Community trajectory analysis

2.1 Trajectory data

To specify community dynamics, we need three data items:

1. A set of community states (i.e. coordinates in a space \(\Omega\)), described using a distance matrix \(d\);
2. A vector specifying the site (i.e. sampling unit) corresponding to each community state;
3. A vector specifying the survey (i.e. time point) corresponding to the sampling of each community state.

In CTA sampling units do not need to be surveyed synchronously nor the same number of times. However, in this example we assume all three sites have been surveyed synchronously four times.

Let us first define the vectors that describe the site and the survey of each community state:

#Description of sites and surveys
sites = c(1,1,1,1,2,2,2,2,3,3,3,3)
surveys=c(1,2,3,4,1,2,3,4,1,2,3,4)

We then build a matrix with the coordinates corresponding to the community states of the three sites during each survey. We assume that the community space \(\Omega\) has two dimensions:

#Raw data table
xy<-matrix(0, nrow=12, ncol=2)
xy[2,2]<-1
xy[3,2]<-2
xy[4,2]<-3
xy[5:6,2] <- xy[1:2,2]
xy[7,2]<-1.5
xy[8,2]<-2.0
xy[5:6,1] <- 0.25
xy[7,1]<-0.5
xy[8,1]<-1.0
xy[9:10,1] <- xy[5:6,1]+0.25
xy[11,1] <- 1.0
xy[12,1] <-1.5
xy[9:10,2] <- xy[5:6,2]
xy[11:12,2]<-c(1.25,1.0)
cbind(sites,surveys,xy)

##       sites surveys          
##  [1,]     1       1 0.00 0.00
##  [2,]     1       2 0.00 1.00
##  [3,]     1       3 0.00 2.00
##  [4,]     1       4 0.00 3.00
##  [5,]     2       1 0.25 0.00
##  [6,]     2       2 0.25 1.00
##  [7,]     2       3 0.50 1.50
##  [8,]     2       4 1.00 2.00
##  [9,]     3       1 0.50 0.00
## [10,]     3       2 0.50 1.00
## [11,]     3       3 1.00 1.25
## [12,]     3       4 1.50 1.00

The matrix of Euclidean distances \(d\) between community states in \(\Omega\) is then:

#Distance matrix
D = dist(xy)
D

##            1         2         3         4         5         6         7
## 2  1.0000000                                                            
## 3  2.0000000 1.0000000                                                  
## 4  3.0000000 2.0000000 1.0000000                                        
## 5  0.2500000 1.0307764 2.0155644 3.0103986                              
## 6  1.0307764 0.2500000 1.0307764 2.0155644 1.0000000                    
## 7  1.5811388 0.7071068 0.7071068 1.5811388 1.5206906 0.5590170          
## 8  2.2360680 1.4142136 1.0000000 1.4142136 2.1360009 1.2500000 0.7071068
## 9  0.5000000 1.1180340 2.0615528 3.0413813 0.2500000 1.0307764 1.5000000
## 10 1.1180340 0.5000000 1.1180340 2.0615528 1.0307764 0.2500000 0.5000000
## 11 1.6007811 1.0307764 1.2500000 2.0155644 1.4577380 0.7905694 0.5590170
## 12 1.8027756 1.5000000 1.8027756 2.5000000 1.6007811 1.2500000 1.1180340
##            8         9        10        11
## 2                                         
## 3                                         
## 4                                         
## 5                                         
## 6                                         
## 7                                         
## 8                                         
## 9  2.0615528                              
## 10 1.1180340 1.0000000                    
## 11 0.7500000 1.3462912 0.5590170          
## 12 1.1180340 1.4142136 1.0000000 0.5590170

CTA is based on the analysis of information in \(d\). Therefore, it does not require explicit coordinates. This is an advantage because it allows the analysis to be conducted on arbitrary metric (or semi-metric) spaces. The choice of \(d\) is left to the user and will depend on the problem at hand.

2.2 Displaying trajectories

To begin our analysis of the three trajectories, we display them in an ordination space, using function trajectoryPCoA. Since \(\Omega\) has only two dimensions in this example, the Principal Coordinates Analysis (PCoA) on \(d\) displays the complete space:

par(mar=c(4,4,1,1))
trajectoryPCoA(D, sites, surveys, traj.colors = c("black","red", "blue"), lwd = 2)
legend("topleft", col=c("black","red", "blue"), 
       legend=c("Trajectory 1", "Trajectory 2", "Trajectory 3"), bty="n", lty=1, lwd = 2)

While trajectory of site ‘1’ (black arrows) is made of three segments of the same length and direction, trajectory of site ‘2’ (red arrows) has a second and third segments that bend and are shorter than that of the segond segment of site ‘1’. Trajectory of site ‘3’ includes a stronger change in direction and shorter segments.

