Fuzzy prediction method for non-stationary time series

Abstract

This application relates to the field of stationary time series analysis and prediction technology, and discloses a fuzzy prediction method for non-stationary time series, including: determining the fuzzy posterior probability corresponding to the current sequence; updating the fuzzy parameters of the fuzzy set based on the fuzzy posterior probability; generating a rule base associated with the updated fuzzy parameters; performing probabilistic defuzzification prediction based on the rule base to obtain the prediction result; updating the model parameters of the Gaussian mixture model based on the prediction error set corresponding to the prediction result; determining the KL divergence corresponding to the prediction error set based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window; wherein the current window consists of a first half window and a second half window; the prediction error set consists of the first half error set and the second half error set; and performing prediction based on the KL divergence to obtain the output sequence. This solution can meet the practical prediction needs of non-stationary time series.

Description

Technical Field

[0001] This application relates to the field of stationary time series analysis and forecasting technology, specifically to, but not limited to, a fuzzy forecasting method for non-stationary time series. Background Technology

[0002] The potential changes in the distribution of data streams over time in non-stationary time series enable them to match data changes related to seasonality, periodicity, and sudden events. Therefore, non-stationary time series have wide applications in various scenarios and fields. Consequently, accurate prediction of non-stationary time series has become an important prerequisite for their application in these scenarios and fields.

[0003] To address the aforementioned technical issues, related technologies have provided time-varying fuzzy time series prediction frameworks. However, these frameworks can only predict non-stationary time series related to training samples. When faced with application scenarios that require processing new data points, the training process needs to be re-executed.

[0004] Therefore, the prediction framework provided by the relevant technologies cannot meet the prediction needs of non-stationary time series. Summary of the Invention

[0005] Based on the above technical problems, this application provides a fuzzy prediction method for non-stationary time series, which can meet the actual prediction needs of non-stationary time series.

[0006] The technical solution provided in this application is as follows: This application provides a fuzzy prediction method for non-stationary time series, including: Determine the fuzzy posterior probability corresponding to the current sequence; wherein, the current sequence includes the proper subset sequence of the input sequence corresponding to the current window; the input sequence includes a non-stationary time series within at least one time period; The fuzzy parameters of the fuzzy set are updated based on the fuzzy posterior probability; wherein the fuzzy set corresponds to the input sequence; Generate a rule base that is associated with the updated fuzzy parameters; Based on the rule base, probabilistic defuzzification prediction is performed to obtain the prediction result; Based on the prediction error set corresponding to the prediction result, the model parameters of the Gaussian mixture model are updated; wherein, the model parameters include the k-th mean, k-th variance of the k-th Gaussian component in the Gaussian mixture model, and the k-th weight of the k-th Gaussian component in the Gaussian mixture model; k is an integer greater than or equal to 1; Based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window, the KL divergence corresponding to the prediction error set is determined; wherein, the current window is composed of the first half window and the second half window; the prediction error set is composed of the first half error set and the second half error set; Prediction is performed based on the KL divergence to obtain the output sequence.

[0007] The fuzzy prediction method for non-stationary time series provided in this application has at least the following beneficial effects: The fuzzy prediction method for non-stationary time series provided in this application determines the fuzzy posterior probability corresponding to the current sequence. The current sequence includes a proper subset sequence of the input sequence corresponding to the current window. The input sequence includes a non-stationary time series within at least one time period. Thus, through the fuzzy posterior probability, flexible fuzzy partitioning of the data in the current sequence can be achieved. Furthermore, the fuzzy parameters of the fuzzy set are updated based on the fuzzy posterior probability, thereby achieving targeted updates to the fuzzy parameters. On this basis, a rule base associated with the updated fuzzy parameters is generated. Based on the rule base, probabilistic defuzzification prediction is performed to obtain the prediction result, achieving follow-up updates to the rule base. Moreover, the prediction result can integrate the transformation relationships between fuzzy states carried by the rule base, which can improve the stationarity of the prediction result and reduce the uncertainty of the prediction result. At the same time, based on the prediction error set corresponding to the prediction result, the model parameters of the Gaussian mixture model are updated. The model parameters include the k-th mean, k-th variance, and k-th variance of the k-th Gaussian component in the Gaussian mixture model. By assigning the k-th weight, the system can adaptively fit the multi-form, time-varying error distribution characteristics of non-stationary time series. Furthermore, based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window, the KL divergence corresponding to the prediction error set is determined. The current window consists of the first half window and the second half window, and the prediction error set consists of the first half error set and the second half error set. Thus, combining the previously mentioned steps of obtaining prediction results and updating the Gaussian mixture model parameters, this scheme constructs a closed-loop linkage mechanism for probabilistic defuzzification prediction, prediction error set determination, and KL divergence determination. This improves the performance of the Gaussian mixture model in adaptively processing non-stationary time series with complex time-varying data. Based on this, prediction is performed based on the KL divergence to obtain the output sequence, realizing the correlation between the output sequence and the KL divergence. This improves the non-stationary change characteristics of the output sequence, increases its accuracy, and ultimately meets the actual prediction needs of non-stationary time series. Attached Figure Description

[0008] Figure 1 A flowchart illustrating the fuzzy prediction method for non-stationary time series provided in this application embodiment; Figure 2 This is another schematic diagram of the fuzzy prediction method provided in the embodiments of this application; Figure 3 A visualization diagram of a real dataset provided in the embodiments of this application; Figure 4 A visual illustration of the synthetic dataset provided in the embodiments of this application; Figure 5A Statistical results of the root mean square error of the prediction results of the synthetic dataset provided in this application's embodiments and the comparative scheme respectively; Figure 5B Statistical results of the root mean square error of the prediction results of the present solution and the comparative solution provided in the embodiments of this application on the real dataset; Figure 6A Statistical results of the time utility scores of the prediction results of the synthetic dataset provided in the embodiments of this application and the comparative scheme respectively; Figure 6B The statistical results of the time utility scores of the prediction results of the present solution and the comparative solution provided in the embodiments of this application on the real dataset. Detailed Implementation

[0009] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings.

