider system 104 and / or the user device 106 via the communication network 108. For example, the time series analysis system 102 may include servers, server clusters, and / or other similar apparatuses. In some non-limiting embodiments or aspects, the time series analysis system 102 may be associated with the transaction service provider system 104 as described herein. Additionally or alternatively, the time series analysis system 102 may generate (e.g., train, validate, retrain, etc.), store, and / or implement one or more machine learning models (e.g., operate one or more machine learning models, provide input to one or more machine learning models, and / or provide output from one or more machine learning models, etc.). In some non-limiting embodiments or aspects, the time series analysis system 102 may communicate with a data storage device that may be local or remote to the time series analysis system 102. In some non-limiting embodiments or aspects, the time series analysis system 102 is capable of receiving information from the data storage device, storing information in the data storage device, sending information to the data storage device, and / or searching for information stored in the data storage device.

[0072] The transaction service provider system 104 may include one or more devices configured to communicate with the time series analysis system 102 and / or the user device 106 via the communication network 108. For example, the transaction service provider system 104 may include computing devices such as servers, server clusters, and / or other similar devices. In some non-limiting embodiments or aspects, the transaction service provider system 104 may be associated with the transaction service provider described herein. In some non-limiting embodiments or aspects, the time series analysis system 102 may be a component of the transaction service provider system 104.

[0073] The user device 106 may include one or more devices configured to communicate with the time series analysis system 102 and / or the transaction service provider system 104 via the communication network 108. For example, the user device 106 may include computing devices such as desktop computers, portable computers (e.g., tablets, laptops, etc.), mobile devices (e.g., cellular phones, smartphones, personal digital assistants, wearable devices, etc.), and / or other similar devices. In some non-limiting embodiments or aspects, the user device 106 may be associated with a user (e.g., an individual operating the user device 106).

[0074] The communication network 108 may include one or more wired and / or wireless networks. For example, the communication network 108 may include cellular networks (e.g., Long-Term Evolution (LTE) networks, Third Generation (3G) networks, Fourth Generation (4G) networks, Fifth Generation networks (5G) networks, Code Division Multiple Access (CDMA) networks, etc.), Public Land Mobile Networks (PLMNs), Local Area Networks (LANs), Wide Area Networks (WANs), Metropolitan Area Networks (MANs), telephone networks (e.g., Public Switched Telephone Networks (PSTNs), etc.), private networks, ad-hoc networks, intranets, the Internet, fiber-based networks, cloud computing networks, etc., and / or combinations of these or other types of networks.

[0075] Provided by way of example Figure 1 the number and arrangement of the devices and networks shown in Figure 1 There may be additional devices and / or networks, fewer devices and / or networks, different devices and / or networks, or differently arranged devices and / or networks compared to those shown in Figure 1 Two or more of the devices shown in Figure 1 may be implemented within a single device, or

[0076] Now refer to Figure 2 , Figure 2A diagram of an example component of device 200. Device 200 may correspond to time series analysis system 102 (e.g., one or more devices of time series analysis system 102), transaction service provider system 104 (e.g., one or more devices of transaction service provider system 104), and / or user device 106. In some non-limiting embodiments or aspects, time series analysis system 102, transaction service provider system 104, and / or user device 106 may include at least one device 200 and / or at least one component of device 200. As Figure 2 shown, device 200 may include bus 202, processor 204, memory 206, storage component 208, input component 210, output component 212, and communication interface 214.

[0077] Bus 202 may include components that permit communication between the components of device 200. In some non-limiting embodiments or aspects, processor 204 may be implemented in hardware, software, or a combination of hardware and software. For example, processor 204 may include a processor (e.g., central processing unit (CPU), graphics processing unit (GPU), accelerated processing unit (APU), etc.), a microprocessor, a digital signal processor (DSP), and / or any processing component that can be programmed to perform functions (e.g., field programmable gate array (FPGA), application specific integrated circuit (ASIC), etc.). Memory 206 may include random access memory (RAM), read only memory (ROM), and / or another type of dynamic or static memory that stores information and / or instructions for use by processor 204 (e.g., flash memory, magnetic memory, optical memory, etc.).

[0078] Storage component 208 may store information and / or software associated with the operation and use of device 200. For example, storage component 208 may include a hard disk (e.g., magnetic disk, optical disk, magneto-optical disk, solid state disk, etc.), a compact disc (CD), a digital versatile disc (DVD), a floppy disk, a cassette tape, a magnetic tape, and / or another type of computer-readable medium, as well as a corresponding drive.

[0079] Input component 210 may include components that permit device 200 to receive information, for example, via user input (e.g., touch screen display, keyboard, keypad, mouse, button, switch, microphone, etc.). Additionally or alternatively, input component 210 may include sensors for sensing information (e.g., global positioning system (GPS) component, accelerometer, gyroscope, actuator, etc.). Output component 212 may include components that provide output information from device 200 (e.g., display, speaker, one or more light emitting diodes (LEDs), etc.).

[0080] The communication interface 214 may include transceiver-like components (e.g., transceivers, separate receivers and transmitters, etc.), which enable the device 200 to communicate with other devices via, for example, a wired connection, a wireless connection, or a combination of a wired connection and a wireless connection. The communication interface 214 may permit the device 200 to receive information from another device and / or provide information to another device. For example, the communication interface 214 may include an Ethernet interface, an optical interface, a coaxial interface, an infrared interface, a radio frequency (RF) interface, a universal serial bus (USB) interface, interface, a cellular network interface, etc.

[0081] The device 200 may perform one or more of the processes described herein. The device 200 may perform these processes based on software instructions stored by a computer-readable medium such as, for example, the memory 206 and / or the storage component 208 and executed by the processor 204. A computer-readable medium (e.g., a non-transitory computer-readable medium) is defined herein as a non-transitory memory device. A memory device includes a memory space located within a single physical storage device or a memory space extending across multiple physical storage devices.

[0082] The software instructions may be read into the memory 206 and / or the storage component 208 from another computer-readable medium or from another device via the communication interface 214. When executed, the software instructions stored in the memory 206 and / or the storage component 208 may cause the processor 204 to perform one or more of the processes described herein. Additionally or alternatively, hardwired circuitry may be used in place of or in combination with software instructions to perform one or more of the processes described herein. Thus, the embodiments described herein are not limited to any particular combination of hardware circuitry and software.

[0083] Figure 2 The number and arrangement of the components shown are provided as an example. In some non-limiting embodiments or aspects, the device 200 may include additional components, fewer components, different components, or components arranged in a different manner compared to those shown in Figure 2 In addition or alternatively, a set of components (e.g., one or more components) of the device 200 may perform one or more functions described as being performed by another set of components of the device 200.

[0084] Now refer to Figure 3 , Figure 3FIG. 300 is a flowchart of a non - limiting embodiment or aspect of a process 300 for analyzing a multivariate time series using a machine learning model including a convolutional Fourier network. In some non - limiting embodiments or aspects, one or more steps of process 300 may be performed by the time series analysis system 102 (e.g., one or more devices of the time series analysis system 102) (e.g., fully, partially, etc.). In some non - limiting embodiments or aspects, one or more steps of process 300 may be performed by another device or another set of devices (e.g., fully, partially, etc.) that are separate from or include the foregoing components, such as the time series analysis system 102 (e.g., one or more devices of the time series analysis system 102), the transaction service provider system 104 (e.g., one or more devices of the transaction service provider system 104), and / or the user device 106.

[0085] As Figure 3 shown, at step 302, process 300 includes receiving a time series of historical data points. For example, the time series analysis system 102 may receive a time series of historical data points from the transaction service provider system 104. In some non - limiting embodiments or aspects, the time series of historical data points may include data (e.g., transaction data) associated with historical payment transactions made using one or more payment processing networks (e.g., one or more payment processing networks associated with the transaction service provider system 104).

[0086] In some non - limiting embodiments or aspects, the time series of historical data points may include a multivariate time series. In some non - limiting embodiments or aspects, a multivariate time series may be a series of values based on multiple time - dependent variables, where each variable depends on past values of the variable and also on other time - dependent variables among the multiple time - dependent variables.

[0087] As Figure 3 shown, at step 304, process 300 includes generating a machine learning model including a convolutional Fourier network. For example, the time series analysis system 102 may generate a machine learning model including a convolutional Fourier network based on the time series of historical data points. In some non - limiting embodiments or aspects, the time series analysis system 102 may generate the machine learning model by training the machine learning model based on a portion of the time series of historical data points and validating the machine learning model based on another portion of the time series of historical data points. In some non - limiting embodiments or aspects, the machine learning model may be configured to provide an output of a predicted time series including data points associated with contemporary target transaction metrics.

[0088] As Figure 3As shown, at step 306, process 300 includes performing an action using a machine learning model. For example, time series analysis system 102 may perform an action using a machine learning model. In some non-limiting embodiments or aspects, time series analysis system 102 may generate a prediction using a machine learning model. For example, time series analysis system 102 may receive an input (e.g., a real-time input) and generate an output of the machine learning model based on the input. In some non-limiting embodiments or aspects, the output may include a prediction. In some non-limiting embodiments or aspects, the prediction may include one or more future transaction metrics (e.g., one or more contemporary target transaction metrics) associated with one or more entities based on the historical transaction behavior of the one or more entities.

