In the present paper, the method of minimum dimension analysis (MDA) is given with examples. This method belongs to the same genre as smallest space analysis (SSA) by L. Guttman. Let Rij be the relation between i-element and j-element, i, j = 1, 2, ..., N. If Rij is given in the form of rank order, SSA gives the configuration of the elements with less inconsistancy in Euclidean space having smallest dimension. MDA reveals the configuration of those in the case where Rij is given in the form of rank ordered group. The author gives the algorithm to solve this problem using correlation ratio the main process of which is based on repeated linear programming method and eij-type quantification method.
Covariation analysis is a simple method of analysing the structure of data in the form of frequency tables. In this paper, the argument is limited within the context of 2×2 frequency tables for ease of introducing the basic principles. In this context, it is shown that the deviation of the observed frequency (in any cell) from its marginal estimate, the familiar quantity in the 2×2 X2-test, can be interpreted as an estimate of covariation of the two conditional probabilities corresponding to the pair of observed binary random variables. By using this value as an indicator of structural information contained in the data, the contribution of each of the condition variables to the determination of the nature of the given data can effectively be evaulated. This is demonstrated in this paper both in theory and through an example of actual analysis.
It was proposed that people could evaluate the other person's evaluation scheme in terms of the similarity (or dissimilarity) of the other person's evaluation scheme to their own schemes. The metric for the dissimilarity of evaluation schemes was proposed to be the Minkowski-p metric for the utilities evaluated under different schemes. This model provides a basis for allocating importance to utility functions, where importance is to be interpreted as the similarity of the integrated scheme to the schemes represented by the utility functions. An experiment was done to investigate whether people's intuitive judgments for the dissimilarity of evaluation schemes could be described by the Minkowski-p metric; the results generally supported the model and also suggested that the metric would be the “city-block” metric, i.e., p=1 in the Minkowski-p metric.
In this paper new clustering methods are presented. The essence of these methods is based on the approximation to probability density function or to cumulative distribution function. One group of clustering methods is aimed at approximating histograms or empirical distribution functions. The other group is aimed at approximating cumulative distribution functions. Using these methods one can cluster a sample data into some cluster in the sense of minimum-error-rate decision.
We propose a practical method to test the hypothesis of multivariate normality. The multivariate normal distribution is characterized by the fact that all its joint cumulants of order higher than two are zero, so that a simple and natural way to test the hypothesis of multivariate normality will be to use sample cumulants as test statistics. In view of this, we suggest to compute exact conditional moments of sample cumlants given sample variances and covariances. And by applying asymptotic normality, we calculate exact second order conditional moments of the third cumulants, which will be used to test the normality.