Functions providing access to the Log Likelihood, Gradient, and Hessian
of the collapsed pibble model. Note: These are convenience functions
but are not as optimized as direct coding of the PibbleCollapsed
C++ class due to a lack of Memoization. By contrast function optimPibbleCollapsed
is much more optimized and massively cuts down on repeated calculations.
A more efficient Rcpp module based implementation of these functions
may following if the future. For model details see optimPibbleCollapsed
documentation
loglikPibbleCollapsed(Y, upsilon, ThetaX, KInv, AInv, eta, sylv = FALSE)
gradPibbleCollapsed(Y, upsilon, ThetaX, KInv, AInv, eta, sylv = FALSE)
hessPibbleCollapsed(Y, upsilon, ThetaX, KInv, AInv, eta, sylv = FALSE)D x N matrix of counts
(must be > D)
D-1 x N matrix formed by Theta*X (Theta is Prior mean for regression coefficients)
Inverse of K for LTP (for Pibble defined as KInv = solve(Xi))
Inverse of A for LTP (for Pibble defined as AInv = solve(diag(N)+ X'GammaX) )
matrix (D-1)xN of parameter values at which to calculate quantities
(default:false) if true and if N < D-1 will use sylvester determinant identity to speed computation
see below
loglikPibbleCollapsed - double
gradPibbleCollapsed - vector
hessPibbleCollapsed- matrix
D <- 10
Q <- 2
N <- 30
# Simulate Data
Sigma <- diag(sample(1:8, D-1, replace=TRUE))
Sigma[2, 3] <- Sigma[3,2] <- -1
Gamma <- diag(sqrt(rnorm(Q)^2))
Theta <- matrix(0, D-1, Q)
Phi <- Theta + t(chol(Sigma))%*%matrix(rnorm(Q*(D-1)), nrow=D-1)%*%chol(Gamma)
X <- matrix(rnorm(N*(Q-1)), Q-1, N)
X <- rbind(1, X)
Eta <- Phi%*%X + t(chol(Sigma))%*%matrix(rnorm(N*(D-1)), nrow=D-1)
Pi <- t(alrInv(t(Eta)))
Y <- matrix(0, D, N)
for (i in 1:N) Y[,i] <- rmultinom(1, sample(5000:10000), prob = Pi[,i])
# Priors
upsilon <- D+10
Xi <- Sigma*(upsilon-D)
# Precompute
KInv <- solve(Xi)
AInv <- solve(diag(N)+ t(X)%*%Gamma%*%X)
ThetaX <- Theta%*%X
loglikPibbleCollapsed(Y, upsilon, ThetaX, KInv, AInv, Eta)
#> [1] -115558.2
gradPibbleCollapsed(Y, upsilon, ThetaX, KInv, AInv, Eta)[1:5]
#> [1] 4.7440863 -2.0141509 -0.6579329 -5.0887964 -0.9877219
hessPibbleCollapsed(Y, upsilon, ThetaX, KInv, AInv, Eta)[1:5,1:5]
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] -1454.9780390 1.23680090 0.2229720363 1.450126e+03 0.0531649019
#> [2,] 1.2368009 -3.57827522 -0.2111650936 6.691041e-01 0.0287388480
#> [3,] 0.2229720 -0.21116509 -0.5506432005 1.461724e-01 -0.0003659692
#> [4,] 1450.1262939 0.66910413 0.1461724285 -1.453056e+03 -0.0140213131
#> [5,] 0.0531649 0.02873885 -0.0003659692 -1.402131e-02 -0.4031252063