Continuous measure of significant linear dimensionality of a waveform set

• Published in: Computational statistics & data analysis Vol. 35; no. 1; pp. 1 - 10
• Main Author: Shimansky, Yury P
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 28-11-2000; Elsevier Science; Elsevier
• Series: Computational Statistics & Data Analysis
• Subjects: Correlation analysis; Eigenvalue; Exact sciences and technology; Kinematics; Linear dimensionality; Mathematics; Multivariate analysis; Principal component analysis; Probability and statistics; Sciences and techniques of general use; Statistics; Waveform set; Eigenvalue; Waveform set; Correlation analysis; Linear dimensionality; Kinematics; Principal component analysis; Correlation coefficient; Waveform; Data analysis; Degree of linear interdependence; Correlation matrix; Independence; Data processing; Statistical association; Continuous function; Multivariate analysis; Continuous significant dimensionality; Continuous measure(mathematics); Linear algebra; Cross correlation; Independence test; Correlation function; Measurement method
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/S0167-9473(00)00005-0

A novel method for measuring the degree of mutual linear independence between the waveforms of a given set is introduced as a natural result of combining principal component and correlation analyses. The proposed measure,
continuous significant dimensionality (CSD), inherits its properties from both the dimensionality of a vector set as defined in linear algebra and the correlation between two waveforms. A simple formula for CSD based on the eigenvalues of the correlation matrix is suggested. It is shown that CSD is a continuous function of the cross-correlation coefficients, and it can be viewed as interpolating the dimensionality of a waveform set between cases in which the set contains an orthogonal basis. There are two important advantages of the CSD measure over the known integer estimate of significant linear dimensionality. First, the CSD formula does not depend on any subjective parameter such as “significance level”. Second, the CSD allows the detection of relatively small differences in the degree of linear independence of waveforms between different sets. A complementary measure of the
degree of linear interdependence between the waveforms of a given set also is described. The application of these measures is illustrated in the example of kinematic data processing.