As this example has two dimensions and we used Euclidean distance, the same plot (albeit rotated) can be straightforwardly obtained using matrix xy and function trajectoryPlot:

par(mar=c(4,4,1,1))
trajectoryPlot(xy, sites, surveys, traj.colors = c("black","red", "blue"), lwd = 2)
legend("topright", col=c("black","red", "blue"), 
       legend=c("Trajectory 1", "Trajectory 2", "Trajectory 3"), bty="n", lty=1, lwd = 2)

While trajectoryPCoA uses PCoA (also known as classical Multidimensional Scaling) to display trajectories, users can display community trajectories using other ordination techniques such as metric Multidimensional Scaling (mMDS; see function mds of package smacof) or non-metric MDS (nMDS; see function metaMDS in package vegan or function isoMDS in package MASS). Function trajectoryPlot will help drawing arrows between segments to represent trajectories on the ordination space given by any of these methods.

Functions trajectoryPCoAand trajectoryPlot allow selecting the subset of trajectories to be displayed, so that the representation can focus on particular pathways (see also function trajectorySelection).

2.3 Trajectory lengths, angles and overall directionality

One may be interested in studying the geometric properties of particular trajectories. For example, one can obtain the length of trajectory segments and the total pathway length using function trajectoryLengths:

trajectoryLengths(D, sites, surveys)

##   S1       S2        S3 Trajectory
## 1  1 1.000000 1.0000000   3.000000
## 2  1 0.559017 0.7071068   2.266124
## 3  1 0.559017 0.5590170   2.118034

In CTA, angles are measured using triplets of time-ordered community states (a pair of consecutive segments is an example of such triplets). As \(d\) may represent a space of multiple dimensions, angles cannot be calculated with respect to a single plane. Instead, each angle is measured on the plane defined by each triplet. Zero angles indicate that the three points (e.g. the two consecutive segments) are in a straight line. The larger the angle value, the more is trajectory changing in direction. Mean and standard deviation statistics of angles are calculated according to circular statistics (see R package circular). Function trajectoryAngles allows calculating the angles between consecutive segments:

trajectoryAngles(D, sites, surveys)

##      S1-S2    S2-S3     mean         sd       rho
## 1  0.00000  0.00000  0.00000 0.00000000 1.0000000
## 2 26.56505 18.43495 22.50000 0.07097832 0.9974842
## 3 63.43495 53.13010 58.28253 0.08998746 0.9959593

While site ‘1’ follows a straight pathway, angles are > 0 for trajectories of site ‘2’ and ‘3’, denoting the change in direction. In this case, the same information could be obtained by inspecting the previous plots, but in a case where \(\Omega\) has many dimensions, the representation will correspond to a reduced (ordination) space and hence, angles and lengths in the plot will not correspond exactly to those of functions trajectoryLengths and trajectoryAngles, which take into account the complete \(\Omega\) space.

Angles can be calculated not only for all consecutive segments but for all four triplets of ordered community states, whether of consecutive segments or not (i.e., between points 1-2-3, 1-2-4, 1-3-4 and 2-3-4). This is done by specifying all=TRUE in trajectoryAngles:

trajectoryAngles(D, sites, surveys, all=TRUE)

##         A1      A2       A3       A4     mean        sd       rho
## 1  0.00000  0.0000  0.00000  0.00000  0.00000 0.0000000 1.0000000
## 2 26.56505 36.8699 35.53768 18.43495 29.36033 0.1300790 0.9915754
## 3 63.43495 90.0000 94.76364 53.13010 75.36015 0.3078934 0.9537066

The mean resultant length of circular statistics (column rho of the previous result), which takes values between 0 and 1, can be used to assess the degree of homogeneity of angle values and it will take a value of 1 if all angles are the same. This approach, however, uses only angular information and does not take into account the length of segments.

To measure the overall directionality of a community trajectory (i.e. if the pathway consistently follows the same direction in \(\Omega\) ), we recommend using another statistic that is sensitive to both angles and segment lengths and is implemented in function trajectoryDirectionality:

trajectoryDirectionality(D, sites, surveys)

##         1         2         3 
## 1.0000000 0.8274026 0.5620859

As known from previous plots, trajectory of site ‘2’ is less straight than trajectory of site ‘1’ and that of site ‘3’ has the lowest directionality value. Note that function only computes a descriptive statistic, i.e. it does not perform any statistical test on directionality.

