Point Pattern Analysis

Data

Minnesota has the highest volume of ash trees in the U.S. The presence of EAB were discovered in May 2009 in the South Saint Anthony Park neighborhood in St. Paul, MN within Ramsey County. In response, the Minnesota Department of Agriculture has placed regulatory quarantines on a county-by-county basis (Minnesota Department of Agriculture, 2018). Determining spatial patterns present at more localized scales could still aid in decision making towards where to apply insecticide treatments, remove trees, and introduce biological controls with finite funding and resources.

Data for this project include a subset of point locations of ash trees where EAB were identified to be present through larval signs underneath ash tree bark and trapping across the state of Minnesota (Figure 1). Data have been collected via GPS in tree inventories and published by the Minnesota Department of Agriculture (2014) from 2009 to present and are updated during the EAB active season from May to September of each year (Figure 2).

Methods

The projected coordinate system used for this analysis is the NAD 1983 UTM Zone 15N. As Ramsey County is located near the center of this UTM Zone, distortion should be approximately 1 part in 1000. The original data were collected through GPS with an accuracy of ±3 ft, so projecting to a more localized zone such as the Minnesota State Plane South system would not necessarily result in noticeably less distortion for this analysis.

Each of the 2,876 points included in this analysis represent an ash tree in the study area where the presence of EAB has been confirmed during a tree inventory by the Minnesota Department of Agriculture. A rectangular extent (20 km wide by 26 km long) was selected to closely align with the northern and eastern boundaries of Ramsey County as well as to aid in quadrat delineation. The study area thus extends slightly westward into the twin city of Minneapolis, where EAB have also been identified.

Quadrat Count Method

The quadrat count is a density-based method to describe a point pattern in which the location of each event might be affected by an underlying attribute at that location. It involves creating a regular grid of quadrats across the study area (Figure 3) and then counting the number of events that occur in each quadrat (Figure 4). Squares with side lengths of 2 km were used to generate 130 quadrats across the study area. The frequency distribution of the quadrats and the number of EAB trees they contain was then tabulated (Figure 5) to calculate the mean, the observed variance, and the variance-mean ratio (VMR) to signify clustering, randomness, or dispersion present (O’Sullivan & Unwin, 2014).

Average Nearest Neighbor Method

The average nearest neighbor is a distance-based method to describe point patterns that might be affected by the location of each event depending on the location of other events. The distance from each event to the nearest event that is also in the point pattern is calculated and then averaged across the study area (Figure 6). This average is then compared to the expected mean distance for a hypothetical random pattern of events. From this distribution, the significance level (p-value) and critical value (z-score) statistics can be calculated to indicate any patterns present (O’Sullivan & Unwin, 2014).

Ripley's K Function Method

Ripley’s K function is another distance-based measure of point patterns. It utilizes a series of distances to indicate whether clustering or dispersion is present as it is based on the ability of patterns to vary at different distance and scales. A series of distances are placed as circles around each point, and then the number of events inside each circle is counted so that the mean count for all events can be divided by the study area event density to give Ripley’s K as a function of distance (Figure 7). An expected K function based on a random pattern is also calculated for a number of permutations to generate a confidence envelope to indicate where clustering or dispersion may be occurring at different distances (O’Sullivan & Unwin, 2014). 

Assumptions

The quadrat size of 4 km2 was chosen for ease of calculation, as this resulted in a grid of 10 by 13 quadrats, but also to reduce the number of quadrats that contained zero events. However, this still resulted in 41 out of 130 quadrats that contained no EAB tree events. For the average nearest neighbor method, Euclidean distances were used as the effects of the Earth's curvature would not be pronounced at this scale. As the observed Ripley’s K function was already indicating significant clustering at smaller distances, calculating the K function at distances greater than 500 m did not seem to provide additional information about the potential pattern. 

Results

Quadrat Count Method

The quadrat count calculations are given in Table 1. As the VMR is much greater than 1, this indicates that significant clustering is present. 89 of the 130 quadrats contained trees with EAB present, and slightly over half of the quadrats contained more than one tree. Only 28 of the 130 quadrats contained greater than the mean number of EAB trees, indicating that the distribution is positively skewed. The 7 quadrats with the greatest number of events contained over half of the total EAB trees in the study area. 

Average Nearest Neighbor Method

The summary statistics from the average nearest neighborhood method are shown in Figure 8. As the nearest neighbor ratio of 0.27 is less than 1, this indicates that clustering is present. The p-value equaling 0 suggests that this distribution is unlikely to occur due to a random process. The highly large, negative z-score also indicates that significant clustering is present and is not likely to be a result of random chance.

Ripley's K Function Method

The observed Ripley’s K function is plotted over distance in Figure 9. This also suggests that significant clustering is present, as the observed K function is greater than the expected K function as well as the upper confidence envelope across the entirety of the 500 m distance.

Conclusion

All three of the point pattern analyses conducted in this study indicate that clustering is present among the EAB trees identified in Ramsey County, MN. This corresponds with what would be expected in the natural world. First-order effects are present, in that trees cannot be located everywhere in the study area. Trees are much more likely to be in parks, natural areas, and along neighborhood streets than in buildings, water bodies, or on impervious cover. A serious limitation is in the second-order effects, as ash trees near other ash trees facilitate the spread of EAB. A comprehensive tree inventory of all ash trees in the study area, not just those already infected, could thus be highly useful towards slowing the spread of EAB.

The quadrat method indicates that EAB trees are more densely clustered in several key areas, such as the park where EAB were first discovered, along rows on neighborhood streets, and the banks of the Mississippi River where trees are more likely to be located. The results of the nearest neighbor method suggest that the distances between trees are small, which again attests to how EAB spread from tree to tree. The Ripley’s K function illustrates that the EAB trees are also clustered across larger distances. This could be a result of EAB spreading over the 15 years of data collection, or it could also indicate the transport of infested firewood which would be more difficult to track.

There are several other limitations that affect this analysis. Edge effects at the boundary of the study area were not accounted for. EAB do not conform to county lines or any other artificial boundaries. Extending the study area would affect all three of the point pattern methods, as there are additional infected trees along all sides of the study area. Similarly, the size of the study area may be arbitrarily large. As counties in Minnesota are already quarantined at the county level, it may be more of interest to analyze EAB tree distributions at smaller scales to assess if applied treatments and controls are effectively slowing the growth and spread of EAB. To accomplish this, the timing of EAB spread would need to be considered. While this study explored patterns across all 15 years of data collection, further analysis could include examining how EAB has spread on a yearly basis.

The null hypothesis for this study can be rejected, in that the distribution of ash trees where EAB are present within Ramsey County, MN are not spatially random but instead exhibit clustered patterns.

References

Minnesota Department of Agriculture. (2014). Emerald Ash Borer Trees [Data set]. Minnesota Geospatial Commons. https://gisdata.mn.gov/dataset/env-emerald-ash-borer

Minnesota Department of Agriculture. (2018). Guidelines to slow the growth and spread of Emerald Ash Borer. Minnesota Department of Agriculture. https://www.mda.state.mn.us/sites/default/files/inline-files/EAB%20Management%20Guidelines%202018%20WEB.pdf

O’Sullivan, D., & Unwin, D. J. (2014). Geographic information analysis (2nd ed.). John Wiley & Sons. https://doi.org/10.1002/9780470549094