The M4 Competition: 100,000 time series and 61 forecasting methods

The M4 Competition follows on from the three previous M competitions, the purpose of which was to learn from empirical evidence both how to improve the forecasting accuracy and how such learning could be used to advance the theory and practice of forecasting. The aim of M4 was to replicate and extend the three previous competitions by: (a) significantly increasing the number of series, (b) expanding the number of forecasting methods, and (c) including prediction intervals in the evaluation process as well as point forecasts. This paper covers all aspects of M4 in detail, including its organization and running, the presentation of its results, the top-performing methods overall and by categories, its major findings and their implications, and the computational requirements of the various methods. Finally, it summarizes its main conclusions and states the expectation that its series will become a testing ground for the evaluation of new methods and the improvement of the practice of forecasting, while also suggesting some ways forward for the field.     

Sensitivity to autocorrelation in judgmental time series forecasting

How well can people use autocorrelation information when making judgmental forecasts? In Experiment 1, participants forecast from 12 series in which the autocorrelation varied within subjects. The participants showed a sensitivity to the degree of autocorrelation. However, their forecasts indicated that they implicitly assumed positive autocorrelation in uncorrelated time series. Experiments 2 and 2a used a one-shot single-trial between-subjects design and obtained similar results. Experiment 3 investigated the way in which the between-trials context influenced forecasting. The results showed that forecasts are affected by the characteristics of previous series, as well as those of the series from which forecasts are to be made. Our findings can be accommodated within an adaptive approach. Forecasters base their initial expectations of series characteristics on their past experience and modify these expectations in a pseudo-Bayesian manner on the basis of their analysis of those characteristics in the series to be forecast.     

Investigating the accuracy of cross-learning time series forecasting methods

The M4 competition identified innovative forecasting methods, advancing the theory and practice of forecasting. One of the most promising innovations of M4 was the utilization of cross-learning approaches that allow models to learn from multiple series how to accurately predict individual ones. In this paper, we investigate the potential of cross-learning by developing various neural network models that adopt such an approach, and we compare their accuracy to that of traditional models that are trained in a series-by-series fashion. Our empirical evaluation, which is based on the M4 monthly data, confirms that cross-learning is a promising alternative to traditional forecasting, at least when appropriate strategies for extracting information from large, diverse time series data sets are considered. Ways of combining traditional with cross-learning methods are also examined in order to initiate further research in the field.     

Model selection in univariate time series forecasting using discriminant analysis

When a large number of time series are to be forecast on a regular basis, as in large scale inventory management or production control, the appropriate choice of a forecast model is important as it has the potential for large cost savings through improved accuracy. A possible solution to this problem is to select one best forecast model for all the series in the dataset. Alternatively one may develop a rule that will select the best model for each series. Fildes (1989) calls the former an aggregate selection rule and the latter an individual selection rule. In this paper we develop an individual selection rule using discriminant analysis and compare its performance to aggregate selection for the quarterly series of the M-Competition data. A number of forecast accuracy measures are used for the evaluation and confidence intervals for them are constructed using bootstrapping. The results indicate that the individual selection rule based on discriminant scores is more accurate, and sometimes significantly so, than any aggregate selection method.     

Confidence intervals: An empirical investigation of the series in the M-competition

This paper empirically evaluates the uncertainty of forecasts. It does so using the 1001 series of the M-Competition. The study indicates that although, in model fitting the percentage of observations outside the confidence intervals is close to that postulated theoretically, this is not true for forecasting. In the latter case the percentage of observations outside the confidence intervals is much higher than that postulated theoretically. This is so for the great majority of series, forecasting horizonts, and methods. In addition to evaluating the extent of uncertainty, we provide tables to help users to construct more realistic confidence intervals for their forecasts.     

Correlated daily time series and forecasting in the M4 competition

We participated in the M4 competition for time series forecasting and here describe our methods for forecasting daily time series. We used an ensemble of five statistical forecasting methods and a method that we refer to as the correlator. Our retrospective analysis using the ground truth values published by the M4 organisers after the competition demonstrates that the correlator was responsible for most of our gains over the naïve constant forecasting method. We identify data leakage as one reason for its success, due partly to test data selected from different time intervals, and partly to quality issues with the original time series. We suggest that future forecasting competitions should provide actual dates for the time series so that some of these leakages could be avoided by participants.     