[0010] It should be understood that the specific embodiments described herein are merely illustrative of this application and are not intended to limit this application.

[0011] The potential changes in the distribution of data streams over time in non-stationary time series enable them to match data changes related to seasonality, periodicity, and sudden events. Therefore, non-stationary time series have wide applications in various scenarios and fields. Consequently, accurate prediction of non-stationary time series has become an important prerequisite for their application in these scenarios and fields.

[0012] To address the aforementioned technical issues, related technologies have provided time-varying fuzzy time series prediction frameworks. However, these frameworks can only predict non-stationary time series related to training samples. When faced with application scenarios that require processing new data points, the training process needs to be re-executed.

[0013] Therefore, the prediction framework provided by the relevant technologies cannot meet the prediction needs of non-stationary time series.

[0014] Based on the above technical problems, this application provides a fuzzy prediction method for non-stationary time series. Figure 1This is a flowchart illustrating the fuzzy prediction method for non-stationary time series provided in an embodiment of this application. Figure 1 As shown, the method may include the following steps: Step 101: Determine the fuzzy posterior probability corresponding to the current sequence.

[0015] The current sequence includes the proper subset sequence of the input sequence corresponding to the current window; the input sequence includes a non-stationary time series within at least one time period.

[0016] In some embodiments, at least one time period may include at least one historical time period; for example, the time period corresponding to the current window may be a proper subset of at least one historical time period.

[0017] In some embodiments, the current window can slide within a time range corresponding to at least one time period; therefore, the current window can also be called a sliding window.

[0018] In some embodiments, the fuzzy posterior probability can characterize the conditional probability that a data point in a non-stationary time series belongs to a certain fuzzy set; specifically, the fuzzy posterior probability can be determined in the following ways: First, initialize the fuzzy set and the probabilistic fuzzification distribution. Second, calculate the posterior probability of each data point in the current sequence to the fuzzy set based on the probabilistic fuzzification distribution.

[0019] For example, initializing the fuzzy set and the probabilistic fuzzification distribution can be achieved in the following way: First, construct fuzzy sets; second, count the number of data points in the current sequence that fall into a certain fuzzy set, and determine the prior weight of each fuzzy set based on the number of data points; third, calculate the fuzzy posterior probability of each data point in the current sequence to the fuzzy set based on the prior weight of each fuzzy set.

[0020] Specifically, when a certain fuzzy set is the k-th fuzzy set, the k-th prior weight of the k-th fuzzy set can be calculated using equation (1): (1); Where K is an integer greater than 1, used to represent the total number of fuzzy sets. Let k be the k-th prior weight of the k-th fuzzy set. This represents the number of data points in the current sequence that fall into the k-th fuzzy set. This is an indicator function; its value is 1 when the condition is true and 0 when the condition is false. Let the center value of the k-th fuzzy set be... The length or width of the k-th fuzzy set. Let k be the kth observation or the kth data point in the current sequence, where k and j are both integers greater than or equal to 1 and less than or equal to K.

[0021] Specifically, the fuzzy posterior probability of the k-th data point in the current sequence to the k-th fuzzy set can be calculated using equation (2): (2); in, For the j-th data point For the k-th fuzzy set The fuzzy posterior probability, For the conditional expectation operator, For the j-th data point to the k-th fuzzy set membership degree , , These are the initial fuzzy parameters for the k-th fuzzy set.

[0022] Step 102: Update the fuzzy parameters of the fuzzy set based on the fuzzy posterior probability.

[0023] The fuzzy set corresponds to the input sequence.

[0024] In some embodiments, the fuzzy parameters of each fuzzy set can be updated according to the fuzzy posterior probability of each fuzzy set; for example, the fuzzy parameters may include estimates of the fuzzy set center, the fuzzy set width, and the fuzzy set prior weights, respectively.

[0025] Specifically, the estimated values ​​of the center of the k-th fuzzy set, the width of the k-th fuzzy set, and the prior weight of the k-th fuzzy set can be calculated using equation (3): (3); in, The estimated value of the center of the k-th fuzzy set. Let be the estimated width of the k-th fuzzy set. is the estimated value of the k-th prior weight, and N is the total number of data points in the current sequence.

[0026] Step 103: Generate a rule base associated with the updated fuzzy parameters.

[0027] In some embodiments, the rule base may include state transition relationships between different fuzzy states in a fuzzy set; for example, the rule base may be generated in the following manner: First, based on the updated fuzzy parameters, the fuzzy set to which the data points in the current sequence belong is determined. Then, the state transition relationships between different fuzzy sets are analyzed to generate a rule base. Specifically, the rule base can be calculated using equation (4): (4); in, For the rule base, Let be the fuzzy state at time t. The fuzzy state at time t-1 Let j represent the state transition relationship between the fuzzy state at time t-1 and the fuzzy state at time t, where j is the data point. The index of the final fuzzy set to which it falls, where T is an integer greater than 1, used to represent the number of fuzzy states.

[0028] Step 104: Perform probabilistic defuzzification prediction based on the rule base to obtain the prediction result.

