31,932 research outputs found
A Robust Statistics Approach to Minimum Variance Portfolio Optimization
We study the design of portfolios under a minimum risk criterion. The
performance of the optimized portfolio relies on the accuracy of the estimated
covariance matrix of the portfolio asset returns. For large portfolios, the
number of available market returns is often of similar order to the number of
assets, so that the sample covariance matrix performs poorly as a covariance
estimator. Additionally, financial market data often contain outliers which, if
not correctly handled, may further corrupt the covariance estimation. We
address these shortcomings by studying the performance of a hybrid covariance
matrix estimator based on Tyler's robust M-estimator and on Ledoit-Wolf's
shrinkage estimator while assuming samples with heavy-tailed distribution.
Employing recent results from random matrix theory, we develop a consistent
estimator of (a scaled version of) the realized portfolio risk, which is
minimized by optimizing online the shrinkage intensity. Our portfolio
optimization method is shown via simulations to outperform existing methods
both for synthetic and real market data
Nonlinear shrinkage estimation of large-dimensional covariance matrices
Many statistical applications require an estimate of a covariance matrix
and/or its inverse. When the matrix dimension is large compared to the sample
size, which happens frequently, the sample covariance matrix is known to
perform poorly and may suffer from ill-conditioning. There already exists an
extensive literature concerning improved estimators in such situations. In the
absence of further knowledge about the structure of the true covariance matrix,
the most successful approach so far, arguably, has been shrinkage estimation.
Shrinking the sample covariance matrix to a multiple of the identity, by taking
a weighted average of the two, turns out to be equivalent to linearly shrinking
the sample eigenvalues to their grand mean, while retaining the sample
eigenvectors. Our paper extends this approach by considering nonlinear
transformations of the sample eigenvalues. We show how to construct an
estimator that is asymptotically equivalent to an oracle estimator suggested in
previous work. As demonstrated in extensive Monte Carlo simulations, the
resulting bona fide estimator can result in sizeable improvements over the
sample covariance matrix and also over linear shrinkage.Comment: Published in at http://dx.doi.org/10.1214/12-AOS989 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Regularized Block Toeplitz Covariance Matrix Estimation via Kronecker Product Expansions
In this work we consider the estimation of spatio-temporal covariance
matrices in the low sample non-Gaussian regime. We impose covariance structure
in the form of a sum of Kronecker products decomposition (Tsiligkaridis et al.
2013, Greenewald et al. 2013) with diagonal correction (Greenewald et al.),
which we refer to as DC-KronPCA, in the estimation of multiframe covariance
matrices. This paper extends the approaches of (Tsiligkaridis et al.) in two
directions. First, we modify the diagonally corrected method of (Greenewald et
al.) to include a block Toeplitz constraint imposing temporal stationarity
structure. Second, we improve the conditioning of the estimate in the very low
sample regime by using Ledoit-Wolf type shrinkage regularization similar to
(Chen, Hero et al. 2010). For improved robustness to heavy tailed
distributions, we modify the KronPCA to incorporate robust shrinkage estimation
(Chen, Hero et al. 2011). Results of numerical simulations establish benefits
in terms of estimation MSE when compared to previous methods. Finally, we apply
our methods to a real-world network spatio-temporal anomaly detection problem
and achieve superior results.Comment: To appear at IEEE SSP 2014 4 page
OptShrink: An algorithm for improved low-rank signal matrix denoising by optimal, data-driven singular value shrinkage
The truncated singular value decomposition (SVD) of the measurement matrix is
the optimal solution to the_representation_ problem of how to best approximate
a noisy measurement matrix using a low-rank matrix. Here, we consider the
(unobservable)_denoising_ problem of how to best approximate a low-rank signal
matrix buried in noise by optimal (re)weighting of the singular vectors of the
measurement matrix. We exploit recent results from random matrix theory to
exactly characterize the large matrix limit of the optimal weighting
coefficients and show that they can be computed directly from data for a large
class of noise models that includes the i.i.d. Gaussian noise case.
Our analysis brings into sharp focus the shrinkage-and-thresholding form of
the optimal weights, the non-convex nature of the associated shrinkage function
(on the singular values) and explains why matrix regularization via singular
value thresholding with convex penalty functions (such as the nuclear norm)
will always be suboptimal. We validate our theoretical predictions with
numerical simulations, develop an implementable algorithm (OptShrink) that
realizes the predicted performance gains and show how our methods can be used
to improve estimation in the setting where the measured matrix has missing
entries.Comment: Published version. The algorithm can be downloaded from
http://www.eecs.umich.edu/~rajnrao/optshrin
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