This document contains the code we used to conduct multi-scale pattern analysis (MSPA) following Jombart et al. (2009) and the results of the analysis. MSPA graphically depicts the correlation structure among species, environmental variables (fire severity in our analyses), and the previously calculated Moran’s Eigenvector Matrixes to assess correlation at all possible scales of variation.

The best Spatial Weighting Matrix (SWM) according to optimisation by maximizing adjusted R-squared (Bauman et al. 2018) with a Sidak (Sidak 1967) correction is the Relative Neighbour connection matrix calculated with a concave-up function.

## select SWM
## create list of candidate models
listw_pr=listw.candidates(
  xyir,
  style = "B",
  nb = c( "rel", "gab","del","mst"),
  weights = c( "binary", "flin", "fup", "fdown"))

## optimisation for selecting best SWM
W_sel=listw.select(
  datInvWide[,7:9],
  listw_pr,
  MEM.autocor = c("positive"),
  method = c("FWD"),
  MEM.all = FALSE,
  nperm = 999,
  alpha = 0.05,
  p.adjust = TRUE,
  verbose = TRUE
)

# W_sel shows best candidates - Relative_UP_0.5 is best with adjR2=0.52
##                     R2Adj      Pvalue N.var R2Adj.select
## Delaunay_Binary 0.3593277 0.001598801    66    0.3349287
## Delaunay_Linear 0.3629217 0.001598801    71    0.3380280
## Delaunay_Down_5 0.3600747 0.001598801    74    0.3407920
## Delaunay_Up_0.5 0.4871527 0.001598801   129    0.4867280
## Gabriel_Binary  0.4694276 0.001598801   114    0.4192142
## Gabriel_Linear  0.4724079 0.001598801   124    0.4556797
## Gabriel_Down_5  0.4710575 0.001598801   130    0.4428817
## Gabriel_Up_0.5  0.5285943 0.001598801   154    0.5139826
## Relative_Binary 0.4964577 0.001598801   122    0.4565401
## Relative_Linear 0.4982042 0.001598801   130    0.4675188
## Relative_Down_5 0.4960503 0.001598801   126    0.4605035
## Relative_Up_0.5 0.5279846 0.001598801   155    0.5244301
## MST_Binary      0.5130199 0.001598801   137    0.4746827
## MST_Linear      0.5124459 0.001598801   129    0.4732863
## MST_Down_5      0.5119097 0.001598801   138    0.4765375
## MST_Up_0.5      0.5200716 0.001598801   145    0.5100793
## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 3437 
## Number of nonzero links: 7482 
## Percentage nonzero weights: 0.06333717 
## Average number of links: 2.176898 
## 
## Weights style: B 
## Weights constants summary:
##      n       nn       S0       S1       S2
## B 3437 11812969 1244.872 445.9385 1821.882

We applied a Hellinger transformation to species abundance data to minimize the effects of different sample abundances.

## hellinger transformation
hellTrans <- function(X){
  if (!( is.matrix(X) | is.data.frame(X) )) stop("Object is not a matrix.")  
  if (any(is.na(X))) stop("na entries in table.")
  
  sumRow <- apply(X,1,sum)
  Y <- X/sumRow
  Y <- sqrt(Y)
  
  return(Y)
}

Spatial Structuring of Invasive Species Occurrence

Species abundances were weakly structured at a range of small to moderate scales (Figure 1). All three species clustered close to the centroid indicating that spatial structuring was not strong. However, they were not located at 0.0, indicating some spatial structuring. Smooth brome was structured at slightly larger scales than cheatgrass and Kentucky bluegrass (larger MEMs as indicated by the size of the red circles are influential in separation of smooth brome along axis 2 in the MSPA for species abundance (Figure 1. Kentucky bluegrass was structured at smaller scale (co-located in ordination space with smaller MEMs along axis 1 and 2). Cheatgrass fell closest to the centroid on both axes, indicating that it’s abundance was the least spatially structured of the three species.

