Classification of Events Using Local Pair Correlation Functions for Spatial Point Patterns

Abstract

Spatial point pattern analysis usually concerns identifying features in an observation window where there is also noise. This identification traditionally begins with studying the second-order properties of the point pattern, and it may be done locally by using local second-order characteristics (LISA). Some properties of this local structure solve the problem of classification into feature and clutter points. This paper proposes an estimator for local pair correlation LISA functions, discusses some of its properties and considers a particular distance to measure dissimilarities. Two classification procedures to separate feature from clutter points are described. One of them adopts multidimensional scaling and support vector machines, and the other employs bagged clustering. Simulations demonstrate the performance of the method, and it is applied to a dataset concerning earthquakes in a seismic nest located in Colombia.

Appendices

Appendix A. Expected Value of Pair Correlation LISA Functions

Theorem 1

For an inhomogeneous Poisson point process with intensity $\lambda ({\mathbf {u}})$, and $\Lambda (A):= \int _A \lambda ({\mathbf {u}})\mathrm {d}{\mathbf {u}}$ for any $A\subset W$, the expected value with respect to the reduced Palm process (conditional on observing a point ${\mathbf {u}}_i$) of a pair correlation LISA function ${\hat{g}}_{\epsilon }^{(i)}(r)$ is

$$\begin{aligned} {\mathbb {E}}_{!}\left[ {\hat{g}}_{\epsilon } ^{(i)}(r)\right] = \frac{\Lambda (W) + 1}{\lambda ({\mathbf {u}}_{i})|W|}. \end{aligned}$$

Proof

We follow the general developments of (Cressie and Collins 2001a, b). For any $A\subset W$, we have ${\mathbb {E}}_![N(A)-1]=\Lambda (A)$, and ${\mathbb {E}}_![(N(A)-1)^2]=\Lambda ^2(A) + \Lambda (A).$ Conditional on observing an event at ${\mathbf {u}}_i\in X$, the expectation with respect to the reduced Palm process is

$$\begin{aligned} {\mathbb {E}}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right] = \frac{1}{2\pi |W|r}{{\mathbb {E}}_{!}\left[ {\mathbb {E}}\left\{ (n-1) \left. \sum _{j\ne i}\frac{\kappa _{\epsilon }\left( \left\| {\mathbf {u}}_{i}-{\mathbf {u}}_{j}\right\| -r\right) 2\pi \Vert {\mathbf {u}}_{i}-{\mathbf {u}}_{j}\Vert }{{\lambda }({\mathbf {u}}_{i}){\lambda }({\mathbf {u}}_{j}) \left| \partial b\left( {\mathbf {u}}_{i},\Vert {\mathbf {u}}_{i}-{\mathbf {u}}_{j}\Vert \right) \cap W\right| } \right| N(W) = n \right\} \right] }.\nonumber \\ \end{aligned}$$

(9)

We bear in mind that for an integrable function $\varphi ({\mathbf {u}})$ on ${\mathbb {R}}^2$, and n independent points observed in W,

$$\begin{aligned} {\mathbb {E}}\left[ \frac{1}{n-1} \sum _{i=1}^{n-1} \frac{\varphi ({\mathbf {u}}_i)}{f({\mathbf {u}}_i)}\right] = \int _W \varphi ({\mathbf {s}})\mathrm {d}{\mathbf {s}}, \end{aligned}$$

where f is the probability density function of the points. Given that the density function of an inhomogeneous Poisson point process in an individual point is given by

$$\begin{aligned} f({\mathbf {u}}_i)=\frac{\lambda ({\mathbf {u}}_i)}{\Lambda (W)}, \quad \text {for any }{\mathbf {u}}_i \in W, \end{aligned}$$

we have that

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^{n-1} \frac{\varphi ({\mathbf {u}}_i)}{\lambda ({\mathbf {u}}_i)}\right] = \frac{n-1}{\Lambda (W)}\int _W \varphi ({\mathbf {s}})\mathrm {d}{\mathbf {s}}. \end{aligned}$$