2.4 Relative positions within trajectories

Community states occupy a position within their trajectory that depends on the total pathway length of the trajectory. By adding the length of segments prior to a given state and dividing the sum by the total length of the trajectory we obtain the relative position of the community state. Function trajectoryProjection allows obtaining the relative position of each point of a trajectory. To use it for this purpose one should use as parameters the distance matrix between state and the indices that conform the trajectory. For example for the two example trajectories we would have:

trajectoryProjection(D, 1:4, 1:4)

##   distanceToTrajectory segment relativePosition
## 1                    0       1        0.0000000
## 2                    0       1        0.3333333
## 3                    0       2        0.6666667
## 4                    0       3        1.0000000

trajectoryProjection(D, 5:8, 5:8)

##   distanceToTrajectory segment relativePosition
## 5                    0       1        0.0000000
## 6                    0       1        0.4412822
## 7                    0       2        0.6879664
## 8                   NA      NA               NA

Because the trajectory of site ‘2’ as the second and third segments that are longer than the first, the relative position of state ‘6’ is lower than 1/3 and that of state ‘7’ is lower than 2/3.

Function trajectoryProjection can also be used to project arbitrary community states onto a given reference trajectory. For example we can study the projection of third state of the trajectory of site ‘1’ (i.e. state 3) onto the trajectory of site ‘2’ (i.e. states 5 to 8):

trajectoryProjection(D, 3, 5:8)

##   distanceToTrajectory segment relativePosition
## 3            0.7071068       3        0.6879664

2.5 Trajectory convergence

When trajectories have been sampled the same number of times, function trajectoryConvergence allows performing tests of convergence based on the trend analysis of the sequences of distances between points of the two trajectories (i.e. first-first, second-second, …):

trajectoryConvergence(D, sites, surveys, symmetric = TRUE)

## $tau
##           1         2         3
## 1        NA 0.9128709 0.9128709
## 2 0.9128709        NA 0.9128709
## 3 0.9128709 0.9128709        NA
## 
## $p.value
##           1         2         3
## 1        NA 0.1485617 0.1485617
## 2 0.1485617        NA 0.1485617
## 3 0.1485617 0.1485617        NA

The function performs the Mann-Whitney trend test. Values of the statistic (‘tau’) larger than 0 correspond to trajectories that are diverging, whereas values lower than 0 correspond to trajectories that are converging. By setting symmetric = FALSE the convergence test becomes asymmetric. In this case the sequence of distances between every point of one trajectory and the other:

trajectoryConvergence(D, sites, surveys, symmetric = FALSE)

## WARNING: Error exit, tauk2. IFAULT =  12

## $tau
##           1         2         3
## 1        NA 0.9128709 0.9128709
## 2 0.9128709        NA 0.8164966
## 3 0.9128709 0.6666666        NA
## 
## $p.value
##           1         2         3
## 1        NA 0.1485617 0.1485617
## 2 0.1485617        NA 1.0000000
## 3 0.1485617 0.3081795        NA

The asymmetric test is useful to determine if one trajectory is becoming closer to the other or if it is departing from the other.

2.6 Distances between segments and trajectories

To start comparing trajectories between sites (i.e. between sampling units), one important step is the calculation of distances between directed segments, which can be obtained by calling function segmentDistances:

Ds = segmentDistances(D, sites, surveys)$Dseg
Ds

##           1[1-2]    1[2-3]    1[3-4]    2[1-2]    2[2-3]    2[3-4]
## 1[2-3] 1.0000000                                                  
## 1[3-4] 2.0000000 1.0000000                                        
## 2[1-2] 0.2500000 1.0307764 2.0155644                              
## 2[2-3] 1.0307764 0.7071068 1.5811388 1.0000000                    
## 2[3-4] 1.5811388 1.0000000 1.4142136 1.5206906 0.7071068          
## 3[1-2] 0.5000000 1.1180340 2.0615528 0.2500000 1.0307764 1.5000000
## 3[2-3] 1.1180340 1.1180340 2.0124612 1.0307764 0.5590170 0.7500000
## 3[3-4] 1.6007811 1.5590170 2.0155644 1.4577380 1.1180340 1.0606602
##           3[1-2]    3[2-3]
## 1[2-3]                    
## 1[3-4]                    
## 2[1-2]                    
## 2[2-3]                    
## 2[3-4]                    
## 3[1-2]                    
## 3[2-3] 1.0000000          
## 3[3-4] 1.5590170 0.5590170

Distances between segments are affected by differences in both position, size and direction. Hence, among the six segments of this example, the distance is maximum between the last segment of trajectory ‘1’ and the first segment of trajectory ‘3’.