[0029] For example, for fuzzy set time series ( ) and its prediction results In terms of the prediction results of probabilistic defuzzification prediction It can be calculated using equation (5): (5); in, To optimize the problem The solution is used to characterize all elements in the current sequence that are blurred into... The point where the Euclidean distance of the data is minimized. The current sequence is blurred into The data points are 1, where 1 is a vector with all components equal to 1. express Norm, This indicates that the prediction result is . The probability, its expression can be shown in equation (6): (6); In summary, probabilistic defuzzification is an optimization problem to be solved as shown in equation (7): (7); in, This is the defuzzified prediction result corresponding to time t+1. For the variable whose predicted value is to be determined, For fuzzy sets The set of data indices Used to characterize the fuzzy state at time t The state transition relationship between the fuzzy states before time t and the state transition relationship between the fuzzy states. The calculated fuzzy state of a future time at time t.

[0030] Step 105: Update the model parameters of the Gaussian mixture model based on the set of prediction errors corresponding to the prediction results.

[0031] The model parameters include the k-th mean, k-th variance, and k-th weight of the k-th Gaussian component in the Gaussian mixture model; k is an integer greater than or equal to 1.

[0032] In some embodiments, the prediction error set may include the data points in the prediction results and the value differences between the data points at the corresponding time points in the non-stationary time series.

[0033] In some embodiments, a Gaussian mixture model may include multiple Gaussian components, and different Gaussian components may represent different Gaussian distributions; for example, the mean and / or variance of different Gaussian components may be different.

[0034] In some embodiments, model parameters can be updated in the following ways: First, determine the posterior probability that the error in the prediction error set belongs to the k-th Gaussian component in the Gaussian mixture model, and then update the k-th mean, k-th variance, and k-th weight based on the above posterior probability; specifically as shown in equation (8): (8); in, The k-th mean, Let be the k-th variance. For the k-th weight, Let be the posterior probability that the j-th error in the prediction error set belongs to the k-th Gaussian component in the Gaussian mixture model. The mixing weights for the k-th Gaussian component. For the k-th Gaussian component at the j-th error The probability density at that location, These are the initial parameters for the k-th Gaussian component, specifically including the initial mean and initial variance of the k-th Gaussian component. The set of parameters used to characterize each Gaussian component in a Gaussian mixture model. The mixture weight vector used to characterize the Gaussian mixture model. It is the original mean value of the k-th Gaussian component.

[0035] Step 106: Based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window, determine the KL divergence corresponding to the prediction error set.

[0036] The current window consists of a front half window and a back half window; the prediction error set consists of a front half error set and a back half error set.

[0037] In some embodiments, the front half window can be the first half of the current window, and the back half window can be the second half of the current window. The front half window and the back half window can be obtained by dividing the current window equally. Accordingly, the front half error set can include the set of differences between the predicted value and the data point at the same time within the time range corresponding to the front half window, and the back half error set can include the set of differences between the predicted value and the data point at the same time within the time range corresponding to the back half window.

[0038] In some embodiments, KL divergence can be used to assess the degree of difference between the first error distribution represented by the first half of the error set and the second error distribution represented by the second half of the error set. Thus, KL divergence can be used to determine whether concept drift has occurred between the first sequence corresponding to the first half of the window and the second sequence corresponding to the second half of the window in the current sequence.

[0039] In some embodiments, the KL divergence can be determined in the following ways: Random sampling is performed on the first half of the error set to obtain the first sample set, and random sampling is performed on the second half of the error set to obtain the second sample set. Then, log-likelihood processing is performed on the first sample set based on the parameters in the model parameters to obtain the first processing result, and log-likelihood processing is performed on the second sample set based on the parameters in the model parameters to obtain the second processing result. Finally, the first processing result and the second processing result are combined to obtain the KL divergence.

[0040] Step 107: Perform prediction based on KL divergence to obtain the output sequence.

[0041] In some embodiments, the output sequence may include precise values ​​for future time periods obtained by performing defuzzification prediction on the input sequence.

[0042] In some embodiments, the output sequence can be obtained in the following way: If the KL divergence is less than or equal to the divergence threshold, it indicates that no concept drift has occurred between the first sequence corresponding to the first half window and the second sequence corresponding to the second half window in the current sequence. In this case, the prediction result can be determined as the output sequence.

[0043] If the KL divergence is greater than the divergence threshold, it indicates that a concept drift has occurred between the first sequence corresponding to the first half window and the second sequence corresponding to the second half window in the current sequence. At this time, the rule base can be updated, and probabilistic defuzzification prediction can be performed based on the updated rule base to obtain a new prediction result, which is then determined as the output sequence.

[0044] As can be seen from the above, the fuzzy prediction method for non-stationary time series provided in this application determines the fuzzy posterior probability corresponding to the current sequence. The current sequence includes the proper subset sequence corresponding to the current window in the input sequence, and the input sequence includes non-stationary time series within at least one time period. Thus, through the fuzzy posterior probability, flexible fuzzy partitioning of the data in the current sequence can be achieved. Furthermore, the fuzzy parameters of the fuzzy set are updated based on the fuzzy posterior probability, thus achieving targeted updates of the fuzzy parameters. On this basis, a rule base associated with the updated fuzzy parameters is generated, and the prediction result is obtained by performing probabilities-based defuzzification prediction based on the rule base, achieving follow-up updates of the rule base. Moreover, the prediction result can integrate the transformation relationship between the fuzzy states carried by the rule base, which can improve the stationarity of the prediction result and reduce the uncertainty of the prediction result. At the same time, based on the prediction error set corresponding to the prediction result, the model parameters of the Gaussian mixture model are updated. The model parameters include the k-th mean and k-th mean of the k-th Gaussian component in the Gaussian mixture model. By defining the variance and the k-th weight, the model can adaptively fit the multi-morphological and time-varying error distribution characteristics of non-stationary time series. Furthermore, based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window, the KL divergence corresponding to the prediction error set is determined. The current window consists of the first half window and the second half window, and the prediction error set consists of the first half error set and the second half error set. Thus, combining the previously mentioned steps of obtaining prediction results and updating the Gaussian mixture model parameters, this scheme constructs a closed-loop linkage mechanism for probabilistic defuzzification prediction, prediction error set determination, and KL divergence determination. This improves the performance of the Gaussian mixture model in adaptively processing non-stationary time series with complex time-varying data. Based on this, prediction is performed based on the KL divergence to obtain the output sequence, realizing the correlation between the output sequence and the KL divergence. This improves the non-stationary change characteristics of the output sequence, increases the accuracy of the output sequence, and ultimately meets the actual prediction needs of non-stationary time series.

