| Type: | Package |
| Title: | Tropical Geometry Tools for Machine Learning |
| Version: | 2.3.0 |
| Description: | Suite of tropical geometric tools for use in machine learning applications. These methods may be summarized in the following references: Yoshida, et al. (2022) <doi:10.2140/astat.2023.14.37>, Barnhill et al. (2023) <doi:10.48550/arXiv.2303.02539>, Barnhill and Yoshida (2023) <doi:10.3390/math11153433>, Aliatimis et al. (2023) <doi:10.1007/s11538-024-01327-8>, Yoshida et al. (2022) <doi:10.1109/TCBB.2024.3420815>, and Yoshida et al. (2019) <doi:10.1007/s11538-018-0493-4>. |
| License: | MIT + file LICENSE |
| Maintainer: | David Barnhill <david.barnhill@nps.edu> |
| URL: | https://github.com/barnhilldave/TML |
| BugReports: | https://github.com/barnhilldave/TML/issues |
| Depends: | R (≥ 3.5.0) |
| Imports: | MASS, Matrix, RcppAlgos, Rfast, combinat, gtools, lpSolve, lpSolveAPI, miscTools, phangorn, rcdd, rgl, ape, phytools, maps, cluster, ROCR, stats |
| Encoding: | UTF-8 |
| LazyData: | true |
| RoxygenNote: | 7.3.1 |
| Suggests: | testthat (≥ 3.0.0) |
| Config/testthat/edition: | 3 |
| NeedsCompilation: | no |
| Packaged: | 2024-07-29 15:51:17 UTC; davidbarnhill |
| Author: | David Barnhill 
[aut, cre],
Ruriko Yoshida [aut],
Georgios Aliatimis [aut],
Keiji Miura [aut] |
| Repository: | CRAN |
| Date/Publication: | 2024-07-29 16:10:02 UTC |
TML: Tropical Geometry Tools for Machine Learning
Description
Suite of tropical geometric tools for use in machine learning applications. These methods may be summarized in the following references: Yoshida, et al. (2022) doi:10.2140/astat.2023.14.37, Barnhill et al. (2023) doi:10.48550/arXiv.2303.02539, Barnhill and Yoshida (2023) doi:10.3390/math11153433, Aliatimis et al. (2023) doi:10.1007/s11538-024-01327-8, Yoshida et al. (2022) doi:10.1109/TCBB.2024.3420815, and Yoshida et al. (2019) doi:10.1007/s11538-018-0493-4.
Author(s)
Maintainer: David Barnhill david.barnhill@nps.edu (ORCID)
Authors:
Ruriko Yoshida
Georgios Aliatimis
Keiji Miura
See Also
Useful links:
Modified Fermat-Weber point numerical solver for ultrametrics
Description
Returns a modified Fermat-Weber point of N points using a gradient based numerical method This method is appropriate for points coming from ultrametrics. The algorithm tries to find a point that minimizes the sum of tropical distances from the samples, but also also tries to find a point that is as close as possible to the space of ultrametrics. The tradeoff between these two objectives is controlled by the penalty parameter. If penalty=0, the method is identical to FWpoint_numerical; it finds the Fermat-Weber point, which may not be an ultrametric. If penalty is very large, the algorithm is trying to find the Fermat-Weber point in the space of ultrametrics.
Usage
FWpoint.num.w.reg(datamatrix, penalty = 0)
Arguments
datamatrix |
matrix of dimension N*e, where N is the number of observations which lie in R^e |
penalty |
positive real number; the regularization rate |
Value
vector; Fermat-Weber point approximation
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
Examples
D = matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE)
FWpoint.num.w.reg(D,1e4) # (0,2,5/3) not ultrametric
FWpoint.num.w.reg(D,1e4) # (0,5/3,5/3) ultrametric
Fermat-Weber point numerical solver
Description
Returns the Fermat-Weber point of N points using a gradient based numerical method
Usage
FWpoint.numerical(datamatrix)
Arguments
datamatrix |
matrix of dimension N*e, where N is the number of observations which lie in R^e. |
Value
Fermat-Weber point approximation (vector in R^e)
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
Examples
D = matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE)
FWpoint.numerical(D)
Uniformly sample from a max-plus tropical line segment
Description
This function uses a hit-and-run sampler to uniformly sample from a max-plus tropical line segment
Usage
HAR.TLineSeg(D1, D2, tadd = max)
Arguments
D1 |
point in the tropical projective torus |
D2 |
point in the tropical projective torus |
tadd |
string; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
point on the line segment defined by D1 and D2
Author(s)
Ruriko Yoshida ryoshida@nps.edu
References
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
Examples
D1 <-c(0,4,2)
D2 <- c(0,7,-1)
HAR.TLineSeg(D1, D2,tadd=max)
HAR.TLineSeg(D1, D2,tadd=min)
Gaussian-like Sampling on a max- or min-plus tropical line segment
Description
This function samples points on a tropical line segment about a location parameter for a given scale parameter defined in terms of tropical distance
Usage
HAR.TLineSeg.centroid(D1, D2, m, s, tadd = max)
Arguments
D1 |
point in the tropical projective torus |
D2 |
point in the tropical projective torus |
m |
location parameter |
s |
scale parameter |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
point on the line segment defined by D1 and D2 sampled about mu
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
D1 <-c(0,4,2)
D2 <- c(0,7,-1)
m<-c(0,7,2)
s<-1
HAR.TLineSeg.centroid(D1, D2,m,s)
HAR.TLineSeg.centroid(D1, D2,m,s,tadd=min)
Sample k equally spaced points on a max- or min-plus tropical line segment
Description
This function calculates k equally spaced points on a tropical line segment
Usage
Points.TLineSeg(D1, D2, k = 20, tadd = max)
Arguments
D1 |
point in the tropical projective torus |
D2 |
point in the tropical projective torus |
k |
number of points |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
matrix of k equally spaced points on a tropical line segment
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
D1 <-c(0,4,2)
D2 <- c(0,7,-1)
Points.TLineSeg(D1, D2, k = 5)
Points.TLineSeg(D1, D2, k = 5,tadd=min)
Six data sets of phylogenetic trees data simulated from the Coalescant model.