One can display distances between segments in two dimensions using mMDS.

mMDS = mds(Ds)
mMDS

## 
## Call:
## mds(delta = Ds)
## 
## Model: Symmetric SMACOF 
## Number of objects: 9 
## Stress-1 value: 0.062 
## Number of iterations: 15

xret = mMDS$conf
par(mar=c(4,4,1,1))
plot(xret, xlab="axis 1", ylab = "axis 2", asp=1, pch=21,
     bg=c(rep("black",3), rep("red",3), rep("blue",3)), 
     xlim=c(-1.5,1), ylim=c(-1,1.5))
text(xret, labels=rep(paste0("s",1:3),3), pos=1)
legend("topleft", pt.bg=c("black","red","blue"), pch=21, bty="n", legend=c("Trajectory 1", "Trajectory 2", "Trajectory 3"))

Distances between segments are internally calculated when comparing whole trajectories using function trajectoryDistances. Here we show the dissimilarity between the two trajectories as assessed using either the Hausdorff distance (equal to the maximum distance between directed segments) or the directed segment path distance (an average of distances between segments):

trajectoryDistances(D, sites, surveys, distance.type = "Hausdorff")

##          1        2
## 2 2.015564         
## 3 2.061553 1.500000

trajectoryDistances(D, sites, surveys, distance.type = "DSPD")

##           1         2
## 2 0.7214045          
## 3 1.1345910 0.5714490

DSPD is a symmetrized distance. To calculate non-symmetric distances one uses:

trajectoryDistances(D, sites, surveys, distance.type = "DSPD", symmetrization = NULL)

##           1         2         3
## 1 0.0000000 0.7904401 1.2101651
## 2 0.6523689 0.0000000 0.5196723
## 3 1.0590170 0.6232257 0.0000000

3.1 Display trajectories in PCoA

The distance matrix avoca_D_man conforms our definition of \(\Omega\). We use trajectoryPCoA to display the relations between forest plot states in this space and to draw the trajectory of each plot:

par(mar=c(4,4,1,1))
trajectoryPCoA(avoca_D_man,  avoca_sites, avoca_surveys,
               traj.colors = brewer.pal(8,"Accent"), 
               axes=c(1,2), length=0.1, lwd=2)
legend("topright", bty="n", legend = 1:8, col = brewer.pal(8,"Accent"), lwd=2)

Note that in this case, the full \(\Omega\) includes more than two dimensions, and PCoA is representing 43% of total variance (correction for negative eigenvalues is included in the call to cmdscale from trajectoryPCoA), so one has to be careful when interpreting trajectories visually.

Another option is to use mMDS to represent trajectories, which in this case produces a similar result:

mMDS = mds(avoca_D_man)
mMDS

## 
## Call:
## mds(delta = avoca_D_man)
## 
## Model: Symmetric SMACOF 
## Number of objects: 72 
## Stress-1 value: 0.114 
## Number of iterations: 49

par(mar=c(4,4,1,1))
trajectoryPlot(mMDS$conf,  avoca_sites, avoca_surveys,
               traj.colors = brewer.pal(8,"Accent"), 
               axes=c(1,2), length=0.1, lwd=2)
legend("topright", bty="n", legend = 1:8, col = brewer.pal(8,"Accent"), lwd=2)

One can inspect specific trajectories using argument selection in function trajectoryPCoA. This allows getting a better view of particular trajectories, here that of forest plot ‘3’:

par(mfrow=c(1,2))
trajectoryPCoA(avoca_D_man,  avoca_sites, avoca_surveys,
               selection= 3,
               length=0.1, lwd=2)
plotTrajDiamDist(3)

In the right hand, we added a representation of the change in the mountain beech tree diameter distribution through time for trajectory of forest plot ‘3’. The dynamics of this plot include mostly growth, which results in individuals moving from one diameter class to the other. The whole trajectory looks mostly directional. Let’s now inspect the trajectory of forest plot ‘4’:

par(mfrow=c(1,2))
trajectoryPCoA(avoca_D_man,  avoca_sites, avoca_surveys,
               selection= 4,
               length=0.1, lwd=2)
plotTrajDiamDist(4)

This second trajectory is less straight and seems to include a turn by the end of the sampling period, corresponding to the recruitment of new saplings.

