Title:	Mapping Averaged Pairwise Information
Version:	1.1.4
Description:	Mapping Averaged Pairwise Information (MAPI) is an exploratory method providing graphical representations summarizing the spatial variation of pairwise metrics (eg. distance, similarity coefficient, ...) computed between georeferenced samples.
License:	GPL (≥ 3)
URL:	https://www1.montpellier.inrae.fr/CBGP/software/MAPI/
Encoding:	UTF-8
Language:	en-US
RoxygenNote:	7.3.2
Depends:	R (≥ 4.1)
LinkingTo:	Rcpp
Imports:	data.table, doParallel, fmesher, foreach, Rcpp, s2, sf
Suggests:	dggridR, ggplot2, latticeExtra, progress, sp, testthat (≥ 3.0.0)
Config/testthat/edition:	3
LazyData:	true
NeedsCompilation:	yes
Packaged:	2025-07-03 07:01:09 UTC; piry
Author:	Sylvain Piry [aut, cre], Thomas Campolunghi [aut], Florent Cestier [aut], Karine Berthier [aut]
Maintainer:	Sylvain Piry <sylvain.piry@inrae.fr>
Repository:	CRAN
Date/Publication:	2025-07-03 12:40:33 UTC

MAPI, general presentation

Description

MAPI is an exploratory method providing graphical representations of the spatial variation of pairwise metrics (eg. distance, similarity coefficient, ...) computed between georeferenced samples.

Principle

As schematically illustrated Figure 1, MAPI relies on spatial joins between a hexagonal grid and a network of georeferenced samples connected by ellipses, i.e. polygons with 32 segments approaching an elliptical shape.

The shape of the ellipses can be controlled through the eccentricity value and the sample locations can be "blurred" by applying an error circle of a given radius on the geographic coordinates. Each elliptical polygon is associated to 1) the value of the pairwise metric computed between the samples it connects and 2) a weight corresponding to the inverse of its area (i.e. larger ellipses have lower weights).

Each cell of the grid receives the weighted mean of the pairwise metric values associated to the ellipses intersecting the cell.

Figure 1: Schematic principle of the MAPI method from Piry et al. 2016.

Input data

The analysis requires two tables (data.frame or data.table):

1. Information on samples: table with three mandatory columns and column names: 'ind' (sample name), 'x' and 'y' (projected coordinates). An optional column 'errRad' (radius of error circle on sample coordinates) can be provided.

MAPI requires cartesian coordinates (ie. projected, such as UTM or Lambert) NOT (yet?) angular coordinates (eg. latitude/longitude). The package sf provides the st_transform function for coordinates transformation and projection. GIS software such as QGis can also help with datum transformation.

Example of 'samples' data:

1. Values of the pairwise metric computed between samples provided, either, as a complete matrix with the same number of columns and rows (column and row names must match the sample names provided in the 'samples' data) or as a table with three mandatory columns and column names: 'ind1', 'ind2' (sample names) and 'value' (pairwise metric values).

Example of 'metric' data:

Try it

Using the test dataset ('samples' and 'metric') included in the package, let's run an (almost) automatic MAPI analysis

Test data result from population genetic simulations in which two panmictic populations are separated by a barrier to dispersal. As we use dummy coordinates, there is no appropriate crs, so we just use 'crs=3857' (a pseudo-mercator projection). Of course, in this case, sample position on earth is meaningless. For a real dataset, 'crs' must be the EPSG code of the projection of your cartesian coordinates.

 # Load the package
 library(mapi)
 
 # Load 'samples' data
 data("samples")
 
 # Load 'metric' data. For our simulated data set the parwise metric 
 # computed between samples is the individual genetic distance â of Rousset (2000).
 data("metric")
 
 # Run MAPI the lazy way (automatic) with 1000 permutations 
 # for detection of significant (dis)continuous areas.
 # As crs must be set, we go with crs=3857 even if we use dummy coordinates.
 # Of course, this have no geographical meaning.
 # As we have a regular sampling, we use beta=0.5
 my.results <- MAPI_RunAuto(samples, metric, crs=3857, beta=0.5, nbPermuts=1000)
 
 # Get significant areas with a FDR control at alpha=0.05 (5%, by default)
 my.tails <- MAPI_Tails(my.results, alpha=0.05)
 
 # Look at the result Figure 2.
 MAPI_Plot2(my.results, tails=my.tails)
 

Spatial variation of the genetic distance is represented with a color scale from dark brown (lowest values) to dark blue (higher value). The central blue area identified as a significant area of discontinuity corresponds to the position of the simulated barrier. Note that due to the permutation procedure, delineation of the significant areas may vary slightly among runs.

