---
title: "Defining a latent model in `R`: The `rgeneric` model"
author: "Haavard Rue `(hrue@r-inla.org)`"
date: "Nov 22th 2017"
output: rmarkdown::pdf_document
vignette: >
  %\VignetteIndexEntry{rgeneric model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8} 
---

```{r setup, include=FALSE}
set.seed(123)
library(INLA)
library(knitr)
library(rmarkdown)
inla.setOption(num.threads=1)
knitr::opts_chunk$set(echo=TRUE, cache=FALSE, message=FALSE, warning=FALSE)
```

# Introduction

This is a class of generic models allows the user to define latent
model component in `R`, for cases where the requested model is not yet
implemented in `INLA`, and do the Bayesian inference using `INLA`.
Since the model component has to be interpreted in `R`, it will run
slower compared to a similar model implemented in `INLA`.


# Defining a latent model in `R`

A `rgeneric` model is defined in a function `rmodel` (to be defined
later), and the usage is quite simple. 
First we need to define a `inla-rgeneric` object
```{r eval=FALSE}
model = inla.rgeneric.define(rmodel, ...)
```
with additional variables/functions/etc in `...` that 
we might use in `rmodel`. This
can be the size, prior parameters, covariates, external functions 
and so on.
This object can then be used to define a normal model component in
`INLA` using `f()`,
```{r eval=FALSE}
y ~ ... + f(idx, model=model, ...)
```
where `idx` can take values $1, 2, \ldots, n$ where `n` is the size of
`model`. 
 All additional features for `f()` will still be valid. 


# Example: The AR1 model

The function `rmodel` needs to follow some rules to provide the required
features. We explain this while 
demonstrating how to implement the AR1-model.
This model already exists, see
`inla.doc("ar1")`. With the parmeterisation we use,
the AR1-model is defined as 
\begin{displaymath}
    x_{1} \;\sim\; {\mathcal N}(0, \tau)
\qquad\text{and}\qquad
    x_{t} \mid x_{1}, \ldots, x_{t-1} \;\sim\; {\mathcal N}(\rho
    x_{t-1}, \tau_{I}), \qquad t=2, \ldots,n.
\end{displaymath}
where $\tau_I = \tau/(1-\rho^2)$.
The scale-parameter is the *marginal precision* $\tau$, **not** the
commonly used innovation precision $\tau_I$.
The joint density of $x$ is Gaussian
\begin{displaymath}
    \pi(x|\rho,\tau) = \left(\frac{1}{\sqrt{2\pi}}\right)^{n}
    \tau_{I}^{n/2} (1-\rho^{2})^{1/2}
    \exp\left(-\frac{\tau_{I}}{2} x^{T} R x\right)
\end{displaymath}
where the precision-matrix is
\begin{displaymath}
    Q = \tau_I R = 
    \tau_I
    \begin{bmatrix}
        1 & -\rho &&&& \\
        -\rho & 1+\rho^{2}& -\rho &&& \\
        &-\rho & 1+\rho^{2}& -\rho && \\
        && \ddots& \ddots& \ddots& \\
        &       &       &       -\rho & 1 + \rho^{2} & -\rho\\
        & & & & -\rho & 1
    \end{bmatrix}
\end{displaymath}
There are two (hyper-)parameters for this model: the marginal
precision $\tau$ and the lag-one correlation $\rho$.
We will 
reparameterise these as
\begin{displaymath}
    \tau = \exp(\theta_1), \qquad\text{and}\qquad
    \rho = 2\frac{\exp(\theta_{2})}{1+\exp(\theta_{2})} - 1.
\end{displaymath}
It is required that the parameters $\theta = (\theta_{1}, \theta_{2})$
have support on $\Re^2$ and the priors for $\tau$ and $\rho$ are given
as the corresponding priors for $\theta_{1}$ and $\theta_{2}$.

A good re-parametersation is required for `INLA` to work well. A good
parmeterisation makes, ideally, the *Fisher information matrix* of
$\theta$ constant with respect to to $\theta$. It is sufficient to
check this in a frequentistic setting with data directly from the
AR$(1)$ model, in this case. `INLA` will provide the posterior
marginals for $\theta$, but `inla.tmarginal()` can be used to convert
it to the appropriate marginals for $\rho$ and $\tau$.

