Forecasting strong seasonal time series with artificial neural networks

Transcription

1 Journal of Scientific ADHIKARI & Industrial & Research AGRAWAL : FORECASTING SEASONAL TIME SERIES NEURAL NETWORKS Vol. 7, October 202, pp Forecasting strong seasonal time series with artificial neural networks Ratnadip Adhikari * and R. K. Agrawal School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi-0067, INDIA Received 5 May 202 ; revised 6 August 202 ;accepted 6 September 202 Many practical time series often exhibit trends and seasonal patterns. The traditional statistical models eliminate the effect of seasonality from a time series before making future forecasts. As a result, the computational complexities are increased together with substantial reductions in overall forecasting accuracies. This paper comprehensively explores the outstanding ability of Artificial Neural Networks (ANNs) in recognizing and forecasting strong seasonal patterns without removing them from the raw data. Six real-world time series having dominant seasonal fluctuations are used in our work. The performances of the fitted ANN for each of these time series are compared with those of three traditional models both manually as well as through a non-parametric statistical test. The empirical results show that the properly designed ANNs are remarkably efficient in directly forecasting strong seasonal variations as well as outperform each of the three statistical models for all six time series. A robust algorithm together with important practical guidelines is also suggested for ANN forecasting of strong seasonal data. Keywords: Time series forecasting; Seasonal time series; Artificial neural networks; Holt-Winters, Box-Jenkins model, Support vector machine. Introduction Time series modeling and forecasting has received indispensable importance in various areas of science and engineering. Forecasting is performed by carefully analyzing the past observations to develop a proper model which in turn is used to generate future values. Seasonality is a special property which is frequently observed in many economic and financial time series. These are the fluctuations within a year which repeat in each season. Often the seasonal patterns occur with a quarterly, bimonthly or monthly period. A number of models have been derived to routinely analyze and forecast seasonal data. Some of them are the Seasonal autoregressive Integrated Moving Average (SARIMA) models 2, the Holt-Winters (HW) models 3, the Periodic Autoregressive Moving Average (PARMA) models, etc. Recently, Support Vector Machine (SVM) 4-6 has also found notable applications in forecasting seasonal data. These traditional models use either deseasonalization or differencing in order to remove or adjust the seasonal factors. However, several researchers have recently criticized the approach of seasonal adjustment and removal. Miller and Williams 7 observed that the traditional deseasonalization approaches are biased in presence of different types of trends and may even lead to overestimation *Author for correspondence adhikari.ratan@gmail.com of the seasonal components. Ghysels et al. 8 pointed that univariate time series data often show adverse nonlinear properties due to seasonal adjustments. Another critical issue encountered with such techniques is the increased model complexity. These problems are even more apparent when the seasonal pattern of the associated time series is of dominant nature. Thus, it is evident that a model which can directly track the seasonal variations will be more appealing as well as effective. Unfortunately, till now no such adequate statistical model exists in literature. During the last two decades Artificial Neural Networks (ANNs) have been evolved as an attractive and efficient alternative tool for time series forecasting. Several distinguishing properties of ANNs made them extremely popular in the forecasting domain. Their most remarkable feature is the nonlinear, nonparametric, data-driven and selfadaptive nature 9-. Although