Figure 2: MAPI graphical Output produced using the MAPI_Plot2 function.

To go deeper

MAPI_RunAuto is a wrapper which calls three other functions: MAPI_CheckData, MAPI_GridAuto and MAPI_RunOnGrid.

MAPI_GridAuto is itself another wrapper around MAPI_EstimateHalfwidth and MAPI_GridHexagonal.

Typically, a "manual" MAPI analysis will involve the following ordered steps:

1. MAPI_CheckData

2. MAPI_EstimateHalfwidth

3. MAPI_GridHexagonal

4. MAPI_RunOnGrid

5. MAPI_Tails

6. MAPI_Plot2

Within this general framework, you may, for example:

• set your own value for 'halfwidth' (ignore step 2)

• use your own grid, or reuse one from another run (ignore steps 2 & 3)

• tweak some MAPI parameters (such as dMin or dMax for filtering on geographic distances between samples)

• discard poorly supported cells prior detecting significant areas of (dis)continuity (parameter minQ) and/or change significance level (parameter alpha in MAPI_Tails)

• build your MAPI maps with a GIS software (ignore step 6). See 'Export results' section below

Export results

Output tables (weighted mean of the pairwise metric within cell and polygons delineating significant areas of (dis)continuity) are spatial objects built using the package sf. Refer to sf documentation to export MAPI results in various format. Below is an example of how MAPI results can be exported as ESRI Shapefiles:

library(sf)
# Export results for our test dataset
st_write(my.results, dsn=".", layer="myFirstMapiResult", 
   driver="ESRI Shapefile", update=TRUE, delete_layer=TRUE)
st_write(my.tails, dsn=".", layer="myFirstMapiResultTails", 
   driver="ESRI Shapefile", update=TRUE, delete_layer=TRUE)

You may now open these files ‘myFirstMapiResult.shp’ and ‘myFirstMapiResultTails.shp’ in a GIS software such as QGis and customize the layout.

Overlaying MAPI results with landscape layouts can help in analyzing the relationship between environmental features and spatial genetic patterns (eg. Piry et al., 2016; Piry et al., 2018).

References

Description of the MAPI method

Piry S., Chapuis M.-P., Gauffre B., Papaïx J., Cruaud A. and Berthier K. (2016). Mapping Averaged Pairwise Information (MAPI): a new exploratory tool to uncover spatial structure. Methods in Ecology and Evolution 7:(12), 1463–1475. doi:10.1111/2041-210X.12616

Applications of MAPI in Landscape Genetics

• Larson S., Gagne RB et al. 2021 Translocations maintain genetic diversity and increase connectivity in sea otters, Enhydra lutris Marine Mammal Science doi:10.1111/mms.12841

• Stragier C., Piry S., et al. 2020. Interplay between historical and current features of the cityscape in shaping the genetic structure of the house mouse (Mus musculus domesticus) in Dakar (Senegal, West Africa) bioRxiv ; Version 4 of this preprint has been peer-reviewed and is recommended by Peer Community In Ecology (DOI:10.24072/pci.ecology.100044) doi:10.1101/557066

• Piry S., Berthier K., Streiff R., Cros-Arteil S., Tatin L., Foucart A., Bröder L., Hochkirch A., and Chapuis M.-P. (2018). Fine-scale interactions between habitat quality and genetic variation suggest an impact of grazing on the critically endangered Crau Plain grasshopper (Pamphagidae: Prionotropis rhodanica). Journal of Orthoptera Research 27, 61–73. doi:10.3897/jor.27.15036

• Dellicour S, Prunier JG, Piry S, et al. (2019) Landscape genetic analyses of Cervus elaphus and Sus scrofa: comparative study and analytical developments. Heredity. doi:10.1038/s41437-019-0183-5

Function MAPI_CheckData

Description

Check the validity of the 'samples' and 'metric' data loaded.
Missing data are removed from 'metric', samples with missing coordinates are removed and samples that are not present in both dataset ('samples' and 'metric') are discarded.

