Type:	Package
Title:	Learning with Data on Riemannian Manifolds
Version:	0.1.4
Description:	We provide a variety of algorithms for manifold-valued data, including Fréchet summaries, hypothesis testing, clustering, visualization, and other learning tasks. See Bhattacharya and Bhattacharya (2012) <doi:10.1017/CBO9781139094764> for general exposition to statistics on manifolds.
License:	MIT + file LICENSE
Imports:	CVXR, Rcpp (≥ 1.0.5), Rdpack, RiemBase, Rdimtools, T4cluster, DEoptim, lpSolve, Matrix, maotai (≥ 0.2.2), stats, utils
Encoding:	UTF-8
URL:	https://kisungyou.com/Riemann/
BugReports:	https://github.com/kisungyou/Riemann/issues
LinkingTo:	Rcpp, RcppArmadillo
RoxygenNote:	7.1.2
RdMacros:	Rdpack
Depends:	R (≥ 2.10)
Suggests:	R.rsp, knitr, rmarkdown
VignetteBuilder:	R.rsp, knitr
LazyData:	true
NeedsCompilation:	yes
Packaged:	2022-02-28 19:30:40 UTC; kisung
Author:	Kisung You [aut, cre]
Maintainer:	Kisung You <kisungyou@outlook.com>
Repository:	CRAN
Date/Publication:	2022-02-28 20:30:02 UTC

Data : EEG Covariances for Event-Related Potentials

Description

This dataset delivers 216 covariance matrices from EEG ERPs with 4 different known classes by types of sources. Among 60 channels, only 32 channels are taken and sample covariance matrix is computed for each participant. The data is taken from a Python library mne's sample data.

Usage

data(ERP)

Format

a named list containing

covariance

an (32\times 32\times 216) array of covariance matrices.

label

a length-216 factor of 4 different classes.

See Also

wrap.spd

Examples

## LOAD THE DATA AND WRAP AS RIEMOBJ
data(ERP)
myriem = wrap.spd(ERP$covariance)

Angular Central Gaussian Distribution

Description

For a hypersphere \mathcal{S}^{p-1} in \mathbf{R}^p, Angular Central Gaussian (ACG) distribution ACG_p (A) is defined via a density

f(x\vert A) = |A|^{-1/2} (x^\top A^{-1} x)^{-p/2}

with respect to the uniform measure on \mathcal{S}^{p-1} and A is a symmetric positive-definite matrix. Since f(x\vert A) = f(-x\vert A), it can also be used as an axial distribution on real projective space, which is unit sphere modulo \lbrace{+1,-1\rbrace}. One constraint we follow is that f(x\vert A) = f(x\vert cA) for c > 0 in that we use a normalized version for numerical stability by restricting tr(A)=p.

Usage

dacg(datalist, A)

racg(n, A)

mle.acg(datalist, ...)

Arguments

Value

dacg gives a vector of evaluated densities given samples. racg generates unit-norm vectors in \mathbf{R}^p wrapped in a list. mle.acg estimates the SPD matrix A.

References

Tyler DE (1987). “Statistical analysis for the angular central Gaussian distribution on the sphere.” Biometrika, 74(3), 579–589. ISSN 0006-3444, 1464-3510.

Mardia KV, Jupp PE (eds.) (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley \& Sons, Inc., Hoboken, NJ, USA. ISBN 978-0-470-31697-9 978-0-471-95333-3.

Examples

# -------------------------------------------------------------------
#          Example with Angular Central Gaussian Distribution
#
# Given a fixed A, generate samples and estimate A via ML.
# -------------------------------------------------------------------
## GENERATE AND MLE in R^5
#  Generate data
Atrue = diag(5)          # true SPD matrix
sam1  = racg(50,  Atrue) # random samples
sam2  = racg(100, Atrue)

#  MLE
Amle1 = mle.acg(sam1)
Amle2 = mle.acg(sam2)

#  Visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(Atrue[,5:1], axes=FALSE, main="true SPD")
image(Amle1[,5:1], axes=FALSE, main="MLE with n=50")
image(Amle2[,5:1], axes=FALSE, main="MLE with n=100")
par(opar)

Data : Populated Cities in the U.S.

Description

As of January 2006, there are 60 cities in the contiguous U.S. with population size larger than 300000. We extracted information of the cities from the data delivered by maps package. Variables coord and cartesian are two identical representations of locations, which can be mutually converted by sphere.convert.

Usage

data(cities)

Format

a named list containing

names

a length-60 vector of city names.

coord

a (60\times 2) matrix of latitude and longitude.

cartesian

a (60\times 3) matrix of cartesian coordinates on the unit sphere.

population

a length-60 vector of cities' populations.

See Also

wrap.sphere

Examples

## LOAD THE DATA AND WRAP AS RIEMOBJ
data(cities)
myriem = wrap.sphere(cities$cartesian)

## COMPUTE INTRINSIC/EXTRINSIC MEANS
intmean = as.vector(riem.mean(myriem, geometry="intrinsic")$mean)
extmean = as.vector(riem.mean(myriem, geometry="extrinsic")$mean)

## CONVERT TO GEOGRAPHIC COORDINATES (Lat/Lon)
geo.int = sphere.xyz2geo(intmean)
geo.ext = sphere.xyz2geo(extmean)

S3 method for mixture model : evaluate density

Description

Compute density for a fitted mixture model.

Usage

density(object, newdata)

Arguments

Value

a length-n vector of class labels.