Description
Six data sets of 1000 gene trees simulated from the Coalescant model based on a specified species with each data set possessing a ratio of species depth to effective population of 0.25, 0.5, 1, 2, 5, and 10.
Usage
Sim_Trees1025
Sim_Trees105
Sim_Trees11
Sim_Trees12
Sim_Trees15
Sim_Trees110
Format
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
Six data sets of phylogenetic trees data simulated from the Coalescant model.
Description
Six data sets of 1000 gene trees simulated from the Coalescant model based on a specified species with each data set possessing a ratio of species depth to effective population of 0.25, 0.5, 1, 2, 5, and 10.
Usage
Sim_Trees2025
Sim_Trees205
Sim_Trees21
Sim_Trees22
Sim_Trees25
Sim_Trees210
Format
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
An object of class matrix (inherits from array) with 1000 rows and 45 columns.
Simulated points over the tropical projective torus
Description
150 points generated using Gaussian-like Hit-and-Run sampling with three separate pairs of location and scale parameters
Usage
Sim_points
Format
Sim_points A 150 x 3 matrix where each row is a point in the
tropical projective torus
K-means clustering over the tropical projective torus
Description
This function performs k-means clustering over the tropical projective torus
Usage
TKmeans(A, C, M)
Arguments
A |
matrix of points defining a tropical polytope; rows are the tropical points |
C |
number of clusters |
M |
maximum number of iterations of algorithm to find cluster centroids |
Value
list with matrix of observation classified by centroid; matrix of centroid coordinates; number of iterations used
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
P <-Sim_points
C<-3
M<-10
res<-TKmeans(P,C,M)
try<-res[[1]]
cen<-res[[2]]
plot(try[,2],try[,3],col=try[,4],asp=1)
plot(try[,2],try[,3],col=try[,4],asp=1,xlab='x2',ylab='x3')
points(cen[,2],cen[,3],col=c('purple','hotpink','orange'),pch=19)
Construct a max- or min-plus tropical line segment between two points
Description
This function constructs a max-plus tropical line segment between two points
Usage
TLineSeg(D1, D2, tadd = max)
Arguments
D1 |
point in the tropical projective torus |
D2 |
point in the tropical projective torus |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
list of points defining the tropical line segment
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
D1 <-c(0,4,2)
D2 <- c(0,7,-1)
TLineSeg(D1, D2)
TLineSeg(D1, D2,tadd=min)
Tropical Machine Learning in R
Description
TML provides a suite of tools for machine learning application on data over the tropical semiring
Visualize a Tropical ball in 2D or 3D
Description
This function constructs a visualization of a 2D or 3D tropical ball
Usage
Trop_ball(
v,
d,
a = 1,
cls = "black",
cent.col = "black",
fil = TRUE,
plt = TRUE,
bord = "black"
)
Arguments
v |
center of tropical ball; numeric vector of length 3 or 4 |
d |
radius of tropical ball |
a |
shading level; 1 is opaque |
cls |
string indicating color of interior of ball |
cent.col |
string indicating color of center point |
fil |
logical for 3D plots; if TRUE 2D facets of 3D ball fill in color of cls parameter |
plt |
logical; indicates plot a new object; defaults to TRUE; if FALSE, overlays the ball on existing plot |
bord |
string indicating color of border of ball (only for 2D plots) |
Value
2D or 3D visualization of tropical ball
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
v <-c(0,0,0)
d <- 2
Trop_ball(v,d,a=.1,cls='white',cent.col='black',fil=TRUE,plt=TRUE,bord='black')
v <-c(0,0,0,0)
d <- 2
Trop_ball(v,d,a=1,cls='red',cent.col='black',fil=FALSE,plt=TRUE)
Agglomerative (AGNES) tropical hierarchical clustering
Description
This function performs agglomerative (AGNES) hierarchical clustering over the space of ultrametrics defining the space of equidistant trees
Usage
Tropical.HC.AGNES(D, method = mean)
Arguments
D |
matrix of points defining a tropical polytope. Rows are the tropical points |
method |
linkage method: mean, min, or max |
Value
list of distances in when merges occur; list of indices of points in each cluster
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
P <-Sim_points
Tropical.HC.AGNES(P, method=mean)
Hit-and-Run Sampler for the space of ultrametrics
Description
This sampler samples a point in the space of ultrametrics where each point represents an equidistant tree on n leaves
