Time Series Forecasting Methods:

Transcription

1 Corso di TRASPORTI E TERRITORIO Time Series Forecasting Methods: EXPONENTIAL SMOOTHING DOCENTI Agostino Nuzzolo (nuzzolo@ing.uniroma2.it) Antonio Comi (comi@ing.uniroma2.it) 1

2 Classificazione dei metodi di previsione Metodi semplificati di previsione Metodi di previsione con decomposizione (classica o STL) Metodi di previsione exponential smoothing Semplice exp. smoothing Holt Winters exponential smoothing Innovations state space models - ETS Modelli ARIMA Modelli Neural Networks 2

3 Exponential Smoothing ForecastingMethods Component form The component form representation of exponential smoothing forecasting methods comprises: - a smoothing equation, that gives the smoothed value at each t, for each of the components included in the method (residuals, trend, seasonality) - a forecast equation: the forecasted value at time t+1 is the smoothed value at time t.

4 Exponential smoothing [1/2] Simple exponential smoothing forecast SES The Simple Exponential Smoothing - SES can be applied to temporal series with no trend or seasonal pattern (or to de-trended and seasonality adjusted component, that is to residuals). In the case of simple exponential smoothing, the component form is given by: t 1 t t 1 Smoothing equation lt yt 1 ll Forecasting equation where l t is the level (or the smoothed value) of the series at time t and 0 α 1 is the smoothing parameter. ŷ l The forecast equation shows that the forecasted value at time t+1 is the estimated level at time t. 4

5 Exponential smoothing [2/2] Simple exponential smoothing forecast: weighted average form Applying the forecast equation for time T gives, the most recent estimated level. T 1 T T If in the smoothing equation, we replace l t by ŷt and l t 1 by ŷ 1 t t t 1 we will recover the weighted average form of simple exponential smoothing equation: for t=1,,t, where 0 α 1 is the smoothing parameter ŷ ˆy y 1 ˆy t 1 t t t t 1 l 5

6 Exponential smoothing Simple exponential smoothing forecasting The weights decrease exponentially as observations come from further in the past --- the smallest weights are associated with the oldest observations: ŷ y y y... from which the name of the method T 1 T T T 1 T 2 The one-step-ahead forecast for time T+1 is a weighted average of all the observations in the series y 1,,y T. The rate at which the weights decrease is controlled by the parameter α. 6

7 Exponential smoothing Simple exponential smoothing It is the optimal estimated value 7

8 Exponential smoothing [1/3] Holt-Winters seasonal method The Holt-Winters seasonal method comprises the forecast equation and three smoothing equations: one for the level l t, one for trend b t, one for the seasonal component s t, with smoothing parameters α, β and γ. 8

9 Exponential smoothing [2/3] Holt-Winters seasonal method We use m to denote the period of the seasonality. The component form for the additive method is: ŷ l h b s t 1 t t t t m h m 1 l y s l b t t t m t 1 t 1 1 b l l b * * t t t 1 t 1 1 s y l b s t t t 1 t 1 t m where h, which ensures that the estimates of the seasonal indices m h 1 modm 1 used for forecasting come from the final year of the sample. (The notation u means the largest integer not greater than u. 9

10 Exponential smoothing [3/3] Holt-Winters seasonal method The level equation shows a weighted average between the seasonally adjusted observation (y t s t m ) and the non-seasonal forecast (l t 1 +b t 1 ) for time t. The trend equation shows a weighted average between the difference of level (l t l t-1 ), that is the new b t, and the previous b t-1. The seasonal equation shows a weighted average between the current seasonal index, (y t l t 1 b t 1 ), and the seasonal index of the same season last «year» (i.e., m time periods ago). 10

