Mimicking Complexity of Structured Data Matrix’s Information Content: Categorical Exploratory Data Analysis

• Main Authors: Fushing Hsieh, Elizabeth P. Chou, Ting-Li Chen
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Entropy
• Subjects: contingency-kD-lattice; high order structural dependency; heterogeneity; mutual conditional entropy matrix; principal component analysis (PCA)
• Online Access: https://www.mdpi.com/1099-4300/23/5/594

We develop Categorical Exploratory Data Analysis (CEDA) with mimicking to explore and exhibit the complexity of information content that is contained within any data matrix: categorical, discrete, or continuous. Such complexity is shown through visible and explainable serial multiscale structural dependency with heterogeneity. CEDA is developed upon all features’ categorical nature via histogram and it is guided by all features’ associative patterns (order-2 dependence) in a mutual conditional entropy matrix. Higher-order structural dependency of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>(</mo><mo>≥</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> features is exhibited through block patterns within heatmaps that are constructed by permuting contingency-<i>k</i>D-lattices of counts. By growing <i>k</i>, the resultant heatmap series contains global and large scales of structural dependency that constitute the data matrix’s information content. When involving continuous features, the principal component analysis (PCA) extracts fine-scale information content from each block in the final heatmap. Our mimicking protocol coherently simulates this heatmap series by preserving global-to-fine scales structural dependency. Upon every step of mimicking process, each accepted simulated heatmap is subject to constraints with respect to all of the reliable observed categorical patterns. For reliability and robustness in sciences, CEDA with mimicking enhances data visualization by revealing deterministic and stochastic structures within each scale-specific structural dependency. For inferences in Machine Learning (ML) and Statistics, it clarifies, upon which scales, which covariate feature-groups have major-vs.-minor predictive powers on response features. For the social justice of Artificial Intelligence (AI) products, it checks whether a data matrix incompletely prescribes the targeted system.