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Extra resources for Discriminant Analysis and Applications
3 . Now if the points x. were uniformly distributed in RA and if x^ is sufficiently far from the origin so that the distance to the r t n nearest neighbours of x. may be assumed to be independent of the distance to the s t n nearest neighbour of o then the probability P (exactly r points are closer to x^ than the s*-*1 closest point is to o) n! n! (r+s-1)! (2n-r-s)! r! (n-r)! (s-1)! (n-s)! 2n! which for large n is approximately equal to (r+s-1)! r! (s-1)! r Lr+s V Now for most non-uniform distributions the density of points decreases monotonically away from ρ and thus the variable r is stochastically less than the distribution derived above would indicate.
X^-x^Xx^-x^r j=l J J N + I' (xf2)-x(2))(xf2)-x(2))· . J j-i 3 and (3) n = N-. + Np - 2. The rule is to classify x as coming from Niy^ 1 ), Σ) if W > c and from Ν(μ^ 2) , Σ) if W <_c, where c may be a constant, particularly 0, or a function of x , Έ , and S. The squared Mahalanobis distance is a = ( H a)_ H (2) ) t ç-i (Hd) . H(2)}) (4) which can be estimated by a= (^ (1) -x (2) )· S"1 ( I ( 1 ) - i ( 2 ) ) . W. ). Okamoto's expansion of the probability distribution [(1963), Corollary 1] to terms of order n is 2" Δ \ PrJ , (1) ±xx μ=μ ν ; = Φ(υ) + ±φ(ιι) !
A function is plotted as a character string made up of units of length ten. The first three characters give the sequence number of the observation in the previous table. The next 3 characters indicate the group or cluster. The last 4 characters are just dots. The functions plotted were based on the mapping x -> f (t) = x sin(t) + x2cos(t) + x«sin(2t) + . The clusters appear as relatively tight bands, the thinner the band, the smaller the size of the cluster. 12 appear less dense than many of the others and the presence of some holes in the center of these cluster bands suggests that they might be better divided and assigned to two or more different clusters.
Discriminant Analysis and Applications by T. Cacoullos