We assign Gamma prior $\Gamma(.; a,b)$ (with mean $a/b$ and
variance $a/b^{2}$) for $\tau$ and a Gaussian prior
${\mathcal N}(\mu,\kappa)$ for $\theta_{2}$, so the joint prior for
$\theta$ becomes
\begin{displaymath}
    \pi(\theta) = \Gamma(\exp(\theta_1); a,b) \exp(\theta_1) \;\times\;
    {\mathcal N}(\theta_{2}; \mu, \kappa).
\end{displaymath}
The extra term, $\exp(\theta_1)$ is the Jacobian for the change of
variable from $\tau$ to $\theta_1$. We will in this example use
$a=b=1$, $\mu=0$ and $\kappa=1$.


In order to define the AR1-model, we need to make `R`-functions that
returns

*    the graph,
*    the precision matrix $Q(\theta)$,
*    the zero mean,
*    the initial values of $\theta$,
*    the log-normalising constant, and
*    the log-prior

We need to incorporate these functions into `rmodel`, in
the following way
```{r eval=FALSE}
inla.rgeneric.ar1.model = function(
        cmd = c("graph", "Q", "mu", "initial", "log.norm.const",
                "log.prior", "quit"),
        theta = NULL)
{
    # for reference and potential storage for objects to
    # cache, this is the environment of this function
    # which holds arguments passed as `...` in
    # `inla.rgeneric.define()`.
    envir = environment(sys.call()[[1]])

    graph = function(){ <to be completed> }
    Q = function() { <to be completed> }
    mu = function() { <to be completed> }
    log.norm.const = function() { <to be completed> }
    log.prior = function() { <to be completed> }
    initial = function() { <to be completed> }
    quit = function() { <to be completed> }

    # sometimes this is useful, as argument 'graph' and 'quit'
    # will pass theta=NULL as the values of theta are not
	# required for defining the graph. however, this statement
    # will ensure that theta is always defined.
    if (is.null(theta)) theta = initial()

    val = do.call(match.arg(cmd), args = list())
    return (val)
}
```
The input parameters are

* **cmd** What to return
* **theta**  The values of the $\theta$-parameters

Other parameters in the model definition, like $n$ and possibly the
parameters of the prior, goes into the `...` part of
`inla.rgeneric.define()`, like
```{r eval=FALSE}
model = inla.rgeneric.define(inla.rgeneric.ar1.model, n = 100)
```
and is assigned in the environment of
`inla.rgeneric.ar1.model`. Using
variable `n` inside this function will then
return `100`. This environment can also be accessed as `envir` as
defined
in the function skeleton. Sometimes this is useful, to hold static
varibles or to cache intermediate calculations.

Our next task, is to *fill in the blanks* and define the
functions required. To help us, we will add a function that return
a list of the **real** parameters in the model from $\theta$,
```{r eval=FALSE}
interpret.theta = function() {
    return(list(prec = exp(theta[1L]), 
                rho = 2 * exp(theta[2L])/(1 + exp(theta[2L])) - 1))
}
```
Since `theta` exist already within `inla.rgeneric.ar1.model` we do
not need to pass it as an argument.

We also assume that variable `n` is defined as an argument in
`inla.rgeneric.define()`.

## Function `graph()`

This is normally an easy function to add, as it is essentially the
matrix $Q$. One can construct cases where this is not
so\footnote{Depending on $\theta$ an element $Q_{ij}$ might be exactly
zero}, and for this reason it exists as a separate function. The only
thing that matter is if the elements are zero or non-zero. Also, it
should return a `sparse matrix` as we do not want to pass $n^2$
elements when ${\mathcal O}(n)$ are sufficient. Also, only the upper
triangular matrix (diagonal included) are actually used, since the
graph must be symmetric.
```{r eval=FALSE}
graph = function() {
    return (Q())
}
```

## function `Q()`

This is normally the most tricky function, as we need to return
the precision matrix (as a sparse matrix) for the given values of
$\theta$.  Only the upper triangular matrix (diagonal included) are
read.

A *dense matrix* version is as follows, and is easier to read
```{r eval=FALSE}
Q = function() {
    p = interpret.theta()
    Q = p$prec/(1 - p$rho^2) *
	    toeplitz(c(1 + p$rho^2, -p$rho, rep(0, n - 2L)))
    Q[1, 1] = Q[n, n] = p$prec/(1 - p$rho^2)
    return (inla.as.sparse(Q))
}
```
The function `inla.as.sparse()` convert a matrix or sparse matrix,
into the appropriate sparse matrix format used internally in `INLA`.
This version of `Q()` creates a dense matrix and then make it sparse,
and is not the way to do it.
The better way, is to define the (upper triangular)
sparse matrix directly using `sparseMatrix`.
```{r eval=FALSE}
Q = function() {
    p = interpret.theta()
    i = c(1L, n, 2L:(n - 1L), 1L:(n - 1L))
    j = c(1L, n, 2L:(n - 1L), 2L:n)
    x = p$prec/(1 - p$rho^2) *
	    c(1L, 1L, rep(1 + p$rho^2, n - 2L),
	      rep(-p$rho, n - 1L))
     return (sparseMatrix(i = i, j = j, x = x, giveCsparse = FALSE))
}
```
This is both faster and requires less memory, but it gets somewhat
unreadable and hard to debug. The dense matrix version above, is at
least easier to debug against for reasonable values of $n$.