ANNs have found numerous applications in seasonal time series forecasting, but opinions widely vary regarding their efficiency. Some researchers have claimed that ANNs cannot adequately track the seasonal patterns without removing them 2,3. On the other hand, many studies have favored the just opposite conclusion, i.e. ANNs with seasonally readjusted or differenced data can achieve reasonably better forecasting accuracies than those with raw data 4,5. Moreover, the appropriate network structures and training algorithms must be selected with utmost care for having successful forecasting performances

2 658 J SCI IND RES VOL 7 OCTOBER 202 with ANNs 9,0. At present no rigorous theoretical procedure exists to resolve these ANN model designing issues. In this paper, we comprehensively analyze the effectiveness of ANNs in tracking dominant seasonal patterns. Six practical time series (three monthly and three quarterly) with strong seasonal fluctuations are used to empirically compare the forecasting performances of ANNs with those of SARIMA, HW and SVM methods. The sensitiveness of ANNs to the choice of architectures and training algorithms are studied using different number of hidden nodes and six different versions of the backpropagation training method. We further present an effective ANN forecasting algorithm for univariate seasonal time series. Materials and methods Statistical Models for Seasonal Time Series Forecasting In general the seasonality in a time series can be of two types, viz. additive and multiplicative. In the additive case, the series shows steady seasonal fluctuations whereas in the multiplicative case, the size of seasonal fluctuations varies with the means 2,6. An important tool for studying seasonal components is provided by the sample autocorrelation coefficients,6, defined as: r k = N k t= ( yt y )( yt+ k y) N t= ( yt y) ( N ) k = 0,, K,. 2 where, {y, y 2,...,y N } is the time series with mean y and k denotes the time lag. A graph which shows the successive autocorrelation values against the time lags is known as a correlogram and is indeed very helpful for understanding the intrinsic properties of a time series. The correlogram of a seasonal data exhibits the same kind of oscillations at the seasonal lags 6. Seasonal Autoregressive Integrated Moving Average (SARIMA) Model It is the most widely used statistical technique, developed by Box and Jenkins 2 for seasonal time series forecasting and is in fact a generalization of the wellknown Autoregressive Integrated Moving Average (ARIMA) model. The SARIMA model attempts to capture the seasonal and nonseasonal relationships () among the successive observations in a time series through sequences of ordinary as well as seasonal differencing of the series. This model is mathematically represented as: φ s s ( ) ( ) θ ( ) ( ) B Φ B W = B Θ B Z (2) p P t q Q t Here, B is the lag or backshift operator, defined as By t =y t- ; φ,,, p ΦP θ q Θ Q are the lagged polynomials in B of orders p, P, q and Q, respectively; Z t is a series of purely random errors and W t is the stationary nonseasonal series which is obtained after the differencing processes, i.e. t d s ( ) ( ) D W = B B y (3) t This model is commonly known as the SARIMA(p,d,q) (P,D,Q) s model. The parameters (p, P), (q, Q) stand for the autoregressive and moving average processes respectively while (d, D) specify the degrees of ordinary and seasonal differencing. The appropriate SARIMA model for a time series is usually selected through the three step iterative model-building procedure, suggested by Box and Jenkins 2. Holt-Winters (HW) Model It is a generalization of the well-known exponential smoothing technique to deal with trend and seasonal properties of a time series 3,6. In this method, the local mean level (L t ), trend (T t ) and seasonal index (I t ) of a time series are iteratively updated on the basis of the successive observations as follows: ( ) ( α)( ) ( ) ( γ) ( ) ( δ) Lt = α yt It s + Lt + It Tt = γ Lt Lt + Tt It = δ yt Lt + It s t = 0,,2, K, N where, α,?, and d are the smoothing parameters and s is the seasonal period. After specifying the model structure, the k-step ahead forecast of y t made at time t is given by: ( ) (4) yˆ = L + kt I,( k =,2, K, s) (5) t t t t s+ k There are available analogous formulae for additive case also. The HW method is quite popular for seasonal time series forecasting due to its simplicity, reduced computational

