Quadrat Counts Methods

1. \(m\) subregions \(B_i\) \(i=1, ...,m\) of area \(|B|\) are constituted
2. A regular tessellation of the study region is performed: usually square cells of side \(a\) such that \(|B|= a^2\)
3. For each \(B_i\) the number of events is counted and the ratio is calculated \[ \hat \lambda_i = \frac{N(B_i)}{|B|} \]

With this method we manage to create a vector of process intensity estimates. If the process is non-stationary the vector can represent an estimate of \(\lambda (s)\); in the case, instead, of a stationary process we can calculate the summary measure \(\hat \lambda = \frac{\sum N(B_i)}{m \space |B|}\)

Kernel Estimation of Intensity

The problem of the quadrat counts method constitutes (as for CSR estimation through graphical methods) the choice of area size for tessellation, in fact few \(B_i\) areas of large dimensions tend to smooth too much the intensity surface while many areas of small dimensions tend to be sparse and irregular.

A possible solution that this method proposes is to smooth local intensities through Kernel.

Therefore \(s_1, ...,s_n\) locations of W where events occur are identified (with s we mean the location on which the intensity is estimated where \(s \in W\) )

… A first solution:

A circle \(B(s,h)\) centered at \(s\) with radius \(h\) is created

Then a local estimation of intensity is performed \[\hat \lambda(s) = \frac{N(B(s,h))}{|B(s,h)|} = \frac{\sum I(\mid\mid s - s_i\mid\mid \le h)}{h^2\space \pi} \\ where \\ I(\mid\mid s - s_i\mid\mid \le h) = \begin{cases} 0, & \mbox{if } \mid\mid s - s_i\mid\mid > h \\ 1, & \mbox{if } \mid\mid s - s_i\mid\mid \le h \end{cases} \]

Problem of this first solution:

Edge effect part of the weight can "disperse" beyond the boundaries of the study region if this is limited: points outside the window W cannot be included

Correction:

Calculate the area by removing the extra part when it goes out:

\[\hat \lambda(s) = \frac{N(B(s,h))}{|W \cap B(s,h)|}\]

Problem of this solution:

Effect of correction: "inflate" the numerator, reducing the area at the denominator, to compensate for the potential reduction of information for "unregistered" events outside the study area when estimating intensity near the region boundary

Final correction:

A system of "weights" is used that is not constant but determined based on a kernel function

\[\hat \lambda(s) = \frac{1}{p_k(s)} \sum_{i=1}^n \frac{1}{h^2}\space k \space \bigg(\frac{s-s_i}{h} \bigg)\]

• where: \(p_k(s)\) constitutes a correction factor for the edge effect: \[p_k(s)= \int_W \frac{1}{h^2}\space k \space \bigg(\frac{s-t}{h} \bigg) \space dt\]
• where: \(k(u) : R^2 \rightarrow R\) (kernel)
• where: \(h\) is a smoothing parameter

Parametric Methods

Suppose to parameterize the intensity of an IHPP as \(\lambda(s, \theta)\) known except for the parameter \(\theta\).

\[ L(\theta | s_1, ...,s_n,n) \propto \prod_{i=1}^n \lambda(s_i, \theta) \exp(-\mu(W,\theta)) \]

In these cases it is always advisable to resort to log-linear specifications to guarantee the positivity of the intensity function

\[ l(\theta) = l(\theta | s_1, ...,s_n,n) = \sum_{i=1}^n \log \bigg(\lambda(s_i, \theta) \bigg) -\mu(W,\theta)\]

The log-likelihood is maximized for the parameter \(\theta\)

\[ \hat \theta_{ML} = \max_\theta l(\theta) \]