Usage
Ultrametrics.HAR(x0, n, I = 1, h = 1)
Arguments
x0 |
an equidistant tree defined as ultrametric |
n |
number of leaves for the equidistant tree |
I |
number of states in the Markov chain |
h |
height of phylogenetic tree |
Value
point in the space of ultrametrics over n leaves
Author(s)
Ruriko Yoshida ryoshida@nps.edu
References
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
Examples
x0 <-Sim_Trees15[1,]
n<-10
Ultrametrics.HAR(x0, n, I = 50, h = 1)
Vertex HAR with extrapolation (VHE) MCMC with uniform target distribution
Description
This function samples points uniformly the space defined by a tropical simplex
Usage
VE.HAR(D_s, x0, I = 1, tadd = max)
Arguments
D_s |
matrix of vertices of a tropical simplex; each row is a vertex |
x0 |
initial point for sampler, numeric vector |
I |
number of states in Markov chain |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
next sampled point from the tropical polytope
Author(s)
David Barnhill david.barnhill@nps.edu
References
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
Examples
D_s <-matrix(c(0,0,0,0,10,0,0,0,10),3,3,TRUE)
x0 <- c(0,0,0)
VE.HAR(D_s, x0, I = 50)
VE.HAR(D_s, x0, I = 50,tadd=min)
Nearest neighbor bandwidth calculation
Description
This function finds the bandwidth for an ultrametric based on the tropical distance of the nearest point. The function provides the bandwidth input to trop.KDE and was originally used in the KDETrees package.
Usage
bw.nn(x, prop = 0.2, tol = 1e-06)
Arguments
x |
matrix; dissimilarity matrix between points in a data set |
prop |
proportion of observations that defines neighborhood of a point |
tol |
tolerance for zero bandwidth check |
Value
a vector of bandwidths for each tree (row) in x
Author(s)
Ruriko Yoshida ryoshida@nps.edu
References
Weyenberg, G., Huggins, P., Schardl, C., Howe, D. K., & Yoshida, R. (2014). kdetrees: Nonparametric Estimation of Phylogenetic Tree Distributions. In Bioinformatics.
https://github.com/grady/kdetrees/blob/master/R/bw.R
Examples
T1<-Sim_Trees15
T2<-Sim_Trees25
D <- rbind(T1, T2[1,])
M <- pw.trop.dist(D, D)
bw.nn(M)
Ratio of within and between tropical measures for tropical hierarchical clusters
Description
Ratio of within and between cluster tropical measures for a set hierarchical clusters
Usage
cluster.ratio_HC(A, V, method = mean)
Arguments
A |
matrix of tropical points; rows are points |
V |
list of clusters where each cluster is defined as a matrix |
method |
method to use for within cluster measure; mean or max |
Value
vector of ratios for each cluster
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
har<-rbind(Sim_points[1:20,],Sim_points[51:70,])
V<-Tropical.HC.AGNES(har, method=mean)
inds<-V[[2]][[38]]
cluster.ratio_HC(har,inds,method=mean)
Ratio of within and between tropical measures for k-means clusters
Description
Ratio of within and between cluster tropical measures for k-means derived clusters
Usage
cluster.ratio_KM(A, C, method = mean)
Arguments
A |
matrix of tropical points; rows are points |
C |
number of clusters |
method |
method to use for within cluster measure; mean or max |
Value
vector of ratios for each cluster
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
hars<-Sim_points
cls<-c(rep(1,50),rep(2,50),rep(3,50))
cl_pt<-cbind(hars,cls)
C<-3
cluster.ratio_KM(cl_pt,C,method=mean)
Create a phylogenetic tree from an ultrametric
Description
This function constructs a phylogenetic tree from an ultrametric.
Usage
convert.to.tree(n, L, u)
Arguments
n |
is the number of leaves |
L |
is a vector of labels (strings) of leaves |
u |
is an ultrametric |
Value
A phylogenetic tree of class phylo
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
um<-Sim_Trees21[1,]
ll <- 10
L <- LETTERS[1:10]
tr<-convert.to.tree(ll, L, um)
Draw a 2-D or 3-D tropical polytope
Description
This command draws a three dimensional tropical polytope
Usage
draw.tpolytope.3d(D, col_lines, col_verts, plot = TRUE, tadd = max)
draw.tpolytope.2d(D, col_lines, col_verts, plot = TRUE, tadd = max)
Arguments
D |
matrix of vertices of a tropical polytope; rows are the vertices |
col_lines |
string; color to render the polytope. |
col_verts |
string; color to render the vertices. |
plot |
logical; initiate new plot visualization or not. |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
2-D or 3-D rendering of a tropical polytope.
