A simple combination of univariate models

This paper describes the approach that we implemented for producing the point forecasts and prediction intervals for our M4-competition submission. The proposed simple combination of univariate models (SCUM) is a median combination of the point forecasts and prediction intervals of four models, namely exponential smoothing, complex exponential smoothing, automatic autoregressive integrated moving average and dynamic optimised theta. Our submission performed very well in the M4-competition, being ranked 6^th for the point forecasts (with a small difference compared to the 2^nd submission) and prediction intervals and 2^nd and 3^rd for the point forecasts of the weekly and quarterly data respectively.     

Combining exponential smoothing forecasts using Akaike weights

Simple forecast combinations such as medians and trimmed or winsorized means are known to improve the accuracy of point forecasts, and Akaike’s Information Criterion (AIC) has given rise to so-called Akaike weights, which have been used successfully to combine statistical models for inference and prediction in specialist fields, e.g., ecology and medicine. We examine combining exponential smoothing point and interval forecasts using weights derived from AIC, small-sample-corrected AIC and BIC on the M1 and M3 Competition datasets. Weighted forecast combinations perform better than forecasts selected using information criteria, in terms of both point forecast accuracy and prediction interval coverage. Simple combinations and weighted combinations do not consistently outperform one another, while simple combinations sometimes perform worse than single forecasts selected by information criteria. We find a tendency for a longer history to be associated with a better prediction interval coverage.     

Improving time series forecasting: An approach combining bootstrap aggregation,clusters and exponential smoothing

Some recent papers have demonstrated that combining bagging (bootstrap aggregating) with exponential smoothing methods can produce highly accurate forecasts and improve the forecast accuracy relative to traditional methods. We therefore propose a new approach that combines the bagging, exponential smoothing and clustering methods. The existing methods use bagging to generate and aggregate groups of forecasts in order to reduce the variance. However, none of them consider the effect of covariance among the group of forecasts, even though it could have a dramatic impact on the variance of the group, and therefore on the forecast accuracy. The proposed approach, referred to here as Bagged.Cluster.ETS, aims to reduce the covariance effect by using partitioning around medoids (PAM) to produce clusters of similar forecasts, then selecting several forecasts from each cluster to create a group with a reduced variance. This approach was tested on various different time series sets from the M3 and CIF 2016 competitions. The empirical results have shown a substantial reduction in the forecast error, considering sMAPE and MASE.     

GROEC: Combination method via Generalized Rolling Origin Evaluation

Combination methods have performed well in time series forecast competitions. This study proposes a simple but general methodology for combining time series forecast methods. Weights are calculated using a cross-validation scheme that assigns greater weights to methods with more accurate in-sample predictions. The methodology was used to combine forecasts from the Theta, exponential smoothing, and ARIMA models, and placed fifth in the M4 Competition for both point and interval forecasting.     

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.     

The tourism forecasting competition

We evaluate the performances of various methods for forecasting tourism data. The data used include 366 monthly series, 427 quarterly series and 518 annual series, all supplied to us by either tourism bodies or academics who had used them in previous tourism forecasting studies. The forecasting methods implemented in the competition are univariate and multivariate time series approaches, and econometric models. This forecasting competition differs from previous competitions in several ways: (i) we concentrate on tourism data only; (ii) we include approaches with explanatory variables; (iii) we evaluate the forecast interval coverage as well as the point forecast accuracy; (iv) we observe the effect of temporal aggregation on the forecasting accuracy; and (v) we consider the mean absolute scaled error as an alternative forecasting accuracy measure. We find that pure time series approaches provide more accurate forecasts for tourism data than models with explanatory variables. For seasonal data we implement three fully automated pure time series algorithms that generate accurate point forecasts, and two of these also produce forecast coverage probabilities which are satisfactorily close to the nominal rates. For annual data we find that Naïve forecasts are hard to beat.     

Improving forecasting by estimating time series structural components across multiple frequencies

Identifying the most appropriate time series model to achieve a good forecasting accuracy is a challenging task. We propose a novel algorithm that aims to mitigate the importance of model selection, while increasing the accuracy. Multiple time series are constructed from the original time series, using temporal aggregation. These derivative series highlight different aspects of the original data, as temporal aggregation helps in strengthening or attenuating the signals of different time series components. In each series, the appropriate exponential smoothing method is fitted and its respective time series components are forecast. Subsequently, the time series components from each aggregation level are combined, then used to construct the final forecast. This approach achieves a better estimation of the different time series components, through temporal aggregation, and reduces the importance of model selection through forecast combination. An empirical evaluation of the proposed framework demonstrates significant improvements in forecasting accuracy, especially for long-term forecasts.     