Usage

MAPI_CheckData(
  samples,
  metric,
  isMatrix = all(("matrix" %in% class(metric)), (nrow(metric) == ncol(metric)))
)

Arguments

Value

a list of two data.table objects corresponding to 'samples' and 'metric' after cleaning.

Examples

## Not run: 
data("samples")
data("metric")
# remove first sample in order to force warning
samples <- samples[-c(1) , ]
clean.list <- MAPI_CheckData(samples, metric)
checked.samples <- clean.list[['samples']]
checked.metric <- clean.list[['metric']]

## End(Not run)

Function MAPI_EstimateHalfwidth

Description

This function computes the side length (= halfwidth) of the hexagonal cells. Halfwidth value can be further used to build a MAPI grid.

Usage

MAPI_EstimateHalfwidth(samples, crs, beta = 0.25)

Arguments

Details

h_w = \frac{\beta \sqrt{A/N}}{\sqrt{2.5980}}, where A is the study area (convex hull of sampling points) and N the number of samples. Parameter beta allows to respect the Nyquist-Shannon sampling theorem depending on sampling regularity.

Value

halfwidth cell value (side length of hexagonal cells).

References

Hengl, T. (2006) Finding the right pixel size. Computers & Geosciences, 32, 1283–1298.

Examples

data(samples)
# Computes hexagonal cell halfwidth for the 'samples' dataset using beta=0.5
hw <- MAPI_EstimateHalfwidth(samples, crs=3857, beta=0.5)

Function MAPI_GridAuto

Description

Wrapper that computes cell halfwidth for a given beta value, and then builds a grid of hexagonal cells (call to MAPI_GridHexagonal). If coordinates are in angular units (eg. longitude/latitude) MAPI will try to build a worldwide "discrete global grid system" using package 'dggridR'. As the 'dggridR' package is no more available on CRAN, this feature have been removed and an error is thrown in this case. See https://github.com/r-barnes/dggridR for more informations. As an alternative, grids (levels 1 to 11) are downloadable from MAPI website.

Usage

MAPI_GridAuto(samples, crs, beta = 0.25, buf = 0)

Arguments

Details

The halfwidth cell value used to build the grid is computed as h_w = \frac{\beta \sqrt{A/N}}{\sqrt{2.5980}}, where A is the study area (convex hull of sampling points) and N the number of samples. Parameter beta allows to respect the Nyquist-Shannon sampling theorem depending on sampling regularity (call to MAPI_EstimateHalfwidth). If coordinates are provided as longitude/latitude (eg. crs=4326) MAPI will try to build a worldwide grid. See MAPI_EstimateHalfwidth and MAPI_GridHexagonal for more information.

Value

a spatial object of class 'sf' including the x and y coordinates of cell centers, cell geometry (polygons) and cell id (gid).

Examples

data("samples")
grid <- MAPI_GridAuto(samples, crs=3857, beta=0.5)

Function MAPI_GridHexagonal

Description

Build a grid of hexagonal cells according to samples coordinates and a given halfwidth cell value provided by users (can be computed using MAPI_EstimateHalfwidth). If coordinates are in angular units (eg. longitude/latitude) MAPI will try to build a worldwide "discrete global grid system" using package 'dggridR'. As the 'dggridR' package is no more available on CRAN, this feature have been removed and an error is thrown in this case. See https://github.com/r-barnes/dggridR for more informations. As an alternative, grids (levels 1 to 11) are downloadable from MAPI website.

Usage

MAPI_GridHexagonal(samples, crs, hw, buf = 0)

Arguments

Value

a spatial object of class 'sf' including the x and y coordinates of cell centers, cell geometry (polygons) and cell id (gid).

Examples

data("samples")
# Builds a grid of hexagonal cells according to samples coordinates (columns x and y) 
# using the EPSG:3857 projection and an halfwidth cell value of hw=250m.
grid <- MAPI_GridHexagonal(samples, crs=3857, hw=250)

Function MAPI_Plot

Description

Plot a MAPI analysis result

Usage

MAPI_Plot(
  resu,
  tails = NULL,
  samples = NULL,
  pal = c("#994000", "#CC5800", "#FF8F33", "#FFAD66", "#FFCA99", "#FFE6CC", "#FBFBFB",
    "#CCFDFF", "#99F8FF", "#66F0FF", "#33E4FF", "#00AACC", "#007A99"),
  shades = 20,
  main = NA,
  upper = TRUE,
  lower = TRUE,
  upper.border = "black",
  lower.border = "gray"
)

Arguments

Value

Returns the "trellis" object.

