Summary

The package distantia allows to measure the dissimilarity between multivariate ecological time-series (METS hereafter). The package assumes that the target sequences are ordered along a given dimension, being depth and time the most common ones, but others such as latitude or elevation are also possible. Furthermore, the target METS can be regular or irregular, and have their samples aligned (same age/time/depth) or unaligned (different age/time/depth). The only requirement is that the sequences must have at least two (but ideally more) columns with the same name and units representing different variables relevant to the dynamics of an ecological system.

In this document I explain the logics behind the method, show how to use it, and demonstrate how the distantia package introduces useful tools to compare multivariate time-series. The topics covered in this document are:

  • Installation of the package.
  • Comparing two irregular METS.
  • Comparing multiple irregular METS.
  • Comparing regular and aligned METS.
  • Restricted permutation test to assess the significance of dissimilarity scores
  • Assessing the contribution of every variable to the dissimilarity between two METS.
  • Partial matching: finding the section in a long METS more similar to a given shorter one.
  • Sequence slotting: combining samples of two METS into a single composite sequence.
  • Tranferring an attribute from one METS to another: direct and interpolated modes.

Installation

You can install the released version of distantia (currently v1.0.0) from CRAN with:

install.packages("distantia")

And the development version (currently v1.0.1) from GitHub with:

install.packages("devtools")
library(devtools)
devtools::install_github("BlasBenito/distantia")

Loading the library, plus other helper libraries:

library(distantia)
library(ggplot2)
library(viridis)
library(kableExtra)
library(qgraph)
library(tidyr)

Comparing two irregular METS

In this section I will use two example datasets based on the Abernethy pollen core (Birks and Mathewes, 1978) to fully explain the logical backbone of the dissimilarity analyses implemented in distantia.

#loading sequences
data(sequenceA)
data(sequenceB)

#showing first rows
kable(sequenceA[1:15, ], caption = "Sequence A")
Sequence A
betula pinus corylu junipe empetr gramin cypera artemi rumex
79 271 36 0 4 7 25 0 0
113 320 42 0 4 3 11 0 0
51 420 39 0 2 1 12 0 0
130 470 6 0 0 2 4 0 0
31 450 6 0 3 2 3 0 0
59 425 12 0 0 2 3 0 0
78 386 29 2 0 0 2 0 0
71 397 52 2 0 6 3 0 0
140 310 50 2 0 4 3 0 0
150 323 34 2 0 11 2 0 0
175 317 37 2 0 11 3 0 0
181 345 28 3 0 7 3 0 0
153 285 36 2 0 8 3 0 1
214 315 54 2 1 13 5 0 0
200 210 41 6 0 10 4 0 0
kable(sequenceB[1:15, ], caption = "Sequence B")
Sequence B
betula pinus corylu junipe gramin cypera artemi rumex
19 175 NA 2 34 39 1 0
18 119 28 1 36 44 0 4
30 99 37 0 2 20 0 1
26 101 29 0 0 18 0 0
31 99 30 0 1 10 0 0
24 97 28 0 2 9 0 0
23 105 34 0 1 6 0 0
48 112 46 0 0 12 0 0
29 108 16 0 6 3 0 0
23 110 21 0 2 11 0 1
5 119 19 0 1 1 0 0
30 105 NA 0 9 7 0 0
22 116 17 0 1 7 0 0
24 115 20 0 2 4 0 0
26 119 23 0 4 0 0 0

Data preparation

Notice that sequenceB has a few NA values (that were introduced to serve as an example). The function prepareSequences gets them ready for analysis by matching colum names and handling empty data. It allows to merge two or more METS into a single dataframe ready for further analyses. Note that, since the data represents pollen abundances, a Hellinger transformation (square root of the relative proportions of each taxa, it balances the relative abundances of rare and dominant taxa) is applied. This transformation balances the relative importance of very abundant versus rare taxa. The function prepareSequences will generally be the starting point of any analysis performed with the distantia package.

Sequences A and B ready for analysis.
id betula pinus corylu junipe empetr gramin cypera artemi rumex
A 0.4327 0.8014 0.2921 0.0002 0.0974 0.1288 0.2434 2e-04 0.0002
A 0.4788 0.8057 0.2919 0.0001 0.0901 0.0780 0.1494 1e-04 0.0001
A 0.3117 0.8944 0.2726 0.0001 0.0617 0.0436 0.1512 1e-04 0.0001
A 0.4609 0.8763 0.0990 0.0001 0.0001 0.0572 0.0808 1e-04 0.0001
A 0.2503 0.9535 0.1101 0.0001 0.0778 0.0636 0.0778 1e-04 0.0001
A 0.3432 0.9210 0.1548 0.0001 0.0001 0.0632 0.0774 1e-04 0.0001
A 0.3962 0.8813 0.2416 0.0634 0.0001 0.0001 0.0634 1e-04 0.0001
A 0.3657 0.8647 0.3129 0.0614 0.0001 0.1063 0.0752 1e-04 0.0001
A 0.5245 0.7804 0.3134 0.0627 0.0001 0.0886 0.0768 1e-04 0.0001
A 0.5361 0.7866 0.2552 0.0619 0.0001 0.1452 0.0619 1e-04 0.0001
A 0.5667 0.7627 0.2606 0.0606 0.0001 0.1421 0.0742 1e-04 0.0001
A 0.5650 0.7800 0.2222 0.0727 0.0001 0.1111 0.0727 1e-04 0.0001
A 0.5599 0.7642 0.2716 0.0640 0.0001 0.1280 0.0784 1e-04 0.0453
A 0.5952 0.7222 0.2990 0.0575 0.0407 0.1467 0.0910 1e-04 0.0001
A 0.6516 0.6677 0.2950 0.1129 0.0001 0.1457 0.0922 1e-04 0.0001

The computation of dissimilarity between the datasets A and B requires several steps.

