  The restricted maximum likelihood method enhances popularity of maximum
likelihood methods for variance component analysis on large scale unbalanced
data. As the high throughput biological data sets and the emerged science on
uncertainty quantification, such a method receives increasing attention.
Estimating the unknown variance parameters with restricted maximum likelihood
method usually requires an nonlinear iterative method. Therefore proper
formulae for the log-likelihood function and its derivatives play an essential
role in practical algorithm design. It is our aim to provide a mathematical
introduction to this method, and supply a self-contained derivation on some
available formulae used in practical algorithms. Some new proof are supplied.
