MH4522 Kernel Estimation for Poisson Point Processes – Spatial Data Science Assignment

The classical kernel estimator¹ of the probability density function ϕ(x) of a random variable X is defined by

where x_i, i = 1, …, n, are n independent samples of X. Here, h > 0 is a positive parameter called the bandwidth, and φ is a bounded probability density function, such that

N=1; h=0.1; z =seq(0,1,0.01); kernel=function(z){dnorm(z,0,h/4)};
x=runif(N); kdensity=function(z){sum(as.numeric(lapply(z-x,kernel))/length(x)}
plot(0, xlab = "", ylab = "", type = "l", xlim = c(0,1), col = 0,
ylim=c(0,max(as.numeric(lapply(z,kdensity))),xaxt='n',yaxt='n')
axis(1, at=c(), xlab = "", lwd=2,labels=c(), pos=0,lwd.ticks=2)
axis(2, lwd=2, at = c(1,axTicks(4)), lwd.ticks=2); points(x, rep(0,N), pch=3, lwd = 3, col = "blue")
lines(density(x,width=h),col="purple",lwd=3); lines(z,dunif(z),col="black",lwd=3);
lines(z,as.numeric(lapply(z,kdensity)),col="red",lwd=2,type='l')

The aim of this assignment is to implement a kernel estimation for the intensity of a Poisson point process η on ℝ^d, d ≥ 1. We assume that the intensity measure µ of η has a C²_b density ρ : ℝ^d → ℝ⁺ with respect to the Lebesgue measure on (ℝ^d, B(ℝ^d)), i.e. µ(dx) = ρ(x)dx, and

We also let

denote the Euclidean norm in ℝ^d, and we denote by φ_h the Gaussian kernel

with variance h > 0. The following questions are interdependent and should be treated in sequence.

1)

Show that for all x ∈ ℝ^d we have

2)

Show that the estimator

of the density ρ(x) is asymptotically unbiased, i.e. we have

3)

Show that the asymptotic variance of the estimator ρ̂_h,0 satisfies

i.e.

4)

Given a domain A ⊂ ℝ^d such that 0 < µ(A) < ∞ and f ∈ L¹(A, µ), compute the expectation

5)

Given a domain A ⊂ ℝ^d such that 0 < µ(A) < ∞ and f ∈ L¹(A, µ) ∩ L²(A, µ), compute the variance

using the quantity

6)

Show that for any domain A ⊂ ℝ^d such that 0 < µ(A) < ∞, we have

7)

For any h > 0, let A_h ⊂ ℝ^d denote a domain of finite Lebesgue measure in ℝ^d, and consider the estimator ρ̂_h,1 of the probability density ρ(x)/µ(A_h) defined by

Show that ρ̂_h,1(x) is asymptotically unbiased in the sense that

as h → 0, i.e.

provided that µ(A_h) → ∞ as h → 0.

8)

Show that the variance of ρ̂_h,1(x) satisfies

provided that µ(A_h)⁻¹ = o(h^d).

9)

Show that for any domain A ⊂ ℝ^d such that 0 < ℓ(A) < ∞ we have

10)

Based on a dataset of your choice on a domain A, find the value of h > 0 that minimizes the quantity

and compare the estimations of the density ρ(x) obtained from ρ̂_h,0 and ρ̂_h,1 (graphs are welcome).

Examples of datasets include:

• Simulated datasets;
• The spatstat package; see https://cran.r-project.org/web/packages/spatstat/vignettes/datasets.pdf;
• The scikit-learn package in Python, see https://scikit-learn.org/stable/datasets/real_world.html and this example.

See also:

• P. Moraga. Geospatial Health Data – Modeling and Visualization with R-INLA and Shiny. Chapman & Hall/CRC Biostatistics Series. CRC Press, 2020.
• P. Moraga. Spatial Statistics for Data Science – Theory and Practice with R. Chapman & Hall/CRC Data Science Series. CRC Press, 2024.