Dunnett's procedure is a pairwise comparison procedure for comparing several treatment effects with a control in a one-way layout, under the normality assumption about the data. In such a Dunnett type multiple comparison problem, this paper supposes that we can utilize the prior information of an umbrella ordering in the treatment effects. This ordering represents a pattern of treatment effects that increase up to a certain group, which is called the peak of the umbrella, and conversely decrease beyond the group. We propose a pairwise comparison procedure using test statistics based on the difference between the maximum likelihood estimators of the treatment means under the umbrella restriction and the sample mean of the control. Critical values for the proposed procedure are determined to guarantee the type I FEW (familywise error rates) requirement. Furthermore, all-pairs powers are compared between the proposed procedure and Dunnett's procedure by means of a Monte Carlo simulation.

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For discriminating functional data, several linear methods have been proposed, such as, filtering method, regularization method and functional linear discriminant analysis. However, if the size of training data is small or if it can not be assumed that the data in each class are sampled from multivariate normal populations with equal covariance matrices, there is no guarantee that linear methods are always available. In this paper, nonlinear discriminant analysis methods are introduced, based on principal component expansion of functional data. They are called functional subspace methods, which include functional subspace method and functional CLAFIC method. These procedures do not require the knowledge on the underlying distributions of populations, but linear discriminant analysis methods do. Furthermore, our procedures can be carried out more rapidly than the traditional subspace methods, because they require only small size of training data and they can be computed with lower dimensions. In order to show that the functional subspace methods are effective, they are compared with filtering method. As a result, the functional subspace methods produce good discrimination. Particularly, when the size of training data is small they can give more stable results than the filtering method.

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Shinmura (1998) proposed Optimal Linear Discriminant Function using integer programming (IP-OLDF). Although IP-OLDF enters the black hall of explosion of calculation, it reveals some new facts about discriminant analysis by the optimal convex polyhedron and MMN. But, IP-OLDF has weakness when it is applied for the tiny data that violates Haar condition. In this paper, Revised IP-OLDF is proposed. This method can find true MMN for the data that violate Haar condition. And it can resolve several following problems. 1) it can find minimum linearly separable feature space, 2) it can obtain optimal discriminant function by one step, although IP-OLDF needs two steps, 3) it can examine evaluation data using by this optimal discriminant hyperplane. Swiss bank notes data is used as training data. Twenty thousand random numbers that have as same means and variance-covariance matrices as this data are generated for evaluation data. Revised IP-OLDF, LDF and nominal logistic model are applied for this data, and 63 discriminate functions are obtained for all combinations of independent variables. Next, these models are applied for evaluation data.

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The advancement of the computer systems enables researchers to process a huge amount of data for various fields at a reasonable cost. However, the data which should be processed and the calculation performance are also increased rapidly. The "ALL-IP concept" is the idea to connect all computer resources flatly with high performance IP network, and high performance computing system for huge data is achieved by the idea. A high performance IP switch, new storage architecture with autonomic function and grid computing technology are key technologies for the new computer systems with the "ALL-IP concept".

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