1. Computation of a distance matrix among the samples of both sequences.

It is computed by the distanceMatrix function, which allows the user to select a distance metric (so far the ones implemented are manhattan, euclidean, chi, and hellinger). The function plotMatrix allows an easy visualization of the resulting distance matrix.

Distance matrix between the samples of two irregular multivariate sequences. Darker colors indicate higher distance between samples.

Distance matrix between the samples of two irregular multivariate sequences. Darker colors indicate higher distance between samples.

2. Computation of the least-cost path within the distance matrix.

This step uses a dynamic programming algorithm to find the least-cost path that connnects the cell 1,1 of the matrix (lower left in the image above) and the last cell of the matrix (opposite corner). This can be done via in two different ways.

  • an orthogonal search by moving either one step on the x axis or one step on the y axis at a time (see Equation 1).

Equation 1 \[AB_{between} = 2 \times (D(A_{1}, B_{1}) + \sum_{i=1}^{m}\sum_{j=1}^{n} min\left(\begin{array}{c}D(A_{i}, B_{j+1}), \\ D(A_{i+1}, B_{j}) \end{array}\right))\]

  • an orthogonal and diagonal search (a.k.a diagonal) which includes the above, plus a diagonal search.

Equation 2 \[AB_{between} = 2 \times (D(A_{1}, B_{1}) + \sum_{i=1}^{m}\sum_{j=1}^{n} min\left(\begin{array}{c}D(A_{i}, B_{j+1}), \\ D(A_{i+1}, B_{j} \\ D(A_{i+1}, B_{j+1}) \end{array}\right))\]

Where:

  • \(m\) and \(n\) are the number of samples of the multivariate time-series \(A\) and \(B\).
  • \(i\) and \(j\) are the indices of the samples in \(A\) and \(B\) being considered on each step of the recursive algorithm.
  • \(D\) is a function that returns the multivariate distance (i.e. Manhattan) between any given pair of samples of \(A\) and \(B\).
  • \(min\) is a function returning the minimum distance of a subset of distances defined in the neigborhood of the given indices \(i\) and \(j\)

The equation returns \(AB_{between}\), which is the double of the sum of distances that lie within the least-cost path, and represent the distance between the samples of A and B. The value of \(AB_{between}\) is computed by using the functions leastCostMatrix, which computes the partial solutions to the least-cost problem, leastCostPath, which returns the best global solution, and leastCost function, which sums the distances of the least-cost path and multiplies them by 2.

The code below performs these steps according to both equations

Least-cost path plotted on the least-cost matrix. Left solution is computed with an orthogonal search path, while the right one includes diagonals.

Least-cost path plotted on the least-cost matrix. Left solution is computed with an orthogonal search path, while the right one includes diagonals.

Computing \(AB_{between}\) from these solutions is straightforward with the function leastCost

Which returns a value for \(AB_{between}\) of 33.7206 for the orthogonal solution, and 22.7596 for the diagonal one. Diagonal solutions always yield lower values for \(AB_{between}\) than orthogonal ones.

Notice the straight vertical and horizontal lines that show up in some regions of the least cost paths shown in the figure above. These are blocks, and happen in dissimilar sections of the compared sequences. Also, an unbalanced number of rows in the compared sequences can generate long blocks. Blocks inflate the value of \(AB_{between}\) because the distance to a given sample is counted several times per block. This problem often leads to false negatives, that is, to the conclusion that two sequences are statistically different when actually they are not.

This package includes an algorithm to remove blocks from the least cost path, which offers more realistic values for \(AB_{between}\). The function leastCostPathNoBlocks reads a least cost path, and removes all blocks as follows.

Which now yields 11.2975 for the orthogonal solution, and 16.8667 for the diagonal one. Notice how now the diagonal solution has a higher value, because by default, the diagonal method generates less blocks. That is why each measure of dissimilarity (orthogonal, diagonal, orthogonal no-blocks, and diagonal no-blocks) lies within a different comparative framework, and therefore, outputs from different methods should not be compared.

Hereafter only the diagonal no-blocks option will be considered in the example cases, since it is the most general and safe solution of the four mentioned above.

3. Autosum, or sum of the distances among adjacent samples on each sequence.

This step requires to compute the distances between adjacent samples in each sequence and sum them, as shown in Equation 3.

Equation 3 \[AB_{within} = \sum_{i=1}^{m} D(A_{i }, A_{i + 1}) + \sum_{i=1}^{n} D(B_{i }, B_{i + 1})\]

This operation is performed by the autoSum function shown below.

4. Compute dissimilarity score \(\psi\).

The dissimilarity measure \(\psi\) was first described in the book “Numerical methods in Quaternary pollen analysis” (Birks and Gordon, 1985). Psi is computed as shown in Equation 4a:

Equation 4a \[\psi = \frac{AB_{between} - AB_{within}}{AB_{within}}\]

This equation has a particularity. Imagine two identical sequences A and B, with three samples each. In this case, \(AB_{between}\) is computed as

\(AB_{between} = 2 \times (D(A_{1}, B_{1}) + D(A_{1}, B_{2}) + D(A_{2}, B_{2}) + D(A_{2}, B_{3}) + D(A_{3}, B_{3}))\)

Since the samples of each sequence with the same index are identical, this can be reduced to

\(AB_{between} = 2 \times (D(A_{1}, B_{2}) + D(A_{2}, B_{3})) = AB_{within}\)

which in turn equals \(AB_{within}\) as shown in Equation 4, yielding a \(\psi\) value of 0.

This equality does not work in the same way when the least-cost path search-method includes diagonals. When the sequenes are identical, diagonal methods yield an \(AB_{between}\) of 0, leading to a \(\psi\) equal to -1. To fix this shift, this package uses Equation 4b instead when \(diagonal = TRUE\) is selected, which adds 1 to the final solution.