Temporal Fusion Transformers for interpretable multi-horizon time series forecasting

Multi-horizon forecasting often contains a complex mix of inputs – including static (i.e. time-invariant) covariates, known future inputs, and other exogenous time series that are only observed in the past – without any prior information on how they interact with the target. Several deep learning methods have been proposed, but they are typically ‘black-box’ models that do not shed light on how they use the full range of inputs present in practical scenarios. In this paper, we introduce the Temporal Fusion Transformer (TFT) – a novel attention-based architecture that combines high-performance multi-horizon forecasting with interpretable insights into temporal dynamics. To learn temporal relationships at different scales, TFT uses recurrent layers for local processing and interpretable self-attention layers for long-term dependencies. TFT utilizes specialized components to select relevant features and a series of gating layers to suppress unnecessary components, enabling high performance in a wide range of scenarios. On a variety of real-world datasets, we demonstrate significant performance improvements over existing benchmarks, and highlight three practical interpretability use cases of TFT.     

Principles and algorithms for forecasting groups of time series: Locality and globality

Global methods that fit a single forecasting method to all time series in a set have recently shown surprising accuracy, even when forecasting large groups of heterogeneous time series. We provide the following contributions that help understand the potential and applicability of global methods and how they relate to traditional local methods that fit a separate forecasting method to each series:

• •Global and local methods can produce the same forecasts without any assumptions about similarity of the series in the set.
• •The complexity of local methods grows with the size of the set while it remains constant for global methods. This result supports the recent evidence and provides principles for the design of new algorithms.
• •In an extensive empirical study, we show that purposely naïve algorithms derived from these principles show outstanding accuracy. In particular, global linear models provide competitive accuracy with far fewer parameters than the simplest of local methods.

     

Shrinkage estimators of time series seasonal factors and their effect on forecasting accuracy

This paper shows that forecasting accuracy can be improved through better estimation of seasonal factors under conditions for which relatively simple methods are preferred, such as relatively few historical data, noisy data, and/or a large number of series to be forecasted. In such situations, the preferred method of seasonal adjustment is often ratio-to-moving-averages (classical) decomposition. This paper proposes two shrinkage estimators to improve the accuracy of classical decomposition seasonal factors. In a simulation study, both of the proposed estimators provided consistently greater accuracy than classical decomposition, with the improvement sometimes being dramatic. The performances of the two estimators depended on characteristics of the series, and guidelines were developed for choosing one of them under a given set of conditions. For a set of monthly, M-competition series, greater forecasting accuracy was achieved when either of the proposed methods was used for seasonal adjustment rather than classical decomposition, and the greatest accuracy was achieved by following the guidelines for choosing a method.     

Forecasting from time series subject to sporadic perturbations: Effectiveness of different types of forecasting support

How effective are different approaches for the provision of forecasting support? Forecasts may be either unaided or made with the help of statistical forecasts. In practice, the latter are often crude forecasts that do not take sporadic perturbations into account. Most research considers forecasts based on series that have been cleansed of perturbation effects. This paper considers an experiment in which people made forecasts from time series that were disturbed by promotions. In all conditions, under-forecasting occurred during promotional periods and over-forecasting during normal ones. The relative sizes of these effects depended on the proportions of periods in the data series that contained promotions. The statistical forecasts improved the forecasting accuracy, not because they reduced these biases, but because they decreased the random error (scatter). The performance improvement did not depend on whether the forecasts were based on cleansed series. Thus, the effort invested in producing cleansed time series from which to forecast may not be warranted: companies may benefit from giving their forecasters even crude statistical forecasts. In a second experiment, forecasters received optimal statistical forecasts that took the effects of promotions into account fully. This increased the accuracy because the biases were almost eliminated and the random error was reduced by 20%. Thus, the additional effort required to produce forecasts that take promotional effects into account is worthwhile.     

Consistent estimation of a general nonparametric regression function in time series

We propose an estimator of the conditional distribution of X_t|X_t−1,X_t−2,…, and the corresponding regression function , where the conditioning set is of infinite order. We establish consistency of our estimator under stationarity and ergodicity conditions plus a mild smoothness condition.     