[0045] Based on the foregoing embodiments, the fuzzy prediction method for non-stationary time series provided in this application determines the KL divergence corresponding to the prediction error set based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window. This can be achieved in the following way: Based on the first set of parameters and the first half of the error set, the first average log-likelihood result is determined; based on the second set of parameters and the second half of the error set, the second average log-likelihood result is determined; based on the first average log-likelihood result and the second average log-likelihood result, the KL divergence is determined.

[0046] The first parameter set includes the parameter set of the Gaussian component corresponding to the first half window in the model parameters; the second parameter set includes the parameter set of the Gaussian component corresponding to the second half window in the model parameters; the first time period corresponding to the first half window is earlier than the second time period corresponding to the second half window.

[0047] In some embodiments, the first mean log-likelihood result can be determined in the following manner: Determine the probability density values ​​of the errors in the first half of the error set as represented by the first Gaussian model in the first parameter set, and statistically average the logarithms of the above probability density values ​​to obtain the first average log-likelihood result.

[0048] Accordingly, the second mean log-likelihood result can be determined in the following way: The probability density values ​​of the errors in the second half error set are determined by the second Gaussian model characterized by the second parameter set, and the logarithms of the above probability density values ​​are statistically averaged to obtain the second average log-likelihood result.

[0049] For example, a Gaussian mixture model may include a first Gaussian model and a second Gaussian model.

[0050] For example, the KL divergence can be obtained in the following way: The difference between the second mean log-likelihood result and the first mean log-likelihood result is determined as the KL divergence; specifically, it can be shown in equation (9): (9); in, Let KL divergence be the KL divergence. This is the first mean log-likelihood result. This is the second mean log-likelihood result. For the i-th error in the first half of the error set, For the i-th error in the second half of the error set, For the first parameter set, Let w be the second set of parameters, where w represents the number of prediction errors contained in the first half and the second half of the error set, respectively. Let be the probability density value of the i-th error in the second half error set, as represented by the second parameter set of the second Gaussian model. The error in the first half of the error set is the probability density value of the first Gaussian model characterized by the first parameter set.

[0051] As can be seen from the above, the fuzzy prediction method for non-stationary time series provided in this application determines the first average log-likelihood result based on the first parameter set and the first half error set, and determines the second average log-likelihood result based on the second parameter set and the second half error set. The first parameter set includes the set of parameters of the Gaussian component corresponding to the first half window in the model parameters, and the second parameter set includes the set of parameters of the Gaussian component corresponding to the second half window in the model parameters. The first time period corresponding to the first half window is earlier than the second time period corresponding to the second half window. In this way, the matching degree between the error series corresponding to different time windows in the current window and the distribution represented by the Gaussian component in the Gaussian mixture model is quantified, and the unbiased estimation of the set of observed differences within the first half window and the second half window range is also achieved.

[0052] Based on the foregoing embodiments, the fuzzy prediction method for non-stationary time series provided in this application, which performs prediction based on KL divergence to obtain the output sequence, can be implemented in the following way: The window state of the current window is determined based on at least the KL divergence. Prediction is then performed based on the window state to obtain the output sequence.

[0053] The window state is used to indicate whether the current window should be updated.

[0054] In some embodiments, the window state may include whether to update the duration covered by the current window. For example, if the window state indicates that the duration should be updated, the first duration covered by the current window can be extended to a second duration, and the second duration may be greater than the first duration.

[0055] In some embodiments, the window state may also indicate whether to replace at least a portion of the above time periods with a new time period.

[0056] In some embodiments, the window state can be determined in the following ways: If the KL divergence is greater than or equal to the divergence threshold, the window state is determined to be: update the current window; if the KL divergence is less than the divergence threshold, the window state is determined to be: do not update the current window.

[0057] For example, a KL divergence greater than or equal to a divergence threshold can directly indicate that the errors in the prediction error set have undergone concept drift, and indirectly indicate that the non-stationary time series have undergone concept drift.

[0058] In some embodiments, the output sequence can be implemented in the following ways: If the window state indicates that the current window is not updated, the output sequence is determined as the prediction result; if the window state indicates that the current window is updated, the current window is updated to obtain a new current window, and the method provided in the foregoing embodiment is repeated according to the new current sequence corresponding to the new current window. If it is determined that the window state is that the new current window is not updated, the output sequence corresponding to the new current window is determined as the prediction result.

[0059] As can be seen from the above, the fuzzy prediction method for non-stationary time series provided in this application determines the window state of the current window based at least on KL divergence, performs prediction based on the window state, and obtains the output sequence. The window state is used to indicate whether to update the current window. In this way, the correlation between the window state and KL divergence is realized, thereby improving the accuracy of the window state; and, performing prediction based on the window state can also improve the dynamics and accuracy of the output sequence.

