Treating and Pruning: New approaches to forecasting model selection and combination using prediction intervals

We propose a new way of selecting among model forms in automated exponential smoothing routines, consequently enhancing their predictive power. The procedure, here addressed as treating, operates by selectively subsetting the ensemble of competing models based on information from their prediction intervals. By the same token, we set forth a pruning strategy to improve the accuracy of both point forecasts and prediction intervals in forecast combination methods. The proposed approaches are respectively applied to automated exponential smoothing routines and Bagging algorithms, to demonstrate their potential. An empirical experiment is conducted on a wide range of series from the M-Competitions. The results attest that the proposed approaches are simple, without requiring much additional computational cost, but capable of substantially improving forecasting accuracy for both point forecasts and prediction intervals, outperforming important benchmarks and recently developed forecast combination methods.     

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.     

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.     

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.     

A hybrid method of exponential smoothing and recurrent neural networks for time series forecasting

This paper presents the winning submission of the M4 forecasting competition. The submission utilizes a dynamic computational graph neural network system that enables a standard exponential smoothing model to be mixed with advanced long short term memory networks into a common framework. The result is a hybrid and hierarchical forecasting method.     

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.     

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.     

Incorporating a tracking signal into a state space model

It is a common practice to complement a forecasting method such as simple exponential smoothing with a monitoring scheme to detect those situations where forecasts have failed to adapt to structural change. It will be suggested in this paper that the equations for simple exponential smoothing can be augmented by a common monitoring statistic to provide a method that automatically adapts to structural change without human intervention. The resulting method, which turns out to be a restricted form of damped trend corrected exponential smoothing, is compared with related methods on the annual data from the M3 competition. It is shown to be better than simple exponential smoothing and more consistent than traditional damped trend exponential smoothing.     

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.     

Data transforms with exponential smoothing methods of forecasting

In this paper, transforms are used with exponential smoothing, in the quest for better forecasts. Two types of transforms are explored: those which are applied to a time series directly, and those which are applied indirectly to the prediction errors. The various transforms are tested on a large number of time series from the M3 competition, and ANOVA is applied to the results. We find that the non-transformed time series is significantly worse than some transforms on the monthly data, and on a distribution-based performance measure for both annual and quarterly data.     

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.     

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.     

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.     

Macroeconomic forecasting in Austria: An analysis of accuracy

In the present paper, we attempt a critical evaluation of macroeconomic forecasting in Austria. For this purpose, we calculate conventional magnitude measures of accuracy as well as probabilities of correctly predicting directional change for the forecasts made by two Austrian institutions (WIFO and IHS) and by the OECD. ARIMA models and Holt-Winters exponential smoothing serve as benchmarks for comparison.     

Forecast Encompassing and Parameter Estimation*

A desirable property of a forecast is that it encompasses competing predictions, in the sense that the accuracy of the preferred forecast cannot be improved through linear combination with a rival prediction. In this paper, we investigate the impact of the uncertainty associated with estimating model parameters in‐sample on the encompassing properties of out‐of‐sample forecasts. Specifically, using examples of non‐nested econometric models, we show that forecasts from the true (but estimated) data generating process (DGP) do not encompass forecasts from competing mis‐specified models in general, particularly when the number of in‐sample observations is small. Following this result, we also examine the scope for achieving gains in accuracy by combining the forecasts from the DGP and mis‐specified models.     

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.     

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.     

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.     

Does Forecast Combination Improve Norges Bank Inflation Forecasts?*

We develop a system that provides model‐based forecasts for inflation in Norway. We recursively evaluate quasi out‐of‐sample forecasts from a large suite of models from 1999 to 2009. The performance of the models are then used to derive quasi real time weights that are used to combine the forecasts. Our results indicate that a combination forecast improves upon the point forecasts from individual models. Furthermore, a combination forecast outperforms Norges Bank's own point forecast for inflation. The beneficial results are obtained using a trimmed weighted average. Some degree of trimming is required for the combination forecasts to outperform the judgmental forecasts from the policymaker.