We propose a new multidimensional projection index for relative projection pursuit (RPP; Mizuta, 2002). RPP is a dimension reduction method that is an extension of conventional projection pursuit (Friedman and Tukey, 1974). Conventional projection pursuit finds 'interesting' structures which differ from the normal distribution. RPP finds structures that differ from a reference data set predefined by the user as having 'uninteresting' structure. We have already proposed a one-dimensional projection index for RPP, the area index, which measures the difference between target data and reference data as a degree of 'Interestingness'. However, it cannot be applied when a user wants to reduce high dimensional data into spaces of more than one dimension. Therefore, we extend the area index so that it can be applied even when the target data set is projected into multidimensional space. In addition, we develop a new index for RPP, which is based on the Hall index (Hall, 1989), called the Hall type relative projection index.
We demonstrate the effectiveness of multidimensional RPP using artificial and actual data. In the numerical example with artificial data, it is shown that with the Hall type relative projection index we can detect more 'interesting' multidimensional spaces than that with Area index. When we apply multidimensional RPP to actual data, we can obtain 'interesting' structures of high dimensional data that cannot be derived using conventional projection pursuit.

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Recently, functional data analysis (FDA) has received considerable attention in many fields including climatology, electromyography, and signal processing. A number of successful applications have been reported (see, e.g., Ramsay and Silverman, 1997, 2002). The basic idea behind FDA is to express observed data in the form of a function, and then draw information from a collection of functional data.
In this paper, we consider the problem of constructing functional regression models, using radial basis function networks (RBFNs) along with the technique of regularization. RBFNs combine the ideas of basis expansion and kernel smoothing methods, and may be described as a linear combination of radially symmetric nonlinear basis functions. An advantage of our approach to FDA lies in more flexibility transforming observations on each individual into a functional form.
In practice, individuals are measured at possibly differing sets of time points, so the amount of smoothness imposed on a set of discrete data could differ among subjects. Hence, in constructing functional regression models there remains the problem of how to determine the number of basis functions and an appropriate value of the regularization parameter. Cross-validation and generalized cross-validation are often referred as in the literature. However, computational effort can be enormous and there can be large variation with a tendency to undersmooth when applied to the analysis of functional data, because the selectors are repeatedly applied.
We present an information-theoretic criterion for evaluating models estimated by the method of regularization in the context of functional regression analysis. The proposed criterion is applied to choosing smoothing parameters and the number of basis functions. We analyze a set of data on climate in Canada. Bootstrap simulations were conducted to examine the performance of our modeling technique compared to the classical cross-validation method.

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Because a particular feature of longitudinal data is the repeated measurement of subjects over time, it inevitably produces correlation among observations. Therefore, it is not appropriate to use procedures developed under an assumption of independence. In the conduct of actual pharmaceutical clinical trials, due to problems of ethics, cost to the maker, etc. it is necessary to limit the scale of the pilot study by, for example, collecting repeated observations on the same subjects. Therefore, procedures for longitudinal data analysis are important in practice. In this paper, we consider a method of analysis for multivariate longitudinal measurements on multiple subjects in a fixed time period. We treat the individuals as random effects, as the purpose of the analysis is not to make inference about each individual per se but rather the population. For such a model, we give the analysis of variance and describe the parameter-estimation method to examine whether there are effects of group, phase and item.

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