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
D <-matrix(c(0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1),4,4,TRUE)
col_lines<-'blue'
col_verts<-'red'
draw.tpolytope.3d(D,col_lines,col_verts,plot=TRUE)
draw.tpolytope.3d(D,col_lines,col_verts,plot=TRUE,tadd=min)
D <- matrix(c(0,-2,2,0,-2,5,0,2,1,0,1,-1),4,3,TRUE)
col_lines <- 'blue'
col_verts <- 'red'
draw.tpolytope.2d(D,col_lines,col_verts,plot=TRUE)
draw.tpolytope.2d(D,col_lines,col_verts,plot=TRUE,tadd=min)
2D or 3D rendering of max-plus or min-plus tropical hyperplane
Description
This function renders a 2D or 3D max-plus or min-plus tropical hyperplane
Usage
draw.thyper(D, ext, min.ax, max.ax, plot = FALSE, tadd = max)
Arguments
D |
point in the tropical projective torus representing the apex of the hyperplane |
ext |
scalar; indicates how far the hyperplane should extend |
min.ax |
scalar; value applied to define the minimum limits of the axes of the plot |
max.ax |
scalar; value applied to define the maximum limits of the axes of the plot |
plot |
logical; if true produces a new plot otherwise overlays tropical hyperplane on existing plot |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
2D or 3D rendering of max-plus or min-plus tropical hyperplane
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
# 2D Example
D <-c(0,0,0)
ext<-4
min.ax<- 5
max.ax<- 5
draw.thyper(D,ext,min.ax,max.ax,plot=TRUE)
# 3D Example
D <-c(0,0,0,0)
ext<-4
min.ax<- 5
max.ax<- 5
draw.thyper(D,ext,min.ax,max.ax,plot=TRUE)
draw.thyper(D,ext,min.ax,max.ax,plot=TRUE,tadd=min)
Phylogenetic trees based on lung fish data
Description
1290 (non-equidistant) gene trees with 45 leaves originating from lung fish data in matrix form. Also we provide a vector of strings consisting of leaf labels for each species associated with the data set.
Usage
lung_fish
Format
An object of class matrix (inherits from array) with 1290 rows and 45 columns.
Source
Liang D, Shen XX, Zhang P. One thousand two hundred ninety nuclear genes from a genome-wide survey support lungfishes as the sister group of tetrapods. Mol Biol Evol. 2013 Aug;30(8):1803-7. doi: 10.1093/molbev/mst072. Epub 2013 Apr 14. PMID: 23589454.
Calculate the center point and radius of the maximum inscribed ball for a tropical simplex
Description
This function calculates the center point and radius of the maximum inscribed ball for a max- or min-plus tropical simplex
Usage
max_ins.ball(A, tadd = max)
Arguments
A |
matrix of points defining a tropical polytope; rows are the points |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
list containing the radius and center point of a maximum inscribed ball
Author(s)
David Barnhill david.barnhill@nps.edu
References
Barnhill, David, Ruriko Yoshida and Keiji Miura (2023). Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling.
Examples
P<-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE)
max_ins.ball(P)
max_ins.ball(P,tadd=min)
Calculate a minimum enclosing ball for a tropical polytope
Description
This function constructs a minimum enclosing ball for a set of points defining a tropical polytope.
Usage
min_enc.ball(A)
Arguments
A |
matrix of points defining a tropical polytope. Rows are the points. |
Value
list containing center point and radius of minimum enclosing ball of P
Author(s)
David Barnhill david.barnhill@nps.edu
References
Barnhill, David, Ruriko Yoshida and Keiji Miura (2023). Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling.
Examples
P <-matrix(c(0,0,0,0,3,1,0,2,5),3,3,TRUE)
min_enc.ball(P)
Normalize a phylogenetic tree
Description
This function normalizes the height of a phylogenetic tree
Usage
normaliz.tree(D, h = 1)
Arguments
D |
numeric vector; ultrametric equidistant tree |
h |
desired height; defaults to 1 |
Value
normalized equidistant tree
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
D <-c(4,4,2)
normaliz.tree(D, h=1)
Normalize a point or set of points in the tropical projective torus
Description
This function normalizes a point or set of points in the tropical projective torus by making the first coordinate zero
Usage
normaliz.vector(D)
normaliz.vectors(D)
normaliz.polytope(D)
normaliz.ultrametrics(D)
Arguments
D |
numeric vector in the tropical projective torus or a matrix of points in the tropical projective torus; for matrices, rows are the points |
Value
a single or set of normalized points with the first coordinate zero
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
D <-c(8,4,2)
normaliz.vector(D)
P <-matrix(c(8,4,2,10,1,3,7,2,1),3,3,TRUE)
normaliz.vectors(P)
M<-matrix(c(2,2,2,3,6,4,2,4,7),3,3,TRUE)
normaliz.polytope(M)
M <- Sim_Trees15[1:3,]
normaliz.ultrametrics(M)
Tropical cluster betweeness measure for each cluster in a set of hierarchical clusters
Description
This function calculates an overall betweenness measure based on tropical distance between a set of clusters derived from tropical hierarchical clustering
Usage
over_bet_HC(A, V)
Arguments
A |
