Time sequence point process event prediction method based on monotone alternating spline

Abstract

The invention discloses a time sequence point process event prediction method based on a monotone alternating spline. The method comprises the following steps: acquiring historical information of an event sequence; generating a historical embedded vector using an encoder; parameters of a monotonic alternating spline are determined, an interpolation function and an extrapolation function are determined through the parameters of the monotonic alternating spline, the monotonic alternating spline comprises an interpolation part and an extrapolation part, the interpolation part utilizes any monotonic spline to enhance fitting performance, and the extrapolation part adopts any monotonic function to ensure global monotonicity; forming a conditional cumulative intensity function model by splicing the interpolation part and the extrapolation part; training the conditional cumulative intensity function model by using maximum likelihood estimation to optimize model parameters; and predicting the occurrence probability of the next event. According to the method, the monotone alternating spline is combined with the interpolation part and the extrapolation part, various monotone splines and extrapolation functions are compatible, various event sequence types and point process models are adapted, and wider applicability and framework flexibility are provided.

Description

Technical Field

[0001] This invention relates to the field of computer technology, and more specifically to a method for predicting time-series process events based on monotonically alternating splines. Background Technology

[0002] These sequences are typically characterized by asynchronous timestamps over continuous time, distinguishing them from traditional synchronous time series data. Understanding the dynamics within these sequences helps in predicting future events. Temporal Point Processes (TPPs), as a class of stochastic process models, have wide applications across various fields, providing a statistical framework for modeling such event sequences.

[0003] The key to characterizing the TPP lies in the distribution at the next timestamp given historical conditions. One common method for parameterizing the TPP is to specify the Conditional Intensity Function (CIF). Many works employ deep learning methods to model the CIF and recover the model parameters through Maximum Likelihood Estimation (MLE). However, CIF-based modeling has a significant drawback: the likelihood function of the TPP includes a time integral of the CIF. Numerical integration inevitably introduces approximation errors and can significantly reduce computational efficiency. To avoid the efficiency and accuracy degradation caused by modeling the CIF, researchers have proposed another approach—modeling the Conditional Cumulative Intensity Function (CCIF) of the TPP, instead of directly modeling the CIF. The CIF can be recovered by differentiating the CCIF from the integral of the CIF, and the differentiation process can be achieved using the automatic differentiation (Autograd) function built into frameworks like PyTorch. Compared to approximate numerical integration, modeling the CCIF eliminates approximation errors and improves computational efficiency through Autograd.

[0004] Despite the advantages of modeling CCIF, existing methods for parameterizing CCIF remain scarce. Most past methods have simply used monotonically increasing monotonically increasing neural networks (MNNs) to ensure the monotonically increasing nature of CCIF. MNNs are typically multilayer perceptrons (MLPs) with positive weights. Such designs have limited architectural flexibility and often neglect other requirements when modeling CCIF.

[0005] Therefore, how to provide a parameterization method for the conditional cumulative intensity function of time-series process points that is highly versatile, flexible, computationally efficient, and balances fitting ability and generalization performance is a problem that urgently needs to be solved by those skilled in the art. Summary of the Invention

[0006] In view of the above problems, the present invention provides a method for predicting time-series process events based on monotonically alternating splines to overcome or at least partially solve the above problems; To achieve the above objectives, the present invention adopts the following technical solution: This invention provides a method for predicting time-series process events based on monotonically alternating splines, comprising the following steps: S1. Obtain historical information of the event sequence, which may be one of the following: online shopping event sequence, forwarding event sequence, taxi event sequence, or earthquake event sequence; S2. Based on historical information, use an encoder to generate historical embedding vectors; S3. Based on the historical embedding vector, determine the parameters of the monotonic alternating spline, and determine the interpolation function and extrapolation function through the parameters of the monotonic alternating spline. The monotonic alternating spline includes an interpolation part and an extrapolation part. The interpolation part uses arbitrary monotonic splines to enhance the fitting performance, and the extrapolation part uses arbitrary monotonic functions to ensure global monotonicity. S4. A conditional cumulative intensity function model is formed by splicing the interpolation part and the extrapolation part; S5. Use maximum likelihood estimation to train the conditional cumulative intensity function model to optimize the model parameters; S6. Based on the trained conditional cumulative intensity function model, predict the probability of the next event.

[0007] Furthermore, the event sequence in step S1 includes timestamps and event types for multiple events.

