The embodiments relate generally to time series data processing and machine learning systems, and more specifically to compositional seasonal-trend representations for time-series forecasting.
Time series constitutes a prevalent form of data whose analysis has several important applications in areas, such as business, medicine, aerospace, and information systems. For example, time-series analysis is often applied in anomaly detection. Classical time series analysis often processes the time series data by considering trend and seasonal components. Some existing systems adopt deep learning to time series analysis task such as forecasting and anomaly detection. But most such systems learn the relevant models in a supervised end-to-end fashion.
As time-series data is a high dimensional and complex form of data, it is often challenging and expensive to label time-series data in order to perform supervised learning tasks, such as time-series classification. For example, in the medical setting, the task of identifying cardiac abnormalities from electrocardiogram (ECG) data requires expert knowledge and thus manual annotation can be costly. Another example is emotion recognition through electroencephalogram (EEG) and ECG signals, in which elaborate experiments are carried out to collect labelled data. Unlabeled data, on the other hand, is usually cheaply available.
Therefore, there is a need for a mechanism for improved time-series forecasting including making use of the unlabeled data.
In the figures, elements having the same designations have the same or similar functions.
As used herein, the term “network” may comprise any hardware or software-based framework that includes any artificial intelligence network or system, neural network or system and/or any training or learning models implemented thereon or therewith.
As used herein, the term “module” may comprise hardware or software-based framework that performs one or more functions. In some embodiments, the module may be implemented on one or more neural networks.
Time series forecasting may be applied to various domains, such as electricity pricing, demand forecasting, capacity planning and management, and anomaly detection. Time series data can often take a complex form and thus make labeling task challenging and expensive. For example, in the medical setting, the task of identifying cardiac abnormalities from ECG data requires expert knowledge. Another example is emotion recognition through EEG and ECG signals, in which elaborate experiments have to be carried out to collect labelled data.
In various embodiments, deep learning may be applied for forecasting. Owing to the increase in data availability and computational resources, these approaches with deep learning may promising performance over conventional methods for forecasting. In various embodiments, the methods applying deep learning for forecasting may be jointly learn feature representations and the prediction function (or forecasting function) by stacking a series of nonlinear layers to perform feature extraction, followed by a regression layer focused on forecasting. However, jointly learning these layers end-to-end from observed data may lead to the model over-fitting and capturing spurious correlations of the unpredictable noise contained in the observed data. In some embodiments, this situation is exacerbated when the learned representations are entangled. In some entangled learned representations, a single dimension of the feature representation may encode information from multiple local independent modules of the data-generating process, and a local independent module may experience a distribution shift. For example, an observed time series is generated by a seasonal module and nonlinear trend module. If it is known that the seasonal module has experienced a distribution shift, a reasonable prediction may be made based on the invariant trend module. However, if an entangled feature representation (e.g., encoding information from both the seasonal module and the trend module) is learned from the observed data, it would be challenging for the learned model to handle this distribution shift, even if it only happens in a local component of the data-generating process. As such, the learned representations and prediction associations from the end-to-end training approach may be unable to transfer or generalize well when the data is generated from a nonstationary environment, a very common scenario in the time series analysis.
Therefore, as discussed in detail below, learning disentangled seasonal-trend representations may improve the performance for time series forecasting. A time series model may formulate time series as a sum of trend, seasonal and error variables, and exploit such prior knowledge to learn time series representations. Learning disentangled seasonal-trend representation are robust to interventions on the error variable. In various embodiments, interventions on the error variable are introduced via for example data augmentations, and the disentangled seasonal-trend representations are learned via for example contrastive learning. In various embodiments, the system may leverage inductive biases in the model architecture to learn disentangled seasonal-trend representations, efficiently learn trend representations, mitigating the problem of look-back window selection by introducing a mixture of auto-regressive experts. It may learn more powerful seasonal representations by leveraging a learnable Fourier layer which enables intra-frequency interactions. Both trend and seasonal representations are learned via contrastive loss functions. The trend representations are learned in the time domain, whereas the seasonal representations are learned via a novel frequency domain contrastive loss which encourages discriminative seasonal representations and side steps the issue of determining the period of seasonal patterns present in the data. Such a system is robust to various choices of backbone encoders, as well as downstream regressors.