11 Exponential smoothing [1/2] Initialisation 1 The application of every exponential smoothing method requires the initialisation of the smoothing process. For simple exponential smoothing we need to specify an initial value for the level, l 0, which appears in the last term of equation T 1 j 1 1 ŷ y l T 1 T T j 0 j 0 Hence l 0 plays a role in all forecasts generated by the process. In general, the weight attached to l 0 is small. However, in the case that α is small and/or the time series is relatively short, the weight may be large enough to have a noticeable effect on the resulting forecasts. T 11

12 Exponential smoothing [2/2] Initialisation 2 Therefore, selecting suitable initial values can be quite important. A common approach is to set l 0 =y 1 (recall that l ˆy ) Other exponential smoothing methods that also involve a trend and/or a seasonal component require initial values for these components also. In the book are tabulated common strategies for selecting initial values. An alternative approach is to use optimization: the unknown parameters and the initial values for any exponential smoothing method can be estimated by minimizing the SSE. 12

13 Exponential smoothing [1/2] Error correction form 1 Another form of exponential smoothing equation is obtained by rearranging the level equation in to get what we refer to as the error correction form. For the simple exp. smoothing: l t = l t 1 +α (y t l t 1 ) = l t 1 +α e t where e y l y ˆy t t t 1 t t t 1 for t = 1,...,T That is, e t is the one-step error at time t computed on the training data. The training data errors lead to the adjustment/correction of the estimated level throughout the smoothing process for t=1,,t. 13

14 Exponential smoothing [2/2] Error correction form 2 For example, if the error at time t is negative, then the level at time t 1 has been over-estimated. ŷ t t 1 y t and so The new level l t is then the previous level l t 1 adjusted downwards. The closer α is to one, the rougher the estimate of the level (large adjustments take place). The smaller the α, the smoother the level (small adjustments take place) 14

15 Exponential Smoothing Method Error Component form Using the error component form, the error e t is given by the measurement equation: y ˆy e ovvero e y ˆy t t t t t t and the smoothing equations can be re-arranged, giving the transition equations at each t, that describe how the unobserved components or states (level, trend, seasonal) change over time. Now we can introduce a new form of exponential smoothing models: the Innovation State Space models

16 Innovations state space (ISS) models [1/2] Each ISS model consists of: a measurement equation that describes the observed data some transition equations that describe how the unobserved components or states (level, trend, seasonal) change over time. Hence these are referred to as state space models. We label each state space model as ETS(,, ) for (Error, Trend, Seasonal). This label can also be thought of as ExponenTial Smoothing. The possibilities for each state or component, are: Error ={A,M}, Trend ={N,A,M} and Seasonal ={N,A,M}, con N=Nulle; A= additive; M= Moltiplicative. 16

17 Innovations state space (ISS) models [2/2] These models derive from the previous exponential smoothing method, specifying the probability distribution for e t. For a model with additive errors, we assume that one-step forecast errors e t are normally distributed white noise with mean 0 and variance σ 2. A short-hand notation for this is e t = ε t NID(0,σ 2 ) NID stands for normally and independently distributed. The term innovations comes from the fact that all equations in this type of specification use the same random error process, ε t. 17

18 ETS (A,A,A) - Holt-Winters In the case of ETS (A;A;A) - Holt-Winters, the formulation is: y l b s t t 1 t 1 t m t l l b t t 1 t 1 t b s b t t 1 t s t t m t We estimate the smoothing parameters α, β, γ and the initial states l 0, b 0, s 0, s 1,, s m+1 by maximizing the likelihood. The possible values that the smoothing parameters can take is restricted. Traditionally, the parameters have been constrained to lie between 0 and 1 so that the equations can be interpreted as weighted averages. That is, 0 < α, β, γ < 1. For the state space models, we have set β=αβ and γ=(1 α)γ. Therefore, the traditional restrictions translate to 0<α<1, 0<β<α and 0<γ<1 α. 18