3 ADHIKARI & AGRAWAL : FORECASTING SEASONAL TIME SERIES NEURAL NETWORKS 659 Fig. Architectures of: a) a typical MLP ANN model, b) an SANN model time and reasonably good forecasting accuracy 6. A detailed study about the application of the HW method was carried out by Chatfield and Yar 3 and in this paper we implement it by precisely following their guidelines. Support Vector Machine (SVM) Model SVM is based on the Structural Risk Minimization (SRM) principle and its objective is to find a decision rule with good generalization ability through selecting some special training data points, known as the support vectors 4. Time series forecasting problem is a branch of Support Vector Regression (SVR) in which a maximum margin hyperplane is constructed for correctly classifying real-valued outputsma: 5,7. Given a training data set of N points { i, yi} i= N x with n x i R, yi R, SVM attempts to approximate the unknown data generation function in a linear form. Using Vapnik s e-insensitive loss function, the SVM regression is converted to a Quadratic Programming Problem (QPP) to minimize the empirical risk. Solving the QPP, the optimal decision hyperplane is given by N s x = xx + opt (6) i= * ( ) ( αi αi ) (, i) y K b Here, N s is the number of support vectors, α i and * α (i=, 2,..., N i s ) are the Lagrange multipliers, b opt is the optimal bias and K(x, x i ) is the kernel function. There are different choices for the SVM kernel function and a widely popular one is the Radial Basis Function (RBF) kernel, defined as K(x, y) =exp( x y 2 D 2s 2 ) where s is a tuning parameter. This kernel is used in this paper and the SVM parameters are estimated by cross-validations 5,6,7. Artificial Neural Networks for Seasonal Time Series Forecasting ANNs are a class of flexible computing frameworks which try to mimic the intelligent working paradigm of the human brain for solving a broad range of nonlinear problems. The most widely used ANN model for time series forecasting is the Multilayer Perceptron (MLP) 0, which is characterized by a feedforward architecture of an input layer, one or more hidden layers, and an output layer. Each layer contains a number of nodes which are connected to those in the immediate next layer by acyclic links. Usually, a single hidden layer is sufficient for most applications. An example of an MLP with one hidden layer is depicted in Fig. a. In a typical MLP with p input, h hidden, and a single output node, the relationship between the inputs y t-i (i=, 2,..., p) and the output y t is given by the formula: h p yt = G α0 + α jf β0 j + βijyt i j= i= (7) where a j, ß ij (i=, 2,..., p; j =, 2,..., h) are the connection weights, a 0, ß 0j are the bias terms and F, G are hidden and output layer activation functions respectively. The MLP, given by (7) is commonly referred as a (p, h, ) ANN model which performs a one-step ahead forecasting. Similarly, a (p, h, q) ANN structure can be used for q-step ahead forecasting where q is the number of output nodes.

4 660 J SCI IND RES VOL 7 OCTOBER 202 Over the years, ANNs have found numerous applications for seasonal time series forecasting. However, there are notable variations among the opinions of researchers regarding their efficiency 2-4. Recently, Hamzaçebi 8 has proposed the Seasonal ANN (SANN) model which is shown to be quite effective in forecasting seasonal data. In this model, the number of input and output neurons is represented by the seasonal period s of the time series, as shown in Fig. b. Thus, the SANN model is similar to a typical (s, h, s) ANN where h is the number of hidden nodes. Due to the implementation