Bootstraps Regularize Singular Correlation Matrices

Christian Bongiorno (1)

1 MICS - Mathématiques et Informatique pour la Complexité et les Systèmes

I show analytically that the average of $k$ bootstrapped correlation matrices rapidly becomes positive-definite as $k$ increases, which provides a simple approach to regularize singular Pearson correlation matrices. If $n$ is the number of objects and $t$ the number of features, the averaged correlation matrix is almost surely positive-definite if $k> \frac{e}{e-1}\frac{n}{t}\simeq 1.58\frac{n}{t}$ in the limit of large $t$ and $n$. The probability of obtaining a positive-definite correlation matrix with $k$ bootstraps is also derived for finite $n$ and $t$. Finally, I demonstrate that the number of required bootstraps is always smaller than $n$. This method is particularly relevant in fields where $n$ is orders of magnitude larger than the size of data points $t$, e.g., in finance, genetics, social science, or image processing.

Domaines: Applications [stat.AP]; Génétique humaine; Analyse de données, Statistiques et Probabilités [physics.data-an]; Informatique; Finance quantitative [q-fin.CP]

Christian Bongiorno. Bootstraps Regularize Singular Correlation Matrices. 2020. ⟨hal-02536278⟩