Examples

# ---------------------------------------------------- #
#            FIT A MODEL & APPLY THE METHOD
# ---------------------------------------------------- #
# Load the 'city' data and wrap as 'riemobj'
data(cities)
locations = cities$cartesian
embed2    = array(0,c(60,2)) 
for (i in 1:60){
   embed2[i,] = sphere.xyz2geo(locations[i,])
}

# Fit a model
k3 = moSN(locations, k=3)

# Evaluate 
newdensity = density(k3, locations)

Data : Gorilla Skull

Description

This is 29 male and 30 female gorillas' skull landmark data where each individual is represented as 8-ad/landmarks in 2 dimensions. This is a re-arranged version of the data from shapes package.

Usage

data(gorilla)

Format

a named list containing

male

a 3d array of size (8\times 2\times 29)

female

a 3d array of size (8\times 2\times 30)

References

Dryden IL, Mardia KV (2016). Statistical shape analysis with applications in R, Wiley series in probability and statistics, Second edition edition. John Wiley \& Sons, Chichester, UK ; Hoboken, NJ. ISBN 978-1-119-07251-5 978-1-119-07250-8.

Reno PL, Meindl RS, McCollum MA, Lovejoy CO (2003). "Sexual dimorphism in Australopithecus afarensis was similar to that of modern humans." Proceedings of the National Academy of Sciences, 100(16):9404–9409.

See Also

wrap.landmark

Examples

data(gorilla)                               # load the data
riem.female = wrap.landmark(gorilla$female) # wrap as RIEMOBJ
opar <- par(no.readonly=TRUE)
for (i in 1:30){
  if (i < 2){
    plot(riem.female$data[[i]], cex=0.5, 
         xlim=c(-1,1)/2, ylim=c(-2,2)/5,
         main="30 female gorilla skull preshapes",
         xlab="dimension 1", ylab="dimension 2")
    lines(riem.female$data[[i]])
  } else {
    points(riem.female$data[[i]], cex=0.5)
    lines(riem.female$data[[i]])
  }
}
par(opar)

Estimation of Distribution Algorithm with MACG Distribution

Description

For a function f : Gr(k,p) \rightarrow \mathbf{R}, find the minimizer and the attained minimum value with estimation of distribution algorithm using MACG distribution.

Usage

grassmann.optmacg(func, p, k, ...)

Arguments

Value

a named list containing:

cost

minimized function value.

solution

a (p\times k) matrix that attains the cost.

Examples

#-------------------------------------------------------------------
#               Optimization for Eigen-Decomposition
#
# Given (5x5) covariance matrix S, eigendecomposition is can be 
# considered as an optimization on Grassmann manifold. Here, 
# we are trying to find top 3 eigenvalues and compare.
#-------------------------------------------------------------------

## PREPARE
A = cov(matrix(rnorm(100*5), ncol=5)) # define covariance
myfunc <- function(p){                # cost function to minimize
  return(sum(-diag(t(p)%*%A%*%p)))
} 

## SOLVE THE OPTIMIZATION PROBLEM
Aout = grassmann.optmacg(myfunc, p=5, k=3, popsize=100, n.start=30)

## COMPUTE EIGENVALUES
#  1. USE SOLUTIONS TO THE ABOVE OPTIMIZATION 
abase   = Aout$solution
eig3sol = sort(diag(t(abase)%*%A%*%abase), decreasing=TRUE)

#  2. USE BASIC 'EIGEN' FUNCTION
eig3dec = sort(eigen(A)$values, decreasing=TRUE)[1:3]

## VISUALIZE
opar <- par(no.readonly=TRUE)
yran = c(min(min(eig3sol),min(eig3dec))*0.95,
         max(max(eig3sol),max(eig3dec))*1.05)
plot(1:3, eig3sol, type="b", col="red",  pch=19, ylim=yran,
     xlab="index", ylab="eigenvalue", main="compare top 3 eigenvalues")
lines(1:3, eig3dec, type="b", col="blue", pch=19)
legend(1.55, max(yran), legend=c("optimization","decomposition"), col=c("red","blue"),
       lty=rep(1,2), pch=19)
par(opar)

Generate Uniform Samples on Grassmann Manifold

Description

It generates n random samples from Grassmann manifold Gr(k,p).

Usage

grassmann.runif(n, k, p, type = c("list", "array", "riemdata"))

Arguments

Value

an object from one of the above by type option.

References

Chikuse Y (2003). Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-00160-9 978-0-387-21540-2.

See Also

stiefel.runif, wrap.grassmann

Examples

#-------------------------------------------------------------------
#                 Draw Samples on Grassmann Manifold 
#-------------------------------------------------------------------
#  Multiple Return Types with 3 Observations of 5-dim subspaces in R^10
dat.list = grassmann.runif(n=3, k=5, p=10, type="list")
dat.arr3 = grassmann.runif(n=3, k=5, p=10, type="array")
dat.riem = grassmann.runif(n=3, k=5, p=10, type="riemdata")

Test of Uniformity on Grassmann Manifold

Description

Given the data on Grassmann manifold Gr(k,p), it tests whether the data is distributed uniformly.

Usage

grassmann.utest(grobj, method = c("Bing", "BingM"))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

p-value under H_0.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Chikuse Y (2003). Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-00160-9 978-0-387-21540-2.

Mardia KV, Jupp PE (eds.) (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley \& Sons, Inc., Hoboken, NJ, USA. ISBN 978-0-470-31697-9 978-0-471-95333-3.

