Input-variable Specification for Neural Networks An Analysis of Forecasting Low and High Time Series Frequency
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1 Universität Hamburg Institut für Wirtschaftsinformatik Prof. Dr. D.B. Preßmar Input-variable Specification for Neural Networks An Analysis of Forecasting Low and High Time Series Frequency Dr. Sven F. Crone Nikolaos Kourentzes LUMS Department of Management Science
2 Agenda I. Motivation IJCNN 2009 i. NN in business forecasting II. III. ii. iii. Why focus on the input variables? Forecasting high frequency time series Experimental Design I. The experiment II. The dataset Results & Conclusions I. Forecasting errors II. What are the challenges for HF-data?
3 Motivation Why Neural Networks? NN in Business Time Series Forecasting Promising performance 77% (out of 84) articles found NN outperforming benchmarks(73% according to Adya& Collopy, 98) Large scale studies (100+ time series)nn at least as good as benchmarks [Hilletal.,96, Liao&Fildes,05] Forecasting Competitions (M3) NN lower accuracy than statistical models [Makridakis& Hibbon, 00] NN produce unreliable forecasts criticised to offer little promise even after much research [Armstrong, 06] Why? What does NN research suggest?
4 Motivation Focus on the input vector Modelling complexity gives rise to the problems Problem caused by inconsistent trial and error modelling approaches [Zhang et al., 98] Input variable selection The most important issue in forecasting with NN [Zhang,01,Zhangetal.,01,Zhangetal., 98,DarbellayandSlama,00] No widely accepted methodology on how to select the input variables [Anders andkorn,99,zhangetal.,98] 70.5% of papers are based on trial and error approaches to model the input vector! Problem in NN forecasting literature... Focus on the input variables selection methodology!
5 Motivation High Frequency Data Forecasting business time series Traditionally companies forecasted annual, quarterly or monthly time series Advances in computer technology weekly, daily, hourly, etc data Typically companies forecast 1000s of time series Increased frequency of time series allows more frequent forecasts Very large volume of time series to be forecasted Need for fast automated forecasting What makes high frequency data different?
6 Motivation High Frequency Data x 10 7 High frequency daily data Weekly Annual 1.7 x 1010 Low frequency monthly data Several modelling problems: a. vast amounts of data daily data: 30 times longer than monthly 1.2 b. multiple seasonalities intraday, intraweek c. increased detail increased noise level d. high computational requirements Conventional statistical methods difficult to interpret/use information [Granger, 98] Demand a different approach to forecasting [Taylor et al., 06]
7 Universität Hamburg Institut für Wirtschaftsinformatik Prof. Dr. D.B. Preßmar
8 Experimental Design Objectives & Setup Investigate how the modelling complexity changes as the time series frequency increases Use a consistent modelling approach across three different frequency domains Daily, Weekly and Monthly Experimental design and setup: Based on NN5 competition (Daily cash machine time series) Forecast horizon Daily: Forecast next 56 days, F t+h t, h = 1, 2, Weekly: Forecast next 8 weeks, F t+h t, h = 1, 2,... 8 Monthly: Forecast next 2months, F t+h t, h = 1, 2 n 1 X t Ft SMAPE, rolling origin evaluation SMAPE = 100 n = 1 ( ) / 2 t X t + Ft X: time series F: forecast
9 Daily time series Experimental Design 84 days NN5-035 Time Series 84 days High frequency Summed into calendar weeks Summed into calendar months May Aug Nov Jan May Jul Oct Jan-98 Christmas and New Year effect Weekly time series Monthly time series 8 weeks 8 weeks Outliers - No trend (Phillips-Perron Perron test) 2 months 2 months Seasonality Daily & Annual (insufficient data << 3 years) 699 days Medium frequency 100 weeks Low frequency 23 months
10 Neural Networks Setup Standard MLP fixed training parameters Simple back-propagation with (constant) momentum of 0.4 Experimental Design Initial learning rate η=0.5 cooling factor η = 0.01 per epoch Train for 1000 epochs Early stopping criterion MSE evaluated every epoch Linear scaling of the data [-0.6, 0.6] 40 random weight initialisations per MLP error distribution / statistical tests Single hidden layer. Number of hidden nodes grid search from 1 to 12 hidden nodes [1:1:12] NN model
11 Selecting the input variables for NN Experimental Design Input Variable Selection Filters Wrappers Extensively used in time series forecasting literature Several alternatives: Partial Autocorrelation Function (PACF) Autocorrelation Function (ACF) Regression analysis Correntropy Mutual Information...