[0089] Now referring to Figure 4 , Figure 4 is a flowchart of a non-limiting embodiment or aspect of process 400 for generating a machine learning model that includes a convolutional Fourier network (e.g., in the same or similar manner as step 304 of process 300). In some non-limiting embodiments or aspects, one or more steps of process 400 may be performed by time series analysis system 102 (e.g., one or more devices of time series analysis system 102) (e.g., fully, partially, etc.). In some non-limiting embodiments or aspects, one or more steps of process 400 may be performed by another device or another group of devices (e.g., fully, partially, etc.) that are separate from or include the foregoing components of time series analysis system 102 (e.g., one or more devices of time series analysis system 102), transaction service provider system 104 (e.g., one or more devices of transaction service provider system 104), and / or user device 106.

[0090] As Figure 4 further shown, at step 402, process 400 includes processing a time series using a fast Fourier transform (FFT). For example, time series analysis system 102 may process a time series using an FFT. In some non-limiting embodiments or aspects, time series analysis system 102 may use a discrete Fourier transform, such as a real-valued FFT (RFFT), to process the time series. In this way, by using the RFFT, time series analysis system 102 may determine different patterns in the time series based on the results of the RFFT. This may be particularly true for a time series that has a pattern during a time interval (e.g., a sub-period) of a total time period, such as a seasonal pattern in the time series.

[0091] In some non - limiting embodiments or aspects, the time - series analysis system 102 can receive a time - series of historical data points. In some non - limiting embodiments or aspects, the data points can include values of multiple transaction characteristics for multiple time intervals during an initial time period (e.g., transaction characteristics of data associated with multiple historical payment transactions involving one or more entities, such as values of transaction metrics associated with multiple historical payment transactions, measured values of transaction metrics associated with multiple historical payment transactions, etc.). In some non - limiting embodiments or aspects, the time - series analysis system 102 can determine a historical time period that includes the initial time period and a contemporary time period that includes the initial time period and is after the historical time period. In some non - limiting embodiments or aspects, the historical time period can be longer than the contemporary time period. Additionally or alternatively, the historical time period and the contemporary time period can be separated by an amount of time within the initial time period. For example, the historical time period and the contemporary time period can be separated by a time gap that includes the amount of time within the initial time period.

[0092] In some non - limiting embodiments or aspects, the time - series analysis system 102 can determine a first time - series of data points associated with historical transaction metrics from the historical time period; determine a second time - series of data points associated with historical target transaction metrics from the historical time period; and / or determine a third time - series of data points associated with contemporary transaction metrics from the contemporary time period.

[0093] In some non - limiting embodiments or aspects, the time - series analysis system 102 can train a machine - learning model based on the first time - series of data points associated with historical transaction metrics, the second time - series of data points associated with historical target transaction metrics, and / or the third time - series of data points associated with contemporary transaction metrics. In some non - limiting embodiments or aspects, the time - series analysis system 102 can train the machine - learning model by providing the first time - series of data points associated with historical transaction metrics, the second time - series of data points associated with historical target transaction metrics, and the third time - series of data points associated with contemporary transaction metrics as inputs to a processing layer. In some non - limiting embodiments or aspects, the processing layer can include an RFFT layer.

[0094] In some non - limiting embodiments or aspects, when a first time series of data points associated with historical transaction metrics, a second time series of data points associated with historical target transaction metrics, and a third time series of data points associated with contemporary transaction metrics are provided as inputs to a processing layer of an input machine learning model, the time series analysis system 102 can provide each feature in a first plurality of features of the first time series of data points associated with historical transaction metrics as an input to the RFFT layer to generate a real part and an imaginary part of each feature in the first plurality of features. Additionally or alternatively, the time series analysis system 102 can provide each feature in a second plurality of features of the second time series of data points associated with historical target transaction metrics as an input to the RFFT layer to generate a real part and an imaginary part of each feature in the second plurality of features. Additionally or alternatively, the time series analysis system 102 can provide each feature in a third plurality of features of the third time series of data points associated with contemporary transaction metrics as an input to the RFFT layer to generate a real part and an imaginary part of each feature in the third plurality of features.

[0095] In some non - limiting embodiments or aspects, the time series analysis system 102 can combine the real parts of each feature in the first plurality of features to generate a combined real part of the first plurality of features, combine the imaginary parts of each feature in the first plurality of features to generate a combined imaginary part of the first plurality of features, combine the real parts of each feature in the second plurality of features to generate a combined real part of the second plurality of features, combine the imaginary parts of each feature in the second plurality of features to generate a combined imaginary part of the second plurality of features, combine the real parts of each feature in the third plurality of features to generate a combined real part of the third plurality of features, and / or combine the imaginary parts of each feature in the third plurality of features to generate a combined imaginary part of the third plurality of features. In some non - limiting embodiments or aspects, the output of the processing layer of the machine learning model can include a combined real part of the first plurality of features, a combined imaginary part of the first plurality of features, a combined real part of the second plurality of features, a combined imaginary part of the second plurality of features, a combined real part of the third plurality of features, and a combined imaginary part of the third plurality of features.

[0096] As Figure 4As further shown in FIG. 404, at step 404, process 400 includes performing a feature extraction technique. For example, time series analysis system 102 may perform a feature extraction technique on the output of the processing layer of a machine learning model. In some non-limiting embodiments or aspects, time series analysis system 102 may provide the output of the processing layer as an input to the feature extraction component of the machine learning model. In some non-limiting embodiments or aspects, time series analysis system 102 may provide the combined real part of a first plurality of features as an input to a one-dimensional feature extraction convolutional layer, provide the combined imaginary part of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer, provide the combined real part of a second plurality of features as an input to the one-dimensional feature extraction convolutional layer, provide the combined imaginary part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer, provide the combined real part of a third plurality of features as an input to the one-dimensional feature extraction convolutional layer, and / or provide the combined imaginary part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer.

[0097] In some non-limiting embodiments or aspects, time series analysis system 102 may provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features as an input to a dropout layer to generate a real contemporary transaction metric feature vector, provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector, provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector, provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features as an input to the dropout layer to generate an imaginary contemporary transaction metric feature vector, provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features as an input to the dropout layer to generate an imaginary historical transaction metric feature vector, and / or provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features as an input to the dropout layer to generate an imaginary historical target transaction metric feature vector.

[0098] In some non-limiting embodiments or aspects, time series analysis system 102 may generate the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer. In some non-limiting embodiments or aspects, the output of the feature extraction component of the machine learning model may include a real contemporary transaction metric feature vector, a real historical transaction metric feature vector, a real historical target transaction metric feature vector, an imaginary contemporary transaction metric feature vector, an imaginary historical transaction metric feature vector, and / or an imaginary historical target transaction metric feature vector.

[0099] As Figure 4As further shown in , at step 406, process 400 includes performing an attention technique. For example, time series analysis system 102 may perform an attention technique. In some non-limiting embodiments or aspects, time series analysis system 102 may provide the output of the feature extraction component of the machine learning model as the input to the dual attention component of the machine learning model. In some non-limiting embodiments or aspects, the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part. In some non-limiting embodiments or aspects, the real input part includes a first real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features; a second real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features; and / or a third real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features. In some non-limiting embodiments or aspects, the imaginary input part includes a first imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features; a second imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features; and / or a third imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features.

[0100] In some non-limiting embodiments or aspects, time series analysis system 102 may generate a real component attention matrix based on the real input part and / or generate an imaginary component attention matrix based on the imaginary input part. In some non-limiting embodiments or aspects, the first attention matrix has a plurality of attention vectors equal to the number of features among the plurality of transaction features. In some non-limiting embodiments or aspects, the second attention matrix has a plurality of attention vectors equal to the number of features among the plurality of transaction features.

[0101] In some non - limiting embodiments or aspects, the time - series analysis system 102 may generate a real input portion of the input to the dual - attention component of the machine - learning model and / or generate an imaginary input portion of the input to the dual - attention component of the machine - learning model. In some non - limiting embodiments or aspects, when generating the real input portion of the input to the dual - attention component of the machine - learning model, the time - series analysis system 102 may provide the output of the one - dimensional feature - extraction convolutional layer for the combined real portion of the first plurality of features as the input to the dropout layer to generate a real contemporary transaction metric feature vector as the first real input; provide the output of the one - dimensional feature - extraction convolutional layer for the combined real portion of the second plurality of features as the input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and / or provide the output of the one - dimensional feature - extraction convolutional layer for the combined real portion of the third plurality of features as the input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input. In some non - limiting embodiments or aspects, when generating the imaginary input portion of the input to the dual - attention component of the machine - learning model, the time - series analysis system 102 may provide the output of the one - dimensional feature - extraction convolutional layer for the combined imaginary portion of the first plurality of features as the input to the dropout layer to generate an imaginary contemporary transaction metric feature vector as the first imaginary input; provide the output of the one - dimensional feature - extraction convolutional layer for the combined imaginary portion of the second plurality of features as the input to the dropout layer to generate an imaginary historical transaction metric feature vector as the second imaginary input; and / or provide the output of the one - dimensional feature - extraction convolutional layer for the combined imaginary portion of the third plurality of features as the input to the dropout layer to generate an imaginary historical target transaction metric feature vector as the third imaginary input.

[0102] In some non - limiting embodiments or aspects, when generating the first attention matrix, the time - series analysis system 102 may generate the dot - product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector, provide the dot - product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as the input to the softmax layer to generate a real attention feature vector, and / or generate the dot - product of the real attention feature vector and the real historical target transaction metric feature vector to generate a real component attention matrix.

[0103] In some non - limiting embodiments or aspects, when generating the first attention matrix, the time - series analysis system 102 may perform an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector, provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as the input to the tanh activation function, provide the output of the tanh activation function as the input to the softmax layer to generate a real attention feature vector; and / or generate the dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate a real component attention matrix.