[0008] Furthermore, the determination of the monotonically alternating spline parameters in step S3 includes mapping the historical embedding vector to parameters using a multilayer perceptron and a softplus function.

[0009] Furthermore, the process of determining the interpolation function in step S3 includes: Determine the specific form of the interpolation function; The time domain Divided into M+1 segments

[0010] in, Let M be the timestamp of the nth event, and M be the number of segments in the interpolation interval. For the time domain The Mth interpolation node on; Parameterized interpolation function.

[0011] Furthermore, the interpolation function is implemented using monotonic splines, which are selected from piecewise rational quadratic splines, rational cubic splines, and rational linear splines.

[0012] Furthermore, the process of determining the extrapolation function in step S3 includes: Determine the specific form of the extrapolation function; Parameterized extrapolation function; Ensure a smooth connection with the interpolation function.

[0013] Furthermore, the process of forming the conditional cumulative intensity function model by splicing the interpolation part and the extrapolation part in step S4 specifically includes: By concatenating the interpolation and extrapolation parts using an indicator function, a complete conditional cumulative intensity function model is formed, the expression of which is:

[0014] in, As an indicator function, only when The value is 1 in the case of a specific condition and 0 in the other cases. For the set of model parameters, For the time domain The m-th interpolation function on, For the time domain Extrapolation function on.

[0015] Furthermore, the constraints satisfied by the model include: Global monotonicity; C1 continuity between timestamps; C0 continuity at timestamps.

[0016] Furthermore, the prediction formula in step S6 is:

[0017]

[0018] in, For time t n Historical information of the event set up to date. For conditional intensity function, For the predicted timestamp of the next event, For the predicted type of the next event, This represents the intensity function over event type k. The overall intensity function satisfies .

[0019] As can be seen from the above technical solution, compared with the prior art, the present invention discloses a method for predicting time-series process events based on monotonically alternating splines, which has the following beneficial effects: 1. This invention combines monotonically alternating splines with interpolation and extrapolation components, making it compatible with various monotonically alternating splines and extrapolation functions, adaptable to various event sequence types and point process models, and providing wider applicability and architectural flexibility; 2. This invention directly models the conditional cumulative intensity function, avoiding the approximation errors caused by numerical integration in traditional methods, and improves computational efficiency through analytical derivatives. Simultaneously, the interpolation part of the monotone alternating splines enhances the fitting ability, while the extrapolation part ensures generalization performance, thus achieving higher accuracy and lower computational cost in event prediction. Attached Figure Description

[0020] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0021] Figure 1 This is a flowchart of the time-series process event prediction method provided in the embodiments of the present invention; Figure 2 This is a schematic diagram of the cumulative intensity function of the modeling conditions provided in the embodiments of the present invention; Figure 3 This is a parameterized schematic diagram of a monotonically alternating spline provided in an embodiment of the present invention; Figure 4 This is a schematic diagram of a monotonic spline and a monotonic alternating spline provided in an embodiment of the present invention; Figure 5 The image shows the fitting plots of the monotonically alternating splines provided in this embodiment of the invention on the Hawkes1 dataset for the conditional intensity function (left) and the conditional cumulative intensity function (right). Figure 6 The image shows the fitting plots of the monotonically alternating splines provided in this embodiment of the invention on the self-correcting dataset for the conditional intensity function (left) and the conditional cumulative intensity function (right). Figure 7 This is a comparison chart of the efficiency of monotonic alternating splines, FullyNN, and EMTPP provided in the embodiments of the present invention. Detailed Implementation

[0022] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0023] This invention discloses a method for predicting time-series process events based on monotonically alternating splines; such as... Figure 1 As shown, the specific steps include: S1. Obtain historical information of the event sequence, wherein the event sequence is one of the following: online shopping event sequence, forwarding event sequence, taxi event sequence, and earthquake event sequence; S2. Based on the historical information, use an encoder to generate a historical embedding vector; S3. Based on the historical embedding vector, determine the parameters of the monotonic alternating spline, and determine the interpolation function and extrapolation function through the parameters of the monotonic alternating spline. The monotonic alternating spline includes an interpolation part and an extrapolation part. The interpolation part uses arbitrary monotonic splines to enhance the fitting performance, and the extrapolation part uses arbitrary monotonic functions to ensure global monotonicity. S4. A conditional cumulative intensity function model is formed by splicing the interpolation part and the extrapolation part; S5. Train the conditional cumulative intensity function model using maximum likelihood estimation to optimize the model parameters; S6. Based on the trained conditional cumulative intensity function model, predict the probability of the next event.