Equation 4b \[\psi = \frac{AB_{between} - AB_{within}}{AB_{within}} + 1\]

In any case, the psi function only requires the least-cost, and the autosum of both sequences to compute \(\psi\). Since we are working with a diagonal search, 1 has to be added to the final solution.

Which yields a psi equal to 1.7131. The output of psi is a list, that can be transformed to a dataframe or a matrix by using the formatPsi function.

A B psi
A B 1.7131

workflowPsi: doing it all at once

All the steps required to compute psi, including the format options provided by formatPsi are wrapped together in the function workflowPsi. It includes options to switch to a diagonal method, and to ignore blocks, as shown below.

The function allows to exclude particular columns from the analysis (argument exclude.columns), select different distance metrics (argument method), use diagonals to find the least-cost path (argument diagonal), or measure psi by ignoring blocks in the least-cost path (argument ignore.blocks). Since we have observed several blocks in the least-cost path, below we compute psi by ignoring them.

The package can work seamlessly with any given number of sequences, as long as there is memory enough available (but check the new function workflowPsiHP, it can work with up to 40k sequences, if you have a cluster at hand, and a few years to waste). To do so, almost every function uses the packages “doParallel” and “foreach”, that together allow to parallelize the execution of the distantia functions by using all the processors in your machine but one.

The example dataset sequencesMIS contains 12 sections of the same sequence belonging to different marine isotopic stages identified by a column named “MIS”. MIS stages with odd numbers are generally interpreted as warm periods (interglacials), while the odd ones are interpreted as cold periods (glacials). In any case, this interpretation is not important to illustrate this capability of the library.

data(sequencesMIS)
kable(head(sequencesMIS, n=15), digits = 4, caption = "Header of the sequencesMIS dataset.")
Header of the sequencesMIS dataset.
MIS Quercus Betula Pinus Alnus Tilia Carpinus
MIS-1 55 1 5 3 4 5
MIS-1 86 21 35 8 0 10
MIS-1 120 15 8 1 0 1
MIS-1 138 16 12 6 1 3
MIS-1 130 12 17 2 1 1
MIS-1 128 0 6 4 2 2
MIS-1 140 0 19 9 4 0
MIS-1 113 0 15 12 2 5
MIS-1 98 0 27 2 2 0
MIS-1 92 1 16 7 3 0
MIS-1 73 3 22 3 0 0
MIS-1 91 1 21 3 7 0
MIS-1 148 1 22 1 4 0
MIS-1 148 0 1 7 13 0
MIS-1 149 1 2 5 4 0

The dataset is checked and prepared with prepareSequences.

The dissimilarity measure psi can be computed for every combination of sequences through the function workflowPsi shown below.

Psi values between pairs of MIS periods.
A B psi
24 MIS-3 MIS-6 0.8476
65 MIS-10 MIS-12 0.8622
61 MIS-9 MIS-10 0.8978
59 MIS-8 MIS-11 0.9027
66 MIS-11 MIS-12 0.9165
30 MIS-3 MIS-12 0.9373
40 MIS-5 MIS-7 0.9834
60 MIS-8 MIS-12 0.9887
62 MIS-9 MIS-11 0.9958
64 MIS-10 MIS-11 1.0191
43 MIS-5 MIS-10 1.0294
45 MIS-5 MIS-12 1.0343
63 MIS-9 MIS-12 1.0395
56 MIS-7 MIS-12 1.0408
28 MIS-3 MIS-10 1.0513
42 MIS-5 MIS-9 1.0553
32 MIS-4 MIS-6 1.0628
51 MIS-6 MIS-12 1.0692
53 MIS-7 MIS-9 1.0727
26 MIS-3 MIS-8 1.0827
54 MIS-7 MIS-10 1.0946
58 MIS-8 MIS-10 1.0994
49 MIS-6 MIS-10 1.1007
21 MIS-2 MIS-12 1.1073
27 MIS-3 MIS-9 1.1122
15 MIS-2 MIS-6 1.1148
12 MIS-2 MIS-3 1.1170
52 MIS-7 MIS-8 1.1320
57 MIS-8 MIS-9 1.1386
13 MIS-2 MIS-4 1.1551
47 MIS-6 MIS-8 1.1719
29 MIS-3 MIS-11 1.1725
23 MIS-3 MIS-5 1.1972
19 MIS-2 MIS-10 1.2080
22 MIS-3 MIS-4 1.2388
44 MIS-5 MIS-11 1.2731
41 MIS-5 MIS-8 1.2745
48 MIS-6 MIS-9 1.2799
55 MIS-7 MIS-11 1.3560
50 MIS-6 MIS-11 1.3871
25 MIS-3 MIS-7 1.4027
34 MIS-4 MIS-8 1.4384
39 MIS-5 MIS-6 1.4490
36 MIS-4 MIS-10 1.4555
38 MIS-4 MIS-12 1.4911
17 MIS-2 MIS-8 1.5888
46 MIS-6 MIS-7 1.6028
18 MIS-2 MIS-9 1.6722
20 MIS-2 MIS-11 1.6951
14 MIS-2 MIS-5 1.7162
6 MIS-1 MIS-7 1.7311
4 MIS-1 MIS-5 1.8062
3 MIS-1 MIS-4 1.8454
33 MIS-4 MIS-7 1.8768
16 MIS-2 MIS-7 2.0543
35 MIS-4 MIS-9 2.0668
11 MIS-1 MIS-12 2.0765
8 MIS-1 MIS-9 2.1592
5 MIS-1 MIS-6 2.2248
7 MIS-1 MIS-8 2.2759
10 MIS-1 MIS-11 2.3033
1 MIS-1 MIS-2 2.3077
2 MIS-1 MIS-3 2.3355
31 MIS-4 MIS-5 2.4154
37 MIS-4 MIS-11 2.4798
9 MIS-1 MIS-10 2.5562

A dataframe like this can be transformed into a matrix to be plotted as an adjacency network with the qgraph package.