12 ACF/PACF models Experimental Design Input Variable Selection ACF: ˆ( ρ X ( t), X ( t k)) = Cov( X ( t), X ( t Var( X ( t)) k)) Var( X ( t k)) X: time series k: k th lag ˆ = ˆ π ˆ π ˆ k+ 1, j k, j k + 1, k + 1π k, k j + 1 π j = 1...k PACF: ˆ π k+ 1 ˆ π k+ 1, k+ 1 = ˆ ρ k+ 1 1 k j= 1 k j= 1 ˆ π ˆ π m, j k, j ˆ ρ ˆ ρ k+ 1 j j Yule-Walker (recursive) estimation Common in NN forecasting modelling [Ghiassi et al., 05] Also, Burg estimation Better estimation of true PACF for AR processes [McCullough, 98], but not used in NN literature Lachtermarcher & Fuller (95) suggest a combination of ACF and PACF
13 ACF/PACF models Experimental Design Input Variable Selection Input Vector t -1 Neural Network t -12 t -13 t For combinations of models all identified significant lags are used
14 Stepwise regression Experimental Design Input Variable Selection x Time Series Regression Input Vector t -1 Neural Network t t -35 t Common methodology to specify the input variables [Swanson & White, 97, Qi & Maddala, 99, Dahl & Hylleberg, 04]
15 Universität Hamburg Institut für Wirtschaftsinformatik Prof. Dr. D.B. Preßmar
16 Two sets of inputs Number of inputs and number of hidden units a. S suffix inputs Force maximum lag in As the frequency becomes higher: a. Longer Input vectors b. Number of hidden units uncorrelated? Monthly time series barely any autoregressive information Results Input / Hidden Layer Daily Number of Inputs S Hidden Units PACF:Yule PACF:Burg ACF-PACF:Yule Regression PACF:Yule S PACF:Burg S ACF-PACF:Yule S Regression S Weekly # S Hidden Units PACF: Yule PACF: Burg ACF-PACF:Yule Regression PACF:Yule S PACF:Burg S ACF-PACF:Yule S Regression S Monthly # S Hidden Units PACF:Yule PACF:Burg ACF-PACF:Yule Regression PACF:Yule S PACF:Burg S ACF-PACF:Yule S Regression S - 7 -
17 Model Monthly EXSM Naive NN -PACF: Yule S NN -PACF: Burg S NN - ACF-Yule S NN - PACF: Yule NN - PACF: Burg NN - ACF-Yule Legend: NN models Statistical Benchmarks Red: Best models on validation set Test set SMAPE Model Weekly NN -PACF: Yule S NN - ACF-Yule S NN - PACF: Yule NN - ACF-Yule NN - PACF: Burg NN -PACF: Burg S EXSM Naive Model Results Daily NN -PACF: Yule S NN -PACF: Burg S NN - Regression NN - Regression S EXSM Naive NN - PACF: Yule EXSM NN - PACF: Burg Naive Naive NN - ACF-Yule NN - ACF-Yule S Errors NN outperform statistical benchmarks in weekly and daily time series
18 Use Friedman and Nemenyi nonparametric statistical tests for comparison [Demsar Mean Median Benchmark Best Model Results Demsar, 06] Daily PACF:Yule PACF:Burg ACF-PACF:Yule (S) Regression (S) PACF:Yule S PACF:Burg S Rank Model SMAPE Regression (S) PACF:Yule S PACF:Burg S Friedman p-value: PACF:Yule CritDist: 7.5 PACF:Burg ACF-PACF:Yule (S) PACF:Yule Rank Weekly ACF-PACF:Yule PACF:Burg PACF:Yule S ACF-PACF:Yule S PACF:Burg S Model -- PACF:Yule S ACF-PACF:Yule S PACF:Yule ACF-PACF:Yule PACF:Burg S PACF:Burg Friedman p-value: CritDist: 7.5 Monthly All models All models S Rank Model All models S All models SMAPE Friedman p-value: CritDist: 2.0
19 Accuracy comparisons across frequencies - SMAPE Results Bottom-up Results Time Series Daily time series Weekly time series Monthly time series Daily Model used to create forecasts Weekly Monthly Increased frequency forecasts provide increase in accuracy! Differences in accuracy between weekly and daily effect of outlier coding
20 Outlier coding binary and integer coding insufficient Challenges NN can forecast high frequency data What are the challenges? Input vector identification methods problems with confidence intervals Effect of sample size on confidence interval observations observations Confidence Interval Sample size Sample Autocorrelation Lag Non significant Significant CI Too many significant lags (inputs) NN training problems
21 Computational resources implications for wrapper methods Challenges NN can forecast high frequency data What are the challenges? Computational Time (secs) % se ec % Monthly Weekly Daily Filters vs Wrappers Importance of input vector Computer: Core-Duo 2.2Ghz 3mb Ram 32 bit OS
22 Nikolaos Kourentzes Lancaster University Management School Centre for Forecasting Lancaster, LA1 4YX, UK Tel. +44 (0)
