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The forecasting with time series imaging paper is accepted in Expert Systems with Applications

Our foresting paper with time series imaing approach is accepted in Expert Systems with Applications.

Xixi Li, Yanfei Kang, and Feng Li*. (2020). Forecasting with time series imaging, Expert Systems with Applications. (In Press)

[Working Paper | Software]

Feature-based time series representations have attracted substantial attention in a wide range of time series analysis methods. Recently, the use of time series features for forecast model averaging has been an emerging research focus in the forecasting community. Nonetheless, most of the existing approaches depend on the manual choice of an appropriate set of features. Exploiting machine learning methods to extract features from time series automatically becomes crucial in state-of-the-art time series analysis. In this paper, we introduce an automated approach to extract time series features based on time series imaging. We first transform time series into recurrence plots, from which local features can be extracted using computer vision algorithms. The extracted features are used for forecast model averaging. Our experiments show that forecasting based on automatically extracted features, with less human intervention and a more comprehensive view of the raw time series data, yields highly comparable performances with the best methods in the largest forecasting competition dataset (M4) and outperforms the top methods in the Tourism forecasting competition dataset.

Feature-based time series forecasting has been proved highly promising, primarily through the extraction and selection of an appropriate set of features. Nonetheless, traditional time series feature extraction requires manual design of feature metrics, which is typically complicated to time series forecasting practitioners. Known features used in time series forecasting literature are global characteristics of a time series, which may ignore important local patterns. Evidence from the literature further indicates that feature-based forecast combination might not perform as well as simple averaging when the feature extraction and selection are not properly conducted.

We propose an automated time series imaging feature extraction approach with computer vision algorithms, and our experiment results show that our approach works well for forecast combination. An innovative point of our approach over other feature-based time series forecasting methods is that time series features are extracted automatically from time series imaging, which are obtained using recurrence plots. In principle, any image feature extraction algorithm is applicable to our proposed framework. We employ two widely used algorithms to extract features from time series images, namely the spatial bag-of-features (SBoF) model and the deep convolutional neural networks (CNN).

The SBoF model, combining the scale-invariant feature transform (SIFT) algorithm, the locality constrained linear coding (LLC) method, and spatial pyramid matching (SPM) and max pooling, can capture both global and local characteristics of images. The traditional SBoF model is a fast industry level model in computer vision applications. One may notice that the features extracted based on the traditional SIFT model performs better than the deep CNN model in some scenarios with our testing data. But it is worth to mention that SIFT method is not a fully automated image feature extraction processing because it requires a careful specification of four steps, namely (1) detecting extreme values in the scale spaces, (2) finding the key points, (3) assigning feature directions, and (4) describing key points. Moreover, SIFT algorithm is patent protected, which means other open source program could not incorporate it without the patent owner’s permission. Having an alternative approach with highly comparable performance but without patent restrictions is important to time series forecasters.

The alternative feature extraction algorithm based on deep CNN is an automated process once the source task is confirmed. We use transfer learning to borrow the information of well pre-trained neural network models for imaging classification, which can avoid the complication of settings the network structure and tuning the hyper-parameters. Unlike traditional CNN tasks that require the fine-tuning and massive computation, we transfer the convolutional layers and fully-connected lays from the ImageNet competition results to our task. Hence only one new adaption layer needs to train, which significantly saves the computational power.

Although the aims of source task in ImageNet and the target task of time series forecasting are naturally different, the image features generated from time series share similar shapes and angles with the image of real objects. This explains why we could transfer a different task to time series forecasting. In practice, the forecasting practitioners may train a customized CNN model to further improve the forecasting performance if a rich collection of time series is available.

Another significant merit of using deep CNN and transfer learning for time series feature extraction is that, the pre-trained neural network models (e.g., on ImageNet) are continuously updated and improved in the image processing literature. Thus, we believe that this line of automated time series feature extraction approaches has great potential in the future.

In this paper, we use the features extracted from recurrence plots to reveal the characteristics of the corresponding time series. The recurrence plot for a given time series displays its dynamics based on the distance correlations within the time series. However, other features such as cross-correlation coefficients can also be used to generate cross-correlation recurrence plots. Thus, multi-channel images, with more comprehensive information, can be obtained for each time series, which can potentially improve the feature extraction and feature-based forecast combination performances. Therefore, time series forecasting based on multi-channel imaging can be one potential extension of our current work.

The forecasting framework based on time series image features is in line with the work in the feature-based forecast (FFORMA), where they use 42 manual time series features and nine forecasting methods to optimize the weights for forecast combination. FFORMA won the second place in the M4 competition. To be consistent and comparable, in our study, we employ the same set of forecasting methods in the M4 dataset. However, we want to mention that the choice of candidate forecasting methods for forecast combination also requires expert knowledge and practical experience. The performance of forecast combinations depends on the accuracy of individual forecasting methods and the diversity among them since the merits of forecast combination stem from the independent information across multiple forecasts. How to automatically select an appropriate set of candidate methods for combination is another interesting direction for future research.

In our experiments, all the time series are independent data. Therefore we treat the time series features as independent images and apply them to the CNN framework which is also used for classifying objects in ImageNet. A further extension of our work is to extend time series forecasting with imaging to (1) forecasting with time varying image features, and (2) hierarchical time series or multivariate time series with recurrent dependence. In both scenes, the hierarchical image classification framework mixtures with CNN and RNN could be further explored.

We make our code publicly available at https://github.com/lixixibj/forecasting-with-time-series-imaging. Making it open-source can enrich the toolboxes of forecasting support systems by providing a competitive alternative to the existing feature-based time series forecasting methods.

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