simplicity and efficiency of the SANN model, we use it as our base network structure in this paper and explore its forecasting strengths for time series with dominant seasonal fluctuations. ANN Model Selection Issues To achieve satisfactory accuracies with ANNs, the adequate model must have to be carefully designed 0,9. The benefit of using the (s, h, s) SANN model is that both the number of input and output nodes are already specified in terms of the seasonal period s and only an appropriate number of hidden nodes are to be determined. For this purpose, one of the following four well-known network selection criteria are often used 9,20 : Akaike Information Criterion (AIC),Bayesian Information Criterion (BIC), Schwarz s Bayesian Criterion (SBC) and Bias-corrected AIC (AIC c ). For a seasonal time series with period s, we select the maximum number of hidden nodes as: s for quarterly data max_hid_nodes = (8) s/2 for monthly data For each seasonal time series, the (s, h, s) SANN model is successively trained on the training set by varying the number of hidden nodes h from to max_hid_nodes and the network selection criteria are calculated. The optimal number of hidden nodes is then selected as that h which minimizes the value of the Bias-corrected AIC (AIC c ). The activation functions of a neural network determine the relationships between the input and output nodes and also introduce some important degree of nonlinearity to the network 0. Among various choices, we use the widely popular logistic function (F) and identity function (G) as the hidden and output layer activation functions, respectively. These are defined as: F( x) = x + e G( x) = x After selecting the proper architecture and activation functions, the next major task in ANN model designing is to select the appropriate training algorithm. So far, the most popular training method is the classic backpropagation (BP) algorithm 2. Despite its simplicity and popularity, the major drawbacks of the BP algorithm are its slow convergence rate and the risk of getting stuck at local minima. Due to these limitations, various modified BP methods have been developed in literature. Some important among them include the Levenberg- Marquardt (LM) 22, Resilient Propagation (RP) 23, Scaled Conjugate Gradient (SCG) 24, One Step Secant (OSS) 25 and Broyden-Fletcher-Goldfarb-Shanno (BFGS) 26 algorithms. Recently, the Particle Swarm Optimization (PSO) 27 has also been successfully applied as an alternative to the BP training. Adhikari and Agrawal 28 have applied PSO-Trelea and PSO-Trelea2 29 for training MLP networks to forecast three seasonal time series and have empirically demonstrated their superior performances over a BP (the RPROP) algorithm. In spite of various modified training algorithms, some crucial limitations of the standard BP technique are still remained unresolved 0. For example, none of the available training algorithms can ensure a unique global optimal solution in general. Thus, it is reasonable to train a neural network with different algorithms and then suitably combine their outputs. This practice aggregates the strengths of individual training algorithms and increases the ANN forecasting accuracy. Here, we train the SANN model for each data with six training algorithms: LM, RPROP, SCG, OSS, PSO-Trelea and PSO-Trelea2 and combine their forecasting results through arithmetic mean and median. At this point, we suggest the following robust algorithm in order to achieve efficient forecasting performances through ANNs for time series with strong seasonal fluctuations. (9) Algorithm: ANN forecasting of time series with strong seasonality Inputs: The dataset: Y=[y, y 2,..., y N ] T of a seasonal time series and the forecast horizon H. Output: The desired forecasted dataset: [ y y K y ] T Y ˆ =,,,. N+ N+ 2 N+ H

