The long horizon forecasting (LHF) problem has come up in the time series literature for over the last 35 years or so. This review covers aspects of LHF in this period and how deep learning has incorporated variants of trend, seasonality, fourier and wavelet transforms, misspecification bias reduction and bandpass filters while contributing using convolutions, residual connections, sparsity reduction, strided convolutions, attention masks, SSMs, normalization methods, low-rank approximations and gating mechanisms. We highlight time series decomposition techniques, input data preprocessing and dataset windowing schemes that improve performance. Multi-layer perceptron models, recurrent neural network hybrids, self-attention models that improve and/or address the performances of the LHF problem are described, with an emphasis on the feature space construction. Ablation studies are conducted over the ETTm2 dataset in the multivariate and univariate high useful load (HUFL) forecasting contexts, evaluated over the last 4 months of the dataset. The heatmaps of MSE averages per time step over test set series in the horizon show that there is a steady increase in the error proportionate to its length except with xLSTM and Triformer models and motivate LHF as an error propagation problem. The trained models are available here: https://bit.ly/LHFModelZoo
Accurate evaluation of forecasting models is essential for ensuring reliable predictions. Current practices for evaluating and comparing forecasting models focus on summarising performance into a single score, using metrics such as SMAPE. While convenient, averaging performance over all samples dilutes relevant information about model behavior under varying conditions. This limitation is especially problematic for time series forecasting, where multiple layers of averaging--across time steps, horizons, and multiple time series in a dataset--can mask relevant performance variations. We address this limitation by proposing ModelRadar, a framework for evaluating univariate time series forecasting models across multiple aspects, such as stationarity, presence of anomalies, or forecasting horizons. We demonstrate the advantages of this framework by comparing 24 forecasting methods, including classical approaches and different machine learning algorithms. NHITS, a state-of-the-art neural network architecture, performs best overall but its superiority varies with forecasting conditions. For instance, concerning the forecasting horizon, we found that NHITS (and also other neural networks) only outperforms classical approaches for multi-step ahead forecasting. Another relevant insight is that classical approaches such as ETS or Theta are notably more robust in the presence of anomalies. These and other findings highlight the importance of aspect-based model evaluation for both practitioners and researchers. ModelRadar is available as a Python package.
Pre-trained foundation models (FMs) have shown exceptional performance in univariate time series forecasting tasks. However, several practical challenges persist, including managing intricate dependencies among features and quantifying uncertainty in predictions. This study aims to tackle these critical limitations by introducing adapters; feature-space transformations that facilitate the effective use of pre-trained univariate time series FMs for multivariate tasks. Adapters operate by projecting multivariate inputs into a suitable latent space and applying the FM independently to each dimension. Inspired by the literature on representation learning and partially stochastic Bayesian neural networks, we present a range of adapters and optimization/inference strategies. Experiments conducted on both synthetic and real-world datasets confirm the efficacy of adapters, demonstrating substantial enhancements in forecasting accuracy and uncertainty quantification compared to baseline methods. Our framework, AdaPTS, positions adapters as a modular, scalable, and effective solution for leveraging time series FMs in multivariate contexts, thereby promoting their wider adoption in real-world applications. We release the code at https://github.com/abenechehab/AdaPTS.
Transfer Learning (TL) is an emerging field in modeling building thermal dynamics. This method reduces the data required for a data-driven model of a target building by leveraging knowledge from a source building. Consequently, it enables the creation of data-efficient models that can be used for advanced control and fault detection & diagnosis. A major limitation of the TL approach is its inconsistent performance across different sources. Although accurate source-building selection for a target is crucial, it remains a persistent challenge. We present GenTL, a general transfer learning model for single-family houses in Central Europe. GenTL can be efficiently fine-tuned to a large variety of target buildings. It is pretrained on a Long Short-Term Memory (LSTM) network with data from 450 different buildings. The general transfer learning model eliminates the need for source-building selection by serving as a universal source for fine-tuning. Comparative analysis with conventional single-source to single-target TL demonstrates the efficacy and reliability of the general pretraining approach. Testing GenTL on 144 target buildings for fine-tuning reveals an average prediction error (RMSE) reduction of 42.1 % compared to fine-tuning single-source models.
