Accurate uncertainty measurement is a key step to building robust and reliable machine learning systems. Conformal prediction is a distribution-free uncertainty quantification algorithm popular for its ease of implementation, statistical coverage guarantees, and versatility for underlying forecasters. However, existing conformal prediction algorithms for time series are limited to single-step prediction without considering the temporal dependency. In this paper we propose a Copula Conformal Prediction algorithm for multivariate, multi-step Time Series forecasting, CopulaCPTS. We prove that CopulaCPTS has finite sample validity guarantee. On several synthetic and real-world multivariate time series datasets, we show that CopulaCPTS produces more calibrated and sharp confidence intervals for multi-step prediction tasks than existing techniques.
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A methodology for high dimensional causal inference in a time series context is introduced. It is assumed that there is a monotonic transformation of the data such that the dynamics of the transformed variables are described by a Gaussian vector autoregressive process. This is tantamount to assume that the dynamics are captured by a Gaussian copula. No knowledge or estimation of the marginal distribution of the data is required. The procedure consistently identifies the parameters that describe the dynamics of the process and the conditional causal relations among the possibly high dimensional variables under sparsity conditions. The methodology allows us to identify such causal relations in the form of a directed acyclic graph. As illustrative applications we consider the impact of supply side oil shocks on the economy, and the causal relations between aggregated variables constructed from the limit order book on four stock constituents of the S&P500.
Spain is the third-largest producer of pork meat in the world, and many farms in several regions depend on the evolution of this market. However, the current pricing system is unfair, as some actors have better market information than others. In this context, historical pricing is an easy-to-find and affordable data source that can help all agents to be better informed. However, the time lag in data acquisition can affect their pricing decisions. In this paper, we study the effect that data acquisition delay has on a price prediction system using multiple prediction algorithms. We describe the integration of the best proposal into a decision support system prototype and test it in a real-case scenario. Specifically, we use public data from the most important regional pork meat markets in Spain published by the Ministry of Agriculture with a two-week delay and subscription-based data of the same markets obtained on the same day. The results show that the error difference between the best public and data subscription models is 0.6 Euro cents in favor of the data without delay. The market dimension makes these differences significant in the supply chain, giving pricing agents a better tool to negotiate market prices.
This paper proposes a novel approach to predict epidemiological parameters by integrating new real-time signals from various sources of information, such as novel social media-based population density maps and Air Quality data. We implement an ensemble of Convolutional Neural Networks (CNN) models using various data sources and fusion methodology to build robust predictions and simulate several dynamic parameters that could improve the decision-making process for policymakers. Additionally, we used data assimilation to estimate the state of our system from fused CNN predictions. The combination of meteorological signals and social media-based population density maps improved the performance and flexibility of our prediction of the COVID-19 outbreak in London. While the proposed approach outperforms standard models, such as compartmental models traditionally used in disease forecasting (SEIR), generating robust and consistent predictions allows us to increase the stability of our model while increasing its accuracy.
Conformal prediction has emerged as a rigorous means of providing deep learning models with reliable uncertainty estimates and safety guarantees. Yet, its performance is known to degrade under distribution shift and long-tailed class distributions, which are often present in real world applications. Here, we characterize the performance of several post-hoc and training-based conformal prediction methods under these settings, providing the first empirical evaluation on large-scale datasets and models. We show that across numerous conformal methods and neural network families, performance greatly degrades under distribution shifts violating safety guarantees. Similarly, we show that in long-tailed settings the guarantees are frequently violated on many classes. Understanding the limitations of these methods is necessary for deployment in real world and safety-critical applications.
Inclinometer probes are devices that can be used to measure deformations within earthwork slopes. This paper demonstrates a novel application of Bayesian techniques to real-world inclinometer data, providing both anomaly detection and forecasting. Specifically, this paper details an analysis of data collected from inclinometer data across the entire UK rail network. Practitioners have effectively two goals when processing monitoring data. The first is to identify any anomalous or dangerous movements, and the second is to predict potential future adverse scenarios by forecasting. In this paper we apply Uncertainty Quantification (UQ) techniques by implementing a Bayesian approach to anomaly detection and forecasting for inclinometer data. Subsequently, both costs and risks may be minimised by quantifying and evaluating the appropriate uncertainties. This framework may then act as an enabler for enhanced decision making and risk analysis. We show that inclinometer data can be described by a latent autocorrelated Markov process derived from measurements. This can be used as the transition model of a non-linear Bayesian filter. This allows for the prediction of system states. This learnt latent model also allows for the detection of anomalies: observations that are far from their expected value may be considered to have `high surprisal', that is they have a high information content relative to the model encoding represented by the learnt latent model. We successfully apply the forecasting and anomaly detection techniques to a large real-world data set in a computationally efficient manner. Although this paper studies inclinometers in particular, the techniques are broadly applicable to all areas of engineering UQ and Structural Health Monitoring (SHM).
