Directed Principal Component Analysis

We consider a problem involving estimation of a high-dimensional covariance matrix that is the sum of a diagonal matrix and a low-rank matrix, and making a decision based on the resulting estimate. Such problems arise, for example, in portfolio management, where a common approach employs principal c...

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Bibliographic Details
Published in:Operations research 2014-07, Vol.62 (4), p.957-972
Main Authors: Kao, Yi-Hao, Van Roy, Benjamin
Format: Article
Language:English
Subjects:
Analysis
Analytical estimating
convex optimization
Covariance
Covariance matrices
covariance matrix estimation
Datasets
decision analysis
Decision-making
Eigenvalues
Eigenvectors
Estimating techniques
Estimation methods
high-dimensional data
Mathematical optimization
Mathematical vectors
Matrices
METHODS
Portfolio management
principal component analysis
Principal components analysis
Robust optimization
stochastic programming
Studies
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Summary:We consider a problem involving estimation of a high-dimensional covariance matrix that is the sum of a diagonal matrix and a low-rank matrix, and making a decision based on the resulting estimate. Such problems arise, for example, in portfolio management, where a common approach employs principal component analysis (PCA) to estimate factors used in constructing the low-rank term of the covariance matrix. The decision problem is typically treated separately, with the estimated covariance matrix taken to be an input to an optimization problem. We propose directed PCA , an efficient algorithm that takes the decision objective into account when estimating the covariance matrix. Directed PCA effectively adjusts factors that would be produced by PCA so that they better guide the specific decision at hand. We demonstrate through computational studies that directed PCA yields significant benefit, and we prove theoretical results establishing that the degree of improvement over conventional PCA can be arbitrarily large.
ISSN:0030-364X
1526-5463
DOI:10.1287/opre.2014.1290