The decomposition of time series into components is an important task that helps to understand time series and can enable better forecasting. Nowadays, with high sampling rates leading to high-frequency data (such as daily, hourly, or minutely data), many real-world datasets contain time series data that can exhibit multiple seasonal patterns. Although several methods have been proposed to decompose time series better under these circumstances, they are often computationally inefficient or inaccurate. In this study, we propose Multiple Seasonal-Trend decomposition using Loess (MSTL), an extension to the traditional Seasonal-Trend decomposition using Loess (STL) procedure, allowing the decomposition of time series with multiple seasonal patterns. In our evaluation on synthetic and a perturbed real-world time series dataset, compared to other decomposition benchmarks, MSTL demonstrates competitive results with lower computational cost. The implementation of MSTL is available in the R package forecast.

Forecast reconciliation ensures forecasts of time series in a hierarchy adhere to aggregation constraints, enabling aligned decision making. While forecast reconciliation can enhance overall accuracy in a hierarchical or grouped structure, it can lead to worse forecasts for certain series, with the greatest gains typically seen in series that originally have poorly performing base forecasts. In practical applications, some series in a structure often produce poor base forecasts due to model misspecification or low forecastability. To mitigate their negative impact, we propose two categories of forecast reconciliation methods that incorporate automatic time series selection based on out-of-sample and in-sample information, respectively. These methods keep “poor” base forecasts unused in forming reconciled forecasts, while adjusting the weights assigned to the remaining series accordingly when generating bottom-level reconciled forecasts. Additionally, our methods ameliorate disparities stemming from varied estimators of the base forecast error covariance matrix, alleviating challenges associated with estimator selection. Empirical evaluations through two simulation studies and applications using Australian labour force and domestic tourism data demonstrate the potential of the proposed methods to exclude series with high scaled forecast errors and show promising results.

A useful approach to exploratory functional data analysis is to work in the lower-dimensional principal component space rather than in the original functional data space. I demonstrate this approach by finding anomalies in age-specific US mortality rates between 1933 and 2022. The same approach can be employed for many other standard data analysis tasks and has the advantage that it allows immediate use of the vast array of multivariate data analysis tools that already exist, rather than having to develop new tools for functional data.

Organizations worldwide that rely on data-driven approaches regularly employ forecasting methods to enhance their planning and decision-making processes. While extensive research has examined the harms associated with traditional machine learning applications, relatively little attention has been given to the ethical implications of time series forecasting. However, forecasting presents distinct ethical challenges due to its diverse organizational applications, varied objectives, and unique data processing, model development, and evaluation workflows. These distinctions complicate the direct application of existing machine learning harm taxonomies to common forecasting scenarios. To address this gap, we conduct multiple interviews with industry experts and academic researchers, systematically identifying and analyzing underexplored domains, use cases, and potential risks associated with forecasting. Our objective is to develop a novel taxonomy of forecasting-specific harms. Drawing inspiration from Microsoft Azure taxonomy for responsible innovation, we integrate a human-led inductive coding approach with AI-driven analysis to extract key categories of harm in forecasting. This taxonomy aims to support researchers and practitioners by fostering ethical reflection on their decision-making throughout the forecasting process. Additionally, we seek to establish a research agenda focused on identifying measures to mitigate potential harms in forecasting. By highlighting unique risks within forecasting, our work contributes to the broader discourse on machine learning ethics.

Forecast reconciliation is a post-forecasting process that involves transforming a set of incoherent forecasts into coherent forecasts which satisfy a given set of linear constraints for a multivariate time series. In this paper we extend the current state-of-the-art cross-sectional probabilistic forecast reconciliation approach to encompass a cross-temporal framework, where temporal constraints are also applied. Our proposed methodology employs both parametric Gaussian and non-parametric bootstrap approaches to draw samples from an incoherent cross-temporal distribution. To improve the estimation of the forecast error covariance matrix, we propose using multi-step residuals, especially in the time dimension where the usual one-step residuals fail. To address high-dimensionality issues, we present four alternatives for the covariance matrix, where we exploit the two-fold nature (cross-sectional and temporal) of the cross-temporal structure, and introduce the idea of overlapping residuals. We assess the effectiveness of the proposed cross-temporal reconciliation approaches through a simulation study that investigates their theoretical and empirical properties and two forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, the optimal cross-temporal reconciliation approaches significantly outperform the incoherent base forecasts in terms of the Continuous Ranked Probability Score and the Energy Score. Overall, the results highlight the potential of the proposed methods to improve the accuracy of probabilistic forecasts and to address the challenge of integrating disparate scenarios while coherently taking into account short-term operational, medium-term tactical, and long-term strategic planning.

