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).
In my study site, 2 fires burned. Once in 1829, burning most of the stand, and then again in 1892, burning everywhere except the fire refugia (polygons filled in blue). This created a multi-storied forest with remnant trees located in the fire refugias. One component of the remnant forest are infected western hemlocks. These remnant hemlocks serve as the source of inoculum for the hemlocks regenerating after the 1892 fire.
For Exercise 2, my research question was: How does the spatial pattern of fire refugia affect the spatial pattern of western hemlock dwarf mistletoe?
I predicted that a cluster of infected western hemlocks are more likely to be next to a fire refugia than a cluster of uninfected trees. In order to assess this relationship, I used the geographically weighted regression tool in ArcMap.
Geographically Weighted Regression
Geographically weight regression (GWR) works by creating a local regression equation for each feature in a data set you want to analyze, using an explanatory variable(s) to predict values for the response variable, using the least squares method. The Ordinary Least Squares (OLS) tool differs from GWR because OLS creates a global regression model (one model for all features) whereas GWR creates local models (one model per feature) to account for the spatial relationship of the features to each other. Because the method of least squares is still used, assumptions should still be met for statistically rigorous testing. The output of the GWR tool is a feature class of the same type as the input, with a variety of attributes for each feature. These attributes summarize the ability of the local regression model to predict the actual observed value at that feature’s location. If you have an explanatory variable that explains a significant amount of the variation of the response variable, this is useful for seeing how its coefficient varies spatially.
Execution of GWR
To use this tool, I quantified the relationship between the trees and the fire refugia. I used the “Near” tool for this to calculate the nearest distance to a fire refugia polygon’s edge. This was my explanatory variable. My response variable was the z-score that was output for each tree from the Optimized Hot Spot Analysis. Then I ran the GWR tool. I then used the Moran’s I tool to check for spatial autocorrelation of the residuals. This is to check the clustering of residuals. Clustering indicates I may have left out a key explanatory variable. The figure below displays my process.
I tested the relationship between nearest distance to a fire refugia polygon’s edge and the z-score that was output for each tree from the Optimized Hot Spot Analysis using OLS, which is necessary to develop a well specified model. My R2 value for this global model was 0.005, which is incredibly small. Normally I would have stopped here and sought out other variables to explain this pattern, but for this exercise I continued the process.
Results
This GWR produced a high global R2 value of 0.98 (Adj R2 0.98) indicating that distance to refugia does a good job of explaining variance in the spatial pattern of infected and uninfected trees. However, examining the other metrics for the local model performance gives a different picture of model performance.
Map 2 displays results for the coefficients for the explanatory variable of distance to nearest refugia. As this variable changes, the z-score increases or decreases. These changes in z-scores indicate a clustering of high or low values. From examining the range of coefficient values, the range is quite small, -0.513 to 0.953. This means that across my study site, the coefficient only changes slightly from positive to negative. In the north western corner, we see a cluster of positive coefficient values. Here, as distance to refugia increases, the z-score of trees increases, predicting a clustering of infected trees. These values are associated with high local R2 values (Map 4). In other places of the stand we see slight clustering of negative coefficients, indicating distance to refugia decreases the z-score of trees, predicting a clustering of uninfected trees.
Map 3 displays the standardized residuals for each tree. Blue values indicate where the local model over-predicted what the actual observed value was, and red values are under-predictions. When residuals from the local regression models are distributed randomly (i.e. not clustered or dispersed) over the study area, then the geographically weighted regression model is fit well, or well specified. The residuals of the local regression models were significantly clustered. (Moran’s Index of 0.265, p-value of 0.000, z-score of 24.344). Because we can observe clustering in my study area of residuals, there is another phenomenon driving the changes in z-scores; in other words, driving the clustering of infected and uninfected trees.
From the previous two map evaluations I saw that the distance of a tree to fire refugia was not the only explanatory variable necessary to explain why infected and uninfected trees clustered. Map 4 displays the local R2 values for each feature. The areas in red are high local R2 values. We see the northwestern corner has a large number of large values which correspond to a cluster of small residuals and positive coefficients. Here, distance to fire refugia explains the clustering of infected trees well. The reverse is observed in several other places (clusters of blue) where distance to fire refugia does not explain why infected or uninfected trees cluster. In fact the majority of observations had a local R2 of 0.4 or less. From this evaluation, I believe this GWR model using distance to refugia does a good job of explaining the clustering of infected trees, but not much else.
Critique
GWR is useful for determining how the coefficient of an explanatory variable can change across an area. One feature in a specified area may have a slightly different coefficient from another feature, indicating these two features are experiencing different conditions in space. This allows the user to make decisions about where the explanatory has the most positive or negative impact. This result is not something you can derive from a simple OLS global model. This local regression process is something you could do manually but the tool in ArcMap makes this process easy. The output of GWR is also easy to interpret visually.
Some drawbacks are that you need to run the OLS model first for your data to determine which variables are significant in determining your response variable. If not, then a poorly specified model can lead to inappropriate conclusions about the explanatory variable (i.e. high R2 values). Also, the evaluation of how the features interact in space is not totally clear. The features are evaluated within a fixed distance or number of neighbors, but there is no description for how weights are applied to each neighboring feature. Lastly, for incidence data, this tool is much harder to use if you want to determine what is driving the spatial pattern of your incidence data. Some other continuous metric (in my case a z-score) must be used as the response variable, making results harder to interpret.
Model Results Follow-Up
After finding that distance to a refugia was not a significant driver for the majority of trees, I examined my data for other spatial relationships. After a hotspot analysis on solely the infected trees, I found that the dispersal of infected trees slightly lined up with the fire refugia drawn on the map (Map 5).
Among other measures, forest structure was used to determine where fire refugia were located. Old forest structure is typically more diverse vertically and less clustered spatially. Also infected western hemlocks are good indicators of fire refugia boundaries because as a fire sensitive tree species, they would not survive most fire damage and the presence of dwarf mistletoe indicates they have been present on the landscape for a while. From the map we can see that the dispersal of infected trees only lines up with the refugia in a few places. This mis-drawing of fire refguia bounds may be a potential explanation for under-performance of the GWR model.
Stephen, excellent work. Good understanding of GWR and appropriate use and interpretations. I think you are moving toward identifying other factors, such as tree species (potential for infection), to include in your spatial models, as well as focusing on the tree level (not the polygon) data.