Exponential smoothing with a damped multiplicative trend

Multiplicative trend exponential smoothing has received very little attention in the literature. It involves modelling the local slope by smoothing successive ratios of the local level, and this leads to a forecast function that is the product of level and growth rate. By contrast, the popular Holt method uses an additive trend formulation. It has been argued that more real series have multiplicative trends than additive. However, even if this is true, it seems likely that the more conservative forecast function of the Holt method will be more robust when applied in an automated way to a large batch of series with different types of trend. In view of the improvements in accuracy seen in dampening the Holt method, in this paper we investigate a new damped multiplicative trend approach. An empirical study, using the monthly time series from the M3-Competition, gave encouraging results for the new approach at a range of forecast horizons, when compared to the established exponential smoothing methods.     

fETSmcs: Feature-based ETS model component selection

The well-developed ETS (ExponenTial Smoothing, or Error, Trend, Seasonality) method incorporates a family of exponential smoothing models in state space representation and is widely used for automatic forecasting. The existing ETS method uses information criteria for model selection by choosing an optimal model with the smallest information criterion among all models fitted to a given time series. The ETS method under such a model selection scheme suffers from computational complexity when applied to large-scale time series data. To tackle this issue, we propose an efficient approach to ETS model selection by training classifiers on simulated data to predict appropriate model component forms for a given time series. We provide a simulation study to show the model selection ability of the proposed approach on simulated data. We evaluate our approach on the widely used M4 forecasting competition dataset in terms of both point forecasts and prediction intervals. To demonstrate the practical value of our method, we showcase the performance improvements from our approach on a monthly hospital dataset.     

Bayesian forecast combination using time-varying features

In this work, we propose a novel framework for density forecast combination by constructing time-varying weights based on time-varying features. Our framework estimates weights in the forecast combination via Bayesian log predictive scores, in which the optimal forecast combination is determined by time series features from historical information. In particular, we use an automatic Bayesian variable selection method to identify the importance of different features. To this end, our approach has better interpretability compared to other black-box forecasting combination schemes. We apply our framework to stock market data and M3 competition data. Based on our structure, a simple maximum-a-posteriori scheme outperforms benchmark methods, and Bayesian variable selection can further enhance the accuracy for both point forecasts and density forecasts.     

Seasonal exponential smoothing with damped trends : An application for production planning

This paper reviews a spreadsheet-based forecasting approach which a process industry manufacturer developed and implemented to link annual corporate forecasts with its manufacturing/distribution operations. First, we consider how this forecasting system supports overall production planning and why it must be compatible with corporate forecasts. We then review the results of substantial testing of variations on the Winters three-parameter exponential smoothing model on 28 actual product family time series. In particular, we evaluate whether the use of damping parameters improves forecast accuracy. The paper concludes that a Winters four-parameter model (i.e. the standard Winters three-parameter model augmented by a fourth parameter to damp the trend) provides the most accurate forecasts of the models evaluated. Our application confirms the fact that there are situations where the use of damped trend parameters in short-run exponential smoothing based forecasting models is beneficial.     

An empirical analysis of neural network memory structures for basin water quality forecasting

This research investigates the cumulative multi-period forecast accuracy of a diverse set of potential forecasting models for basin water quality management. The models are characterized by their short-term (memory by delay or memory by feedback) and long-term (linear or nonlinear) memory structures. The experiments are conducted as a series of forecast cycles, with a rolling origin of a constant fit size. The models are recalibrated with each cycle, and out-of-sample forecasts are generated for a five-period forecast horizon. The results confirm that the JENN and GMNN neural network models are generally more accurate than competitors for cumulative multi-period basin water quality prediction. For example, the JENN and GMNN models reduce the cumulative five-period forecast errors by as much as 50%, relative to exponential smoothing and ARIMA models. These findings are significant in view of the increasing social and economic consequences of basin water quality management, and have the potential for extention to other scientific, medical, and business applications where multi-period predictions of nonlinear time series are critical.     

Predicting housing prices in China based on modified Holt's exponential smoothing incorporating whale optimization algorithm

The forecast of the real estate market is an important part of studying the Chinese economic market. Most existing methods have strict requirements on input variables and are complex in parameter estimation. To obtain better prediction results, a modified Holt's exponential smoothing (MHES) method was proposed to predict the housing price by using historical data. Unlike the traditional exponential smoothing models, MHES sets different weights on historical data and the smoothing parameters depend on the sample size. Meanwhile, the proposed MHES incorporates the whale optimization algorithm (WOA) to obtain the optimal parameters. Housing price data from Kunming, Changchun, Xuzhou and Handan were used to test the performance of the model. The housing prices results of four cities indicate that the proposed method has a smaller prediction error and shorter computation time than that of other traditional models. Therefore, WOA-MHES can be applied efficiently to housing price forecasting and can be a reliable tool for market investors and policy makers.     

