In BlasBenito/distantia: Assessing Dissimilarity Between Multivariate Time Series
distantia: an R package to assess dissimilarity between multivariate time-series
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>",
fig.path = "man/figures/README-",
out.width = "100%",
eval = FALSE
)
#installing required libraries
list.of.packages <- c(
"ggplot2",
"viridis",
"fields",
"foreach",
"parallel",
"doParallel",
"qgraph",
"tidyr",
"devtools"
)
packages.to.install <- setdiff(
x = list.of.packages,
y = installed.packages()[,"Package"]
)
if(length(packages.to.install) > 0){
install.packages(packages.to.install, dep=TRUE)
}
rm(list.of.packages, packages.to.install)
Summary
The package distantia helps quantify 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 logic 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:
library(remotes)
remotes::install_github(
repo = "BlasBenito/distantia",
ref = "development"
)
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
kbl(sequenceA[1:15, ], caption = "Sequence A")
kbl(sequenceB[1:15, ], caption = "Sequence B")
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.
#preparing sequences
AB.sequences <- prepareSequences(
sequence.A = sequenceA,
sequence.A.name = "A",
sequence.B = sequenceB,
sequence.B.name = "B",
merge.mode = "complete",
if.empty.cases = "zero",
transformation = "hellinger"
)
#showing first rows of the transformed data
kbl(
AB.sequences[1:15, ],
digits = 4,
caption = "Sequences A and B ready for analysis."
)
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.
#computing distance matrix
AB.distance.matrix <- distanceMatrix(
sequences = AB.sequences,
method = "euclidean"
)
#plotting distance matrix
plotMatrix(
distance.matrix = AB.distance.matrix,
color.palette = "viridis",
margins = rep(4,4)
)
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
#ORTHOGONAL SEARCH
#computing least-cost matrix
AB.least.cost.matrix <- leastCostMatrix(
distance.matrix = AB.distance.matrix,
diagonal = FALSE
)
#extracting least-cost path
AB.least.cost.path <- leastCostPath(
distance.matrix = AB.distance.matrix,
least.cost.matrix = AB.least.cost.matrix,
diagonal = FALSE
)
#DIAGONAL SEARCH
#computing least-cost matrix
AB.least.cost.matrix.diag <- leastCostMatrix(
distance.matrix = AB.distance.matrix,
diagonal = TRUE
)
#extracting least-cost path
AB.least.cost.path.diag <- leastCostPath(
distance.matrix = AB.distance.matrix,
least.cost.matrix = AB.least.cost.matrix.diag,
diagonal = TRUE
)
#plotting solutions
plotMatrix(
distance.matrix = list(
'A|B' = AB.least.cost.matrix[[1]],
'A|B' = AB.least.cost.matrix.diag[[1]]
),
least.cost.path = list(
'A|B' = AB.least.cost.path[[1]],
'A|B' = AB.least.cost.path.diag[[1]]
),
color.palette = "viridis",
margin = rep(4,4),
plot.rows = 1,
plot.columns = 2
)
Computing $AB_{between}$ from these solutions is straightforward with the function leastCost
#orthogonal solution
AB.between <- leastCost(
least.cost.path = AB.least.cost.path
)
#diagonal solution
AB.between.diag <- leastCost(
least.cost.path = AB.least.cost.path.diag
)
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.
#ORTHOGONAL SOLUTION
#removing blocks from least cost path
AB.least.cost.path.nb <- leastCostPathNoBlocks(
least.cost.path = AB.least.cost.path
)
#computing AB.between again
AB.between.nb <- leastCost(
least.cost.path = AB.least.cost.path.nb
)
#DIAGONAL SOLUTION
#removing blocks
AB.least.cost.path.diag.nb <- leastCostPathNoBlocks(
least.cost.path = AB.least.cost.path.diag
)
#diagonal solution without blocks
AB.between.diag.nb <- leastCost(
least.cost.path = AB.least.cost.path.diag.nb
)
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.
#changing names of the selected solutions
AB.least.cost.path <- AB.least.cost.path.diag.nb
AB.between <- AB.between.diag.nb
#removing unneeded objects
rm(AB.between.diag, AB.between.diag.nb, AB.between.nb, AB.distance.matrix, AB.least.cost.matrix, AB.least.cost.matrix.diag, AB.least.cost.path.diag, AB.least.cost.path.diag.nb, AB.least.cost.path.nb, sequenceA, sequenceB)
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.