## Function `mu()`

This function must return the mean which might depend on $\theta$.
The convention, is that if `numeric(0)` is returned, then the mean is
zero.
```{r eval=FALSE}
mu = function() {
    return(numeric(0))
}
```



## Function `log.norm.const()`

This function must return the log of the normalising constant. For the
AR1-model the normalising constant is
\begin{displaymath}
    \left(\frac{1}{\sqrt{2\pi}}\right)^{n} \tau_{I}^{n/2} (1-\rho^{2})^{1/2}
\end{displaymath}
where
\begin{displaymath}
    \tau_{I} = \tau/(1-\rho^{2}).
\end{displaymath}
The function can then be implemented as
```{r eval=FALSE}
log.norm.const = function() {
   p = interpret.theta()
   prec.i  = p / (1.0 - p$rho^2)
   val = n * (- 0.5 * log(2*pi) + 0.5 * log(prec.i)) +
         0.5 * log(1.0 - p$rho^2)
   return (val)
}
```
Since the normalizing constant is known, we can ask `INLA` to evaluate
$$-\frac{n}{2}\log(2\pi) + \frac{1}{2} \log(|Q(\theta)|)$$ 
and $\log|Q(\theta)|$ can be computed from the
sparse Cholesky factorisation of $Q(\theta)$. In this case 
we can return `numeric(0)` (which is a code for "`compute it yourself,
please!`")
```{r eval=FALSE}
log.norm.const = function() {
   return (numeric(0))
}
```
Unless the log-normalizing constant is known analytically (and the
precision matrix depends on $\theta$) it is both better, and
easier, just to return `numeric(0)`.


## Function `log.prior()`

This function must return the (log-)prior of the prior density for
$\theta$. For the AR1-model, we have for simplicity chosen this prior
\begin{displaymath}
    \pi(\theta) = \Gamma(\exp(\theta_1); a,b) \exp(\theta_1) \;\times\;
    {\mathcal N}(\theta_{2}; \mu, \kappa)
\end{displaymath}
so we can implement this as with our choices $a=b=1$, $\mu=0$ and
$\kappa=1$ as
```{r eval=FALSE}
log.prior = function() {
    p = interpret.theta()
    val = dgamma(p$prec, shape = 1, rate = 1, log=TRUE) + theta[1L] + 
          dnorm(theta[2L], mean = 0, sd = 1, log=TRUE)
    return (val)
}
```
The parameters in the joint prior can also be defined in the 
`inla.rgeneric.define()` call, by adding arguments `a=1`, `b=1` and so on.

**Note** that `log.prior()` must return the log prior for $\theta$, and
not the prior for the more natural parameters defined in
`interpret.theta()`.


## Function `initial()`

This function returns the initial values for $\theta$, like
```{r eval=FALSE}
initial = function() {
    return (rep(1, 2))
}
```
or `numeric(0)` is there are no $\theta$'s. For a precision parameters
it is generally advisable to chose a high precision as the initial
value, as this helps the optimizer. `INLA` generally use initial value
$4$ for log precisions. 


## Function `quit()`

This function is called when all the computations are done and before
exit-ing the \texttt{C}-program. If there is some cleanup to do, you
can do this here. In our example, there is nothing do to.
```{r eval=FALSE}
quit = function() {
    return (invisible())
}
```

## Example of usage

Here is an example of use. The function `inla.rgeneric.ar1.model()`
contains the functions given above, and can be used directly like
this.
```{r}
n = 100
rho=0.9
x = arima.sim(n, model = list(ar = rho)) * sqrt(1-rho^2)
y = x + rnorm(n, sd = 0.1)
model = inla.rgeneric.define(inla.rgeneric.ar1.model, n=n)
formula = y ~ -1 + f(idx, model=model)
r = inla(formula, data = data.frame(y, idx = 1:n))
```
We can also compare with the buildt-in version, if we make sure to use
the same priors
```{r}
fformula = y ~ -1 +
	f(idx, model = "ar1",
	  hyper = list(prec = list(prior = "loggamma", param = c(1,1)),
      rho = list(prior = "normal", param = c(0,1))))
rr = inla(fformula, data = data.frame(y, idx = 1:n))
```	  
and plot the hyperparameters in the same scale
```{r}
plot(inla.smarginal(rr$marginals.hyperpar[[2]]), 
	 type="l", lwd=5, col="red", xlab="stdev", ylab="density")
lines(inla.tmarginal(exp, r$internal.marginals.hyperpar[[2]]), 
     col="yellow")
```
```{r}
plot(inla.smarginal(rr$marginals.hyperpar[[3]]), 
	 type="l", lwd=5, col="red", xlab="rho", ylab="density")
lines(inla.tmarginal(function(x) 2*exp(x)/(1+exp(x))-1,
                     r$internal.marginals.hyperpar[[3]]), 
     col="yellow")
```
The running time will of'course be quite different
```{r}
round(rbind(native = rr$cpu.used,  
            rgeneric = r$cpu.used), digits = 3)
```