matrix of tropical points; rows are points with the last column representing a numbered cluster assignment |
V |
list of clusters defined as matrices derived from agglomerative or divisive hierarchical clustering |
Value
vector of betweenness cluster measures
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
har<-rbind(Sim_points[1:20,],Sim_points[51:70,])
V<-Tropical.HC.AGNES(har, method=mean)
inds<-V[[2]][[38]]
over_bet_HC(har,inds)
Tropical cluster betweeness measure for a each of a set of k-means derived set of clusters
Description
This function calculates an overall betweenness measure between a set of clusters derived from tropical k-means clustering
Usage
over_bet_KM(A, C)
Arguments
A |
matrix of tropical points; rows are points with the last column representing a numbered cluster assignment |
C |
number of clusters |
Value
betweenness cluster measure
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
hars<-Sim_points
cls<-c(rep(1,50),rep(2,50),rep(3,50))
cl_pt<-cbind(hars,cls)
C<-3
over_bet_KM(cl_pt,C)
Projections of points onto a tropical triangle
Description
This function produces the a matrix of points projected onto a tropical triangle defined by the column space of a matrix
Usage
pre.pplot.pro(S, D)
Arguments
S |
matrix of points representing a tropical polytope; rows are the vertices |
D |
data points in the tropical projective torus |
Value
matrix of points representing projections of the points in D (row vectors) onto S
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
s <- 3 #number of vertices. Here it is a tropical triangle
d <- 3 ## dimension
N <- 100 ## sample size
D <- matrix(rep(0, N*d), N, d)
D[, 1] <- rnorm(N, mean = 5, sd = 5)
D[, 2] <- rnorm(N, mean = -5, sd = 5)
D[, 3] <- rnorm(N, mean = 0, sd = 5)
index <- sample(1:N, s)
S <- D[index,]
DD <- pre.pplot.pro(S, D)
Estimated probability for binary class assignment
Description
Estimates the probability that an observation x belongs to class 1.
Usage
prob.class(pars, x)
Arguments
pars |
vector of parameters, which can be decomposed as two normal vectors and two scaling parameters and has dimension 2*e+2 |
x |
vector of dimension e |
Value
real number
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
Examples
library(ROCR)
T0 = Sim_Trees15
T1 = Sim_Trees25
D = rbind(T0,T1)
Y = c(rep(0,dim(T0)[1]), rep(1,dim(T1)[1]))
N = length(Y)
set.seed(1)
train_set = sample(N,floor(0.8 * N)) ## 80/20 train-test split
pars <- trop.logistic.regression(D[train_set,],Y[train_set], penalty=1e4)
test_set = (1:N)[-train_set]
Y.hat <- rep(0, length(test_set))
for(i in 1:length(test_set)) Y.hat[i] <- prob.class(pars, D[test_set[i],])
Logit.ROC <- performance(prediction(Y.hat, Y[test_set]), measure="tpr", x.measure="fpr")
plot(Logit.ROC, lwd = 2, main = "ROC Curve for Logistic Regression Model")
print(paste("Logit.AUC=", performance(prediction(Y.hat, Y[test_set]), measure="auc")@y.values))
Project a point on the tropical projective torus onto a tropical polytope
Description
This function projects points in the tropical projective torus onto a max- or min-plus tropical polytope based on tropical distance
Usage
project.pi(D_s, D, tadd = max)
Arguments
D_s |
matrix where each row is a point defining a tropical polytope |
D |
point to be projected onto D_s |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max. |
Value
projection of point D onto the tropical polytope defined by D_s
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
D_s <-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE)
D <- c(0,7,-1)
project.pi(D_s,D)
project.pi(D_s,D,tadd=min)
Constructs the dissimilarity matrix for a set of ultrametrics
Description
Constructs the dissimilarity matrix based on the tropical distance between points in a dataset
Usage
pw.trop.dist(D1, D2)
Arguments
D1 |
matrix of ultrametrics |
D2 |
matrix of ultrametrics |
Value
matrix; dissimilarity matrix showing the tropical pairwise distance between each point
Author(s)
Ruriko Yoshida ryoshida@nps.edu
References
Weyenberg, G., Huggins, P., Schardl, C., Howe, D. K., & Yoshida, R. (2014). kdetrees: Nonparametric Estimation of Phylogenetic Tree Distributions. In Bioinformatics.
Yoshida, Ruriko, David Barnhill, Keiji Miura and Daniel Howe (2022). Tropical Density Estimation of Phylogenetic Trees.
https://github.com/grady/kdetrees/blob/master/R/dist.diss.R
Examples
T1<-Sim_Trees15
T2<-Sim_Trees25
D <- rbind(T1, T2[1,])
pw.trop.dist(D, D)
Remove all tentacles from a tropical simplex
Description
This function removes all tentacles from a tropical simplex. The remaining portion is a full-dimensional tropical polytope known as the trunk of the tropical polytope.
Usage
rounding(P)
Arguments
P |
matrix of points defining a tropical simplex. Rows are the points |
Value
matrix of points defining only the full-dimensional element (the trunk) of a tropical polytope; rows are points
Author(s)
David Barnhill david.barnhill@nps.edu
References
Barnhill, David, Ruriko Yoshida and Keiji Miura (2023). Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling.
