Introduction to point patterns

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# Introduction to point pattern analysis

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**Session 25**

.light[HES597: Introduction to Spatial Data in R<br>
Boise State University Human-Environment Systems<br>
Fall 2021]

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# Goals for today

* Define point patterns and their utility spatial analysis

* Introduce methods for describing point patterns

* Describe several commmon methods for the statistical analysis of point patterns 
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# What is a point pattern?

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* _Point pattern_: A __set__ of __events__ eithin a study region (i.e., a _window_)

* __Set__: A collection of mathematical __events__

* __Events__: The existence of a point object of the type we are interested in at a particular location in the study region

* A _marked point pattern_ refers to a point pattern where the events have additional descriptors
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__Some notation:__

* `$S$`: refers to the entire set

* `$\mathbf{s_i}$` denotes the vector of data describing point `$s_i$` in set `$S$`

* `$\#(S \in A )$` refers to the number of points in `$S$` within study area `$A$`
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# Requirements for a set to be considered a point pattern

* The pattern must be mapped on a plane to preserve distance

* The study area, `$A$`, should be objectively determined

* There should be a `$1:1$` correspondence between objects in `$A$` and events in the pattern

* Events must be _proper_ i.e., refer to actual locations of the event

* For some analyses the pattern should be a census of the relevant events

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# Describing point patterns
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# Describing point patterns

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* _Density-based metrics_: the `$\#$` of points within area, `$a$`, in study area `$A$`

* _Distance-based metrics_: based on nearest neighbor distances or the distance matrix for all points

* _First order_ effects reflect variation in __intensity__ due to variation in the 'attractiveness' of locations

* _Second order_ effects reflect variation in __intensity__ due to the presence of points themselves
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<img src="13-slides_files/figure-html/clust-1.png" width="504" style="display: block; margin: auto;" />
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# Centrography
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* _Mean center_: the point, `$\hat{\mathbf{s}}$`, whose coordinates are the average of all events in the pattern
* _Standard distance_: a measure of the dispersion of points around the _mean center_
* _Standard ellipse_: dispersion in one dimension
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<figure>
  <img src="img/13/centrography.png" alt="ZZZ" title="ZZZ" width="100%">
</figure>
.caption[
From Manuel Gimond
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# Density based methods
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* The overall _intensity_ of a point pattern is a crude density estimate
`$$\hat{\lambda} = \frac{\#(S \in A )}{a}$$`
* Local density = quadrat counts
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<img src="13-slides_files/figure-html/qct-1.png" width="504" style="display: block; margin: auto;" />
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# Density based methods
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* __kernel density estimates__: similar to quadrat counts, but using a _moving window_ for the subregion

* __kernel__: defines the shape, size, and weight assigned to observations in the window

* weights often assigned based on distance from the window center
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<img src="13-slides_files/figure-html/krn-1.png" width="504" style="display: block; margin: auto;" />
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# Distance based metrics
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* Provide an estimate of the _second order_ effects

* _Mean nearest-neighbor distance_:
`$$\hat{d_{min}} = \frac{\sum_{i = 1}^{m} d_{min}(\mathbf{s_i})}{n}$$`

* _G_ function: the cummulative frequency distribution of the nearest neighbor distances

* _F_ function: similar to _G_ but based on randomly located points

* Clustering with _G_
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<img src="13-slides_files/figure-html/cumdist-1.png" width="504" style="display: block; margin: auto;" />
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# Distance based metrics

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* Nearest neighbor methods throw away a lot of information

* The _K_ function is an alternative, based on a series of circles with increasing radius
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<img src="13-slides_files/figure-html/Kfun-1.png" width="504" style="display: block; margin: auto;" />
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# Statistical analysis of point patterns
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# Simple tests

* Distributional assumptions and heuristics
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# Expected Values

* We can use the `$\chi^{2}$` distribution to compare an observed point pattern to one generated by an expectation

* Often CSR is not interesting so we can use _Monte Carlo Methods_ to generate custom expectations

* We'll do this on Thursday

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# Estimating the parameters of covariate effects

* Often we are more interested in the effects of specific covariates on the resulting pattern

* Spatial regression models

* These are the foundation of "event distribution models"

* More on these after break