discrimquad(groups, y), factor or vector of positive integers groups, REAL matrix y with nrows(y) = length(groups) |

discrimquad(groups, y), where groups is a factor or an integer vector, and y is a REAL data matrix, computes the coefficients of quadratic discriminant functions that can be used to classify an observation into one of the populations specified by argument groups. It is an error if the smallest group has p or fewer members or if y has any MISSING elements. When there are g = max(groups) populations, and p = ncols(y) variables, the value returned is structure(Q:q, L:l, addcon:c, grandmean:ybar), where the components are as follows: q structure(Q1,Q2,...Qg), each Qj a REAL p by p matrix l structure(L1,L2,...Lg), each L2 a REAL vector of length p c vector(c1,c2,...cg), cj REAL scalars ybar vector(ybar1,...ybarp), the vector of column means When x is a vector of length p to be classified, the quadratic score for group j is qs[j] = (x-ybar)' %*% q[j] %*% (x-ybar) + (x-ybar)' %*% l[j] + c[j] The functions are optimal in the case when the distribution in each population is multivariate normal with no assumption that the variance- covariance matrices are the same for all populations. When P = vector(P1,P2,...,Pg) is a vector of prior probabilities a randomly selected case comes from the various populations, then the posterior probabilities the elements of the vector P*exp(qs)/sum(P*exp(qs)) = P*exp(qs - qs[1])/sum(P*exp(qs - qs[1]) The latter form is usually preferable since it is possible for exp(qs[1]) to be so large as to be uncomputable. These probabilities can be computed using macro probsquad(). NOTE: It is well known that posterior probabilities computed for a case that is in "training set", the data set from which a classification method was estimated, are biased in an "optimistic" direction: The estimated posterior probability for its actual population is biased upward. For this reason posterior probabilities should be estimated only for cases that are not in the training set. See also discrim() and probsquad().

Gary Oehlert 2003-01-15