Examples
P<-matrix(c(0,-1,1,0,0,0,0,1,-1),3,3,TRUE)
BP<-min_enc.ball(P)
RP<-rounding(P)
BRP<-min_enc.ball(RP)
Sigmoid function
Description
Returns the sigmoid function valuation
Usage
sigmoid(x)
Arguments
x |
real number |
Value
sigmoid function value at x
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
Examples
sigmoid(0) # 0.5
Calculate the tropical determinant of a square matrix.
Description
This function calculates the tropical determinant (or singularity) of a square matrix
Usage
tdets(P, tadd = max)
Arguments
P |
matrix of points defining a tropical polytope. Rows are the points |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
list containing the value of the determinant and reordered matrix P
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
P<-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE)
tdets(P)
tdets(P,tadd=min)
Phylogenetic tree to vector
Description
A tree is converted to a vector of pairwise distances between leaves. Distance between leaves is defined as the cophenetic distance between them. Normalization is applied so that the maximum distance in the vector output is 1.
Usage
tree.to.vector(tree, normalization = TRUE)
Arguments
tree |
phylogenetic tree |
normalization |
logical; normalize the tree if TRUE |
Value
vector of pairwise distances in R^(m choose 2), where m is the number of leaves
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci, James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
Examples
tree <- ape::read.tree(text='((A:1, B:1):2, (C:1.5, D:1.5):1.5);')
tree.to.vector(tree)
Calculate the tropical Fermat-Weber point
Description
This function calculates the Fermat-Weber point for a tropical polytope
Usage
trop.FW(A)
Arguments
A |
matrix with normalized tropical points as rows |
Value
numeric vector providing the tropical Fermat-Weber point for the tropical polytope
Author(s)
David Barnhill david.barnhill@nps.edu
References
Lin, Bo and Ruriko Yoshida (2016). Tropical Fermat-Weber Points. SIAM J. Discret. Math. 32: 1229-1245.
Examples
P <-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE)
trop.FW(P)
Estimate the volume of a tropical polytope
Description
This function uses tropical HAR with a uniform target distribution to estimate the volume of a tropical polytope
Usage
trop.Volume(B, P, x0, s, I, r)
Arguments
B |
matrix of points defining a minimum enclosing ball for a polytope P; rows are the points |
P |
matrix of points defining a tropical polytope; rows are the points |
x0 |
initial point used for the HAR sampler |
s |
number of points to sample from the minimum enclosing ball |
I |
number of iterations for the HAR sampler |
r |
radius of the minimum enclosing tropical ball |
Value
list containing ratio of points falling in P; volume of the tropical ball; volume estimate of P
Author(s)
David Barnhill david.barnhill@nps.edu
References
Barnhill, David, Ruriko Yoshida and Keiji Miura (2023). Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling.
Examples
P <-matrix(c(0,0,0,0,3,1,0,2,5),3,3,TRUE)
BR<-min_enc.ball(P)
B<-trop.bal.vert(BR[[1]],BR[[2]])
x0<-c(0,1.5,.4)
S<-200
I<-50
R<-BR[[2]]
trop.Volume(B,P,x0,S,I,R)
Calculate the minimum or entire generating vertex set of a tropical ball using a max- or min-plus algebra
Description
This function calculates the coordinates of the minimum or entire vertex set of a tropical ball in terms of either a max- or min-plus algebra for a given a center point
Usage
trop.bal.vert(x, d, tadd = max)
trop.bal.all_vert(x, d)
Arguments
x |
matrix where each row is a point defining a tropical polytope |
d |
radius of the tropical ball in terms of tropical distance |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition, 'all' indicates all vertices. Defaults to 'max' |
Value
matrix of normalized tropical points defining the tropical ball. Rows are the points
Author(s)
David Barnhill david.barnhill@nps.edu
References
Barnhill, David, Ruriko Yoshida and Keiji Miura (2023). Maximum Inscribed and Minimum Enclosing Tropical Balls of Tropical Polytopes and Applications to Volume Estimation and Uniform Sampling.
Examples
x <-c(0,3,7,5)
d <- 2
trop.bal.vert(x,d)
trop.bal.vert(x,d,tadd=min)
trop.bal.all_vert(x,d)
Compute the tropical distance
Description
This function computes the tropical distance between two points in the tropical projective torus
Usage
trop.dist(D1, D2)
Arguments
D1 |
point in the tropical projective torus |
D2 |
point in the tropical projective torus |
Value
tropical distance between D1 and D2
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
D1 <-c(0,4,2)
D2 <- c(0,7,-1)
trop.dist(D1, D2)
Calculate the tropical distance to a max-tropical hyperplane
Description
Calculate the tropical distance to a max-tropical hyperplane
Usage
trop.hyper.dist(O, x0, tadd = max)
Arguments
O |
normal vector of a tropical hyperplane; numeric vector |
x0 |
point of interest; numeric vector |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
tropical distance to max-plus tropical hyperplane
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
O <-c(0,-1,-1)
x0 <- c(0,-2,-8)
trop.hyper.dist(O,x0)
trop.hyper.dist(O,x0,tadd=min)
Tropical Logistic Regression
Description
Performs tropical logistic regression, by finding the optimal statistical parameters for the training dataset (D,Y), where D is the matrix of covariates and Y is the binary response vector
Usage
trop.logistic.regression(D, Y, penalty = 0, model_type = "two_species")
Arguments
D |
matrix of dimension N*e, where N is the number of observations which lie in R^e |
Y |
binary vector with dimension N, with each component corresponding to an observation |
penalty |
scalar; positive real number |
model_type |
string; options are "two-species" (default), "one-species", "general" |
Value
vector; optimal model parameters (two normal vectors and two scaling factors)
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees.
