NOTES_LCG_function

Here is a prototype call to our possible function LCG() "Latent Gaussian Counts":

LGC(x=data,
    count.family = "Poisson", "mixed-Poisson", "binomial",
    gauss.series = "AR", "farima"
    estim.method = "gaussianLik, particleMLE",
    max.terms = 30, [30 has proven to be enough. \infty\approx 30]
    p=NULL, d=NULL, q=NULL, n.mix=NULL)     these are the additional model params
                                            that might be needed for a specific
                                            count.family or gauss.series. We will
                                            check for them when needed using the
                                            is.null() function.
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The argument count.family gives the desired marginal distribution for the
count series. It is a discrete distribution.

The argument gauss.series gives the type of latent gaussian process that will
be used.

estimate.method gives the estimation method used

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We have to decide how to pass the parameter vector around.

For example if we choose "mixed-Poisson", "AR1", n.mixed=2 then we would have
theta = (lam1, lam2, phi)

If we choose "Poisson", "farima", p=1, d=1, q=0 then we would have
theta = (lam, phi, d)

Notice the placement changes. In the former the first two parameters govern the
count distribution. In the later only the first parameter governs the count.

My initial thought is to have a parameter vector theta and an index pointer that
tells how many variables are in the count.family, say cfi. So when we call the hermite.coeff
function, which only needs the count.family params, we have:
hermite.coef(theta[1:cfi])
Then when we call the ACVF of X, gamX(theta[cfi+1, length(theta)])

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For each count.family we need to define the CDF (to get hermite coefs)

For example:

if(count.family = "mixed-poisson", n.mixed=2)
 if(is.null(n.mixed)) stop("you number specify the number of Poissons to mix, n.mixed
                            when using the count.family=mixed.Poisson option")
 IMPORTANT - pass paramters as a vector. Then we remain consistant with this structure.
 cdf = function(x, theta){
    theta[1]*ppois(x,theta[2]) + (1-theta[1])*ppois(x,theta[3])
 }

if(count.family=="Poisson")
  cdf = function(x, lam=lam){ dpois(x, lambda=lam) }

if(count.family="Binomial")
 if(is.null(n)) stop("you must specify the number of trials, n, to use count.family=Binomial")
 cdf = function(p, n=n){ dbinom(p, n=n) }

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For each gauss.series we need to define the ACVF function.

For example,

if(gauss.series="AR", p=1)
  if(is.null(p)) stop("you must specify the AR order, p, to use gauss.series=AR")
  if(p>1) stop("the ACVF is not coded for AR models of order higher than 1 currently")
  gamz = function(h, theta){
    theta^h / (1-theta^2)
  }