Use of the multinomial jack-knife and bootstrap in generalized non-linear canonical correlation analysis

In this paper we discuss the estimation of mean and standard errors of the eigenvalues and category quantifications in generalized non-linear canonical correlation analysis (OVERALS). Starting points are the delta method equations, but the jack-knife and bootstrap are used to provide finite difference approximations to the derivatives

Homogeneity analysis with k sets of variables: An alternating least squares method with optimal scaling features

Homogeneity analysis, or multiple correspondence analysis, is usually applied to k separate variables. In this paper, it is applied to sets of variables by using sums within sets. The resulting technique is referred to as OVERALS. It uses the notion of optimal scaling, with transformations that can be multiple or single. The single transformations consist of three types: (1) nominal; (2) ordinal; and (3) numerical. The corresponding OVERALS computer program minimizes a least squares loss function by using an alternating least squares algorithm. Many existing linear and non-linear multivariate analysis techniques are shown to be special cases of OVERALS. Disadvantages of the OVERALS method include the possibility of local minima in some complicated special cases, a lack of information on the stability of results, and its inability to handle incomplete data matrices. Means of dealing with some of these problems are suggested (i.e., an alternating least squares algorithm to solve the minimization problem). An application of the method to data from an epidemiological survey is provided

Nonlinear canonical correlation analysis with k sets of variables

The multivariate technique OVERALS is introduced as a non-linear generalization of canonical correlation analysis (CCA). First, two sets CCA is introduced. Two sets CCA is a technique that computes linear combinations of sets of variables that correlate in an optimal way. Two sets CCA is then expanded to generalized (or k sets) CCA. The formulation for the OVERALS technique fits well in the general tradition of "k" sets methods. The formulation is based on a minimization of the loss between object scores and canonical variates of all sets together, but is expanded with optimal scaling and the method of copies. Single and multiple transformations are discussed. The method is illustrated using data from an American consumer report giving the characteristics of 33 popular cars and 3 sets of data. Three tables and seven graphs present the data from the application study

Non-linear canonical correlation

Non-linear canonical correlation analysis is a method for canonical correlation analysis with optimal scaling features. The method fits many kinds of discrete data. The different parameters are solved for in an alternating least squares way and the corresponding program is called CANALS. An application of CANALS is discussed and also a study of the stability of the scaling results

Nonlinear redundancy analysis

A non-linear version of redundancy analysis is introduced. The technique is called REDUNDALS. It is implemented within the computer program for canonical correlation analysis called CANALS. The REDUNDALS algorithm is of an alternating least square (ALS) type. The technique is defined as minimization of a squared distance between criterion variables and weighted predictor variables. With the help of optimal scaling, the variables are non-linearly transformed. An application of the REDUNDALS technique used data from a survey conducted with members of the Dutch parliament who gave their opinions on seven issues and their preference votes for political parties. This example illustrates that the non-linear redundancy analysis corresponds to a multivariate multiple regression with optimal scaling. In the case of the Dutch parliamentary data, the REDUNDALS results are mostly comparable with the numerical CANALS analysis. The programs are combined, but CANALS finds directions in both sets of variables that correlate maximally, independent of how much variance is explained, while REDUNDALS explains as much variance as possible in every criterion direction. Two tables provide information about the parliamentary study, and a figure illustrates the monotone transformations of the variables

Non-linear canonical correlation

Non-linear canonical correlation analysis is a method for canonical correlation analysis with optimal scaling features. The method fits many kinds of discrete data. The different parameters are solved for in an alternating least squares way and the corresponding program is called CANALS. An application of CANALS is discussed and also a study of the stability of the scaling results

Análisis de K-conjuntos longitudinales mediante variables retardadas

We present an application of nonlinear Generalised Canonical Analysis (GCA) for analysing longitudinal data. The application uses lagged versions of variables to accomodate the time-dependence in the measurements. The usefulness of the proposed method is illustrated in an example from developmental psychology, in which we explore the relationship between mother and child dyadic interaction during the first six months after birth, demonstrating how child behaviour can elicit mother behaviour. We discuss the relationship between our proposed method and the most closely resembling SERIALS (Van Buuren, 1990) method for nonlinear time series analysis

Análisis de K-conjuntos longitudinales mediante variables retardadas

We present an application of nonlinear Generalised Canonical Analysis (GCA) for analysing longitudinal data. The application uses lagged versions of variables to accomodate the time-dependence in the measurements. The usefulness of the proposed method is illustrated in an example from developmental psychology, in which we explore the relationship between mother and child dyadic interaction during the first six months after birth, demonstrating how child behaviour can elicit mother behaviour. We discuss the relationship between our proposed method and the most closely resembling SERIALS (Van Buuren, 1990) method for nonlinear time series analysis

Análisis de K-conjuntos longitudinales empleando una variable temporal ficticia (dummy)

We present an application of nonlinear Generalised Canonical Analysts (GCA) for analysing longitudinal data. The application uses lagged versions of variables to accommodate the time-dependence in the measurements. The usefulness of the proposed method is illustrated in an example from developmental psychology, in which we explore the relationship between mother and child dyadic interaction during the first six months after birth, demonstrating how child behaviour can elicit mother behaviour. We discuss the relationship between our proposed method and the most closely resembling SERIALS (Van Buuren, 1990) method for nonlinear time series analysis