[0060] Based on the foregoing embodiments, the fuzzy prediction method for non-stationary time series provided in this application, which determines the window state of the current window based at least on KL divergence, can be implemented in the following ways: If the KL divergence is less than the divergence threshold, and the time period length corresponding to the current window is less than the time period threshold, the window state is determined as follows: the current window is expanded to obtain a new current window.

[0061] Accordingly, if the KL divergence is greater than or equal to the divergence threshold, or the current time period is greater than or equal to the time period threshold, then the operation of determining the window state to expand the window can be omitted.

[0062] In some embodiments, the divergence threshold and the time period threshold can be predetermined or adjusted, and this application does not limit this.

[0063] In some embodiments, a KL divergence less than a divergence threshold can indicate that no concept drift has occurred for the current sequence.

[0064] In some embodiments, the current window can be expanded in the following ways: The time period length corresponding to the current window is incremented to obtain a new current window. For example, the increment operation may include incrementing the time period length corresponding to the current window by one hour, and correspondingly, the current sequence is also incremented by one data point.

[0065] As can be seen from the above, the fuzzy prediction method for non-stationary time series provided in this application determines the window state as follows: if the KL divergence is less than the divergence threshold and the time period length corresponding to the current window is less than the time period threshold, the current window is expanded to obtain a new current window. In this way, the interrelationship between the KL divergence, the time period length corresponding to the current window, and the expansion of the current window is realized, thereby enabling targeted expansion of the current window and improving the accuracy of the window state.

[0066] Based on the foregoing embodiments, the fuzzy prediction method for non-stationary time series provided in this application, which determines the window state of the current window based at least on KL divergence, can also be implemented in the following ways: If the KL divergence is greater than or equal to the divergence threshold, the window state is determined as follows: remove the data in the current sequence that corresponds to the first half of the window, and construct a new current window based on the second half of the window and the candidate window.

[0067] The current window consists of a first half window and a second half window; the first time period corresponding to the first half window is earlier than the second time period corresponding to the second half window; the third time period corresponding to the candidate window is adjacent to the second time period, and the third time period is later than the second time period.

[0068] Accordingly, if the KL divergence is less than the divergence threshold, the above operations to determine the window state can be omitted.

[0069] In some embodiments, if the KL divergence is greater than or equal to the divergence threshold, it can indicate that a concept drift has occurred between the set of data points corresponding to the first half of the current window and the set of data points corresponding to the second half of the current window.

[0070] In some embodiments, the length of the third time period may be the same as or different from the length of the first time period; however, this application does not limit this.

[0071] In some embodiments, if the current window is updated according to the above window state, the latter half of the window can be the intersection between the current window and the new current window.

[0072] In some embodiments, constructing a new current window based on the latter half of the window and candidate windows can be achieved in the following ways: The second half of the window is designated as the new first half of the window, the candidate window is designated as the new second half of the window, and the window formed by the new first half of the window and the new second half of the window is designated as the new current window.

[0073] As can be seen from the above, the fuzzy prediction method for non-stationary time series provided in this application determines the window state as follows if the KL divergence is greater than or equal to the divergence threshold: The method involves designing data in the current sequence corresponding to the first half of the window, constructing a new current window based on the second half of the window and candidate windows. The current window consists of the first half of the window and the second half of the window. The first time period corresponding to the first half of the window is earlier than the second time period corresponding to the second half of the window. The third time period corresponding to the candidate window is adjacent to the second time period, and the third time period is later than the second time period. Thus, by associating the window state with the KL divergence, the accuracy of the window state can be improved. Furthermore, updating the current window in the above manner can improve the continuity between the time series corresponding to the current window and the new current window.

[0074] Based on the foregoing embodiments, the fuzzy prediction method for non-stationary time series provided in this application, which performs prediction based on window states, can be implemented in the following ways: If the window state is: remove the data corresponding to the first half of the window from the current sequence, construct a new current window based on the second half of the window and the candidate window, and perform the fuzzy prediction method based on the new current sequence corresponding to the new current window.

[0075] The current window consists of a first half window and a second half window. The first time period corresponding to the first half window is earlier than the second time period corresponding to the second half window. The third time period corresponding to the candidate window is adjacent to the second time period and is later than the second time period.

[0076] In some embodiments, the method provided in the foregoing embodiments can be used to determine a new current sequence corresponding to a new current window, then determine a new fuzzy posterior probability corresponding to the new current sequence, update the fuzzy parameters of the new fuzzy set based on the new fuzzy posterior probability, generate a new rule base associated with the updated fuzzy parameters, perform probabilistic defuzzification prediction based on the new rule base to obtain a new prediction result, update the model parameters of the Gaussian mixture model based on the new prediction error set corresponding to the new prediction result, determine a new KL divergence corresponding to the new prediction error based on the parameters in the updated model parameters, the first half error set corresponding to the first half window in the new current window, and the second half error set corresponding to the second half window in the new current window, and perform prediction based on the new KL divergence to obtain an output sequence.

[0077] As can be seen from the above, the fuzzy prediction method for non-stationary time series provided in this application embodiment, if the window state is: removing data corresponding to the first half of the window in the current sequence, constructing a new current window based on the second half of the window and candidate windows, and executing the fuzzy prediction method based on the new current sequence corresponding to the new current window, where the current window consists of a first half of the window and a second half of the window, the first time period corresponding to the first half of the window is earlier than the second time period corresponding to the second half of the window, the third time period corresponding to the candidate window is adjacent to the second time period, and the third time period is later than the second time period. Thus, after switching the current window to a new current window, the recursive execution of the fuzzy prediction method provided in this application embodiment is realized. When the current window is a sliding window, through the above operations, tracking prediction of non-stationary time series in the time dimension can be achieved, thereby improving the flexibility and robustness of the fuzzy prediction method.

