HMMEsolver (version 0.1.2)

SolveHMME: Solve Henderson's Mixed Model Equation.

Description

Consider a linear mixed model with normal random effects, $$Y_{ij} = X_{ij}^T\beta + v_i + \epsilon_{ij}$$ where \(i=1,\ldots,n,\quad j=1,\ldots,m\), or it can be equivalently expressed using matrix notation, $$Y = X\beta + Zv + \epsilon$$ where \(Y\in \mathrm{R}^{nm}\) is a known vector of observations, \(X \in \mathrm{R}^{nm\times p}\) and \(Z \in \mathrm{R}^{nm\times n} \) design matrices for \(\beta\) and \(v\) respectively, \(\beta \in \mathrm{R}^p\) and \(v\in \mathrm{R}^n\) unknown vectors of fixed effects and random effects where \(v_i \sim N(0,\lambda_i)\), and \(\epsilon \in \mathrm{R}^{nm}\) an unknown vector random errors independent of random effects. Note that \(Z\) does not need to be provided by a user since it is automatically created accordingly to the problem specification.

Usage

SolveHMME(X, Y, Mu, Lambda)

Arguments

X

an \((nm\times p)\) design matrix for \(\beta\).

Y

a length-\(nm\) vector of observations.

Mu

a length-\(nm\) vector of initial values for \(\mu_i = E(Y_i)\).

Lambda

a length-\(n\) vector of initial values for \(\lambda\), variance of \(v_i \sim N(0,\lambda_i)\)

Value

a named list containing

beta

a length-\(p\) vector of BLUE \(\hat{beta}\).

v

a length-\(n\) vector of BLUP \(\hat{v}\).

leverage

a length-\((mn+n)\) vector of leverages.

References

henderson_estimation_1959HMMEsolver

robinson_that_1991HMMEsolver

mclean_unified_1991HMMEsolver

kim_fast_2017HMMEsolver

Examples

# NOT RUN {
## small setting for data generation
n = 100; m = 2; p = 2
nm = n*m;   nmp = n*m*p

## generate artifical data
X = matrix(rnorm(nmp, 2,1), nm,p) # design matrix
Y = rnorm(nm, 2,1)                # observation

Mu = rep(1, times=nm)
Lambda = rep(1, times=n)

## solve
ans = SolveHMME(X, Y, Mu, Lambda)


# }