cov_shrink_diag {sparsediscrim} | R Documentation |
Computes a shrunken version of the maximum likelihood estimator for the sample covariance matrix under the assumption of multivariate normality.
Description
For a sample matrix, x
, we compute the sample covariance matrix as the
maximum likelihood estimator (MLE) of the population covariance matrix and
shrink it towards its diagonal.
Usage
cov_shrink_diag(x, gamma = 1)
Arguments
x |
data matrix with |
gamma |
the shrinkage parameter. Must be between 0 and 1, inclusively. By default, the shrinkage parameter is 1, which simply yields the MLE. |
Details
Let \widehat{\Sigma}
be the MLE of the covariance matrix \Sigma
.
Then, we shrink the MLE towards its diagonal by computing
\widehat{\Sigma}(\gamma) = \gamma \widehat{\Sigma} + (1 - \gamma)
\widehat{\Sigma} \circ I_p,
where \circ
denotes the Hadamard product
and \gamma \in [0,1]
.
For \gamma < 1
, the resulting shrunken covariance matrix estimator is
positive definite, and for \gamma = 1
, we simply have the MLE, which can
potentially be positive semidefinite (singular).
The estimator given here is based on Section 18.3.1 of the Hastie et al. (2008) text.
Value
shrunken sample covariance matrix of size p \times p
References
Hastie, T., Tibshirani, R., and Friedman, J. (2008), "The Elements of Statistical Learning: Data Mining, Inference, and Prediction," 2nd edition. http://web.stanford.edu/~hastie/ElemStatLearn/