See Also

wrap.grassmann

Examples

#-------------------------------------------------------------------
#   Compare Bingham's original and modified versions of the test
# 
# Test 1. sample uniformly from Gr(2,4)
# Test 2. use perturbed principal components from 'iris' data in R^4
#         which is concentrated around a point to reject H0.
#-------------------------------------------------------------------
## Data Generation
#  1. uniform data
myobj1 = grassmann.runif(n=100, k=2, p=4)

#  2. perturbed principal components
data(iris)
irdat = list()
for (n in 1:100){
   tmpdata    = iris[1:50,1:4] + matrix(rnorm(50*4,sd=0.5),ncol=4)
   irdat[[n]] = eigen(cov(tmpdata))$vectors[,1:2]
}
myobj2 = wrap.grassmann(irdat)

## Test 1 : uniform data
grassmann.utest(myobj1, method="Bing")
grassmann.utest(myobj1, method="BingM")

## Tests : iris data
grassmann.utest(myobj2, method="bINg")   # method names are 
grassmann.utest(myobj2, method="BiNgM")  # CASE - INSENSITIVE !

Data : Left Hands

Description

This dataset contains 10 shapes of 4 subjects's left hands where each shape is represented by 56 landmark points. For each person, first six shapes are equally spaced sequence from maximally to minimally spread fingures. The rest are arbitrarily chosen with two constraints; (1) the palm should face the support and (2) the contour should contain no crossins.

Usage

data(hands)

Format

a named list containing

data

an (56\times 2\times 40) array of landmarks for 40 subjects.

person

a length-40 vector of subject indices.

References

Stegmann M, Gomez D (2002) "A Brief Introduction to Statistical Shape Analysis." Informatics and Mathematical Modelling, Technical University of Denmark, DTU.

See Also

wrap.landmark

Examples

## LOAD THE DATA 
data(hands)

## VISUALIZE 6 HANDS OF PERSON 1
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,3))
for (i in 1:6){
  xx = hands$data[,1,i]
  yy = hands$data[,2,i]
  plot(xx,yy,"b", cex=0.9)
}
par(opar)

S3 method for mixture model : predict labels

Description

Given a fitted mixture model of K components, predict labels of observations accordingly.

Usage

label(object, newdata)

Arguments

Value

a length-n vector of class labels.

Examples

# ---------------------------------------------------- #
#            FIT A MODEL & APPLY THE METHOD
# ---------------------------------------------------- #
# Load the 'city' data and wrap as 'riemobj'
data(cities)
locations = cities$cartesian
embed2    = array(0,c(60,2)) 
for (i in 1:60){
   embed2[i,] = sphere.xyz2geo(locations[i,])
}

# Fit a model
k3 = moSN(locations, k=3)

# Evaluate
newlabel = label(k3, locations)

S3 method for mixture model : log-likelihood

Description

Given a fitted mixture model f(x) and observations x_1, \ldots, x_n \in \mathcal{M}, compute the log-likelihood

L = \log \prod_{i=1}^n f(x_i) = \sum_{i=1}^n \log f(x_i)

.

Usage

loglkd(object, newdata)

Arguments

Value

the log-likelihood.

Examples

# ---------------------------------------------------- #
#            FIT A MODEL & APPLY THE METHOD
# ---------------------------------------------------- #
# Load the 'city' data and wrap as 'riemobj'
data(cities)
locations = cities$cartesian
embed2    = array(0,c(60,2)) 
for (i in 1:60){
   embed2[i,] = sphere.xyz2geo(locations[i,])
}

# Fit a model
k3 = moSN(locations, k=3)

# Evaluate
newloglkd = round(loglkd(k3, locations), 3)
print(paste0("Log-likelihood for K=3 model fit : ", newloglkd))

Matrix Angular Central Gaussian Distribution

Description

For Stiefel and Grassmann manifolds St(r,p) and Gr(r,p), the matrix variant of ACG distribution is known as Matrix Angular Central Gaussian (MACG) distribution MACG_{p,r}(\Sigma) with density

f(X\vert \Sigma) = |\Sigma|^{-r/2} |X^\top \Sigma^{-1} X|^{-p/2}

where \Sigma is a (p\times p) symmetric positive-definite matrix. Similar to vector-variate ACG case, we follow a convention that tr(\Sigma)=p.

Usage

dmacg(datalist, Sigma)

rmacg(n, r, Sigma)

mle.macg(datalist, ...)

Arguments

Value

dmacg gives a vector of evaluated densities given samples. rmacg generates (p\times r) orthonormal matrices wrapped in a list. mle.macg estimates the SPD matrix \Sigma.

References

Chikuse Y (1990). “The matrix angular central Gaussian distribution.” Journal of Multivariate Analysis, 33(2), 265–274. ISSN 0047259X.

Mardia KV, Jupp PE (eds.) (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley \& Sons, Inc., Hoboken, NJ, USA. ISBN 978-0-470-31697-9 978-0-471-95333-3.

Kent JT, Ganeiber AM, Mardia KV (2013). "A new method to simulate the Bingham and related distributions in directional data analysis with applications." arXiv:1310.8110.