Similarity between MIS sequences represented as a network. More similar sites are closer, and linked by a wider edge. Note that glacials are colored in blue and interglacials in green

Similarity between MIS sequences represented as a network. More similar sites are closer, and linked by a wider edge. Note that glacials are colored in blue and interglacials in green

Or as a matrix with ggplot2.

#ordering factors to get a triangular matrix
MIS.psi$A <- factor(MIS.psi$A, levels=unique(sequencesMIS$MIS))
MIS.psi$B <- factor(MIS.psi$B, levels=unique(sequencesMIS$MIS))

#plotting matrix
ggplot(data=na.omit(MIS.psi), aes(x=A, y=B, size=psi, color=psi)) + 
  geom_point() +
  viridis::scale_color_viridis(direction = -1) +
  guides(size = FALSE)
Dissimilarity between MIS sequences. Darker colors indicate a higher dissimilarity.

Dissimilarity between MIS sequences. Darker colors indicate a higher dissimilarity.

The dataframe of dissimilarities between pairs of sequences can be also used to analyze the drivers of dissimilarity. To do so, attributes such as differences in time (when sequences represent different times) or distance (when sequences represent different sites) between sequences, or differences between physical/climatic attributes between sequences such as topography or climate can be added to the table, so models such as \(psi = A + B + C\) (were A, B, and C are these attributes) can be fitted.

Comparing regular aligned METS

The package distantia is also useful to compare synchronic sequences that have the same number of samples. In this particular case, distances to obtain \(AB_{between}\) are computed only between samples with the same time/depth/order, and no distance matrix (nor least-cost analysis) is required. When the argument paired.samples in prepareSequences is set to TRUE, the function checks if the sequences have the same number of rows, and, if time.column is provided, it selects the samples that have valid time/depth columns for every sequence in the dataset.

Here we test these ideas with the climate dataset included in the library. It represents simulated palaeoclimate over 200 ky. at four sites identified by the column sequenceId. Note that this time the transformation applied is “scaled”, which uses the scale function of R base to center and scale the data.

In this case, the argument paired.samples of workflowPsi must be set to TRUE. Additionally, if the argument same.time is set to TRUE, the time/age of the samples is checked, and samples without the same time/age are removed from the analysis.

Psi values between pairs of sequences in the ‘climate’ dataset.
A B psi
2 4 3.4092
4 2 3.4092
1 3 3.5702
3 1 3.5702
3 4 4.1139
4 3 4.1139
1 2 4.2467
2 1 4.2467
2 3 4.6040
3 2 4.6040
1 4 4.8791
4 1 4.8791

Restricted permutation test to assess the significance of dissimilarity values

One question that may arise when comparing time series is “to what extent are dissimilarity values a result of chance?”. Answering this question requires to compare a given dissimilarity value with a distribution of dissimilarity values resulting from chance. However… how do we simulate chance in a multivariate time-series? The natural answer is “permutation”. Since samples in a multivariate time-series are ordered, randomly re-shuffling samples is out of the question, because that would destroy the structure of the data. A more gentler alternative is to randomly switch single data-points (a case of a variable) independently by variable. This kind of permutation is named “restricted permutation”, and preserves global trends within the data, but changes local structure.

A restricted permutation test on psi values requires the following steps:

  • Compute the real psi on two given sequences A and B.
  • Repeat the following steps several times (99 to 999):
    • For each case of each column of A and B, randomly apply one of these actions:
      • Leave it as is.
      • Replace it with the previous case.
      • Replace it with the next case.
    • Compute randomized psi between A and B and store the value.
  • Add real psi to the pool of randomized psi.
  • Compute the proportion of randomized psi that is equal or lower than real psi.

Such a proportion represents the probability of obtaining a value lower than real psi by chance.

Since the restricted permutation only happens at a local scale within each column of each sequence, the probability values returned are very conservative and shouldn’t be interpreted in the same way p-values are interpreted.

The process described above has been implemented in the workflowNullPsi function. We will apply it to three groups of the sequencesMIS dataset.

The computation of the null psi values goes as follows:

Note that the number of repetitions has been set to 9 in order to speed-up execution. The actual number should ideally be 999.

The output is a list with two dataframes, psi and p.

The dataframe psi contains the real and random psi values. The column psi contains the dissimilarity between the sequences in the columns A and B. The columns r1 to r9 contain the psi values obtained from permutations of the sequences.

kable(random.psi$psi, digits = 4, caption = "True and null psi values generated by workflowNullPsi.")
True and null psi values generated by workflowNullPsi.
A B psi r1 r2 r3 r4 r5 r6 r7 r8 r9
MIS-4 MIS-5 3.2077 3.4067 1.5016 2.1582 3.4563 1.6678 2.1579 3.7669 1.4573 2.0739
MIS-4 MIS-6 1.3084 3.5720 1.5761 2.0159 3.4390 1.6080 2.1479 3.4279 1.4289 2.1388
MIS-5 MIS-6 1.8380 3.4589 1.5842 2.2318 3.7984 1.4916 2.2585 3.5364 1.4779 2.1321

The dataframe p contains the probability of obtaining the real psi value by chance for each combination of sequences.

kable(random.psi$p, caption = "Probability of obtaining a given set of psi values by chance.")
Probability of obtaining a given set of psi values by chance.
A B p
MIS-4 MIS-5 0.7
MIS-4 MIS-6 0.1
MIS-5 MIS-6 0.4

Assessing the contribution of a variable to the dissimilarity between two sequences

What variables are more important in explaining the dissimilarity between two sequences?, or in other words, what variables contribute the most to the dissimilarity between two sequences? One reasonable answer is: the one that reduces dissimilarity the most when removed from the data. This section explains how to use the function workflowImportance follows such a principle to evaluate the importance of given variables in explaining differences between sequences.

