probabilistic forecasting

By encoding time series as a string of numerical digits, we can frame time series forecasting as next-token prediction in text. Developing this approach, we find that large language models (LLMs) such as GPT-3 and LLaMA-2 can surprisingly zero-shot extrapolate time series at a level comparable to or exceeding the performance of purpose-built time series models trained on the downstream tasks. To facilitate this performance, we propose procedures for effectively tokenizing time series data and converting discrete distributions over tokens into highly flexible densities over continuous values. We argue the success of LLMs for time series stems from their ability to naturally represent multimodal distributions, in conjunction with biases for simplicity, and repetition, which align with the salient features in many time series, such as repeated seasonal trends. We also show how LLMs can naturally handle missing data without imputation through non-numerical text, accommodate textual side information, and answer questions to help explain predictions. While we find that increasing model size generally improves performance on time series, we show GPT-4 can perform worse than GPT-3 because of how it tokenizes numbers, and poor uncertainty calibration, which is likely the result of alignment interventions such as RLHF.

We propose a framework for probabilistic forecasting of dynamical systems based on generative modeling. Given observations of the system state over time, we formulate the forecasting problem as sampling from the conditional distribution of the future system state given its current state. To this end, we leverage the framework of stochastic interpolants, which facilitates the construction of a generative model between an arbitrary base distribution and the target. We design a fictitious, non-physical stochastic dynamics that takes as initial condition the current system state and produces as output a sample from the target conditional distribution in finite time and without bias. This process therefore maps a point mass centered at the current state onto a probabilistic ensemble of forecasts. We prove that the drift coefficient entering the stochastic differential equation (SDE) achieving this task is non-singular, and that it can be learned efficiently by square loss regression over the time-series data. We show that the drift and the diffusion coefficients of this SDE can be adjusted after training, and that a specific choice that minimizes the impact of the estimation error gives a Föllmer process. We highlight the utility of our approach on several complex, high-dimensional forecasting problems, including stochastically forced Navier-Stokes and video prediction on the KTH and CLEVRER datasets. The code is available at https://github.com/interpolants/forecasting.

Temporal data such as time series can be viewed as discretized measurements of the underlying function. To build a generative model for such data we have to model the stochastic process that governs it. We propose a solution by defining the denoising diffusion model in the function space which also allows us to naturally handle irregularly-sampled observations. The forward process gradually adds noise to functions, preserving their continuity, while the learned reverse process removes the noise and returns functions as new samples. To this end, we define suitable noise sources and introduce novel denoising and score-matching models. We show how our method can be used for multivariate probabilistic forecasting and imputation, and how our model can be interpreted as a neural process.

We present Darts¹, a Python machine learning library for time series, with a focus on forecasting. Darts offers a variety of models, from classics such as ARIMA to state-of-the-art deep neural networks. The emphasis of the library is on offering modern machine learning functionalities, such as supporting multidimensional series, fitting models on multiple series, training on large datasets, incorporating external data, ensembling models, and providing a rich support for probabilistic forecasting. At the same time, great care goes into the API design to make it user-friendly and easy to use. For instance, all models can be used using fit()/predict(), similar to scikit-learn (Pedregosa et al., 2011).

The estimation of time-varying quantities is a fundamental component of decision making in fields such as healthcare and finance. However, the practical utility of such estimates is limited by how accurately they quantify predictive uncertainty. In this work, we address the problem of estimating the joint predictive distribution of high-dimensional multivariate time series. We propose a versatile method, based on the transformer architecture, that estimates joint distributions using an attention-based decoder that provably learns to mimic the properties of non-parametric copulas. The resulting model has several desirable properties: it can scale to hundreds of time series, supports both forecasting and interpolation, can handle unaligned and non-uniformly sampled data, and can seamlessly adapt to missing data during training. We demonstrate these properties empirically and show that our model produces state-of-the-art predictions on multiple real-world datasets.

We introduce Sundial, a family of native, flexible, and scalable time series foundation models. To predict the next-patch's distribution, we propose a TimeFlow Loss based on flow-matching, which facilitates native pre-training of Transformers on continuous-valued time series without discrete tokenization. Conditioned on arbitrary-length time series, our models are pre-trained without specifying any prior distribution and can generate multiple probable predictions, achieving more flexibility in representation learning than using parametric densities. Towards time series foundation models, we leverage minimal but crucial adaptations of Transformers and curate TimeBench with one trillion time points, comprising mostly real-world datasets and synthetic data. By mitigating mode collapse via TimeFlow Loss, we pre-train a family of Sundial models on TimeBench, which achieve unprecedented model capacity and generalization performance. In addition to excellent scalability, Sundial achieves state-of-the-art results on both point and probabilistic forecasting benchmarks with a just-in-time inference speed, i.e., making zero-shot predictions within a few milliseconds. We believe that Sundial's pioneering generative forecasting capability can improve model reliability in real-world decision-making. Code is available at: https://github.com/thuml/Sundial.

