A novel scale-space approach for multinormality testing and the k-sample problem in the high dimension low sample...

Area of Science:

• Multivariate Statistics
• Statistical Computing
• Data Analysis

Background:

• Classical multivariate statistical problems like testing normality and the k-sample problem are challenging, especially in high-dimensional, low-sample-size (HDLSS) scenarios.
• Existing methods often rely on inverting covariance matrices, limiting their applicability when the number of variables (p) exceeds the sample size (n).

Purpose of the Study:

• To develop and present a novel multivariate statistical analysis method that addresses the challenges of the High Dimension Low Sample Size (HDLSS) situation.
• To apply this method to two classical problems: testing multivariate normality and the k-sample problem.
• To demonstrate the method's utility through significance maps and feature selection.

Main Methods:

• A novel analysis approach is employed, simultaneously considering multiple resolutions without inverting estimated covariance matrices.
• The method generates a significance map by performing one-dimensional tests across all resolution/position pairs.
• For multivariate normality testing, the Anderson-Darling test is adapted. For the k-sample problem, the k-sample Anderson-Darling test is used on vectors from different datasets at corresponding resolution/position pairs.

Main Results:

• The developed methodology effectively handles the High Dimension Low Sample Size (HDLSS) situation where n ≤ p.
• Significance maps are generated, visually indicating where null hypotheses are rejected across different resolution and position combinations.
• Successful demonstrations on both artificial and real datasets confirm the method's practical applicability and effectiveness.

Conclusions:

• The novel methodology provides a robust approach for tackling multivariate statistical problems in HDLSS settings.
• The significance map output offers intuitive insights into data structure and deviations from null hypotheses.
• The demonstrated feature selection scheme highlights the method's potential for further data exploration and analysis.