**Overview**

For Exercise 1, I wanted to know about the spatial pattern of western hemlock trees infected with western hemlock dwarf mistletoe. I used a hotspot analysis to determine where clusters of infected and uninfected trees were in my 2.2 ha study area (Map 1). I discovered a hot spot and a cold spot, indicating two clusters, one of high values (infected) and one of low values (uninfected).

For Exercise 2, I wanted to know how the spatial pattern of these clustered infected and uninfected trees were related to the spatial pattern of fire refugia outlined in my study site. I used Geographically weighted Regression to determine the significance of this relationship, however I did not find a significant relationship between a western hemlock, its intensity of clustering and infection status, and it’s distance to its nearest fire refugia polygon.

This result led to the realization that the polygons as they were drawn on the map were not as relevant as the actual “functional refugia”. I hypothesized that, after the 1892 fire, the only way for western hemlock dwarf mistletoe to spread back into the stand would be from the trees that survived that fire, or “remnant” trees. These would then infect the “regeneration” tree that came after the 1892 fire. The functional refugia I am interested in are defined by the location of the remnant western hemlocks. I also hypothesized that the spatial pattern of non-susceptible host tree (trees that were not western hemlocks) would play a role in the distribution of the mistletoe.

**Question Asked**

How are the spatial patterns of remnant western hemlocks related to the spatial patterns of regeneration western hemlocks, uninfected western hemlocks, and non-western hemlock tree species, and how are these relationships related to the spread of western hemlock dwarf mistletoe in the stand?

### The Kcross and Jcross Functions

The cross-type functions (also referred to as multi-type functions) are tools capable of comparing the spatial patterns of two different type events (type i and j) in a similar spatial window, of some point process, X. It does this by assigning labels to the events differentiating the type and summarizing the number or distance, of and between events, at differing spatial scales, or radius circles (r).

The statistic *Kij (r)* , summarizes the number of type j events, around a type i event at a distance of r, or a point process X. Deviations of the observed *Kij(r)* curve from the the Poisson curve, or if type j events are truly randomly distributed, indicates dependence of type j events on type i events. Similar results can be obtained from the regular Ripley’s K: deviations above the curve indicate clustering and deviations below indicate dispersal.

(Incredibly helpful and interactive explanation link: https://blog.jlevente.com/understanding-the-cross-k-function/)

The statistic *Jij(r) = (1 – Gij(r))/(1-Fj(r)) *summarizes the shortest distance between a type i and j event and compares it to the empty space function of type j event. This is another test for inferring independence or dependence of type j events to type i. Deviations of the *Jij(r)* curve the value of 1, indicate levels of dependence of the events to each-other. Specific deviations from 1 can be hard to interpret without an understanding of the *Fj(r)* function so imagining it stationary in the ratio makes it easier. As* Gij(r) *increases, the numerator shrinks, creating a smaller *Jij(r) *statistic. Deviations below 1 indicate that type *i* and *j* events are dependent and that as *r* increases, the shortest distance between points of type *i* and *j* increases. As* Gij(r) *decreases, the numerator grows, creating a larger *Jij(r) *statistic. Deviations above 1 indicate that type *i* and *j* events are dependent and that as *r* increases, the shortest distance between points of type *i* and *j* decreases.

**Methods Overview**

In R, the package “spatstat” provides a suite of spatial statistic functions including the cross-type functions. In order to use these you need to create a “point pattern process” object. These objects incorporate X and Y coordinates, and a frame of reference, or a “window,” and give spatial context top a list of values. Then marks are applied to these points that create the necessary multi-type point pattern process object. These marks serve to distinguish the type *i *and type *j* events described earlier in your analysis. Then running the “Kcross()” or “Jcross()” functions with the specified type events produces a graph that you can interpret, very similar to producing the normal Ripley’s K plot.

- I took my X – Y coordinates of all trees on the stand and added a column called “Status” to serve as my mark for the point pattern analysis.
- The four statuses were “Remnant,” “Regen,” “Uninfected,” and “NonHost” to identify my populations of interest.
- I had access to tree cores, so I identified trees that were older than 170 yrs old and these trees’ diameters served as my cutoff for the “Remnant” diameter class.
- All trees DBH > 39.8 cm.

- Doing this in ArcMap removed steps I would have to have taken when I migrated the dataset to R.
- I removed all the dead trees because I wasn’t concerned with them for my analysis.

- The four statuses were “Remnant,” “Regen,” “Uninfected,” and “NonHost” to identify my populations of interest.
- I exported this attribute table to a csv and loaded it into R Studio.
- I created the boundary window of my study site using the “owin()” function, and the corner points from my study site polygon.
- The function “ppp()” creates the point pattern object and I assigned the marks to the data set using the “Status” column I created in ArcMap
- It’s important your marks are factors otherwise it is not converted into a multi-type point pattern object.