AB.within <- autoSum(
sequences = AB.sequences,
least.cost.path = AB.least.cost.path,
method = "euclidean"
)
AB.within
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.
AB.psi <- psi(
least.cost = AB.between,
autosum = AB.within
)
AB.psi[[1]] <- AB.psi[[1]] + 1
The output of psi is a list, that can be transformed to a dataframe or a matrix by using the formatPsi function.
#to dataframe
AB.psi.dataframe <- formatPsi(
psi.values = AB.psi,
to = "dataframe")
kbl(AB.psi.dataframe, digits = 4)
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.
#checking the help file
help(workflowPsi)
#computing psi for A and B
AB.psi <- workflowPsi(
sequences = AB.sequences,
grouping.column = "id",
method = "euclidean",
format = "list",
diagonal = TRUE,
ignore.blocks = TRUE
)
AB.psi
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.
#cleaning workspace
rm(list = ls())
Comparing multiple irregular METS
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)
kbl(
head(sequencesMIS, n=15),
digits = 4,
caption = "Header of the sequencesMIS dataset."
)
unique(sequencesMIS$MIS)
The dataset is checked and prepared with prepareSequences.
MIS.sequences <- prepareSequences(
sequences = sequencesMIS,
grouping.column = "MIS",
if.empty.cases = "zero",
transformation = "hellinger"
)
The dissimilarity measure psi can be computed for every combination of sequences through the function workflowPsi shown below.
MIS.psi <- workflowPsi(
sequences = MIS.sequences,
grouping.column = "MIS",
method = "euclidean",
diagonal = TRUE,
ignore.blocks = TRUE
)
kbl(
MIS.psi[order(MIS.psi$psi), ],
digits = 4
)
There is also a "high-performance" (HP) version of this function with a much lower memory footprint. It uses the options method = "euclidean", diagonal = TRUE, and ignore.blocks = TRUE by default.
MIS.psi <- workflowPsiHP(
sequences = MIS.sequences,
grouping.column = "MIS"
)
The output of workflowPsi() can transformed into a matrix to be plotted as an adjacency network with the qgraph package.
#psi values to matrix
MIS.psi.matrix <- formatPsi(
psi.values = MIS.psi,
to = "matrix"
)
#dissimilariy to distance
MIS.distance <- 1/MIS.psi.matrix**4
#plotting network
qgraph::qgraph(
MIS.distance,
layout='spring',
vsize=5,
labels = colnames(MIS.distance),
colors = viridis::viridis(2, begin = 0.3, end = 0.8, alpha = 0.5, direction = -1)
)
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) +
theme_bw()
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.
#cleaning workspace
rm(list = ls())
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.
#loading sample data
data(climate)
#preparing sequences
climate <- prepareSequences(
sequences = climate,
grouping.column = "sequenceId",
time.column = "time",
paired.samples = TRUE,
transformation = "scale"
)
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.
#computing psi
climate.psi <- workflowPsi(
sequences = climate,
grouping.column = "sequenceId",
time.column = "time",
method = "euclidean",
paired.samples = TRUE, #this bit is important
same.time = TRUE, #removes samples with unequal time
format = "dataframe"
)
#ordered with lower psi on top
kbl(climate.psi[order(climate.psi$psi), ], digits = 4, row.names = FALSE, caption = "Psi values between pairs of sequences in the 'climate' dataset.")
#cleaning workspace
rm(list = ls())
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.
#getting example data
data(sequencesMIS)
#working with 3 groups (to make this fast)
sequencesMIS <- sequencesMIS[sequencesMIS$MIS %in% c("MIS-4", "MIS-5", "MIS-6"),]
#preparing sequences
sequencesMIS <- prepareSequences(
sequences = sequencesMIS,
grouping.column = "MIS",
transformation = "hellinger"
)
The computation of the null psi values goes as follows:
random.psi <- workflowNullPsi(
sequences = sequencesMIS,
grouping.column = "MIS",
method = "euclidean",
diagonal = TRUE,
ignore.blocks = TRUE,
repetitions = 99, #recommended value: 999
parallel.execution = TRUE
)
#there is also a high-performance version of this function with fewer options (diagonal = TRUE, ignore.blocks = TRUE, and method = "euclidean" are used by default)
random.psi <- workflowNullPsiHP(
sequences = sequencesMIS,
grouping.column = "MIS",
repetitions = 99, #recommended value: 999
parallel.execution = TRUE
)
Note that the number of repetitions has been set to 9 in order to speed-up execution. The actual number would 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.
kbl(
random.psi$psi,
digits = 4
)
The dataframe p contains the probability of obtaining the real psi value by chance for each combination of sequences.
kbl(random.psi$p)
#cleaning workspace
rm(list = ls())
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.
