Multivariate analysis

Principle Components Analysis

Principle components analysis (PCA) was the initial method used to establish prominent relationships in the data. Due to the nature of the analysis, it allows for the possibility of revealing unexpected correlations in the data that might have otherwise been missed (McCune et al., 2002). A diagram of the visual interpretation of the results can be observed in Figure 4.8. The resulting eigenvectors were successful at explaining the variance within the dataset. Axis 1 explained 97% of the variance, while axis 2 explained 3%, leading to a cumulative explanation of 100% of the data along the first 2 axis. Stand contagion is most

Figure 4.8: 2-Dimensional solution for PCA results. ’Richness’ is species richness, ’tpha’ is trees per hectare, ’bioha’ is biomass (kg) per hectare, ’baha’ is basal area (m2_{) per hectare.}

Figure 4.9: 2-D NMDS output

significantly correlated with axis 1 while relative ponderosa pine abundance and Shannon diversity scores are nearly equally and oppositely correlated with axis 2.

Non-metric multidimensional scaling

Non-metric Multidimensional Scaling (NMDS) provides a different perspective from PCA in that it does not assume a linear relationship between response variables. This makes it a useful tool within the multivariate analysis toolkit since it has the potential to reveal oth- erwise hidden relationships, particularly among environmental data (McCune et al., 2002). Figure 4.9 shows a graphical representation of the NMDS results. The NMDS analysis re- ported an overall type 1 stress of 0.036 across 2 dimensions. 33 iterations were performed to reach the final result.

4.4 Discussion

4.4.1 Overview

The spatial interactions between trees in a forest stand plays a significant role in each in- dividual’s growth trajectory. As space is consumed by one stem, another will respond to the unavailability of that space. An example of this is the analysis of distance-dependent relationships in forest stands by Tome and Burkhart (1989). In conjunction with the fact that technologies such as LiDAR are becoming faster and lighter, this means it will be easier to perform spatial analysis on forest stands. This will increase the available bandwidth for research on these types of spatial interactions. In which case, it is beneficial to improve the understanding of how various forest biometrics relate to the spatial function of the stand. Significant correlations between spatial metrics and biometrics will allow for the inference of biometric measurements from spatial data. Additionally, the increased accessibility of spatial data will enhance the understanding of how commonly observed attributes like stand density, species mixture, and QMD influence the spatial organization of the stand.

Therefore, the objective of this analysis is to determine:

1. If any meaningful correlative or causal relationship exists between these two groups of variables.

2. If it is possible to use one of the calculated spatial metrics as a proxy for a biometric. 3. Examine how any the given forest biometrics affect the spatial organization of the

stand.

4.4.2 Spatial Indexes

Four spatial indexes were investigated in this chapter, nearest neighbor, stand contagion, species mingling, and DBH differentiation. Composed of the simplest calculation, nearest

neighbor was the most strongly correlated with the featured stand biometrics. This is sup- ported not only by the linear relationships tested, but from the results of the multivariate analyses. Both the PCA and NMDS indicated a strong correlation between mean nearest neighbor and their respective primary axis. Additionally, species mingling was strongly cor- related with the Shannon diversity index. The two other indexes, stand contagion and DBH differentiation, were not found to be significantly correlated with any biometric.

4.4.3 Nearest Neighbor

The nearest neighbor index was most significantly correlated with trees per hectare as shown in Figure 4.2. This is not a particularly surprising finding on its own as it is a reasonable assumption that as the number of trees in a stand increases, the mean distance between trees is likely to decrease. However, stand density alone does not indicate how stems distribute at varying density levels. They could either evenly distribute or cluster according to some environmental factor. The former would have a more significant impact on the mean nearest neighbor for the stand than the latter. Combined, these two variables can reveal whether or not stems are clustering within the stand.

Operating on the assumption that trees will distribute themselves the maximum possible distance from each other, an ideal relationship can be established between stand density and mean nearest neighbor. This relationship can then be used as a baseline to compare against measured values of these two variables, allowing for the determination of any significant clustering happening within the stand. An example of what this relationship would look like is shown in Figure 4.10. The dashed line indicates the maximum nearest neighbor value at a given stand density under ideal conditions. Further deviation from the dashed line into the area highlighted in red indicates a greater degree of stem clustering within the stand.

Mean nearest neighbor was also found to be significantly correlated to quadratic mean diameter (Figure 4.3). As mean nearest neighbor increases, so does QMD. Furthermore, QMD was related to stand density at near the same magnitude. What is most likely the

Mean

Nearest

Neigh

bor

Stand Density

Figure 4.10: Example of a baseline relationship between stand density and mean nearest neighbor. The dashed line indicates the hypothetical baseline, while the area shaded in red indicates where the mean nearest neighbor would be lower than expected for the stand density, potentially indicating clustering.

Sp

ecies

Mingling

Species Diversity

Figure 4.11: Example of a baseline relationship between species diversity and species min- gling. The dashed line indicates the hypothetical baseline, while the area shaded in red indicates where species mingling would be lower than expected for the stand density, poten- tially indicating species clustering.

underlying cause for this relationship is that QMD and stand density are significantly and inversely related. Therefore a variation in either QMD or stand density is going to affect the other (Reineke, 1933).

4.4.4 Species Mingling

In terms of R2 _{value, species mingling was strongly correlated with the Shannon diversity}

index. The mechanism for this relationship is likely due to the typical spatial distribution of species individuals within a forest stand. As the stand approaches full species evenness, the likelihood that the neighbor of a given stem is of another species increases. This results in a strong, positive correlation between species diversity and species mingling. However, much in the way of how mean nearest neighbor can describe spatial distribution with respect to stand density, species mingling can do so with respect to species diversity.

Similar to the proposed index between stand density and nearest neighbor, Figure 4.11 presents an index to examine clustering within the stand at the species level. If a species mingling value is low relative to the species richness or diversity of the stand, this would indicate a greater degree of spatial clustering across the stand. With the dashed line rep- resenting a perfectly even distribution of species individuals throughout the stand, points within the red region would indicate some amount of species clustering.