The M4 Competition: Results,findings, conclusion and way forward

The M4 competition is the continuation of three previous competitions started more than 45 years ago whose purpose was to learn how to improve forecasting accuracy, and how such learning can be applied to advance the theory and practice of forecasting. The purpose of M4 was to replicate the results of the previous ones and extend them into three directions: First significantly increase the number of series, second include Machine Learning (ML) forecasting methods, and third evaluate both point forecasts and prediction intervals. The five major findings of the M4 Competitions are: 1. Out Of the 17 most accurate methods, 12 were “combinations” of mostly statistical approaches. 2. The biggest surprise was a “hybrid” approach that utilized both statistical and ML features. This method’s average sMAPE was close to 10% more accurate than the combination benchmark used to compare the submitted methods. 3. The second most accurate method was a combination of seven statistical methods and one ML one, with the weights for the averaging being calculated by a ML algorithm that was trained to minimize the forecasting. 4. The two most accurate methods also achieved an amazing success in specifying the 95% prediction intervals correctly. 5. The six pure ML methods performed poorly, with none of them being more accurate than the combination benchmark and only one being more accurate than Naïve2. This paper presents some initial results of M4, its major findings and a logical conclusion. Finally, it outlines what the authors consider to be the way forward for the field of forecasting.     

Beyond point forecasting: Evaluation of alternative prediction intervals for tourist arrivals

This paper evaluates the performances of prediction intervals generated from alternative time series models, in the context of tourism forecasting. The forecasting methods considered include the autoregressive (AR) model, the AR model using the bias-corrected bootstrap, seasonal ARIMA models, innovations state space models for exponential smoothing, and Harvey’s structural time series models. We use thirteen monthly time series for the number of tourist arrivals to Hong Kong and Australia. The mean coverage rates and widths of the alternative prediction intervals are evaluated in an empirical setting. It is found that all models produce satisfactory prediction intervals, except for the autoregressive model. In particular, those based on the bias-corrected bootstrap perform best in general, providing tight intervals with accurate coverage rates, especially when the forecast horizon is long.     

Distributed ARIMA models for ultra-long time series

Providing forecasts for ultra-long time series plays a vital role in various activities, such as investment decisions, industrial production arrangements, and farm management. This paper develops a novel distributed forecasting framework to tackle the challenges of forecasting ultra-long time series using the industry-standard MapReduce framework. The proposed model combination approach retains the local time dependency. It utilizes a straightforward splitting across samples to facilitate distributed forecasting by combining the local estimators of time series models delivered from worker nodes and minimizing a global loss function. Instead of unrealistically assuming the data generating process (DGP) of an ultra-long time series stays invariant, we only make assumptions on the DGP of subseries spanning shorter time periods. We investigate the performance of the proposed approach with AutoRegressive Integrated Moving Average (ARIMA) models using the real data application as well as numerical simulations. Our approach improves forecasting accuracy and computational efficiency in point forecasts and prediction intervals, especially for longer forecast horizons, compared to directly fitting the whole data with ARIMA models. Moreover, we explore some potential factors that may affect the forecasting performance of our approach.     

Improving forecast stability using deep learning

In this paper, we define forecast (in)stability in terms of the variability in forecasts for a specific time period caused by updating the forecast for this time period when new observations become available, i.e., as time passes. We propose an extension to the state-of-the-art N-BEATS deep learning architecture for the univariate time series point forecasting problem. The extension allows us to optimize forecasts from both a traditional forecast accuracy perspective as well as a forecast stability perspective. We show that the proposed extension results in forecasts that are more stable without leading to a deterioration in forecast accuracy for the M3 and M4 data sets. Moreover, our experimental study shows that it is possible to improve both forecast accuracy and stability compared to the original N-BEATS architecture, indicating that including a forecast instability component in the loss function can be used as regularization mechanism.     

Cross-temporal forecast reconciliation: Optimal combination method and heuristic alternatives

Forecast reconciliation is a post-forecasting process aimed to improve the quality of the base forecasts for a system of hierarchical/grouped time series. Cross-sectional and temporal hierarchies have been considered in the literature, but generally, these two features have not been fully considered together. The paper presents two new results by adopting a notation that simultaneously deals with both forecast reconciliation dimensions. (i) The closed-form expression of the optimal (in the least squares sense) point forecasts fulfilling both contemporaneous and temporal constraints. (ii) An iterative procedure that produces cross-temporally reconciled forecasts by alternating forecast reconciliation along one single dimension (either cross-sectional or temporal) at each iteration step. The feasibility of the proposed procedures, along with first evaluations of their performance as compared to the most performing ‘single dimension’ (either cross-sectional or temporal) forecast reconciliation procedures, is studied through a forecasting experiment on the 95 quarterly time series of the Australian Gross Domestic Product from Income and Expenditure sides. For this dataset, the new procedures, in addition to providing fully coherent forecasts in both cross-sectional and temporal dimensions, improve the forecast accuracy of the state-of-the-art point forecast reconciliation techniques.     

CONSTRUCTING OPTIMAL DENSITY FORECASTS FROM POINT FORECAST COMBINATIONS

Decision makers often observe point forecasts of the same variable computed, for instance, by commercial banks, IMF and the World Bank, but the econometric models used by such institutions are frequently unknown. This paper shows how to use the information available on point forecasts to compute optimal density forecasts. Our idea builds upon the combination of point forecasts under general loss functions and unknown forecast error distributions. We use real‐time data to forecast the density of US inflation. The results indicate that the proposed method materially improves the real‐time accuracy of density forecasts vis‐à‐vis those from the (unknown) individual econometric models. Copyright © 2013 John Wiley & Sons, Ltd.