#getting example data
data(sequencesMIS)
#getting three groups only to simplify
sequencesMIS <- sequencesMIS[sequencesMIS$MIS %in% c("MIS-4", "MIS-5", "MIS-6"),]
#preparing sequences
sequences <- prepareSequences(
sequences = sequencesMIS,
grouping.column = "MIS",
merge.mode = "complete"
)
The workflow function is pretty similar to the ones explained above. However, unlike the other functions in the package, that parallelize across the comparison of pairs of sequences, this one parallelizes the computation of psi on combinations of columns, removing one column each time.
psi.importance <- workflowImportance(
sequences = sequencesMIS,
grouping.column = "MIS",
method = "euclidean",
diagonal = TRUE,
ignore.blocks = TRUE
)
There is also a high performance version of this function, but with fewer options (it uses euclidean, diagonal, and ignores blocks by default)
psi.importance <- workflowImportanceHP(
sequences = sequencesMIS,
grouping.column = "MIS"
)
The output is a list with two slots named psi and psi.drop.
The dataframe psi contains psi values for each combination of variables (named in the coluns A and B) computed for all columns in the column All variables, and one column per variable named Without variable_name containing the psi value when that variable is removed from the compared sequences.
kbl(psi.importance$psi, digits = 4,)
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") +
theme_bw()
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.
kbl(psi.importance$psi.drop)
#extracting object
psi.drop.df <- psi.importance$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$B, sep=" - ")
#plot
ggplot(data=psi.drop.df.long, aes(x=variable, y=psi, fill=psi)) +
geom_bar(stat = "identity") +
coord_flip() +
geom_hline(yintercept = 0, size = 0.3) +
facet_wrap("name") +
scale_fill_viridis(direction = -1) +
ggtitle("Drop in dissimilarity when variables are removed.") +
ylab("Drop in dissimilarity (%)") +
labs(fill = "Psi drop (%)") +
theme_bw()
#cleaning workspace
rm(list = ls())
Partial matching: finding the section in a long sequence more similar to a given short sequence
In this scenario the user has one short and one long sequence, and the goal is to find the section in the long sequence that better matches the short one. To recreate this scenario we use the dataset sequencesMIS. The first 10 samples will serve as short sequence, and the first 40 samples as long sequence. These small subsets are selected to speed-up the execution time of this example.
#loading the data
data(sequencesMIS)
#removing grouping column
sequencesMIS$MIS <- NULL
#subsetting to get the short sequence
MIS.short <- sequencesMIS[1:10, ]
#subsetting to get the long sequence
MIS.long <- sequencesMIS[1:40, ]
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.
MIS.short.long <- prepareSequences(
sequence.A = MIS.short,
sequence.A.name = "short",
sequence.B = MIS.long,
sequence.B.name = "long",
grouping.column = "id",
transformation = "hellinger"
)
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.
MIS.psi <- workflowPartialMatch(
sequences = MIS.short.long,
grouping.column = "id",
method = "euclidean",
diagonal = TRUE,
ignore.blocks = TRUE,
min.length = nrow(MIS.short),
max.length = nrow(MIS.short)
)
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.
kbl(MIS.psi[1:15, ], digits = 4)
Subsetting the long sequence to obtain the segment best matching with the short sequence goes as follows.
#indices of the best matching segment
best.match.indices <- MIS.psi[1, "first.row"]:MIS.psi[1, "last.row"]
#subsetting by these indices
best.match <- MIS.long[best.match.indices, ]
#cleaning workspace
rm(list = ls())
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.
#loading the data
data(pollenGP)
#getting first 20 samples
pollenGP <- pollenGP[1:20, ]
#sampling indices
set.seed(10) #to get same result every time
sampling.indices <- sort(sample(1:20, 10))
#subsetting the sequence
A <- pollenGP[sampling.indices, ]
B <- pollenGP[-sampling.indices, ]
#preparing the sequences
AB <- prepareSequences(
sequence.A = A,
sequence.A.name = "A",
sequence.B = B,
sequence.B.name = "B",
grouping.column = "id",
exclude.columns = c("depth", "age"),
transformation = "none"
)
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.
