  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.
