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Mathematical Tools for Applied Multivariate Analysis Index


Index

Adjoint, of square matrix, 137-138 Arbitrary matrix, equivalence transformation of, 184 Associative data analysis, techniques in, 4 - 5 Automatic Interaction Detection, 13 Axis permutation, 151-152 Axis rotations, 111-114, 134

B Baerwaldt, N., 12 Bartlett, M. S., 282-283 Basic structure algebra of, 232-234 conditions for, 234-235 finding of, 235-236 of matrix, 230-240 of sample problem matrix of predictors, 237-238 Basic structure decomposition procedure, 236-237 Basis concept of, 102-103 fixed, 133 orthonormal, 103 Basis vector(s), 80-83 original and new, 112-113 in quadratic forms, 245-246 Basis vector changes, 103-106, 133, 213 arbitrary, 143-146 Basis vector transformations, 133-134, 140-146 eigenstructures in, 203-205 Baumol, W. J., 11 Bilinear forms, 241-242 Bock, R. D., 284 Byrd, J., Jr., 13

Calculus chain rule in, 300-301 function of one argument in, 297-304 function of two arguments in, 304-311 in multivariate analysis, 318-321 Canonical correlation, 12 Carroll, J. D., 12,291-292 Carte sian axes, oblique, 104 Cartesian coordinates, 78-85 Cattell, Raymond, 2-3 Central dilation, of set of points, 152 Centroid of variables, 120 Chain rule, 300-301 Characteristic equation, eigenstructures and, 198-199,213-214 Characteristics, variation in, 3 see also Eigenstructures Cliff, N., 291 Cluster analysis, 6, 12 Cofactors, expansion of determinants by, 60-63 Column vector, 27, 41 Component loadings, in factor analysis problem, 274-275 Component scores, in factor analysis problem, 273-274 Components analysis, other aspects of, 276-278 Composite transformations, geometric effect of, 194 Confidence, statistical, 4 ConformabiUty, in matrix multiplication, 45 Constrained optimization of functions of two arguments, 307-309 Lagrange multipliers in, 309-311 Cooley,W.W., 12, 291 Coordinate systems, 78 Correlation matrices, 71-73 Cosines, law of, 93-94

369

370
Co variance matrix, 71-72 determinant of, 121-123 under linear transformation, 212 transformations of, 207-209 Cross-national comparisons, 12 Cross products, in matrix operations, 70, 118-124

INDEX of nonsymmetric matrices, 247-254 of symmetric matrices, 210-219 of W- A, 249-251 Eigenvalues of C(JO, 214-217 characteristic equation in, 199 defined, 196 zero, 222 Eigenvector basis, distinguishing feature of, 205 Eigenvectors of C(X), 214-217 defined, 196 Eisenbeis, R. A., 284 Elementary matrix, defined, 179 Elementary operations matrix inverse in, 182-184 matrix rank in, 178-180 in simultaneous equations, 176-178, 180-181 Equivalence transformation, of arbitrary matrix, 184 Euclidean distances, 84 Euclidean space, 78-85 defined, 83-85 Euclidean metric, 83 n.

D Data matrix in associative data analysis, 3-4 partitioning of, 6, 9 types of scales in, 7 Decomposing, of matrix transformation, 163 194-256 Decompositions, basic structure of, 230-239 Determinant(s), 58-69 computation of, 60 of covariance matrix, 121 evaluation of by pivotal method, 66-69, 184-187 expansion of by cofactors, 60-63 geometric interpretation of, 118-119 matrix rank and, 173-174 operational definition of, 59-60 properties of, 64-66 Diagonal matrix, 54 Dichotomies, vs. polytomies, 7 Differentiation basic rules of, 299 partial, 304-307 symboUc, 312-316 Dimension-reducing methods, 10 Distributive laws for matrix multiplication, 49 for scalar product, 35-36, 45 Do-it-yourself activities, survey of, 12-13 Dummy variable, 7-8

