By Dr. Patrick L. Odell, Dr. Benjamin S. Duran (auth.)
A super volume of labor has been performed over the past thirty years in cluster research, with an important quantity happening considering the fact that 1960. a considerable component to this paintings has seemed in lots of journals, together with various utilized journals, and a unified ex place is missing. the aim of this monograph is to provide such an exposition by way of featuring a short survey on cluster research. the most cause of the monograph is to provide the reader a short account of the prob lem of cluster research and to reveal to him a few of the features thereof. With this cause in brain a lot aspect has been passed over, fairly in as far as specific examples are thought of. many of the references said in the textual content include examples and the reader can seek advice them for more information on particular subject matters. Efforts have been made to incorporate within the reference part all papers that performed a task in constructing the "theory" of cluster research. Any omission of such references was once now not intentional and we might have fun with realizing approximately them. Many references to papers in utilized journals also are contained, notwithstanding, the list-is faraway from being whole. This monograph has been tremendously encouraged by way of the paintings of many of us, such a lot particularly, J. A. Hartigan, D. Wishart, J. ok. Bryan, R. E. Jensen, H. D. Vinod, and M. R. Rao. a number of parts of the monograph have been stimulated by way of study played below the aid of NASA Manned Spacecraft heart, Earth Observations department, less than agreement NAS 9-12775.
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Extra info for Cluster Analysis: A Survey
J W.. ). ~ J ~ J J ~ The purpose of a dynamic programming scheme is to systematically search for groupings which yield minimum values of the quantity W, eliminating those groupings which do not yield minimum values of Wand also those that are redundant. We now discuss the problem of partitioning n = 6 objects into m = 3 subsets by complete enumeration. This will serve to motivate a dynamic programming scheme for the cluster problem given by Jensen . 11), S (6,3) 1 TI ~ k=O (_l)k (k3) (3-k) 6 = 90.
Stirling's numbers of the second kind arise in the calculus of finite differences and provide a way of expressing powers of x in terms of "falling factorials". 1. The falling factorial is defined by ••. (x-n+l), n 1,2, ••• Stirling's numbers of the second kind are defined to be those numbers S(n,i) such that x n n I i=l S(n,i)x(i)' S(n,O) and S (n,n+k) = ° for k > °, 0. 12) we obtain n L m=O m n [~] x(m) m. x=O From the definition of S(n,i) we then have S(n,m) m n [4-] m. x=O 1 m! It remains to show that m L j=O To show this we make use of a shift operator E = 1 + 6, with Ef(x) f(x+1), Ejf(x) = f(x + j), and 6 = E - 1.
If repetitions are allowed then the enumerator for any object is a series containing a term tk/k! for each k in the specification of repetitions. Thus if unlimited repetition is allowed, then the enumerator for each object is given by 1 + t + 2 t 2T + ... t e. Consequently the number of permutations of n distinct objects taken k at a time is given by the coefficient of tk/k! in the generating function (1 + t + t2 2T + ... ) n e nt I k=O n The required number of permutations is thus nk. k tk k! If unlimited 34 repetition is allowed with each object occurring at least once then the enumerator is given by (t t2 + 2T + ••• ) n n (~) I j=O J (_l)je(n-j)t n tk (~) I kT ( I k=O j=O J .
Cluster Analysis: A Survey by Dr. Patrick L. Odell, Dr. Benjamin S. Duran (auth.)