Sparse multivariate regression with missing values and its application to the prediction of material properties

In the field of materials science and engineering, statistical analysis and machine learning techniques have recently been used to predict multiple material properties from an experimental design. These material properties correspond to response variables in the multivariate regression model. This study conducts a penalized maximum likelihood procedure to estimate model parameters, including the regression coefficients and covariance matrix of response variables. In particular, we employ $l_1$-regularization to achieve a sparse estimation of regression coefficients and the inverse covariance matrix of response variables. In some cases, there may be a relatively large number of missing values in response variables, owing to the difficulty in collecting data on material properties. A method to improve prediction accuracy under the situation with missing values incorporates a correlation structure among the response variables into the statistical model. The expectation and maximization algorithm is constructed, which enables application to a data set with missing values in the responses. We apply our proposed procedure to real data consisting of 22 material properties.Comment: 18 page

Robust relative error estimation

Relative error estimation has been recently used in regression analysis. A crucial issue of the existing relative error estimation procedures is that they are sensitive to outliers. To address this issue, we employ the $\gamma$-likelihood function, which is constructed through $\gamma$-cross entropy with keeping the original statistical model in use. The estimating equation has a redescending property, a desirable property in robust statistics, for a broad class of noise distributions. To find a minimizer of the negative $\gamma$-likelihood function, a majorize-minimization (MM) algorithm is constructed. The proposed algorithm is guaranteed to decrease the negative $\gamma$-likelihood function at each iteration. We also derive asymptotic normality of the corresponding estimator together with a simple consistent estimator of the asymptotic covariance matrix, so that we can readily construct approximate confidence sets. Monte Carlo simulation is conducted to investigate the effectiveness of the proposed procedure. Real data analysis illustrates the usefulness of our proposed procedure.Comment: 34 pages, 6 figure