20_complete-spatial-randomness.Rmd

# Complete spatial randomness

**Learning objective:**

- learn how to test whether a given spatial point pattern deviates from CSR

## Complete spatial randomness {-}

CSR is represented by the homogeneous Poisson process (chapter 19).

It is defined as a Poisson process with spatially constant $\lambda(x) = \lambda$.

## Randomness of a given spatial point pattern {-}

Most processes, hence their patterns, deviate from CSR to some degree.

For a given point pattern we can:

- describe the deviation from CSR using a test statistic;
- then compare the test statistic with its null distribution, i.e. under CSR, to perform a statistical hypothesis test.

## Test statistic {-}

The test statistic relies on _dividing_ the observation window into $m$ subregions.

For each subregion $i$ we compare:

- the observed number of events $n_{i,obs}$
- the -- under CSR -- expected number of events $n_{i,exp}$.\
For subregions of equal size, this is: $n_{i,exp} = n / m$

## Test statistic {-}

The test statistic is defined as:

$$X^2 = \sum_{i = 1}^{m}\frac{(n_{i,obs} - n_{i,exp})^2}{n_{i,exp}}$$

It follows a Chi-square distribution under the null hypothesis, i.e. _under CSR_.

$$X_{CSR}^2 \sim \chi_{m-1}^2$$

## Possible outcomes {-}

The test statistic can be:

- significantly less than expected (one-sided left testing): a **regular** point pattern.
    - i.e. points are more spaced than in a random pattern
- significantly greater than expected (one-sided right testing): a **clustered** point pattern.
    - i.e. points are more aggregated than in a random pattern
- significantly different than expected (two-sided testing): a **non-random** point pattern.
- _not_ significantly different than expected (one- or two-sided testing): the point pattern has not been shown to deviate from a **random** point pattern in one or both directions.

## spatstat functions {-}

- `quadratcount(<ppp>, nx = , ny = )` to define subregions (`nx` columns, `ny` rows) and count events per subregion
- `quadrat.test(<quadratcount>)` for CSR hypothesis testing. Extra arguments:
    - `alternative = `: possible are `"two-sided"` (default), `"clustered"` and `"regular"`
    - `method = `: defaults to `"Chisq"`; `"MonteCarlo"` is also possible (CSR distribution by simulations)

## Example: longleaf {-}

We don't want the marks (tree diameter) in plots, so let's drop those.

```{r message=FALSE}
library(spatstat)
(longleaf2 <- unmark(longleaf))
```

## Example: longleaf {-}

```{r out.width="100%"}
plot(longleaf2)
axis(1)
axis(2)
```

## Example: longleaf {-}

```{r}
qc <- quadratcount(longleaf2, nx = 6, ny = 6)
qc
```

`nx` and `ny` default to 5

## Example: longleaf {-}

```{r out.width="100%"}
plot(longleaf2, cols = "grey60")
plot(qc, add = TRUE, cex = 1.2)
```

## Example: longleaf {-}

```{r}
quadrat.test(qc)
```

## Example: longleaf {-}

```{r}
quadrat.test(qc, alternative = "clustered")
```

## Example: longleaf {-}

```{r}
quadrat.test(qc, alternative = "regular")
```

## Meeting Videos {-}

### Cohort 1 {-}

`r knitr::include_url("https://www.youtube.com/embed/URL")`

<details>
<summary> Meeting chat log </summary>

```
LOG
```
</details>