[0104] In some non - limiting embodiments or aspects, when generating the second attention matrix, the time - series analysis system 102 may generate the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector, provide the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as the input to the softmax layer to generate an imaginary attention feature vector, and / or generate the dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate an imaginary component attention matrix.

[0105] In some non - limiting embodiments or aspects, when generating the second attention matrix, the time - series analysis system 102 may perform an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector, provide the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as the input to the tanh activation function, provide the output of the tanh activation function as the input to the softmax layer to generate an imaginary attention feature vector; and / or generate the dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate an imaginary component attention matrix.

[0106] As Figure 4 Further shown in, at step 408, process 400 includes performing learning and prediction techniques. For example, the time - series analysis system 102 may perform learning and prediction techniques. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the output of the dual - attention component of the machine - learning model as the input to the learning and prediction component of the machine - learning model. In some non - limiting embodiments or aspects, the time - series analysis system 102 may concatenate the first attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation and provide the first concatenation as the input to the feature - learning one - dimensional convolutional layer. In some non - limiting embodiments or aspects, the time - series analysis system 102 may concatenate the second attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation and provide the second concatenation as the input to the feature - learning one - dimensional convolutional layer.

[0107] In some non - limiting embodiments or aspects, the time - series analysis system 102 can provide an input to a reconstruction layer based on the output of a one - dimensional convolutional layer for feature learning for a first connection to generate a real - component feature vector; and / or provide an input to the reconstruction layer based on the output of a one - dimensional convolutional layer for feature learning for a second connection to generate an imaginary - component feature vector.

[0108] In some non - limiting embodiments or aspects, the time - series analysis system 102 can generate the outputs of the learning and prediction components of a machine - learning model. The output of the prediction component of the machine - learning model can include a contemporary target transaction metric, and the contemporary target transaction metric can include the value of the target transaction metric during a target prediction period. In some non - limiting embodiments or aspects, the time - series analysis system 102 can generate a complex - number feature vector based on the real - component feature vector and the imaginary - component feature vector, and provide the complex - number feature vector as an input to an inverse RFFT layer to generate the contemporary target transaction metric.

[0109] Now refer to Figures 5A - 5E , shown are non - limiting embodiments or aspects of a machine - learning model architecture 500 and components of the machine - learning model architecture 500 for a process of analyzing a multivariate time series (e.g., process 300) or generating a machine - learning model including a convolutional Fourier network (e.g., process 400) using a machine - learning model including a convolutional Fourier network.

[0110] As Figure 5A shown, the machine - learning model architecture 500 can include a processing layer 502, a feature extraction component 504, a dual - attention component 506, and a learning and prediction component 508. As Figure 5A further shown, the input to the processing layer 502 can include a time series T of contemporary transaction metrics M , a time series T of historical transaction metrics H , and a time series T of historical target transaction metrics E . As Figure 5A further shown, the output of the learning and prediction component 508 can include a time series T of contemporary target transaction metrics U . By using Fourier transactions (e.g., RFFT), the machine - learning model architecture 500 is a machine - learning model including a convolutional Fourier network. Thus, the machine - learning model architecture 500 involves the principle that multiplication in the frequency domain corresponds to convolution in the time domain.

[0111] In some non - limiting embodiments or aspects, the time - series contemporary transaction metric T M can include a multivariate time series T M ∈R Τ C ×f, where Τ Cis the length of a multivariate time series, and f is the number of features included in the multivariate time series. In some non-limiting embodiments or aspects, at the m-th time, the value of the n-th feature can be represented as T Mn,m . In one example, a contemporary transaction metric can include a sequence of transaction metrics for one year, which includes 14 features, where τ is 8,760 because there are 8,760 hours in a year, and the value of f is 14, representing the number of features used (e.g., the number of transaction metrics).

[0112] In some non-limiting embodiments or aspects, the time series T U of the contemporary target transaction metric can include a univariate time series T U ∈R τ C , where τ C is the length of the univariate time series. In some non-limiting embodiments or aspects, the time series T H of the historical transaction metric can include a multivariate time series T H ∈R Τ h ×f, where Τ h is the length of the multivariate time series, and f is the number of features included in the multivariate time series. In some non-limiting embodiments or aspects, the time series T E of the historical target transaction metric can include a univariate time series T E ∈R Τ h ×f, where Τ h is the length of the time series. In some non-limiting embodiments or aspects, for each time step in the time series T H of the historical transaction metric, there may be a corresponding (e.g., one-to-one mapped to) time step in the time series T E of the historical target transaction metric.

[0113] In some non-limiting embodiments or aspects, the time series T M 、T U 、T H and T E [[ID=4۷]]can be regarded as two separate pairs, where the contemporary transaction metric T M and the contemporary target transaction metric T U form the first pair, and the historical transaction metric T H and the historical target transaction metric T E form the second pair. For all four transaction metrics, T[i:j] can be used to represent the sequence (e.g., a subsequence of the total sequence) starting at the i-th data point (e.g., timestamp) and ending at the j-th data point.

[0114] In some non - limiting embodiments, a time gap g may be provided between the start of the time series of contemporary transaction metrics and the end of the time series of historical transaction metrics. In one example, the start of the time series of contemporary transaction metrics is s, and the start of the time series of historical transaction metrics may be described by the formula: s - g - Τ h .

[0115] In some non - limiting embodiments or aspects, the time series T of contemporary transaction metrics M and the time series T of contemporary target transaction metrics U may have the same length. In some non - limiting embodiments or aspects, the length of each time series may be based on the sampling rate of the time series. In one example, if T M and T U have the same sampling rate, then T M and T U may have the same length. In some non - limiting embodiments or aspects, the time series T of historical transaction metrics H and / or the time series T of historical target transaction metrics E may have a length different from that of T M and / or T U because more historical data points may be available for the attention layer compared to T M and / or T U .

[0116] As Figure 5B shown, the processing layer 502 may include an RFFT layer. As Figure 5B further shown, the input to the processing layer 502 may include input data (e.g., features of the time series T M of data points associated with contemporary transaction metrics, features of the time series T H of data points associated with historical transaction metrics, and / or features of the time series T E of data points associated with historical target transaction metrics). In some non - limiting embodiments or aspects, the time series analysis system 102 may provide the input data to the processing layer 502. In some non - limiting embodiments or aspects, the time series analysis system 102 may provide the input data to the RFFT layer of the processing layer 502. For example, the time series analysis system 102 may provide features of the time series of data points associated with contemporary transaction metrics, features of the time series of data points associated with historical transaction metrics, and / or features of the time series of data points associated with historical target transaction metrics as input to the RFFT layer of the processing layer 502.

[0117] As Figure 5BFurther shown, the time series analysis system 102 may apply the RFFT to each feature individually using the RFFT layer of the processing layer 502. For example, the time series analysis system 102 may apply the RFFT to each feature of the time series of data points associated with contemporary transaction metrics using the RFFT layer of the processing layer 502. The time series analysis system 102 may apply the RFFT to each feature of the time series of data points associated with historical transaction metrics and / or each feature of the time series of data points associated with historical target transaction metrics using the RFFT layer of the processing layer 502. In some non-limiting embodiments or aspects, the time series analysis system 102 may generate an output based on applying the RFFT to each feature individually using the RFFT layer of the processing layer 502. For example, the time series analysis system 102 may generate the real and imaginary parts of each feature of the time series of data points associated with contemporary transaction metrics, each feature of the time series of data points associated with historical transaction metrics, and / or each feature of the time series of data points associated with historical target transaction metrics based on the time series analysis system 102 applying the RFFT to each feature individually using the RFFT layer of the processing layer 502.

[0118] In some non-limiting embodiments or aspects, the time series analysis system 102 may split the output into the real and imaginary parts of each feature. For example, the time series analysis system 102 may split the output of each feature of the time series of data points associated with contemporary transaction metrics, each feature of the time series of data points associated with historical transaction metrics, and / or each feature of the time series of data points associated with historical target transaction metrics into the real and imaginary parts of each feature. In some non-limiting embodiments or aspects, the time series analysis system 102 may individually combine (e.g., group) the real and imaginary parts of each feature (e.g., each feature of the time series of data points associated with contemporary transaction metrics, historical transaction metrics, and / or historical target transaction metrics) based on the time series analysis system 102 splitting the output into the real and imaginary parts of each feature.

[0119] In some non - limiting embodiments or aspects, the time - series analysis system 102 may combine the real parts of each feature to generate a combined real part of the features. For example, the time - series analysis system 102 may combine the real parts of each feature of the time series of data points associated with a contemporary transaction metric to generate a combined real part of the features of the time series of data points associated with the contemporary transaction metric. The time - series analysis system 102 may combine the real parts of each feature of the time series of data points associated with a historical transaction metric to generate a combined real part of the features of the time series of data points associated with the historical transaction metric. The time - series analysis system 102 may combine the real parts of each feature of the time series of data points associated with a historical target transaction metric to generate a combined real part of the features of the time series of data points associated with the historical target transaction metric.