Examples
library(ROCR)
T0 = Sim_Trees15
T1 = Sim_Trees25
D = rbind(T0,T1)
Y = c(rep(0,dim(T0)[1]), rep(1,dim(T1)[1]))
N = length(Y)
set.seed(1)
train_set = sample(N,floor(0.8 * N)) ## 80/20 train-test split
pars <- trop.logistic.regression(D[train_set,],Y[train_set], penalty=1e4)
test_set = (1:N)[-train_set]
Y.hat <- rep(0, length(test_set))
for(i in 1:length(test_set)) Y.hat[i] <- prob.class(pars, D[test_set[i],])
Logit.ROC <- performance(prediction(Y.hat, Y[test_set]), measure="tpr", x.measure="fpr")
plot(Logit.ROC, lwd = 2, main = "ROC Curve for Logistic Regression Model")
print(paste("Logit.AUC=", performance(prediction(Y.hat, Y[test_set]), measure="auc")@y.values))
Plotting PCA-derived tropical triangles
Description
This function conducts tropical PCA to find the best fit tropical triangle given data defined in the tropical projective torus. It employs the vertex HAR with extrapolation sampler to sample points to determine the vertices of the tropical triangle.
Usage
trop.tri.plot.w.pts(S, D)
Arguments
S |
inital set of vertices for the tropical triangle |
D |
matrix of data where each row is an observation in the tropical projective torus |
Value
rendering of tropical triangle saved to current directory
Author(s)
Ruriko Yoshida ryoshida@nps.edu
Examples
s <- 3 #number of vertices. Here it is a tropical triangle
d <- 3 ## dimension
N <- 100 ## sample size
V <- matrix(c(100, 0, 0, 0, 100, 0, 0, 0, 100, -100, 0, 0, 0, -100, 0, 0, 0, -100), 6, 3, TRUE)
D <- matrix(rep(0, N*d), N, d)
D[, 1] <- rnorm(N, mean = 5, sd = 5)
D[, 2] <- rnorm(N, mean = -5, sd = 5)
D[, 3] <- rnorm(N, mean = 0, sd = 5)
index <- sample(1:N, s)
S <- D[index,]
res <- tropical.PCA.Polytope(S, D, V, I = 1000,50)
DD <- pre.pplot.pro(res[[2]], res[[3]])
trop.tri.plot.w.pts(normaliz.ultrametrics(res[[2]]), DD)
Tropical within-cluster measure
Description
This function calculates a within cluster measure by measuring the pairwise tropical distance between points in the cluster.
Usage
trop_wi_dist(D1, method = mean)
Arguments
D1 |
matrix of tropical points; rows are points |
method |
function; metric to measure; mean is the average pairwise tropical distance; max is the maximum pairwise tropical distance |
Value
within cluster measure
Author(s)
David Barnhill david.barnhill@nps.edu
References
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
Examples
D<-Sim_points
avg.m<-trop_wi_dist(D, method=mean)
max.m<-trop_wi_dist(D, method=max)
Tropical Kernel Density Estimation of Phylogenetic Trees
Description
This function calculates a non-parametric density estimate of a tree over the space of phylogenetic trees on m leaves. It mimics classical kernel density estimation by using a Gaussian kernel in conjunction with tropical distance.
Usage
tropical.KDE(D, n, sigma, h = 2)
Arguments
D |
matrix of phylogenetic tree observations as ultrametrics |
n |
number of leaves for each tree |
sigma |
bandwidth parameter based on tropical distance |
h |
height of the tree |
Value
list containing center point and radius of minimum enclosing ball of P
Author(s)
Ruriko Yoshida ryoshida@nps.edu
References
Weyenberg, G., Huggins, P., Schardl, C., Howe, D. K., & Yoshida, R. (2014). kdetrees: Nonparametric Estimation of Phylogenetic Tree Distributions. In Bioinformatics.
Yoshida, Ruriko, David Barnhill, Keiji Miura and Daniel Howe (2022). Tropical Density Estimation of Phylogenetic Trees.
