,
Ruriko Yoshida [aut],
Georgios Aliatimis [aut],
Keiji Miura [aut]| Title: | 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) <arxiv:2209.15045>, Barnhill et al. (2023) <arxiv:2303.02539>, Barnhill and Yoshida (2023) <doi:10.3390/math11153433>, Aliatimis et al. (2023) <arXiv:2306.08796>, Yoshida et al. (2022) <arXiv:2206.04206>, and Yoshida et al. (2019) <doi:10.1007/s11538-018-0493-4>. |
| Authors: | David Barnhill [aut, cre] ,
Ruriko Yoshida [aut],
Georgios Aliatimis [aut],
Keiji Miura [aut] |
| Maintainer: | David Barnhill <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 2.3.0 |
| Built: | 2025-02-24 04:48:41 UTC |
| Source: | https://github.com/barnhilldave/tml |
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.
bw.nn(x, prop = 0.2, tol = 1e-06)bw.nn(x, prop = 0.2, tol = 1e-06)
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 |
a vector of bandwidths for each tree (row) in x
Ruriko Yoshida [email protected]
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
T1<-Sim_Trees15 T2<-Sim_Trees25 D <- rbind(T1, T2[1,]) M <- pw.trop.dist(D, D) bw.nn(M)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 cluster tropical measures for a set hierarchical clusters
cluster.ratio_HC(A, V, method = mean)cluster.ratio_HC(A, V, method = mean)
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 |
vector of ratios for each cluster
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
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)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 cluster tropical measures for k-means derived clusters
cluster.ratio_KM(A, C, method = mean)cluster.ratio_KM(A, C, method = mean)
A |
matrix of tropical points; rows are points |
C |
number of clusters |
method |
method to use for within cluster measure; mean or max |
vector of ratios for each cluster
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
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)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)
This function constructs a phylogenetic tree from an ultrametric.
convert.to.tree(n, L, u)convert.to.tree(n, L, u)
n |
is the number of leaves |
L |
is a vector of labels (strings) of leaves |
u |
is an ultrametric |
A phylogenetic tree of class phylo
Ruriko Yoshida [email protected]
um<-Sim_Trees21[1,] ll <- 10 L <- LETTERS[1:10] tr<-convert.to.tree(ll, L, um)um<-Sim_Trees21[1,] ll <- 10 L <- LETTERS[1:10] tr<-convert.to.tree(ll, L, um)
This command draws a three dimensional tropical polytope
draw.tpolytope.3d(D, col_lines, col_verts, plot = TRUE, tadd = max) draw.tpolytope.2d(D, col_lines, col_verts, plot = TRUE, tadd = max)draw.tpolytope.3d(D, col_lines, col_verts, plot = TRUE, tadd = max) draw.tpolytope.2d(D, col_lines, col_verts, plot = TRUE, tadd = max)
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 |
2-D or 3-D rendering of a tropical polytope.
Ruriko Yoshida [email protected]
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)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)
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.
