2. Simple example
In this section we describe how to study the trajectories of three sites that have been surveyed four times each. We use a small dataset where trajectories occur in a space of two dimensions, so that geometric calculations can be followed more easily.
2.1 Trajectory data
To specify ecosystem dynamics, we need three data items:
- A set of ecosystem states (i.e. coordinates in a space \(\Omega\)), described using a distance matrix \(d\);
- A vector specifying the site (i.e. sampling unit) corresponding to each ecosystem state;
- A vector specifying the survey (i.e. time point) corresponding to the sampling of each ecosystem 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 ecosystem 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 define a matrix whose coordinates correspond to the set of ecosystem states observed. The number of rows in this matrix has to be equal to the length of vectors sites and surveys. We assume that the ecosystem 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 1.00The matrix of Euclidean distances \(d\) between ecosystem 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 0.5590170CTA is based on the analysis of information in the distance matrix \(\Delta = [d]\). 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:
oldpar <- par(mar=c(4,4,1,1))
trajectoryPCoA(D, sites, surveys, traj.colors = c("black","red", "blue"), lwd = 2,
survey.labels = T)
legend("topleft", col=c("black","red", "blue"),
legend=c("Trajectory 1", "Trajectory 2", "Trajectory 3"), bty="n", lty=1, lwd = 2)
par(oldpar)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:
oldpar<-par(mar=c(4,4,1,1))
trajectoryPlot(xy, sites, surveys, traj.colors = c("black","red", "blue"), lwd = 2,
survey.labels = T)
legend("topright", col=c("black","red", "blue"),
legend=c("Trajectory 1", "Trajectory 2", "Trajectory 3"), bty="n", lty=1, lwd = 2)
par(oldpar)While trajectoryPCoA uses PCoA (also known as classical Multidimensional Scaling) to display trajectories, users can display ecosystem 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 2.118034In CTA, angles are measured using triplets of time-ordered ecosystem states (a pair of consecutive segments is an example of such triplets). As matrix \(\Delta\) 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. 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 0.9959593While 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 ecosystem 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 0.9537066The 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 ecosystem 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 0.5620859As 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
Ecosystem states occupy a position within their trajectory that depends on the total pathway length of the trajectory (see Fig. 2 of De Cáceres et al. 2019). 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 ecosystem 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 states and the indices that conform the trajectory, which have to be entered twice. 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 1.0000000If we inspect the relative positions of the points in the trajectory of site ‘2’, we find than the second and third segments have relative positions larger than 1/3 and 2/3, respectively, because the second and third segments are shorter:
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 NAFunction trajectoryProjection can also be used to project arbitrary ecosystem states onto a given reference trajectory. For example we can study the projection of third state of the trajectory of site ‘2’ (i.e. state 7) onto the trajectory of site ‘1’ (i.e. states 1 to 4), which happens to be in the half of the trajectory:
trajectoryProjection(D, 7, 1:4)## distanceToTrajectory segment relativePosition
## 7 0.5 2 0.5If we project the points of the trajectory of site ‘3’ onto the trajectory of site ‘1’ we see how the curved pathway of site ‘3’ projects its fourth point to the same relative position as its second point.
trajectoryProjection(D, 9:12, 1:4)## distanceToTrajectory segment relativePosition
## 9 0.5 1 -3.700743e-17
## 10 0.5 2 3.333333e-01
## 11 1.0 2 4.166667e-01
## 12 1.5 1 3.333333e-012.5 Trajectory convergence
When trajectories have been sampled the same number of times, one can study their symmetric convergence or divergence (see Fig. 3a of De Cáceres et al. 2019). 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 NAThe 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 (see Figs. 3b and 3c of De Cáceres et al. 2019). 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
##
## $p.value
## 1 2 3
## 1 NA 0.1485617 0.1485617
## 2 0.1485617 NA 1.0000000
## 3 0.1485617 0.3081795 NAThe asymmetric test is useful to determine if one trajectory is becoming closer to the other or if it is departing from the other.
2.6 Distances between segments and trajectories
To start comparing trajectories between sites (i.e. between sampling units), one important step is the calculation of distances between directed segments (see Fig. 4 of De Cáceres et al. 2019), which can be obtained by calling function segmentDistances:
Ds = segmentDistances(D, sites, surveys)$Dseg
Ds## 1[1-2] 1[2-3] 1[3-4] 2[1-2] 2[2-3] 2[3-4] 3[1-2]
## 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 1.0000000
## 3[3-4] 1.6007811 1.5590170 2.0155644 1.4577380 1.1180340 1.0606602 1.5590170
## 3[2-3]
## 1[2-3]
## 1[3-4]
## 2[1-2]
## 2[2-3]
## 2[3-4]
## 3[1-2]
## 3[2-3]
## 3[3-4] 0.5590170Distances 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’ (named 1[3-4]) and the first segment of trajectory ‘3’ (named 3[1-2]).
One can display distances between segments in two dimensions using mMDS.
mMDS = smacof::mds(Ds)
mMDS##
## Call:
## smacof::mds(delta = Ds)
##
## Model: Symmetric SMACOF
## Number of objects: 9
## Stress-1 value: 0.062
## Number of iterations: 15xret = mMDS$conf
oldpar<-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"))
par(oldpar)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; see eq. 8 in De Cáceres et al. 2019) or the directed segment path distance (an average of distances between segments; see eq. 13 in De Cáceres et al. 2019):
trajectoryDistances(D, sites, surveys, distance.type = "Hausdorff")## 1 2
## 2 2.015564
## 3 2.061553 1.500000trajectoryDistances(D, sites, surveys, distance.type = "DSPD")## 1 2
## 2 0.7214045
## 3 1.1345910 0.5714490DSPD is a symmetrized distance. To calculate non-symmetric distances one uses (see eq. 11 in De Cáceres et al. 2019):
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