5 ADHIKARI & AGRAWAL : FORECASTING SEASONAL TIME SERIES NEURAL NETWORKS 66 Table Descriptions of the six time series datasets Time series Description Seasonal pattern Size Airline Passengers (APTS) USA Accidental Deaths (USADTS) Monthly number of international airline passengers (in thousands) (January 949 December 960). Monthly number of accidental deaths in USA ( ) Red Wine (RWTS) Sales of Australian red wine (January 980 July 995). Quarterly Sales (QSTS) Quarterly Beer Production (QBPTS) Quarterly export values of a French firm for 6 years. Quarterly beer production in USA (millions of barrels) (First quarter of 975 to fourth quarter of 982). USA expenditure (UETS) Quarterly USA new plant/equipment expenditures ( ). Monthly, multiplicative. Total: 44 Testing: 2 Monthly, additive. Total: 72 Monthly, multiplicative. Quarterly, multiplicative. Quarterly, multiplicative. Quarterly, multiplicative. Testing: 2 Total: 87 Testing: 9 Total: 24 Testing: 4 Total: 32 Testing: 8 Total: 52 Testing: 8. Divide the time series Y into appropriate training and validation subsets Y train and Y val respectively. 2. Calculate the autocorrelation coefficients of Y and plot the correlogram. 3. From the correlogram of Y, determine (or ensure) the period of seasonality s. 4. Select the k network training algorithms: TA, TA 2,, TA k. 5. Set i:=. 6. While (id k ) 7. Determine the number of hidden nodes h of the (s, h, s) SANN model for Y. 8. Train and validate the SANN model on Y train and Y val respectively with the training algorithm TA i and use it to forecast the H future observations. 9. Obtain the forecast vector: ( ) ( ) ( ) ( ) Y ˆ i = y i, i,, i. N+ yn+ 2 y K N+ H 0. i:= i +.. End while. 2. Choose an appropriate function f to combine the k forecast vectors. 3. Obtain the final forecast vector as: ( ) ( 2) ( k) ( K ) Yˆ = f Yˆ, Yˆ,, Yˆ. T Results and discussion Six real-world time series with dominant seasonal fluctuations are collected from the open-source Time Series Data Library (TSDL) repository 30. The descriptions of these time series are presented in Table and their time plots are shown in Fig. 2. In this work, the SARIMA, ANN and SVM models are fitted using MATLAB and the HW models are fitted using the R environment 3. The neural network toolbox of MATLAB 32 is used for implementing the four versions of the BP training algorithms, whereas the PSO toolbox by Birge 33 is used for implementing PSO-Trelea and PSO-Trelea2. Each ANN model is trained for 2000 epochs with each training algorithm. Every dataset is preprocessed with appropriate data transformation before fitting the forecasting models and the obtained outputs are again post-processed afterwards. The forecasting accuracies of the fitted models are evaluated in terms of the Mean Squared Error (MSE) and the Mean Absolute Percentage Error (MAPE) which are defined as: MSE= N N test test t= N ( y yˆ ) test y ˆ t yt MAPE= 00 Ntest t= y t t t 2 (0) where y t and y ˆt are respectively the actual and forecasted outputs and N test is the size of the out-of-sample

6 662 J SCI IND RES VOL 7 OCTOBER 202 Fig. 2 Time plots of: (a) APTS, (b) USADTS, (c) RWTS, (d) QSTS, (e) QBPTS, (f) UETS Fig. 3 Boxplots of MSE: a) APTS, b) USADTS, c) RWTS, d) QSTS, e) QBPTS, f) UETS testing dataset. In forecasting applications, both these error measures are desired to be as small as possible. We fit the SARIMA(0,, ) (0,, ) s models to the six time series, s being the corresponding seasonal period. The proper ANN models for APTS, USADTS, RWTS, QSTS, QBPTS and UETS datasets are respectively determined to be the (2,, 2), (2, 2, 2), (2, 2, 2), (4, 2, 4), (4, 2, 4) and (4,, 4) SANNs. We use boxplots in order to depict the effect of changing the number of hidden nodes on ANN forecasting accuracies. The boxplots of forecast MSE and MAPE values, obtained with all six training algorithms across all the datasets are depicted in Fig. 3 and Fig. 4 respectively. Figs 3 and 4 clearly depict that with a non-optimal selection of the number of hidden nodes, both MSE and MAPE values are drastically increased for each time series which indicates reasonable reductions in the ANN forecasting accuracies. This finding emphasizes the paramount importance of appropriate ANN model designing.