Factor analysis, 6, 10, 12 numerical example of, 20-22 Factor analysis problem, 272-278 basic structure of X^ in, 275-276 component loadings in, 274-275 component scores in, 273-274 other aspects of principal components analysis in, 276-278 Functions of one argument derivatives of, 297-300 differentiation of, 296-304 optimization of, 301-304 Functions of two arguments differentiation of, 304-311 unconstrained optimization of, 307-309

Echelon matrix, 174, 179 Eckart, C , 287 Eigenstructure(s), 195-207 additional properties of, 222-225 characteristic equation in, 198-199, 213-214 defined, 196 finding of, 237 matrix rank and, 225-230 in multiple discrimination analysis problem, 278-282

Generalized inverse computation of A^ in, 339-341 concept of, 334 introductory aspects of, 334-342 left and right, 337-339 linear equations in, 323-350 Penrose conditions for, 336-337

INDEX ^inverse, 343-349 numerical examples of, 345-349 Good, I. J., 334 n. Gram-Schmidt orthonormalization process, 106-107,221,239

371

H Haggard, E. A., 284 Hamm, B. C , 13 Hancock, H., 317 Harman, H. H., 277 Harris, R. J., 283 Haynes, R. D., 13 Horst, P., 5 n., 291-293 Hypervolume, 101

Linear transformation see also Matrix transformation; Transformation(s) arbitrary, 160-163 covariance matrix in, 212 geometric viewpoint for, 127-190 matrix rank and, 169-172 Lohnes, P. R., 12, 291

M McDonald, R. P., 249 n., 292 Mapping concept of, 128 images and preimages in, 128 matrix rank and, 170 Matrix (matrices) addition of, 43-45 adjoint of, 137-138 basic definitions and operations for, 40-52 basic structure of, 230-240 bilinear forms of, 241-242 cofactorsof, 137-138 decomposition of, 230-239 defined, 40 determinants of, 58-69, 137-138, 167 diagonal, 54 echelon, 179 elementary, 177 examples of, 56-57 "exterior" dimension of, 46 horizontal and vertical, 40 identity, 55 "interior" dimension of, 46 linear forms of, 240-241 multiplication of with vector, 47-48 null, 41 pivotal method for, 66-69, 184-187 postmultiplying of, 147-148, 327 pre- and postmultiplication of by diagonal, 55-57 product-moment, 227-229 quadratic form of, 240-247 rank of, see Matrix rank rotation of, 153-154 scalar, 54 sign, 54 special, 52-57 square, see Square matrix subtraction of, 43-44 symmetric, see Symmetric matrix synthetic data, 147 trace of, 222

I Identity matrix, 55, 177 linear equations in, 327 Identity transformation, 289 Image, in mapping, 128 Image space, dimensionality of, 175 Interobject similarity, 8,11 Invariant vectors, under transformation, 196, 201 see also Vector(s) Inverse generalized, see Generalized inverse of inverse, 165 matrix, see Matrix inverse; Matrix inversion Invertible transformation, characteristics of, 166

Kettenring, J. R., 292 Komar, C A., 13

Lagrange multiplier, 213 Law of cosines, 93-94 Level curves, partial differentiation and, 304-307 Linear dependence, of vectors, 101-110 Linear equations generalized inverses and, 323-350 general procedure for solving, 327-329 matrix transformation in, 129-130 Linear forms, 240-241 Linearity in parameters, defined, 7 n. Linear model, forms of, 270-272

