Statistical Distance Matrix Calculation for Local Feature Analysis
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Solution Overview
Problem
Conventional statistical distances, such as Mahalanobis Distance, Bhattacharya distance, Hellinger Distance, Kullback-Leibler Divergence, and Chernoff Distance, provide only scalar outputs for global distances between observation sets, failing to elaborate on local distances between feature elements, and existing methods do not convert these scalar values into matrix forms effectively for machine learning applications.
Innovation Solution
A method to convert scalar-valued statistical distances into statistical distance matrices through a de-trace operation, calculating mean vectors and covariance matrices to derive trace forms of these distances, enabling the creation of distance accumulation matrices and facilitating element clustering.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Device complexity
If scalar-valued statistical distances are used to measure global distance between observation sets, then computational simplicity is maintained, but local distances between feature elements cannot be elaborated
Solution Approach 1:
The patent segments the scalar statistical distance into multiple matrix elements, where each element represents the distance between specific feature elements. This is achieved by computing the statistical distance matrix D where D[i,j] contains the statistical distance between feature element i and feature element j, thereby elaborating local distances while maintaining the mathematical properties of statistical distances
Solution Approach 2:
The patent transitions from a scalar-valued statistical distance (0-dimensional) to a matrix-valued statistical distance (2-dimensional), where the matrix dimensions correspond to the number of feature elements. This dimensional expansion enables the representation of local distances between all pairs of feature elements while preserving the global distance information through the trace of the matrix
2Ease of manufacture
If correlation matrices are used to quantify distance matrices, then information loss occurs from assuming all data follow a probability space
Solution Approach 1:
The patent creates a statistical distance matrix that copies and preserves the probability distribution information from the original statistical distance calculations. Instead of transforming the data into correlation coefficients that lose probability information, the method maintains the original statistical distance values in matrix form, thereby avoiding information loss while still enabling quantitative analysis
Solution Approach 2:
The patent changes the parameter representation from scalar statistical distance to matrix-valued statistical distance, where each element D[i,j] contains the statistical distance between specific feature elements. This parameter transformation allows the preservation of probability distribution information while enabling detailed local distance analysis between feature elements
Data Source
AI summary
The purpose of the present invention is to convert statistical distances into a statistical distance matrix in order to refine the distances. A processing unit calculates first and second mean vectors μμ1 and μμ2, and first and second covariance matrices ΣΣ1 and ΣΣ2 as targets for comparison from first and second vector data aa and bb, or from first and second matrices or sets AA and BB to find a statistical distance matrix DD as a de-trace form of a statistical distance D defined by them. The processing unit accumulates local distances related to the statistical distance matrix DD to find a distance accumulation vector φφ and matrix the distance accumulation vector φφ into a distance accumulation matrix ϕϕ. The processing unit stores, in a storage unit, or displays, on a display unit, or outputs, through an output unit, the distance accumulation matrix ϕϕ or an image by the distance accumulation matrix ϕϕ. The processing unit can execute clustering processing on the statistical distance matrix DD to further assign cluster labels or element labels to the distance accumulation matrix ϕϕ.