Applying this expression in Eq. (9), we obtain

$$\begin{aligned} {\mathbb {E}}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right] =&\frac{{\mathbb {E}}_{!}\left[ (N(W)-1)^2\right] }{\Lambda (W)\lambda ({\mathbf {u}}_{i})|W| 2\pi r} \int _{W}\frac{\kappa _{\epsilon }\left( \left\| {\mathbf {u}}_{i}-{\mathbf {v}}\right\| -r\right) 2\pi \Vert {\mathbf {u}}_{i}-{\mathbf {v}}\Vert }{ \left| \partial b\left( {\mathbf {u}}_{i},\Vert {\mathbf {u}}_{i}-{\mathbf {v}}\Vert \right) \cap W\right| } \mathrm {d}{\mathbf {v}} \\ =&\frac{{\mathbb {E}}_{!}\left[ (N(W)-1)^2\right] }{\Lambda (W)\lambda ({\mathbf {u}}_{i})|W| 2\pi r} \int _{r-\epsilon }^{r+\epsilon }\frac{\kappa _{\epsilon }\left( s-r\right) 2\pi s}{\left| \partial b\left( {\mathbf {u}}_{i},s\right) \cap W\right| } {\left| \partial b\left( {\mathbf {u}}_{i},s\right) \cap W\right| }\,\mathrm {d}s \\ =&\frac{\Lambda ^2(W) + \Lambda (W)}{\Lambda (W) \lambda ({\mathbf {u}}_{i}) |W| 2\pi r} \int _{r-\epsilon }^{r+\epsilon }2\pi s\kappa _{\epsilon }\left( s-r\right) \,\mathrm {d}s\\ =&\frac{\Lambda (W) + 1}{\lambda ({\mathbf {u}}_{i})|W|}. \end{aligned}$$

$\square$

Appendix B. Variance of Pair Correlation LISA Functions

Theorem 2

Consider an inhomogeneous Poisson point process with intensity $\lambda ({\mathbf {u}})$, and $\Lambda (A):= \int _A \lambda ({\mathbf {u}})\mathrm {d}{\mathbf {u}}$ for any $A\subset W$. The variance with respect to the reduced Palm process (conditional on observing a point ${\mathbf {u}}_i$) of a pair correlation LISA function ${\hat{g}}_{\epsilon }^{(i)}(r)$ is

$$\begin{aligned} {\mathbb {V}}\text {ar}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right] =\frac{\Lambda (W) + \Lambda ^{-1}(W) +3}{(\lambda ({\mathbf {u}}_i)|W|)^2 2\pi r^2}\int _W \frac{2\pi (\Vert {\mathbf {u}}_{i}-{\mathbf {s}}\Vert )^2 \kappa ^2_{\epsilon }\left( \left\| {\mathbf {u}}_{i}-{\mathbf {s}}\right\| -r\right) }{\lambda ({\mathbf {s}}) \left| \partial b\left( {\mathbf {u}}_{i},\Vert {\mathbf {u}}_{i}-{\mathbf {s}}\Vert \right) \cap W\right| }\mathrm {d}{\mathbf {s}} + \frac{3\Lambda (W) + 3}{(\lambda ({\mathbf {u}}_i)|W|)^2}.\nonumber \\ \end{aligned}$$

(10)

Proof

Given that point counts are Poisson distributed, we have the following moments:

$$\begin{aligned} {\mathbb {E}}_![N(A)-1]= & {} \Lambda (A), \\ {\mathbb {E}}_![(N(A)-1)^2]= & {} \Lambda ^2(A) + \Lambda (A), \\ {\mathbb {E}}_![(N(A)-1)^3]= & {} \Lambda ^3(A) + 3\Lambda ^2(A) + \Lambda (A), \\ {\mathbb {E}}_![(N(A)-1)^4]= & {} \Lambda ^4(A) + 6\Lambda ^3(A) + 7\Lambda ^2(A) + \Lambda (A). \end{aligned}$$