7 ADHIKARI & AGRAWAL : FORECASTING SEASONAL TIME SERIES NEURAL NETWORKS 663 Fig. 4 Boxplots of MAPE: a) APTS, b) USADTS, c) RWTS, d) QSTS, e) QBPTS, f) UETS Table 2 Forecasting results of the ANNs and the three traditional statistical models Forecasting Method APTS USAD RWTS QSTS QBP UETS MSE MAPE MSE MAPE MSE MAPE MSE MAPE MSE MAPE MSE MAPE ANN (LM) ANN (RPROP) ANN (SCG) ANN (OSS) ANN (PSO Trelea) ANN (PSO Trelea2) ANN (Mean) ANN (Median) SARIMA SVM HW The summary of the obtained results are presented in Table 2. The MSE values for APTS, USADTS, RWTS and QSTS are given in transformed scales (i.e. original MSE=MSE 0 4 ). The optimal HW and SVM parameters are shown in Table 3. The following important observations are evident from Table 2: The forecast errors obtained by the ANN models with all six training algorithms are significantly less than those obtained by SARIMA, SVM and HW methods for each seasonal time series. There are consistently good forecasting performances of the ANN models with each training algorithm for all the six time series. None among the six training algorithms could perform best for all time series. Considerable improvements in the forecasting accuracies are achieved by combining the different ANN outputs through arithmetic mean and median. These empirical findings suggest that with proper network structure and training algorithm, an ANN model is remarkably efficient in recognizing as well as forecasting strong seasonal fluctuation of a time series without removing it. Also, we observe that ANNs notably outperform the traditional SARIMA, SVM and HW methods in terms of the obtained forecasting accuracies for all the six seasonal time series. Our experimental results also favor the strategy of combining the ANN outputs from different training algorithms, instead of selecting solely an individual one. We use the term forecast diagram in order to refer the graph which shows the actual and predicted observations

8 664 J SCI IND RES VOL 7 OCTOBER 202 Table 3 The optimal HW and SVM model parameters for the six time series Model Parameters APTS USADTS RWTS QSTS QBPTS UETS HW α β NA γ SVM a σ C a Note: C is the positive regularization constant which assigns a penalty to misfit Fig. 5 Forecast diagrams: a) APTS, b) USADTS, c) RWTS, d) QSTS, e) QBPTS, f) UETS Fig.6 Friedman test result for: a) MSE, b) MAPE of a time series. The forecast diagrams obtained by the ANN models, combined through arithmetic mean and median for all six seasonal time series are shown in Fig. 5. We also perform the non-parametric Friedman test 34,35 in order to check whether the applied forecasting methods are equally effective or differ significantly. The results of the Friedman tests for MSE and MAPE are depicted in Figs 6a and 6b. In these figures, the mean rank of a forecasting method is pointed by a circle and the horizontal bar across each circle indicates the critical difference. The performances of two methods differ significantly if their mean ranks differ by at least the critical difference, i.e. if their horizontal bars are non-overlapping.

9 ADHIKARI & AGRAWAL : FORECASTING SEASONAL TIME SERIES NEURAL NETWORKS 665 Figs 6a and 6b reveal that the performances of the ANN models with the six training algorithms do not differ significantly among themselves but are reasonably better than the performances of the three traditional statistical methods. Also, it is clearly visible that the best forecasting accuracies in terms of both MSE and MAPE are achieved by arithmetic mean and median which are used to combine the ANN outputs for different training algorithms. Moreover, Fig. 6a shows that the two ANN combination techniques differ significantly from all the three statistical methods in terms of the obtained MSE values. Similarly, Fig. 6b shows that the ANNs with PSO-Trelea and the two combination techniques differ significantly from the SARIMA and SVM methods in terms of the obtained MAPE values. Conclusions Seasonal fluctuation is a distinctive characteristic of many time series, especially those from the economic and financial domains. The main goal of this paper was to meticulously explore the outstanding ability of artificial neural networks to efficiently recognize as well as forecast strong seasonal patterns. It is observed in our study that contrary to the claims of some researchers, ANNs can successfully learn the inherent seasonal structures in time series without removing them from the raw data. The ANN model designing issues, e.g. the selection of proper network structure, activation functions, training algorithm, etc. are discussed and then a robust algorithm is suggested to achieve effective forecasting performances with ANNs for seasonal data. Empirical analysis is conducted on six real-world time series, each containing dominant seasonal fluctuations. For an unbiased study, the ANN model for each time series is trained with six different training algorithms. The results clearly demonstrate that for each seasonal data, ANNs not only achieved consistently good results but also significantly outperformed three other well-known traditional statistical methods, viz. SARIMA, HW and SVM in terms of obtained forecasting accuracies. It is also observed that the ANN forecasting precisions are remarkably increased through combining the outputs, obtained with different training algorithms. Our findings are further statistically justified through the non-parametric Friedman test. Acknowledgements The first author expresses his gratitude to the Council of Scientific and Industrial Research (CSIR), India for the obtained financial support to perform this research work. 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