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Principal component error in high-dimensional factor models

Alex Bernstein, Lisa R. Goldberg, Nicholas Gunther, Alec N. Kercheval, Tian Lan, Yian Lin, 姚大义 · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.20550 · cited by 0 works (OpenAlex) · open-access PDF

Korean

통계적 요소 모델에서, 샘플 코바리언스 매트릭스의 주요 구성 요소 (또는 eigenvectors)는 주 방향의 추정으로, 관찰 변수의 집합의 공동 움직임의 진정한 드라이버로 봉사합니다. 우리는 이러한 추정에서 종종 상당한 오류를 두 개의 해석 가능한 용어의 합으로 작성합니다. 우리는 샘플 크기에 따라 변수의 수가 증가함에 따라 거의 확실한 비합리적인 한계를 보여줍니다. 이 시나리오는 금융 경제, 게놈학, 기계 학습 및 신호 처리에서 일반적입니다. 아웃 스페이스 오류는 추정에서 인구 요소 노출에 의해 확장 된 하위 공간까지의 거리를 측정합니다. 그것은 데이터 측면에서 표현 될 수 있으며, 샘플 크

Machine translation (Meta M2M100); the English original follows.

Abstract

In a statistical factor model, principal components (or eigenvectors) of a sample covariance matrix serve as estimates of principal directions, the true drivers of co-movement of a collection of observed variables. We write the often substantial error in these estimates as a sum of two interpretable terms, which we show have almost sure asymptotic limits as the number of variables grows with sample size bounded. This scenario is commonplace in financial economics, genomics, machine learning and signal processing. Out-of-subspace error measures the distance from an estimate to the subspace spanned by population factor exposures. It can be expressed in terms of data, providing an estimable floor for asymptotic error. In-subspace error is driven by a rotation error within that subspace, which arises from the fixed sample size of the latent factor returns and cannot be estimated from data alone. We illustrate our error analysis with a three-factor simulation of the US public equity market, showing the dependence of the magnitude of the error and its components on dimension and sample size. In that simulation, out-of-subspace error dominates. Researchers who rely on principal component analysis to estimate factor models can use our results to quantify errors in model-based predictions and attributions.

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Topics: Principal component analysis, Factor analysis, Mathematics, Statistics, Covariance matrix, Dimension (graph theory)

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