[0024] This invention combines monotonically alternating splines with interpolation and extrapolation components, enabling compatibility with various monotonically alternating splines and extrapolation functions, adapting to various event sequence types and point process models, and providing broader applicability and architectural flexibility.

[0025] Temporal Point Processes A time-point process (TPP) is a probabilistic model that describes a sequence of events occurring within a time interval [0, T]. A single observation of a multivariate TPP can be represented as an event sequence. , where N is a random variable representing the number of events. The timestamp of the event, and The mark represents the nth event. TPP can be represented by the Conditional Intensity Function (CIF): As depicted, the function is defined in a given history The above represents the instantaneous occurrence rate of event of type k at time t. For simplicity, let's define it as... Recorded as Its strict definition is as follows:

[0026] This represents information prior to time t, excluding historical information up to time t. Represents the nth event, i.e., the timestamp. Historical information.

[0027] The log likelihood of the multivariate TPP can be written as:

[0028] in The overall strength is given. Model parameters are estimated by maximizing the log-likelihood described above. Given a timestamp... Historical information At that time, the next event timestamp The conditional probability density is:

[0029] If the CIF of the model has already been modeled, the timestamp of the next event can be predicted using the following estimation formula. and marking :

[0030] In step S1, the obtained event sequences include taxi event sequences, forwarding event sequences, online shopping event sequences, and earthquake event sequences; Taking a taxi event sequence as an example, the event timestamp refers to the specific moment when each taxi passenger trip begins.

[0031] In a taxi event sequence, the event type refers to the geographical area identifier where each passenger-hailing event occurs. Specifically, the city map is divided into 14 different regions according to administrative divisions or latitude and longitude grids, and each region corresponds to a unique type number. If a taxi-hailing behavior occurs in region one, the event type is recorded as "1".

[0032] The event prediction method of this invention, using Monotonic Alternating Spline (MAS), can accurately and efficiently predict this sequence. First, historical taxi event sequences containing timestamps and geographic location types are collected and input into a time-series process model constructed based on monotonic alternating splines. Next, the model is trained using historical data, and the MAS method is used to directly fit the conditional cumulative intensity function of the events, thereby capturing the spatiotemporal dependence of traffic demand. Finally, real-time observed historical data is input into the trained model, and by analytically calculating the probability distribution of the next event, accurate and efficient predictions of the timing and location of future taxi orders can be made.

[0033] CCIF Parametrics Although CIF-based parametric methods are commonly used for TPP modeling, they have a major drawback: the CIF time integral appears in the log-likelihood. There is usually no closed-form solution, therefore, numerical integration must be introduced. This leads to a decrease in accuracy and efficiency. To avoid this problem, an alternative is to directly modify the Conditional Cumulative Intensity Function (CCIF). Modeling can then be performed. At this point, the log-likelihood can be rewritten as:

[0034] in and express exist The left derivative at the given point. By modeling CCIF, the maximum likelihood estimation (MLE) no longer depends on integral calculations, but only on the left derivative at the given point. Differentiation can be performed precisely and efficiently using automatic differentiation (Autograd).

[0035] Monotonic splines Monotonic splines are a common method for modeling monotonic data and are widely used in modeling the cumulative distribution function (CDF). Various monotonic splines, such as piecewise cubic splines and piecewise rational quadratic splines (RQS), not only preserve the monotonicity and C1 continuity of the modeled data but are also easy to differentiate and have analytic inverses. Other monotonic splines, including rational cubic splines (RCS) and rational linear splines (RLS), also possess this characteristic. An important feature is that piecewise monotonic splines guarantee smoothness at the interpolation endpoints. Taking RQS as an example, given the interpolation interval... The m-th segment of RQS can be written as:

[0036] in ,and Verifiable They are respectively The function values ​​and derivatives at the interpolation endpoints are as follows:

[0037]

[0038] As can be seen from the above expression, within a given interpolation interval At that time, a segment of RQS can be composed of endpoint values and endpoint derivatives Uniquely determined. If two adjacent segments share the same endpoint values ​​and derivatives, they can be seamlessly connected to ensure the smoothness of the spline. RCS and RLS also have similar properties.