See Also

acg

Examples

# -------------------------------------------------------------------
#          Example with Matrix Angular Central Gaussian Distribution
#
# Given a fixed Sigma, generate samples and estimate Sigma via ML.
# -------------------------------------------------------------------
## GENERATE AND MLE in St(2,5)/Gr(2,5)
#  Generate data
Strue = diag(5)                  # true SPD matrix
sam1  = rmacg(n=50,  r=2, Strue) # random samples
sam2  = rmacg(n=100, r=2, Strue) # random samples

#  MLE
Smle1 = mle.macg(sam1)
Smle2 = mle.macg(sam2)

#  Visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(Strue[,5:1], axes=FALSE, main="true SPD")
image(Smle1[,5:1], axes=FALSE, main="MLE with n=50")
image(Smle2[,5:1], axes=FALSE, main="MLE with n=100")
par(opar)

Finite Mixture of Spherical Laplace Distributions

Description

For n observations on a (p-1) sphere in \mathbf{R}^p, a finite mixture model is fitted whose components are spherical Laplace distributions via the following model

f(x; \left\lbrace w_k, \mu_k, \sigma_k \right\rbrace_{k=1}^K) = \sum_{k=1}^K w_k SL(x; \mu_k, \sigma_k)

with parameters w_k's for component weights, \mu_k's for component locations, and \sigma_k's for component scales.

Usage

moSL(
  data,
  k = 2,
  same.sigma = FALSE,
  variants = c("soft", "hard", "stochastic"),
  ...
)

## S3 method for class 'moSL'
loglkd(object, newdata)

## S3 method for class 'moSL'
label(object, newdata)

## S3 method for class 'moSL'
density(object, newdata)

Arguments

Value

a named list of S3 class riemmix containing

cluster

a length-n vector of class labels (from 1:k).

loglkd

log likelihood of the fitted model.

criteria

a vector of information criteria.

parameters

a list containing proportion, location, and scale. See the section for more details.

membership

an (n\times k) row-stochastic matrix of membership.

Parameters of the fitted model

A fitted model is characterized by three parameters. For k-mixture model on a (p-1) sphere in \mathbf{R}^p, (1) proportion is a length-k vector of component weight that sums to 1, (2) location is an (k\times p) matrix whose rows are per-cluster locations, and (3) concentration is a length-k vector of scale parameters for each component.

Note on S3 methods

There are three S3 methods; loglkd, label, and density. Given a random sample of size m as newdata, (1) loglkd returns a scalar value of the computed log-likelihood, (2) label returns a length-m vector of cluster assignments, and (3) density evaluates densities of every observation according ot the model fit.

Examples

# ---------------------------------------------------- #
#                 FITTING THE MODEL
# ---------------------------------------------------- #
# Load the 'city' data and wrap as 'riemobj'
data(cities)
locations = cities$cartesian
embed2    = array(0,c(60,2)) 
for (i in 1:60){
   embed2[i,] = sphere.xyz2geo(locations[i,])
}

# Fit the model with different numbers of clusters
k2 = moSL(locations, k=2)
k3 = moSL(locations, k=3)
k4 = moSL(locations, k=4)

# Visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(embed2, col=k2$cluster, pch=19, main="K=2")
plot(embed2, col=k3$cluster, pch=19, main="K=3")
plot(embed2, col=k4$cluster, pch=19, main="K=4")
par(opar)

# ---------------------------------------------------- #
#                   USE S3 METHODS
# ---------------------------------------------------- #
# Use the same 'locations' data as new data 
# (1) log-likelihood
newloglkd = round(loglkd(k3, locations), 5)
fitloglkd = round(k3$loglkd, 5)
print(paste0("Log-likelihood for K=3 fitted    : ", fitloglkd))
print(paste0("Log-likelihood for K=3 predicted : ", newloglkd))

# (2) label
newlabel = label(k3, locations)

# (3) density
newdensity = density(k3, locations)

Finite Mixture of Spherical Normal Distributions

Description

For n observations on a (p-1) sphere in \mathbf{R}^p, a finite mixture model is fitted whose components are spherical normal distributions via the following model

f(x; \left\lbrace w_k, \mu_k, \lambda_k \right\rbrace_{k=1}^K) = \sum_{k=1}^K w_k SN(x; \mu_k, \lambda_k)

with parameters w_k's for component weights, \mu_k's for component locations, and \lambda_k's for component concentrations.

Usage

moSN(
  data,
  k = 2,
  same.lambda = FALSE,
  variants = c("soft", "hard", "stochastic"),
  ...
)

## S3 method for class 'moSN'
loglkd(object, newdata)

## S3 method for class 'moSN'
label(object, newdata)

## S3 method for class 'moSN'
density(object, newdata)

Arguments

Value

a named list of S3 class riemmix containing

cluster

a length-n vector of class labels (from 1:k).

loglkd

log likelihood of the fitted model.

criteria

a vector of information criteria.

parameters

a list containing proportion, center, and concentration. See the section for more details.

membership

an (n\times k) row-stochastic matrix of membership.

Parameters of the fitted model

A fitted model is characterized by three parameters. For k-mixture model on a (p-1) sphere in \mathbf{R}^p, (1) proportion is a length-k vector of component weight that sums to 1, (2) center is an (k\times p) matrix whose rows are cluster centers, and (3) concentration is a length-k vector of concentration parameters for each component.

Note on S3 methods

There are three S3 methods; loglkd, label, and density. Given a random sample of size m as newdata, (1) loglkd returns a scalar value of the computed log-likelihood, (2) label returns a length-m vector of cluster assignments, and (3) density evaluates densities of every observation according ot the model fit.

References

You K, Suh C (2022). “Parameter Estimation and Model-Based Clustering with Spherical Normal Distribution on the Unit Hypersphere.” Computational Statistics \& Data Analysis, 107457. ISSN 01679473.

