65 research outputs found
Covariance Estimation: The GLM and Regularization Perspectives
Finding an unconstrained and statistically interpretable reparameterization
of a covariance matrix is still an open problem in statistics. Its solution is
of central importance in covariance estimation, particularly in the recent
high-dimensional data environment where enforcing the positive-definiteness
constraint could be computationally expensive. We provide a survey of the
progress made in modeling covariance matrices from two relatively complementary
perspectives: (1) generalized linear models (GLM) or parsimony and use of
covariates in low dimensions, and (2) regularization or sparsity for
high-dimensional data. An emerging, unifying and powerful trend in both
perspectives is that of reducing a covariance estimation problem to that of
estimating a sequence of regression problems. We point out several instances of
the regression-based formulation. A notable case is in sparse estimation of a
precision matrix or a Gaussian graphical model leading to the fast graphical
LASSO algorithm. Some advantages and limitations of the regression-based
Cholesky decomposition relative to the classical spectral (eigenvalue) and
variance-correlation decompositions are highlighted. The former provides an
unconstrained and statistically interpretable reparameterization, and
guarantees the positive-definiteness of the estimated covariance matrix. It
reduces the unintuitive task of covariance estimation to that of modeling a
sequence of regressions at the cost of imposing an a priori order among the
variables. Elementwise regularization of the sample covariance matrix such as
banding, tapering and thresholding has desirable asymptotic properties and the
sparse estimated covariance matrix is positive definite with probability
tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Computation of canonical correlation and best predictable aspect of future for time series
The canonical correlation between the (infinite) past and future of a stationary time series is shown to be the limit of the canonical correlation between the (infinite) past and (finite) future, and computation of the latter is reduced to a (generalized) eigenvalue problem involving (finite) matrices. This provides a convenient and essentially, finite-dimensional algorithm for computing canonical correlations and components of a time series. An upper bound is conjectured for the largest canonical correlation
Applications of a finite-dimensional duality principle to some prediction problems
Some of the most important results in prediction theory and time series
analysis when finitely many values are removed from or added to its infinite
past have been obtained using difficult and diverse techniques ranging from
duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear
regression in statistics (Box and Tiao, 1975). We unify these results via a
finite-dimensional duality lemma and elementary ideas from the linear algebra.
The approach reveals the inherent finite-dimensional character of many
difficult prediction problems, the role of duality and biorthogonality for a
finite set of random variables. The lemma is particularly useful when the
number of missing values is small, like one or two, as in the case of
Kolmogorov and Nakazi prediction problems. The stationarity of the underlying
process is not a requirement. It opens up the possibility of extending such
results to nonstationary processes.Comment: 15 page
Regularized Multivariate Regression Models with Skew-\u3cem\u3et\u3c/em\u3e Error Distributions
We consider regularization of the parameters in multivariate linear regression models with the errors having a multivariate skew-t distribution. An iterative penalized likelihood procedure is proposed for constructing sparse estimators of both the regression coefficient and inverse scale matrices simultaneously. The sparsity is introduced through penalizing the negative log-likelihood by adding L1-penalties on the entries of the two matrices. Taking advantage of the hierarchical representation of skew-t distributions, and using the expectation conditional maximization (ECM) algorithm, we reduce the problem to penalized normal likelihood and develop a procedure to minimize the ensuing objective function. Using a simulation study the performance of the method is assessed, and the methodology is illustrated using a real data set with a 24-dimensional response vector
The intersection of past and future for multivariate stationary processes
We consider an intersection of past and future property of multivariate
stationary processes which is the key to deriving various representation
theorems for their linear predictor coefficient matrices. We extend useful
spectral characterizations for this property from univariate processes to
multivariate processes.Comment: 8 page
Robust Estimation of the Correlation Matrix of Longitudinal Data
We propose a double-robust procedure for modeling the correlation matrix of a longitudinal dataset. It is based on an alternative Cholesky decomposition of the form Σ=DLL ⊤ D where D is a diagonal matrix proportional to the square roots of the diagonal entries of Σ and L is a unit lower-triangular matrix determining solely the correlation matrix. The first robustness is with respect to model misspecification for the innovation variances in D, and the second is robustness to outliers in the data. The latter is handled using heavy-tailed multivariate t-distributions with unknown degrees of freedom. We develop a Fisher scoring algorithm for computing the maximum likelihood estimator of the parameters when the nonredundant and unconstrained entries of (L,D) are modeled parsimoniously using covariates. We compare our results with those based on the modified Cholesky decomposition of the form LD 2 L ⊤ using simulations and a real dataset
MLE of Jointly Constrained Mean-Covariance of Multivariate Normal Distributions
Estimating the unconstrained mean and covariance matrix is a popular topic in
statistics. However, estimation of the parameters of under
joint constraints such as has not received much attention. It
can be viewed as a multivariate counterpart of the classical estimation problem
in the distribution. In addition to the usual inference
challenges under such non-linear constraints among the parameters (curved
exponential family), one has to deal with the basic requirements of symmetry
and positive definiteness when estimating a covariance matrix. We derive the
non-linear likelihood equations for the constrained maximum likelihood
estimator of and solve them using iterative methods. Generally,
the MLE of covariance matrices computed using iterative methods do not satisfy
the constraints. We propose a novel algorithm to modify such (infeasible)
estimators or any other (reasonable) estimator. The key step is to re-align the
mean vector along the eigenvectors of the covariance matrix using the idea of
regression. In using the Lagrangian function for constrained MLE (Aitchison et
al. 1958), the Lagrange multiplier entangles with the parameters of interest
and presents another computational challenge. We handle this by either
iterative or explicit calculation of the Lagrange multiplier. The existence and
nature of location of the constrained MLE are explored within a data-dependent
convex set using recent results from random matrix theory. A simulation study
illustrates our methodology and shows that the modified estimators perform
better than the initial estimators from the iterative methods
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