NUMERICAL ANALYSIS OF BIOLOGICAL AND ….ppt

NUMERICAL ANALYSIS OF BIOLOGICAL AND ENVIRONMENTAL DATA;Discriminant analysis of two groups Assumptions of discriminant analysis - Multivariate normality Homogeneity Comparison of properties of two groups Identification of unknowns - Picea pollen Canonical variates analysis (= multiple discriminant analysis) of three or more groups Discriminant analysis in the framework of regression Discriminant analysis and artificial neural networks Niche analysis of species Relation of canonical correspondence analysis (CCA) to canonical variates analysis (CVA) Generalised distance-based canonical variates analysis Discriminant analysis and classification trees Software;IMPORTANCE OF CONSIDERING GROUP STRUCTURE;Taxonomy – species discrimination e.g. Iris setosa, I. virginica 2. Pollen analysis – pollen grain separation 3. Morphometrics – sexual dimorphism 4. Geology – distinguishing rock samples;Plot of two bivariate distributions, showing overlap between groups A and B along both variables X1 and X2. Groups can be distinguished by projecting members of the two groups onto the discriminant function line. z = b1x1 + b2x2 ;Can generalise for three or more variables;SIMPLE EXAMPLE OF LINEAR DISCRIMINANT ANALYSIS ;Covariance matrix for group A (SA) = 0.00092 -0.00489 -0.00489 0.07566 ?and for group B (SB) = 0.00138 -0.00844 -0.00844 0.10700;To solve [SW] [?] = [D] ?we need the inverse of SW = SW-1 = 59112.280 4312.646 4312.646 747.132;MATRIX INVERSION;If matrix A is 4 10 10 30 ?to find its inverse we first place an identity matrix [I] next to it. 4 10 . 1 0 10 30 0 1;To reduce element a12 to zero, we now subtract 2.5 times row one to give 1 0 1.5 -0.5 0 1 -0.5 0.2 The inverse of A is thus 1.5 -0.5 -0.5 0.2 This can be checked by multiplying [A] by [A]-1 which should yield the identity matrix I i.e. 1.5 -0.5 . 4 10 = 1 0 -0.5 0.2 10 30 0 1 ? ;R.A. Fisher;Can position the means of group A and of group B