Covariance Matrix Delta Method F1 Variance Method

A Covariance Matrix Delta Method F1 Variance Method is a multivariate variance estimation method that uses full covariance matrix Cov(X) with gradient vectors to compute F1 score variance accounting for count correlations.

• AKA: Full Covariance F1 Variance Method, Multivariate Delta Method F1 SE, Correlated Counts F1 Variance Method, Matrix-Based F1 SE Method.
• Context:
  • It can typically compute Var(F1) = ∇G(μ)ᵀ Cov(X) ∇G(μ) using full covariance structure.
  • It can typically model off-diagonal terms Cov(TP,FP), Cov(TP,FN), Cov(FP,FN) explicitly.
  • It can typically capture negative correlations between counts (e.g., TP vs FN trade-off).
  • It can often use multinomial covariance structure for fixed sample size n.
  • It can often provide more accurate variance than diagonal covariance assumptions.
  • It can often handle stratified sampling and clustered observations.
  • It can range from being a Multinomial Covariance Matrix Delta Method F1 Variance Method to being a Empirical Covariance Matrix Delta Method F1 Variance Method, depending on its covariance source.
  • It can range from being a Full Covariance Matrix Delta Method F1 Variance Method to being a Block-Diagonal Covariance Matrix Delta Method F1 Variance Method, depending on its matrix structure.
  • It can range from being a Parametric Covariance Matrix Delta Method F1 Variance Method to being a Non-Parametric Covariance Matrix Delta Method F1 Variance Method, depending on its distribution assumption.
  • It can range from being a Fixed-n Covariance Matrix Delta Method F1 Variance Method to being a Variable-n Covariance Matrix Delta Method F1 Variance Method, depending on its sample size model.
  • ...
• Example(s):
  • Multinomial Covariance Structures, such as:
    • n=100, 4 outcomes: Cov(TP,FP) = -n·p(TP)·p(FP) = -100·0.6·0.1 = -6.
    • Diagonal assumption SE=0.04 vs Full covariance SE=0.035 (12.5% difference).
    • Negative covariances reduce overall variance estimate.
  • Multi-Class Covariances, such as:
    • 3-class problem: 9×9 covariance matrix for all count pairs.
    • Block structure: within-class and between-class covariances.
    • Macro-F1 variance using full 9×9 matrix multiplication.
  • Clustered Data Applications, such as:
    • Hospital clusters: within-cluster correlation inflates variance.
    • Cov matrix incorporates intraclass correlation coefficient.
    • Effective sample size reduction from clustering.
  • ...
• Counter-Example(s):
  • Independent Counts Assumption Method, which uses diagonal covariance.
  • Poisson Approximation for Count Variance Method, which assumes Var=Mean.
  • Bootstrap Covariance Estimation, which estimates empirically.
• See: Variance Estimation Method, Covariance Matrix, Delta Method, Multivariate Statistics, Delta-Method F1 Standard Error Estimation Method, Multinomial Distribution, Gradient Vector, Matrix Multiplication, Correlation Structure, Clustered Data Analysis, Variance-Covariance Matrix.

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