Constraints in MARSS (R package) - r

Question

I would like to estimate (MLE) this model using MARSS (or another package in R)
x_t=x_{t-1}+w_t , with w_t ~ N(0,q)
y_t= d1_t + \alpha d2_t + \beta (d3_t -x_{t-1}) + v_t, with v_t ~ N(0,6*q)
where the first line is the transition equation and the second, the observation one.
I managed to write it in form accepted by MARSS (R-package), as below:
[x1_t,x2_{t-1}]= [1,0;1,0][x1_{t-1},x2_{t-2}]+[w1_t,w2_t], with w1_t ~ N(0,q) and w2_t ~ N(0,0)
y_t= D d_t+Z x_t , with v_t ~ N(0,6*q)
where
x_t=[x1_t,x2_{t-1}]
D=[1,\alpha,\beta]
Z=[0,\beta]
d_t=[d1_t,d2_t, d3_t]
The problem is that I couldn't make the constraint work properly. When I run this system, R considers the \beta in Z matrix separately of the \beta in D matrix. All the examples that I saw on internet show a linear restriction using Z matrix only (or just D only). The same occurs in the variances that I would like to be multiples.
Anyone could help me with this?
Here's a toy data:
B <- matrix(list(1,0,1,0),2,2,byrow=TRUE)
U <- matrix(0,2,1)
C <- matrix(0,2,1)
G <- matrix(list(1,0,0,0),2,2,byrow=TRUE)
Q <- matrix(list('d',0,0,0),2,2,byrow=TRUE)
Z <- matrix(list(0,'b'),1,2)
A <- matrix(0)
D <- matrix(list(1,'a','b'),1,3)
H <- matrix(1)
R=matrix(list('6*d'))
dt<-matrix(rnorm(300),3,100)
y<-rnorm(100)
x0=matrix(list(0.094,0.094),2,1)
V0=matrix(list(0.001,0,0,0.001),2,2)
model.list = list(B=B, U=U, C=C, Q=Q, Z=Z, A=A, D=D, d=dt, H=H, R=R,x0=x0,V0=V0)
kemfit = MARSS(y, model=model.list, control=list(maxit=100,conv.test.slope.tol=0.1,abstol=0.1),method='kem')

Answer 1

The EM algorithm in MARSS only allows constraints (like setting values equal) within the same matrices. Setting constraints across A & D or U & C is easy but across D & Z or R & Q requires rewriting your model in a weird way where your covariates (dt) appears as dummy states (x's). So you don't want to do that.
You can just write a function to return the negative log-likelihood of your state-space model and then minimize that with optim(). I would do this with the KFAS package using the SSCustom() function because that will be fast. However, here is how to do this with the MARSS package just to show you the concept. As the author of MARSS, I can write this down immediately whereas with the KFAS package (which I also use), I'd need to look up how to do the covariates.
# Set up the parts that don't change
dt<-matrix(rnorm(300),3,100)
y<-rnorm(100)
x0=matrix(list(0.094,0.094),2,1)
V0=matrix(list(0.001,0,0,0.001),2,2)
B <- matrix(list(1,0,1,0),2,2,byrow=TRUE)
U <- A <- "zero"
# Put the parameters you will estimate into a vector
pars <- c(a=0.1624, b=-0.1, d=sqrt(0.2))
# Write a function to return the negative log-likelihood
negloglik <- function(pars){
Q <- matrix(list(pars["d"]^2,0,0,0),2,2,byrow=TRUE)
Z <- matrix(list(0, pars["b"]),1,2)
D <- matrix(list(1, pars["a"], pars["b"]),1,3)
R <- matrix(6*pars["d"]^2)
model.list = list(B=B, U=U, Q=Q, Z=Z, A=A, D=D, d=dt, R=R, x0=x0, V0=V0)
-1*MARSS(y, model=model.list, control=list(maxit=100,conv.test.slope.tol=0.1,abstol=0.1),method='kem', silent=TRUE)$logLik
}
optim(pars, negloglik, method="BFGS")
Using the MARSS() function to get the logLik is a bit silly here since that is a fitting function but with all the parameters fixed, it will just return the logLik without fitting.
If you want to see what your KFAS model should look like, you can do this:
kfas.model <- MARSSkfas(kemfit, return.kfas.model=TRUE, return.lag.one=FALSE)$kfas.model
Then
library(KFAS)
logLik(kfas.model)
will get you the log-likelihood. But how the covariates are entering the KFAS model is a little non-intuitive. They appear in the kfas.model$Z element as a time-varying Z. I am sure the KFAS package has some helper function to construct models with covariates. I always construct KFAS models from matrices (no helper functions) so I am not familiar with those, but I know they exist.