Time series forecasting, while vital in various applications, often employs complex models that are difficult for humans to understand. Effective explainable AI techniques are crucial to bridging the gap between model predictions and user understanding. This paper presents a framework - TSFeatLIME, extending TSLIME, tailored specifically for explaining univariate time series forecasting. TSFeatLIME integrates an auxiliary feature into the surrogate model and considers the pairwise Euclidean distances between the queried time series and the generated samples to improve the fidelity of the surrogate models. However, the usefulness of such explanations for human beings remains an open question. We address this by conducting a user study with 160 participants through two interactive interfaces, aiming to measure how individuals from different backgrounds can simulate or predict model output changes in the treatment group and control group. Our results show that the surrogate model under the TSFeatLIME framework is able to better simulate the behaviour of the black-box considering distance, without sacrificing accuracy. In addition, the user study suggests that the explanations were significantly more effective for participants without a computer science background.
The effectiveness of univariate forecasting models is often hampered by conditions that cause them stress. A model is considered to be under stress if it shows a negative behaviour, such as higher-than-usual errors or increased uncertainty. Understanding the factors that cause stress to forecasting models is important to improve their reliability, transparency, and utility. This paper addresses this problem by contributing with a novel framework called MAST (Meta-learning and data Augmentation for Stress Testing). The proposed approach aims to model and characterize stress in univariate time series forecasting models, focusing on conditions where they exhibit large errors. In particular, MAST is a meta-learning approach that predicts the probability that a given model will perform poorly on a given time series based on a set of statistical time series features. MAST also encompasses a novel data augmentation technique based on oversampling to improve the metadata concerning stress. We conducted experiments using three benchmark datasets that contain a total of 49.794 time series to validate the performance of MAST. The results suggest that the proposed approach is able to identify conditions that lead to large errors. The method and experiments are publicly available in a repository.
Accurate evaluation of forecasting models is essential for ensuring reliable predictions. Current practices for evaluating and comparing forecasting models focus on summarising performance into a single score, using metrics such as SMAPE. We hypothesize that averaging performance over all samples dilutes relevant information about the relative performance of models. Particularly, conditions in which this relative performance is different than the overall accuracy. We address this limitation by proposing a novel framework for evaluating univariate time series forecasting models from multiple perspectives, such as one-step ahead forecasting versus multi-step ahead forecasting. We show the advantages of this framework by comparing a state-of-the-art deep learning approach with classical forecasting techniques. While classical methods (e.g. ARIMA) are long-standing approaches to forecasting, deep neural networks (e.g. NHITS) have recently shown state-of-the-art forecasting performance in benchmark datasets. We conducted extensive experiments that show NHITS generally performs best, but its superiority varies with forecasting conditions. For instance, concerning the forecasting horizon, NHITS only outperforms classical approaches for multi-step ahead forecasting. Another relevant insight is that, when dealing with anomalies, NHITS is outperformed by methods such as Theta. These findings highlight the importance of aspect-based model evaluation.
Most forecasting methods use recent past observations (lags) to model the future values of univariate time series. Selecting an adequate number of lags is important for training accurate forecasting models. Several approaches and heuristics have been devised to solve this task. However, there is no consensus about what the best approach is. Besides, lag selection procedures have been developed based on local models and classical forecasting techniques such as ARIMA. We bridge this gap in the literature by carrying out an extensive empirical analysis of different lag selection methods. We focus on deep learning methods trained in a global approach, i.e., on datasets comprising multiple univariate time series. The experiments were carried out using three benchmark databases that contain a total of 2411 univariate time series. The results indicate that the lag size is a relevant parameter for accurate forecasts. In particular, excessively small or excessively large lag sizes have a considerable negative impact on forecasting performance. Cross-validation approaches show the best performance for lag selection, but this performance is comparable with simple heuristics.
