Number of cases (objects) Number of variables Number of dimensions. n-vector with categorical observations

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1 PRINCALS Notation The PRINCALS algorithm was first described in Van Rickevorsel and De Leeuw (1979) and De Leeuw and Van Rickevorsel (1980); also see Gifi (1981, 1985). Characteristic features of PRINCALS are the ability to specify any of a number of measurement levels for each variable separately and the treatment of missing values by setting weights in the loss function equal to 0. The following notation is used throughout this chapter unless otherwise noted: n m p Number of cases (obects) Number of variables Number of dimensions For variable, h = 1, K, m n-vector with categorical observations k Number of categories (distinct values) of variable G Indicator matrix for variable, of order n k The elements of G are defined as i = 1, K, n; r = 1, K, k 16 = % & ' 1 g ir 0 when the ith obect is in the rth category of variable when the ith obect is not in the rth category of variable M Binary, diagonal n n matrix The diagonal elements of M are i k m16 % 1 when the th observation is within the range [ 1, ] ii= & ' 0 when the ith observation is outside the range [ 1, k ] D Diagonal matrix containing the univariate marginals; that is, the column sums of G 1

2 2 PRINCALS The quantification matrices and parameter vectors are: X Obect scores, of order n p Y Multiple category quantifications, of order k p y a Single category quantifications, of order k Variable weights (equal to component loadings) of order p Q Transformed data matrix of order n m with columns q = G y Y Collection of multiple and single category quantifications Note: The matrices G, M, and D are exclusively notational devices; they are stored in reduced form, and the program fully profits from their sparseness by replacing matrix multiplications with selective accumulation. Obective Function Optimization The PRINCALS obective is to find obect scores X and a set of Y (for = 1, K, m) the underlining indicates that they are possibly restricted so that the function σ XY ; = 1 m tr X G Y M X GY is minimal, under the normalization restriction XM X = M = M mn I, where the matrix, and I is the p p identity matrix. The inclusion of M in σ1xy ; 6 ensures that there is no influence of data values outside the range [ 1, k ], which may be really missing or merely regarded as such; M contains the number of active data values for each obect. The obect scores are also centered; that is, they satisfy um X = 0, with u denoting an n-vector with ones.

3 PRINCALS 3 The following measurement levels are distinguished in PRINCALS: Multiple Nominal Y = Y (unrestricted) Single Nominal Y = y a (rank-one restrictions only) (Single) Ordinal Y = y a and y C (rank-one and montonicity restrictions) (Single) Numerical Y = y a and y L (rank-one and linearity restrictions) For each variable, these levels can be chosen independently. The general requirement in the single options is Y = y a ; that is, Y is of rank one; for identification purposes, y is always normalized so that y Dy = n, which implies that the variance of the quantified variable q = G y is 1. In the ordinal case, the additional restriction y C means that y must be located in the convex cone of all k -vectors with nondecreasing elements. In the numerical case, the additional restriction y L means that y must be located in the subspace of all k -vectors that are a linear transformation of the vector consisting of k successive integers. Optimization is achieved through the following iteration scheme twice: (1) Initialization (a) or (b) (2) Update obect scores

4 4 PRINCALS (3) Orthonormalization (4) Update category quantifications (5) Convergence test: repeat (2) (4) or continue (6) Rotation The first time (for the initial configuration) initialization (a) described below is used and all single variables are temporarily treated as single numerical. Multiple nominal variables are treated as multiple nominal. The second time (for the final configuration) initialization (b) is used. Steps (1) through (6) are explained below. (1) Initialization (a) The obect scores X are initialized with random numbers, which are normalized so that um X = 0 and XM X = mn I, yielding X ~. For multiple variables, the initial category quantifications are obtained as ~ 1 Y D G X ~ =. For single variables, the initial category quantifications are defined as the first k successive integers, normalized so that ud ~ y = 0 and ~ ydy ~ = n, and the initial variable weights are calculated as the vector ~ ~ a X G ~ = y, rescaled to unit length. (b) All relevant quantities are copied from the results of the first cycle. (2) Update obect scores First, the auxiliary score matrix Z is computed as Z ~ M G Y and centered with respect to M : ~ Z M M uu M u M u Z 1 6 < A. These two steps yield locally the best updates when there are no orthogonality constraints. (3) Orthonormalization The orthonormalization problem is to find an M -orthonormal X + that is closest to Z ~ in the least squares sense. In PRINCALS, this is done by setting + 12 X M 12 m M 12~ GRAM Z 4 9 which is equal to the genuine least squares estimate up to a rotation see (6). The notation GRAM( ) is used to denote the Gram-Schmidt transformation (Börk and Golub, 1973).

