COMPSTAT 2010 Robust forecasting of non-stationary time series - PowerPoint PPT Presentation

Introduction Model Estimation Bandwidth Data example Conclusion COMPSTAT 2010 Robust forecasting of non-stationary time series ∗ Koen Mahieu K.U.Leuven ∗ joint work with Christophe Croux, Roland Fried and Ir` ene Gijbels COMPSTAT 2010 Robust forecasting of non-stationary time series 1 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Goal Forecasting of time series : non-stationarity non-parametric robust heteroscedasticity COMPSTAT 2010 Robust forecasting of non-stationary time series 2 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Example Temperature Data in Dresden, Germany 20 15 Temperature 10 5 0 Dec Jan Feb Mar Apr 2007 COMPSTAT 2010 Robust forecasting of non-stationary time series 3 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Idea → local polynomial regression non-stationarity → M-estimation robust → MM-estimation heteroscedasticity COMPSTAT 2010 Robust forecasting of non-stationary time series 4 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Model Y t = m ( t ) + σ ( t ) ǫ t , Y t the observed time series, m ( t ) the signal, σ ( t ) the scale and i.i.d ǫ t the error term ∼ N (0 , 1). COMPSTAT 2010 Robust forecasting of non-stationary time series 5 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Prediction Based on the data y 1 , . . . , y T , predict y t 0 , t 0 > T . The signal is approximated locally by a polynomial of degree p : p � β j ( t − t 0 ) j , m ( t ) = j =0 where β = ( β 0 , . . . , β p ) ′ is to be estimated. Then m ( t 0 ) = ˆ ˆ y t 0 := ˆ β 0 . COMPSTAT 2010 Robust forecasting of non-stationary time series 6 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Local polynomial regression The OLS estimator of local polynomial regression for β minimizes T � 2 � t − t 0 � � ′ x t , t 0 � Y t − ˆ , β K h t =1 with x t , t 0 =(1 , ( t − t 0 ) , ( t − t 0 ) 2 , . . . , ( t − t 0 ) p ) ′ , K an asymmetric kernel and h the bandwidth. COMPSTAT 2010 Robust forecasting of non-stationary time series 7 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Local polynomial regression The MM estimator of local polynomial regression for β minimizes ′ x t , t 0 T � � Y t − ˆ β � t − t 0 � � ρ , K σ ( t 0 ) ˆ h t =1 where ρ is a loss function and ˆ σ ( t 0 ) is the S-estimator of scale. COMPSTAT 2010 Robust forecasting of non-stationary time series 8 / 16

Introduction Model Estimation Bandwidth Data example Conclusion � 2 � � Y t − ˆ ′ x t , t 0 ′ x t , t 0 � β Y t − ˆ β → ρ σ ( t 0 ) ˆ 6 5 ρ ( x ) 4 3 2 1 0 −2 −1 0 1 2 x COMPSTAT 2010 Robust forecasting of non-stationary time series 9 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Algorithm Iteratively weighted least squares: Weights = kernel weights × robustness weights β start = Local least absolute deviation regression (LAD) σ start = Locally weighted median absolute deviation from zero of the residuals of LAD-regression. COMPSTAT 2010 Robust forecasting of non-stationary time series 10 / 16

Introduction Model Estimation Bandwidth Data example Conclusion “Parameters to choose” degree of the polynomial ( p = 1) kernel function ( K ( x ) = exp( x ) I { x < 0 } ) breakdown point (50%) bandwidth h COMPSTAT 2010 Robust forecasting of non-stationary time series 11 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Bandwidth selection Select h such that the trimmed mean squared standardized forecast error is minimal: � e t ⌊ 0 . 8 T ⌋ � 2 1 � h opt = argmin , ⌊ 0 . 8 T ⌋ ˆ σ ( t ) h ( i ) i =1 where e t / ˆ σ ( t ) are the standardized forecast errors. COMPSTAT 2010 Robust forecasting of non-stationary time series 12 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Example Forecast Temperature Data LPR MM 20 15 Temperature 10 5 0 Dec Jan Feb Mar Apr 2007 COMPSTAT 2010 Robust forecasting of non-stationary time series 13 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Example Forecast accuracy 20% right trimmed means of the squared forecast errors of 50 one-step ahead forecasts. LPR WRM M MM TMSFE 7.37 8.31 7.03 6.88 COMPSTAT 2010 Robust forecasting of non-stationary time series 14 / 16

Introduction Model Estimation Bandwidth Data example Conclusion Conclusion Forecasting method for time series short-term non-parametric robust allow for non-stationarity and heteroscedasticity. COMPSTAT 2010 Robust forecasting of non-stationary time series 15 / 16

Introduction Model Estimation Bandwidth Data example Conclusion References Croux, C., Gelper, S. and Mahieu, K. (2010). Robust exponential smoothing of multivariate time series. Computational Statistics & Data Analysis , 54 (12). Grillenzoni, C. (2009). Robust non-parametric smoothing of non-stationary time series. Journal of Statistical Computation and Simulation , 79 (4). COMPSTAT 2010 Robust forecasting of non-stationary time series 16 / 16