Examples

# ---------------------------------------------------- #
#                 FITTING THE MODEL
# ---------------------------------------------------- #
# Load the 'city' data and wrap as 'riemobj'
data(cities)
locations = cities$cartesian
embed2    = array(0,c(60,2)) 
for (i in 1:60){
   embed2[i,] = sphere.xyz2geo(locations[i,])
}

# Fit the model with different numbers of clusters
k2 = moSN(locations, k=2)
k3 = moSN(locations, k=3)
k4 = moSN(locations, k=4)

# Visualize
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(embed2, col=k2$cluster, pch=19, main="K=2")
plot(embed2, col=k3$cluster, pch=19, main="K=3")
plot(embed2, col=k4$cluster, pch=19, main="K=4")
par(opar)

# ---------------------------------------------------- #
#                   USE S3 METHODS
# ---------------------------------------------------- #
# Use the same 'locations' data as new data 
# (1) log-likelihood
newloglkd = round(loglkd(k3, locations), 3)
print(paste0("Log-likelihood for K=3 model fit : ", newloglkd))

# (2) label
newlabel = label(k3, locations)

# (3) density
newdensity = density(k3, locations)

Data : Normal Vectors to the Orbital Planes of the 9 Planets

Description

The 9 planets in our solar system are evolving the sun via their own orbits. This data provides normal vector of the orbital planes. Normal vectors are unit-norm vectors, so that they are thought to reside on 2-dimensional sphere.

Usage

data(orbital)

Format

an (9\times 3) matrix where each row is a normal vector for a planet.

See Also

wrap.sphere

Examples

## LOAD THE DATA AND WRAP AS RIEMOBJ
data(orbital)
myorb = wrap.sphere(orbital)

## VISUALIZE
mds2d = riem.mds(myorb)$embed
opar <- par(no.readonly=TRUE)
plot(mds2d, main="9 Planets", pch=19, xlab="x", ylab="y")
par(opar)

Data : Passiflora Leaves

Description

Passiflora is a genus of about 550 species of flowering plants. This dataset contains 15 landmarks in 2 dimension of 3319 leaves of 40 species. Papers listed in the reference section analyzed the data and found 7 clusters.

Usage

data(passiflora)

Format

a named list containing

data

a 3d array of size (15\times 2\times 3319).

species

a length-3319 vector of 40 species factors.

class

a length-3319 vector of 7 cluster factors.

References

Chitwood DH, Otoni WC (2017). “Divergent leaf shapes among Passiflora species arise from a shared juvenile morphology.” Plant Direct, 1(5), e00028. ISSN 24754455.

Chitwood DH, Otoni WC (2017). “Morphometric analysis of Passiflora leaves: the relationship between landmarks of the vasculature and elliptical Fourier descriptors of the blade.” GigaScience, 6(1). ISSN 2047-217X.

See Also

wrap.landmark

Examples

data(passiflora)                         # load the data
riemobj = wrap.landmark(passiflora$data) # wrap as RIEMOBJ
pga2d   = riem.pga(riemobj)$embed        # embedding via PGA

opar <- par(no.readonly=TRUE)            # visualize
plot(pga2d, col=passiflora$class, pch=19, cex=0.7,
     main="PGA Embedding of Passiflora Leaves",
     xlab="dimension 1", ylab="dimension 2")
par(opar)

Prediction for Manifold-to-Scalar Kernel Regression

Description

Given new observations X_1, X_2, \ldots, X_M \in \mathcal{M}, plug in the data with respect to the fitted model for prediction.

Usage

## S3 method for class 'm2skreg'
predict(object, newdata, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a length-M vector of predictted values.

See Also

riem.m2skreg

Examples

#-------------------------------------------------------------------
#                    Example on Sphere S^2
#
#  X : equi-spaced points from (0,0,1) to (0,1,0)
#  y : sin(x) with perturbation
#
#  Our goal is to check whether the predict function works well
#  by comparing the originally predicted values vs. those of the same data.
#-------------------------------------------------------------------
# GENERATE DATA
npts = 100
nlev = 0.25
thetas = seq(from=0, to=pi/2, length.out=npts)
Xstack = cbind(rep(0,npts), sin(thetas), cos(thetas))

Xriem  = wrap.sphere(Xstack)
ytrue  = sin(seq(from=0, to=2*pi, length.out=npts))
ynoise = ytrue + rnorm(npts, sd=nlev)

# FIT & PREDICT
obj_fit   = riem.m2skreg(Xriem, ynoise, bandwidth=0.01)
yval_fits = obj_fit$ypred
yval_pred = predict(obj_fit, Xriem)

# VISUALIZE
xgrd <- 1:npts
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(xgrd, yval_fits, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="original fit")
lines(xgrd, ytrue, col="red", lwd=1.5)
plot(xgrd, yval_pred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="from 'predict'")
lines(xgrd, ytrue, col="red", lwd=1.5)
par(opar)

Competitive Learning Riemannian Quantization

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, perform clustering via Competitive Learning Riemannian Quantization (CLRQ). Originally, the algorithm is designed for finding voronoi cells that are used in domain quantization. Given the discrete measure of data, centers of the cells play a role of cluster centers and data are labeled accordingly based on the distance to voronoi centers. For an iterative update of centers, gradient descent algorithm adapted for the Riemannian manifold setting is used with the gain factor sequence

\gamma_t = \frac{a}{1 + b \sqrt{t}}

where two parameters a,b are represented by par.a and par.b. For initialization, we provide k-means++ and random seeding options as in k-means.

Usage

riem.clrq(riemobj, k = 2, init = c("plus", "random"), gain.a = 1, gain.b = 1)

Arguments

Value

a named list containing

centers

a 3d array where each slice along 3rd dimension is a matrix representation of class centers.

cluster

a length-N vector of class labels (from 1:k).