5 PRINCALS 5 (4) Update category quantifications; loop across variable = 1, K, m (a) For multiple nominal variables, the new category quantifications are computed as: + 1 ~ Y = D G X (b) For single variables, first an unconstrained update is computed in the same way: ~ 1 + Y = D G X Next, one cycle of an ALS algorithm (De Leeuw et al., 1976) is executed for computing a rank-one decomposition of Y ~, with restrictions on the left-hand vector. This cycle starts from the previous single quantifications ~ y with a + ~ = Y D ~ y When the current variable is numerical, we are ready; otherwise we compute y = Y ~ a + + Now, when the current variable is single nominal you can simply obtain y by normalizing y in the way indicated below; otherwise the variable must be ordinal, and you have to insert the weighted monotonic regression process y WMON4y 9 which make y monotonically increasing. The weights used are the diagonal elements of D, and the subalgorithm used is the up-and-down-blocks minimum violators algorithm (Kruskal, 1964; Barlow et al., 1972). The result is normalized: y = n y y D y Finally, we set Y + = y + a +

6 6 PRINCALS (5) Convergence test The difference between consecutive values of the quantity 16 16! TFIT = 1 m y sd y s+ a a s J J where y16 s denotes the sth column of Y and where J is an index set recording which variables are multiple, is compared with the user-specified convergence criterion e a small positive number. It can be shown that TFIT = p σ 1X; Y6. Steps (2) through (4) are repeated as long as the loss difference exceeds ε. (6) Rotation As remarked in (3), during iteration the orientation of X and Y with respect to the coordinate system is not necessarily correct; this also reflects that σ1xy ; 6 is invariant under simultaneous rotations of X and Y. From the theory of principal components, it is known that if all variables would be single, the matrix A which can be formed by stacking the row vectors a has the property that AA is diagonal. Therefore you can rotate so that the matrix 1 m AA = 1 m a a = 1 m YD Y " # $ # Diagnostics becomes diagonal. The corresponding eigenvalues are printed after the convergence message of the program. The calculation involves tridiagonalization with Householder transformations followed by the implicit QL algorithm (Wilkinson, 1965). Maximum Rank (may be issued as a warning when exceeded) The maximum rank p max indicates the maximum number of dimensions that can be computed for any data set. In general

7 PRINCALS 7 % ( &K J )K, 'K *K pmax = min n 1, k + m max m, max, m 0 1 where m 1 is the number of multiple variables with no missing values, m 2 is the number of single variables, and J is an index set recording which variables are multiple. Although the number of nontrivial dimensions may be less than p max when m = 2, PRINCALS does allow dimensionalities all the way up to p max. When, due to empty categories in the actual data, the rank deteriorates below the specified dimensionality, the program stops. Marginal Frequencies The frequencies table gives the univariate marginals and the number of missing values (that is, values that are regarded as out of range for the current analysis) for each variable. These are computed as the column sums of D and the total sum of M. Fit and Loss Measures When the HISTORY option is in effect, the following fit and loss measures are reported: (a) Total fit. This is the quantity TFIT defined in (5). (b) Total loss. This is σ XY ; loss defined below. (c) Multiple loss. This measure is computed as TMLOSS = p 1 m tr YD Y 1 6, computed as the sum of multiple loss and single (d) Single loss. This measure is computed only when some of the variables are single: SLOSS = 1 m tr YD Y + a a J J

8 8 PRINCALS Eigenvalues and Correlations between Optimally Scaled Variables References If there are no missing data, the eigenvalues printed by PRINCALS are those of 1 mr1q6, where RQ 1 6 denotes the matrix of correlations between the optimally scaled variables in the columns of Q. For multiple variables, q is defined here as 1 6 itself is also printed. G y161. When all variables are single or when p = 1, RQ If there are missing data, then the eigenvalues are those of the matrix with elements qm 1 q 1, which is not necessarily a correlation matrix, although it is positive semidefinite. Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D Statistical inference under order restrictions. New York: John Wiley & Sons, Inc. Börk, A., and Golub, G. H Numerical methods for computing angles between linear subspaces. Mathematics of Computation, 27: De Leeuw, J., and Van Rickevorsel, J HOMALS and PRINCALS Some generalizations of principal components analysis. In: Data Analysis and Informatics, E. Diday et al, eds. Amsterdam: North-Holland. De Leeuw, J., Young, F. W., and Takane, Y Additive structure in qualitative data: an alternating least squares method with optimal scaling features. Psychometrika, 31: Gifi, A Nonlinear multivariate analysis. Leiden: Department of Data Theory. Gifi, A PRINCALS. Leiden: Department of Data Theory, Internal Report UG Kruskal, J. B Nonmetric multidimensional scaling: a numerical method. Psychometrika, 29: Van Rickevorsel, J., and De Leeuw, J An outline of PRINCALS. Leiden: Internal Report RB , Department of Data Theory.

9 PRINCALS 9 Wilkinson, J. H The algebraic eigenvalue problem. Oxford: Clarendon Press.