First, we prepare the data. It is again sequencesMIS, but with only three groups selected (MIS 4 to 6) to simplify the analysis.

The workflow function is pretty similar to the ones explained above. However, unlike the other functions in the package, that parallelize across the comparison of pairs of sequences, this one parallelizes the computation of psi on combinations of columns, removing one column each time.

WARNING: the argument ‘exclude.columns’ of ‘workflowImportance’ does not work in version 1.0.0 (available in CRAN), but the bug is fixed in version 1.0.1 (available in GitHub). If you are using 1.0.0, I recommend you to subset ‘sequences’ so only the grouping column and the numeric columns to be compared are available for the function.

The output is a list with two slots named psi and psi.drop.

The dataframe psi contains psi values for each combination of variables (named in the coluns A and B) computed for all columns in the column All variables, and one column per variable named Without variable_name containing the psi value when that variable is removed from the compared sequences.

kable(psi.importance$psi, digits = 4, caption = "Psi values with all variables (column 'All variables'), and without one variable at a time.")
Psi values with all variables (column ‘All variables’), and without one variable at a time.
A B All variables Without Carpinus Without Tilia Without Alnus Without Pinus Without Betula Without Quercus
MIS-4 MIS-5 3.8917 3.8939 3.9036 3.9056 5.5347 3.9122 0.9693
MIS-4 MIS-6 1.0253 1.0251 1.0250 1.0191 1.3828 1.0209 0.9563
MIS-5 MIS-6 1.6496 1.6499 1.6500 1.6467 1.0680 1.6481 0.9129

This table can be plotted as a bar plot as follows:

#extracting object
psi.df <- psi.importance$psi

#to long format
psi.df.long <- tidyr::gather(psi.df, variable, psi, 3:ncol(psi.df))

#creating column with names of the sequences
psi.df.long$name <- paste(psi.df.long$A, psi.df.long$B, sep=" - ")

#plot
ggplot(data=psi.df.long, aes(x=variable, y=psi, fill=psi)) + 
  geom_bar(stat = "identity") + 
  coord_flip() + 
  facet_wrap("name") +
  scale_fill_viridis(direction = -1) +
  ggtitle("Contribution of separated variables to dissimilarity.") +
  labs(fill = "Psi")
Variable importance analysis of three combinations of sequences. The plot suggest that MIS-4 and MIS-6 are more similar (both are glacial periods), and that the column Quercus is the one with a higher contribution to dissimilarity between sequences.

Variable importance analysis of three combinations of sequences. The plot suggest that MIS-4 and MIS-6 are more similar (both are glacial periods), and that the column Quercus is the one with a higher contribution to dissimilarity between sequences.

The second table, named psi.drop describes the drop in psi values, in percentage, when the given variable is removed from the analysis. Large positive numbers indicate that dissimilarity drops (increase in similarity) when the given variable is removed, confirming that the variable is important to explain the dissimilarity between both sequences. Negative values indicate an increase in dissimilarity between the sequences when the variable is dropped.

In summary:

  • High psi-drop value: variable contributes to dissimilarity.
  • Low or negative psi-drop value: variable contributes to similarity.
kable(psi.importance$psi.drop, caption = "Drop in psi, as percentage of the psi value obtained when using all variables. Positive values indicate that the sequences become more similar when the given variable is removed (contribution to dissimilarity), while negative values indicate that the sequences become more dissimilar when the variable is removed (contribution to similarity).")
Drop in psi, as percentage of the psi value obtained when using all variables. Positive values indicate that the sequences become more similar when the given variable is removed (contribution to dissimilarity), while negative values indicate that the sequences become more dissimilar when the variable is removed (contribution to similarity).
A B Carpinus Tilia Alnus Pinus Betula Quercus
MIS-4 MIS-5 -0.06 -0.31 -0.36 -42.22 -0.53 75.09
MIS-4 MIS-6 0.01 0.02 0.61 -34.87 0.43 6.73
MIS-5 MIS-6 -0.02 -0.02 0.18 35.26 0.09 44.66
#extracting object
psi.drop.df <- psi.importance$psi.drop

#to long format
psi.drop.df.long <- tidyr::gather(psi.drop.df, variable, psi, 3:ncol(psi.drop.df))

#creating column with names of the sequences
psi.drop.df.long$name <- paste(psi.drop.df.long$A, psi.drop.df.long$B, sep=" - ")

#plot
ggplot(data=psi.drop.df.long, aes(x=variable, y=psi, fill=psi)) + 
  geom_bar(stat = "identity") + 
  coord_flip() + 
  facet_wrap("name") +
  scale_fill_viridis() +
  ggtitle("Drop in dissimilarity when variables are removed.") +
  ylab("Drop in dissimilarity (%)") +
  labs(fill = "Psi drop (%)")
Drop in psi values, represented as percentage, when a variable is removed from the analysis. Negative values indicate a contribution to similarity, while positive values indicate a contribution to dissimilarity. The plot suggest that Quercus is the variable with a higher contribution to dissimilarity, while Pinus has the higher contribution to similarity.

Drop in psi values, represented as percentage, when a variable is removed from the analysis. Negative values indicate a contribution to similarity, while positive values indicate a contribution to dissimilarity. The plot suggest that Quercus is the variable with a higher contribution to dissimilarity, while Pinus has the higher contribution to similarity.

Partial matching: finding the section in a long sequence more similar to a given short sequence

In this scenario the user has one short and one long sequence, and the goal is to find the section in the long sequence that better matches the short one. To recreate this scenario we use the dataset sequencesMIS. The first 10 samples will serve as short sequence, and the first 40 samples as long sequence. These small subsets are selected to speed-up the execution time of this example.