[0078] Based on the foregoing embodiments, the fuzzy prediction method for non-stationary time series provided in this application, which performs prediction based on window state to obtain the output sequence, can also be implemented in the following ways: If the window state includes: expanding the current window to obtain a new current window, performing probabilistic defuzzification prediction based on the rule base to obtain a new prediction result, updating the model parameters of the Gaussian mixture model based on the new prediction error set corresponding to the new prediction result, determining the new KL divergence corresponding to the new current window based on the parameters in the model parameters, the new first half error set corresponding to the new first half window, and the second half error set corresponding to the new second half window, performing prediction based on the new KL divergence to obtain a new output sequence.

[0079] The new current window consists of a new front half window and a new back half window; the new prediction error set consists of a new front half error set and a new back half error set.

[0080] In some embodiments, if the window state includes: expanding the current window to obtain a new current window, which can represent that the new current window only adds one data point from a non-stationary time series compared to the current window, then the rule base does not need to be updated, and the subsequent steps of generating the rule base in the fuzzy prediction method can be performed directly based on the already determined rule base.

[0081] As can be seen from the above, in the fuzzy prediction method for non-stationary time series provided in this application embodiment, if the window state includes: expanding the current window to obtain a new current window, performing probabilistic defuzzification prediction based on the rule base to obtain a new prediction result, updating the model parameters of the Gaussian mixture model based on the new prediction result and the new prediction error set, determining the new KL divergence corresponding to the new current window based on the parameters in the model parameters, the new first half error set corresponding to the new first half window, and the new second half error set corresponding to the new second half window, performing prediction based on the new KL divergence to obtain a new output sequence, and the new current window is composed of the new first half window and the new second half window, and the new prediction error set is composed of the new first half error set and the new second half error set. Thus, automated iterative recursive execution of some steps in the fuzzy prediction method is achieved according to the window state, thereby improving the automation level of the fuzzy prediction method and further enhancing its flexibility and robustness.

[0082] Figure 2 This is another flowchart illustrating the fuzzy prediction method provided in an embodiment of this application. Figure 2 As shown, the process may include the following steps: Step 201: Input the time series.

[0083] For example, a non-stationary time series can be input.

[0084] Step 202: Initialize parameters.

[0085] For example, parameters such as the number of fuzzy sets can be initialized.

[0086] Step 203: Define the window.

[0087] For example, you can define the time period length corresponding to the current window, or you can define a new time period length corresponding to the new current window.

[0088] Step 204: Probabilistic fuzzification and parameter estimation.

[0089] Specifically, step 204 may include the following sub-steps: Step 204-1: Initialize the fuzzy set and the probabilistic fuzzy distribution.

[0090] For example, initializing a fuzzy set may include the number of fuzzy sets and the membership degree of each fuzzy set.

[0091] For example, the probabilistic fuzzification distribution may include calculating the k-th prior weight of the k-th fuzzy set, which can be specifically calculated using equation (1).

[0092] Step 204-2: Calculate the posterior probability of each data point for the fuzzy set.

[0093] For example, the posterior probability of each data point to the fuzzy set may include the fuzzy posterior probability of the kth data point in the current sequence to the kth fuzzy set, which can be specifically calculated by equation (2).

[0094] Step 204-3: Update the fuzzy set parameters by maximizing the likelihood function.

[0095] For example, the fuzzy set parameters may include the fuzzy parameters of the fuzzy set; specifically, Equation (3) can be obtained by maximizing the likelihood function as shown in Equation (10), and the fuzzy parameters can be updated by Equation (3).

[0096] (10); in, , , used to represent the N data points in the current sequence; N is an integer greater than 1.

[0097] Step 204-4: Determine whether to continue iterating.

[0098] For example, if If the number of iterations is greater than M, then it can be determined not to continue iterating and to execute step 205; otherwise, it can continue iterating and execute step 204-1.

[0099] in, This is used to represent the parameter vector obtained after the (r+1)th iteration. It can include the centers of all fuzzy sets, the width of each fuzzy set, and the estimated values ​​of the prior weights of the fuzzy sets. M is used to represent the parameter vector obtained after the r-th iteration; M can be preset to represent the maximum number of iterations.

[0100] Step 205: Generate a fuzzy state sequence.

[0101] Step 206: Generate a rule base.

[0102] For example, the fuzzy set to which the data points in the current sequence belong can be determined by equation (4), and the data points in the sequence to which the fuzzy set belongs can be determined as a fuzzy state sequence. Equation (4) can also be used to generate a rule base.

[0103] Step 207: Probabilistic defuzzification.

[0104] For example, probabilistic defuzzification prediction can be performed using equation (5) to obtain the prediction result.

[0105] Step 208: Calculate the prediction error.

[0106] For example, the difference between the data points in the prediction result and the corresponding data points in the non-stationary time series can be determined as the prediction error; for example, the prediction error can correspond to the prediction error set in the foregoing embodiments.

[0107] Step 209: Error Modeling and Concept Drift Detection.

[0108] For example, step 209 may include the following sub-steps: Step 209-1: Initialize the Gaussian mixture model parameters.

[0109] For example, the mean and variance of each Gaussian component in the parameters of a Gaussian mixture model can be initialized.

[0110] Step 209-2: Calculate the posterior probability of the prediction error with respect to the Gaussian component.