FWpoint.num.w.reg(datamatrix, penalty = 0)FWpoint.num.w.reg(datamatrix, penalty = 0)
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 |
vector; Fermat-Weber point approximation
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
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) ultrametricD = 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
Returns the Fermat-Weber point of N points using a gradient based numerical method
FWpoint.numerical(datamatrix)FWpoint.numerical(datamatrix)
datamatrix |
matrix of dimension N*e, where N is the number of observations which lie in R^e. |
Fermat-Weber point approximation (vector in R^e)
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
D = matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE) FWpoint.numerical(D)D = matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE) FWpoint.numerical(D)
This function uses a hit-and-run sampler to uniformly sample from a max-plus tropical line segment
HAR.TLineSeg(D1, D2, tadd = max)HAR.TLineSeg(D1, D2, tadd = max)
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 |
point on the line segment defined by D1 and D2
Ruriko Yoshida [email protected]
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
D1 <-c(0,4,2) D2 <- c(0,7,-1) HAR.TLineSeg(D1, D2,tadd=max) HAR.TLineSeg(D1, D2,tadd=min)D1 <-c(0,4,2) D2 <- c(0,7,-1) HAR.TLineSeg(D1, D2,tadd=max) HAR.TLineSeg(D1, D2,tadd=min)
This function samples points on a tropical line segment about a location parameter for a given scale parameter defined in terms of tropical distance
HAR.TLineSeg.centroid(D1, D2, m, s, tadd = max)HAR.TLineSeg.centroid(D1, D2, m, s, tadd = max)
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 |
point on the line segment defined by D1 and D2 sampled about mu
David Barnhill [email protected]
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)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)
This function renders a 2D or 3D max-plus or min-plus tropical hyperplane
draw.thyper(D, ext, min.ax, max.ax, plot = FALSE, tadd = max)draw.thyper(D, ext, min.ax, max.ax, plot = FALSE, tadd = max)
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 |
2D or 3D rendering of max-plus or min-plus tropical hyperplane
David Barnhill [email protected]
# 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)# 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)
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.
lung_fishlung_fish
An object of class matrix (inherits from array) with 1290 rows and 45 columns.
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.
This function calculates the center point and radius of the maximum inscribed ball for a max- or min-plus tropical simplex
max_ins.ball(A, tadd = max)max_ins.ball(A, tadd = max)
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 |
list containing the radius and center point of a maximum inscribed ball
David Barnhill [email protected]
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.
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)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)
This function constructs a minimum enclosing ball for a set of points defining a tropical polytope.
min_enc.ball(A)min_enc.ball(A)
A |
matrix of points defining a tropical polytope. Rows are the points. |
list containing center point and radius of minimum enclosing ball of P
David Barnhill [email protected]
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.
P <-matrix(c(0,0,0,0,3,1,0,2,5),3,3,TRUE) min_enc.ball(P)P <-matrix(c(0,0,0,0,3,1,0,2,5),3,3,TRUE) min_enc.ball(P)
This function normalizes the height of a phylogenetic tree
normaliz.tree(D, h = 1)normaliz.tree(D, h = 1)
D |
numeric vector; ultrametric equidistant tree |
h |
desired height; defaults to 1 |
normalized equidistant tree
Ruriko Yoshida [email protected]
D <-c(4,4,2) normaliz.tree(D, h=1)D <-c(4,4,2) normaliz.tree(D, h=1)
This function normalizes a point or set of points in the tropical projective torus by making the first coordinate zero
normaliz.vector(D) normaliz.vectors(D) normaliz.polytope(D) normaliz.ultrametrics(D)normaliz.vector(D) normaliz.vectors(D) normaliz.polytope(D) normaliz.ultrametrics(D)
D |
numeric vector in the tropical projective torus or a matrix of points in the tropical projective torus; for matrices, rows are the points |
a single or set of normalized points with the first coordinate zero
Ruriko Yoshida [email protected]
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)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)
This function calculates an overall betweenness measure based on tropical distance between a set of clusters derived from tropical hierarchical clustering
over_bet_HC(A, V)over_bet_HC(A, V)
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 |
vector of betweenness cluster measures
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
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)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)
This function calculates an overall betweenness measure between a set of clusters derived from tropical k-means clustering
over_bet_KM(A, C)over_bet_KM(A, C)
A |
matrix of tropical points; rows are points with the last column representing a numbered cluster assignment |
C |
number of clusters |
betweenness cluster measure
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
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)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)
This function calculates k equally spaced points on a tropical line segment
Points.TLineSeg(D1, D2, k = 20, tadd = max)Points.TLineSeg(D1, D2, k = 20, tadd = max)
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 |
matrix of k equally spaced points on a tropical line segment
Ruriko Yoshida [email protected]
D1 <-c(0,4,2) D2 <- c(0,7,-1) Points.TLineSeg(D1, D2, k = 5) Points.TLineSeg(D1, D2, k = 5,tadd=min)D1 <-c(0,4,2) D2 <- c(0,7,-1) Points.TLineSeg(D1, D2, k = 5) Points.TLineSeg(D1, D2, k = 5,tadd=min)
This function produces the a matrix of points projected onto a tropical triangle defined by the column space of a matrix
pre.pplot.pro(S, D)pre.pplot.pro(S, D)
S |
matrix of points representing a tropical polytope; rows are the vertices |
D |
data points in the tropical projective torus |
matrix of points representing projections of the points in D (row vectors) onto S
Ruriko Yoshida [email protected]
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)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)
Estimates the probability that an observation x belongs to class 1.
