tessdev: Calculate tessellation deviance residuals
In stppResid: Perform residual analysis on space-time point process models.

Description Usage Arguments Details Value Author(s) See Also Examples

tessdev divides the space-time window into cells using a Voronoi tessellation and calculates the deviance residuals within each cell between two competing conditional intensity models.

tessdev(X, cifunction1, cifunction2, theta1 = NULL, theta2 = NULL, 
lambda1 = NULL, lambda2 = NULL, algthm1 = c("mc", "miser", "none"), 
algthm2 = c("mc", "miser", "none"), n = 100, n1.miser = 10000, 
n2.miser = 10000, ints1 = NULL, ints2 = NULL)

`X`	A “`stpp`” object.
`cifunction1`	A function returning estimates of the conditional intensity at all points in `X`, according to model 1 (`cifunction1`). The function should take arguments `X` and an optional vector of parameters `theta1`.
`cifunction2`	A function returning estimates of the conditional intensity at all points in `X`, according to model 2 (`cifunction2`) which should be different than model 1 (`cifunction1`). The function should take arguments `X` and an optional vector of parameters `theta2`.
`theta1`	Optional: A vector of parameters to be passed to `cifunction1`.
`theta2`	Optional: A vector of parameters to be passed to `cifunction2`.
`lambda1`	Optional: A vector of conditional intensities based on `cifunction1` at each point in `X`.
`lambda2`	Optional: A vector of conditional intensities based on `cifunction2` at each point in `X`.
`algthm1`	The algorithm used for estimating the integrals in each cell for model 1. The three algorithms are “`mc`”, “`miser`”, and “`none`”
`algthm2`	The algorithm used for estimating the integrals in each cell for model 2. The three algorithms are “`mc`”, “`miser`”, and “`none`”
`n`	Initial number of sample points in each cell for approximating integrals. The number of sample points are iteratively increased by `n` until some accuracy threshold is reached.
`n1.miser`	The total number of sample points for estimating all integrals for model 1 if the `miser` algorithm is selected.
`n2.miser`	The total number of sample points for estimating all integrals for model 2 if the `miser` algorithm is selected.
`ints1`	An optional vector of integrals for model 1. Must be the same length as the number of tessellation cells, and each element of `ints1` should correspond to each cell in the `tile.list` that is returned using the `deldir` function, which can be called separately.
`ints2`	An optional vector of integrals for model 2. Must be the same length as the number of tessellation cells, and each element of `ints2` should correspond to each cell in the `tile.list` that is returned using the `deldir` function, which can be called separately.

The tessellation deviance residuals are the differences in the tessellation residuals of model 1 vs. model 2 within each Voronoi tessellation cell, denoted here as V_i. The tessellation deviance residual is given by

R_{TD}(V_{i}) = ≤ft(1 - \int_{V_{i}}\hat{λ}_{1}(x)dx\right)/√{\int_{V_{i}}\hat{λ}_{1}(x)dx}-≤ft(1 - \int_{V_{i}}\hat{λ}_{2}(x)dx\right)/√{\int_{V_{i}}\hat{λ}_{2}(x)dx},

where lambda_hat is the fitted conditional intensity model.

The conditional intensity functions, cifunction1 and cifunction2, should take X as their first argument, and an optional theta as their second argument, and return a vector of conditional intensity estimates with length equal to the number of points in X, i.e. the length of X$x. Both cifunction1 and cifunction2 are required. lambda1 and lambda2 are optional, and if passed will eliminate the need for devresid to calculate the conditional intensities at each observed point in X.

The integrals in R_{TD}(V_{i}) are approximated using one of two algorithms: a simple Monte Carlo (mc) algorithm, or the MISER algorithm. The simple Monte Carlo iteratively adds n sample points to each tessellation cell to approximate the integral, and the iteration stops when some threshold in the accuracy of the approximation is reached. The MISER algorithm samples a total number of n.miser points in a recursive way, sampling the points in locations that have the highest variance. This part can be very slow and the approximations can be very inaccurate. For highest accuracy these algorithms will require a very large n or n.miser depending on the complexity of the conditional intensity functions (some might say ~1 billion sample points are needed for a good approximation).

Passing the arguments ints1 and/or ints2 eliminates the need for approximating the integrals using either of the two algorithms here. However, the tile.list must first be obtained in order to assure that each element of ints1 and/or ints2 corresponds to the correct cell. The tile.list can be obtained, either by using the deldir function separately, or by using tessresid with one of the included algorithms first (the tile.list is returned along with the residuals). tessresid can then be called again with ints1 and/or ints2 included and algthm = “none”.

Note that if miser is selected, and if the points in the point pattern are very densely clustered, the integral in some cells may end up being approximated based on only the observed point in the point pattern that is contained in that cell. This happens because the cells in these clusters of points will be very small, and so it may be likely that sampled points based on the MISER algorithm will miss these cells entirely. For this reason, the simple Monte Carlo algorithm might be preferred.

Prints to the screen the number of simulated points used to approximate the integrals.

Outputs an object of class “tessdev”, which is a list of

`X`	An object of class “`stpp`”.
`tile.list`	An object of class “`tile.list`”.
`residuals`	A vector of tessellation deviance residuals. The order of the residuals corresponds with the order of the cells (tiles) in `tile.list`.

If algthm = “mc”, then a list of the following elements are also included for each model that uses the mc algorithm:

`n`	Vector of the total number of points used for approximating integrals.
`integral`	Vector of actual integral approximations in each grid cell.
`mean.lambda`	Vector of the approximate final mean of lambda in each grid cell.
`sd.lambda`	Vector of the approximate standard deviation of lambda in each grid cell.

If the miser algorithm is selected, then a list of the following elements are also included for each model that uses the miser algorithm:

`n`	Total number of points used for approximating integrals.
`integral`	Vector of actual integral approximations in each grid cell.
`mean.lambda`	Vector of the approximate final mean of lambda in each grid cell.
`sd.lambda`	Vector of the approximate standard deviation of lambda in each grid cell.
`app.pts`	A data frame of the x,y, and t coordinates of a sample of 10,000 of the sampled points for integral approximation, along with the value of lambda (l).

Robert Clements

tessresid

#===> load simulated data <===#
data(simdata)
X <- stpp(simdata$x, simdata$y, simdata$t)

#===> define two conditional intensity functions <===#
ci1 <- function(X, theta){theta*exp(-2*X$x - 2*X$y - 2*X$t)} #correct model

ci2 <- function(X, theta = NULL){rep(250, length(X$x))} #homogeneous Poisson model

## Not run: 
deviance <- tessdev(X, ci1, ci2, theta1 = 3000)
#===> plot results <===#
plot(deviance)

## End(Not run)