AB.combined <- workflowSlotting(
sequences = AB,
grouping.column = "id",
time.column = "age",
exclude.columns = "depth",
method = "euclidean",
plot = TRUE
)
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.
kbl(
AB.combined[1:15,1:10],
digits = 4,
caption = "Combination of sequences A and B."
)
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.
#color palette
cols <- viridis::viridis(3, begin = 0.2, end = 0.8)
#column to plot
taxon <- "Abies"
if(!(taxon %in% colnames(AB))){
warning("This taxon is not in AB, plotting 'Abies' instead")
taxon <- "Abies"
}
#setting y limits
y.lim <- range(c(AB[AB$id == "A", taxon], AB[AB$id == "B", taxon]))
#multipanel and margins
par(mfrow=c(2,1), mar = c(4,2,2,1))
#plotting separated sequences
plot(AB[AB$id == "A", taxon],
type="l",
ylim = y.lim,
xlab = "",
ylab = "",
col = cols[1],
lwd = 2,
main = paste("Column '", taxon, "' of the sequences A and B", sep="")
)
lines(AB[AB$id == "B", taxon],
col = cols[3],
lwd = 2
)
text(x = 1:10,
y = AB[AB$id == "B", taxon],
labels = 1:10,
pos = 3,
col = cols[3])
text(x = 1:10,
y = AB[AB$id == "A", taxon],
labels = 1:10,
pos = 1,
col = cols[1])
#plotting composite sequence
plot(AB.combined[, taxon],
type="l",
ylim = y.lim,
xlab = "Sample order",
ylab = "",
col = cols[2],
lwd = 2,
main = paste("Column '", taxon, "' of the composite sequence", sep="")
)
points(AB.combined[, taxon],
col = cols[c(1,3)][AB.combined$id],
pch = 19
)
text(x = 1:20,
y = AB.combined[, taxon],
labels = AB.combined[, "original.index"],
pos = 3,
col = cols[c(1,3)][AB.combined$id])
#cleaning workspace
rm(list = ls())
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"
#loading sample dataset
data(pollenGP)
#subset pollenGP to make a shorter dataset
pollenGP <- pollenGP[1:50, ]
#generating a subset of pollenGP
set.seed(10)
pollenX <- pollenGP[sort(sample(1:50, 40)), ]
#we separate the age column
pollenX.age <- pollenX$age
#and remove the age values from pollenX
pollenX$age <- NULL
pollenX$depth <- NULL
#removing some samples from pollenGP
#so pollenX is not a perfect subset of pollenGP
pollenGP <- pollenGP[-sample(1:50, 10), ]
#prepare sequences
GP.X <- prepareSequences(
sequence.A = pollenGP,
sequence.A.name = "GP",
sequence.B = pollenX,
sequence.B.name = "X",
grouping.column = "id",
time.column = "age",
exclude.columns = "depth",
transformation = "none"
)
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 $mode = "direct"$.
#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 = "direct"
)
kbl(X.new[1:15, ], digits = 4)
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.
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 $Bk$.
#loading data
data(pollenGP)
#samples in A
Ai <- pollenGP[1, 3:ncol(pollenGP)]
Aj <- pollenGP[4, 3:ncol(pollenGP)]
#ages of the samples in A
Ati <- pollenGP[1, "age"]
Atj <- pollenGP[4, "age"]
#sample in B
Bk <- pollenGP[2, 3:ncol(pollenGP)]
#computing distances between Bk, Ai, and Aj
DBkAi <- distance(Bk, Ai)
DBkAj <- distance(Bk, Aj)
#normalizing the distances to 1
wi <- DBkAi / (DBkAi + DBkAj)
wj <- DBkAj / (DBkAi + DBkAj)
#computing Btk
Btk <- wi * Ati + wj * Atj
The table below shows the observed versus the predicted values for $Btk$.
temp.df <- data.frame(Observed = pollenGP[3, "age"], Predicted = Btk)
kbl(t(temp.df), digits = 4, caption = "Observed versus predicted value in the interpolation of an age based on similarity between samples.")
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.
#loading sample dataset
data(pollenGP)
#subset pollenGP to make a shorter dataset
pollenGP <- pollenGP[1:50, ]
#generating a subset of pollenGP
set.seed(10)
pollenX <- pollenGP[sort(sample(1:50, 40)), ]
#we separate the age column
pollenX.age <- pollenX$age
#and remove the age values from pollenX
pollenX$age <- NULL
pollenX$depth <- NULL
#removing some samples from pollenGP
#so pollenX is not a perfect subset of pollenGP
pollenGP <- pollenGP[-sample(1:50, 10), ]
#prepare sequences
GP.X <- prepareSequences(
sequence.A = pollenGP,
sequence.A.name = "GP",
sequence.B = pollenX,
sequence.B.name = "X",
grouping.column = "id",
time.column = "age",
exclude.columns = "depth",
transformation = "none"
)
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"
)
kbl(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)
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.