References

Le Brigant A, Puechmorel S (2019). “Quantization and clustering on Riemannian manifolds with an application to air traffic analysis.” Journal of Multivariate Analysis, 173, 685–703. ISSN 0047259X.

Bonnabel S (2013). “Stochastic Gradient Descent on Riemannian Manifolds.” IEEE Transactions on Automatic Control, 58(9), 2217–2229. ISSN 0018-9286, 1558-2523.

See Also

riem.kmeans

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## CLRQ WITH K=2,3,4
clust2 = riem.clrq(myriem, k=2)
clust3 = riem.clrq(myriem, k=3)
clust4 = riem.clrq(myriem, k=4)

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
plot(mds2d, pch=19, main="true label", col=mylabs)
plot(mds2d, pch=19, main="K=2", col=clust2$cluster)
plot(mds2d, pch=19, main="K=3", col=clust3$cluster)
plot(mds2d, pch=19, main="K=4", col=clust4$cluster)
par(opar)

Build Lightweight Coreset

Description

Given manifold-valued data X_1,X_2,\ldots,X_N \in \mathcal{M}, this algorithm finds the coreset of size M that can be considered as a compressed representation according to the lightweight coreset construction scheme proposed by the reference below.

Usage

riem.coreset18B(
  riemobj,
  M = length(riemobj$data)/2,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

a named list containing

coreid

a length-M index vector of the coreset.

weight

a length-M vector of weights for each element.

References

Bachem O, Lucic M, Krause A (2018). “Scalable k -Means Clustering via Lightweight Coresets.” In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery \& Data Mining, 1119–1127. ISBN 978-1-4503-5552-0.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# * 10 perturbed data points near (1,0,0) on S^2 in R^3
# * 10 perturbed data points near (0,1,0) on S^2 in R^3
# * 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)

## MDS FOR VISUALIZATION
embed2 = riem.mds(myriem, ndim=2)$embed

## FIND CORESET OF SIZES 3, 6, 9
core1 = riem.coreset18B(myriem, M=3)
core2 = riem.coreset18B(myriem, M=6)
core3 = riem.coreset18B(myriem, M=9)

col1 = rep(1,30); col1[core1$coreid] = 2
col2 = rep(1,30); col2[core2$coreid] = 2
col3 = rep(1,30); col3[core3$coreid] = 2

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
plot(embed2, pch=19, col=col1, main="coreset size=3")
plot(embed2, pch=19, col=col2, main="coreset size=6")
plot(embed2, pch=19, col=col3, main="coreset size=9")
par(opar)

Distance between Two Curves on Manifolds

Description

Given two curves \gamma_1, \gamma_2 : I \rightarrow \mathcal{M}, we are interested in measuring the discrepancy of two curves. Usually, data are given as discrete observations so we are offering several methods to perform the task. See the section below for detailed description.

Usage

riem.distlp(
  riemobj1,
  riemobj2,
  vect = NULL,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

the distance value.

Default Method

Trapezoidal Approximation Assume \gamma_1 (t_i) = X_i and \gamma_2 (t_i) = Y_i for i=1,2,\ldots,N. In the Euclidean space, L_p distance between two scalar-valued functions is defined as

L_p^p (\gamma_1 (x), \gamma_2 (x) = \int_{\mathcal{X}} |\gamma_1 (x) - \gamma_2 (x)|^p dx

. We extend this approach to manifold-valued curves

L_p^p (\gamma_1 (t), \gamma_2 (t)) = \int_{t\in I} d^p (\gamma_1 (t), \gamma_2 (t)) dt

where d(\cdot,\cdot) is an intrinsic/extrinsic distance on manifolds. With the given representations, the above integral is approximated using trapezoidal rule.

Examples

#-------------------------------------------------------------------
#                          Curves on Sphere
#
#  curve1 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve2 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve3 : y = 0.5*sin(x) on the tangent space at (0,0,1)
#
# * distance between curve1 & curve2 should be close to 0.
# * distance between curve1 & curve3 should be large.
#-------------------------------------------------------------------
## GENERATION
vecx  = seq(from=-0.9, to=0.9, length.out=50)
vecy1 = 0.5*cos(vecx) + rnorm(50, sd=0.05)
vecy2 = 0.5*cos(vecx) + rnorm(50, sd=0.05)
vecy3 = 0.5*sin(vecx) + rnorm(50, sd=0.05)

## WRAP AS RIEMOBJ
mat1 = cbind(vecx, vecy1, 1); mat1 = mat1/sqrt(rowSums(mat1^2))
mat2 = cbind(vecx, vecy2, 1); mat2 = mat2/sqrt(rowSums(mat2^2))
mat3 = cbind(vecx, vecy3, 1); mat3 = mat3/sqrt(rowSums(mat3^2))

rcurve1 = wrap.sphere(mat1)
rcurve2 = wrap.sphere(mat2)
rcurve3 = wrap.sphere(mat3)

## COMPUTE DISTANCES
riem.distlp(rcurve1, rcurve2, vect=vecx)
riem.distlp(rcurve1, rcurve3, vect=vecx)

Dynamic Time Warping Distance

Description

Given two time series - a query X = (X_1,X_2,\ldots,X_N) and a reference Y = (Y_1,Y_2,\ldots,Y_M), riem.dtw computes the most basic version of Dynamic Time Warping (DTW) distance between two series using a symmetric step pattern, meaning no window constraints and others at all. Although the scope of DTW in Euclidean space-valued objects is rich, it is scarce for manifold-valued curves. If you are interested in the topic, we refer to dtw package.