Examples

## Not run: 
data("metric")
data("samples")
resu <- MAPI_RunAuto(samples, metric, crs=3857, nbPermuts = 1000)
tails <- MAPI_Tails(resu)
pl <- MAPI_Plot(resu, tails=tails, samples=samples)
# Open png driver
png("mapiPlotOutput.png", width=1000, type="cairo-png")
print(pl) # Do plot in file
dev.off() # Close driver

## End(Not run)

Function MAPI_Plot2

Description

Plot a MAPI analysis result with ggplot

Usage

MAPI_Plot2(
  resu,
  tails = NULL,
  samples = NULL,
  pal = c("#994000", "#CC5800", "#FF8F33", "#FFAD66", "#FFCA99", "#FFE6CC", "#FBFBFB",
    "#CCFDFF", "#99F8FF", "#66F0FF", "#33E4FF", "#00AACC", "#007A99"),
  shades = 20,
  main = "",
  upper = TRUE,
  lower = TRUE,
  upper.border = "black",
  lower.border = "gray"
)

Arguments

Value

Returns the ggplot object.

Examples

## Not run: 
library(ggplot2)
data("metric")
data("samples")
resu <- MAPI_RunAuto(samples, metric, crs=3857, nbPermuts = 1000)
tails <- MAPI_Tails(resu)
pl <- MAPI_Plot2(resu, tails=tails, samples=samples)
# Save to image
ggsave("mapiPlotOutput.png", plot=pl)

## End(Not run)

Function MAPI_RunAuto

Description

This function is a wrapper allowing to run a complete MAPI analysis.

Usage

MAPI_RunAuto(
  samples,
  metric,
  crs,
  isMatrix = all(class(metric) == "matrix", nrow(metric) == ncol(metric)),
  beta = 0.25,
  ecc = 0.975,
  buf = 0,
  errRad = 10,
  nbPermuts = 0,
  dMin = 0,
  dMax = Inf,
  nbCores = ifelse(requireNamespace("parallel", quietly = TRUE), parallel::detectCores()
    - 1, 1),
  N = 8
)

Arguments

Details

Following functions are called by MAPI_RunAuto in following order:

• MAPI_CheckData cleans the dataset;

• MAPI_GridAuto generates a grid of hexagons by calling MAPI_EstimateHalfwidth then MAPI_GridHexagonal;

• MAPI_RunOnGrid performs the MAPI analysis.

NOTE: The call to MAPI_Tails is not included. It should be done afterwards on the object returned by MAPI_RunAuto.

Value

a spatial object of class 'sf' providing for each cell:

• gid: Cell ID
  

• x and y coordinates of cell center
  

• nb_ell: number of ellipses used to compute the weighted mean
  

• avg_value: weighted mean of the pairwise metric
  

• sum_wgts: sum of weights of ellipses used to compute the weighted mean
  

• w_stdev: weighted standard deviation of the pairwise metric
  

• swQ: percentile of the sum of weights
  

• geometry
  When permutations are performed:
  

• proba: proportion of the permuted weighted means below the observed weighted mean
  

• ltP: lower-tail p-value adjusted using the FDR procedure of Benjamini and Yekutieli
  

• utP: upper-tail p-value adjusted using the FDR procedure of Benjamini and Yekutieli
  

References

Benjamini, Y. & Yekutieli, D. (2001) The control of the false discovery rate in multiple testing under dependency. Annals of Statistics, 29, 1165–1188.

Examples

## Not run: 
data("metric")
data("samples")
# Run a MAPI analysis without permutation
my.results <- MAPI_RunAuto(samples, metric, crs=3857, beta=0.5, nbPermuts=0)

# eg. Export results to shapefile "myFirstMapiResult" in current directory
# to further visualize and customize the MAPI plot in SIG software.
library(sf)
st_write(my.results, dsn=".", layer="myFirstMapiResult", driver="ESRI Shapefile")

## End(Not run)

Function MAPI_RunOnGrid

Description

Launch a MAPI analysis for a given grid computed with MAPI_GridAuto or MAPI_GridHexagonal or provided by users.

