## 第8回 統計的機械学習セミナー (2012.7.12)

第8回 統計的機械学習セミナー／The 8th Statistical Machine Learning Seminar

（主催：統計数理研究所 統計的機械学習NOE・統計的機械学習研究センター)

**日時** 2012年7月12日（木） 13:30-17:45

**会場** 統計数理研究所 セミナー室5（3階D313）

* Arthur Gretton (Gatsby Computational Neuroscience Unit, University College

London, UK)

Title:

**Consistent Nonparametric Tests of Independence: L1, Log-Likelihood and Kernel**

Abstract:

Three simple and explicit procedures for testing the independence of two

multi-dimensional random variables are described. Two of the associated test

statistics (L1, log-likelihood) are defined when the empirical distribution

of the variables is restricted to finite partitions. A third test statistic

is defined as a kernel-based independence measure. Two kinds of tests are

provided.

Distribution-free strong consistent tests are derived on the basis of large

deviation bounds on the test statistics: these tests make almost surely no

Type I or Type II error after a random sample size.

Asymptotically alpha-level tests are obtained from the limiting distribution

of the test statistics. For the latter tests, the Type I error converges to

a fixed non-zero value alpha, and the Type II error drops to zero, for

increasing sample size. All tests reject the null hypothesis of independence

if the test statistics become large. The performance of the tests is

evaluated experimentally on benchmark data.

–

* Subhajit Dutta (Indian Statistical Institute, India)

Title :

**Classification using Localized Spatial Depth with Multiple Localization**

Abstract :

In the recent past, data depth has been considered by several authors as an

effective methodology for supervised and unsupervised classification

problems. However, most of the depth based classifiers studied in the

literature require the population distributions to be elliptic and unimodal

differing only in their locations in order to have satisfactory performance.

Another limitation of such classifiers is that they usually require equal

prior probabilities for the populations. Further, for many choices of the

well-known depth function, practical implementation of depth based

classifiers becomes computationally prohibitive even for moderately large

dimensional data. In this talk, we propose a new classifier based on spatial

depth, which can be used for high-dimensional. The main idea behind the

construction of the proposed classifier is based on fitting generalized

additive models to the posterior probabilities corresponding to different

classes. In order to cope with possible multimodal and/or non-elliptic

nature of the population distributions, we develop a localized version of

spatial depth and use that with varying degrees of localization to build the

classifier. Our classifier is formed by aggregation of several classifiers

each of which is based on spatial depth with a fixed level of localization.

This new classifier can be conveniently used for high-dimensional data, and

its possess good discriminatory power for such data. Using some real

benchmark data sets, the proposed classifier is shown to have competitive

performance when compared with well-known and widely used classifiers like

those based on nearest-neighbors, kernel density estimates, support vector

machines, classification trees, artificial neural nets, etc.

(This is a joint work with Prof. P. Chaudhuri and Dr. A. K. Ghosh)

–

* Su-Yun Huang (Institute of Statistical Science, Academia Sinica, Taiwan)

Titile:

**Multilinear Principal Component Analysis -Asymptotic Theory**

Abstract:

Principal component analysis is commonly used for dimension reduction in

analyzing high dimensional data. Multilinear principal component analysis

aims to serve a similar function for analyzing tensor structure data, and

has empirically been shown effective in reducing dimensionality. In this

paper, we investigate its statistical properties and demonstrate its

advantages. Conventional principal component analysis, which vectorizes the

tensor data, may lead to inefficient and unstable prediction due to the

often extremely large dimensionality involved. Multilinear principal

component analysis, in trying to preserve the data structure, searches for

low-dimensional projections and, thereby, decreases dimensionality more

efficiently.

Asymptotic theory of order-two multilinear principal component analysis,

including asymptotic efficiency and distributions of principal components,

associated projections, and the explained variance, is developed. A test of

dimensionality is also proposed.

Finally, multilinear principal component analysis is shown to improve

conventional principal component analysis in analyzing the Olivetti faces

data set, which is achieved by extracting a more modularly-oriented basis in

reconstructing test faces.

(joint with Hung Hung, Pei-Shien Wu and I-Ping Tu)

–

* Hung Hung (Institute of Epidemiology and Preventive Medicine, National

Taiwan University, Taiwan)

Title:

**Matrix Variate Logistic Regression Model with Application to EEG Data**

Abstract:

Logistic regression has been widely applied in the field of biomedical

research for a long time. In some applications, covariates of interest have

a natural structure, such as being a matrix, at the time of collection. The

rows and columns of the covariate matrix then have certain physical

meanings, and they must contain useful information regarding the response.

If we simply stack the covariate matrix as a vector and fit a conventional

logistic regression model, relevant information can be lost, and the problem

of inefficiency will arise. Motivated from these reasons, we propose in this

paper the matrix variate logistic (MV-logistic) regression model. Advantages

of MV-logistic regression model include the preservation of the inherent

matrix structure of covariates and the parsimony of parameters needed. In

the EEG Database Data Set, we successfully extract the structural effects of

covariate matrix, and a high classification accuracy is achieved. (Joint

with Chen-Chien Wang) MV-logistic regression belongs to the class of tensor

regression, which now attracts the attention of statisticians. In this talk,

I will also introduce some our recent developments of tensor regression, and

will focus on its application to the detection of gene-gene interactions.