Mean vector and covariance matrix
Definition taught in Covariance and the multivariate normal, part of Statistical learning
Mean vector and covariance matrix
| Type | Definition |
| Taught in | Covariance and the multivariate normal |
| Part | Statistical learning |
| Questions using it | 39 |
The expected value of \mathbf{X} is the vector of coordinate expectations,
\mathbb{E}(\mathbf{X})=\boldsymbol{\mu}=\bigl(\mathbb{E} X_1,\ldots,\mathbb{E} X_p\bigr)^\top.
The covariance of coordinates j and k is
\sigma_{jk}=\operatorname{Cov}(X_j,X_k)=\mathbb{E}\bigl[(X_j-\mathbb{E} X_j)(X_k-\mathbb{E} X_k)\bigr]=\mathbb{E}(X_j X_k)-\mathbb{E}(X_j)\mathbb{E}(X_k).
The covariance matrix is the p\times p matrix of these entries,
\Sigma=\operatorname{Var}(\mathbf{X})=\operatorname{Cov}(\mathbf{X},\mathbf{X}^\top)=\mathbb{E}\bigl[(\mathbf{X}-\boldsymbol{\mu})(\mathbf{X}-\boldsymbol{\mu})^\top\bigr].
The diagonal entry \sigma_{jj}=\sigma_j^2 is the variance of X_j, and \sigma_j is its standard deviation.
Used in
- Taught in Covariance and the multivariate normal, where the derivation and examples are.
- Also used in Estimation and inference for the multivariate normal · Principal component analysis · Mixtures and the EM algorithm · Model selection for stochastic models · Hierarchical clustering and kernel density estimation · Statistical classifiers and discriminant analysis · Logistic regression · Ensembles, random forests, and boosting · High-dimensional analysis · Graphical models
- Questions that use it (39) — Variances of linear combinations · Uncorrelated but dependent (answer) · Normal marginals without joint normality (answer) · The bivariate conditional (answer) · A Schur-complement check (answer) · Conditional normal prediction · Where the n-1 comes from · Pooling two groups · A small T^2 test (answer) · A two-sample statistic (answer) · Whitening and its limits · Normal conditioning and whitening · A transformation is not a distributional assumption · A two-variable analysis (answer) · Properties of the components · Principal components through the singular values · How many components · A low-variance predictor (answer) · Principal components as a modelling step (answer) · Euclidean clusters and mixture components · Counting parameters · Label switching (answer) · Mixtures versus K-means · Mixture selection and uncertainty (answer) · Parameter counts · Fisher’s direction (answer) · The fitted LDA model (answer) · LDA and principal components (answer) · Linear discriminant analysis implies a logistic model · A logistic rule in an unfamiliar setting (answer) · Variance of an average (answer) · Rank · When ordinary linear discrimination is not defined · Principal components before linear discrimination · Precision to graph · Conditional independence from a partitioned covariance · How many graphs? (answer) · A tempting wrong turn (answer) · Read a Gaussian graphical model
- All 8 questions on Covariance and the multivariate normal
See also
Expectation vector and covariance matrix · Covariance · Expectation · Variance · Behrens–Fisher problem · Correlation · Eigenvalues and eigenvectors · Factor analysis