Usage

MAPI_RunOnGrid(
  samples,
  metric,
  grid,
  isMatrix = FALSE,
  ecc = 0.975,
  errRad = 10,
  nbPermuts = 0,
  dMin = 0,
  dMax = Inf,
  use_s2 = FALSE,
  nbCores = parallel::detectCores() - 1,
  N = 8,
  ignore.weights = FALSE
)

Arguments

Details

To test whether the pairwise metric values associated with the ellipses are independent of the sample locations, those are permuted 'nbPermuts' times. At each permutation, new cell values are computed and stored to build a cumulative null distribution for each cell of the grid. Each cell value from the observed data set is then ranked against its null distribution. For each cell, the proportion of permuted values that are smaller or greater than the observed value provides a lower-tailed (ltP) and upper-tailed (utP) test p-value.

A false discovery rate (FDR) procedure (Benjamini and Yekutieli, 2001) is applied to account for multiple testing (number of cells) under positive dependency conditions (spatial autocorrelation). An adjusted p-value is computed for each cell using the function p.adjust from the 'stats' package with the method 'BY'.

Value

a spatial object of class 'sf' providing for each cell:

• gid: Cell ID
  

• x and y coordinates of cell center
  

• nb_ell: number of ellipses used to compute the weighted mean
  

• avg_value: weighted mean of the pairwise metric
  

• sum_wgts: sum of weights of ellipses used to compute the weighted mean
  

• w_stdev: weighted standard deviation of the pairwise metric
  

• swQ: percentile of the sum of weights
  

• geometry
  When permutations are performed:
  

• proba: proportion of the permuted weighted means below the observed weighted mean
  

• ltP: lower-tail p-value adjusted using the FDR procedure of Benjamini and Yekutieli
  

• utP: upper-tail p-value adjusted using the FDR procedure of Benjamini and Yekutieli
  

References

Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics 29, 1165–1188.

Examples

## Not run: 
data(metric)
data(samples)
my.grid <- MAPI_GridHexagonal(samples, crs=3857, 500) # 500m halfwidth

# Note: 10 permutations is only for test purpose, increase to >=1000 in real life!
my.results <- MAPI_RunOnGrid(samples, metric, grid=my.grid, nbPermuts=10, nbCores=1)

# eg. Export results to shapefile "myFirstMapiResult" in current directory
sf::st_write(my.results, dsn=".", layer="myFirstMapiResult", driver="ESRI Shapefile")

## End(Not run)

Function MAPI_Tails

Description

Determine significant continuous and discontinuous areas from the result of a MAPI analysis when run with permutations.

Usage

MAPI_Tails(resu, minQ = 0, alpha = 0.05)

Arguments

Details

When permutations are performed, in MAPI_RunOnGrid for each cell, the proportion of permuted values that are smaller or greater than the observed value provides a lower-tailed (ltP) and upper-tailed (utP) test p-value. A false discovery rate (FDR) procedure (Benjamini and Yekutieli, 2001) is applied to account for multiple testing (number of cells) under positive dependency conditions (spatial autocorrelation). An adjusted p-value is computed for each cell using the function p.adjust from the 'stats' package with the method 'BY'. The significance level at which FDR is controlled is set through the parameter alpha. For example, when alpha is set to 0.05, this means that 5\

Significant cells belonging to the lower (or upper) tail that are spatially connected are aggregated together to form the significant areas with the lowest (or greater) average values of the pairwise metric analyzed.

Value

a spatial object of class 'sf' with the area and geometry of the polygons delineating the significant areas. A column provides the tail for each polygon (upper or lower).

References

Benjamini, Y. and Yekutieli, D. (2001). The control of the false discovery rate in multiple testing under dependency. Annals of Statistics 29, 1165–1188.

Examples

## Not run: 
data("metric")
data("samples")
# Run MAPI computation
resu <- MAPI_RunAuto(samples, metric, crs=3857, nbPermuts=1000)
# Discards the 10% cells with the smallest sum-of-weights 
#    and aggregates adjacent cells belonging to the same tail 
#    at a 5% significance level
tails <- MAPI_Tails(resu, minQ=10, alpha=0.05)

## End(Not run)

Function MAPI_Varicell

Description

Alter grid by changing polygons size according to samples density.