372

INDEX geometric relationships involving, 147-156 invertible, 163-175 matrix rank and, 163-175 orthogonal, 130-134 quadratic forms and, 246-247 reflection, 150-151 rotation, 153-154 rotation followed by reflection, 157 rotation followed by stretch, 159-160 rotation-stretch-rotation, 160 shear, 154-156 simultaneous equations and, 128-136 singular matrices and, 172-173 stretch, 152-153, 159-160 stretch followed by rotation, 157-159 translation, 149-150 Matrix transpose, 42-43 in matrix multiplication, 50 MDA, see Multiple discriminant analysis Mean-corrected (SSCP) matrix, 70-71 see also Sums of squares and cross products matrix Minkowski metric, 83 Mitra, S. K., 349 Model, public and private, 5 Moore, E. H., 336 Moore-Penrose inverse, 323, 334, 336-337, 343 properties of, 342 Morgan, J., 12 Multidimensional scaling, 10 Multiple correlation coefficient, 262 Multiple criterion, in multiple predictor association, 10 Multiple discriminant analysis, 278-286 calculus in, 320 eigenstructure in, 278-282 example of, 22-23 other aspects of, 284-286 preUminary calculations for, 250 statistical significance and classification in, 282-284 Multiple predictor association multiple criterion in, 10 single criterion in, 8-10 Multiple regression, 18-20, 260-270 alternative representation of problem in, 268-270 defined, 20 geometric aspects of problem in, 266-268 least squares principle in, 18-20 matrix inversion in, 187-189 other formulations of problem in, 265-266 overall relationship and statistical significance in, 262-264 pivotal method in, 189

transformation, see Matrix transformation; Transformation matrix transpose of, 4 2 - 4 3 , 50 vector multiplication with, 47-48 Matrix addition, 43-45 Matrix algebra, 22-23, 51 Matrix concepts, from geometric viewpoint, 77-124 Matrix division, 51 Matrix eigenstructures see also Eigenstructure(s) overview of, 195-207 properties of, 219-225 Matrix equality, 43-44 Matrix inverse elementary operations and, 182-184 point transformations and, 139-140 properties of, 165-166 Matrix inversion, 51,127, 136-147 methods for, 176-189 in multiple regression, 187-189 Matrix matching internal criterion in, 292-293 in multivariate technique classification, 290-292 Matrix multiplication conformability in, 45 order in, 47 properties of, 4 9 - 5 0 triple product in, 48-49 Matrix operations covariance and correlation matrices in, 71-73 examples of, 5 1 - 5 2 in statistical data, 59-73 Matrix product, of two vectors, 48 Matrix rank, 65, 127, 167-168 determinants and, 173-174 determination of, 176-189 eigenstructures and, 225-230 elementary operations and, 178-180 invertible transformations and, 163-175 use of in matrix algebra, 174-175 Matrix relations, vs. scalar, 50 Matrix representation, 40 Matrix transformation see also Linear transformation; Point transformation; Transformation matrix; Transformations arbitrary linear, 160-163 axis permutation and, 151-152 central dilation and, 152 composite, 156-163 decomposing of, 163, 194-256 elementary row operations in, 176-178

INDEX purpose of, 268-269 response surface model in, 266-268 selected output from, 263 Student t value in, 264 Multiple regression equation, 18-20, 260-262, 318-319 Multiple variates, association among, 2 Multivariate analysis characteristics in, 2, 5-8 data matrix in, 3 defined, 2 dimension-reducing methods in, 10 eigenstructures of nonsymmetric matrices in, 247-254 illustrative applications of, 11-14 interobject similarity in, 8, 11 linear transformations in, 127 major subdivisions of, 8 mixed scales in, 7 prior judgments or presuppositions in, 6 quadratic forms in, 240-247 translations in, 149-150 types of associations in, 7 types of scales in, 7 vector and matrix operations in, 26-74 Multivariate data, applying tools to, 259-293 Multivariate data analysis see also Multivariate analysis nature of, 1-25 numerical examples of, 14-23 Multivariate functions, symbolic differentiation and optimization in, 295-322 Multivariate technique classification, 286-293 construction of, 289-293 matrix matching in, 290-292 transformation types in, 287-289 vector-matrix matching in, 290 Multivariate techniques see also Multivariate analysis characteristics of, 2 linear transformations in, 14 personnel data as illustration of, 15 in scientific research, 1-4 two types of variables in, 8-9 uses of, 1 N Neter, J., 271 Nonsingular matrix, 177 Nonsymmetric matrix, eigenstructures of, 247-254 -space, basis of, 103 Null matrix, 41, 45 Null vector, 28 O