3.2 Trajectory lengths, angles and overall directionality

While trajectory lengths and angles can be inspected visually in ordination diagrams, it is better to calculate them using the full \(\Omega\) space (i.e., from matrix avoca_D_man). Using function trajectoryLengths we can see that the trajectory of forest plot ‘4’ is lengthier than that of plot ‘3’, mostly because includes a lengthier last segment (i.e. the recruitment of new individuals):

trajectoryLengths(avoca_D_man, avoca_sites, avoca_surveys)

##          S1        S2        S3        S4        S5        S6        S7
## 1 1.2173214 1.5911988 1.0976965 2.1173501 0.5238760 1.5863283 1.5750365
## 2 0.5971165 1.7469687 0.9866591 0.9413060 1.3687614 0.6280231 1.4565581
## 3 1.1480971 1.2404953 0.6508116 0.4371405 0.5098385 1.2078811 1.6241741
## 4 0.7932307 1.8572629 0.7268623 0.8348635 3.0560437 1.9863939 0.9905892
## 5 1.7769875 0.3367341 0.7283030 0.6515714 1.2660552 0.9544933 1.2605333
## 6 2.1891568 0.5312711 1.0035212 0.4113220 2.1394743 1.0482871 1.4924056
## 7 0.2919002 0.8857645 1.0908604 0.5601649 2.0600208 0.3712067 1.1014563
## 8 0.1909713 1.2000266 2.3211891 0.6539882 2.7807668 0.8170202 1.2072425
##          S8 Trajectory
## 1 0.6277940  10.336602
## 2 1.1232798   8.848673
## 3 1.4536390   8.272077
## 4 3.8794520  14.124698
## 5 0.5842314   7.558909
## 6 1.9623777  10.777816
## 7 1.7518489   8.113223
## 8 1.5107357  10.681940

If we calculate the angles between consecutive segments (using function trajectoryLengths) we see that indeed the trajectory of ‘3’ is rather directional, but the angles of trajectory of ‘4’ are larger, on aveerage:

avoca_ang <- trajectoryAngles(avoca_D_man, avoca_sites, avoca_surveys)
avoca_ang

##       S1-S2        S2-S3        S3-S4     S4-S5     S5-S6     S6-S7
## 1  41.74809 8.669806e+01 7.401875e+01  26.94138 113.40657 100.67068
## 2  68.14891 3.466506e+01 8.537736e-07   0.00000  25.97111   0.00000
## 3  87.24519 3.088828e+01 1.207418e-06   0.00000  50.90743  48.12854
## 4  37.65736 8.537736e-07 8.537736e-07  36.25240  55.53607  74.21065
## 5  42.02156 7.166400e+01 1.207418e-06  49.95436  65.75897  65.82083
## 6  41.69894 4.611675e+01 5.669641e+01 135.84929   0.00000   0.00000
## 7  53.63254 1.152378e+02 6.519921e+01  60.71442   0.00000  56.25733
## 8 180.00000 0.000000e+00 9.213121e+01 132.36445  71.33948  36.43189
##          S7-S8     mean        sd       rho
## 1 1.021996e+02 78.82477 0.5334615 0.8673692
## 2 1.207418e-06 17.61550 0.4257423 0.9133572
## 3 4.568199e+01 37.39925 0.5018798 0.8816663
## 4 4.980332e+01 36.44717 0.4531030 0.9024417
## 5 1.061250e+02 57.89321 0.5239476 0.8717431
## 6 0.000000e+00 34.34150 0.7777287 0.7390195
## 7 1.207418e-06 49.96352 0.6562139 0.8062928
## 8 1.207418e-06 66.68343 1.1601974 0.5101610

By calling function trajectoryDirectionality we can confirm that the trajectory for site ‘4’ is less straight than that of site ‘3’:

avoca_dir <- trajectoryDirectionality(avoca_D_man, avoca_sites, avoca_surveys)
avoca_dir

##         1         2         3         4         5         6         7 
## 0.6781369 0.6736490 0.8651467 0.5122482 0.6677116 0.7058465 0.7391775 
##         8 
## 0.5254225

The following code displays the relationship between the statistic in trajectoryDirectionality and the mean resultant vector length that uses angular information only and assesses the constancy of angle values:

avoca_rho = trajectoryAngles(avoca_D_man, avoca_sites, avoca_surveys, all=TRUE)$rho
par(mar=c(4,4,1,1))
plot(avoca_rho, avoca_dir, xlab = "rho(T)", ylab = "dir(T)", type="n")
text(avoca_rho, avoca_dir, as.character(1:8))

