OPTIMIZATION BASED ON ODIF Sari Peltonen

In this paper we use a recently introduced method called output distributional influence function (ODIF) in the optimization of filters. The ODIF gives information about the robustness of finite length filters and thus it can be used in real filtering situations to find filters having desired robustness properties. We consider possible optimization criteria based on the ODIFs for the expectation and variance which could be used alone or combined with other types of optimization criteria. We use in this paper L-filters as examples of the filter optimization since that filter class includes filters having wide range of different robustness properties. 1. INFLUENCE FUNCTION Influence function (IF) is a useful heuristic tool of robust statistics introduced by Hampel [1, 2] under the name influence curve (IC) for studying the performance of filters under noisy conditions. Definition 1. The IF of estimatorT at underlying probability distributionF is given by IF(y) = lim t! 0+ T ((1 t)F + t y) T (F ) t for thosey where this limit exists. In this definition y is the probability measure which puts mass 1 at the point y. The IF gives the effect that an infinitesimal contamination at point y has on the estimator T when divided by the mass of the contamination. So the IF gives asymptotic bias caused by the contamination and thus characterizes properties of the estimator as the number of observations approaches infinity. 2. CHANGE-OF-VARIANCE FUNCTION The IF gives only one aspect of robustness of an estimator, namely local robustness of the asymptotic value of the estimator. Another important aspect is the local robustness of the asymptotic variance. The asymptotic variance of estimatorT atF denoted byV (T; F ) is defined to be the variance of p N [T (FN ) T (F )] asN ! 1, where FN is the empirical distribution of sample (X1; X2; : : : ; XN ). Local robustness of the asymptotic variance can be characterized by the change-of-variance function (CVF) defined as follows [3]. Definition 2. The CVF of estimator T atF is defined as CVF(y) = lim t! 0+ V (T; (1 t)F + t y) V (T; F ) t for thosey where this limit exists. 3. OUTPUT DISTRIBUTIONAL INFLUENCE FUNCTION Since the IF is an asymptotic measure, it describes properties of infinite length filters which may differ from those of finite length filters used in the real world filtering applications. It would be more useful and more interesting to examine properties of these finite length filters rather than the asymptotic properties. In the case where the output distribution of a filter can be expressed in a closed form as a function of the distribution functions of the input samples we introduced in [4] output distributional influence function (ODIF) for analyzing the robustness of the finite length filters. We assume here that the input samples are independent and identically distributed (i.i.d.) random variables. First we need a way to denote the output distribution function of a filter when a fraction" of the input samples has different distribution than the rest of the samples. We denote byH(1 ")F+"Gy( ) the output distributionHF ( ) of the filter where every occurrence of the common distribution functionF of the input samples is replaced by(1 ")F +"Gy andGy is a distribution function with meany. As usual, we defineh(1 ")F+"Gy(x) = @ @x H(1 ")F+"Gy(x). We gave the following definition for the ODIF for the distribution function in [4]. Definition 3 . Let the output distribution function of a filter be HF ( ) whereF ( ) is the common distribution function of the input samples and let Gy( ) be a distribution function having meany. Then the ODIF for the distribution function ( ) is (x; y) = lim "! 0+ H(1 ")F+"Gy(x) HF (x) " for thosex andy where this limit exists. In [4] we defined the ODIF in the same way as for the distribution function in Definition 3 also for the density function and moments. The ODIFs for the density function, expectation and variance were derived to be ! (x; y) = @ @x (x; y);