Kernel density estimate¶
There are different ways to estimate the intensity for an IPP. One of them is a non-parametrically estimation by means of kernel smoothing. If we observe $n$ points $\{x_i\}_{i=1}^n$ then the form of a kernel smoothing estimator is
$$\hat{\lambda}(x) = \frac{1}{h^2}\sum_{i=1}^n\frac{\kappa(\frac{\Vert x-x_i\Vert}{h})}{q(\Vert x \Vert)}\text{,} $$
where $x_i \in \{x_1, x_2,..., x_n\}$ is an observed point, $h$ is the bandwidth, $q(\Vert x \Vert)$ is a border correction to compensate for observations missing due to edge effects, and $\kappa(u)$ is a bivariate and symmetrical kernel function.
R currently implements a two-dimensional quartic kernel function:
$$
\kappa(u) =
\begin{cases}
\frac{3}{\pi}(1-\Vert u \Vert^2)^2 & \text{if } u \in (-1, 1) \\
0 & \text{otherwise}
\end{cases}
$$
where $\Vert u \Vert^2=u_1^2+u_2^2$ is the squared norm of point $u=(u_1, u_2)$.
There is no general rule for selecting the bandwidth $h$, which governs the level of smoothing. Small bandwidths result in spikier maps and large bandwidths result in smoother map. It is reasonable to use several values depending on the process under consideration, and choose a value that seems plausible (Bivand et al. 2008).
Kernel density plot (2-D)¶
A kernel density plot is the graphical representation of kernel smoothing. When using the Python the seaborn package provides a nice implementation for visualizing spatial kernel density plots. Here we use the kdeplot() function.
The seaborn.kdeplot() funciton provides an estimate of the intensity function of the point process that generated the point pattern data. The argument bw_method will determine the smoothing bandwidth to use, we will pass this argument as sigma. Sigma is taken as the standard deviation of the isotropic Gaussian kernel, corresponding to the bandwidth parameter. A small standard deviation results in spikier density plot and a large standard deviation results in smoother density plot.
We code a loop to apply the sns.kdeplot() function on our three data sets, berlin_locations, berlin_regular, and berlin_random with tree different values for sigma $(0.2, 0.4,0.9)$.