3.3 Distances between trajectories

We can calculate the resemblance between forest plot trajectories using trajectoryDistances:

##       1     2     3     4     5     6     7
## 2 2.405                                    
## 3 6.805 5.773                              
## 4 6.123 6.646 5.225                        
## 5 6.020 5.541 3.235 4.966                  
## 6 9.490 8.866 3.436 6.043 4.505            
## 7 4.024 3.291 4.365 4.993 4.389 6.205      
## 8 9.797 9.747 5.591 4.611 6.383 5.043 6.949

The closest trajectories are those of plots ‘1’ and ‘2’. They looked rather close in position in the PCoA ordination of \(\Omega\) with all trajectories, so probably it is position, rather than shape which has influenced this low value. The next pair of similar trajectories are those of the ‘3’-‘5’ pair. We can again use mMDS to produce an ordination of resemblances between trajectories:

mMDS<-mds(avoca_D_traj_man)
mMDS

## 
## Call:
## mds(delta = avoca_D_traj_man)
## 
## Model: Symmetric SMACOF 
## Number of objects: 8 
## Stress-1 value: 0.091 
## Number of iterations: 25

x<-mMDS$conf[,1]
y<-mMDS$conf[,2]
par(mar=c(4,4,1,1))
plot(x,y, type="p", asp=1, xlab=paste0("Axis 1"), 
     ylab=paste0("Axis 2"), col="black",
     bg= brewer.pal(8,"Accent"), pch=21)
text(x,y, labels=1:8, pos=1)

Distances between trajectories can be calculated after centering them (i.e. after bringing all trajectories to the center of the \(\Omega\) space). This is done using function centerTrajectories, which returns a new dissimilarity matrix. Centering removes differences in position between trajectories, which is useful in cases where one wants to focus on spatio-temporal interaction while discarding spatial patterns that are constant in time. We do no illustrate centering here, because in is not interesting for this particular example.

4.1 Introduction

Some dissimilarity coefficients that are popular in community ecology, such as the percentage difference (alias Bray-Curtis), have the drawback of being a non-Euclidean (dissimilarity matrices do not allow a representation in a Cartesian space), or even semi-metric (i.e. triangle inequality is not ensured). In order to use these coefficients in multivariate analyses that require these properties a transformation of the original space will normally be in order. In this section, we compare different alternatives and provide some recommendations on this issue.

4.2 Effect of square root on a simple directional trajectory

Here, we use an example of a single synthetic community to illustrate the effect of square root transformation on a community trajectory. We begin by defining the species data of the trajectory itself. The dataset consists of four rows (i.e. surveys) and four columns (species). The dynamics in this example consist in an constant increase in the number of individuals of the first species and a corresponding decrease of the others, while keeping a total abundance of 100 individuals:

sites = rep(1,4)
surveys=1:4
spdata = rbind(c(35,30,20,15),
               c(50,25,15,10),
               c(65,20,10,5),
               c(80,15,5,0))

We now use function vegdist from package vegan to calculate the Bray-Curtis coefficient:

D = vegan::vegdist(spdata, "bray")
is.metric(D)

## [1] TRUE

D

##      1    2    3
## 2 0.15          
## 3 0.30 0.15     
## 4 0.45 0.30 0.15

This dissimilarity matrix is a metric, so one would not need any transformation for CTA. However, it is a good example to illustrate the effect of the square root transformation.

If we draw the resemblance space corresponding to this dissimilarity matrix we see a straight trajectory:

trajectoryPCoA(D,sites,surveys)

Here we see that Bray-Curtis dissimilarity responds linearly to the proposed sequence of community dynamics. To confirm this geometry, we can calculate the geometric properties of the trajectory (i.e. length, angle between consecutive segments and overall directionality):

trajectoryLengths(D,sites,surveys)

##     S1   S2   S3 Trajectory
## 1 0.15 0.15 0.15       0.45

trajectoryAngles(D,sites,surveys)

##   S1-S2 S2-S3 mean sd rho
## 1     0     0    0  0   1

trajectoryDirectionality(D,sites,surveys)