19 ISS model selection A great advantage of the ETS statistical framework is that information criteria can be used for model selection. The AIC, AIC c and BIC can be used to determine which model is most appropriate for a given time series. For ETS models, Akaike s Information Criterion (AIC) is defined as AIC= 2log(L)+2k, where L is the likelihood of the model and k is the total number of parameters and initial states that have been estimated (including the residual variance). The AIC corrected for small sample bias is AIC c =AIC+[2k(k+1)] / (T k 1), and the Bayesian Information Criterion (BIC) is BIC=AIC+k[log(T) 2]. 19

20 Forecasting using STL + ETS(A,N,N) (Simple Exponential Smoothing su serie dei residui ottenuti con STL da time 1 a time 7,5) Forecast method: Simple exponential smoothing Model Information: Simple exponential smoothing Call: ses(y = ts.re75, h = 90, initial = "optimal") Smoothing parameters: alpha = Initial states: l = sigma: AIC AICc BIC Error measures: ME RMSE MAE MPE MAPE MASE ACF1 Training set

21 Forecasting using STL and ETS(A,N,N) Example: forecasted and observed values time interval trend component seasonal compone nt STL forecast data observed STL Errors Error forecasts STL + ETS forecasts STL + ETS e i ,9 32,7 2447,6 2802,0 354,4 105,3 2552,9 249, ,9 74,0 2488,9 2589,0 100,1 105,3 2594,1-5, ,9 97,3 2512,2 2446,0-66,2 64,8 2577,0-131, ,9 257,1 2671,9 2549,0-122,9-96,4 2575,6-26, ,9 387,1 2801,9 2643,0-158,9-55,3 2746,7-103, ,9 405,2 2820,1 2612,0-208,1-82,5 2737,6-125, ,9 197,4 2612,3 2572,0-40,3-119,6 2492,7 79, ,9 310,8 2725,6 2551,0-174,6 23,0 2748,6-197, ,9 203,0 2617,9 2433,0-184,9-121,5 2496,4-63, ,9 275,0 2689,9 2552,0-137,9-100,3 2589,6-37, ,9 193,1 2608,0 2453,0-155,0-44,7 2563,3-110, ,9-48,4 2366,5 2257,0-109,5-90,3 2276,2-19, ,9-431,3 1983,5 1845,0-138,5-44,2 1939,3-94, ,9-541,7 1873,2 1843,0-30,2-73,7 1799,6 43, ,9-641,4 1773,4 1730,0-43,4 5,6 1779,0-49, ,9-728,1 1686,8 1617,0-69,8-23,6 1663,2-46, ,9-784,5 1630,4 1575,0-55,4-47,0 1583,4-8, ,9-842,8 1572,1 1531,0-41,1-18,8 1553,3-22, ,9-887,4 1527,4 1712,0 184,6-18,5 1508,9 203, ,9-804,1 1610,7 1696,0 85,3 141,1 1751,8-55, ,9-470,1 1944,8 1905,0-39,8 15,3 1960,2-55, ,9-153,5 2261,3 2272,0 10,7-55,3 2206,0 66,0 21

22 Travel time [seconds] Forecasting using STL and ETS Example forecasted and observed values 4500,0 forecasts data observed 4000,0 3500,0 3000,0 2500,0 2000,0 1500,0 1000, minutes time interval [from 14:15 of Wednesday to 22:45 of Friday in the week 8 22

23 seconds Residual diagnostics Example of error diagnostic for forecasting with STL and ETS Average e i s 16,9 209,2 1200,0 Errors e i (difference between observed and forecast values) 1000,0 800,0 600,0 400,0 200,0 0, ,0-400,0-600,0-800,0 30-minutes time interval MAE 137,8 RMSE 209,8 MAPE 5,5% 23

24 Comparison Classical, STL and STL + ETS forecasting Classical Average e i s -61,1 278,1 STL Average e i s -31,7 222,7 STL + ETS Average e i s 19,9 209,2 MAE 234,4 RMSE 284,8 MAPE 10% MAE 164,9 RMSE 225,0 MAPE 6,5% MAE 137,8 RMSE 209,8 MAPE 5,5% 24