This paper deals with the selection of variables in multivariate linear regression models with fewer observations than the dimension by using Akaike's information criterion (AIC). It is well known that the AIC cannot be defined when the dimension of an observation is larger than the sample size, since an ordinary estimator of the covariance matrix becomes singular. By replacing the ordinary estimator of the covariance matrix with its ridge-type estimator, we propose a new AIC for selecting variables of multivariate linear regression models even though the dimension of an observation is larger than the sample size. The bias correction term of AIC is evaluated from a remarkable asymptotic theory based on the dimension and the sample size approaching to ∞ simultaneously. By conducting numerical studies, we verify that our new criteria perform well.

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We discuss an extension of Functional Principal Component Analysis (Functional PCA) to Symbolic Data Analysis (SDA).
SDA proposed by Diday is a new approach for analyzing datasets which are too large and complex to handle with conventional methods. In SDA, an observation is represented by symbolic concept including numerical, interval-valued and modal-valued data. Symbolic PCA methods have been studied as dimension reduction techniques, which are mainly applied to interval-valued data.
Another approach for a huge variety of datasets is Functional Data Analysis (FDA), developed by Ramsay. In FDA, each data is characterized by real-valued functions, rather than by a vector and/or a matrix whose components are real-values. We can analyze datasets effectively with FDA if observations are identified as discretized functions. We can apply FDA, for instance, to time series, spectrometric data, weather data, etc.
In this paper, we introduce an idea of interval-valued functional data with a pair of functions, an upper function and a lower function, and extend an FDA method to the framework of SDA. In particular, we propose an interval-valued functional PCA method based on interval-valued PCA methods. We apply our method to actual data and show its effectiveness.

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