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.
#cleaning workspace
rm(list = ls())
BlasBenito/distantia documentation built on Nov. 17, 2023, 11:06 p.m.
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distantia: an R package to assess dissimilarity between multivariate time-series
knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.path = "man/figures/README-", out.width = "100%", eval = FALSE )
#installing required libraries list.of.packages <- c( "ggplot2", "viridis", "fields", "foreach", "parallel", "doParallel", "qgraph", "tidyr", "devtools" ) packages.to.install <- setdiff( x = list.of.packages, y = installed.packages()[,"Package"] ) if(length(packages.to.install) > 0){ install.packages(packages.to.install, dep=TRUE) } rm(list.of.packages, packages.to.install)
Summary
The package distantia helps quantify 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 logic 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:
library(remotes) remotes::install_github( repo = "BlasBenito/distantia", ref = "development" )
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 kbl(sequenceA[1:15, ], caption = "Sequence A") kbl(sequenceB[1:15, ], caption = "Sequence B")
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.
#preparing sequences AB.sequences <- prepareSequences( sequence.A = sequenceA, sequence.A.name = "A", sequence.B = sequenceB, sequence.B.name = "B", merge.mode = "complete", if.empty.cases = "zero", transformation = "hellinger" ) #showing first rows of the transformed data kbl( AB.sequences[1:15, ], digits = 4, caption = "Sequences A and B ready for analysis." )
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.
#computing distance matrix AB.distance.matrix <- distanceMatrix( sequences = AB.sequences, method = "euclidean" ) #plotting distance matrix plotMatrix( distance.matrix = AB.distance.matrix, color.palette = "viridis", margins = rep(4,4) )
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
#ORTHOGONAL SEARCH #computing least-cost matrix AB.least.cost.matrix <- leastCostMatrix( distance.matrix = AB.distance.matrix, diagonal = FALSE ) #extracting least-cost path AB.least.cost.path <- leastCostPath( distance.matrix = AB.distance.matrix, least.cost.matrix = AB.least.cost.matrix, diagonal = FALSE ) #DIAGONAL SEARCH #computing least-cost matrix AB.least.cost.matrix.diag <- leastCostMatrix( distance.matrix = AB.distance.matrix, diagonal = TRUE ) #extracting least-cost path AB.least.cost.path.diag <- leastCostPath( distance.matrix = AB.distance.matrix, least.cost.matrix = AB.least.cost.matrix.diag, diagonal = TRUE ) #plotting solutions plotMatrix( distance.matrix = list( 'A|B' = AB.least.cost.matrix[[1]], 'A|B' = AB.least.cost.matrix.diag[[1]] ), least.cost.path = list( 'A|B' = AB.least.cost.path[[1]], 'A|B' = AB.least.cost.path.diag[[1]] ), color.palette = "viridis", margin = rep(4,4), plot.rows = 1, plot.columns = 2 )
Computing $AB_{between}$ from these solutions is straightforward with the function leastCost
#orthogonal solution AB.between <- leastCost( least.cost.path = AB.least.cost.path ) #diagonal solution AB.between.diag <- leastCost( least.cost.path = AB.least.cost.path.diag )
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.
#ORTHOGONAL SOLUTION #removing blocks from least cost path AB.least.cost.path.nb <- leastCostPathNoBlocks( least.cost.path = AB.least.cost.path ) #computing AB.between again AB.between.nb <- leastCost( least.cost.path = AB.least.cost.path.nb ) #DIAGONAL SOLUTION #removing blocks AB.least.cost.path.diag.nb <- leastCostPathNoBlocks( least.cost.path = AB.least.cost.path.diag ) #diagonal solution without blocks AB.between.diag.nb <- leastCost( least.cost.path = AB.least.cost.path.diag.nb )
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.
#changing names of the selected solutions AB.least.cost.path <- AB.least.cost.path.diag.nb AB.between <- AB.between.diag.nb #removing unneeded objects rm(AB.between.diag, AB.between.diag.nb, AB.between.nb, AB.distance.matrix, AB.least.cost.matrix, AB.least.cost.matrix.diag, AB.least.cost.path.diag, AB.least.cost.path.diag.nb, AB.least.cost.path.nb, sequenceA, sequenceB)
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.