[0039] Modeling CCIF A CCIF It can be broken down into:

[0040] in It is an indicator function: when It takes the value 1 if the condition is met, and 0 otherwise. This is the integral over CIF:

[0041] Its domain is , representing the CCIF after the nth event, where it is specified It is worth noting that, The domain is This is because, during the prediction phase, the next event... It is unknown, and it may happen in At any time; while during the training phase, since it is known Indicator functions can be used Will Cut off to Subsequently, and To piece together to construct a complete ,like Figure 2 As shown. The goal of this work is to propose a novel, flexible, and efficient method for modeling. .

[0042] Modeling CCIF requirements As mentioned earlier, modeling the CCIF of TPP simplifies maximum likelihood estimation and improves efficiency. However, modeling CCIF is not straightforward because it requires satisfying various properties. The following theorem summarizes... The conditions that must be met.

[0043] (1) Global monotonicity: For any ,have

[0044] (2) C1 continuity between timestamps: for any , It is continuous in this interval, and its derivative is... It exists and continues.

[0045] (3) C0 continuity at timestamps: at each timestamp , But the derivative and They don't have to be equal.

[0046] Conditions (2) and (3) guarantee that the CIF is continuous between two adjacent timestamps, but at timestamps... There may be abrupt changes, which is common in historically dependent TPPs (such as Hawkes processes, self-correcting processes, etc.). Furthermore, if each segment... If both monotonicity and C1 continuity are guaranteed, then by splicing... That can make all The conditions in the above theorem are satisfied.

[0047] Most current CCIF-based methods use monotonic neural networks (MNNs) to model CCIF, thus ensuring its monotonicity; however, these methods have several limitations. First, while monotonicity is guaranteed, MNNs may ignore other requirements of CCIF. For example, some methods do not enforce... Or, an inappropriate activation function, such as the ReLU function, was used, which disrupted the time-stamp intervals. The C1 continuity. Finally, the architectural flexibility of monotonic neural networks is limited: almost all current methods for modeling CCIF implement MNNs using only a simple positive weighted MLP. However, transforming more complex network structures into monotonic networks is very complicated, and current research on MNNs is still very limited, which restricts the development of methods for modeling CCIF using MNNs.

[0048] Monotonic alternating splines (MAS) For complex TPPs, the corresponding It could have extremely high complexity. A natural approach is to... The model is divided into multiple smaller sub-intervals, making it more feasible to fit the model with a simple fundamental function within each sub-interval. Based on this idea, a monotonically alternating spline (MAS) approach is proposed for modeling. .

[0049] MAS consists of two parts: an interpolation part and an extrapolation part.

[0050] Among them Divided into M+1 segments .

[0051] function In the interval The above represents the interpolation function. exist The above represents the extrapolation function. These are the model parameters. The specific implementations of each part of MAS are as follows: MAS interpolation section: To make To satisfy monotonicity and C1 continuity, each segment of the function must... extrapolation function Within its defined interval, it is monotonically increasing and has C1 continuity, and the connection condition must be satisfied at the piecewise nodes. A typical monotonic spline exhibits good monotonicity at interpolation points and C1 properties at connections. Based on the general form of a spline, given the interpolation nodes... Function values ​​at nodes and the derivative at the node After that, the interpolation spline is uniquely determined.

[0052] In MAS, these parameters are determined by historical information. First, a Transformer encoder is used, with the following parameters: Compressing history into embedded Then use an MLP (with parameters as follows) The `softplus` function will... Mapping to the required parameters described above, thus obtaining... ,in Let two adjacent segments and Derivatives sharing the same endpoint This is to ensure the continuity of C1. It also sets... = To ensure that The function is concatenated with two consecutive function segments, such as... Figure 3 As shown.

[0053] MAS interpolation fitting ability: If the MAS segmentation is finer and the interpolation interval is shorter, then the MAS will naturally approximate the reality more accurately. Assume the MAS interpolation segment can extend to infinity, i.e. Then for any real There exists a MAS representation. , making in , It is the length of the maximum interpolation interval. It is a constant independent of the data distribution. Furthermore, there also exists... , making

[0054] This shows that as long as the interpolation interval is small enough, MAS can fit any real CIF and CCIF, and its fitting performance can rival any other TPP parameterization method.