# Example: The iid-model

The following function defines the \texttt{iid}-model, see
`inla.doc("iid")`, which we give without further comments. To
run this model in `R`, you may run `demo(rgeneric)`.

```{r}
inla.rgeneric.iid.model
```


# Example: A model for the mean structure

Up to now, we have assumed zero mean. In this example, we will
illustrate how to add a non-zero mean model, focusing on the mean
model only. We can of'course have a mean model and a non-trivial
precision matrix together.
```{r, eval=FALSE}
## In this example we do linear regression using 'rgeneric'.
## The regression model is y = a + b*x + noise,  and we
## define 'a + b*x + tiny.noise' as a latent model.
## The dimension is length(x) and number of hyperparameters
## is 2 ('a' and 'b'). 
```
```{r}
rgeneric.linear.regression =
    function(cmd = c("graph", "Q", "mu", "initial", "log.norm.const", 
                     "log.prior", "quit"),
             theta = NULL)
{
    envir = environment(sys.call()[[1]])

    ## artifical high precision to be added to the mean-model
    prec.high = exp(15)
    
    interpret.theta = function() {
        return(list(a = theta[1L], b = theta[2L]))
    }
    
    graph = function() {
        G = Diagonal(n = length(x), x=1)
        return(G)
    } 
    
    Q = function() {
        Q = prec.high * graph()
        return(Q)
    }
    
    mu = function() {
        par = interpret.theta()
        return(par$a + par$b * x)
    }

    log.norm.const = function() {
        return(numeric(0))
    }

    log.prior = function() {
        par = interpret.theta()
        val = (dnorm(par$a, mean=0, sd=1, log=TRUE) +
               dnorm(par$b, mean=0, sd=1, log=TRUE))
        return(val)
    }

    initial = function() {
        return(rep(0, 2))
    }
   
    quit = function() {
        return(invisible())
    }

    val = do.call(match.arg(cmd), args = list())
    return(val)
}
```
and we can run this as
```{r}
a = 1
b = 2
n = 50
x = rnorm(n)
eta = a + b*x
s = 0.25
y = eta + rnorm(n, sd=s)

rgen = inla.rgeneric.define(model = rgeneric.linear.regression, x=x)
r = inla(y ~ -1 + f(idx, model=rgen),
         data = data.frame(y, idx = 1:n))
rr = inla(y ~ 1 + x,
          data = data.frame(y, x),
          control.fixed = list(prec.intercept = 1, prec = 1))
```
and we can compare the results with the native model
```{r}
plot(inla.smarginal(r$marginals.hyperpar[['Theta1 for idx']]),
     type="l", lwd=5, col="red", xlab="Intercept", ylab="density")
lines(inla.smarginal(rr$marginals.fixed$'(Intercept)'), col="yellow")
```
```{r}
plot(inla.smarginal(r$marginals.hyperpar[['Theta2 for idx']]),
     type="l", lwd=5, col="red", xlab="Slope", ylab="density")
lines(inla.smarginal(rr$marginals.fixed$x), col="yellow")
```
The tiny in-accuacy is due to treatment of `a` and `b` as
hyperparameters in the `rgeneric` model.
We can improve the estimates using `inla.hyperpar()` as usual, 
```{r}
r = inla.hyperpar(r)
```
Replotting the results shows improvement:
```{r}
plot(inla.smarginal(r$marginals.hyperpar[['Theta1 for idx']]),
     type="l", lwd=5, col="red", xlab="Intercept", ylab="density")
lines(inla.smarginal(rr$marginals.fixed$'(Intercept)'), col="yellow")
```
```{r}
plot(inla.smarginal(r$marginals.hyperpar[['Theta2 for idx']]),
     type="l", lwd=5, col="red", xlab="Slope", ylab="density")
lines(inla.smarginal(rr$marginals.fixed$x), col="yellow")
```
