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Probability mass function

About: Probability mass function is a(n) research topic. Over the lifetime, 2853 publication(s) have been published within this topic receiving 101710 citation(s). The topic is also known as: pmf. more

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01 Jan 1967-
Abstract: The main purpose of this paper is to describe a process for partitioning an N-dimensional population into k sets on the basis of a sample. The process, which is called 'k-means,' appears to give partitions which are reasonably efficient in the sense of within-class variance. That is, if p is the probability mass function for the population, S = {S1, S2, * *, Sk} is a partition of EN, and ui, i = 1, 2, * , k, is the conditional mean of p over the set Si, then W2(S) = ff=ISi f z u42 dp(z) tends to be low for the partitions S generated by the method. We say 'tends to be low,' primarily because of intuitive considerations, corroborated to some extent by mathematical analysis and practical computational experience. Also, the k-means procedure is easily programmed and is computationally economical, so that it is feasible to process very large samples on a digital computer. Possible applications include methods for similarity grouping, nonlinear prediction, approximating multivariate distributions, and nonparametric tests for independence among several variables. In addition to suggesting practical classification methods, the study of k-means has proved to be theoretically interesting. The k-means concept represents a generalization of the ordinary sample mean, and one is naturally led to study the pertinent asymptotic behavior, the object being to establish some sort of law of large numbers for the k-means. This problem is sufficiently interesting, in fact, for us to devote a good portion of this paper to it. The k-means are defined in section 2.1, and the main results which have been obtained on the asymptotic behavior are given there. The rest of section 2 is devoted to the proofs of these results. Section 3 describes several specific possible applications, and reports some preliminary results from computer experiments conducted to explore the possibilities inherent in the k-means idea. The extension to general metric spaces is indicated briefly in section 4. The original point of departure for the work described here was a series of problems in optimal classification (MacQueen [9]) which represented special more

22,533 citations

Journal ArticleDOI
Abstract: : Given a sequence of independent identically distributed random variables with a common probability density function, the problem of the estimation of a probability density function and of determining the mode of a probability function are discussed. Only estimates which are consistent and asymptotically normal are constructed. (Author) more

9,261 citations

29 Mar 1977-

6,168 citations

Journal ArticleDOI
Lotfi A. Zadeh1Institutions (1)
TL;DR: In probability theory, an event, A, is a member of a a-field, CY, of subsets of a sample space ~2, where CY is any collection of disjoint events. more

Abstract: In probability theory [I], an event, A, is a member of a a-field, CY, of subsets of a sample space ~2. A probability measure, P, is a normed measure over a measurable space (Q, GY); that is, P is a real-valued function which assigns to every A in Gk’ a probability, P(A), such that (a) P(A) > 0 for all A E a, (b) P(Q) = 1; and (c) P is countably additive, i.e., if {Ai} is any collection of disjoint events, then more

2,268 citations

01 Jan 1988-

1,472 citations

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No. of papers in the topic in previous years

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Topic's top 5 most impactful authors

Fadoua Balabdaoui

6 papers, 26 citations

Mitsuo Ohta

6 papers, 3 citations

Uwe D. Hanebeck

5 papers, 33 citations

Edward R. Dougherty

5 papers, 58 citations

Themistoklis P. Sapsis

4 papers, 82 citations