Time series are generated in diverse domains such as economic, traffic, health, and energy, where forecasting of future values has numerous important applications. Not surprisingly, many forecasting methods are being proposed. To ensure progress, it is essential to be able to study and compare such methods empirically in a comprehensive and reliable manner. To achieve this, we propose TFB, an automated benchmark for Time Series Forecasting (TSF) methods. TFB advances the state-of-the-art by addressing shortcomings related to datasets, comparison methods, and evaluation pipelines: 1) insufficient coverage of data domains, 2) stereotype bias against traditional methods, and 3) inconsistent and inflexible pipelines. To achieve better domain coverage, we include datasets from 10 different domains: traffic, electricity, energy, the environment, nature, economic, stock markets, banking, health, and the web. We also provide a time series characterization to ensure that the selected datasets are comprehensive. To remove biases against some methods, we include a diverse range of methods, including statistical learning, machine learning, and deep learning methods, and we also support a variety of evaluation strategies and metrics to ensure a more comprehensive evaluations of different methods. To support the integration of different methods into the benchmark and enable fair comparisons, TFB features a flexible and scalable pipeline that eliminates biases. Next, we employ TFB to perform a thorough evaluation of 21 Univariate Time Series Forecasting (UTSF) methods on 8,068 univariate time series and 14 Multivariate Time Series Forecasting (MTSF) methods on 25 datasets. The benchmark code and data are available at https://github.com/decisionintelligence/TFB.
This article combines wavelet analysis techniques with machine learning methods for univariate time series forecasting, focusing on three main contributions. Firstly, we consider the use of Daubechies wavelets with different numbers of vanishing moments as input features to both non-temporal and temporal forecasting methods, by selecting these numbers during the cross-validation phase. Secondly, we compare the use of both the non-decimated wavelet transform and the non-decimated wavelet packet transform for computing these features, the latter providing a much larger set of potentially useful coefficient vectors. The wavelet coefficients are computed using a shifted version of the typical pyramidal algorithm to ensure no leakage of future information into these inputs. Thirdly, we evaluate the use of these wavelet features on a significantly wider set of forecasting methods than previous studies, including both temporal and non-temporal models, and both statistical and deep learning-based methods. The latter include state-of-the-art transformer-based neural network architectures. Our experiments suggest significant benefit in replacing higher-order lagged features with wavelet features across all examined non-temporal methods for one-step-forward forecasting, and modest benefit when used as inputs for temporal deep learning-based models for long-horizon forecasting.
We propose Feature-aligned N-BEATS as a domain-generalized time series forecasting model. It is a nontrivial extension of N-BEATS with doubly residual stacking principle (Oreshkin et al. [45]) into a representation learning framework. In particular, it revolves around marginal feature probability measures induced by the intricate composition of residual and feature extracting operators of N-BEATS in each stack and aligns them stack-wise via an approximate of an optimal transport distance referred to as the Sinkhorn divergence. The training loss consists of an empirical risk minimization from multiple source domains, i.e., forecasting loss, and an alignment loss calculated with the Sinkhorn divergence, which allows the model to learn invariant features stack-wise across multiple source data sequences while retaining N-BEATS's interpretable design and forecasting power. Comprehensive experimental evaluations with ablation studies are provided and the corresponding results demonstrate the proposed model's forecasting and generalization capabilities.
Correlated time series (CTS) forecasting plays an essential role in many practical applications, such as traffic management and server load control. Many deep learning models have been proposed to improve the accuracy of CTS forecasting. However, while models have become increasingly complex and computationally intensive, they struggle to improve accuracy. Pursuing a different direction, this study aims instead to enable much more efficient, lightweight models that preserve accuracy while being able to be deployed on resource-constrained devices. To achieve this goal, we characterize popular CTS forecasting models and yield two observations that indicate directions for lightweight CTS forecasting. On this basis, we propose the LightCTS framework that adopts plain stacking of temporal and spatial operators instead of alternate stacking that is much more computationally expensive. Moreover, LightCTS features light temporal and spatial operator modules, called L-TCN and GL-Former, that offer improved computational efficiency without compromising their feature extraction capabilities. LightCTS also encompasses a last-shot compression scheme to reduce redundant temporal features and speed up subsequent computations. Experiments with single-step and multi-step forecasting benchmark datasets show that LightCTS is capable of nearly state-of-the-art accuracy at much reduced computational and storage overheads.