We consider the problem of evaluating forecasts of binary events whose predictions are consumed by rational agents who take an action in response to a prediction, but whose utility is unknown to the forecaster. We show that optimizing forecasts for a single scoring rule (e.g., the Brier score) cannot guarantee low regret for all possible agents. In contrast, forecasts that are well-calibrated guarantee that all agents incur sublinear regret. However, calibration is not a necessary criterion here (it is possible for miscalibrated forecasts to provide good regret guarantees for all possible agents), and calibrated forecasting procedures have provably worse convergence rates than forecasting procedures targeting a single scoring rule. Motivated by this, we present a new metric for evaluating forecasts that we call U-calibration, equal to the maximal regret of the sequence of forecasts when evaluated under any bounded scoring rule. We show that sublinear U-calibration error is a necessary and sufficient condition for all agents to achieve sublinear regret guarantees. We additionally demonstrate how to compute the U-calibration error efficiently and provide an online algorithm that achieves $O(\sqrt{T})$ U-calibration error (on par with optimal rates for optimizing for a single scoring rule, and bypassing lower bounds for the traditionally calibrated learning procedures). Finally, we discuss generalizations to the multiclass prediction setting.
Forecasting has always been at the forefront of decision making and planning. The uncertainty that surrounds the future is both exciting and challenging, with individuals and organisations seeking to minimise risks and maximise utilities. The large number of forecasting applications calls for a diverse set of forecasting methods to tackle real-life challenges. This article provides a non-systematic review of the theory and the practice of forecasting. We provide an overview of a wide range of theoretical, state-of-the-art models, methods, principles, and approaches to prepare, produce, organise, and evaluate forecasts. We then demonstrate how such theoretical concepts are applied in a variety of real-life contexts. We do not claim that this review is an exhaustive list of methods and applications. However, we wish that our encyclopedic presentation will offer a point of reference for the rich work that has been undertaken over the last decades, with some key insights for the future of forecasting theory and practice. Given its encyclopedic nature, the intended mode of reading is non-linear. We offer cross-references to allow the readers to navigate through the various topics. We complement the theoretical concepts and applications covered by large lists of free or open-source software implementations and publicly-available databases.
The accurate and interpretable prediction of future events in time-series data often requires the capturing of representative patterns (or referred to as states) underpinning the observed data. To this end, most existing studies focus on the representation and recognition of states, but ignore the changing transitional relations among them. In this paper, we present evolutionary state graph, a dynamic graph structure designed to systematically represent the evolving relations (edges) among states (nodes) along time. We conduct analysis on the dynamic graphs constructed from the time-series data and show that changes on the graph structures (e.g., edges connecting certain state nodes) can inform the occurrences of events (i.e., time-series fluctuation). Inspired by this, we propose a novel graph neural network model, Evolutionary State Graph Network (EvoNet), to encode the evolutionary state graph for accurate and interpretable time-series event prediction. Specifically, Evolutionary State Graph Network models both the node-level (state-to-state) and graph-level (segment-to-segment) propagation, and captures the node-graph (state-to-segment) interactions over time. Experimental results based on five real-world datasets show that our approach not only achieves clear improvements compared with 11 baselines, but also provides more insights towards explaining the results of event predictions.
Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.
Multivariate time series forecasting is extensively studied throughout the years with ubiquitous applications in areas such as finance, traffic, environment, etc. Still, concerns have been raised on traditional methods for incapable of modeling complex patterns or dependencies lying in real word data. To address such concerns, various deep learning models, mainly Recurrent Neural Network (RNN) based methods, are proposed. Nevertheless, capturing extremely long-term patterns while effectively incorporating information from other variables remains a challenge for time-series forecasting. Furthermore, lack-of-explainability remains one serious drawback for deep neural network models. Inspired by Memory Network proposed for solving the question-answering task, we propose a deep learning based model named Memory Time-series network (MTNet) for time series forecasting. MTNet consists of a large memory component, three separate encoders, and an autoregressive component to train jointly. Additionally, the attention mechanism designed enable MTNet to be highly interpretable. We can easily tell which part of the historic data is referenced the most.
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