AB.within <- autoSum( sequences = AB.sequences, least.cost.path = AB.least.cost.path, method = "euclidean" ) AB.within
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.
AB.psi <- psi( least.cost = AB.between, autosum = AB.within ) AB.psi[[1]] <- AB.psi[[1]] + 1
The output of psi is a list, that can be transformed to a dataframe or a matrix by using the formatPsi function.
#to dataframe AB.psi.dataframe <- formatPsi( psi.values = AB.psi, to = "dataframe") kbl(AB.psi.dataframe, digits = 4)
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.
#checking the help file help(workflowPsi) #computing psi for A and B AB.psi <- workflowPsi( sequences = AB.sequences, grouping.column = "id", method = "euclidean", format = "list", diagonal = TRUE, ignore.blocks = TRUE ) AB.psi
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.
#cleaning workspace rm(list = ls())
Comparing multiple irregular METS
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) kbl( head(sequencesMIS, n=15), digits = 4, caption = "Header of the sequencesMIS dataset." ) unique(sequencesMIS$MIS)
The dataset is checked and prepared with prepareSequences.
MIS.sequences <- prepareSequences( sequences = sequencesMIS, grouping.column = "MIS", if.empty.cases = "zero", transformation = "hellinger" )
The dissimilarity measure psi can be computed for every combination of sequences through the function workflowPsi shown below.
MIS.psi <- workflowPsi( sequences = MIS.sequences, grouping.column = "MIS", method = "euclidean", diagonal = TRUE, ignore.blocks = TRUE ) kbl( MIS.psi[order(MIS.psi$psi), ], digits = 4 )
There is also a "high-performance" (HP) version of this function with a much lower memory footprint. It uses the options method = "euclidean", diagonal = TRUE, and ignore.blocks = TRUE by default.
MIS.psi <- workflowPsiHP( sequences = MIS.sequences, grouping.column = "MIS" )
The output of workflowPsi() can transformed into a matrix to be plotted as an adjacency network with the qgraph package.
#psi values to matrix MIS.psi.matrix <- formatPsi( psi.values = MIS.psi, to = "matrix" ) #dissimilariy to distance MIS.distance <- 1/MIS.psi.matrix**4 #plotting network qgraph::qgraph( MIS.distance, layout='spring', vsize=5, labels = colnames(MIS.distance), colors = viridis::viridis(2, begin = 0.3, end = 0.8, alpha = 0.5, direction = -1) )
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) + theme_bw()
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.
#cleaning workspace rm(list = ls())
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.
#loading sample data data(climate) #preparing sequences climate <- prepareSequences( sequences = climate, grouping.column = "sequenceId", time.column = "time", paired.samples = TRUE, transformation = "scale" )
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.
#computing psi climate.psi <- workflowPsi( sequences = climate, grouping.column = "sequenceId", time.column = "time", method = "euclidean", paired.samples = TRUE, #this bit is important same.time = TRUE, #removes samples with unequal time format = "dataframe" ) #ordered with lower psi on top kbl(climate.psi[order(climate.psi$psi), ], digits = 4, row.names = FALSE, caption = "Psi values between pairs of sequences in the 'climate' dataset.")
#cleaning workspace rm(list = ls())
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.
- For each case of each column of A and B, randomly apply one of these actions:
- 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.
#getting example data data(sequencesMIS) #working with 3 groups (to make this fast) sequencesMIS <- sequencesMIS[sequencesMIS$MIS %in% c("MIS-4", "MIS-5", "MIS-6"),] #preparing sequences sequencesMIS <- prepareSequences( sequences = sequencesMIS, grouping.column = "MIS", transformation = "hellinger" )
The computation of the null psi values goes as follows:
random.psi <- workflowNullPsi( sequences = sequencesMIS, grouping.column = "MIS", method = "euclidean", diagonal = TRUE, ignore.blocks = TRUE, repetitions = 99, #recommended value: 999 parallel.execution = TRUE ) #there is also a high-performance version of this function with fewer options (diagonal = TRUE, ignore.blocks = TRUE, and method = "euclidean" are used by default) random.psi <- workflowNullPsiHP( sequences = sequencesMIS, grouping.column = "MIS", repetitions = 99, #recommended value: 999 parallel.execution = TRUE )
Note that the number of repetitions has been set to 9 in order to speed-up execution. The actual number would 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.
kbl( random.psi$psi, digits = 4 )
The dataframe p contains the probability of obtaining the real psi value by chance for each combination of sequences.
kbl(random.psi$p)
#cleaning workspace rm(list = ls())
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.
