MTV and MGV: Two Criteria for Nonlinear PCA [chapter]

Tatsuo Otsu, Hiroko Matsuo
2002 Recent Advances in Statistical Research and Data Analysis  
MTV (Maximizing Total Variance) and MGV (Minimizing Generalized Variance) are popular criteria for PCA with optimal scaling. They are adopted by the SAS-PRINQUAL procedure and OSMOD (Saito and Otsu,1988). MTV is an intuitive generalization of linear PCA criterion. We will show some properties of nonlinear PCA with these criteria in an application to the data of NLSY79 (Zagorsky,1997), a large panel survey in the U.S., conducted over twenty years. We will show the following. (1) The
more » ... of PCA with optimal scaling as a tool for large social research data analysis. We can obtain useful results when it complements analyses by regression models. (2) Features of MTV and MGV, especially their abilities and deficiencies in real data analysis. on rigid mathematics, and its naive correlation estimation between categorical variables often shows under-estimation. In spite of these unsophisticated features, PCAOS can obtain useful insights from real social/behavioral data. In this article, at first we point out some problems of MCA, then explain the model of PCAOS. The most important feature is the model estimation criterion. Some properties of two criteria, MTV and MGV, are explained. An example of OSMOD analysis on large survey data (NLSY79) is shown. Inadequate Solutions by Multiple Correspondence Analysis Here we consider some properties of MCA. For a demonstration example, we use the following artificial data. 1. Generate 100 samples of 5-dimensional multivariate normal distribution that has the covariance matrix shown in Tab. 1. 2. For each variable, categorize their values into 5 categories by rank. The samples of rank 1 to 20 are given category 1, and the samples of rank 21 to 40 are given category 2, and so on. 3. Therefore we obtain 5 discrete variables, each of them having 5 categories. Make 0-1 valued 25 dummy variables from these 5 variables. These 25 dummy variables are analyzed by MCA. The estimated correlations (singular values) by MCA are shown in Tab. 2. Configurations of category scores are shown in Fig.1 and Fig.2. In the figures, symbols such that 1A show the variables and categories. Numbers show the items (variables) and alphabetical letters show the categories. The first PCA component of the covariance matrix is obtained as the first component of the MCA analysis. But the second component of the MCA solution is the quadratic polynomial of the first component. The second PCA component of the original covariance structure is embedded in the higher order MCA solutions. In this case, the solution of MCA might lead the analyst into the wrong interpretation. We usually interpret MCA solutions by adopting a few score dimensions. The dimension adoption is based on the values of the correlations (singular values). The adopted dimensions correspond to the largest correlations. In the above example, the adopted two dimensions show inadequate latent data structures. In this example, we know the latent structure. Therefore the inadequacy of the method can be recognized. But in real applications, these insights may be difficult.
doi:10.1007/978-4-431-68544-9_4 fatcat:3l44vfp7hfa5lnqlu6ojrnckyu