Furthermore, decoupling the representation learning and supervised downstream tasks may improve the performance of time series learning. For example, for the forecasting task of the data-generating process to be a hidden Markov model, a supervised forecasting task p(xt+1|xt) relies on spurious correlations via the latent confounders. While this strategy may work well for the in-distribution supervised learning setting, when dealing with nonstationary time series where out-of-distribution future time steps are predicted, this may lead to catastrophic results. Thus, by decoupling the representation and downstream forecasting task, representations of the direct causes for the desired dependent variable are first learnt, resulting in a regression task which follows a causal mechanism.
Thus, embodiments described herein provide an improved framework for time series forecasting by learning disentangled seasonal and trend representations of time series.
Referring to
As shown in
Given the problem formation, instead of jointly learning the representation and prediction association through g(*), a system may focus on learning feature representations from observed data, with the goal of improving predictive performance. Specifically, a nonlinear feature embedding function V=f(X) is learned to project m-dimensional raw signals into a d-dimensional latent space for each timestamp. Subsequently, the learned representation of the final timestamp vh is used as inputs for the downstream regressor of the forecasting task.
In various embodiments, complex data may arise from the rich interaction of multiple sources. A goal of the representation is to disentangle the various explanatory sources, making it robust to complex and richly structured variations. Not doing so may otherwise lead to capturing spurious features that do not transfer well under non-independent and identically distributed (i. i. d.) data distribution settings. To achieve this goal, structural priors for time series is introduced. As illustrated in the causal graph 150 in
In some embodiments, end-to-end deep forecasting methods, apart from modeling multivariate interactions, directly model the time-lagged relationship along the observed data X 152. However, in those embodiments, each X 152 includes unpredictable noise E 154, which might lead to capturing spurious correlations. Thus, to address this issue and improve performance, methods for learning the error-free latent variable X*156 may be used.
In various embodiments, the seasonal and trend modules do not influence or inform each other. Therefore, even if one mechanism changes due to a distribution shift, the other remains unchanged. Accordingly, disentangling seasonality and trend leads to better transfer, or generalization in nonstationary environments. Furthermore, independent seasonal and trend mechanisms can be learned independently and be flexibly re-used and re-purposed.
Further, interventions on E does not influence the conditional distribution P(X*|T,S), i.e. Pdo(E=ei)(X*|T,S)=Pdo(E=ej)(X*|T,S), for any ei and ej in the domain of E. Thus, S and T are invariant under changes in E. Learning representations for S and T allows to find a stable association with the optimal prediction (of X*) in terms of various types of errors. Since the targets X*are unknown, a proxy contrastive learning task may be constructed. For example, data augmentation 162 may be used as interventions on the error E. For further example, invariant representations of T and S may be learned via representation learning 164, e.g., contrastive learning. While it may be impossible to generate all possible variations of errors, the data augmentations may include various augmentations including for example, scale, shift, jitter, any other suitable augmentations, and/or a combination thereof, which can simulate a large and diverse set of errors, beneficial for learning better representations.
Referring to
In the system 200, a backbone encoder 202 maps observations to a latent space, e.g., projecting m-dimensional raw signals into a d-dimensional latent space for each timestep. The backbone encoder may use various types of encoders, including for example, a Temporal Convolution Network. Various representations (e.g., trend representations, seasonality representations, any other suitable representations) may be constructed from the intermediate representations 203 generated by the backbone encoder 202. For example, trend feature disentangler 204 (also referred to as a trend feature extractor 204) may extract the trend representations (e.g., via a mixture of auto-regressive experts), and may be learnt via a time domain contrastive loss 208 (denoted as Ltime) using contrastive learning. The trend representations are disentangled from the seasonal representations, and do not include seasonal features for the seasonal component. For further example, seasonal feature disentangler 206 (also referred to as a seasonal feature extractor 206) may extract the seasonal representations (e.g., via a learnable Fourier layer), and may be learned by a frequency domain contrastive loss 210 using contrastive learning. The frequency domain contrastive loss 210 may include an amplitude contrastive loss 212 (denoted as Lamp), a phase contrastive loss 214 (denoted as Lphase), any other suitable frequency domain contrastive loss components, and/or a combination thereof. The neural network model of system 200 may then be learnt in an end-to-end fashion, with an overall loss function L that is generated based on the time domain contrastive loss 208, the frequency domain contrastive loss 210, any other suitable losses, and/or a combination thereof. In an example, the overall loss function L may be provided as follows:
where α a hyper-parameter which balances the trade-off between trend and seasonal factors. The trend feature representations from the trend feature disentangler 204 and the seasonal feature representations from the seasonal feature disentangler 206 may be concatenated to generate the final output representations.
