Correlation
Definition taught in Covariance and the multivariate normal, part of Statistical learning
Correlation
| Type | Definition |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
| Questions using it | 21 |
With positive finite variances, the Pearson correlation is the scale-free covariance
\rho_{jk}=\operatorname{corr}(X_j,X_k)=\frac{\operatorname{Cov}(X_j,X_k)}{\sqrt{\operatorname{Var}(X_j)\operatorname{Var}(X_k)}}=\frac{\sigma_{jk}}{\sigma_j\sigma_k}\in[-1,1].
The correlation matrix is P=(\rho_{jk}), with ones on the diagonal. If D=\operatorname{diag}(\sigma_1,\ldots,\sigma_p), then
P=D^{-1}\Sigma D^{-1}, \qquad \Sigma=DPD.
Used in
- Taught in Covariance and the multivariate normal, where the derivation and examples are.
- Also used in Joint distributions · Functions of random variables and limit theorems · Estimation · Linear regression · Descriptive statistics · Markov chains, Gibbs sampling, and Metropolis–Hastings · Categorical responses · Estimation and inference for the multivariate normal · Principal component analysis · Nearest neighbours · Ensembles, random forests, and boosting · High-dimensional analysis · Graphical models · Random intercepts · Random slopes, and a binary mixed model · Marginal models, and two responses at once · Forecasting
- Questions that use it (21) — Reading a correlation · A bivariate-normal Gibbs sampler · Why a covariance is positive semi-definite · The effect of correlation · LDA and principal components (answer) · Normalisation and nearest neighbours (answer) · Variance of an average · A tempting wrong turn (answer) · Selection in many dimensions (answer) · Precision to graph (answer) · A tempting wrong turn (answer) · Read a Gaussian graphical model · Conditional association and sparsity · Q9 (answer) · Q33 (answer) · Question 20 (answer) · Question 22 (answer) · C36 (answer) · C39 (answer) · C45 (answer) · C49 (answer)
- All 8 questions on Covariance and the multivariate normal
See also
Sample correlation coefficient · Partial correlation · Variance · Mean vector and covariance matrix · Autocorrelation function · Intraclass correlation · Partial autocorrelation · Scaling