#getting example data data(sequencesMIS) #getting three groups only to simplify sequencesMIS <- sequencesMIS[sequencesMIS$MIS %in% c("MIS-4", "MIS-5", "MIS-6"),] #preparing sequences sequences <- prepareSequences( sequences = sequencesMIS, grouping.column = "MIS", merge.mode = "complete" )
The workflow function is pretty similar to the ones explained above. However, unlike the other functions in the package, that parallelize across the comparison of pairs of sequences, this one parallelizes the computation of psi on combinations of columns, removing one column each time.
psi.importance <- workflowImportance( sequences = sequencesMIS, grouping.column = "MIS", method = "euclidean", diagonal = TRUE, ignore.blocks = TRUE )
There is also a high performance version of this function, but with fewer options (it uses euclidean, diagonal, and ignores blocks by default)
psi.importance <- workflowImportanceHP( sequences = sequencesMIS, grouping.column = "MIS" )
The output is a list with two slots named psi and psi.drop.
The dataframe psi contains psi values for each combination of variables (named in the coluns A and B) computed for all columns in the column All variables, and one column per variable named Without variable_name containing the psi value when that variable is removed from the compared sequences.
kbl(psi.importance$psi, digits = 4,)
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") + theme_bw()
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.
kbl(psi.importance$psi.drop)
#extracting object psi.drop.df <- psi.importance$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$B, sep=" - ") #plot ggplot(data=psi.drop.df.long, aes(x=variable, y=psi, fill=psi)) + geom_bar(stat = "identity") + coord_flip() + geom_hline(yintercept = 0, size = 0.3) + facet_wrap("name") + scale_fill_viridis(direction = -1) + ggtitle("Drop in dissimilarity when variables are removed.") + ylab("Drop in dissimilarity (%)") + labs(fill = "Psi drop (%)") + theme_bw()
#cleaning workspace rm(list = ls())
Partial matching: finding the section in a long sequence more similar to a given short sequence
In this scenario the user has one short and one long sequence, and the goal is to find the section in the long sequence that better matches the short one. To recreate this scenario we use the dataset sequencesMIS. The first 10 samples will serve as short sequence, and the first 40 samples as long sequence. These small subsets are selected to speed-up the execution time of this example.
#loading the data data(sequencesMIS) #removing grouping column sequencesMIS$MIS <- NULL #subsetting to get the short sequence MIS.short <- sequencesMIS[1:10, ] #subsetting to get the long sequence MIS.long <- sequencesMIS[1:40, ]
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.
MIS.short.long <- prepareSequences( sequence.A = MIS.short, sequence.A.name = "short", sequence.B = MIS.long, sequence.B.name = "long", grouping.column = "id", transformation = "hellinger" )
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.
MIS.psi <- workflowPartialMatch( sequences = MIS.short.long, grouping.column = "id", method = "euclidean", diagonal = TRUE, ignore.blocks = TRUE, min.length = nrow(MIS.short), max.length = nrow(MIS.short) )
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.
kbl(MIS.psi[1:15, ], digits = 4)
Subsetting the long sequence to obtain the segment best matching with the short sequence goes as follows.
#indices of the best matching segment best.match.indices <- MIS.psi[1, "first.row"]:MIS.psi[1, "last.row"] #subsetting by these indices best.match <- MIS.long[best.match.indices, ]
#cleaning workspace rm(list = ls())
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.
#loading the data data(pollenGP) #getting first 20 samples pollenGP <- pollenGP[1:20, ] #sampling indices set.seed(10) #to get same result every time sampling.indices <- sort(sample(1:20, 10)) #subsetting the sequence A <- pollenGP[sampling.indices, ] B <- pollenGP[-sampling.indices, ] #preparing the sequences AB <- prepareSequences( sequence.A = A, sequence.A.name = "A", sequence.B = B, sequence.B.name = "B", grouping.column = "id", exclude.columns = c("depth", "age"), transformation = "none" )
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.
AB.combined <- workflowSlotting( sequences = AB, grouping.column = "id", time.column = "age", exclude.columns = "depth", method = "euclidean", plot = TRUE )
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.
kbl( AB.combined[1:15,1:10], digits = 4, caption = "Combination of sequences A and B." )
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.