Usage

riem.dtw(riemobj1, riemobj2, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

the distance value.

Examples

#-------------------------------------------------------------------
#                          Curves on Sphere
#
#  curve1 : y = 0.5*cos(x) on the tangent space at (0,0,1)
#  curve2 : y = 0.5*sin(x) on the tangent space at (0,0,1)
# 
#  we will generate two sets for curves of different sizes.
#-------------------------------------------------------------------
## GENERATION
clist = list()
for (i in 1:10){ # curve type 1
  vecx = seq(from=-0.9, to=0.9, length.out=sample(10:50, 1))
  vecy = 0.5*cos(vecx) + rnorm(length(vecx), sd=0.1)
  mats = cbind(vecx, vecy, 1)
  clist[[i]] = wrap.sphere(mats/sqrt(rowSums(mats^2)))
}
for (i in 1:10){ # curve type 2
  vecx = seq(from=-0.9, to=0.9, length.out=sample(10:50, 1))
  vecy = 0.5*sin(vecx) + rnorm(length(vecx), sd=0.1)
  mats = cbind(vecx, vecy, 1)
  clist[[i+10]] = wrap.sphere(mats/sqrt(rowSums(mats^2)))
}

## COMPUTE DISTANCES
outint = array(0,c(20,20))
outext = array(0,c(20,20))
for (i in 1:19){
  for (j in 2:20){
    outint[i,j] <- outint[j,i] <- riem.dtw(clist[[i]], clist[[j]], 
                                           geometry="intrinsic")
    outext[i,j] <- outext[j,i] <- riem.dtw(clist[[i]], clist[[j]],
                                           geometry="extrinsic")
  }
}

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
image(outint[,20:1], axes=FALSE, main="intrinsic DTW Distance")
image(outext[,20:1], axes=FALSE, main="extrinsic DTW Distance")
par(opar)

Fréchet Analysis of Variance

Description

Given sets of manifold-valued data X^{(1)}_{1:{n_1}}, X^{(2)}_{1:{n_2}}, \ldots, X^{(m)}_{1:{n_m}}, performs analysis of variance to test equality of distributions. This means, small p-value implies that at least one of the equalities does not hold.

Usage

riem.fanova(..., maxiter = 50, eps = 1e-05)

riem.fanovaP(..., maxiter = 50, eps = 1e-05, nperm = 99)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

p-value under H_0.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Dubey P, Müller H (2019). “Fréchet analysis of variance for random objects.” Biometrika, 106(4), 803–821. ISSN 0006-3444, 1464-3510.

Examples

#-------------------------------------------------------------------
#            Example on Sphere : Uniform Samples
#
#  Each of 4 classes consists of 20 uniform samples from uniform 
#  density on 2-dimensional sphere S^2 in R^3.
#-------------------------------------------------------------------
## PREPARE DATA OF 4 CLASSES
ndata  = 200
class1 = list()
class2 = list()
class3 = list()
class4 = list()
for (i in 1:ndata){
  tmpxy = matrix(rnorm(4*2, sd=0.1), ncol=2)
  tmpz  = rep(1,4)
  tmp3d = cbind(tmpxy, tmpz)
  tmp  = tmp3d/sqrt(rowSums(tmp3d^2))
  
  class1[[i]] = tmp[1,]
  class2[[i]] = tmp[2,]
  class3[[i]] = tmp[3,]
  class4[[i]] = tmp[4,]
}
obj1 = wrap.sphere(class1)
obj2 = wrap.sphere(class2)
obj3 = wrap.sphere(class3)
obj4 = wrap.sphere(class4)

## RUN THE ASYMPTOTIC TEST
riem.fanova(obj1, obj2, obj3, obj4)


## RUN THE PERMUTATION TEST WITH MANY PERMUTATIONS
riem.fanovaP(obj1, obj2, obj3, obj4, nperm=999)

Hierarchical Agglomerative Clustering

Description

Given N observations X_1, X_2, \ldots, X_M \in \mathcal{M}, perform hierarchical agglomerative clustering with fastcluster package's implementation.

Usage

riem.hclust(
  riemobj,
  geometry = c("intrinsic", "extrinsic"),
  method = c("single", "complete", "average", "mcquitty", "ward.D", "ward.D2",
    "centroid", "median"),
  members = NULL
)

Arguments

Value

an object of class hclust. See hclust for details.

References

Müllner D (2013). “fastcluster : Fast Hierarchical, Agglomerative Clustering Routines for R and Python.” Journal of Statistical Software, 53(9). ISSN 1548-7660.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)

## COMPUTE SINGLE AND COMPLETE LINKAGE
hc.sing <- riem.hclust(myriem, method="single")
hc.comp <- riem.hclust(myriem, method="complete")

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(hc.sing, main="single linkage")
plot(hc.comp, main="complete linkage")
par(opar)

Geodesic Interpolation

Description

Given 2 observations X_1, X_2 \in \mathcal{M}, find the interpolated point of a geodesic \gamma(t) for t \in (0,1) which assumes two endpoints \gamma(0)=X_1 and \gamma(1)=X_2.

Usage

riem.interp(riemobj, t = 0.5, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

an interpolated object in matrix representation on \mathcal{M}.

