# Determine rate of change in dissimilarity (distance)?

I have repeated measures plant abundance data for 36 forest plots, across 80 years involving 50+ species of trees.

• The data are structured as:

• Columns = different species,
• Rows = separate samples [Plot-Year combos],
• Each cell = abundance (i.e., basal area) of the the given species in the given sample.

Simplified Example (from here):

> abund.data

Plot Year Sp1 Sp2 Sp3 Sp4
1   P1    1   1   2   0   0
2   P2    1   1   0   3   2
3   P3    1   0   2   1   0
4   P1    2   1   2   0   0
5   P2    2   1   0   3   2
6   P3    2   0   2   1   0


I've calculated a Bray-Curtis dissimilarity (distance) matrix from these data.

Continuing the example:
library(ecodist)
distance(abun.data[,-c(1:2)], 'bray')

1         2         3         4         5
2 0.7777778
3 0.3333333 0.7777778
4 0.0000000 0.7777778 0.3333333
5 0.7777778 0.0000000 0.7777778 0.7777778
6 0.3333333 0.7777778 0.0000000 0.3333333 0.7777778


I want to calculate the rate at which plots change in community composition over time.

I had originally run a non-metric multidimensional scaling (NMDS) ordination and wanted to simply calculate changes in NMDS space.

• i.e., I wanted to create change vectors between plot points in subsequent years (I did so here) and then compare the lengths between years using some sort of regression....

ChangeVectorLength ~ Time | Plot

However, I don't think this is valid because of the rank-oriented construction of NMDS ordination.

Is there a way I could do something similar but using the "raw" distance (dissimilarity) values??

• For example (using the example data above): I want to quantify how much the community of species (as a whole) in Plot P1 has changed from Year 1 to Year 2.

• However, because the distance matrix represents -- well -- a matrix of pairwise distances bewteen all points, I'm not sure how to go about quantifying change in "distance space."
• So you have a number of distance matrices - between the species, one matrix per date - and you want to visualize the differences or trend between the matrices? Is that what you want? – ttnphns Jun 8 '18 at 1:51
• @ttnphns Nope. I have a single distance matrix where each "unit" I've calculated distances for is the abundance of a given species in a given plot in a given year. The resulting distance matrix has dimensions = dim(abun.data)[1] * [(dim(abund.data)[1] - 1) / 2]. – theforestecologist Jun 8 '18 at 2:07
• @ttnphns As for purpose: I do not want to visualize the differences as that is what NMDS ordination allows me to do. I want to be able to determine the rate of change each sample has undergone between sampling periods. For example, I need to quantify the change in Plot 4 in 2000 vs 2010 (perhaps being represented by row 510 and 511 in my abund.data that informed the distance matrix. However, because the distance matrix represents , well, a matrix of pairwise distances bewteen all points, I'm not sure how to go about quantifying change in "distance space" – theforestecologist Jun 8 '18 at 2:10
• There's lots of ways to approach this. Aggarwal and Reddy's book Data Clustering: Algorithms and Applications covers the waterfront...it's on Amazon. If you want to retain the distance approach there are information theoretic models which use a distance function defined by Kullback-Lieber such as Brandmaier's pairwise permutation distance clustering of time series. jstatsoft.org/article/view/v067i05 describes his R module. Eamonn Keogh's SAX routines are another while many on CV recommend dynamic time warping. Finally there are hidden markov models that cluster ... ctd>> – DJohnson Jun 8 '18 at 20:16
• ctd... Steve Scott has several articles exploring Bayesian methods for this sites.google.com/site/stevethebayesian/… Another guy is Oded Netzer's HMM for customer segmentation ... columbia.edu/~on2110/Papers/… Finally there's proprietary software from Statistical Innovations that has ready to use tools for HMMs...the product is called Latent Gold and costs about \$1,000 or so for a license. Might be worth the investment vs trying to develop a bespoke approach. – DJohnson Jun 8 '18 at 20:20

First an outline of the understanding of the question(s), then the answer.

Questions: Determine rate of change in dissimilarity (distance)?

I have repeated measures plant abundance data for 37 forest plots, across 80 years involving 50+ species of trees.

I want to calculate the rate at which plots change in community composition over time.

Is there a way I could do something similar but using the "raw" distance (dissimilarity) values?

• For example (using the example data above): I want to quantify how much the community of species (as a whole) in Plot P1 has changed from Year 1 to Year 2.

• However, because the distance matrix represents -- well -- a matrix of pairwise distances bewteen all points, I'm not sure how to go about quantifying change in "distance space"

@ttnphns As for purpose: I do not want to visualize the differences as that is what NMDS ordination allows me to do. I want to be able to determine the rate of change each sample has undergone between sampling periods. For example, I need to quantify the change in Plot 4 in 2000 vs 2010 (perhaps being represented by row 510 and 511 in my abund.data that informed the distance matrix. However, because the distance matrix represents, well, a matrix of pairwise distances bewteen all points, I'm not sure how to go about quantifying change in "distance space" – theforestecologist Jun 8 at 2:10

I'm not sure I completely understand the question but based on my understanding of MDS I doubt that it lends itself to an answer. First, are the 80 years of data annual and are all species measured consistently, i.e., is the panel balanced across plots, species and time? Next, why not treat it as a hierarchical and longitudinal growth model? This would involve restructuring your data matrix such that each record is plot-species-year-abundance giving about 150,000 records (37x80x50). Lots of lit on this topic, e.g., Singer's paper "Using SAS PROC MIXED to Fit Multilevel Models, Hierarchical Models, and Individual Growth Models" (.PDF) is a good intro. – DJohnson Jun 8 at 14:13

@DJohnson Thanks for the link and suggestion. To answer your questions: 1. No, the plots are not sampled regularly or even in the same years, but each plot was sampled 12-16 times. This shouldn't matter since I know the length of time between samples and can adjust results accordingly to put into "per annum" scale. 2. I don't think the approach you mention will work for me because I am not interested in a per-species change, but rather, I'm interested in a whole community change (i.e., considering all species together). This is what drew me initially to NMDS – theforestecologist Jun 8 at 14:27

...

@DJohnson, so you're right, that my most basic unit in my analysis is abundance for a species in a year of sampling in a given plot. However, as noted in my comment above, I am not sure how to express "community" other than using a dissimilarity matrix approach. – theforestecologist Jun 8 at 16:15

[For the sake of brevity, as best as it was possible, only comments that seemed to affect the questionS were quoted.]

There's some software specifically designed for forestry:

Read some of the free online information, example: "Manual of forest inventory - FAO" (.PDF).

You can expand your limited data to create a larger set before distilling.

Tree allometry establishes quantitative relations between some key characteristic dimensions of trees (usually fairly easy to measure) and other properties (often more difficult to assess). To the extent these statistical relations, established on the basis of detailed measurements on a small sample of typical trees, hold for other individuals, they permit extrapolations and estimations of a host of dendrometric quantities on the basis of a single (or at most a few) measurements.

The study of allometry is extremely important in dealing with measurements and data analysis in the practice of forestry. Allometry studies the relative size of organs or parts of organisms. Tree allometry narrows the definition to applications involving measurements of the growth or size of trees. Allometric relationships often are used to estimate difficult tree measurements, such as volume, from an easily measured attribute such as diameter at breast height (DBH).

The use of allometry is widespread in forestry and forest ecology. In order to develop an allometric relationship there must be a strong relationship and an ability to quantify this relationship between the parts of the subject measured and the other quantities of interest. Also when developing this equation one must play in factors which affect tree growth such as age, species, site location, etc. Once all these guidelines are met, one may attempt to develop an allometric equation.