Quantifying Spatial Patterns

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# Quantifying Spatial Patterns

.class-info[

**Session 23**

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

]
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# Today's objectives

* Introduce the concept of spatial patterns as realizations of _spatial processes_

* Demonstrate the use of expected values for evaluating a hypothesized process

* Introduce measures for characterizing spatial patterns

* Explore the concept of __scale__ and its implications for understanding spatial processes

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# Patterns as process
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# Description vs. process?

.pull-left[

* Vizualization and the detection of patterns

* The challenge of geographic data

* Implications for analysis 
]

.pull-right[
<figure>
  <img src="img/12/gini-map-census.png" alt="ZZZ" title="ZZZ" width="100%">
</figure>
.caption[
Inequality in the United States: Quintiles of Gini Index by County: 2006–2010. The greater the Gini index, the more unequal a county’s income distribution is.
]
]
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# Patterns as realizations of spatial processes

* A __spatial process__ is a description of how a spatial pattern might be _generated_

* __Generative models__

* An observed pattern as a _possible realization_ of an hypothesized process
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# Deterministic vs. stochastic processes

.pull-left[
* Deterministic processes: always produces the same outcome

$$
z = 2x + 3y
$$
* Results in a spatially continuous field
]
.pull-right[
<img src="12-slides_files/figure-html/detproc-1.png" width="504" style="display: block; margin: auto;" />
]
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# Deterministic vs. stochastic processes

.pull-left[

* Stochastic processes: variation makes each realization difficult to predict

$$
z = 2x + 3y + d
$$
* The _process_ is random, not the result (!!)
* Measurement error makes deterministic processes appear stochastic
]
.pull-right[
<img src="12-slides_files/figure-html/stocproc-1.png" width="504" style="display: block; margin: auto;" />
]
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<img src="12-slides_files/figure-html/stocproc2-1.png" width="504" style="display: block; margin: auto;" />
}
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# Expected values and hypothesis testing

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* Considering each outcome as the realization of a process allows us to generate expected values

* The simplest spatial process is Completely Spatial Random (CSR) process

* __First Order__ effects: any event has an equal probability of occuring in a location

* __Second Order__ effects: the location of one event is independent of the other events
]
.pull-right[
<figure>
  <img src="img/12/IRP_CSR.png" alt="ZZZ" title="ZZZ" width="100%">
</figure>
.caption[
From Manuel Gimond
]
]
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# Generating expactations for CSR

.pull-left[
<img src="12-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" /><img src="12-slides_files/figure-html/unnamed-chunk-1-2.png" width="504" style="display: block; margin: auto;" />
]
.pull-right[
* We can use quadrat counts to estimate the expected number of events in a given area

* The probability of each possible count is given by:
$$
P(n,k) = {n \choose x}p^k(1-p)^{n-k}
$$

* This becomes computationally exhausting...

]
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name: tobler
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# Tobler's Law
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> ‘everything is usually related to all else but those which are near to each other
>are more related when compared to those that are further away’.
> <footer>Waldo Tobler</footer>

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# Spatial autocorrelation

<figure>
  <img src="img/12/Random_maps.png" alt="ZZZ" title="ZZZ" width="100%">
</figure>
.caption[
From Manuel Gimond
]
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# The challenge of areal data

* Spatial autocorrelation threatens _second order_ randomness

* Areal data means an infinite number of potential distances

* Neighbor matrices, `$\boldsymbol W$`, allow different characterizations

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# (One) Measure of autocorrelation
.pull-left[
* Moran's I

<figure>
  <img src="img/12/MI.png" alt="ZZZ" title="ZZZ" width="100%">
</figure>
]

.pull-right[
<figure>
  <img src="img/12/mI2.png" alt="ZZZ" title="ZZZ" width="100%">
</figure>
]
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# The importance of scale

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# What do we mean by __scale__?

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# Why might we care about scale?

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# Implications of scale for understanding spatial processes