Analytic Solution of the Regularized Latent Truth Model for Binary Maps

Consider two maps having the same spatial extent and the same mapping categories but where each map is subject to classi ̄cation error. An overlay of the maps yields a (dis)similarity matrix whose (i; j)-entry is the areal proportion placed into category i by the ̄rst map and into category j by the second map. Patil and Taillie (2003) have proposed a latent truth model which speci ̄es the dissimilarity matrix in terms of the true (but unknown) proportions for the mapping categories and the unknown error rates for the two maps. The number of parameters in the model exceeds the degrees of freedom in the dissimilarity matrix. However, a method of regularization is applied to e®ectively reduce the dimension of the parameter space and to permit model ̄tting. From the ̄tted model, one obtains estimates for the true mapping proportions as well as estimated error matrices for each of the maps. This paper considers binary maps and obtains explicit expressions for the ̄tted parameters of the regularized latent truth model.