- The last step is running the “Kcross()” and “Jcross” to compare the “Remnant” population to the “Regen,” “Uninfected,” and “NonHost” populations.
- This produced 6 plots, 3 of each type of cross-type analysis.
- Compare these easily using the “par()” function, for example:

par(mfrow = c(1,3))

plot(Ex3.Kcross1)

plot(Ex3.Kcross2)

plot(Ex3.Kcross3)

This produces the three plots in a single row and three columns.

**Results**

Because I am assuming the remnant, infected western hemlock trees are one of the main factors for the spread of western hemlock dwarf mistletoe and that they are the center of new infection centers on the study site, I did all my analysis centered on the remnant trees (points with status = “Remnant” treated as event type i).

**1) i = Remnant, j = Regen**

The first analysis between remnant and regeneration trees demonstrate that there is dependence on the two events to each other. At fairly small distances, or values of r, infected western hemlocks that have regenerated after the 1892 fire cluster around infected remnant western hemlocks that survived the 1892 fire. This stands to reason because we assume that infected trees will be near other infected trees, and that infection centers start usually with a “mother tree.” In this case the remnant trees serve as the start of the new infection centers. The Jcross output also shows me that the two types of trees are clustered using the frequency of the shortest distances. After ~8 meters the two tree types exhibit definite clustering. In terms of the function, the Gij(r) in the numerator of the Jij(r) function is approaching 1, or the highest frequency of very short distances.

**2) i = Remnant, j = Uninfected**

The Kcross plot from the second set of analyses between remnant and uninfected trees demonstrates that there is independence between the two events up to ~15 meters. After that, the trees exhibit slight clustering effects. The lack of dispersal tendencies is strange for these two types of trees because we expect uninfected trees to be furthest away from the center of infection centers. The presence of clustering may be indicative of the small spatial scale of my study site. It may also be that the size of the infection centers are only about 15 meters (if we assume that remnant trees are the center). The Jcross plot shows something similar: at small distances the types of trees seem independent and then around 8 meters they exhibit clustering.

**3) i = Remnant, j = NonHost**

The Kcross from the last set of analyses between the remnant trees and the non-hosts demonstrates a similar pattern exhibited by the regeneration trees. After about 4 meters, the trees tend to be clustered. This is an interesting find because if the non-hosts cluster to remnant trees but uninfected trees are independent, then the non-hosts may be playing a role in this. The Jcross plot shows the same: the two types of trees exhibit clustering.

**4) Comparing Kcross Functions with eval.fv()**

A useful way to compare patterns of Kcross functions is using the eval.fv function. The titles of each plot tell which Kcross was subtracted from which; note the difference in scales. The first plot shows that the regenerating trees’ spatial pattern as related to remnant trees is very different from the uninfected trees’ pattern. The regenerating trees’ spatial pattern is much more similar to the non-hosts’ spatial pattern at short distances, until about 15 meters. Then the patterns differ with the regenerating trees exhibiting more of a clustering tendency. However the scale is much smaller than the other two graphs. Lastly, the third plot shows the difference between the non-host trees’ spatial pattern and the uninfected trees’ spatial pattern. There appears to be a stepwise relationship where, at very near and very far distances the non-host trees are much more clustered, but at moderate distances the differences may be less dramatic.

**Critique of Cross-Type Functions**

The amount of easily interpretable literature on the spatstat package as a whole is sparse, although a wealth of very technical information exists. The function was easy to use and execute though and so was the process of creating the point pattern object. These two functions can clearly show how the spatial patterns of the two types of events change with scale. It would be helpful if there was a way to compare three or more types of events. The last drawback is that there is a lack of specific information for each point on your map or study site. This pattern that is generalizable to a whole set of points may not be as useful when trying to put together a story, such as the story of a stand’s development through time.

### Additional Raster Analysis

The last critique of the cross-type functions led me to attempt a visualization of these patterns on my stand. Very briefly, I determined densities of the infected, uninfected, and the non-host trees using the Kernel Density function in ArcMap. Then I classified these densities using natural breaks and coded these for raster addition. After adding all three density rasters together, I coded each unique density classification combination to tell me how the densities of the populations appeared in the study site.

It appears that there are distinct patches of high density separated by areas of low density. On the eastern side of my study site, it appears that the high density areas of infected trees cluster with the remnant infected trees. An interesting interaction is occurring between the high density patches of uninfected trees and infected trees in the western portion of my study site. The mechanism for the seemingly clear divide may be the non-TSHE trees.