## SPECIES DATA ##
## PCA of species abundances, after Hellinger transformation
pcaFau <- dudi.pca(hellTrans(datInvWide_env[,3:5]),scale=FALSE,scannf=FALSE)

## detrending of this PCA
pcaFau.detr <- pcaivortho(pcaFau,xyir,scannf=FALSE)

## MSPA of the detrended analysis
mspaFau <- mspa(pcaFau.detr,listw_relUp,scannf=FALSE,nf=2)
MSPA for species abundances. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each species in ordination space indicates the strength of the spatial structuirng for that species. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Figure 1: MSPA for species abundances. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each species in ordination space indicates the strength of the spatial structuirng for that species. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Table 1: Influential MEMs for invasive species spatial structuring, their position in ordination space on axis 1 and axis 2, and the overall difference from the centroid of each (diff)
MEM CS1 CS2 diff
MEM165 -0.253 0.360 0.612
MEM13 -0.280 -0.277 0.557
MEM8 -0.283 -0.232 0.516
MEM164 -0.201 0.304 0.505
MEM3 -0.219 -0.214 0.433
MEM23 -0.311 -0.115 0.426
MEM63 -0.207 -0.203 0.410
MEM117 -0.205 0.194 0.398
MEM216 -0.164 0.224 0.388
MEM7 -0.198 -0.178 0.376
MEM50 -0.182 -0.151 0.333
MEM114 -0.112 0.178 0.290
MEM166 -0.125 -0.122 0.248
MEM28 -0.096 0.148 0.244
MEM20 -0.127 -0.113 0.240
MEM1 -0.119 0.095 0.214

Fire severity was also weakly spatially structured (all 4 severity classes clustered near the centroid). It was structured at a range of small to moderate scales also (Figure 2). Patterns of high-severity fire occurrence were structured at slightly larger scales than unburned areas. Low and moderate burn patters were the least spatially structured (closest to the centroid).

## FIRE SEVERITY DATA ##
## Hill and Smith analysis for environmental variables

hsEnv <- dudi.hillsmith(datInvWide_env[,c(6:9)],scannf=FALSE)

## detrending of the analysis (residuals of regression onto xy coordinates)
hsEnv.detr <- pcaivortho(hsEnv,xyir,scannf=FALSE)

## MSPA of the detrended analysis
mspaEnv <- mspa(hsEnv.detr,listw_relUp,scannf=FALSE,nf=2)
MSPA for fire severity. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each severity class in ordination space indicates the strength of the spatial structuirng for that severity. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Figure 2: MSPA for fire severity. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each severity class in ordination space indicates the strength of the spatial structuirng for that severity. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Table 2: Influential MEMs for fire severity spatial structuring, their position in ordination space on axis 1 and axis 2, and the overall difference from the centroid of each (diff)
MEM CS1 CS2 diff
MEM80 -0.375 -0.106 0.481
MEM35 -0.367 -0.075 0.442
MEM79 -0.286 -0.098 0.384
MEM250 -0.031 0.330 0.361
MEM87 -0.238 -0.057 0.295
MEM23 -0.219 -0.058 0.277
MEM253 -0.020 0.240 0.260
MEM97 -0.151 0.084 0.236
MEM21 -0.204 -0.025 0.229
MEM411 -0.024 0.205 0.229
MEM2 -0.174 -0.054 0.227
MEM214 -0.020 0.201 0.221
MEM128 -0.181 -0.020 0.201

We performed multiple regression on the results of the PCA of species by fire severity and submitted the regression results to MSPA to determine important spatial scales for structuring the correlation between fire severity and the distribution of invasive species abundance (Figure 3). Species were again clustered near to the centroid (o.o,o.o), indicating weak spatial structuring of the correlation between fire severity and invasive species abundance. The correlation was structured at a range of small and intermediate scales for all species. The correlation between cheatgrass and fire severity showed the least spatial structuring among invasive species (closest to the centroid). There was little differences in the scales that were influential to the correlation between Kentucky bluegrass and fire severity and those that were influential to the correlation between smooth brome and fire severity.

## CANONICAL MSPA ##
## RDA species ~ envir
## (species abundances predicted by fire severity)
## note: RDA = 'PCAIV' (PCA with Instrumental Variables)
rda1 <- pcaiv(dudi=pcaFau.detr, df=datInvWide_env[,6:9],scannf=FALSE)