Due to the high temporal uncertainty and low signal-to-noise ratio, transfer learning for univariate time series forecasting remains a challenging task. In addition, data scarcity, which is commonly encountered in business forecasting, further limits the application of conventional transfer learning protocols. In this work, we have developed, DATSING, a transfer learning-based framework that effectively leverages cross-domain time series latent representations to augment target domain forecasting. In particular, we aim to transfer domain-invariant feature representations from a pre-trained stacked deep residual network to the target domains, so as to assist the prediction of each target time series. To effectively avoid noisy feature representations, we propose a two-phased framework which first clusters similar mixed domains time series data and then performs a fine-tuning procedure with domain adversarial regularization to achieve better out-of-sample generalization. Extensive experiments with real-world datasets have demonstrated that our method significantly improves the forecasting performance of the pre-trained model. DATSING has the unique potential to empower forecasting practitioners to unleash the power of cross-domain time series data.
Despite the notable advancements in numerous Transformer-based models, the task of long multi-horizon time series forecasting remains a persistent challenge, especially towards explainability. Focusing on commonly used saliency maps in explaining DNN in general, our quest is to build attention-based architecture that can automatically encode saliency-related temporal patterns by establishing connections with appropriate attention heads. Hence, this paper introduces Temporal Saliency Detection (TSD), an effective approach that builds upon the attention mechanism and applies it to multi-horizon time series prediction. While our proposed architecture adheres to the general encoder-decoder structure, it undergoes a significant renovation in the encoder component, wherein we incorporate a series of information contracting and expanding blocks inspired by the U-Net style architecture. The TSD approach facilitates the multiresolution analysis of saliency patterns by condensing multi-heads, thereby progressively enhancing the forecasting of complex time series data. Empirical evaluations illustrate the superiority of our proposed approach compared to other models across multiple standard benchmark datasets in diverse far-horizon forecasting settings. The initial TSD achieves substantial relative improvements of 31% and 46% over several models in the context of multivariate and univariate prediction. We believe the comprehensive investigations presented in this study will offer valuable insights and benefits to future research endeavors.
Deep learning utilizing transformers has recently achieved a lot of success in many vital areas such as natural language processing, computer vision, anomaly detection, and recommendation systems, among many others. Among several merits of transformers, the ability to capture long-range temporal dependencies and interactions is desirable for time series forecasting, leading to its progress in various time series applications. In this paper, we build a transformer model for non-stationary time series. The problem is challenging yet crucially important. We present a novel framework for univariate time series representation learning based on the wavelet-based transformer encoder architecture and call it W-Transformer. The proposed W-Transformers utilize a maximal overlap discrete wavelet transformation (MODWT) to the time series data and build local transformers on the decomposed datasets to vividly capture the nonstationarity and long-range nonlinear dependencies in the time series. Evaluating our framework on several publicly available benchmark time series datasets from various domains and with diverse characteristics, we demonstrate that it performs, on average, significantly better than the baseline forecasters for short-term and long-term forecasting, even for datasets that consist of only a few hundred training samples.
Time series models often deal with extreme events and anomalies, both prevalent in real-world datasets. Such models often need to provide careful probabilistic forecasting, which is vital in risk management for extreme events such as hurricanes and pandemics. However, it is challenging to automatically detect and learn to use extreme events and anomalies for large-scale datasets, which often require manual effort. Hence, we propose an anomaly-aware forecast framework that leverages the previously seen effects of anomalies to improve its prediction accuracy during and after the presence of extreme events. Specifically, the framework automatically extracts anomalies and incorporates them through an attention mechanism to increase its accuracy for future extreme events. Moreover, the framework employs a dynamic uncertainty optimization algorithm that reduces the uncertainty of forecasts in an online manner. The proposed framework demonstrated consistent superior accuracy with less uncertainty on three datasets with different varieties of anomalies over the current prediction models.