Examples
T1<-Sim_Trees15
T2<-Sim_Trees25
D <- rbind(T1, T2[1,])
T <- dim(D)[1]
X <- 1:T
M <- pw.trop.dist(D, D)
sigma <- bw.nn(M)
P_5 <- tropical.KDE(D, n, sigma, h = 2)
Q5 <- P_5[T]
Tropical principal component analysis (PCA) on over tropical projective torus
Description
This function conducts tropical PCA to find the best fit tropical triangle given data defined in the tropical projective torus. It employs the vertex HAR with extrapolation sampler to sample points to determine the vertices of the tropical triangle.
Usage
tropical.PCA.Polytope(S, D, V, I = 1, k)
Arguments
S |
inital set of vertices for the tropical triangle |
D |
matrix of data where each row is an observation in the tropical projective torus |
V |
matrix of vertices defining a polytope encompassing D |
I |
number of iterations to perform |
k |
number of iterations for the HAR sampler |
Value
list with the sum of residuals
Author(s)
Ruriko Yoshida ryoshida@nps.edu
References
Page, Robert and others (2020), Tropical principal component analysis on the space of phylogenetic trees, Bioinformatics, Volume 36, Issue 17, Pages 4590–4598.
Yoshida, R., Zhang, L. & Zhang, X (2019). Tropical Principal Component Analysis and Its Application to Phylogenetics. Bull Math Biol 81, 568–597.
Examples
s <- 3 #number of vertices. Here it is a tropical triangle
d <- 3 ## dimension
N <- 100 ## sample size
V <- matrix(c(100, 0, 0, 0, 100, 0, 0, 0, 100, -100, 0, 0, 0, -100, 0, 0, 0, -100), 6, 3, TRUE)
D <- matrix(rep(0, N*d), N, d)
D[, 1] <- rnorm(N, mean = 5, sd = 5)
D[, 2] <- rnorm(N, mean = -5, sd = 5)
D[, 3] <- rnorm(N, mean = 0, sd = 5)
index <- sample(1:N, s)
S <- D[index,]
DD <- pre.pplot.pro(S, D)
for(i in 1:N)
DD[i, ] <- normaliz.vector(DD[i, ])
res <- tropical.PCA.Polytope(S, D, V, I = 1000,50)
DD <- pre.pplot.pro(res[[2]], res[[3]])
trop.tri.plot.w.pts(normaliz.ultrametrics(res[[2]]), DD)
Tropical centroid-based sampling about a center of mass
Description
This function is a centroid-based HAR sampler about a center of mass denoted by a location parameter with scale parameter in terms of the tropical distance
Usage
VE.HAR.centroid(D_s, x0, I = 1, m, s, tadd = max)
Arguments
D_s |
matrix of vertices of a tropical simplex; each row is a vertex |
x0 |
initial point for sampler, numeric vector |
I |
number of states in Markov chain |
m |
location parameter; numeric vector indicating centroid |
s |
scale parameter; in terms of tropical distance |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
next sampled point from the tropical polytope
Author(s)
David Barnhill david.barnhill@nps.edu
Examples
D_s <-matrix(c(0,10,10,0,10,0,0,0,10),3,3,TRUE)
x0 <- c(0,0,0)
m <- c(0,5,5)
s <- 1
VE.HAR.centroid(D_s, x0, I = 50,m,s)
VE.HAR.centroid(D_s, x0, I = 50,m,s,tadd=min)
Centroid-based sampling using Metropolis filter
Description
This function samples points on a tropical line segment about a location parameter for a given scale parameter defined in terms of tropical distance
Usage
trop.centroid.MH(D, x0, m, s, n, I = 50, tadd = max)
trop.centroid.MH.square(D, x0, m, s, n, I = 50, tadd = max)
Arguments
D |
matrix of vertices of a tropical polytope; each row is a vertex |
x0 |
initial point for sampler, numeric vector |
m |
location parameter; numeric vector |
s |
scale parameter; scalar |
n |
number of points to sample |
I |
states in Markov chain |
tadd |
function; max indicates max-plus addition, min indicates min-plus addition. Defaults to max |
Value
matrix of n sampled points where each point is a row
Author(s)
David Barnhill david.barnhill@nps.edu
References
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
Examples
D1 <-matrix(c(0,0,0,0,10,0,0,0,10),3,3,TRUE)
D2 <-matrix(c(0,10,10,0,10,0,0,0,10),3,3,TRUE)
x0 <- c(0,0,0)
m1<-c(0,5,5)
m2<-c(0,-1,1)
s<-1
n<-10
trop.centroid.MH(D1, x0, m1, s, n, I=50)
trop.centroid.MH.square(D1, x0,m1, s, n, I=50)
trop.centroid.MH(D2, x0, m1, s, n, I=50,tadd=min)
trop.centroid.MH.square(D2, x0,m2, s, n, I=50,tadd=min)
Vector to equidistant tree
Description
A vector of pairwise distances is used to reconstruct the corresponding equidistant tree
Usage
vector.to.equidistant.tree(vec)
Arguments
vec |
vector of pairwise distances in R^(m choose 2), where m is the number of leaves |
Value
equidistant phylogenetic tree
Author(s)
Georgios Aliatimis g.aliatimis@lancaster.ac.uk
References
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
Examples
vec = c(1/3,1,1,1,1,1/3)
tree = vector.to.equidistant.tree(vec)
plot(tree)