[0111] Step 209-3: Update the model parameters of the Gaussian mixture model.

[0112] For example, the above model parameters can be updated using equation (6).

[0113] Step 209-4: Determine whether to continue updating.

[0114] For example, if the infinite norm of the difference between the model parameters obtained in two adjacent iterations is less than the target threshold, or if the number of iterations of the model parameters is greater than the number threshold, it can be determined that the model parameters will no longer be updated, and step 210 can be executed; otherwise, step 209-1 can be executed.

[0115] Step 210: Calculate the KL divergence.

[0116] For example, the KL divergence can be calculated using the methods provided in the foregoing embodiments.

[0117] Step 211: Determine whether to update the window.

[0118] For example, the window state can be determined by the method provided in the foregoing embodiments, and whether to update the window can be determined based on the window state.

[0119] For example, if the window state includes removing the data corresponding to the first half of the window in the current sequence and constructing a new current window based on the second half of the window and the candidate window, then step 203 can be executed; while if the window state includes expanding the current window to obtain a new current window, then step 207 can be executed.

[0120] For example, if the window state includes not updating the current window, step 212 can be performed.

[0121] Step 212: Output the final prediction result.

[0122] For example, the final prediction result can be the output sequence in the aforementioned embodiments.

[0123] Through the above process, the probabilistic fuzzification and parameter estimation, error modeling and concept drift detection, and the output of the final prediction result are organically integrated, thereby enabling targeted and accurate prediction of the input time series.

[0124] In order to evaluate the predictive effect of the technical solutions provided in the embodiments of this application, the embodiments of this application provide Figure 3 and Figure 4 The dataset shown consists of non-stationary time series, and the proposed method and comparative methods are used to analyze it. Figure 3 and Figure 4 Make predictions using the dataset shown.

[0125] The comparison schemes can include the first scheme and the second scheme; the first scheme includes the traditional probabilistic fuzzy time series (PFTS) method, which relies on a static rule base to perform predictions and does not include adaptive updates to the rule base; the second scheme includes the time-varying probabilistic fuzzy time series (TVFTS) method, which only performs adaptive updates to the fuzzy set and does not include the determination of concept drift; correspondingly, this scheme can be called the adaptive probabilistic fuzzy time series (TVPFTS).

[0126] Figure 3 A visual illustration of a real dataset provided for embodiments of this application. Figure 3 The real-world datasets shown include weekly new COVID-19 case data released by the World Health Organization (previous 100 weeks), monthly milk production data from a classic R language dataset (168 months), highway diesel retail price data from the Computational Intelligent Time Series Forecasting Competition (527 weeks), daily closing price data of the Dow Jones Industrial Average from 2020 to 2023 (1006 trading days), monthly average relative sunspot number from 1749 to 1983 (2820 months), and daily maximum temperature data of Melbourne from 1981 to 1990 (3650 days).

[0127] Figure 4 This is a visual illustration of the synthetic dataset provided in an embodiment of this application. Figure 4The synthetic datasets shown may include a stationary signal dataset, a dataset corresponding to a stationary signal with spike interference, a dataset corresponding to a signal with abrupt changes in mean, a dataset corresponding to a signal with abrupt changes in variance, a dataset with periodic mean changes, a dataset with periodic variance changes, a dataset with asymptotic mean drift, and a dataset with asymptotic variance drift.

[0128] Figure 5A The statistical results of the root mean square error of the prediction results of the synthetic dataset provided in the embodiments of this application and the comparative scheme are presented. Figure 5B The statistical results of the root mean square error of the prediction results of the present solution and the comparative solution provided in the embodiments of this application on the real dataset.

[0129] It should be noted that, Figure 5A The horizontal axis represents SS, SB, SSM, SSV, SPM, SPV, SIM, and SIV, respectively, and... Figure 4 The datasets shown correspond to the following: stationary signal dataset, stationary signal dataset with spike interference dataset, mean abrupt change dataset, variance abrupt change dataset, periodic mean change dataset, periodic variance change dataset, asymptotic mean drift dataset, and asymptotic variance drift dataset. Figure 5B The horizontal axis represents WWC, MMP, RPDP, DIAI, ZMSN, and MDMT, respectively, and... Figure 3 The data includes weekly new confirmed cases of COVID-19, monthly milk production, highway diesel retail prices, daily closing prices of the Dow Jones Industrial Average, monthly average relative sunspots, and daily high temperatures in Melbourne.

[0130] from Figure 5A and Figure 5B As can be seen, on synthetic datasets, the TVPFTS method provided in this application consistently outperforms the comparative schemes, demonstrating that the technical solution provided in this application can effectively handle various non-stationary time series. On real datasets, except for the weekly COVID-19 new cases dataset where the root mean square index is relatively weaker than the comparative scheme, this scheme significantly outperforms the comparative scheme on other datasets. It should be noted that the weekly COVID-19 new cases dataset has the fewest observation samples; therefore, this means that the predictive performance of this scheme will decrease when the number of observation samples is limited.

[0131] Figure 6A The statistical results of the time utility scores of the prediction results of the synthetic dataset provided in the embodiments of this application and the comparative scheme are respectively. Figure 6BThe statistical results of the time utility scores of the prediction results of the present solution and the comparative solution provided in the embodiments of this application on the real dataset.

[0132] from Figures 6A to 6B It can be seen that, whether on synthetic or real datasets, the time utility score of this solution is improved compared to the comparison solution.

[0133] In summary, the technical solution provided by the embodiments of this application can efficiently and accurately achieve fuzzy prediction of non-stationary time series, thereby meeting the prediction needs for non-stationary time series.