373

Oblique Cartesian axes, 104 Oblique coordinate systems, scalar products in, 109-110 Optimization of functions involving multivariate arguments, 316-317 in matrix notation, 314-316 Orthogonal complement, 108 Orthogonal matrix, properties of, 116-118 Orthogonal transformations, 111-118 Orthonormal basis, 103-104 finding of, 106-108 three-dimensional, 108

Pair of variables, correlation and covariance of, 121 Parallelogram law, 104 Parameters, linearity in, 7 Partial differentiation, level curves and, 304-307 Partitioned subsets, number of variables in, 6-7 Partitionings, of data matrix, 9 Penrose, R. A., 323, 336 Penrose conditions, for generalized inverse, 336-341 Permutation, of set of points, 151-152 Permutation transformation, 289 Perry, M., 13 Personnel data, multivariate methods in, 14-17 Pivotal methods evaluation of determinant by, 66-69, 184-187 in sample regression problem, 189 Point rotations, 111-114, 130-133 Points permutation of, 151 - 1 5 2 reflection of, 150-151 rigid rotation of set by, 111 stationary vs. extreme, 317 Point transformations, 132 eigenstructures ai^, 203-205 fixed basis vectors in, 135 matrix inverse ar^, 139-140, 145 Polytomies, vs. dichotomies, 7 Predictor variables, basic structure and, 237-238 Prefactor, in matrix multipUcation, 46 Preimages, in mapping, 128 Premultiplying, of vectors, 154 Principal components analysis, calculus in, 319 Pringle, R. M., 349

374
Product information, personal vs. impersonal sources of, 13 Production jobs, sequencing of, 13 Product-moment matrices, 255 special characteristics of, 227-229 Project TALENT, 12 Pseudoinverse, 334 Pythagorean theorem, 94

INDEX

Quadratic forms, 240-247 basis vectors and, 245-246 defined, 242 diagonalizing of, 251-254 illustrative problem in, 243-244 matrix transformations and, 246-247 ratio of, 249 Quandt, R. E., 11 R Rank, of matrix, 167-168, 174-176 see also Matrix rank Rao, C R., 66, 284, 323, 349 Ratio scales, 5 Rayner, A. A., 349 Rectangular matrix eigenstructure of, 226-227 rank of, 226-227 Reflection, of set of points, 150-151 Regression problem, defined, 3 see also Multiple regression Research focus of interest in, 6 multivariate methods in, 2-4 objectives and predictive statements in, 5-6 Response surface model, variations in, 273 Rorer, 11 Rotation(s) followed by reflection, 157 followed by stretch, 159-160 higher-dimensional, 115-116 improper, 117 of matrix, 153-154 point, 111-114, 130-133 three-dimensional, 116 trigonometry of, 114-115 two-dimensional, 111-115 Rotation problem, matrix eigenstructure in, 228-229 Rotation transformation, 288 Row vector, 27 Rummel, R. J., 12

Scalar arithmetic, vs. matrix arithmetic, 50 Scalar matrix, 54 Scalar multipUcation, 43-44 properties of, 45 Scalar product, 35-38 defined, 35 distributive laws for, 35-36, 45 examples of, 39 in oblique coordinate systems, 109-110 special cases of, 36-38 of two vectors, 96-97 vector projections and, 97-100 Scatter plot in factor analysis, 21 in multivariate analysis, 16-17 Schatzoff, 284 Schonemann, 291 Scientific research, multivariate techniques in, 2-6 Set of points central dilation and, 152 permutation of, 151-152 rigid rotation of, 111 stretch transformation of, 152-153 Shear transformation, 155-156 Sign matrix, 54 Sign vector, 28 Simultaneous equations elementary operations and, 180-181 in matrix form, 129-130 matrix of coefficients and, 182-184 matrix transformations and, 128-136 Simultaneous linear equations, 324-334 homogeneous equations and, 331-334 Single criterion, in multiple predictor association, 9-10 Sirageldin, I., 12 Skew symmetric matrix, 54 Spherizing, 252 Square matrix, 40-41 adjoint of, 137 cofactor of entry of, 61 eigenstructure of, 195-206, 225 Hermite form of, 343-345 inverse of, 136 minor of an entry of, 61 singular and nonsingular, 65 symmetric, 4 2 - 4 3 , 53 vs. triangular matrix, 219 n. SSCP, see Sums of squares and cross products matrix Stationary point, as extreme point, 317 Statistical data, matrix operations in, 69-73

