# Data Analysis in Astronomy by Mike Disney (auth.), V. Di Gesù, L. Scarsi, P. Crane, J. H.

By Mike Disney (auth.), V. Di Gesù, L. Scarsi, P. Crane, J. H. Friedman, S. Levialdi (eds.)

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Al. (1983)1 Thus, we shall assume that the underlying Markov process has c states WI, ••• ,We, and, for the purpose of the present discussion, we shall make the simplifying assumption that the observations are drawn from a finite alphabet Y1 , ••• , YN (we shall briefly return to the case of normal mixtures at the end of this section). The Markov chain is specified in terms of an initial state distribution Pi = P(w l = wd-where for any 1 :5 r :5 T, w1' Wi denotes that the process is in state Wi at time rand a matrix of state transition probabilities Pij P(w1'H Wj I w1' wd, 1 :5 i, j :5 c, Vr, 1 :5 r :5 T -1.

Special Issue on Fast Transform and its application ~o Digital Filtering and Spectral Analysis. D. Sound Vib. W. Math. , Sacco B. COMSTAT 82, p. A. et a1. B. Astron. V. "Statistics of directional data", New York, Academic Press, 1972 - Scarsi L. et a1. Proc. 12th ESLAB Symp. N. et al. Ap. , 243, L69, 1981 27 CLUSTER ANALYSIS BY MIXTURE IDENTIFICATION Pierre A. Devijver Philips Research Laboratory Ave. Em. Van Becelaere 2, Box 8 Brussels, Belgium 1. Introduction Clusters analysis is frequently defined as the problem of partitioning a collection of objects into groups of similar objects according to some numerical measure of similarity.

For a Gaussian mixture, it is known that third and higher unconditional moments can be expressed as functions of second and lower conditional ones. For instance, with P(Wl) = P(W2), we have (16) (17) where, clearly, left hand sides are unconditional moments that can be estimated from the data while the right hand sides involve some of the desired conditional ones. From (16) and (11), ±ti = 3(Sii -1) [3(Sii -1)22- (u. - 3)] 1/4 This equation can be solved for Sii for known (estimated) ti and Uj.