## canonical MSPA (species predicted by environment)
mspaCan1 <- mspa(dudi=rda1, lw=listw_relUp, scannf=FALSE, nf=3)
Canonical MSPA for species abundances predicted by fire severity. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each species in ordination space indicates the strength of the spatial structuirng for the correlation between that species and the categorical variable, fire severity. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Figure 3: Canonical MSPA for species abundances predicted by fire severity. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each species in ordination space indicates the strength of the spatial structuirng for the correlation between that species and the categorical variable, fire severity. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Table 3: Influential MEMs for spatial structuring of the correlation between fire severity and invasive species distribution, their position in ordination space on axis 1 and axis 2, and the overall difference from the centroid of each (diff)
MEM CS1 CS2 CS3 diff
MEM98 0.415 -0.526 0.400 0.941
MEM35 0.312 0.304 0.152 0.616
MEM26 0.276 0.266 -0.040 0.542
MEM103 0.166 -0.184 -0.138 0.350
MEM87 0.173 0.176 0.037 0.349
MEM69 0.147 -0.201 -0.030 0.348
MEM29 0.158 0.158 0.043 0.316
MEM21 0.153 0.146 0.095 0.299
MEM54 0.134 0.140 0.034 0.274
MEM86 0.137 0.114 -0.098 0.251
MEM250 0.103 -0.138 -0.073 0.241
MEM411 0.094 -0.126 -0.078 0.220
MEM49 0.107 0.110 0.060 0.217
MEM60 0.128 0.084 0.157 0.212
MEM38 0.101 0.104 0.052 0.205
MEM37 0.101 0.089 -0.051 0.190
MEM3 0.114 -0.070 0.203 0.184
MEM117 0.092 0.091 -0.002 0.182
MEM2 0.089 0.092 0.016 0.181
MEM68 0.098 0.082 -0.066 0.180
MEM154 0.093 0.083 0.089 0.176
MEM62 0.080 0.084 0.020 0.164
MEM214 0.069 -0.090 -0.077 0.159
MEM253 0.067 -0.090 -0.039 0.158
MEM27 0.118 -0.039 -0.178 0.156
MEM244 0.069 -0.087 -0.049 0.156
MEM57 0.138 0.016 -0.225 0.154

We performed partial RDA on the results of the regression of invasive species by fire severity and submitted the regression results to MSPA to determine important spatial scales for structuring invasive species abundances while controlling for the variation resulting from fire severity. The results of this depict the influence of fire severity on spatial structuring of invasive species abundances. The resulting ordination is very similar to the ordination of invasive species spatial structuring without controlling for the effects of fire severity (Figure 1, suggesting fire severity is not a strong driver of invasive species spatial structuring (Figure 4). In both MSPAs, species are only weakly spatially structured (all close to the centrid) and Kentucky bluegrass is structured at smaller spatial scales than smooth brome as indicated by larger MEMs co-located with smooth brome along axis 2 and smaller MEMs co-located with Kentucky bluegrass along both axes.

## PARTIAL CANONICAL MSPA ##
## partial RDA species ~ envir
## (species abundances not predicted by fire severity)
rda2 <- pcaivortho(dudi=pcaFau.detr,df=datInvWide_env[,6:9],scannf=FALSE,nf=2)

## partial canonical MSPA
mspaCan2 <- mspa(dudi=rda2, lw=listw_relUp, scannf=FALSE, nf=2)
Partial canonical MSPA for species abundances controlled for fire severity. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each species in ordination space indicates the strength of the spatial structuirng for that species after controlling for the influence of fire severity on spatial structuring. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Figure 4: Partial canonical MSPA for species abundances controlled for fire severity. Red circles indicate influential MEMs as indicated by scree plots of overall distance from the ordination centroid. Other MEMs are represented with light blue circles. The size of the circle is an indicator of the relative scale of each MEM. Scale ranges in size from 25m (the smallest circle), to ~70 km (the largest circle). The distance from the centroid of each species in ordination space indicates the strength of the spatial structuirng for that species after controlling for the influence of fire severity on spatial structuring. Eigenvalues (percent variance explained) for the first two axes are listed in the axis labels.

Table 4: Influential MEMs for spatial structuring of invasive species distribution when controlling for fire severity effects on spatial structure, their position in ordination space on axis 1 and axis 2, and the overall difference from the centroid of each (diff)
MEM CS1 CS2 diff
MEM165 -0.258 -0.352 0.610
MEM13 -0.270 0.283 0.552
MEM8 -0.279 0.240 0.519
MEM164 -0.197 -0.289 0.486
MEM23 -0.330 0.119 0.450
MEM63 -0.201 0.207 0.408
MEM216 -0.177 -0.230 0.407
MEM3 -0.193 0.196 0.388
MEM7 -0.186 0.183 0.369
MEM117 -0.200 -0.168 0.367
MEM50 -0.168 0.145 0.313
MEM114 -0.112 -0.178 0.291

References

Bauman, D., Drouet, T., Fortin, M. J., & Dray, S. (2018). Optimizing the choice of a spatial weighting matrix in eigenvector‐based methods. Ecology, 99(10), 2159-2166.

Šidák, Z. (1967). Rectangular confidence regions for the means of multivariate normal distributions. Journal of the American Statistical Association, 62(318), 626-633.

Jombart, T., Dray, S., & Dufour, A. B. (2009). Finding essential scales of spatial variation in ecological data: a multivariate approach. Ecography, 32(1), 161-168.