The sequences have to be prepared and transformed. For simplicity, the sequences are named short and long, and the grouping column is named id, but the user can name them at will. Since the data represents community composition, a Hellinger transformation is applied.

The function workflowPartialMatch shown below is going to subset the long sequence in sizes between min.length and max.length. In the example below this search space has the same size as MIS.short to speed-up the execution of this example, but wider windows are possible. If left empty, the length of the segment in the long sequence to be matched will have the same number of samples as the short sequence. In the example below we look for segments of the same length, two samples shorter, and two samples longer than the shorter sequence.

The function returns a dataframe with three columns: first.row (first row of the matched segment of the long sequence), last.row (last row of the matched segment of the long sequence), and psi (ordered from lower to higher). In this case, since the long sequence contains the short sequence, the first row shows a perfect match.

kable(MIS.psi[1:15, ], digits = 4, caption = "First and last row of a section of the long sequence along with the psi value obtained during the partial matching.")
First and last row of a section of the long sequence along with the psi value obtained during the partial matching.
first.row last.row psi
1 10 0.0000
3 12 0.2707
4 13 0.2815
2 11 0.3059
6 15 0.4343
5 14 0.5509
9 18 1.3055
10 19 1.3263
8 17 1.3844
7 16 1.3949
15 24 1.4428
12 21 1.4711
17 26 1.4959
11 20 1.5101
18 27 1.5902

Subsetting the long sequence to obtain the segment best matching with the short sequence goes as follows.

Sequence slotting: combining samples of two sequences into a single composite sequence

Under this scenario, the objective is to combine two sequences into a single composite sequence. The basic assumption followed by the algorithm building the composite sequence is most similar samples should go together, but respecting the original ordering of the sequences. Therefore, the output will contain the samples in both sequences ordered in a way that minimizes the multivariate distance between consecutive samples. This scenario assumes that at least one of the sequences do not have a time/age/depth column, or that the values in such a column are uncertain. In any case, time/age/depth is not considered as a factor in the generation of the composite sequence.

The example below uses the pollenGP dataset, which contains 200 samples, with 40 pollen types each. To create a smalle case study, the code below separates the first 20 samples of the sequence into two different sequences with 10 randomly selected samples each. Even though this scenario assumes that these sequences do not have depth or age, these columns will be kept so the result can be assessed. That is why these columns are added to the exclude.columns argument. Also, note that the argument transformation is set to “none”, so the output is not transformed, and the outcome can be easily interpreted. This will give more weight to the most abundant taxa, which will in fact guide the slotting.

Once the sequences are prepared, the function workflowSlotting will allow to combine (slot) them. The function computes a distance matrix between the samples in both sequences according to the method argument, computes the least-cost matrix, and generates the least-cost path. Note that it only uses an orthogonal method considering blocks, since this is the only option really suitable for this task.

Distance matrix and least-cost path of the example sequences 'A' and 'B'..

Distance matrix and least-cost path of the example sequences ‘A’ and ‘B’..

The function reads the least-cost path in order to find the combination of samples of both sequences that minimizes dissimilarity, constrained by the order of the samples on each sequence. The output dataframe has a column named original.index, which has the index of each sample in the original datasets.

kable(AB.combined[1:15,1:10], digits = 4, caption = "Combination of sequences A and B.")
Combination of sequences A and B.
id original.index depth age Abies Juniperus Hedera Plantago Boraginaceae Crassulaceae
1 A 1 3 3.97 11243 0 5 12 21 95
11 B 1 1 3.92 11108 0 7 0 5 20
12 B 2 2 3.95 11189 0 3 3 15 47
13 B 3 4 4.00 11324 0 8 43 60 65
2 A 2 6 4.05 11459 0 20 73 94 1
14 B 4 5 4.02 11378 0 44 76 110 0
3 A 3 7 4.07 11514 0 20 80 100 0
4 A 4 8 4.10 11595 0 34 80 155 0
5 A 5 9 4.17 11784 0 22 44 131 0
15 B 5 13 4.27 12054 0 37 30 150 0
6 A 6 10 4.20 11865 0 35 30 112 0
7 A 7 11 4.22 11919 0 30 45 150 0
8 A 8 12 4.25 12000 0 44 35 150 0
9 A 9 15 4.32 12189 0 43 17 120 0
16 B 6 14 4.30 12135 0 50 10 120 0

Note that several samples show inverted ages with respect to the previous samples. This is to be expected, since the slotting algorithm only takes into account distance/dissimilarity between adjacent samples to generate the ordering.

Sequences A (green) and B (blue) with their ordered samples (upper panel), and the composite sequence resulting from them (lower panel) after applying the sequence slotting algorithm. Notice that the slotting takes into account all columns in both datasets, and therefore, a single column, as shown in the plot, might not be totally representative of the slotting solution.

Sequences A (green) and B (blue) with their ordered samples (upper panel), and the composite sequence resulting from them (lower panel) after applying the sequence slotting algorithm. Notice that the slotting takes into account all columns in both datasets, and therefore, a single column, as shown in the plot, might not be totally representative of the slotting solution.

Transferring an attribute from one sequence to another

This scenario assumes that the user has two METS, one of them with a given attribute (age/time) that needs to be transferred to the other sequence by using similarity/dissimilarity (constrained by sample order) as a transfer criterion. This case is relatively common in palaeoecology, when a given dataset is dated, and another taken at a close location is not.

The code below prepares the data for the example. The sequence pollenGP is the reference sequence, and contains the column age. The sequence pollenX is the target sequence, without an age column. We generate it by taking 40 random samples between the samples 50 and 100 of pollenGP. The sequences are prepared with prepareSequences, as usual, with the identificators “GP” and “X”

The transfer of “age” values from GP to X can be done in two ways, both constrained by sample order:

  • Direct: each sample in X gets the age of its most similar sample in GP.
  • Interpolated: each sample in X gets an age interpolated from the ages of the two most similar samples in GP. The interpolation is weighted by the similarity between the samples.