Referring to
Referring to
Extracting the underlying trend is crucial for modeling time series. Auto-regressive filtering may be to capture time-lagged causal relationships from past observations. One challenge is to select the appropriate look-back window: a smaller window leads to under-fitting, while a larger model leads to over-fitting and over-parameterization issues. In some examples, this hyper-parameter is optimized by grid search on the training or validation loss, but such an approach is too computationally expensive. In examples like those illustrated in
In various embodiments, contrastive learning is used. Contrastive learning via the instance discrimination task is a powerful approach for self-supervised learning. Firstly, a family of data augmentations is defined. Given a single sample of data xi, two data augmentation operators a and a′ are sampled, where qi=f(a(xi)) is referred to as the query representation with encoder f, and ki=f(a′(xi)) is the positive key representation. Finally, the loss function may be provided as follows:
Where τ is the temperature hyper-parameter, kj are negative key representations, and K is the total number of negative samples. In some examples, an efficient mechanism may be used to obtain negative samples—by simply treating all other samples in the mini-batch as negative samples, i.e. K=N−1. In some embodiments, a queue of size K (a hyper-parameter) may be used to obtain negative samples. At each iteration of training, simply pop N samples from the queue, and push the N representations form the current mini-batch.
A contrastive loss in the time domain (e.g., time domain contrastive loss 208) may be used to learn discriminative trend representations. For example, a momentum encoder may be used to obtain representations of the positive pair, and a dynamic dictionary with a queue may be used to obtain negative pairs. Then, given N samples and K negative samples, the time domain contrastive loss 208 may be provided as follows:
where given a sample V (T), a random time step t is selected for the contrastive loss, and a projection head is applied, which is a one-layer MLP to obtain q, and k is respectively the augmented version of the corresponding sample from the momentum encoder/dynamic dictionary.
In various embodiments, spectral analysis in the frequency domain is used in seasonality detection, and the seasonal feature disentangler 350 handles the learning of seasonal representations the frequency domain. Seasonal feature disentangler 350 address provides support for intra-frequency interactions (between feature dimensions), which allows the representations to encode periodic information more easily. For example, seasonal feature disentangler 350, by using learnable Fourier layer 354, captures intra-frequency level interaction. Then, to learn these seasonal features without prior knowledge of the periodicity, a frequency domain contrastive loss 210 is introduced for each frequency.
As illustrated in
where A and B are parameters of the learnable Fourier layer 354.
As illustrated in the example of
where Fi,(j) is the j-th sample in a mini-batch, and (Fi,(j)) is the augmented version of that sample.
Referring to
The representation learning method 402 may include block 404, where intermediate representations of a data sample is generated using an encoder (backbone encoder 202) of a representation learning model (e.g., representation learning model 200).
The representation learning method 402 may proceed to block 404, where a trend feature disentangler is used to generate trend feature representations from the intermediate representations. The representation learning method 402 may proceed to block 408, where a time domain contrastive loss is generated based on the trend feature representations. The representation learning method 402 may proceed to block 410, where a seasonal feature disentangler is used to generate seasonal feature representations. The representation learning method 402 may proceed to block 412, where a frequency domain contrastive loss is generated based on the seasonal feature representations. The representation learning method 402 may proceed to block 414, where a total loss is generated based on the time domain contrastive loss and the frequency domain contrastive loss. The representation learning model is trained using the total loss.