[0055] MAS extrapolation part The above assumes that the interpolation segment can extend to infinity wMn→∞; however, this is impossible in practice. Most monotonic splines (such as polynomial splines) do not guarantee monotonicity outside the interpolation interval and may even exhibit a decreasing trend or negative values. For example, when the slope parameter is small, RQS may monotonically decrease when t is large. Therefore, in addition to the interpolation function, an appropriate extrapolation function form must be chosen to guarantee Λθ. The global monotonicity of nt over its domain, the difference between simple monotonic splines and MAS, as follows: Figure 4 As shown; In implementation, It can take linear form It can also be taken in exponential form. , where a, b > 0. Its parameters a, b are also derived from historical embeddings. Decision. Similarly, via the encoder (parameters) )get Then use MLP (parameters) Adding softplus maps to a, b. When performing interpolation and extrapolation, to ensure that at the connection point... Smoothness setting:

[0056] This ensures the C1 continuity of MAS. Additionally, This is a hyperparameter that determines the length of the interpolation interval. If we take... Then MAS degenerates into a classical point process model. For example, for a constant intensity... The homogeneous Poisson process can be represented by a single... ,and The MAS can be used to represent this. For example, for a Hawkes process with an exponential kernel, the MAS can also be represented by... It is represented in exponential form. In summary, MAS is compatible with many classic point procedures.

[0057] Application of MAS in Multivariate TPP MAS can be generalized to multivariate TPPs. If a TPP contains K event types, a MAS model can be built for each type to represent the corresponding CCIF. Given an event history... Using Transformer (parameters) The encoding history is embedded Then use K different MLPs (parameters) Add softplus mapping to the MAS parameters required for each type, thus obtaining... The overall model parameters are: The model was then trained using the log-likelihood of CCIF and the corresponding prediction formula was used to predict the next timestamp and event type. It is worth emphasizing that these K MAS models share the same historical embedding. Therefore, these K CCIFs have mutual influences, which is the key characteristic of multivariate TPP.

[0058] Generalization of MAS Assume the training set consists of Z elements within a time interval Independent and identically distributed point process sequences Composition. The nth timestamp of the z-th sequence is denoted as... For simplicity, considering the univariate case, the maximum sequence length is... Without loss of generality, let the interpolation interval be a constant. Loss of experience Defined as the average log-likelihood over all sequences: Although by minimizing While training the model is important, what's even more crucial is the model's expected performance on the data distribution, i.e., the expectation that the training loss will define. So, in terms of probability... The probability of this being true is:

[0059] In the above theorem, , and This is a constant independent of the data distribution. Apart from the probability error term, the remaining terms can be divided into three categories, arising from interpolation, extrapolation, and model complexity, respectively. First, interpolation error... Interpolation interval The decrease is due to the reduction in size, because of the smaller... This helps to more accurately fit the true CCIF. Secondly, extrapolation error. The error decreases as the total interpolation length L increases, because the influence of the extrapolation part weakens as the contribution of the interpolation part increases. Finally, the complexity error... Follow The decrease in L and the increase in Z cause the rise, which can be alleviated by increasing the sample size Z. If L is increased and L is decreased... (i.e., enhanced interpolation) increases the number of parameters in MAS, thereby increasing model complexity, and therefore requires more samples to ensure generalization performance.

[0060] The above theorem provides guidance for hyperparameter selection: with a fixed sample size Z, two key hyperparameters... L can be estimated by minimizing the upper right bound. Theoretically, the optimal MAS should be in A balance is sought between MAS and L. However, extensive experiments in the next chapter show that even without fine hyperparameter tuning, MAS outperforms most baseline models; and as the interpolation interval shrinks, MAS does not exhibit excessive overfitting.

[0061] Baseline model MAS was compared with nine representative baselines. First, the simple non-homogeneous Poisson process IPP was compared. Next, three types of models based on the conditional intensity function (CIF) were compared: RMTPP based on RNNs, THP based on attention models, and SAHP. Additionally, several models based on the cumulative intensity function (CCIF) were compared: FullyNN, TriTPP, and EMTPP. Finally, the non-intensity function-based model IFTPP and the score-matching-based integrator-free method WSM were also considered.

[0062] Since some baseline models, such as IPP, FullyNN, and TriTPP, only consider the univariate case in their original implementations, this experiment is divided into two parts: first, the performance of MAS and all baselines is compared on univariate synthetic data; then, the performance is compared with baselines that support multivariate configuration (RTPP, THP, SAHP, EMTPP, IFTPP, WSM) on multivariate real data.