An empirical investigation of water consumption forecasting methods

Many regions on earth face daily limitations in the quantity and quality of the water resources available. As a result, it is necessary to implement reliable methodologies for water consumption forecasting that will enable the better management and planning of water resources. This research analyses, for the first time, a large database containing data from 2 million water meters in 274 unique postal codes, in one of the most densely populated areas of Europe, which faces issues of droughts and overconsumption in the hot summer months. Using the R programming language, we built and tested three alternative forecasting methodologies, employing univariate forecasting techniques including a machine-learning algorithm, with very promising results.     

Prediction intervals in conditionally heteroscedastic time series with stochastic components

Differencing is a very popular stationary transformation for series with stochastic trends. Moreover, when the differenced series is heteroscedastic, authors commonly model it using an ARMA-GARCH model. The corresponding ARIMA-GARCH model is then used to forecast future values of the original series. However, the heteroscedasticity observed in the stationary transformation should be generated by the transitory and/or the long-run component of the original data. In the former case, the shocks to the variance are transitory and the prediction intervals should converge to homoscedastic intervals with the prediction horizon. We show that, in this case, the prediction intervals constructed from the ARIMA-GARCH models could be inadequate because they never converge to homoscedastic intervals. All of the results are illustrated using simulated and real time series with stochastic levels.     

Comparison of intraday probabilistic forecasting of solar irradiance using only endogenous data

Accurate solar forecasts are necessary to improve the integration of solar renewables into the energy grid. In recent years, numerous methods have been developed for predicting the solar irradiance or the output of solar renewables. By definition, a forecast is uncertain. Thus, the models developed predict the mean and the associated uncertainty. Comparisons are therefore necessary and useful for assessing the skill and accuracy of these new methods in the field of solar energy.The aim of this paper is to present a comparison of various models that provide probabilistic forecasts of the solar irradiance within a very strict framework. Indeed, we consider focusing on intraday forecasts, with lead times ranging from 1 to 6 h. The models selected use only endogenous inputs for generating the forecasts. In other words, the only inputs of the models are the past solar irradiance data. In this context, the most common way of generating the forecasts is to combine point forecasting methods with probabilistic approaches in order to provide prediction intervals for the solar irradiance forecasts. For this task, we selected from the literature three point forecasting models (recursive autoregressive and moving average (ARMA), coupled autoregressive and dynamical system (CARDS), and neural network (NN)), and seven methods for assessing the distribution of their error (linear model in quantile regression (LMQR), weighted quantile regression (WQR), quantile regression neural network (QRNN), recursive generalized autoregressive conditional heteroskedasticity (GARCHrls), sieve bootstrap (SB), quantile regression forest (QRF), and gradient boosting decision trees (GBDT)), leading to a comparison of 20 combinations of models.None of the model combinations clearly outperform the others; nevertheless, some trends emerge from the comparison. First, the use of the clear sky index ensures the accuracy of the forecasts. This derived parameter permits time series to be deseasonalized with missing data, and is also a good explanatory variable of the distribution of the forecasting errors. Second, regardless of the point forecasting method used, linear models in quantile regression, weighted quantile regression and gradient boosting decision trees are able to forecast the prediction intervals accurately.     

Forecast combinations of computational intelligence and linear models for the NN5 time series forecasting competition

In this work we introduce the forecasting model with which we participated in the NN5 forecasting competition (the forecasting of 111 time series representing daily cash withdrawal amounts at ATM machines). The main idea of this model is to utilize the concept of forecast combination, which has proven to be an effective methodology in the forecasting literature. In the proposed system we attempted to follow a principled approach, and make use of some of the guidelines and concepts that are known in the forecasting literature to lead to superior performance. For example, we considered various previous comparison studies and time series competitions as guidance in determining which individual forecasting models to test (for possible inclusion in the forecast combination system). The final model ended up consisting of neural networks, Gaussian process regression, and linear models, combined by simple average. We also paid extra attention to the seasonality aspect, decomposing the seasonality into weekly (which is the strongest one), day of the month, and month of the year seasonality.     