## 1 
## 1

Angles are 0 degrees and overall directionality is maximum (i.e. 1), in accordance with the plot and the data. We now proceed to take the square root of the dissimilarity values, as would be necessary to achieve a metric (and Euclidean) space in a more complex data set:

sqrtD = sqrt(D)
sqrtD

##           1         2         3
## 2 0.3872983                    
## 3 0.5477226 0.3872983          
## 4 0.6708204 0.5477226 0.3872983

The transformation increases all dissimilarity values (because the original values are smaller than 1), but the increase is larger for smaller values, so the ratio between large dissimilarities and small dissimilarities decreases. This has an effect on the overall shape of the trajectory, which surprisingly now looks like:

trajectoryPCoA(sqrtD,sites,surveys)

In addition to the distortion observed, the number of dimensions of the data have increased (i.e the sum of variance explained by the two axes is 88% < 100%), so we cannot be sure that the angles are well represented. If we re-calculate the properties of the trajectory taking into account all dimensions we obtain:

trajectoryLengths(sqrtD,sites,surveys)

##          S1        S2        S3 Trajectory
## 1 0.3872983 0.3872983 0.3872983   1.161895

trajectoryAngles(sqrtD,sites,surveys)

##   S1-S2 S2-S3 mean sd rho
## 1    90    90   90  0   1

trajectoryAngles(sqrtD,sites,surveys, all=TRUE)

##   A1 A2 A3 A4 mean sd rho
## 1 90 90 90 90   90  0   1

trajectoryDirectionality(sqrtD,sites,surveys)

##   1 
## 0.5

The length of segments and the trajectory have increased, but all segments are of the same length, in agreement with the original trajectory. In contrast, the angles are now 90 degrees and the overall directionality has decreased substantially.

4.3 Effect of different transformations on more complex trajectories

Here we use simulated data to compare four transformation approaches:

1. Local transformation of semi-metric dissimilarities (such as the percentage difference) in every triplet when the triangle inequality is required. This is done by default in the CTA functions of package vegclust.
2. Global transformation of dissimilarities, using the square root.
3. Global transformation of dissimilarities by using Principal Coordinates Analysis (classical multidimensional scaling) with eigenvalue correction.
4. Global transformation of dissimilarities by using metric multidimensional scaling (metric MDS).

We use simulated dynamics to build another trajectory with more species (20) and more time steps. We begin by setting the number of time steps (50) and the size of the community (50 individuals):

Nsteps = 50
CC = 50
Nreplace <- CC*0.05

Nreplace is the number of individuals to be replaced each time step (5%). Now we define the initial community vector and the vector with the probabilities of offspring for each species according to some ecological conditions:

x <- c(0, 1, 0, 67, 1, 3, 0, 2, 2, 2, 1, 6, 2, 0, 0, 2, 5, 1, 6, 0)
poffspring <- c(0, 0, 0.002, 0.661 ,0 ,0, 0.037, 0.281, 0, 0, 0, 0.008, 0, 0, 0.005, 0.003, 0, 0, 0, 0)

We can now simulate the dynamics by sequentially applying stochastic deaths and recruitment:

m <- matrix(0, nrow=Nsteps+1, ncol=length(x))
m[1, ] = x
for(k in 1:Nsteps) {
  pdeath <-x/sum(x) #Equal probability of dying
  deaths<-rmultinom(1,Nreplace, pdeath)
  x <- pmax(x - deaths,0)
  offspring = rmultinom(1,Nreplace, as.vector(poffspring))
  x <- x + offspring
  m[k+1, ]<-x
}

Then we decide how frequently (with respect to the simulated step) a sample of the community is taken, here every four steps:

Sj <- seq(1,Nsteps+1, by=4) #Sample every four steps
mj <- m[Sj,]
surveys = 1:length(Sj)
sites = rep(1,length(Sj))

Now we are ready to calculate the Bray-Curtis dissimilarity:

D <- vegan::vegdist(mj,"bray")

In this more complex trajectory, some triangles may not obey the triangle inequality (depending on the simulation). This can be inspected using function is.metric:

is.metric(D, tol=0.0000001)

## [1] TRUE

Deviations from a metric space, if they exist, will be small, so that local transformation of triangles will be very small.

