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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

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Introduction Model Estimation Bandwidth Data example Conclusion

Goal

Forecasting of time series :

non-stationarity non-parametric robust heteroscedasticity

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Example Temperature Data in Dresden, Germany

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Introduction Model Estimation Bandwidth Data example Conclusion

Idea

non-stationarity robust heteroscedasticity

→ local polynomial regression → M-estimation → MM-estimation

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Introduction Model Estimation Bandwidth Data example Conclusion

Model

Yt = m(t) + σ(t)ǫt, Yt the observed time series, m(t) the signal, σ(t) the scale and ǫt the error term

i.i.d

∼ N(0, 1).

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Prediction

Based on the data y1, . . . , yT, predict yt0, t0 > T. The signal is approximated locally by a polynomial of degree p: m(t) =

p

• j=0

βj(t − t0)j, where β = (β0, . . . , βp)′ is to be estimated. Then ˆ yt0 := ˆ m(t0) = ˆ β0.

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Local polynomial regression

The OLS estimator of local polynomial regression for β minimizes

T

• t=1
• Yt − ˆ

β

′xt,t0

2 K t − t0 h

• ,

with xt,t0 =(1,(t−t0),(t−t0)2, . . . ,(t−t0)p)′ , K an asymmetric kernel and h the bandwidth.

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Introduction Model Estimation Bandwidth Data example Conclusion

Local polynomial regression

The MM estimator of local polynomial regression for β minimizes

T

• t=1

ρ

• Yt − ˆ

β

′xt,t0

ˆ σ(t0)

• K

t − t0 h

• ,

where ρ is a loss function and ˆ σ(t0) is the S-estimator of scale.

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• Yt − ˆ

β

′xt,t0

2 → ρ

• Yt− ˆ

β

′xt,t0

ˆ σ(t0)

• −2

−1 1 2 1 2 3 4 5 6

x ρ (x)

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Introduction Model Estimation Bandwidth Data example Conclusion

Algorithm

Iteratively weighted least squares:

Weights = kernel weights × robustness weights βstart = Local least absolute deviation regression (LAD) σstart = Locally weighted median absolute deviation from zero of the residuals of LAD-regression.

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“Parameters to choose”

degree of the polynomial (p = 1) kernel function (K(x) = exp(x)I{x<0}) breakdown point (50%) bandwidth h

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Bandwidth selection

Select h such that the trimmed mean squared standardized forecast error is minimal: hopt = argmin

h

1 ⌊0.8T⌋

⌊0.8T⌋

• i=1

et ˆ σ(t) 2

(i)

, where et/ˆ σ(t) are the standardized forecast errors.

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Example Forecast Temperature Data

5 10 15 20

2007 Temperature

Dec Jan Feb Mar Apr

LPR MM 13 / 16 COMPSTAT 2010 Robust forecasting of non-stationary time series

Introduction Model Estimation Bandwidth Data example Conclusion

Example Forecast accuracy

20% right trimmed means of the squared forecast errors of 50

• ne-step ahead forecasts.

LPR WRM M MM TMSFE 7.37 8.31 7.03 6.88

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Conclusion

Forecasting method for time series

short-term non-parametric robust allow for non-stationarity and heteroscedasticity.

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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).

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