Direct transfer

A direct transfer of an attribute from the samples of one sequence to the samples of another requires to compute a distance matrix between samples, the least-cost matrix and its least-cost path (both with the option diagonal activated), and to parse the least-cost path file to assign attribute values. This is done by the function workflowTransfer with the option \(mode = "direct"\).

id depth age Abies Juniperus Hedera Plantago Boraginaceae Crassulaceae Pinus Ranunculaceae Rhamnus Caryophyllaceae Dipsacaceae Betula Acer Armeria Tilia Hippophae Salix Labiatae Valeriana Nymphaea Umbelliferae Sanguisorba_minor Plantago.lanceolata Campanulaceae Asteroideae Gentiana Fraxinus Cichorioideae Taxus Rumex Cedrus Ranunculus.subgen..Batrachium Cyperaceae Corylus Myriophyllum Filipendula Vitis Rubiaceae Polypodium
41 X 0 3.92 11108 0 7 0 5 20 0 13 0 2 1 0 2 41 0 0 0 0 0 0 0 1 0 8 0 0 0 0 0 0 0 0 0 0 0 60 2 0 0
42 X 0 4.00 11324 0 8 43 60 65 0 10 0 0 2 4 0 0 0 0 0 0 2 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 0 2 0
43 X 0 4.02 11378 0 44 76 110 0 0 0 0 0 2 11 0 0 0 0 3 1 3 0 0 5 0 0 1 0 6 0 0 1 0 1 0 0 0 0 1 1 1
44 X 0 4.05 11459 0 20 73 94 1 0 0 0 0 1 10 0 2 0 0 3 1 3 0 0 3 0 0 0 0 3 0 0 2 0 0 0 0 0 0 1 0 0
45 X 0 4.07 11514 0 20 80 100 0 0 0 0 0 10 4 0 0 1 0 0 0 3 0 0 1 0 0 0 0 8 1 0 0 0 0 0 1 2 1 0 1 0
46 X 0 4.07 11595 0 34 80 155 0 0 0 2 0 2 13 0 0 0 0 1 0 2 0 0 2 0 0 0 0 6 0 0 0 0 0 0 1 1 0 0 1 0
47 X 0 4.17 11784 0 22 44 131 0 0 0 0 0 1 13 0 0 0 0 5 0 5 0 0 3 0 0 0 0 6 0 0 0 0 2 0 1 0 0 2 0 0
48 X 0 4.20 11865 0 35 30 112 0 0 0 0 0 4 8 0 0 0 2 2 1 2 0 0 7 0 1 0 0 10 1 0 0 0 1 0 1 1 0 2 1 0
49 X 0 4.22 11919 0 30 45 150 0 0 0 0 0 4 11 0 0 0 0 2 3 1 0 0 1 0 0 0 0 13 1 0 0 0 0 0 0 0 0 2 1 0
50 X 0 4.25 12000 0 44 35 150 0 0 0 0 0 2 8 0 0 0 0 3 1 0 0 0 5 0 0 0 0 6 3 0 0 0 0 0 0 2 0 1 0 0
51 X 0 4.25 12054 0 37 30 150 0 0 1 0 0 6 10 0 0 0 0 7 2 2 0 0 6 0 0 0 0 7 1 0 0 0 0 0 0 3 0 4 0 0
52 X 0 4.32 12135 0 50 10 120 0 0 0 0 0 1 7 0 0 0 0 1 0 1 0 0 8 0 0 0 0 8 2 0 2 0 0 0 0 0 0 1 0 0
53 X 0 4.32 12189 0 43 17 120 0 0 0 0 0 2 15 0 0 1 1 2 0 2 0 0 5 1 0 0 0 6 2 0 2 0 1 0 2 1 0 0 0 0
54 X 0 4.40 12324 0 50 11 86 0 0 0 0 0 2 15 0 0 2 1 4 5 3 0 0 6 0 0 0 0 5 1 0 2 0 0 1 1 0 0 3 1 0
55 X 0 4.40 12405 0 51 6 70 0 0 0 0 0 1 16 0 0 4 1 2 4 2 0 0 5 2 0 0 0 1 2 0 1 0 0 0 0 0 0 0 3 1

The algorithm finds the most similar samples, and transfers attribute values directly between them. This can result in duplicated attribute values, as highlighted in the table above. The Pearson correlation between the original ages (stored in pollenX.age) and the assigned ones is 0.9996, so it can be concluded that in spite of its simplicity, this algorithm yields accurate results.

Interpolated transfer

If we consider:

  • Two samples \(A_{i}\) and \(A_{j}\) of the sequence \(A\).
  • Each one with a value the attribute \(t\), \(At_{i}\) and \(At_{j}\).
  • One sample \(B_{k}\) of the sequence \(B\).
  • With an unknown attribute \(Bt_{k}\).
  • The multivariate distance \(D_{B_{k}A_{i}}\) between the samples \(A_{i}\) and \(B_{k}\).
  • The multivariate distance \(D_{B_{k}A_{j}}\) between the samples \(A_{j}\) and \(B_{k}\).
  • The weight \(w_{i}\), computed as \(D_{B_{k}A_{i}} / (D_{B_{k}A_{i}} + D_{B_{k}A_{j}})\)
  • The weight \(w_{j}\), computed as \(D_{B_{k}A_{j}} / (D_{B_{k}A_{i}} + D_{B_{k}A_{j}})\)

The unknwon value \(Bt_{k}\) is computed as:

\[Bt_{k} = w_{i} \times At_{i} + w_{j} \times At_{j}\]

The code below exemplifies the operation, using the samples 1 and 4 of the dataset pollenGP as \(Ai\) and \(Aj\), and the sample 3 as \(Bk\).