prob.class(pars, x)prob.class(pars, x)
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 |
real number
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
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))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))
This function projects points in the tropical projective torus onto a max- or min-plus tropical polytope based on tropical distance
project.pi(D_s, D, tadd = max)project.pi(D_s, D, tadd = max)
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. |
projection of point D onto the tropical polytope defined by D_s
David Barnhill [email protected]
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)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 based on the tropical distance between points in a dataset
pw.trop.dist(D1, D2)pw.trop.dist(D1, D2)
D1 |
matrix of ultrametrics |
D2 |
matrix of ultrametrics |
matrix; dissimilarity matrix showing the tropical pairwise distance between each point
Ruriko Yoshida [email protected]
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
T1<-Sim_Trees15 T2<-Sim_Trees25 D <- rbind(T1, T2[1,]) pw.trop.dist(D, D)T1<-Sim_Trees15 T2<-Sim_Trees25 D <- rbind(T1, T2[1,]) pw.trop.dist(D, D)
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.
rounding(P)rounding(P)
P |
matrix of points defining a tropical simplex. Rows are the points |
matrix of points defining only the full-dimensional element (the trunk) of a tropical polytope; rows are points
David Barnhill [email protected]
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.
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)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)
Returns the sigmoid function valuation
sigmoid(x)sigmoid(x)
x |
real number |
sigmoid function value at x
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
sigmoid(0) # 0.5sigmoid(0) # 0.5
150 points generated using Gaussian-like Hit-and-Run sampling with three separate pairs of location and scale parameters
Sim_pointsSim_points
Sim_points A 150 x 3 matrix where each row is a point in thetropical projective torus
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.
Sim_Trees1025 Sim_Trees105 Sim_Trees11 Sim_Trees12 Sim_Trees15 Sim_Trees110Sim_Trees1025 Sim_Trees105 Sim_Trees11 Sim_Trees12 Sim_Trees15 Sim_Trees110
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 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.
Sim_Trees2025 Sim_Trees205 Sim_Trees21 Sim_Trees22 Sim_Trees25 Sim_Trees210Sim_Trees2025 Sim_Trees205 Sim_Trees21 Sim_Trees22 Sim_Trees25 Sim_Trees210
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.
This function calculates the tropical determinant (or singularity) of a square matrix
tdets(P, tadd = max)tdets(P, tadd = max)
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 |
list containing the value of the determinant and reordered matrix P
David Barnhill [email protected]
P<-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE) tdets(P) tdets(P,tadd=min)P<-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE) tdets(P) tdets(P,tadd=min)
This function performs k-means clustering over the tropical projective torus
TKmeans(A, C, M)TKmeans(A, C, M)
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 |
list with matrix of observation classified by centroid; matrix of centroid coordinates; number of iterations used
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
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)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)
This function constructs a max-plus tropical line segment between two points
TLineSeg(D1, D2, tadd = max)TLineSeg(D1, D2, tadd = max)
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 |
list of points defining the tropical line segment
Ruriko Yoshida [email protected]
D1 <-c(0,4,2) D2 <- c(0,7,-1) TLineSeg(D1, D2) TLineSeg(D1, D2,tadd=min)D1 <-c(0,4,2) D2 <- c(0,7,-1) TLineSeg(D1, D2) TLineSeg(D1, D2,tadd=min)
TML provides a suite of tools for machine learning application on data over the tropical semiring
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.