An accurate and fully-automated ensemble model for weekly time series forecasting

Many businesses and industries require accurate forecasts for weekly time series nowadays. However, the forecasting literature does not currently provide easy-to-use, automatic, reproducible and accurate approaches dedicated to this task. We propose a forecasting method in this domain to fill this gap, leveraging state-of-the-art forecasting techniques, such as forecast combination, meta-learning, and global modelling. We consider different meta-learning architectures, algorithms, and base model pools. Based on all considered model variants, we propose to use a stacking approach with lasso regression which optimally combines the forecasts of four base models: a global Recurrent Neural Network (RNN) model, Theta, Trigonometric Box–Cox ARMA Trend Seasonal (TBATS), and Dynamic Harmonic Regression ARIMA (DHR-ARIMA), as it shows the overall best performance across seven experimental weekly datasets on four evaluation metrics. Our proposed method also consistently outperforms a set of benchmarks and state-of-the-art weekly forecasting models by a considerable margin with statistical significance. Our method can produce the most accurate forecasts, in terms of mean sMAPE, for the M4 weekly dataset among all benchmarks and all original competition participants.     

Spatio-temporal probabilistic forecasting of wind power for multiple farms: A copula-based hybrid model

Accurate probabilistic forecasting of wind power output is critical to maximizing network integration of this clean energy source. There is a large literature on temporal modeling of wind power forecasting, but considerably less work combining spatial dependence into the forecasting framework. Through the careful consideration of the temporal modeling component, complemented by support vector regression of the temporal model residuals, this work demonstrates that a DVINE copula model most accurately represents the residual spatial dependence. Additionally, this work proposes a complete set of validation mechanisms for multi-h-step forecasts that, when considered together, comprehensively evaluate accuracy. The model and validation mechanisms are demonstrated in two case studies, totaling ten wind farms in the Texas electric grid. The proposed method outperforms baseline and competitive models, with an average Continuous Ranked Probability Score of less than 0.15 for individual farms, and an average Energy Score of less than 0.35 for multiple farms, over the 24-hour-ahead horizon. Results show the model’s ability to replicate the power output dynamics through calibrated and sharp predictive densities.     

Mixture of MLP-experts for trend forecasting of time series: A case study of the Tehran stock exchange

A new method for forecasting the trend of time series, based on mixture of MLP experts, is presented. In this paper, three neural network combining methods and an Adaptive Network-Based Fuzzy Inference System (ANFIS) are applied to trend forecasting in the Tehran stock exchange. There are two experiments in this study. In experiment I, the time series data are the Kharg petrochemical company’s daily closing prices on the Tehran stock exchange. In this case study, which considers different schemes for forecasting the trend of the time series, the recognition rates are 75.97%, 77.13% and 81.64% for stacked generalization, modified stacked generalization and ANFIS, respectively. Using the mixture of MLP experts (ME) scheme, the recognition rate is strongly increased to 86.35%. A gain and loss analysis is also used, showing the relative forecasting success of the ME method with and without rejection criteria, compared to a simple buy and hold approach. In experiment II, the time series data are the daily closing prices of 37 companies on the Tehran stock exchange. This experiment is conducted to verify the results of experiment I and to show the efficiency of the ME method compared to stacked generalization, modified stacked generalization and ANFIS.     

The impact of incentives on the accuracy of subjects in judgmental forecasting experiments

There is an ongoing debate in the social sciences about whether or not financial incentives are needed in order to obtain good performance from experimental subjects. This debate often extends into the research on judgmental forecasting. Thus, an experiment was conducted to assess the effects of financial incentives on time series forecasting accuracy. There was no evidence that financial incentives impacted forecasting accuracy in stable time series. Financial incentives also had no impact immediately after instabilities occurred and no impact once the trend in the data had fully emerged.     

A note on three-stage confidence intervals for the difference of locations: The exponential case

Fixed-width confidence intervals for the difference of location parameters of two independent negative exponential distributions are constructed via triple sampling when the scale parameters are unknown and unequal. The present three-stage estimation methodology is put forth because (i) it is operationally more convenient than the existing purely sequential counterpart, and (ii) the three-stage and the purely sequential estimation techniques have fairly similar asymptotic second-order characteristics.