[0120] In some non - limiting embodiments or aspects, the time - series analysis system 102 may combine the imaginary parts of each feature to generate a combined imaginary part of the features. For example, the time - series analysis system 102 may combine the imaginary parts of each feature of the time series of data points associated with a contemporary transaction metric to generate a combined imaginary part of the features of the time series of data points associated with the contemporary transaction metric. The time - series analysis system 102 may combine the imaginary parts of each feature of the time series of data points associated with a historical transaction metric to generate a combined imaginary part of the features of the time series of data points associated with the historical transaction metric. The time - series analysis system 102 may combine the imaginary parts of each feature of the time series of data points associated with a historical target transaction metric to generate a combined imaginary part of the features of the time series of data points associated with the historical target transaction metric.

[0121] In some non - limiting embodiments or aspects, the time - series analysis system 102 may generate an output of the processing layer 502. For example, the time - series analysis system 102 may generate an output of the processing layer 502, the output including the combined real part and the combined imaginary part of the features of the time series of data points associated with a contemporary transaction metric, the combined real part and the combined imaginary part of the features of the time series of data points associated with a historical transaction metric, and / or the combined real part and the combined imaginary part of the features of the time series of data points associated with a historical target transaction metric.

[0122] In some non - limiting embodiments or aspects, the time - series input T M , having f features, where the k - th feature is denoted as T Mk = [t0, ···, t ΤC , and the discrete Fourier transform will output a series of complex numbers at the m - th time of the k - th feature, as:

[0123]

[0124]

[0125] F* Mk,m The real and imaginary parts of F* are denoted as F R * Mk,m and FI* Mk,m The transformations of Equations 1 and 2 can also be applied to other time series inputs T H and T E .

[0126] Considering the time step, the contemporary transaction metric T M , the historical transaction metric T H and the historical target transaction metric T E , the interval g between the contemporary and historical cycles, the historical length (e.g., the length of T H and T E ) t h , and the prediction length (e.g., the length of T M ) t p , the model F can be written as:

[0127]

[0128] where t a and t b are the left and right boundaries of the target prediction period such that [-t a :+t b is the target prediction period, and v = -t p -g-t h .

[0129] In some non - limiting embodiments or aspects, the optimization objective during training can be defined as follows:

[0130]

[0131] where μ - t p -g - t h ≥0. (3)

[0132] In some non - limiting embodiments or aspects, the input features F M of the time series of the contemporary transaction metric (T M ), the input features F H of the time series of the historical transaction metric (T H ) and the input features F E of the time series of the historical target transaction metric (T EIt can be transformed from the time domain to the frequency domain through a Fourier layer (e.g., a layer including a Fourier transform such as DFT, e.g., FFT, RFFT, etc.). This transformation enables subsequent layers to capture the underlying patterns of trading metric features. Specifically, for F M and F H the k-th feature in:

[0133]

[0134] Specifically, for F E the k-th feature in:

[0135]

[0136] where each complex number in the sequence is transformed by a Fourier transform including a Fourier transform such as DFT, e.g., FFT, RFFT, etc. Since the input features F M 、F H and F E are real numbers, the outputs F* Mk 、F* Hk and F* E of the Fourier transform are symmetric. This means that if F* Mk includes multiple time steps (e.g., 168 time steps), then there will be half of that number of frequencies (e.g., 84 different frequencies) in F* Mk .

[0137] Thereafter, the complex value output is split into a real part and an imaginary part:

[0138]

[0139]

[0140]

[0141] As ​ shown, the feature extraction component 504 may include one or more one-dimensional convolutional layers (e.g., one-dimensional feature extraction convolutional layers) and dropout layers. As ​ shown, the input of the feature extraction component 504 may include features (e.g., features of a time series T M of data points associated with contemporary trading metrics, features of a time series T H of data points associated with historical trading metrics, and / or features of a time series T E of data points associated with historical target trading metrics)The combined real part of the features and the combined imaginary part of the features. In some non-limiting embodiments or aspects, the time series analysis system 102 may provide an input to the feature extraction component 504. In some non-limiting embodiments or aspects, the time series analysis system 102 may provide an input to one or more one-dimensional feature extraction convolutional layers of the feature extraction component 504. For example, the time series analysis system 102 may provide the combined real part of the features and the combined imaginary part of the features of a time series of data points associated with contemporary transaction metrics (e.g., the output of the processing layer 502), the combined real part of the features and the combined imaginary part of the features of a time series of data points associated with historical transaction metrics, and / or the combined real part of the features and the combined imaginary part of the features of a time series of data points associated with historical target transaction metrics, as an input to one or more one-dimensional feature extraction convolutional layers of the feature extraction component 504.

[0142] In some non-limiting embodiments or aspects, the time series analysis system 102 may use the feature extraction component 504 to extract features in the frequency domain. For example, the time series analysis system 102 may use one or more one-dimensional feature extraction convolutional layers of the feature extraction component 504 to extract features of a time series of data points associated with contemporary transaction metrics in the frequency domain. The time series analysis system 102 may use one or more one-dimensional feature extraction convolutional layers of the feature extraction component 504 to extract features of a time series of data points associated with historical transaction metrics in the frequency domain. The time series analysis system 102 may use one or more one-dimensional feature extraction convolutional layers of the feature extraction component 504 to extract features of a time series of data points associated with historical target transaction metrics in the frequency domain.

[0143] In some non-limiting embodiments or aspects, the time series analysis system 102 may generate an output including features extracted from one or more one-dimensional feature extraction convolutional layers of the feature extraction component 504. In some non-limiting embodiments or aspects, the time series analysis system 102 may provide an output including features extracted from one or more one-dimensional feature extraction convolutional layers as an input to the dropout layer of the feature extraction component 504.

[0144] In some non - limiting embodiments or aspects, the time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate real - valued feature vectors (e.g., real - valued contemporary transaction metric feature vectors, real - valued historical transaction metric feature vectors, and / or real - valued historical target transaction metric feature vectors) based on the input provided by the time - series analysis system 102 to the dropout layer (e.g., the output of the one - dimensional feature - extraction convolutional layer for the combined real part of the features). For example, the time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate a real - valued contemporary transaction metric feature vector. The time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate a real - valued historical transaction metric feature vector. The time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate a real - valued historical target transaction metric feature vector. In some non - limiting embodiments or aspects, the use of the dropout layer in the feature extraction component 504 can be restricted.

[0145] In some non - limiting embodiments or aspects, the time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate imaginary - valued feature vectors (e.g., imaginary - valued contemporary transaction metric feature vectors, imaginary - valued historical transaction metric feature vectors, and / or imaginary - valued historical target transaction metric feature vectors) based on the input provided by the time - series analysis system 102 to the dropout layer (e.g., the output of the one - dimensional feature - extraction convolutional layer for the combined imaginary part of the features). For example, the time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate an imaginary - valued contemporary transaction metric feature vector. The time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate an imaginary - valued historical transaction metric feature vector. The time - series analysis system 102 can use the dropout layer of the feature extraction component 504 to generate an imaginary - valued historical target transaction metric feature vector. In some non - limiting embodiments or aspects, the use of the dropout layer in the feature extraction component 504 can be restricted.

[0146] Before being used as input to a one - dimensional convolutional layer (Conv1D), F M and F H All transformed features in can be combined:

[0147]

[0148] The last step in the feature extraction process performed using the feature extraction component 504 can be to use the Conv1D layer to extract features. A convolutional neural network (CNN) including the Conv1D layer can be defined as:

[0149]

[0150] where l≥1 represents the l-th layer, h (l) is the output of the l-th layer, w (l) and b (l) are the weights and biases of the l-th layer. x in the first layer (l = 0) represents F R* M 、F I* M 、F R * H 、F I* H 、F R* E and F I* E 。For the above six inputs, the corresponding outputs of the Conv1D layer are denoted as h (l) MR 、h (l) MI 、h (l) HR 、h (l) HI 、h (l) ER and h (l) EI 。

[0151] As ​ shown, the dual attention component 506 may include one or more dot product layers (e.g., one or more layers including a dot product function that produces a dot product) and one or more softmax layers. In some non-limiting embodiments or aspects, the dual attention component 506 may include an activation function layer instead of a dot product layer. In some non-limiting embodiments or aspects, the activation function layer may include an addition function and / or a tanh activation function.

[0152] As ​Further shown, the input to the dual attention component 506 can include real feature vectors and / or imaginary feature vectors (e.g., real contemporary transaction metric feature vectors, real historical transaction metric feature vectors, and / or real historical target transaction metric feature vectors). In some non-limiting embodiments or aspects, the time series analysis system 102 can provide the input to the dual attention component 506. In some non-limiting embodiments or aspects, the time series analysis system 102 can provide the input to one or more dot product layers of the dual attention component 506. For example, the time series analysis system 102 can provide real contemporary transaction metric feature vectors, real historical transaction metric feature vectors, and / or real historical target transaction metric feature vectors (e.g., the output of the feature extraction component 504) as the input to one or more dot product layers of the dual attention component 506. As another example, the time series analysis system 102 can provide imaginary contemporary transaction metric feature vectors, imaginary historical transaction metric feature vectors, and / or imaginary historical target transaction metric feature vectors (e.g., the output of the feature extraction component 504) as the input to one or more dot product layers of the dual attention component 506. In some non-limiting embodiments or aspects, the dual attention component 506 can process the dominant frequency.

[0153] In some non-limiting embodiments or aspects, the time series analysis system 102 can provide the input to one or more activation function layers of the dual attention component 506. For example, the time series analysis system 102 can provide real contemporary transaction metric feature vectors, real historical transaction metric feature vectors, and / or real historical target transaction metric feature vectors (e.g., the output of the feature extraction component 504) as the input to one or more activation function layers of the dual attention component 506. As another example, the time series analysis system 102 can provide imaginary contemporary transaction metric feature vectors (e.g., the output of the feature extraction component 504), imaginary historical transaction metric feature vectors, and / or imaginary historical target transaction metric feature vectors as the input to one or more activation function layers of the dual attention component 506.