This work studies the threats of adversarial attack on multivariate probabilistic forecasting models and viable defense mechanisms. Our studies discover a new attack pattern that negatively impact the forecasting of a target time series via making strategic, sparse (imperceptible) modifications to the past observations of a small number of other time series. To mitigate the impact of such attack, we have developed two defense strategies. First, we extend a previously developed randomized smoothing technique in classification to multivariate forecasting scenarios. Second, we develop an adversarial training algorithm that learns to create adversarial examples and at the same time optimizes the forecasting model to improve its robustness against such adversarial simulation. Extensive experiments on real-world datasets confirm that our attack schemes are powerful and our defense algorithms are more effective compared with baseline defense mechanisms.
Forecasts help businesses allocate resources and achieve objectives. At LinkedIn, product owners use forecasts to set business targets, track outlook, and monitor health. Engineers use forecasts to efficiently provision hardware. Developing a forecasting solution to meet these needs requires accurate and interpretable forecasts on diverse time series with sub-hourly to quarterly frequencies. We present Greykite, an open-source Python library for forecasting that has been deployed on over twenty use cases at LinkedIn. Its flagship algorithm, Silverkite, provides interpretable, fast, and highly flexible univariate forecasts that capture effects such as time-varying growth and seasonality, autocorrelation, holidays, and regressors. The library enables self-serve accuracy and trust by facilitating data exploration, model configuration, execution, and interpretation. Our benchmark results show excellent out-of-the-box speed and accuracy on datasets from a variety of domains. Over the past two years, Greykite forecasts have been trusted by Finance, Engineering, and Product teams for resource planning and allocation, target setting and progress tracking, anomaly detection and root cause analysis. We expect Greykite to be useful to forecast practitioners with similar applications who need accurate, interpretable forecasts that capture complex dynamics common to time series related to human activity.
The COVID-19 pandemic has significantly impacted the tourism and hospitality sector. Public policies such as travel restrictions and stay-at-home orders had significantly affected tourist activities and service businesses' operations and profitability. To this end, it is essential to develop an interpretable forecast model that supports managerial and organizational decision-making. We developed DemandNet, a novel deep learning framework for predicting time series data under the influence of the COVID-19 pandemic. The framework starts by selecting the top static and dynamic features embedded in the time series data. Then, it includes a nonlinear model which can provide interpretable insight into the previously seen data. Lastly, a prediction model is developed to leverage the above characteristics to make robust long-term forecasts. We evaluated the framework using daily hotel demand and revenue data from eight cities in the US. Our findings reveal that DemandNet outperforms the state-of-art models and can accurately predict the impact of the COVID-19 pandemic on hotel demand and revenues.
Fuzzy Cognitive Maps (FCMs) have emerged as an interpretable signed weighted digraph method consisting of nodes (concepts) and weights which represent the dependencies among the concepts. Although FCMs have attained considerable achievements in various time series prediction applications, designing an FCM model with time-efficient training method is still an open challenge. Thus, this paper introduces a novel univariate time series forecasting technique, which is composed of a group of randomized high order FCM models labeled R-HFCM. The novelty of the proposed R-HFCM model is relevant to merging the concepts of FCM and Echo State Network (ESN) as an efficient and particular family of Reservoir Computing (RC) models, where the least squares algorithm is applied to train the model. From another perspective, the structure of R-HFCM consists of the input layer, reservoir layer, and output layer in which only the output layer is trainable while the weights of each sub-reservoir components are selected randomly and keep constant during the training process. As case studies, this model considers solar energy forecasting with public data for Brazilian solar stations as well as Malaysia dataset, which includes hourly electric load and temperature data of the power supply company of the city of Johor in Malaysia. The experiment also includes the effect of the map size, activation function, the presence of bias and the size of the reservoir on the accuracy of R-HFCM method. The obtained results confirm the outperformance of the proposed R-HFCM model in comparison to the other methods. This study provides evidence that FCM can be a new way to implement a reservoir of dynamics in time series modelling.