#color palette cols <- viridis::viridis(3, begin = 0.2, end = 0.8) #column to plot taxon <- "Abies" if(!(taxon %in% colnames(AB))){ warning("This taxon is not in AB, plotting 'Abies' instead") taxon <- "Abies" } #setting y limits y.lim <- range(c(AB[AB$id == "A", taxon], AB[AB$id == "B", taxon])) #multipanel and margins par(mfrow=c(2,1), mar = c(4,2,2,1)) #plotting separated sequences plot(AB[AB$id == "A", taxon], type="l", ylim = y.lim, xlab = "", ylab = "", col = cols[1], lwd = 2, main = paste("Column '", taxon, "' of the sequences A and B", sep="") ) lines(AB[AB$id == "B", taxon], col = cols[3], lwd = 2 ) text(x = 1:10, y = AB[AB$id == "B", taxon], labels = 1:10, pos = 3, col = cols[3]) text(x = 1:10, y = AB[AB$id == "A", taxon], labels = 1:10, pos = 1, col = cols[1]) #plotting composite sequence plot(AB.combined[, taxon], type="l", ylim = y.lim, xlab = "Sample order", ylab = "", col = cols[2], lwd = 2, main = paste("Column '", taxon, "' of the composite sequence", sep="") ) points(AB.combined[, taxon], col = cols[c(1,3)][AB.combined$id], pch = 19 ) text(x = 1:20, y = AB.combined[, taxon], labels = AB.combined[, "original.index"], pos = 3, col = cols[c(1,3)][AB.combined$id])
#cleaning workspace rm(list = ls())
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"
#loading sample dataset data(pollenGP) #subset pollenGP to make a shorter dataset pollenGP <- pollenGP[1:50, ] #generating a subset of pollenGP set.seed(10) pollenX <- pollenGP[sort(sample(1:50, 40)), ] #we separate the age column pollenX.age <- pollenX$age #and remove the age values from pollenX pollenX$age <- NULL pollenX$depth <- NULL #removing some samples from pollenGP #so pollenX is not a perfect subset of pollenGP pollenGP <- pollenGP[-sample(1:50, 10), ] #prepare sequences GP.X <- prepareSequences( sequence.A = pollenGP, sequence.A.name = "GP", sequence.B = pollenX, sequence.B.name = "X", grouping.column = "id", time.column = "age", exclude.columns = "depth", transformation = "none" )
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 $mode = "direct"$.
#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 = "direct" ) kbl(X.new[1:15, ], digits = 4)
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.
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 $Bk$.
#loading data data(pollenGP) #samples in A Ai <- pollenGP[1, 3:ncol(pollenGP)] Aj <- pollenGP[4, 3:ncol(pollenGP)] #ages of the samples in A Ati <- pollenGP[1, "age"] Atj <- pollenGP[4, "age"] #sample in B Bk <- pollenGP[2, 3:ncol(pollenGP)] #computing distances between Bk, Ai, and Aj DBkAi <- distance(Bk, Ai) DBkAj <- distance(Bk, Aj) #normalizing the distances to 1 wi <- DBkAi / (DBkAi + DBkAj) wj <- DBkAj / (DBkAi + DBkAj) #computing Btk Btk <- wi * Ati + wj * Atj
The table below shows the observed versus the predicted values for $Btk$.
temp.df <- data.frame(Observed = pollenGP[3, "age"], Predicted = Btk) kbl(t(temp.df), digits = 4, caption = "Observed versus predicted value in the interpolation of an age based on similarity between samples.")
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.
#loading sample dataset data(pollenGP) #subset pollenGP to make a shorter dataset pollenGP <- pollenGP[1:50, ] #generating a subset of pollenGP set.seed(10) pollenX <- pollenGP[sort(sample(1:50, 40)), ] #we separate the age column pollenX.age <- pollenX$age #and remove the age values from pollenX pollenX$age <- NULL pollenX$depth <- NULL #removing some samples from pollenGP #so pollenX is not a perfect subset of pollenGP pollenGP <- pollenGP[-sample(1:50, 10), ] #prepare sequences GP.X <- prepareSequences( sequence.A = pollenGP, sequence.A.name = "GP", sequence.B = pollenX, sequence.B.name = "X", grouping.column = "id", time.column = "age", exclude.columns = "depth", transformation = "none" )
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" ) kbl(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)
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.
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.
#cleaning workspace rm(list = ls())
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