For simplicity, we define

$$\begin{aligned} \phi ({\mathbf {u}}_{i},{\mathbf {u}}_{j},r)=\frac{\kappa _{\epsilon }\left( \left\| {\mathbf {u}}_{i}-{\mathbf {u}}_{j}\right\| -r\right) 2\pi \Vert {\mathbf {u}}_{i}-{\mathbf {u}}_{j}\Vert }{\left| \partial b\left( {\mathbf {u}}_{i},\Vert {\mathbf {u}}_{i}-{\mathbf {u}}_{j}\Vert \right) \cap W\right| }. \end{aligned}$$

The variance with respect to the reduced Palm process is

$$\begin{aligned} {\mathbb {V}}\text {ar}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right] = \frac{1}{(|W|2\pi r)^2}{\mathbb {E}}_{!}\left[ {\mathbb {E}}\left\{ (n-1)^2 \left. \left( \sum _{j\ne i} \frac{ \phi ({\mathbf {u}}_{i},{\mathbf {u}}_{j},r)}{\lambda ({\mathbf {u}}_{i}){\lambda }({\mathbf {u}}_{j})}\right) ^2 \right| N(W) = n \right\} \right] - \left( \frac{\Lambda (W) + 1}{\lambda ({\mathbf {u}}_{i})|W|}\right) ^2.\nonumber \\ \end{aligned}$$

(11)

For solving the first term of Eq. (11), we use the identity

$$\begin{aligned} \left( \sum _{j\ne i}\frac{ \phi ({\mathbf {u}}_{i},{\mathbf {u}}_{j},r)}{\lambda ({\mathbf {u}}_{i}){\lambda }({\mathbf {u}}_{j})}\right) ^2 =\sum _{j\ne i}\frac{\phi ^2({\mathbf {u}}_{i},{\mathbf {u}}_{j},r)}{\lambda ^2 ({\mathbf {u}}_{i})\lambda ^2 ({\mathbf {u}}_{j})} + \sum _{j\ne i}\sum _{\begin{array}{c} k\ne i \\ k \ne j \end{array}}\frac{ \phi ({\mathbf {u}}_{i},{\mathbf {u}}_{j},r)\phi ({\mathbf {u}}_{i},{\mathbf {u}}_{k},r)}{\lambda ^2 ({\mathbf {u}}_{i})\lambda ({\mathbf {u}}_{j})\lambda ({\mathbf {u}}_{k})}. \end{aligned}$$

Again, taking advantage of the properties of the expectation, we note that for any integrable function $\varphi ({\mathbf {u}})$ on ${\mathbb {R}}^2$, we have

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^{n-1} \frac{\varphi ^2({\mathbf {u}}_i)}{\lambda ^2({\mathbf {u}}_i)}\right] = \frac{n-1}{\Lambda (W)}\int _W \frac{\varphi ^2({\mathbf {s}})}{\lambda ({\mathbf {s}})}\mathrm {d}{\mathbf {s}}, \end{aligned}$$

and

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{j=1}^{n-1}\sum _{k \ne j} \frac{\varphi ({\mathbf {u}}_j)}{\lambda ({\mathbf {u}}_j)}\frac{\varphi ({\mathbf {u}}_k)}{\lambda ({\mathbf {u}}_k)}\right] = {\mathbb {E}}\left[ \sum _{j=1}^{n-1}\frac{\varphi ({\mathbf {u}}_j)}{\lambda ({\mathbf {u}}_j)}\sum _{k \ne j} \frac{\varphi ({\mathbf {u}}_k)}{\lambda ({\mathbf {u}}_k)}\right] = \frac{(n-1)(n-2)}{\Lambda ^2(W)}\left( \int _W \varphi ({\mathbf {s}})\mathrm {d}{\mathbf {s}}\right) ^2. \end{aligned}$$

