Bartlett DecompositionEdit
Bartlett decomposition is a foundational result in multivariate statistics that provides a constructive way to generate and understand the Wishart distribution, the distribution of sample covariance matrices drawn from multivariate normal data. Named after M. S. Bartlett, the decomposition offers an explicit link between simple, independent random variables and the complex object that arises as a covariance-like matrix in higher dimensions. In practice, it yields a practical sampling scheme and a clear structural view of how a covariance structure emerges from independent components.
The Bartlett decomposition plays a central role in both theoretical work and applied statistics. It is especially useful in Monte Carlo studies, Bayesian analyses with Wishart distribution priors, and any setting where one needs to simulate or manipulate random covariance matrices. The construction makes transparent the way independence and normality at the component level translate into the dependence and symmetry observed in a sample covariance matrix.
Formal statement
Let W be a p×p random matrix with distribution Wishart_p(n, Σ), where n ≥ p and Σ is a real, symmetric positive definite p×p matrix. The matrix W can be generated in a constructive way using a lower-triangular matrix with independent entries.
Write Σ = L L^T for a Cholesky factor L (the Cholesky decomposition of Σ). Then, generate a p×p matrix A with the following independent entries:
- A_{ii} are independent and distributed as sqrt(χ^2_{n − i + 1}) for i = 1, 2, ..., p, where χ^2_k denotes the Chi-squared distribution with k degrees of freedom.
- A_{ij} are independent standard normals N(0, 1) for j < i (i.e., the strictly lower-triangular part).
- A_{ij} = 0 for j > i (i.e., upper-triangular part is zero).
Then the Wishart random matrix has the Bartlett form W = L A A^T L^T, and W ∼ Wishart distribution_p(n, Σ).
In the special case Σ = I_p (the identity matrix), the construction reduces to W = A A^T with A as above, illustrating how the Bartlett decomposition ties a high-dimensional covariance object to simpler, independent random components. This decomposition also clarifies the relationship between the eigenstructure of W and the underlying χ^2 and normal variables.
Properties and commentary
The key ingredients are independent pieces: the diagonal entries from χ^2 distributions and the below-diagonal entries from a standard normal distribution. This independence is what makes Bartlett’s decomposition particularly convenient for simulation and analytic derivations.
The decomposition highlights the Cholesky factor as a bridge from the scale structure Σ to the random fluctuations that create sample variability. Practitioners often exploit the form W = L A A^T L^T to understand how changes in Σ influence the distribution of W and to build efficient sampling schemes.
The construction is compatible with noncentral variants; variants of the method appear in discussions of noncentral Wishart distributions, where adjustments to the underlying random components reflect the shift away from centrality.
Links to broader topics include the normal distribution, the chi-squared distribution, and the matrix decomposition toolkit used in multivariate analysis, as well as the role of the covariance matrix in statistics and data analysis.
Applications
Sampling from the Wishart distribution: The Bartlett decomposition provides a straightforward algorithm for generating random covariance matrices with a specified scale Σ and degrees of freedom n, which is essential in simulations and bootstrapping of multivariate procedures.
Bayesian statistics: In Bayesian frameworks that use a Wishart prior for covariance matrices, Bartlett-type constructions facilitate posterior computations and Monte Carlo methods.
Multivariate techniques: Since the Wishart distribution governs the distribution of sample covariance matrices from multivariate normal samples, Bartlett’s decomposition underpins methods like principal component analysis and other multivariate procedures that rely on covariance estimation.
Theoretical derivations: Researchers use Bartlett’s decomposition to derive moments, density expressions, and asymptotic results for functions of Wishart matrices, including tests and estimators in high dimensions.