[0134] The description of the various embodiments above tends to emphasize the differences between the various embodiments. The similarities or similarities between them can be referred to, and for the sake of brevity, they will not be repeated here.

[0135] The methods disclosed in the various method embodiments provided in this application can be arbitrarily combined to obtain new method embodiments without conflict.

[0136] The features disclosed in the various product embodiments provided in this application can be arbitrarily combined without conflict to obtain new product embodiments.

[0137] The features disclosed in the various method or device embodiments provided in this application can be arbitrarily combined without conflict to obtain new method or device embodiments.

[0138] It should be noted that the aforementioned computer-readable storage media can be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), magnetic random access memory (FRAM), flash memory, magnetic surface memory, optical disc, or compact disc read-only memory (CD-ROM), etc.; or it can be various electronic devices including one or any combination of the above-mentioned memories, such as mobile phones, computers, tablet devices, personal digital assistants, etc.

[0139] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element.

[0140] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0141] Through the above description of the embodiments, those skilled in the art can clearly understand that the methods of the above embodiments can be implemented by means of software plus necessary general-purpose hardware nodes. Of course, they can also be implemented by hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium (such as ROM / RAM, magnetic disk, optical disk) and includes several instructions to cause a terminal device (which may be a mobile phone, computer, server, air conditioner, or network device, etc.) to execute the methods described in the various embodiments of this application.

[0142] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0143] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0144] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0145] The above are merely preferred embodiments of this application and do not limit the patent scope of this application. Any equivalent structural or procedural transformations made using the content of this application's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of this application.

Claims

1. A fuzzy prediction method for non-stationary time series, characterized in that, include: Determine the fuzzy posterior probability corresponding to the current sequence; wherein, the current sequence includes the proper subset sequence of the input sequence corresponding to the current window; the input sequence includes a non-stationary time series within at least one time period; The fuzzy parameters of the fuzzy set are updated based on the fuzzy posterior probability; wherein the fuzzy set corresponds to the input sequence; Generate a rule base that is associated with the updated fuzzy parameters; Based on the rule base, probabilistic defuzzification prediction is performed to obtain the prediction result; Based on the prediction error set corresponding to the prediction result, the model parameters of the Gaussian mixture model are updated; wherein, the model parameters include the k-th mean, k-th variance of the k-th Gaussian component in the Gaussian mixture model, and the k-th weight of the k-th Gaussian component in the Gaussian mixture model; k is an integer greater than or equal to 1; Based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window, the KL divergence corresponding to the prediction error set is determined; wherein, the current window is composed of the first half window and the second half window; the prediction error set is composed of the first half error set and the second half error set; Prediction is performed based on the KL divergence to obtain the output sequence.

2. The method according to claim 1, characterized in that, The step of determining the KL divergence corresponding to the prediction error set based on the parameters in the model parameters, the first half error set corresponding to the first half window, and the second half error set corresponding to the second half window includes: Based on the first parameter set and the first half error set, a first average log-likelihood result is determined; wherein, the first parameter set includes the set of parameters of the Gaussian components in the model parameters corresponding to the first half window; Based on the second parameter set and the second half error set, a second average log-likelihood result is determined; wherein, the second parameter set includes the set of parameters of the Gaussian components in the model parameters corresponding to the second half window; the first time period corresponding to the first half window is earlier than the second time period corresponding to the second half window; The KL divergence is determined based on the first average log-likelihood result and the second average log-likelihood result.

3. The method according to claim 1, characterized in that, The prediction based on the KL divergence to obtain the output sequence includes: The window state of the current window is determined at least based on the KL divergence, and prediction is performed based on the window state to obtain the output sequence; wherein the window state is used to indicate whether to update the current window.

4. The method according to claim 3, characterized in that, Determining the window state of the current window based at least on the KL divergence includes: If the KL divergence is less than the divergence threshold and the time period length corresponding to the current window is less than the time period threshold, the window state is determined to be: expand the current window to obtain a new current window.

5. The method according to claim 3, characterized in that, Determining the window state of the current window based at least on the KL divergence includes: If the KL divergence is greater than or equal to the divergence threshold, the window state is determined as follows: remove the data in the current sequence corresponding to the first half of the window, and construct a new current window based on the second half of the window and the candidate window; wherein, the first time period corresponding to the first half of the window is earlier than the second time period corresponding to the second half of the window; the third time period corresponding to the candidate window is adjacent to the second time period, and the third time period is later than the second time period.

6. The method according to claim 3, characterized in that, The prediction based on the window state includes: If the window state is: remove the data corresponding to the first half of the window in the current sequence, construct a new current window based on the second half of the window and the candidate window, and execute the fuzzy prediction method based on the new current sequence corresponding to the new current window; wherein, the current window is composed of the first half of the window and the second half of the window; the first time period corresponding to the first half of the window is earlier than the second time period corresponding to the second half of the window; the third time period corresponding to the candidate window is adjacent to the second time period, and the third time period is later than the second time period.

7. The method according to claim 3, characterized in that, The prediction based on the window state includes: If the window state includes: expanding the current window to obtain a new current window, performing probabilistic defuzzification prediction based on the rule base to obtain a new prediction result, updating the model parameters of the Gaussian mixture model based on the new prediction error set corresponding to the new prediction result, determining a new KL divergence corresponding to the new current window based on the parameters in the model parameters, the new first half error set corresponding to the new first half window, and the new second half error set corresponding to the new second half window, and performing prediction on the input sequence based on the new KL divergence; wherein, the new current window consists of a new first half window and the second half window; the new prediction error set consists of a new first half error set and a new second half error set.

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