penalty {LaMa} | R Documentation |
Computes penalty based on quadratic form
Description
This function computes quadratic penalties of the form
0.5 \sum_{i} \lambda_i b_i^T S_i b_i,
with smoothing parameters \lambda_i
, coefficient vectors b_i
, and fixed penalty matrices S_i
.
It is intended to be used inside the penalised negative log-likelihood function when fitting models with penalised splines or simple random effects via quasi restricted maximum likelihood (qREML) with the qreml
function.
For qreml
to work, the likelihood function needs to be compatible with the RTMB
R package to enable automatic differentiation.
Usage
penalty(re_coef, S, lambda)
Arguments
re_coef |
coefficient vector/ matrix or list of coefficient vectors/ matrices Each list entry corresponds to a different smooth/ random effect with its own associated penalty matrix in |
S |
fixed penalty matrix or list of penalty matrices matching the structure of |
lambda |
penalty strength parameter vector that has a length corresponding to the total number of random effects/ spline coefficients in E.g. if |
Details
Caution: The formatting of re_coef
needs to match the structure of the parameter list in your penalised negative log-likelihood function,
i.e. you cannot have two random effect vectors of different names (different list elements in the parameter list), combine them into a matrix inside your likelihood and pass the matrix to penalty
.
If these are seperate random effects, each with its own name, they need to be passed as a list to penalty
. Moreover, the ordering of re_coef
needs to match the character vector random
specified in qreml
.
Value
returns the penalty value and reports to qreml
.
References
Koslik, J. O. (2024). Efficient smoothness selection for nonparametric Markov-switching models via quasi restricted maximum likelihood. arXiv preprint arXiv:2411.11498.
See Also
qreml
for the qREML algorithm
Examples
# Example with a single random effect
re = rep(0, 5)
S = diag(5)
lambda = 1
penalty(re, S, lambda)
# Example with two random effects,
# where one element contains two random effects of similar structure
re = list(matrix(0, 2, 5), rep(0, 4))
S = list(diag(5), diag(4))
lambda = c(1,1,2) # length = total number of random effects
penalty(re, S, lambda)
# Full model-fitting example
data = trex[1:1000,] # subset
# initial parameter list
par = list(logmu = log(c(0.3, 1)), # step mean
logsigma = log(c(0.2, 0.7)), # step sd
beta0 = c(-2,-2), # state process intercept
betaspline = matrix(rep(0, 18), nrow = 2)) # state process spline coefs
# data object with initial penalty strength lambda
dat = list(step = data$step, # step length
tod = data$tod, # time of day covariate
N = 2, # number of states
lambda = rep(10,2)) # initial penalty strength
# building model matrices
modmat = make_matrices(~ s(tod, bs = "cp"),
data = data.frame(tod = 1:24),
knots = list(tod = c(0,24))) # wrapping points
dat$Z = modmat$Z # spline design matrix
dat$S = modmat$S # penalty matrix
# penalised negative log-likelihood function
pnll = function(par) {
getAll(par, dat) # makes everything contained available without $
Gamma = tpm_g(Z, cbind(beta0, betaspline)) # transition probabilities
delta = stationary_p(Gamma, t = 1) # initial distribution
mu = exp(logmu) # step mean
sigma = exp(logsigma) # step sd
# calculating all state-dependent densities
allprobs = matrix(1, nrow = length(step), ncol = N)
ind = which(!is.na(step)) # only for non-NA obs.
for(j in 1:N) allprobs[ind,j] = dgamma2(step[ind],mu[j],sigma[j])
-forward_g(delta, Gamma[,,tod], allprobs) +
penalty(betaspline, S, lambda) # this does all the penalization work
}
# model fitting
mod = qreml(pnll, par, dat, random = "betaspline")