So the first term is

$$\begin{aligned}&\frac{1}{(\lambda ({\mathbf {u}}_i)|W|2\pi r)^2} {\mathbb {E}}_{!}\left[ \frac{(N(W)-1)^3}{\Lambda (W)} \int _W \frac{\phi ^2({\mathbf {u}}_{i},{\mathbf {s}},r)}{\lambda ({\mathbf {s}})}\mathrm {d}{\mathbf {s}} \right. \\&\quad + \left. \frac{(N(W)-1)^3(N(W)-2)}{\Lambda ^2(W)}\left( \int _W \phi ({\mathbf {u}}_{i},{\mathbf {s}},r)\mathrm {d}{\mathbf {s}}\right) ^2\right] . \end{aligned}$$

To solve the last integral above, we proceed as in Theorem 1 and get a value of $(2\pi r)^2$. Therefore, collecting all pieces gives

$$\begin{aligned}&{\mathbb {V}}\text {ar}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right] =\frac{\Lambda ^3(W) + 3\Lambda ^2(W) + \Lambda (W)}{\Lambda (W)(\lambda ({\mathbf {u}}_i)|W|2\pi r)^2} \int _W \frac{\phi ^2({\mathbf {u}}_{i},{\mathbf {s}},r)}{\lambda ({\mathbf {s}})}\mathrm {d}{\mathbf {s}} \\&\quad + \frac{\Lambda ^4(W) + 5\Lambda ^3(W) + 4\Lambda ^2(W)}{\Lambda ^2(W)(\lambda ({\mathbf {u}}_i)|W|2\pi r)^2} (2\pi r)^2 - \frac{\Lambda ^2(W) + 2\Lambda (W) + 1}{(\lambda ({\mathbf {u}}_i)|W|)^2} \end{aligned}$$

and simplifying,

$$\begin{aligned} {\mathbb {V}}\text {ar}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right]= & {} \frac{\Lambda ^2(W) + 3\Lambda (W) +1}{(\lambda ({\mathbf {u}}_i)|W|2\pi r)^2} \int _W \frac{\phi ^2({\mathbf {u}}_{i},{\mathbf {s}},r)}{\lambda ({\mathbf {s}})}\mathrm {d}{\mathbf {s}} + \frac{3\Lambda (W) + 3}{(\lambda ({\mathbf {u}}_i)|W|)^2}\\= & {} \frac{1}{(\lambda ({\mathbf {u}}_i)|W|)^2} \left( \frac{\Lambda ^2(W) + 3\Lambda (W) +1}{(2\pi r)^2}\int _W \frac{\phi ^2({\mathbf {u}}_{i},{\mathbf {s}},r)}{\lambda ({\mathbf {s}})}\mathrm {d}{\mathbf {s}} + 3\Lambda (W) + 3\right) , \end{aligned}$$

which finally gives,

$$\begin{aligned} {\mathbb {V}}\text {ar}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right]= & {} \frac{1}{(\lambda ({\mathbf {u}}_i)|W|)^2} \left. \Big (\frac{\Lambda ^2(W) + 3\Lambda (W) +1}{2\pi r^2}\int _W \frac{2\pi (\Vert {\mathbf {u}}_{i}-{\mathbf {s}}\Vert )^2 \kappa ^2_{\epsilon }\left( \left\| {\mathbf {u}}_{i}-{\mathbf {s}}\right\| -r\right) }{\lambda ({\mathbf {s}}) \left| \partial b\left( {\mathbf {u}}_{i},\Vert {\mathbf {u}}_{i}-{\mathbf {s}}\Vert \right) \cap W\right| }\right. \nonumber \\&\quad \mathrm {d}{\mathbf {s}} + \left. 3\Lambda (W) + 3\Big )\right. \end{aligned}$$

(12)

$\square$

Corollary 1

In the homogeneous case,

$$\begin{aligned} {\mathbb {V}}\text {ar}_{!}\left[ {\hat{\rho }}_{\epsilon }^{(i)}(r)\right] =\lambda ^{4}{\mathbb {V}}\text {ar}_{!}\left[ {\hat{g}}_{\epsilon }^{(i)}(r)\right] , \end{aligned}$$

where ${\hat{\rho }}_{\epsilon }^{(i)}(r)$ is the $i\hbox {th}$ LISA function based on the product density and $\lambda$ is the constant intensity.