Multivariate Normal GAM – Differences Between Sigma and Covariance Matrix

generalized-additive-modelmgcvmultivariate normal distributionr

I'm confused about the difference between sigma and the cov-matrix when one fits a multivariate normal gam.

More precise, I used for example mvn(d=2) when fitting a two-dimensional normal GAM to model $Y_1$ and $Y_2$. The GAM fit is denoted by b in the following

From the documentation, I known that the covariance matrix of $Y_1$ and $Y_2$ can be obtained via

covMatrix <- solve(crossprod(b$family$data$R))

(code is taken from the documentation of the "Multivariate normal additive models").

What I was thinking is that $Var(Y_1) =$ covMatrix [1,1] and $Var(Y_2)=$ covMatrix[2,2] hold.

When bringing back everything introduced so far to the usual GAM formula, I would assume that
$$Y_1 = \mu_1 + Var(Y_1) * \epsilon_1, \quad \epsilon_1 \sim N(0,1). (*)$$

Is this so far correct, i.e. does $(*)$ hold?

And if so, what is b$sig2? I fitted various multivariate Normal GAMs and all yielded b$sig2=1 which seems a bit strange to me since the covariance matrices varied greatly.

Best Answer

sig2 is the dispersion parameter $\phi$ in the GLM context. It is set to 1 I believe so as to have no effect in this kind of model, because R has been estimated.

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