[0063] Evaluation indicators Three evaluation metrics were used: negative log-likelihood (NLL), root mean square error (RMSE), and accuracy (ACC). NLL was calculated on the test set using the likelihood expression given above; RMSE was calculated using the formula... Predicted timestamps obtained from the prediction formula With real timestamp The ACC (Accuracy Calculation) is based on the corresponding prediction formula to predict the next event type and calculate the average accuracy.

[0064] Experimental setup In baselines with historical dependencies, some models use RNNs to extract historical embeddings, such as RMTPP, while others use Transformers, such as THP and SAHP. To ensure a fair comparison, the same Transformer is used as the historical encoder for both Transformer-based baselines and MAS. For MAS, the interpolation part uses 10 nodes and RQS as the monotonic splines for interpolation, while the extrapolation part uses an exponential form. (Other splines and extrapolation functions are discussed in the ablation experiments). The interpolation support length L is set to a constant. The hyperparameters of all baseline models use the default settings from their respective public repositories. Event sequences are randomly divided into 60% training set, 20% validation set, and 20% test set. All models are trained on an RTX 4090 (24GB) using the Adam optimizer.

[0065]

[0066] Table 1 - Performance on univariate datasets; The smaller the NLL (negative log-likelihood) and RMSE (root mean square error), the better the model performance. The best data is represented by bold, and the second best data is represented by underline. All data are averaged from three experiments.

[0067]

[0068] Table 2 - Performance on multivariate datasets; The smaller the NLL (Negative Log-Likelihood) and RMSE (Root Mean Square Error), the better the ACC (Accuracy) model performs. The best data is shown in bold, and the second best is shown in underline. All data are averaged from three experiments.

[0069] Univariate synthetic data Several synthetic datasets were used in the univariate experiments (including two types of Hawkes procedures, two types of renewal procedures, and one type of self-correcting procedure). Table 1 shows the performance comparison between each baseline and MAS. MAS achieved best or near-best performance on most datasets and metrics, verifying that by introducing piecewise monotonic splines as interpolation components, MAS can fit diverse real CCIFs derived from different classic TPPs. In contrast, traditional models (such as IPP) or some deep Hawkes models performed poorly in characterizing certain TPPs. Furthermore, MAS also outperformed several models that used monotonic neural networks to parameterize CCIFs in terms of fitting ability. Figure 5 , 6 The results show the predicted CIF and CCIF of MAS on two datasets, demonstrating that MAS can successfully capture the intensity patterns of different TPPs.

[0070] Multivariate real data Four real-world multivariate datasets were used: Taxi, Taobao, Retweet, and Earthquake. The results are shown in Table 2. On real-world data, MAS again achieved excellent results across multiple metrics, further demonstrating its advantages in flexibility and generalization, indicating that MAS can effectively capture the interactions between various variables in multivariate scenarios. The success of MAS can be attributed to two factors: first, the introduction of more interpolation nodes on the timeline makes the model more flexible than the baseline; second, the extrapolation component ensures accurate simulation of CCIF while maintaining good generalization ability. Therefore, MAS significantly improves the model's expressive power without sacrificing generalization.

[0071] Efficiency and Ablation Experiment In terms of efficiency, MAS was compared with two CCIF-based models, EMTPP and FullyNN, on the Hawkes-1 dataset. While keeping the number of parameters in each model approximately the same, the actual time required to run 50 epochs under the same experimental settings was measured. The results are as follows: Figure 7 The MAS shown is significantly faster than the two baseline models, mainly because MAS eliminates the dependence on automatic differentiation.

[0072] Simultaneously, ablation experiments were conducted to verify the contributions of each component of MAS, considering the following factors: (1) interpolation monotonic spline type (RQS / RLS); (2) extrapolation function type (linear / exponential); (3) interpolation support length (L=4 / 6 / 10); (4) number of interpolation nodes (p=5 / 10 / 50 / 100). Experiments were conducted on the Taxi and Earthquake datasets, with NLL and ACC as evaluation metrics. The default settings were RQS interpolation, exponential extrapolation, L=6, and p=10.

