MVAnumber

MVAnumber is a scalar metric used in multivariate analysis (MVA) contexts to summarize the dispersion or variance structure of a multivariate dataset. It is not a fixed, universally standardized term; different authors may define and use it in varying ways depending on the application and field.

Common interpretations include: (a) the largest eigenvalue of the covariance matrix, representing the maximum variance along any unit direction; (b) the proportion of total variance captured by the first principal component, i.e., lambda1 divided by the sum of all eigenvalues; and (c) a normalized trace-based measure, such as the sum of all eigenvalues divided by the data dimensionality. Depending on the context, MVAnumber serves as a compact summary of how variance is distributed across directions.

Computation typically starts with a centered data matrix X of n observations and d variables. The sample covariance matrix S is computed, and eigen-decomposition yields eigenvalues lambda_i. The MVAnumber is then derived using the chosen definition, such as max(lambda_i), or (sum of selected lambda_i) / (sum of all lambda_i), or (sum lambda_i) / d.

Example: if lambda1 = 5, lambda2 = 2, lambda3 = 1, then MVAnumber_max = 5; MVAnumber_explained_1 = 5/8 = 0.625; MVAnumber_trace_norm = 8/3 ≈ 2.67.

Applications and notes: MVAnumber concepts appear in pattern recognition, data compression, and anomaly detection within MVA. Cautions include sensitivity to data scaling and centering, and the fact that MVAnumber definitions vary across literature, so the specific convention should be clarified in use.