Correlation Matrix
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A Correlation Matrix is a matrix of Pearson Product-Moment Correlation Coefficients between each of the random variables in the random vector [math]\displaystyle{ \mathbf{X} }[/math].
- Example(s):
- Counter-Example(s):
- See: Covariance Matrix, Pearson Product-Moment Correlation Coefficient, Diagonal Matrix, Standardized Variable.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/covariance_matrix#Correlation_matrix Retrieved:2014-12-10.
- A quantity closely related to the covariance matrix is the correlation matrix, the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector [math]\displaystyle{ \mathbf{X} }[/math], which can be written :[math]\displaystyle{ \text{corr}(\mathbf{X}) = \left(\text{diag}(\Sigma)\right)^{-\frac{1}{2}} \, \Sigma \, \left(\text{diag}(\Sigma)\right)^{-\frac{1}{2}} }[/math]
where [math]\displaystyle{ \text{diag}(\Sigma) }[/math] is the matrix of the diagonal elements of [math]\displaystyle{ \Sigma }[/math] (i.e., a diagonal matrix of the variances of [math]\displaystyle{ X_i }[/math] for [math]\displaystyle{ i = 1, \dots, n }[/math]).
Equivalently, the correlation matrix can be seen as the covariance matrix of the standardized random variables [math]\displaystyle{ X_i/\sigma(X_i) }[/math] for [math]\displaystyle{ i = 1, \dots, n }[/math].
- A quantity closely related to the covariance matrix is the correlation matrix, the matrix of Pearson product-moment correlation coefficients between each of the random variables in the random vector [math]\displaystyle{ \mathbf{X} }[/math], which can be written :[math]\displaystyle{ \text{corr}(\mathbf{X}) = \left(\text{diag}(\Sigma)\right)^{-\frac{1}{2}} \, \Sigma \, \left(\text{diag}(\Sigma)\right)^{-\frac{1}{2}} }[/math]