[0073] Ablation results (see Table 3) show that as the number of interpolation nodes *p* and the interpolation support length *L* increase, the model's performance on both datasets first improves and then decreases. This verifies the theoretical analysis's conclusion that increasing the interpolation support length and the number of nodes increases complexity error—that is, excessive interpolation increases model complexity and may lead to overfitting, but this overfitting phenomenon is not severe. Furthermore, MAS is not very sensitive to different interpolation and extrapolation functions; under default configuration, MAS can already fit the CCIF of both datasets with relatively fine granularity, so different function forms do not lead to significant differences. Overall, the experimental results demonstrate the balance between robustness, efficiency, and performance of MAS.

[0074]

[0075] Table 3 - Ablation Experiments; This paper proposes a novel method, MAS, for modeling the conditional intensity function (CCIF) of time-series point processes (TPPs). Compared with the MNN model, MAS exhibits greater versatility, flexibility, and efficiency, and possesses solid theoretical guarantees in terms of fitting and generalization. Furthermore, it is demonstrated that the interpolation component in MAS improves the model's fitting performance, while the extrapolation component enhances its generalization ability. Extensive experimental results show that, across various evaluation metrics, MAS outperforms other CCIF- or CIF-based modeling methods on both univariate and multivariate datasets.

[0076] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0077] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for predicting time-series process events based on monotonically alternating splines, characterized in that, Includes the following steps: S1. Obtain historical information of the event sequence, wherein the event sequence is one of the following: online shopping event sequence, forwarding event sequence, taxi event sequence, and earthquake event sequence; S2. Based on the historical information, use an encoder to generate a historical embedding vector; S3. Based on the historical embedding vector, determine the parameters of the monotonic alternating spline, and determine the interpolation function and extrapolation function through the parameters of the monotonic alternating spline. The monotonic alternating spline includes an interpolation part and an extrapolation part. The interpolation part uses arbitrary monotonic splines to enhance the fitting performance, and the extrapolation part uses arbitrary monotonic functions to ensure global monotonicity. S4. A conditional cumulative intensity function model is formed by splicing the interpolation part and the extrapolation part; S5. Train the conditional cumulative intensity function model using maximum likelihood estimation to optimize the model parameters; S6. Based on the trained conditional cumulative intensity function model, predict the probability of the next event.

2. The method for predicting time-series process events based on monotonically alternating splines as described in claim 1, characterized in that, The event sequence in step S1 includes timestamps and event types for multiple events.

3. The method for predicting time-series process events based on monotonically alternating splines as described in claim 2, characterized in that, The determination of the monotonically alternating spline parameters in step S3 includes mapping the historical embedding vector to parameters using a multilayer perceptron and a softplus function.

4. The method for predicting time-series process events based on monotonically alternating splines as described in claim 3, characterized in that, The process of determining the interpolation function in step S3 includes: Determine the specific form of the interpolation function; The time domain Divided into M+1 segments in, Let M be the timestamp of the nth event, and M be the number of segments in the interpolation interval. For the time domain The Mth interpolation node on; Parameterized interpolation function.

5. The method for predicting time-series process events based on monotonically alternating splines as described in claim 4, characterized in that, The interpolation function is implemented using monotonic splines, which are selected from piecewise rational quadratic splines, rational cubic splines, and rational linear splines.

6. The method for predicting time-series process events based on monotonically alternating splines as described in claim 5, characterized in that, The process of determining the extrapolation function in step S3 includes: Determine the specific form of the extrapolation function; Parameterized extrapolation function; Ensure a smooth connection with the interpolation function.

7. The method for predicting time-series process events based on monotonically alternating splines as described in claim 6, characterized in that, The process of forming the conditional cumulative intensity function model by splicing the interpolation part and the extrapolation part in step S4 specifically includes: By concatenating the interpolation and extrapolation parts using an indicator function, a complete conditional cumulative intensity function model is formed, the expression of which is: in, As an indicator function, only when The value is 1 in the case of a specific condition and 0 in the other cases. For the set of model parameters, For the time domain The m-th interpolation function on, For the time domain Extrapolation function on.

8. The method for predicting time-series process events based on monotonically alternating splines as described in claim 7, characterized in that, The constraints satisfied by the model include: Global monotonicity; C1 continuity between timestamps; C0 continuity at timestamps.

9. The method for predicting time-series process events based on monotonically alternating splines as described in claim 1, characterized in that, The prediction formula in step S6 is: in, For time t n Historical information of the event set up to date. For conditional intensity function, For the predicted timestamp of the next event, For the predicted type of the next event, This represents the intensity function over event type k. The overall intensity function satisfies .

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