a non - iterative algorithm to estimate the modes of univariate mixtures with well - separated components

— This paper deals with the estimation of the modes of an univariate mixture when the number of components is known and when the component density are well separated. We propose an algorithm based on the minimization of the " kp " criterion we introduced in a previous work. In this paper we show that the global minimum of this criterion can be reached with a linear least square minimization followed by a roots finding algorithm. This is a major advantage compared to classical iterative algorithms such as K-means or EM which suffer from the potential convergence to some local extrema of the cost function they use. Our algorithm performances are finally illustrated through simulations of a five components mixture. EDICS Category: SAS-STAT I. INTRODUCTION In this paper we focus on the estimation of the modes of an univariate mixture with a known number of components. When the mixture component belongs to a parameterized family known by the estimator (gaussian mixture case for instance), the observation estimated moments can be mapped to the mixture parameters [1]. Yet, when the number of components is high, the relationships between the moments and the mixture parameters are usually too complicated to be analytically solved. Alternately, the Expectation-Maximization (EM) [2] algorithm is the most commonly used method when the mixture densities belong to a parameterized family. It is an iterative algorithm that look for the mixture parameters that maximize the likelihood of the observations. The EM iteration consists of two steps. The Expectation step estimates the probability for each observation to come from each mixture component. During the Maximization step, these estimated probabilities are used to update the estimation of the mixture parameters. If the mixture components do not belong to any parameterized family, or if the parameterized family is not known by the estimator, the moment method and the EM algorithm do not directly apply. Yet, if the mixture components density are identical and quite separated, any clustering methods can be used to cluster the data and calculate the clusters means to reach the mixture modes. A survey of the clustering techniques can be found in [3]. Among them, the K-means algorithm [4] is one of the most popular method.