Bandwidth selection procedure for kernel spectral density estimator

Select the optimal bandwidth for a given dataset by minimizing the spectral divergence between the periodogram $\hat{I}$ and the leave-one-out kernel estimator $\hat{F}^{(-1)}_b$: $$L(b) = \sum_{\boldsymbol\omega}\text{tr}\left\{\hat{I}(\boldsymbol\omega) \hat{F}^{(-1)}_b(\boldsymbol\omega)^{-1}\right\} + \log\text{det}\hat{F}^{(-1)}_b(\boldsymbol\omega).$$

Usage

select_band(
  ppp,
  inten.formula = NULL,
  data.covariate = NULL,
  a = 0.025,
  band.range,
  correct = TRUE,
  A1 = NULL,
  A2 = A1,
  endpt = 1.5,
  equal = TRUE,
  kern = bartlett_uni
)

Arguments

ppp: A point pattern of class "ppp".
inten.formula: A formula syntax in character format specifying the log-liner model for the intensity function, which is passed to ppm(). The default is constant intensity inten.formula = "~1".
data.covariate: Optional. The values of spatial covariates passed to the data argument in ppm().
a: Taper coefficient, a value within $(0,1/2)$. If a = 0, then taper is not applied.
band.range: Numeric vector. Search space for the optimal bandwidth.
correct: Logical. If TRUE (default), conduct edge correction when computing the kernel spectral estimator.
A1, A2: Optional. Side lengths of the observation window.
endpt: A positive value indicating the scale factor of the endpoint frequency.
equal: Logical. Whether to use the same bandwidth for both x and y axis. The default is TRUE.
kern: Univariate scaled kernel function. The default is Barlett kernel.

Value

A list.

Examples

library(spatstat)
lam <- function(x, y, m) {(x^2 + y) * ifelse(m == "A", 2, 1)}
set.seed(227823)
spp <- rmpoispp(lambda = lam, win = square(5), types = c("A","B"))
b <- seq(0.95, 1.4, 0.05) # You may use finer grid to search optimal bandwidth

cv <- select_band(spp, inten.formula = "~ x + y", band.range = b)
cv$OptimalBandwidth # Print the optimal bandwidth
#> Bandwidth 
#>      1.15 
plot(cv$Result[1,], cv$Result[2,], type = "b", pch = 20, las = 1,
     xlab = "Bandwidth", ylab = "Spectral divergence")
abline(v = cv$OptimalBandwidth, col = "blue")