23 References 1. K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators Neural Networks, vol. 2, no. 5, 1989, pp K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks, vol. 4, no. 2, 1991, pp W.J.M. Levelt, Are multilayer feedforward networks effectively Turing Machines?, Psychological Research, vol. 52, no. 2-3, 1990, pp G.P. Zhang, B.E. Patuwo, and M.Y. Hu, A simulation study of artificial neural networks for nonlinear time-series forecasting, Computers & Operations Research, vol. 28, no. 4, 2001, pp G.P. Zhang, An investigation of neural networks for linear time-series forecasting, Computers & Operations Research, vol. 28, no. 12, 2001, pp G.Q. Zhang, B.E. Patuwo, and M.Y. Hu, Forecasting with artificial neural networks: The state of the art, International Journal of Forecasting, vol. 14, no. 1, 1998, pp M. Adya, and F. Collopy, How effective are neural networks at forecasting and prediction? A review and evaluation, Journal of Forecasting, vol. 17, no. 5-6, 1998, pp T. Hill, M. O'Connor, and W. Remus, Neural network models for time series forecasts, Management Science, vol. 42, no. 7, 1996, pp K.P. Liao, and R. Fildes, The accuracy of a procedural approach to specifying feedforward neural networks for forecasting, Computers & Operations Research, vol. 32, no. 8, 2005, pp S. Makridakis, and M. Hibon, The M3-Competition: results, conclusions and implications, International Journal of Forecasting, vol. 16, no. 4, 2000, pp J.S. Armstrong, Findings from evidence-based forecasting: Methods for reducing forecast error, International Journal of Forecasting, vol. 22, no. 3, 2006, pp U. Anders, O. Korn, and C. Schmitt, Improving the pricing of options: A neural network approach, Journal of Forecasting, vol. 17, no. 5-6, 1998, pp R.F. Engle, The econometrics of ultra-high-frequency data, Econometrica, vol. 68, no. 1, 2000, pp C.W.J. Granger, Extracting information from mega-panels and high-frequency data, Statistica Neerlandica, vol. 52, no. 3, 1998, pp
24 References 15. J.W. Taylor, L.M. de Menezes, and P.E. McSharry, A comparison of univariate methods for forecasting electricity demand up to a day ahead, International Journal of Forecasting, vol. 22, no. 1, 2006, pp H.S. Hippert, D.W. Bunn, and R.C. Souza, Large neural networks for electricity load forecasting: Are they overfitted?, International Journal of Forecasting, vol. 21, no. 3, 2005, pp H.R. Kirby, S.M. Watson, and M.S. Dougherty, Should we use neural networks or statistical models for short-term motorway traffic forecasting?, International Journal of Forecasting, vol. 13, no. 1, 1997, pp M.Y. Hu, G.Q. Zhang, C.Z. Jiang, and B.E. Patuwo, A cross-validation analysis of neural network out-of-sample performance in exchange rate forecasting, Decision Sciences, vol. 30, no. 1, 1999, pp I.S. Markham, and T.R. Rakes, The effect of sample size and variability of data on the comparative performance of artificial neural networks and regression, Computers & Operations Research, vol. 25, no. 4, 1998, pp E.J. Gardner, Exponential smoothing: The state of the art--part II, International Journal of Forecasting, vol. 22, no. 4, 2006, pp M. Ghiassi, H. Saidane, and D.K. Zimbra, A dynamic artificial neural network model for forecasting time series events, International Journal of Forecasting, vol. 21, no. 2, 2005, pp B.D. McCullough, Algorithm choice for (partial) autocorrelation functions, Journal of Economic and Social Measurement, vol. 24, 1998, pp G. Lachtermacher, and J.D. Fuller, Backpropagation in Time-Series Forecasting, Journal of Forecasting, vol. 14, no. 4, 1995, pp M. Qi, and G.S. Maddala, Economic factors and the stock market: A new perspective, Journal of Forecasting, vol. 18, no. 3, 1999, pp N.R. Swanson, and H. White, Forecasting economic time series using flexible versus fixed specification and linear versusnonlinear econometric models, International Journal of Forecasting, vol. 13, no. 4, 1997, pp C.M. Dahl, and S. Hylleberg, Flexible regression models and relative forecast performance, International Journal of Forecasting, vol. 20, no. 2, 2004, pp
25 Experimental Design ANN Models Input vector How many lags to evaluate for the input vector? Common practice arbitrary select a number of lags some cases is not even a full season [Zhang and Qi, 05, Curry, 07] Low frequency data trial and error approaches common in literature [Zhang, 98] High frequency data the amount of data makes a detailed trial and error modelling very challenging A methodology that will use the information of the time series is necessary
26 Experimental Design ANN Models Input vector Split time series in different possible seasonality (periodicity) find euclidean distance Seasonal length plot 400 Season: 4 Season: Euclidean distance: Euclidean distance: Season Euclidean Distance Minimise distance seasonality identification important information for the inputs! [Crone, 2005, Curry, 07] Multiples of seasonality minimises deviation guideline how many seasons to use for the variable selection search
27 Challenges ANN can forecast high frequency data What are the challenges? How many lags to evaluate to model the input vector? Connected to computational needs sample size, overlaying seasonalities no guideline in literature! Calendar problems leap years, number of weeks in months/years, etc... problems for statistical methods
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