DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Covariance factor model with diagonal residual covariance

Definition

Let $p, q$ be positive integers with $q < p$ , let $(Ω, F, P)$ be a probability space, and let $S : Ω \to R^{q}$ and $ε : Ω \to R^{p}$ be measurable random vectors with finite second moments. For finite-second-moment random vectors $U, V$ , define

$Cov (U, V) = E [(U - E U) (V - E V)^{T}], Cov (U) = Cov (U, U),$

where expectation is integration with respect to $P$ . A covariance factor model is a representation

$X = A S + ε$

with a fixed matrix $A \in R^{p \times q}$ and assumptions

$E S = 0, E ε = 0, Cov (S) = I_{q}, Cov (ε) = D = diag (d_{1}, \dots, d_{p}), d_{i} > 0, Cov (S, ε) = 0.$

The model assumes only the stated moment and cross-covariance conditions; it does not assume that $S$ and $ε$ are independent.

Definition references

Hastie, Tibshirani, and Friedman, §14.7.1

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Trevor Hastie, Robert Tibshirani, and Jerome Friedman, The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)