# Multivariate Statistical Analysis

## A High-Dimensional Approach

• V. Serdobolskii

Part of the Theory and Decision Library book series (TDLB, volume 41)

1. Front Matter
Pages i-xii
2. V. Serdobolskii
Pages 1-24
3. V. Serdobolskii
Pages 25-39
4. V. Serdobolskii
Pages 40-60
5. V. Serdobolskii
Pages 61-75
6. V. Serdobolskii
Pages 76-86
7. V. Serdobolskii
Pages 87-101
8. V. Serdobolskii
Pages 102-111
9. V. Serdobolskii
Pages 112-130
10. V. Serdobolskii
Pages 131-155
11. V. Serdobolskii
Pages 156-168
12. V. Serdobolskii
Pages 169-186
13. V. Serdobolskii
Pages 187-226
14. V. Serdobolskii
Pages 227-232
15. Back Matter
Pages 233-244

### Introduction

In the last few decades the accumulation of large amounts of in­ formation in numerous applications. has stimtllated an increased in­ terest in multivariate analysis. Computer technologies allow one to use multi-dimensional and multi-parametric models successfully. At the same time, an interest arose in statistical analysis with a de­ ficiency of sample data. Nevertheless, it is difficult to describe the recent state of affairs in applied multivariate methods as satisfactory. Unimprovable (dominating) statistical procedures are still unknown except for a few specific cases. The simplest problem of estimat­ ing the mean vector with minimum quadratic risk is unsolved, even for normal distributions. Commonly used standard linear multivari­ ate procedures based on the inversion of sample covariance matrices can lead to unstable results or provide no solution in dependence of data. Programs included in standard statistical packages cannot process 'multi-collinear data' and there are no theoretical recommen­ dations except to ignore a part of the data. The probability of data degeneration increases with the dimension n, and for n > N, where N is the sample size, the sample covariance matrix has no inverse. Thus nearly all conventional linear methods of multivariate statis­ tics prove to be unreliable or even not applicable to high-dimensional data.

### Keywords

Estimator Linear discriminant analysis Radiologieinformationssystem Regression analysis Variance calculus econometrics learning linear regression machine learning mathematical statistics quality reliability statistical analysis statistics

#### Authors and affiliations

• V. Serdobolskii
• 1
1. 1.Department of Applied MathematicsMoscow Institute of Electronics and MathematicsMoscowRussia

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-9468-4