INDEX Statistical measures, geometry of, 119-122 Statistical variables, generalized variance among, 118 Stretch, 157-160 Stretch transformation, 153-154, 289 Student t value, defined, 264 Sums of squares, in matrix operations, 70 Sums of squares and cross product matrix, 26, 71-73, 120, 150, 252-253, 283, 296, 319-320 Symbolic differentiation, 312-316 Symmetric matrix, 4 2 - 4 3 , 53-54 eigenstructure of, 210-219 orthogonal diagonalization in, 230 properties of, 220-222

375

Tatsuoka, M. M., 5 n. Tiedeman, D. V., 5 n. Transformational geometry, 22-23 Transformation matrix, 119 see also Matrix transformation; Transformations diagonalizing of, 202-203 postmultiplying of, 202 rank of, 175 Transformations see also Matrix transformation arbitrary linear, 160-163 by basis vector change, 140-146 central dilation, 289 composite, 156-163 of covariance matrices, 207-209 general linear (affine), 288 homogeneous linear, 288 identity, 289 invertible, 166 linear, see Linear transformation permutation, 289 rotation, 288 rotation-annihilation, 288 similarity, 288 of singular matrices, 172-173 stretch, 289 types of in multivariate technique classification, 287-289 Translation, of matrices, 149-150 Travel-demand forecasting model, 11 Triple matrix product, 255 Trivial solution of homogeneous equations, 331

Unit vector, 28

Van de Geer, J. P., 277, 292 Variable(s) batteries of, 10 correlation of pairs of, 121 covariance of pairs of, 121 criterion vs. predictor, 3, 5 dependent and independent, 3 dummy, 7-8 latent, 10 linear composites and, 7 scale types and, 8-9 variance of, 120 Variable interdependence, analysis of, 8 Variance measure, generalized, 122-124 Vector(s) addition and subtraction of, 30-31, 38-39 basic definitions and operations in, 27-39 basis, see Basis vector(s) column and row, 27 defined, 27 difference, 90 direction cosines of, 86 distance and angle between, 92-96 equality of, 29, 80 geometric representation of, 77, 8 5 - 100 length and direction angles of, 85-88 linear combination of, 33-34, 39 linear dependence of, 101-110 matrix product of, 48 multiplication with matrix, 47-48 multiplication with scalar, 91 /i-component, 82 nonredundant, 102 null or zero, 28 orthogonal, 94 parallel displacement of, 80 premultiplying of, 154 representation of in three-dimensional space, 81 row, 27 scalar multiplication of, 32-33, 39 scalar product of, 35-38, 96-97 sign, 28 squared distance between, 95 subtraction of, 90 transformation of, 128, 140-146 transpose of, 38 unit, 28 zero-one, 28 Vector addition, 88-92 Vector and matrix operations, 26-74 Vector concepts, from geometric viewpoint, 77-124

376
Vector equality, 29 Vector length direction angles and, 86 scalar product and, 106 Vector-matrix matching, in multivariate technique classification, 290 Vector multiplication, 47-49 geometric aspects of, 88-92 Vector notation, 26 Vector projection geometric interpretation of, 98 scalar products and, 97-100 Vector representation, 27 Vector space basis change in, 213 dimensionality of, 102-103 Vector transformation, 128 basis vector change and, 140-146

INDEX W

Wasserman,W., 271 Wind, Y., 277 Wish, M., 12

Young, G., 287

Zero eigenvalues, 222 Zero-one vectors, 28


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