A New Robust Estimation Method for ARMA Models

The autoregressive moving-average (ARMA) modeling of time series is popular and used in many applications. In this paper, we introduce a new robust method to estimate the parameters of a Gaussian ARMA model contaminated by outliers. This method makes use of the median and is termed ratio-of-medians estimator (RME). The ratios of medians are used to estimate robustly the autocorrelation function and thus estimate the parameters. Its theoretical robustness is analyzed by the computation of robust measures such as maximum bias, breakdown point and influence function. The RME estimator is shown to be asymptotically normal and its asymptotic variance is computed under Gaussian autoregressive models of order p (p ≥ 1). The new method is evaluated and compared with other robust methods via simulations. Its effectiveness in terms of parameter estimation and forecasting is demonstrated on an example of the French daily electricity consumptions. The new approach improves the load forecasting quality for “normal days” and presents several interesting properties such as good robustness, fast execution, simplicity and easy online implementation.

[1]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[2]  V. Yohai,et al.  Robust Statistics: Theory and Methods , 2006 .

[3]  Marianthi Markatou,et al.  Robust Inference : The Approach Based on Influence Functions , 1996 .

[4]  Lamine Mili,et al.  A new robust estimation method for ARMA models , 2009, ICASSP.

[5]  Richard A. Davis,et al.  Time Series: Theory and Methods (2nd ed.). , 1992 .

[6]  Marcelo C. Medeiros,et al.  Robust Statistical Methods for Electricity Load Forecasting , 2006 .

[7]  K. Chanda,et al.  Strong mixing properties of linear stochastic processes , 1974, Journal of Applied Probability.

[8]  P. Panciatici,et al.  Robust short-term load forecasting using projection statistics , 2009, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[9]  K. Athreya,et al.  Mixing properties of harris chains and autoregressive processes , 1986, Journal of Applied Probability.

[10]  R. Martin,et al.  Robust bayesian estimation for the linear model and robustifying the Kalman filter , 1977 .

[11]  Isabel Silva,et al.  Asymptotic distribution of the Yule–Walker estimator for INAR(p) processes , 2006 .

[12]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[13]  Peter J. Rousseeuw,et al.  Robust regression and outlier detection , 1987 .

[14]  Ryozo Yokoyama,et al.  On the Bahadur representation of sample quantiles for mixing processes , 1978 .

[15]  J. Alexander,et al.  Theory and Methods: Critical Essays in Human Geography , 2008 .

[16]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[17]  A. McQuarrie,et al.  Regression and Time Series Model Selection , 1998 .

[18]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[19]  V. Yohai,et al.  Influence Functionals for Time Series , 1986 .

[20]  R. Weron.  Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach , 2006 .

[21]  Richard H. Jones,et al.  Maximum Likelihood Fitting of ARMA Models to Time Series With Missing Observations , 1980 .

[22]  M. Medeiros,et al.  Modeling and forecasting short-term electricity load: A comparison of methods with an application to Brazilian data , 2008 .

[23]  Oscar H. Bustos,et al.  Robust Estimates for ARMA Models , 1986 .

[24]  P. McSharry,et al.  Short-Term Load Forecasting Methods: An Evaluation Based on European Data , 2007, IEEE Transactions on Power Systems.

[25]  L. Denby,et al.  Robust Estimation of the First-Order Autoregressive Parameter , 1979 .

[26]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[27]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[28]  Peter J. Rousseeuw,et al.  Robust Regression and Outlier Detection , 2005, Wiley Series in Probability and Statistics.

[29]  P. Rousseeuw,et al.  Wiley Series in Probability and Mathematical Statistics , 2005 .