We consider the problem of constructing distribution-free prediction intervals for multi-step time series forecasting, with a focus on the temporal dependencies inherent in multi-step forecast errors. We establish that the optimal h-step-ahead forecast errors exhibit serial correlation up to lag (h-1) under a general non-stationary autoregressive data generating process. To leverage these properties, we propose the Autocorrelated Multi-step Conformal Prediction (AcMCP) method, which effectively incorporates autocorrelations in multi-step forecast errors, resulting in more statistically efficient prediction intervals. This method guarantees asymptotic marginal coverage for multi-step prediction intervals, though we note that, for finite samples, the coverage error admits an upper bound that increases with the forecasting horizon. Additionally, we extend several easy-to-implement conformal prediction methods, originally designed for single-step forecasting, to accommodate multi-step scenarios. Through empirical evaluations, including simulations and applications to data, we demonstrate that AcMCP achieves coverage that closely aligns with the target within local windows, while providing adaptive prediction intervals that effectively respond to varying conditions.

Accurate forecasts of ambulance demand are crucial inputs when planning and deploying staff and fleet. Such demand forecasts are required at national, regional, and sub-regional levels and must take account of the nature of incidents and their priorities. These forecasts are often generated independently by different teams within the organization. As a result, forecasts at different levels may be inconsistent, resulting in conflicting decisions and a lack of coherent coordination in the service. To address this issue, we exploit the hierarchical and grouped structure of the demand time series and apply forecast reconciliation methods to generate both point and probabilistic forecasts that are coherent and use all the available data at all levels of disaggregation. The methods are applied to daily incident data from an ambulance service in Great Britain, from October 2015 to July 2019, disaggregated by nature of incident, priority, managing health board, and control area. We use an ensemble of forecasting models and show that the resulting forecasts are better than any individual forecasting model. We validate the forecasting approach using time series cross-validation.

A novel forecast linear augmented projection (FLAP) method is introduced, which reduces the forecast error variance of any unbiased multivariate forecast without introducing bias. The method first constructs new component series which are linear combinations of the original series. Forecasts are then generated for both the original and component series. Finally, the full vector of forecasts is projected onto a linear subspace where the constraints implied by the combination weights hold. It is proven that the trace of the forecast error variance is non-increasing with the number of components, and mild conditions are established for which it is strictly decreasing. It is also shown that the proposed method achieves maximum forecast error variance reduction among linear projection methods. The theoretical results are validated through simulations and two empirical applications based on Australian tourism and FRED-MD data. Notably, using FLAP with Principal Component Analysis (PCA) to construct the new series leads to substantial forecast error variance reduction.

One of the most challenging aspects for managers when building a forecasting system is choosing how to aggregate the data at different levels. This is frequently done without the manager knowing how these choices can compromise the system's accuracy. This article illustrates these compromises by comparing different structures and aggregation criteria. Our article proposes and empirically tests a framework on how to build a coherent and more accurate forecasting system. The framework's first phase compares different time series forecasting methods, including statistical, "standard" machine learning, and deep learning. Results show that one of the statistical methods (autoregressive integrated moving average, or, for short, ARIMA) outperforms machine and deep learning methods. The second phase compares different combinations of aggregation criteria, structures of the forecasting system, and coherent forecast methods (i.e., adjustments to the forecasts at different levels of aggregation). The results show that using different criteria and structures indeed impacts predictions' accuracy. When it is necessary to disaggregate the forecast, our results show that it is best to add more information in a grouped structure, adjusted by a bottom-up method. This combination provides the best performance, that is, the lowest mean absolute-scaled error (MASE) in most nodes, compared to the other structures and coherent forecast methods used. The results also suggest that aggregating the time series further by geographical regions is essential to improve accuracy when forecasting products' and channels' sales.

Detecting anomalies in a temporal sequence of graphs can be applied is areas such as the detection of accidents in transport networks and cyber attacks in computer networks. Existing methods for detecting abnormal graphs can suffer from multiple limitations, such as high false positive rates as well as difficulties with handling variable-sized graphs and non-trivial temporal dynamics. To address this, we propose a technique where temporal dependencies are explicitly modelled via time series analysis of a large set of pertinent graph features, followed by using residuals to remove the dependencies. Extreme Value Theory is then used to robustly model and classify any remaining extremes, aiming to produce low false positives rates. Comparative evaluations on a multitude of graph instances show that the proposed approach obtains considerably better accuracy than TensorSplat and Laplacian Anomaly Detection.

In this paper, we propose a novel approach to improving forecasts of stock market indexes by considering common stock prices as hierarchical time series, combining clustering with forecast reconciliation. We propose grouping the individual stock price series in various ways including via metadata and using unsupervised learning techniques. The proposed approach is applied to the Dow Jones Industrial Average Index and the Standard & Poor 500 Index and their component stocks, and the results obtained with different grouping approaches are compared. The results empirically demonstrate that the combined use of clustering and reconciliation improves the forecast accuracy of the stock market indexes and their constituents.

Manifold learning obtains a low-dimensional representation of an underlying Riemannian manifold supporting high-dimensional data. Kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the Riemannian manifold. We propose a novel distortion-corrected kernel density estimator (DC-KDE) for any manifold learning embedding, with a bandwidth that depends on the estimated Riemannian metric at each data point. Exploiting the geometric information of the manifold leads to more accurate density estimation, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations including one with data lying in a 100 dimensional ambient space. We demonstrate that the proposed DC-KDE improves the density estimates as long as the manifold learning embedding is of sufficient quality, and has higher rank correlations with the true manifold density. Further simulation results are provided via a supplementary R shiny app. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data.