[0154] In some non - limiting embodiments or aspects, the time - series analysis system 102 may perform an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector using the activation function layer of the dual - attention component 506. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to the tanh activation function. In some non - limiting embodiments or aspects, the time - series analysis system 102 may perform the tanh activation function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector using the activation function layer of the dual - attention component 506. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the result of the tanh activation function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to the softmax layer of the dual - attention component 506. For example, the time - series analysis system 102 may provide the output of the tanh activation function as an input to the softmax layer of the dual - attention component 506 to generate a real attention feature vector.

[0155] In some non - limiting embodiments or aspects, the time - series analysis system 102 may generate a dot - product layer of the real attention feature vector and the real historical target transaction metric feature vector. For example, the time - series analysis system 102 may generate a dot - product layer of the real attention feature vector and the real historical target transaction metric feature vector to generate a real component attention matrix.

[0156] In some non - limiting embodiments or aspects, the time - series analysis system 102 may perform an addition function on the virtual contemporary transaction metric feature vector and the virtual historical transaction metric feature vector using the activation function layer of the dual - attention component 506. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the result of the addition function of the virtual contemporary transaction metric feature vector and the virtual historical transaction metric feature vector as an input to the tanh activation function. In some non - limiting embodiments or aspects, the time - series analysis system 102 may perform the tanh activation function on the virtual contemporary transaction metric feature vector and the virtual historical transaction metric feature vector using the activation function layer of the dual - attention component 506. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the result of the tanh activation function of the virtual contemporary transaction metric feature vector and the virtual historical transaction metric feature vector as an input to the softmax layer of the dual - attention component 506. For example, the time - series analysis system 102 may provide the output of the tanh activation function as an input to the softmax layer of the dual - attention component 506 to generate a virtual attention feature vector.

[0157] In some non - limiting embodiments or aspects, the time - series analysis system 102 may use the dot - product layer of the dual - attention component 506 to generate the dot - product layer of the virtual contemporary transaction metric feature vector and the virtual historical transaction metric feature vector. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the dot - product layer of the virtual contemporary transaction metric feature vector and the virtual historical transaction metric feature vector as an input to the softmax layer of the dual - attention component 506. For example, the time - series analysis system 102 may provide the dot - product layer as an input to the softmax layer of the dual - attention component 506 to generate the virtual attention feature vector.

[0158] In some non - limiting embodiments or aspects, the time - series analysis system 102 may generate the dot - product layer of the virtual attention feature vector and the virtual historical target transaction metric feature vector. For example, the time - series analysis system 102 may generate the dot - product layer of the virtual attention feature vector and the virtual historical target transaction metric feature vector to generate the virtual component attention matrix. In some non - limiting embodiments or aspects, the time - series analysis system 102 may generate the real - component attention matrix and / or the virtual - component attention matrix as the output of the dual - attention component 506.

[0159] In some non - limiting embodiments or aspects, the real - component attention matrix may include attention vectors corresponding to each feature (e.g., real attention feature vectors). For example, the real - component attention matrix may include attention vectors corresponding to each feature of the time - series of data points associated with the contemporary transaction metric. The real - component attention matrix may include attention vectors corresponding to each feature of the time - series of data points associated with the historical transaction metric. The real - component attention matrix may include attention vectors corresponding to each feature of the time - series of data points associated with the historical target transaction metric.

[0160] In some non - limiting embodiments or aspects, the virtual - component attention matrix may include attention vectors corresponding to each feature (e.g., virtual attention feature vectors). For example, the virtual - component attention matrix may include attention vectors corresponding to each feature of the time - series of data points associated with the contemporary transaction metric. The virtual - component attention matrix may include attention vectors corresponding to each feature of the time - series of data points associated with the historical transaction metric. The virtual - component attention matrix may include attention vectors corresponding to each feature of the time - series of data points associated with the historical target transaction metric.

[0161] In some non - limiting embodiments or aspects, the inputs to the dual attention component 506 (e.g., real feature vectors and / or imaginary feature vectors) can have the same dimension as the number of features (e.g., the dimension is equal to the number of features). For example, the real contemporary transaction metric feature vector, the real historical transaction metric feature vector, and / or the real historical target transaction metric feature vector can have 14 features and a dimension equal to 14. As another example, the imaginary contemporary transaction metric feature vector, the imaginary historical transaction metric feature vector, and / or the imaginary historical target transaction metric feature vector can have 14 features and a dimension equal to 14.

[0162] In some non - limiting embodiments or aspects, the time - series analysis system 102 can apply an attention mechanism (e.g., the dual attention component 506) to each dimension of the real feature vector and / or the imaginary feature vector. For example, the time - series analysis system 102 can apply the dual attention component 506 to the real contemporary transaction metric feature vector, the real historical transaction metric feature vector, and / or the real historical target transaction metric feature vector for each dimension of the real feature vector. As another example, the time - series analysis system 102 can apply the dual attention component 506 to the imaginary contemporary transaction metric feature vector, the imaginary historical transaction metric feature vector, and / or the imaginary historical target transaction metric feature vector for each dimension of the imaginary feature vector.

[0163] As ​ Further shown, the machine - learning model architecture 500 includes a dual attention component 506, where attention component 506a is for the real part and attention component 506b is for the imaginary part. The goal of using the attention components is that the dual attention component 506 can be generated using the complete frequency - domain historical data (e.g., the time series of the aforementioned historical transaction metrics, the values of the features of the historical transaction metrics, the time series of the historical target transaction metrics, the values of the features of the historical target transaction metrics, etc.), and the dual attention component 506 may be able to guide the prediction of the machine - learning model architecture 500 based on the characteristics of the contemporary data (e.g., the time series of the contemporary transaction metrics, the values of the features of the contemporary transaction metrics, etc.). In some non - limiting embodiments or aspects, if there is an incomplete time period between the historical data and the contemporary data, the time period of the gap can be not used during training (e.g., can be excluded from training). The gap can be relative to the start of the contemporary transaction metric (e.g., the start timestamp), and if the start of the time series of the contemporary transaction metric is different in different time series, the incomplete time period can be different.

[0164] In some non - limiting embodiments or aspects, historical transaction metrics can be used as attention keys, historical target transaction metrics can be used as attention values, and contemporary transaction metrics can be used as queries. For the real and imaginary parts of the input in the machine - learning model architecture 500, the attention component (e.g., the attention layer of the dual - attention component 506) of the dual - attention component 506 can be written as:

[0165]

[0166] As ​ shown, the learning and prediction component 508 can include a connection layer, one or more one - dimensional feature - extraction convolutional layers, a reconstruction layer, a complex layer, and an inverse FFT layer. As ​ further shown, the input to the learning and prediction component 508 can include a real - component attention matrix and / or an imaginary - component attention matrix (e.g., the output of the dual - attention component 506) and a real contemporary transaction metric feature vector and / or an imaginary contemporary transaction metric feature vector. In some non - limiting embodiments or aspects, the learning and prediction component 508 can include a feature - learning component that includes one or more one - dimensional convolutional layers (e.g., feature - learning one - dimensional convolutional layers).

[0167] In some non - limiting embodiments or aspects, the time - series analysis system 102 can provide an input (e.g., the output of the dual - attention component 506) to the learning and prediction component 508. In some non - limiting embodiments or aspects, the time - series analysis system 102 can provide an input to the connection layer of the learning and prediction component 508. For example, the time - series analysis system 102 can provide a real - component attention matrix and a real contemporary transaction metric feature vector as an input to the connection layer of the learning and prediction component 508. As another example, the time - series analysis system 102 can provide an imaginary - component attention matrix and an imaginary contemporary transaction metric feature vector as an input to the connection layer of the learning and prediction component 508.

[0168] In some non - limiting embodiments or aspects, the time - series analysis system 102 may use the learning and prediction component 508 to concatenate inputs (e.g., the output of the dual - attention component 506). For example, the time - series analysis system 102 may concatenate the real - component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation. As another example, the time - series analysis system 102 may concatenate the imaginary - component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the first concatenation as an input to one or more one - dimensional convolutional layers (e.g., the feature - learning one - dimensional convolutional layer). In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the second concatenation as an input to one or more one - dimensional convolutional layers (e.g., the feature - learning one - dimensional convolutional layer). In some non - limiting embodiments or aspects, the time - series analysis system 102 may use one or more feature - learning one - dimensional convolutional layers of the learning and prediction component 508 to learn features. For example, the time - series analysis system 102 may use one or more feature - learning one - dimensional convolutional layers of the learning and prediction component 508 to learn the features of the first concatenation and / or the second concatenation.

[0169] In some non - limiting embodiments or aspects, the time - series analysis system 102 may generate the output of the feature - learning one - dimensional convolutional layer based on the first concatenation and / or the second concatenation. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the output of the feature - learning one - dimensional convolutional layer based on the first concatenation and / or the second concatenation as an input to the reconstruction layer. For example, the time - series analysis system 102 may provide the output of the feature - learning one - dimensional convolutional layer based on the first concatenation to the reconstruction layer to generate a real - component feature vector. As another example, the time - series analysis system 102 may provide the output of the feature - learning one - dimensional convolutional layer based on the second concatenation to the reconstruction layer to generate an imaginary - component feature vector.

