In this paper, we deal with functional principal points of Gaussian random functions and functional clustering. The k principal points of a p-variate random variable X are defined as the k points which minimize the expected squared distance of X from the nearest of the point (Flury, 1990). The concept of principal points can be extended to functional data analysis (Tarpey & Kinateder, 2003), and we call the extended principal points functional principal points. Functional principal points of random functions have a close relation to functional cluster analysis. We derive functional principal points of polynomial random functions using orthogonal basis expansion. For functional data according to Gaussian random functions, we discuss the relation between the optimum clustering of the functional data and the functional principal points. We also investigate the number of local minima of the functional clustering algorithm with numerical experiments.

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In this paper, we propose a functional relative projection pursuit as an extended method of relative projection pursuit (Mizuta, 2002a) for functional data analysis. This method finds an 'interesting' structure of subset data in a low dimensional projection space of the functional data compared with the structure of the superset data. For example, it can detect some different part of functions such as a time cycle and amplitude between the subset functional data and the superset functional data. Projection pursuit (Friedman & Tukey, 1974) is one of the dimension reduction methods to search for an 'interesting' structure in low dimensional space. This method has been already extended for functional data analysis by Nason (1998), called functional projection pursuit. The both methods are powerful, but can only find the different structures from normal distribution, defined as an 'uninteresting' structure. We consider that 'interesting' structures are not always different from normal distribution because an 'interestingness' depends on factors such as purposes of an analysis. The aim of our method is to search for 'interesting' structures far from a distribution of a functional dataset as reference pre-defined by a user. We assess the effectiveness of our method compared with conventional projection pursuit using a numerical example. In addition, we introduce a case study, which is applied our method to the child development data of National study of health and growth (Holland et al., 1999a, b) in United Kingdom.

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The volatility became an important index in Investment Science. Since the volatility attracts attention, much effort has been put into adding the volatility forecast accuracys. The generalized autoregressive conditional heteroskedasticity (GARCH) models are used the volatility forecast widely. They rely on the assumption of distribution function, therefore the volatility forecast may be error if distribution function changes with time. By contrast, Taylor proposed method of volatility forecasts from conditional autoregressive value at risk (CAViaR) models in 2005. Those models need not assume the distribution function. Many kinds of CAViaR models are presented, however the volatility forecast from Asymmetric Slope CAViaR (ASCAViaR) model is the most accurately. In the existing study, ASCAViaR model has constant expected value. This study aimed at adding the volatility forecast accuracy, and proposed changed expected value ASCAViaR model. This model has changeability expected value. This study compared the volatility forecast accuracy from changed expected value ASCAViaR model with those from existing ASCAViaR model and GARCH model. This study used three stock indices (the Japanese JASDAQ, the Japanese TOPIX and the U.S. S & P). For all indices, there were two forecast periods (10days and 20days).

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This paper provides a procedure for monitoring structural changes, which are gradually reflected in observations. Most of the monitoring schemes assume that parameters change at specific points in time series. However structural changes are sometimes reflected slowly rather than rapidly in real time series such as financial returns. This paper proposes a procedure for monitoring such changes by applying a fuzzy trend model to time series. A fuzzy trend model is a model for time series, based on the Takagi-Sugeno's fuzzy system. Simulation studies show the effectiveness of the prosed method. Moreover the proposed method is applied to real financial time series. The result shows the applicability of the fuzzy trend model.

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Recent movements of the studies on smoothing with splines are overviewed. It is remarkable that smoothing (penalized) splines are represented as mixed effect models. Penalized splines with truncated power basis functions are especially useful, because they match the mixed effect model representation and also avoid complicated computation of the splines and their penalty terms. On the selection of smoothing parameters, which control the smoothness of penalized splines, the traditional mainstream was the (generalized) cross-validation. However, the problem results in the estimation of variance parameters in the mixed effect models, so the restricted maximum likelihood (REML), or equivalently, the empirical Bayes method, is more useful and is the current stream. The test for linear regression hypothesis with penalized splines results in a test for the variance of random effects, and the restricted log likelihood ratio statistic is considered to be useful. However, its null distribution is difficult to obtain asymptotically, and so it is reproduced with random numbers. A simulation study shows that the REML estimate itself of the smoothing parameter gives stronger power in some situations. Finally, the penalized splines can extend to a variety of regression models, of which methods of inference are developed with the mixed effect models and Bayesian approaches.

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