tree.to.vector(tree, normalization = TRUE)tree.to.vector(tree, normalization = TRUE)
tree |
phylogenetic tree |
normalization |
logical; normalize the tree if TRUE |
vector of pairwise distances in R^(m choose 2), where m is the number of leaves
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci, James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
tree <- ape::read.tree(text='((A:1, B:1):2, (C:1.5, D:1.5):1.5);') tree.to.vector(tree)tree <- ape::read.tree(text='((A:1, B:1):2, (C:1.5, D:1.5):1.5);') tree.to.vector(tree)
This function constructs a visualization of a 2D or 3D tropical ball
Trop_ball( v, d, a = 1, cls = "black", cent.col = "black", fil = TRUE, plt = TRUE, bord = "black" )Trop_ball( v, d, a = 1, cls = "black", cent.col = "black", fil = TRUE, plt = TRUE, bord = "black" )
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) |
2D or 3D visualization of tropical ball
David Barnhill [email protected]
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)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)
This function calculates a within cluster measure by measuring the pairwise tropical distance between points in the cluster.
trop_wi_dist(D1, method = mean)trop_wi_dist(D1, method = mean)
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 |
within cluster measure
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
D<-Sim_points avg.m<-trop_wi_dist(D, method=mean) max.m<-trop_wi_dist(D, method=max)D<-Sim_points avg.m<-trop_wi_dist(D, method=mean) max.m<-trop_wi_dist(D, method=max)
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
trop.bal.vert(x, d, tadd = max) trop.bal.all_vert(x, d)trop.bal.vert(x, d, tadd = max) trop.bal.all_vert(x, d)
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' |
matrix of normalized tropical points defining the tropical ball. Rows are the points
David Barnhill [email protected]
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.
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)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)
This function computes the tropical distance between two points in the tropical projective torus
trop.dist(D1, D2)trop.dist(D1, D2)
D1 |
point in the tropical projective torus |
D2 |
point in the tropical projective torus |
tropical distance between D1 and D2
Ruriko Yoshida [email protected]
D1 <-c(0,4,2) D2 <- c(0,7,-1) trop.dist(D1, D2)D1 <-c(0,4,2) D2 <- c(0,7,-1) trop.dist(D1, D2)
This function calculates the Fermat-Weber point for a tropical polytope
trop.FW(A)trop.FW(A)
A |
matrix with normalized tropical points as rows |
numeric vector providing the tropical Fermat-Weber point for the tropical polytope
David Barnhill [email protected]
Lin, Bo and Ruriko Yoshida (2016). Tropical Fermat-Weber Points. SIAM J. Discret. Math. 32: 1229-1245.
P <-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE) trop.FW(P)P <-matrix(c(0,0,0,0,2,5,0,3,1),3,3,TRUE) trop.FW(P)
Calculate the tropical distance to a max-tropical hyperplane
trop.hyper.dist(O, x0, tadd = max)trop.hyper.dist(O, x0, tadd = max)
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 |
tropical distance to max-plus tropical hyperplane
David Barnhill [email protected]
O <-c(0,-1,-1) x0 <- c(0,-2,-8) trop.hyper.dist(O,x0) trop.hyper.dist(O,x0,tadd=min)O <-c(0,-1,-1) x0 <- c(0,-2,-8) trop.hyper.dist(O,x0) trop.hyper.dist(O,x0,tadd=min)
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
trop.logistic.regression(D, Y, penalty = 0, model_type = "two_species")trop.logistic.regression(D, Y, penalty = 0, model_type = "two_species")
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" |
vector; optimal model parameters (two normal vectors and two scaling factors)
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees.