[0170] In some non - limiting embodiments or aspects, the time - series analysis system 102 may use the reconstruction layer of the learning and prediction component 508 to reconstruct (e.g., squeeze) the dimensions of the input to the reconstruction layer (e.g., the output of the feature - learning one - dimensional convolutional layer based on the first concatenation and / or the second concatenation), where the dimension of the input has a shape of 1. In some non - limiting embodiments or aspects, the time - series analysis system 102 may provide the output of the reconstruction layer (e.g., the real - component feature vector, the imaginary - component feature vector) as an input to the complex layer of the learning and prediction component 508. In some non - limiting embodiments or aspects, the time - series analysis system 102 may use the output of the reconstruction layer (e.g., the real - component feature vector, the imaginary - component feature vector) to construct (e.g., generate) a complex number.

[0171] In some non - limiting embodiments or aspects, the time - series analysis system 102 can generate complex feature vectors using complex layers of the learning and prediction component 508. For example, the time - series analysis system 102 can generate complex feature vectors using complex layers of the learning and prediction component 508 based on real - component feature vectors and imaginary - component feature vectors. In some non - limiting embodiments or aspects, the time - series analysis system 102 can provide the output of the complex layer (e.g., complex feature vectors) as an input to the inverse FFT layer of the learning and prediction component 508.

[0172] In some non - limiting embodiments or aspects, the time - series analysis system 102 can use the inverse FFT layer of the learning and prediction component 508 to generate contemporary target transaction metrics. For example, the time - series analysis system 102 can use the inverse FFT layer of the learning and prediction component 508 to generate contemporary target transaction metrics by transforming complex feature vectors into predictions for multiple time steps.

[0173] In some non - limiting embodiments or aspects, the context vectors of the real - part input and the imaginary - part input in the machine - learning model architecture 500 (e.g., the first connection generated based on concatenating the real - component attention matrix and the real contemporary transaction metric feature vector or the second connection generated based on concatenating the imaginary - component attention matrix and the imaginary contemporary transaction metric feature vector) can be written as:

[0174]

[0175] where represents the concatenation operation.

[0176] An alternative attention design for the real and imaginary parts of the input can be written as:

[0177]

[0178] In some non - limiting embodiments or aspects, vectors calculated using historical transaction metrics can have a different shape from vectors calculated using contemporary transaction metrics. This can be based on the fact that the time period used for historical transaction metrics is longer than the time period used for contemporary transaction metrics. In this way, in the frequency domain, the vectors calculated using historical transaction metrics can be based on a longer time period than the time period used for contemporary transaction metrics, and the vectors can enable the machine - learning model architecture 500 to better understand what the history (e.g., the historical record of an entity) is like and produce better predictions in the frequency domain.

[0179] After the dual - attention component 506, the last part of the machine - learning model architecture 500 is the learning and prediction component 508. In some non - limiting embodiments or aspects, a Conv1D layer can be used to learn patterns from the previous output. Specifically, h R and hI Can be calculated as:

[0180]

[0181] In the case where both the real part and the imaginary part are transformed, a complex number can be provided as follows:

[0182] h complex = h R + i·h I (15)

[0183] The inverse RFFT can be used to obtain a prediction as a real number (e.g., a prediction of the time series T of contemporary target transaction metrics U ):

[0184]

[0185] Now refer to ​ , ​ is a diagram of a non-limiting example or aspect of implementation 600 of a process for training a machine learning model in a distributed environment.

[0186] As ​As shown, the time series analysis system 102 can retrieve multiple data shards (e.g., data shard 1, data shard 2, data shard 3, and data shard 4) from a data storage device (e.g., a database device, a data storage device, etc.). In some non-limiting embodiments or aspects, the data shards may include indices of time series (e.g., multivariate time series, time series of contemporary transaction metrics, time series of historical transaction metrics, time series of historical target transaction metrics, etc.). For example, the data shards may include indices of time series that can be used to retrieve values of the time series rather than the real-time series. In some non-limiting embodiments or aspects, the time series analysis system 102 can control multiple devices (e.g., device: 1, device: 2, device: 3, and device: 4; each of which is a computing device, such as a server, a central processing unit, a graphics processing unit, etc.) to train a machine learning model based on each of the multiple data shards. Each of the multiple devices may have a copy (e.g., a copy with the same variables) of the machine learning model (e.g., a machine learning model corresponding to the machine learning model architecture 500 or a machine learning model corresponding to a component of the machine learning model architecture 500) stored on the device. The gradients of the machine learning model can be provided by each device, and the gradients (e.g., gradient_1, gradient_2, gradient_3, or gradient_4) can be used to update one or more parameters of each corresponding machine learning model by each corresponding device. In some non-limiting embodiments or aspects, one or more parameters of each corresponding machine learning model can be propagated to each device and used to update (e.g., update with the average of one or more parameters) all copies of the machine learning model stored on all devices. In some non-limiting embodiments, after the time series analysis system 102 compares the performance of the machine learning model using different parameters, the time series analysis system 102 can select the number of devices, the data shards, and / or the size (e.g., batch size) of the machine learning model copies. In this way, the time series analysis system 102 can provide a linear improvement in the speed of training the machine learning model.

[0187] Now refer to ​ , ​ which is a diagram of a non-limiting embodiment or aspect of an implementation 700 of a process for training a machine learning model in a distributed environment. In some non-limiting embodiments or aspects, the steps of implementation 700 may be the same as or similar to the steps of implementation 600.

[0188] As ​As shown, the time series analysis system 102 can retrieve multiple data shards (e.g., data shard 1, data shard 2, data shard 3, data shard 4, data shard 5, data shard 6, data shard 7, and data shard 8) from a data storage device (e.g., a database device, a data storage device, etc.). In some non-limiting embodiments or aspects, the time series analysis system 102 can control multiple devices (e.g., device: 1, device: 2, device: 3, and device: 4; each of which is a computing device, such as a server, a central processing unit, a graphics processing unit, etc.) to train a machine learning model based on each data shard among the multiple data shards. Each device among the multiple devices can have a copy (e.g., copies 1-8) of a machine learning model (e.g., a machine learning model corresponding to the machine learning model architecture 500 or a machine learning model corresponding to a component of the machine learning model architecture 500) stored on the device.

[0189] As ​ Further shown, multiple copies of the machine learning model can be stored on each device, and each device can train multiple copies of the machine learning model based on the multiple data shards. For example, each device can store two copies of the machine learning model, and each device can use two data shards to train each of the two copies of the machine learning model. In some non-limiting embodiments or aspects, the time series analysis system 102 can use a job scheduling algorithm (e.g., a combinatorial optimization algorithm, such as a Glide optimization algorithm) to optimize the training steps of each copy of the machine learning model.

[0190] In some non-limiting embodiments or aspects, and similar to embodiment 600, each device can provide gradients for the machine learning model, and the gradients can be used by each corresponding device to update one or more parameters of each corresponding machine learning model. In some non-limiting embodiments or aspects, one or more parameters of each corresponding machine learning model can be propagated to each device and used to update (e.g., update with the average of one or more parameters) all copies of the machine learning model stored on all devices. In this way, compared with an embodiment that only uses a single data shard to train a single copy of the machine learning model, the time series analysis system 102 can provide an improvement in the speed of training the machine learning model.

[0191] Although the present disclosure has been described in detail for purposes of illustration based on presently considered to be the most practical and preferred embodiments or aspects, it is to be understood that such details are for that purpose only and that the present disclosure is not limited to the disclosed embodiments or aspects, but rather is intended to cover modifications and equivalent arrangements within the spirit and scope of the appended claims. For example, it is to be understood that the present disclosure contemplates, to the extent possible, that one or more features of any embodiment may be combined with one or more features of any other embodiment.

Claims

1. A system, comprising: at least one processor programmed or configured to: receive a time series of historical data points, wherein the historical data points include values of multiple transaction characteristics for multiple time intervals during an initial time period; determine from a historical time period a first time series of data points associated with a historical transaction metric and a second time series of data points associated with a historical target transaction metric, wherein the historical time period includes a first time period of the initial time period; determine from a contemporary time period a third time series of data points associated with a contemporary transaction metric, wherein the contemporary time period includes a second time period of the initial time period that is after the historical time period, and wherein the historical time period is longer than the contemporary time period; wherein the contemporary transaction metric can be used as a query; train a machine learning model, wherein the machine learning model is configured to provide an output including a predicted time series of data points associated with a contemporary target transaction metric, wherein the contemporary target transaction metric includes values of a target transaction metric during a target prediction period, and wherein when training the machine learning model, the at least one processor is programmed or configured to: generate an output of the machine learning model, wherein the output of the machine learning model includes the contemporary target transaction metric, and wherein the contemporary target transaction metric includes values of a target transaction metric during a target prediction period; provide the first time series of data points, the second time series of data points, and the third time series of data points as inputs to a processing layer of the machine learning model, wherein the processing layer includes a fast Fourier transform (FFT) layer; provide an output of the processing layer as an input to a feature extraction component of the machine learning model; provide an output of the feature extraction component of the machine learning model as an input to a dual attention component of the machine learning model; and provide an output of the dual attention component of the machine learning model as an input to a learning and prediction component of the machine learning model; wherein when providing the first time series of data points, the second time series of data points, and the third time series of data points as inputs to the processing layer of the machine learning model, the at least one processor is programmed or configured to: provide each feature of a first plurality of features of the first time series of data points associated with the historical transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the first plurality of features; provide each feature of a second plurality of features of the second time series of data points associated with the historical target transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the second plurality of features; and provide each feature of a third plurality of features of the third time series of data points associated with the contemporary transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the third plurality of features; Combine the real parts of each of the first plurality of features to generate a combined real part of the first plurality of features; Combine the imaginary parts of each of the first plurality of features to generate a combined imaginary part of the first plurality of features; Combine the real parts of each of the second plurality of features to generate a combined real part of the second plurality of features; Combine the imaginary parts of each of the second plurality of features to generate a combined imaginary part of the second plurality of features; Combine the real parts of each of the third plurality of features to generate a combined real part of the third plurality of features; and Combine the imaginary parts of each of the third plurality of features to generate a combined imaginary part of the third plurality of features; and wherein the output of the processing layer of the machine learning model includes the combined real part of the first plurality of features, the combined imaginary part of the first plurality of features, the combined real part of the second plurality of features, the combined imaginary part of the second plurality of features, the combined real part of the third plurality of features, and the combined imaginary part of the third plurality of features.