Deep learning for time series forecasting has traditionally operated within a one-model-per-dataset framework, limiting its potential to leverage the game-changing impact of large pre-trained models. The concept of universal forecasting, emerging from pre-training on a vast collection of time series datasets, envisions a single Large Time Series Model capable of addressing diverse downstream forecasting tasks. However, constructing such a model poses unique challenges specific to time series data: i) cross-frequency learning, ii) accommodating an arbitrary number of variates for multivariate time series, and iii) addressing the varying distributional properties inherent in large-scale data. To address these challenges, we present novel enhancements to the conventional time series Transformer architecture, resulting in our proposed Masked Encoder-based Universal Time Series Forecasting Transformer (Moirai). Trained on our newly introduced Large-scale Open Time Series Archive (LOTSA) featuring over 27B observations across nine domains, Moirai achieves competitive or superior performance as a zero-shot forecaster when compared to full-shot models. Code, data, and model weights can be found at this https URL.

Recurrent Neural Networks (RNN) have become competitive forecasting methods, as most notably shown in the winning method of the recent M4 competition. However, established statistical models such as ETS and ARIMA gain their popularity not only from their high accuracy, but they are also suitable for non-expert users as they are robust, efficient, and automatic. In these areas, RNNs have still a long way to go. We present an extensive empirical study and an open-source software framework of existing RNN architectures for forecasting, that allow us to develop guidelines and best practices for their use. For example, we conclude that RNNs are capable of modelling seasonality directly if the series in the dataset possess homogeneous seasonal patterns, otherwise we recommend a deseasonalization step. Comparisons against ETS and ARIMA demonstrate that the implemented (semi-)automatic RNN models are no silver bullets, but they are competitive alternatives in many situations.

Numerous deep learning architectures have been developed to accommodate the diversity of time series datasets across different domains. In this article, we survey common encoder and decoder designs used in both one-step-ahead and multi-horizon time series forecasting -- describing how temporal information is incorporated into predictions by each model. Next, we highlight recent developments in hybrid deep learning models, which combine well-studied statistical models with neural network components to improve pure methods in either category. Lastly, we outline some ways in which deep learning can also facilitate decision support with time series data.

Time series forecasting is an important problem across many domains, including predictions of solar plant energy output, electricity consumption, and traffic jam situation. In this paper, we propose to tackle such forecasting problem with Transformer [1]. Although impressed by its performance in our preliminary study, we found its two major weaknesses: (1) locality-agnostics: the point-wise dot-product self-attention in canonical Transformer architecture is insensitive to local context, which can make the model prone to anomalies in time series; (2) memory bottleneck: space complexity of canonical Transformer grows quadratically with sequence length $L$, making directly modeling long time series infeasible. In order to solve these two issues, we first propose convolutional self-attention by producing queries and keys with causal convolution so that local context can be better incorporated into attention mechanism. Then, we propose LogSparse Transformer with only $O(L(\log L)^{2})$ memory cost, improving forecasting accuracy for time series with fine granularity and strong long-term dependencies under constrained memory budget. Our experiments on both synthetic data and real-world datasets show that it compares favorably to the state-of-the-art.

Recent progress in neural forecasting accelerated improvements in the performance of large-scale forecasting systems. Yet, long-horizon forecasting remains a very difficult task. Two common challenges afflicting the task are the volatility of the predictions and their computational complexity. We introduce N-HiTS, a model which addresses both challenges by incorporating novel hierarchical interpolation and multi-rate data sampling techniques. These techniques enable the proposed method to assemble its predictions sequentially, emphasizing components with different frequencies and scales while decomposing the input signal and synthesizing the forecast. We prove that the hierarchical interpolation technique can efficiently approximate arbitrarily long horizons in the presence of smoothness. Additionally, we conduct extensive large-scale dataset experiments from the long-horizon forecasting literature, demonstrating the advantages of our method over the state-of-the-art methods, where N-HiTS provides an average accuracy improvement of almost 20% over the latest Transformer architectures while reducing the computation time by an order of magnitude (50 times). Our code is available at bit.ly/3VA5DoT

We focus on solving the univariate times series point forecasting problem using deep learning. We propose a deep neural architecture based on backward and forward residual links and a very deep stack of fully-connected layers. The architecture has a number of desirable properties, being interpretable, applicable without modification to a wide array of target domains, and fast to train. We test the proposed architecture on several well-known datasets, including M3, M4 and TOURISM competition datasets containing time series from diverse domains. We demonstrate state-of-the-art performance for two configurations of N-BEATS for all the datasets, improving forecast accuracy by 11% over a statistical benchmark and by 3% over last year's winner of the M4 competition, a domain-adjusted hand-crafted hybrid between neural network and statistical time series models. The first configuration of our model does not employ any time-series-specific components and its performance on heterogeneous datasets strongly suggests that, contrarily to received wisdom, deep learning primitives such as residual blocks are by themselves sufficient to solve a wide range of forecasting problems. Finally, we demonstrate how the proposed architecture can be augmented to provide outputs that are interpretable without considerable loss in accuracy.