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))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))
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.
trop.tri.plot.w.pts(S, D)trop.tri.plot.w.pts(S, D)
S |
inital set of vertices for the tropical triangle |
D |
matrix of data where each row is an observation in the tropical projective torus |
rendering of tropical triangle saved to current directory
Ruriko Yoshida [email protected]
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)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)
This function uses tropical HAR with a uniform target distribution to estimate the volume of a tropical polytope
trop.Volume(B, P, x0, s, I, r)trop.Volume(B, P, x0, s, I, r)
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 |
list containing ratio of points falling in P; volume of the tropical ball; volume estimate of P
David Barnhill [email protected]
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.
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)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)
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
VE.HAR.centroid(D_s, x0, I = 1, m, s, tadd = max)VE.HAR.centroid(D_s, x0, I = 1, m, s, tadd = max)
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 |
next sampled point from the tropical polytope
David Barnhill [email protected]
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)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)
This function samples points on a tropical line segment about a location parameter for a given scale parameter defined in terms of tropical distance
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)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)
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 |
matrix of n sampled points where each point is a row
David Barnhill [email protected]
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
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)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)
This function performs agglomerative (AGNES) hierarchical clustering over the space of ultrametrics defining the space of equidistant trees
Tropical.HC.AGNES(D, method = mean)Tropical.HC.AGNES(D, method = mean)
D |
matrix of points defining a tropical polytope. Rows are the tropical points |
method |
linkage method: mean, min, or max |
list of distances in when merges occur; list of indices of points in each cluster
David Barnhill [email protected]
David Barnhill, Ruriko Yoshida (2023). Clustering Methods Over the Tropically Convex Sets.
P <-Sim_points Tropical.HC.AGNES(P, method=mean)P <-Sim_points Tropical.HC.AGNES(P, method=mean)
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.
tropical.KDE(D, n, sigma, h = 2)tropical.KDE(D, n, sigma, h = 2)
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 |
list containing center point and radius of minimum enclosing ball of P
Ruriko Yoshida [email protected]
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.
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]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]
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.
tropical.PCA.Polytope(S, D, V, I = 1, k)tropical.PCA.Polytope(S, D, V, I = 1, k)
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 |
list with the sum of residuals
Ruriko Yoshida [email protected]
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.
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)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)
This sampler samples a point in the space of ultrametrics where each point represents an equidistant tree on n leaves
Ultrametrics.HAR(x0, n, I = 1, h = 1)Ultrametrics.HAR(x0, n, I = 1, h = 1)
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 |
point in the space of ultrametrics over n leaves
Ruriko Yoshida [email protected]
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
x0 <-Sim_Trees15[1,] n<-10 Ultrametrics.HAR(x0, n, I = 50, h = 1)x0 <-Sim_Trees15[1,] n<-10 Ultrametrics.HAR(x0, n, I = 50, h = 1)
This function samples points uniformly the space defined by a tropical simplex
VE.HAR(D_s, x0, I = 1, tadd = max)VE.HAR(D_s, x0, I = 1, tadd = max)
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 |
next sampled point from the tropical polytope
David Barnhill [email protected]
Yoshida, Ruriko, Keiji Miura and David Barnhill (2022). Hit and Run Sampling from Tropically Convex Sets.
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)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)
A vector of pairwise distances is used to reconstruct the corresponding equidistant tree
vector.to.equidistant.tree(vec)vector.to.equidistant.tree(vec)
vec |
vector of pairwise distances in R^(m choose 2), where m is the number of leaves |
equidistant phylogenetic tree
Georgios Aliatimis [email protected]
Aliatimis, Georgios, Ruriko Yoshida, Burak Boyaci and James A. Grant (2023). Tropical Logistic Regression on Space of Phylogenetic Trees
vec = c(1/3,1,1,1,1,1/3) tree = vector.to.equidistant.tree(vec) plot(tree)vec = c(1/3,1,1,1,1,1/3) tree = vector.to.equidistant.tree(vec) plot(tree)