After the representation learning model is trained at block 402, the method 400 may proceed to block 416, where learned feature representations including disentangled trend feature representations and seasonal feature representations are generated using the trained representation learning model. The method 400 may proceed to block 418, where a forecasting task is performed based on the learned feature representations.
Memory 520 may be used to store software executed by computing device 500 and/or one or more data structures used during operation of computing device 500. Memory 520 may include one or more types of machine readable media. Some common forms of machine readable media may include floppy disk, flexible disk, hard disk, magnetic tape, any other magnetic medium, CD-ROM, any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, RAM, PROM, EPROM, FLASH-EPROM, any other memory chip or cartridge, and/or any other medium from which a processor or computer is adapted to read.
Processor 510 and/or memory 520 may be arranged in any suitable physical arrangement. In some embodiments, processor 510 and/or memory 520 may be implemented on a same board, in a same package (e.g., system-in-package), on a same chip (e.g., system-on-chip), and/or the like. In some embodiments, processor 510 and/or memory 520 may include distributed, virtualized, and/or containerized computing resources. Consistent with such embodiments, processor 510 and/or memory 520 may be located in one or more data centers and/or cloud computing facilities.
In some examples, memory 520 may include non-transitory, tangible, machine readable media that includes executable code that when run by one or more processors (e.g., processor 510) may cause the one or more processors to perform the methods described in further detail herein. For example, as shown, memory 520 includes instructions for a season-trend representative learning module 530 that may be used to implement and/or emulate the systems and models, and/or to implement any of the methods described further herein. A trained season-trend representative learning module 530 may receive input that includes a time series data 540 via the data interface 515 and generate representations of time series data 550 as output.
In some embodiments, the season-trend representative learning module 530 includes an encoder 531 (e.g., including a backbone encoder 202 of
In various embodiment, the season-trend representative learning module 530 and its submodules 531-535, may be implemented by hardware, software and/or a combination thereof.
Some examples of computing devices, such as computing device 500 may include non-transitory, tangible, machine readable media that include executable code that when run by one or more processors (e.g., processor 510) may cause the one or more processors to perform the processes of method. Some common forms of machine-readable media that may include the processes of method 400 are, for example, floppy disk, flexible disk, hard disk, magnetic tape, any other magnetic medium, CD-ROM, any other optical medium, punch cards, paper tape, any other physical medium with patterns of holes, RAM, PROM, EPROM, FLASH-EPROM, any other memory chip or cartridge, and/or any other medium from which a processor or computer is adapted to read.
This description and the accompanying drawings that illustrate inventive aspects, embodiments, implementations, or applications should not be taken as limiting. Various mechanical, compositional, structural, electrical, and operational changes may be made without departing from the spirit and scope of this description and the claims. In some instances, well-known circuits, structures, or techniques have not been shown or described in detail in order not to obscure the embodiments of this disclosure. Like numbers in two or more figures represent the same or similar elements.
In this description, specific details are set forth describing some embodiments consistent with the present disclosure. Numerous specific details are set forth in order to provide a thorough understanding of the embodiments. It will be apparent, however, to one skilled in the art that some embodiments may be practiced without some or all of these specific details. The specific embodiments disclosed herein are meant to be illustrative but not limiting. One skilled in the art may realize other elements that, although not specifically described here, are within the scope and the spirit of this disclosure. In addition, to avoid unnecessary repetition, one or more features shown and described in association with one embodiment may be incorporated into other embodiments unless specifically described otherwise or if the one or more features would make an embodiment non-functional.
Although illustrative embodiments have been shown and described, a wide range of modification, change and substitution is contemplated in the foregoing disclosure and in some instances, some features of the embodiments may be employed without a corresponding use of other features. One of ordinary skill in the art would recognize many variations, alternatives, and modifications. Thus, the scope of the invention should be limited only by the following claims, and it is appropriate that the claims be construed broadly and in a manner consistent with the scope of the embodiments disclosed herein.
This application claims priority to U.S. Provisional Patent Application No. 63/252,877 filed Oct. 6, 2021, which is incorporated by reference herein in its entireties.
Number | Date | Country | |
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63252877 | Oct 2021 | US |