2. The system according to claim 1, wherein when training the machine learning model, the at least one processor is programmed or configured to: Provide the combined real part of the first plurality of features as an input to a one-dimensional feature extraction convolutional layer; Provide the combined imaginary part of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined real part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined imaginary part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined real part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined imaginary part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; and Generate the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer.

3. The system according to claim 2, wherein the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part, wherein the real input part includes: A first real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features; A second real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features; A third real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features; and wherein the imaginary input part includes: A first imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features; A second imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features; A third imaginary input, based on an output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features; and wherein when training the machine learning model, the at least one processor is programmed or configured to: generate a real-component attention matrix based on the real input part, wherein the real-component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features; and generate an imaginary-component attention matrix based on the imaginary input part, wherein the imaginary-component attention matrix has a plurality of attention vectors equal to the number of features in the plurality of transaction features.

4. The system according to claim 3, wherein the at least one processor is further programmed or configured to: generate the real input part of the input to the dual attention component of the machine learning model; and generate the imaginary input part of the input to the dual attention component of the machine learning model; wherein when generating the real input part of the input to the dual attention component of the machine learning model, the at least one processor is programmed or configured to: provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features as an input to a dropout layer to generate a real contemporary transaction metric feature vector as the first real input; provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input; and wherein when generating the imaginary input part of the input to the dual attention component of the machine learning model, the at least one processor is programmed or configured to: provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features as an input to the dropout layer to generate an imaginary contemporary transaction metric feature vector as the first imaginary input; provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features as an input to the dropout layer to generate an imaginary historical transaction metric feature vector as the second imaginary input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features as an input to the dropout layer to generate an imaginary historical target transaction metric feature vector as the third imaginary input.

5. The system according to claim 4, wherein when generating the real-component attention matrix, the at least one processor is programmed or configured to: generate a dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; Provide the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as the input to the softmax layer to generate a real attention feature vector; and Generate a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein when generating the imaginary component attention matrix, the at least one processor is programmed or configured to: Generate a dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; Provide the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a softmax layer to generate an imaginary attention feature vector; and Generate a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

6. The system according to claim 4, wherein when generating the real component attention matrix, the at least one processor is programmed or configured to: Execute an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; Provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a tanh activation function; Provide the output of the tanh activation function as the input to the softmax layer to generate a real attention feature vector; and Generate a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; wherein when generating the imaginary component attention matrix, the at least one processor is programmed or configured to: Execute an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; Provide the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a tanh activation function; Provide the output of the tanh activation function as an input to a softmax layer to generate an imaginary attention feature vector; and Generate a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

7. The system according to claim 7, wherein when providing the output of the dual attention component of the machine learning model as an input to the learning and prediction component of the machine learning model, the at least one processor is programmed or configured to: Concatenate the real component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation; Provide the first concatenation as an input to a feature learning one-dimensional convolutional layer; Concatenate the imaginary component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation; Provide the second concatenation as an input to the feature learning one-dimensional convolutional layer; Provide an input to a reconstruction layer based on the output of the feature learning one-dimensional convolutional layer for the first concatenation to generate a real component feature vector; and Provide an input to the reconstruction layer based on the output of the feature learning one-dimensional convolutional layer for the second concatenation to generate an imaginary component feature vector.

8. The system according to claim 7, wherein when generating the output of the machine learning model, the at least one processor is programmed or configured to: Generate a complex feature vector based on the real component feature vector and the imaginary component feature vector; and Provide the complex feature vector as an input to an inverse FFT layer to generate the contemporary target transaction metric.

9. A method, comprising: Receiving, by at least one processor, a time series of historical data points, wherein the historical data points include values of a plurality of transaction features for a plurality of time intervals during an initial time period; Determining, by at least one processor, from a historical time period a first time series of data points associated with a historical transaction metric and a second time series of data points associated with a historical target transaction metric, wherein the historical time period includes a first time period of the initial time period; Determining, by at least one processor, from a contemporary time period a third time series of data points associated with a contemporary transaction metric, wherein the contemporary time period includes a second time period of the initial time period that is after the historical time period, wherein the historical time period is longer than the contemporary time period; wherein the contemporary transaction metric can be used as a query; Training, by at least one processor, a machine learning model, wherein the machine learning model is configured to provide an output including a predicted time series of data points associated with a contemporary target transaction metric, wherein training the machine learning model includes: Generating an output of the machine learning model, wherein the output of the machine learning model includes the contemporary target transaction metric, and wherein the contemporary target transaction metric includes values of the target transaction metric during a target prediction period; Providing the first time series of data points, the second time series of data points, and the third time series of data points as inputs to a processing layer of the machine learning model, wherein the processing layer includes a fast Fourier transform (FFT) layer; Providing an output of the processing layer as an input to a feature extraction component of the machine learning model; Providing an output of the feature extraction component of the machine learning model as an input to a dual attention component of the machine learning model; and Providing an output of the dual attention component of the machine learning model as an input to a learning and prediction component of the machine learning model; wherein providing the first time series of data points, the second time series of data points, and the third time series of data points as inputs to the processing layer of the machine learning model includes: Providing each feature of a first plurality of features of the first time series of data points associated with the historical transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the first plurality of features; Providing each feature of a second plurality of features of the second time series of data points associated with the historical target transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the second plurality of features; and Provide each of the third plurality of features of the third time series of the data points associated with the contemporary transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each of the third plurality of features; Combine the real parts of each of the first plurality of features to generate a combined real part of the first plurality of features; Combine the imaginary parts of each of the first plurality of features to generate a combined imaginary part of the first plurality of features; Combine the real parts of each of the second plurality of features to generate a combined real part of the second plurality of features; Combine the imaginary parts of each of the second plurality of features to generate a combined imaginary part of the second plurality of features; Combine the real parts of each of the third plurality of features to generate a combined real part of the third plurality of features; and Combine the imaginary parts of each of the third plurality of features to generate a combined imaginary part of the third plurality of features; and wherein the output of the processing layer of the machine learning model includes the combined real part of the first plurality of features, the combined imaginary part of the first plurality of features, the combined real part of the second plurality of features, the combined imaginary part of the second plurality of features, the combined real part of the third plurality of features, and the combined imaginary part of the third plurality of features.

10. The method according to claim 9, wherein training the machine learning model comprises: Providing the combined real part of the first plurality of features as an input to a one-dimensional feature extraction convolutional layer; Providing the combined imaginary part of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; Providing the combined real part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; Providing the combined imaginary part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; Providing the combined real part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; Providing the combined imaginary part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; and Generating the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer.

11. The method according to claim 10, wherein the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part, wherein the real input part comprises: A first real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features; A second real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features; A third real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features; and wherein the imaginary input part comprises: A first imaginary input based on the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features; A second virtual input based on an output of the one-dimensional feature extraction convolutional layer for the combined virtual part of the second plurality of features; A third virtual input based on an output of the one-dimensional feature extraction convolutional layer for the combined virtual part of the third plurality of features; and wherein training the machine learning model includes: generating a real component attention matrix based on the real input part, wherein the real component attention matrix has a plurality of attention vectors equal to a number of features among the plurality of transaction features; and generating a virtual component attention matrix based on the virtual input part, wherein the virtual component attention matrix has a plurality of attention vectors equal to a number of features among the plurality of transaction features.

12. The method according to claim 11, wherein the method further includes: generating the real input part of the input to the dual attention component of the machine learning model; and generating the virtual input part of the input to the dual attention component of the machine learning model; wherein generating the real input part of the input to the dual attention component of the machine learning model includes: providing the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features as an input to a dropout layer to generate a real contemporary transaction metric feature vector as the first real input; providing the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and providing the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input; and wherein generating the virtual input part of the input to the dual attention component of the machine learning model includes: providing the output of the one-dimensional feature extraction convolutional layer for the combined virtual part of the first plurality of features as an input to the dropout layer to generate a virtual contemporary transaction metric feature vector as the first virtual input; providing the output of the one-dimensional feature extraction convolutional layer for the combined virtual part of the second plurality of features as an input to the dropout layer to generate a virtual historical transaction metric feature vector as the second virtual input; and providing the output of the one-dimensional feature extraction convolutional layer for the combined virtual part of the third plurality of features as an input to the dropout layer to generate a virtual historical target transaction metric feature vector as the third virtual input.