The table below shows the observed versus the predicted values for \(Btk\).

temp.df <- data.frame(Observed = pollenGP[3, "age"], Predicted = Btk)
kable(t(temp.df), digits = 4, caption = "Observed versus predicted value in the interpolation of an age based on similarity between samples.")
Observed versus predicted value in the interpolation of an age based on similarity between samples.
Observed 3.9700
Predicted 3.9735

Below we create some example data, where a subset of pollenGP will be the donor of age values, and another subset of it, named pollenX will be the receiver of the age values.

To transfer attributes from GP to X we use the workflowTransfer function with the option mode = “interpolate”.

#parameters
X.new <- workflowTransfer(
  sequences = GP.X,
  grouping.column = "id",
  time.column = "age",
  method = "euclidean",
  transfer.what = "age",
  transfer.from = "GP",
  transfer.to = "X",
  mode = "interpolated"
  )

kable(X.new[1:15, ], digits = 4, caption = "Result of the transference of an age attribute from one sequence to another. NA values are expected when predicted ages for a given sample yield a higher number than the age of the previous sample.")  %>% 
  row_spec(c(8, 13), bold = T)
Result of the transference of an age attribute from one sequence to another. NA values are expected when predicted ages for a given sample yield a higher number than the age of the previous sample.
id depth age Abies Juniperus Hedera Plantago Boraginaceae Crassulaceae Pinus Ranunculaceae Rhamnus Caryophyllaceae Dipsacaceae Betula Acer Armeria Tilia Hippophae Salix Labiatae Valeriana Nymphaea Umbelliferae Sanguisorba_minor Plantago.lanceolata Campanulaceae Asteroideae Gentiana Fraxinus Cichorioideae Taxus Rumex Cedrus Ranunculus.subgen..Batrachium Cyperaceae Corylus Myriophyllum Filipendula Vitis Rubiaceae Polypodium
41 X 0 3.9497 11108 0 7 0 5 20 0 13 0 2 1 0 2 41 0 0 0 0 0 0 0 1 0 8 0 0 0 0 0 0 0 0 0 0 0 60 2 0 0
42 X 0 3.9711 11324 0 8 43 60 65 0 10 0 0 2 4 0 0 0 0 0 0 2 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 1 11 0 2 0
43 X 0 4.0009 11378 0 44 76 110 0 0 0 0 0 2 11 0 0 0 0 3 1 3 0 0 5 0 0 1 0 6 0 0 1 0 1 0 0 0 0 1 1 1
44 X 0 4.0219 11459 0 20 73 94 1 0 0 0 0 1 10 0 2 0 0 3 1 3 0 0 3 0 0 0 0 3 0 0 2 0 0 0 0 0 0 1 0 0
45 X 0 4.0522 11514 0 20 80 100 0 0 0 0 0 10 4 0 0 1 0 0 0 3 0 0 1 0 0 0 0 8 1 0 0 0 0 0 1 2 1 0 1 0
46 X 0 4.1361 11595 0 34 80 155 0 0 0 2 0 2 13 0 0 0 0 1 0 2 0 0 2 0 0 0 0 6 0 0 0 0 0 0 1 1 0 0 1 0
47 X 0 4.1972 11784 0 22 44 131 0 0 0 0 0 1 13 0 0 0 0 5 0 5 0 0 3 0 0 0 0 6 0 0 0 0 2 0 1 0 0 2 0 0
48 X 0 NA 11865 0 35 30 112 0 0 0 0 0 4 8 0 0 0 2 2 1 2 0 0 7 0 1 0 0 10 1 0 0 0 1 0 1 1 0 2 1 0
49 X 0 4.2027 11919 0 30 45 150 0 0 0 0 0 4 11 0 0 0 0 2 3 1 0 0 1 0 0 0 0 13 1 0 0 0 0 0 0 0 0 2 1 0
50 X 0 4.2237 12000 0 44 35 150 0 0 0 0 0 2 8 0 0 0 0 3 1 0 0 0 5 0 0 0 0 6 3 0 0 0 0 0 0 2 0 1 0 0
51 X 0 4.2999 12054 0 37 30 150 0 0 1 0 0 6 10 0 0 0 0 7 2 2 0 0 6 0 0 0 0 7 1 0 0 0 0 0 0 3 0 4 0 0
52 X 0 NA 12135 0 50 10 120 0 0 0 0 0 1 7 0 0 0 0 1 0 1 0 0 8 0 0 0 0 8 2 0 2 0 0 0 0 0 0 1 0 0
53 X 0 NA 12189 0 43 17 120 0 0 0 0 0 2 15 0 0 1 1 2 0 2 0 0 5 1 0 0 0 6 2 0 2 0 1 0 2 1 0 0 0 0
54 X 0 4.3501 12324 0 50 11 86 0 0 0 0 0 2 15 0 0 2 1 4 5 3 0 0 6 0 0 0 0 5 1 0 2 0 0 1 1 0 0 3 1 0
55 X 0 4.4155 12405 0 51 6 70 0 0 0 0 0 1 16 0 0 4 1 2 4 2 0 0 5 2 0 0 0 1 2 0 1 0 0 0 0 0 0 0 3 1

When interpolated values of the age column (transferred attribute via interpolation) show the value NA, it means that the interpolation yielded an age lower than the previous one. This happens when the same \(Ai\) and \(Aj\) are used to evaluate two or more different samples \(Bk\), and the second \(Bk\) is more similar to \(Ai\) than the first one. These NA values can be removed with na.omit(), or interpolated with the functions imputeTS::na.interpolation or zoo::na.approx.

Without taking into account these NA values, the Pearson correlation of the interpolated ages with the real ones is 0.9985.

IMPORTANT: the interpretation of the interpolated ages requires a careful consideration. Please, don’t do it blindly, because this algorithm has its limitations. For example, significant hiatuses in the data can introduce wild variations in interpolated ages.