Difference between revisions of "Algorithms"
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==Prediction error variance sampling== | ==Prediction error variance sampling== | ||
<ref name="Harville1999" /> | Following the approach of Harville(1999)<ref name="Harville1999" /> {{lmt}} can sample for fixed factors the diagonal elements of the inverse of the mixed model coefficient matrix, for random factors the diagonal blocks of the inverse of the coefficient matrix where the block size is determined by the dimension of the related $$\Sigma$$ matrix, where the blocks are the prediction error co-variance matrices of the factor levels of correlated sub-factors. When sampling prediction error variances {{lmt}} can run many Gibbs chains in parallel allowing to exploit multi-core hardware architecture. However, it is recommended to run not more chains than the number of available '''real''' cores excluding hyper-threading technology. | ||
==REML variance component estimation== | ==REML variance component estimation== | ||
Revision as of 06:02, 5 January 2021
Solving Linear Mixed model Equations
lmt supports two types of solver for solving MME's: a direct solver and an iterative solver
Iterative solver
The iterative solver uses the preconditioned conjugate gradient method and is lmt's default solver. It does not require the explicit construction of any mixed model equation, and is therefore less resource demanding than the direct solver. That is, many models which cannot be solved using the direct solver can still be solved using the iterative solver. Even for small models the iterative solver usually outperforms the direct solver in terms of total processing time.
The iterative solver has converged to a stable solution if $$log_e\left(\sqrt{\frac{(Cx-b)'(Cx-b)}{b'b}}\right)<t$$, where $$C$$ is the mixed-model coefficient matrix, $$x$$ is the solution vector, $$b$$ is the right-hand side and $$t$$ is the convergence threshold. The default convergence threshold is -18.42, which is equivalent to $$ \sqrt{\frac{\sum_{i=1}^n ((Ax)_i-b_i)^2}{n}/\frac{\sum_{i=1}^n b_i^2}{n}}<10^{-8} $$
Direct solver
The direct solver requires the mixed model coefficient matrix to be build and all Kronecker products to be resolved. This can be quite memory demanding and should therefore be used carefully. The direct solver uses a Cholesky decomposition and forward-backward-substitution to solve the mixed model equation system, where especially the decomposition step can be very resource demanding and time consuming.
Gibbs sampling of variance components
Single pass Gibbs sampling
lmt's single pass Gibbs sampling algorithm is described in [1]. In short, all location parameters are drawn from their joint conditional posterior distribution. Note that this requires solving the mixed model equation system once per iteration which usually leads to a substantial increase in processing time.
Blocked Gibbs sampling
For random factors lmt's blocked Gibbs sampler draws correlated location parameters within factor level from their joint conditional posterior distribution. Location parameters of fixed factors are drawn in scalar mode from their fully conditional posterior.
Prediction error variance sampling
Following the approach of Harville(1999)[2] lmt can sample for fixed factors the diagonal elements of the inverse of the mixed model coefficient matrix, for random factors the diagonal blocks of the inverse of the coefficient matrix where the block size is determined by the dimension of the related $$\Sigma$$ matrix, where the blocks are the prediction error co-variance matrices of the factor levels of correlated sub-factors. When sampling prediction error variances lmt can run many Gibbs chains in parallel allowing to exploit multi-core hardware architecture. However, it is recommended to run not more chains than the number of available real cores excluding hyper-threading technology.