13. The method according to claim 12, wherein generating the real component attention matrix includes: generating a dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; providing the dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a softmax layer to generate a real attention feature vector; and Generate a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; where generating the imaginary component attention matrix includes: Generate a dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; Provide the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a softmax layer to generate an imaginary attention feature vector; and Generate a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

14. The method according to claim 12, wherein generating the real component attention matrix includes: Perform an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; Provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a tanh activation function; Provide the output of the tanh activation function as an input to a softmax layer to generate a real attention feature vector; and Generate a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; where generating the imaginary component attention matrix includes: Perform an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; Provide the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a tanh activation function; Provide the output of the tanh activation function as an input to a softmax layer to generate an imaginary attention feature vector; and Generate a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

15. The method according to claim 11, wherein providing the output of the dual attention component of the machine learning model as an input to the learning and prediction component of the machine learning model includes: Concatenate the real component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation; Provide the first concatenation as an input to a feature learning one-dimensional convolutional layer; Concatenate the imaginary component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second concatenation; Provide the second concatenation as an input to the feature learning one-dimensional convolutional layer; Provide an input to a reconstruction layer based on the output of the feature learning one-dimensional convolutional layer for the first concatenation to generate a real component feature vector; and Provide an input to the reconstruction layer based on the output of the feature learning one-dimensional convolutional layer for the second concatenation to generate an imaginary component feature vector.

16. The method according to claim 15, wherein generating the output of the machine learning model includes: Generate a complex feature vector based on the real component feature vector and the imaginary component feature vector; and Provide the plurality of eigenvectors as an input to an inverse FFT layer to generate the contemporary target transaction metric.

17. A computer program product comprising at least one non-transitory computer-readable medium including one or more instructions that, when executed by at least one processor, cause the at least one processor to: Receive a time series of historical data points, where the historical data points include values of a plurality of transaction characteristics for a plurality of time intervals during an initial time period; Determine from a historical time period a first time series of data points associated with a historical transaction metric and a second time series of data points associated with a historical target transaction metric, where the historical time period includes a first time period of the initial time period; Determine a third time series of data points associated with contemporary transaction metrics from a contemporary time period, where the contemporary time period includes a second time period of the initial time period that is after the historical time period, where the historical time period is longer than the contemporary time period; wherein, The contemporary transaction metric can be used as a query; Train a machine learning model, where the machine learning model is configured to provide an output including a predicted time series of data points associated with a contemporary target transaction metric, where the contemporary target transaction metric includes values of a target transaction metric during a target prediction period, and where the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: Generate an output of the machine learning model, where the output of the machine learning model includes the contemporary target transaction metric, and where the contemporary target transaction metric includes values of a target transaction metric during a target prediction period; Provide the first time series of data points, the second time series of data points, and a third time series of data points as inputs to a processing layer of the machine learning model, where the processing layer includes a fast Fourier transform (FFT) layer; Provide an output of the processing layer as an input to a feature extraction component of the machine learning model; Provide an output of the feature extraction component of the machine learning model as an input to a dual attention component of the machine learning model; And Provide an output of the dual attention component of the machine learning model as an input to a learning and prediction component of the machine learning model; Where the one or more instructions that cause the at least one processor to provide the first time series of data points, the second time series of data points, and the third time series of data points as inputs to the processing layer of the machine learning model cause the at least one processor to: Provide each feature of a first plurality of features of the first time series of data points associated with the historical transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the first plurality of features; Provide each feature of a second plurality of features of the second time series of data points associated with the historical target transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each feature of the second plurality of features; and Provide each of the third plurality of features of the third time series of the data points associated with the contemporary transaction metric as an input to the FFT layer to generate a real part and an imaginary part of each of the third plurality of features; Combine the real parts of each of the first plurality of features to generate a combined real part of the first plurality of features; Combine the imaginary parts of each of the first plurality of features to generate a combined imaginary part of the first plurality of features; Combine the real parts of each of the second plurality of features to generate a combined real part of the second plurality of features; Combine the imaginary parts of each of the second plurality of features to generate a combined imaginary part of the second plurality of features; Combine the real parts of each of the third plurality of features to generate a combined real part of the third plurality of features; And Combine the imaginary parts of each of the third plurality of features to generate a combined imaginary part of the third plurality of features; And where the output of the processing layer of the machine learning model includes the combined real part of the first plurality of features, the combined imaginary part of the first plurality of features, the combined real part of the second plurality of features, the combined imaginary part of the second plurality of features, the combined real part of the third plurality of features, and the combined imaginary part of the third plurality of features.

18. The computer program product according to claim 17, wherein the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: Provide the combined real part of the first plurality of features as an input to a one-dimensional feature extraction convolutional layer; Provide the combined imaginary part of the first plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined real part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined imaginary part of the second plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined real part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; Provide the combined imaginary part of the third plurality of features as an input to the one-dimensional feature extraction convolutional layer; And Generate the output of the feature extraction component of the machine learning model based on the output of the one-dimensional feature extraction convolutional layer.

19. The computer program product according to claim 18, wherein the input to the dual attention component of the machine learning model includes a real input part and an imaginary input part, where the real input part includes: A first real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features; A second real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features; A third real input based on the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features; And where the imaginary input part includes: a first imaginary input based on an output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features; a second imaginary input based on an output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features; a third imaginary input based on an output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features; and wherein the one or more instructions that cause the at least one processor to train the machine learning model cause the at least one processor to: generate a real-component attention matrix based on the real input portion, wherein the real-component attention matrix has a plurality of attention vectors equal to a number of features in the plurality of transaction features; and generate an imaginary-component attention matrix based on the imaginary input portion, wherein the imaginary-component attention matrix has a plurality of attention vectors equal to a number of features in the plurality of transaction features.

20. The computer program product of claim 19, wherein the one or more instructions further cause the at least one processor to: generate the real input portion of the input to the dual attention component of the machine learning model; and generate the imaginary input portion of the input to the dual attention component of the machine learning model; wherein the one or more instructions that cause the at least one processor to generate the real input portion of the input to the dual attention component of the machine learning model cause the at least one processor to: provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the first plurality of features as an input to a dropout layer to generate a real contemporary transaction metric feature vector as the first real input; provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the second plurality of features as an input to the dropout layer to generate a real historical transaction metric feature vector as the second real input; and provide the output of the one-dimensional feature extraction convolutional layer for the combined real part of the third plurality of features as an input to the dropout layer to generate a real historical target transaction metric feature vector as the third real input; and wherein the one or more instructions that cause the at least one processor to generate the imaginary input portion of the input to the dual attention component of the machine learning model cause the at least one processor to: provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the first plurality of features as an input to the dropout layer to generate an imaginary contemporary transaction metric feature vector as the first imaginary input; provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the second plurality of features as an input to the dropout layer to generate an imaginary historical transaction metric feature vector as the second imaginary input; and Provide the output of the one-dimensional feature extraction convolutional layer for the combined imaginary part of the third plurality of features as an input to the dropout layer to generate an imaginary historical target transaction metric feature vector as the third imaginary input.

21. The computer program product according to claim 20, wherein the one or more instructions that cause the at least one processor to generate the real component attention matrix cause the at least one processor to: Generate a dot product of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; Provide the dot product of the contemporary transaction metric feature vector and the historical transaction metric feature vector as an input to the softmax layer to generate a real attention feature vector; And Generate a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; Wherein the one or more instructions that cause the at least one processor to generate the imaginary component attention matrix cause the at least one processor to: Generate a dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; Provide the dot product of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a softmax layer to generate an imaginary attention feature vector; And Generate a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

22. The computer program product according to claim 20, wherein the one or more instructions that cause the at least one processor to generate the real component attention matrix cause the at least one processor to: Execute an addition function on the real contemporary transaction metric feature vector and the real historical transaction metric feature vector; Provide the result of the addition function of the real contemporary transaction metric feature vector and the real historical transaction metric feature vector as an input to a tanh activation function; Provide the output of the tanh activation function as the input to the softmax layer to generate a real attention feature vector; And Generate a dot product of the real attention feature vector and the real historical target transaction metric feature vector to generate the real component attention matrix; Wherein the one or more instructions that cause the at least one processor to generate the imaginary component attention matrix cause the at least one processor to: Execute an addition function on the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector; Provide the result of the addition function of the imaginary contemporary transaction metric feature vector and the imaginary historical transaction metric feature vector as an input to a tanh activation function; Provide the output of the tanh activation function as an input to a softmax layer to generate an imaginary attention feature vector; And Generate a dot product of the imaginary attention feature vector and the imaginary historical target transaction metric feature vector to generate the imaginary component attention matrix.

23. The computer program product according to claim 20, wherein the one or more instructions that cause the at least one processor to provide the output of the dual attention component of the machine learning model as an input to the learning and prediction component of the machine learning model cause the at least one processor to: Concatenate the real component attention matrix and the real contemporary transaction metric feature vector to generate a first concatenation; Provide the first connection as an input to the feature learning one-dimensional convolutional layer; Connect the imaginary component attention matrix and the imaginary contemporary transaction metric feature vector to generate a second connection; Provide the second connection as an input to the feature learning one-dimensional convolutional layer; Based on the output of the feature learning one-dimensional convolutional layer for the first connection, provide an input to the reconstruction layer to generate a real component feature vector; And Based on the output of the feature learning one-dimensional convolutional layer for the second connection, provide an input to the reconstruction layer to generate an imaginary component feature vector.

24. The computer program product according to claim 23, wherein the one or more instructions that cause the at least one processor to generate the output of the machine learning model cause the at least one processor to be programmed or configured to: Generate a complex feature vector based on the real component feature vector and the imaginary component feature vector; and Provide the complex feature vector as an input to the inverse FFT layer to generate the contemporary target transaction metric.

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