Usage

MAPI_Varicell(grid, samples, buf = 0, var.coef = 50)

Arguments

Value

a spatial object of class 'sf' including cell area, the x and y coordinates of cell centroids, cell geometry (polygons) and cell id (gid) plus the computed densities and weights.

Examples

## Not run: 
# Dummy example!
require(data.table, sf, mapi, spatstat.geom, spatstat.core)
data("samples")
keep <- c(2,3,6,9,10,11,16,18,19,20,21,23,26,27,29,31,33,34,38,41,46,50,54,58,61,63,65,71,72,73,
76,78,79,81,85,92,93,94,97,98,99,101,103,113,115,119,120,121,124,127,130,134,142,143,151,152,159,
160,176,185,189,191,195,196,197,198,199)
samples2 <- samples[keep, ] # keep only one third in order to create discontinuities
set.seed(1234)
# Builds a grid of hexagonal cells according to samples coordinates (columns x and y) 
# using the EPSG:3857 projection and an halfwidth cell value of hw=250m.
grid <- MAPI_GridHexagonal(samples2, crs=3857, hw=250, buf=1500)
grid.var <- MAPI_Varicell(grid, samples2, buf=250, var.coef=20)
ggplot() + 
    geom_sf(data=grid.var, aes(fill=dens), size=0.1) + 
    geom_sf(data=st_as_sf(samples2, coords=c("x","y"), crs=3857), aes())

## End(Not run)

'metric' test dataset

Description

The individuals were simulated as described in the samples section. The value is a genetic distance (â estimator in Rousset, 2000) computed between the 200 simulated samples using SPAGeDi v1.4 (Hardy & Vekemans, 2002).

Usage

data("metric")

Value

A data.table object with 19900 rows (one per sample pair, non symmetrical) and 3 columns ("ind1", "ind2", "value") containing respectively the first sample name, the second sample name and the value of their relation.

References

Rousset, F. (2000). Genetic differentiation between individuals. Journal of Evolutionary Biology, 13:58–62.

Hardy OJ, Vekemans X (2002) SPAGeDi: a versatile computer program to analyse spatial genetic structure at the individual or population levels. Molecular Ecology Notes 2: 618-620.

Examples

data("metric")

'samples' test dataset

Description

Test dataset provided with the MAPI package. We used generation-by-generation coalescent algorithms (Hudson et al., 1990) to simulate 10 microsatellite genotypes at migration-mutation-drift equilibrium for 200 diploid individuals, distributed on the nodes of a 20x10 lattice.

Mutations for each locus followed a symmetric generalized stepwise model with a variance equal to 0.36 (Estoup et al., 2001) and a maximum range of allelic states of 40. The mutation rate was fixed so that heterozygosity ranged from 0.6 to 0.8 as frequently observed at microsatellite markers (Chapuis et al., 2012).

Two panmictic populations are separated by a barrier. We used Simcoal2 (Laval and Excoffier, 2004) to generate two panmictic populations of equal effective size N_e = 100 and exchanging N_{e}m = 0.1 migrants at each generation. The barrier to gene flow bisected the lattice from north to south in its center.

The differentiation values computed between these samples is described in the metric section.

Usage

data("samples")

Value

A data.table object with 200 simulated individuals (one per row) and 4 columns ("ind", "x", "y", "errRad") including the sample names, coordinates x and y and an error circle radius on coordinates.

References

Chapuis, M.-P., Streiff, R., and Sword, G. (2012). Long microsatellites and unusually high levels of genetic diversity in the orthoptera. Insect Molecular Biology, 21(2):181–186.

Estoup, A., Wilson, I. J., Sullivan, C., Cornuet, J.-M., and Moritz, C. (2001). Inferring population history from microsatellite and enzyme data in serially introduced cane toads, Bufo marinus. Genetics, 159(4):1671–1687.

Hudson, R. R. et al. (1990). Gene genealogies and the coalescent process. Oxford Surveys in Evolutionary Biology, 7(1):44.

Laval, G. and Excoffier, L. (2004). SIMCOAL 2.0: a program to simulate genomic diversity over large recombining regions in a subdivided population with a complex history. Bioinformatics, 20(15):2485–2487.

Examples

data("samples")