Local transformations are not possible to display trajectories. When we plot the trajectory using function trajectoryPCoA the global transformation of principal coordinates analysis (PCoA) with negative eigenvalue correction is performed. This is fine to display trajectories, but has problems when measuring angular properties, as we will see.

pcoa<-trajectoryPCoA(D, sites, surveys, selection=1,length=0.1, axes=c(1,2))

## Warning in cmdscale(D2, eig = TRUE, add = TRUE, k = nrow(as.matrix(D2)) - :
## only 11 of the first 12 eigenvalues are > 0

pcoaD = dist(pcoa$points)

The trajectory has some twists derived from stochasticity in death and recruitment. Let’s now look at the square root of the Bray-Curtis dissimilarity:

sqrtD = sqrt(D)
pcoaSqrt = trajectoryPCoA(sqrtD, sites, surveys, selection=1,length=0.1, axes=c(1,2))

Finally, we also transform dissimilarities using metric multidimensional scaling (mMDS), provided by package smacof:

res <- mds(D, ndim = length(Sj)-1, type = "interval")
mmdsD <- dist(res$conf)
trajectoryPlot(res$conf, sites, surveys, selection=1,length=0.1, axes=c(1,2))

While the three plots look different, the differences are not striking (besides rotation issues). We can compare the stress of the global solutions:

stress0(D,pcoaSqrt$points, type="interval")

## [1] 0.05955285

stress0(D,pcoa$points, type="interval")

## [1] 0.002693561

stress0(D,res$conf, type="interval")

## [1] 0.009140085

Where we see that the square root leads to the strongest alteration of original dissimilarities. If we calculate geometric properties we are not limited by ordination plots and we can take into account all dimensions.

anglesD = trajectoryAngles(D,sites,surveys)
anglesSqrtD = trajectoryAngles(sqrtD,sites,surveys)
anglesPcoaD = trajectoryAngles(pcoaD,sites,surveys)
anglesmmdsD = trajectoryAngles(mmdsD,sites,surveys)

df<-as.data.frame(rbind(anglesD, anglesSqrtD, anglesPcoaD, anglesmmdsD))
row.names(df)<-c("local", "global.sqrt", "global.pcoa", "global.mmds")
round(df,2)

##             S1-S2 S2-S3  S3-S4  S4-S5  S5-S6  S6-S7  S7-S8  S8-S9 S9-S10
## local        0.00  0.00  71.79  82.82  82.82 109.47 109.47  82.82  75.52
## global.sqrt 90.00 90.00 100.18 104.48 104.48 114.09 114.09 104.48 101.78
## global.pcoa 80.59 63.56  91.84 101.21 101.21 113.20 113.20 101.21  95.51
## global.mmds  1.99 11.31  69.35  90.95  83.41 109.04 104.00  91.32  81.25
##             S10-S11 S11-S12   mean   sd  rho
## local          0.00   67.11  64.58 0.71 0.78
## global.sqrt   90.00   99.59 101.20 0.14 0.99
## global.pcoa   70.17   89.17  92.90 0.27 0.96
## global.mmds   42.09   71.40  70.63 0.60 0.84

The first call to trajectoryAngles with matrix D represents the default strategy of transforming triangles locally, which involves the weakest transformation of all and can be taken as reference. Both the square root and PCoA with negative eigenvalue correction induce a strong transformation of trajectory angles. The global solution of metric MDS leads to angles that are more similar to those of the local transformation strategy.

If we inspect the overall directionality, the global solution of metric MDS provides a value that is again closer to that of local transformation, compared to PCoA and the square root:

trajectoryDirectionality(D,sites,surveys)

##        1 
## 0.854166

trajectoryDirectionality(sqrtD,sites,surveys)

##         1 
## 0.4845857

trajectoryDirectionality(pcoaD,sites,surveys)

##         1 
## 0.6480544

trajectoryDirectionality(mmdsD,sites,surveys)

##         1 
## 0.7818728

4.4 Conclusions

In this small study we compared the effect of different solutions to violation of the triangle inequality. If function is.metric returns TRUE for a given data set one should not worry about violations of the triangle inequality. Local solutions are those that imply the smallest number of changes, but these will not be consistent across triplets, so users may desire to apply a global transformation that produces euclidean spaces. This should be done with care. We have shown how the square root transformation distorts the angles and overall directionality of trajectories on the space defined by the percentage difference (alias Bray-Curtis) dissimilarity index. We suspect that this negative effect of square root transformation on angles happens no matter which coefficient is used to derive the initial distance matrix. Therefore, we advocate for avoiding its use when conducting community trajectory analysis (in particular for angles). The global transformation consisting in the application of PCoA with negative eigenvalue correction is less strong than the square root, but it still strongly change the angles between segments. Perhaps, the less harmful global transformation is provided by metric Multidimensional Scaling, but the need to embed distances in an Euclidean space of all three transformations implies a stronger requirement than being a metric, and results in distortions.