Examples

#-------------------------------------------------------------------
#       Geodesic Interpolation between (1,0) and (0,1) in S^1
#-------------------------------------------------------------------
## PREPARE DATA
sp.start = c(1,0)
sp.end   = c(0,1)
sp.data  = wrap.sphere(rbind(sp.start, sp.end))

## FIND THE INTERPOLATED POINT AT "t=0.25"
mid.int = as.vector(riem.interp(sp.data, t=0.25, geometry="intrinsic"))
mid.ext = as.vector(riem.interp(sp.data, t=0.25, geometry="extrinsic"))

## VISUALIZE
#  Prepare Lines and Points
thetas  = seq(from=0, to=pi/2, length.out=100)
quarter = cbind(cos(thetas), sin(thetas))
pic.pts = rbind(sp.start, mid.int, mid.ext, sp.end)
pic.col = c("black","red","green","black")

# Draw
opar <- par(no.readonly=TRUE)
par(pty="s")
plot(quarter, main="two interpolated points at t=0.25",
     xlab="x", ylab="y", type="l")
points(pic.pts, col=pic.col, pch=19)
text(mid.int[1]-0.1, mid.int[2], "intrinsic", col="red")
text(mid.ext[1]-0.1, mid.ext[2], "extrinsic", col="green")
par(opar)

Geodesic Interpolation of Multiple Points

Description

Given 2 observations X_1, X_2 \in \mathcal{M}, find the interpolated points of a geodesic \gamma(t) for t \in (0,1) which assumes two endpoints \gamma(0)=X_1 and \gamma(1)=X_2.

Usage

riem.interps(
  riemobj,
  vect = c(0.25, 0.5, 0.75),
  geometry = c("intrinsic", "extrinsic")
)

Arguments

Value

a 3d array where T slices along 3rd dimension are interpolated objects in matrix representation.

Examples

#-------------------------------------------------------------------
#       Geodesic Interpolation between (1,0) and (0,1) in S^1
#-------------------------------------------------------------------
## PREPARE DATA
sp.start = c(1,0)
sp.end   = c(0,1)
sp.data  = wrap.sphere(rbind(sp.start, sp.end))

## FIND THE INTERPOLATED POINT AT FOR t=0.1, 0.2, ..., 0.9.
myvect  = seq(from=0.1, to=0.9, by=0.1)
geo.int = riem.interps(sp.data, vect=myvect, geometry="intrinsic")
geo.ext = riem.interps(sp.data, vect=myvect, geometry="extrinsic")

geo.int = matrix(geo.int, byrow=TRUE, ncol=2) # re-arrange for plotting
geo.ext = matrix(geo.ext, byrow=TRUE, ncol=2)

## VISUALIZE
#  Prepare Lines and Points
thetas  = seq(from=0, to=pi/2, length.out=100)
quarter = cbind(cos(thetas), sin(thetas))

pts.int = rbind(sp.start, geo.int, sp.end)
pts.ext = rbind(sp.start, geo.ext, sp.end)
col.int = c("black", rep("red",9),  "black")
col.ext = c("black", rep("blue",9), "black")

# Draw
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(quarter, main="intrinsic interpolation", # intrinsic geodesic
     xlab="x", ylab="y", type="l")
points(pts.int, col=col.int, pch=19)
for (i in 1:9){
  text(geo.int[i,1]*0.9, geo.int[i,2]*0.9, 
       paste0(round(i/10,2)), col="red")
}
plot(quarter, main="extrinsic interpolation", # intrinsic geodesic
     xlab="x", ylab="y", type="l")
points(pts.ext, col=col.ext, pch=19)
for (i in 1:9){
  text(geo.ext[i,1]*0.9, geo.ext[i,2]*0.9, 
       paste0(round(i/10,2)), col="blue")
}
par(opar)

Isometric Feature Mapping

Description

ISOMAP - isometric feature mapping - is a dimensionality reduction method to apply classical multidimensional scaling to the geodesic distance that is computed on a weighted nearest neighborhood graph. Nearest neighbor is defined by k-NN where two observations are said to be connected when they are mutually included in each other's nearest neighbor. Note that it is possible for geodesic distances to be Inf when nearest neighbor graph construction incurs separate connected components. When an extra parameter padding=TRUE, infinite distances are replaced by 2 times the maximal finite geodesic distance.

Usage

riem.isomap(
  riemobj,
  ndim = 2,
  nnbd = 5,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

References

Silva VD, Tenenbaum JB (2003). “Global Versus Local Methods in Nonlinear Dimensionality Reduction.” In Becker S, Thrun S, Obermayer K (eds.), Advances in Neural Information Processing Systems 15, 721–728. MIT Press.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## MDS AND ISOMAP WITH DIFFERENT NEIGHBORHOOD SIZE
mdss = riem.mds(myriem)$embed
iso1 = riem.isomap(myriem, nnbd=5)$embed
iso2 = riem.isomap(myriem, nnbd=10)$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
plot(mdss, col=mylabs, pch=19, main="MDS")
plot(iso1, col=mylabs, pch=19, main="ISOMAP:nnbd=5")
plot(iso2, col=mylabs, pch=19, main="ISOMAP:nnbd=10")
par(opar)

K-Means Clustering

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, perform k-means clustering by minimizing within-cluster sum of squares (WCSS). Since the problem is NP-hard and sensitive to the initialization, we provide an option with multiple starts and return the best result with respect to WCSS.

Usage

riem.kmeans(riemobj, k = 2, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a named list containing

cluster

a length-N vector of class labels (from 1:k).

means

a 3d array where each slice along 3rd dimension is a matrix representation of class mean.

score

within-cluster sum of squares (WCSS).

References

Lloyd S (1982). “Least squares quantization in PCM.” IEEE Transactions on Information Theory, 28(2), 129–137. ISSN 0018-9448.

MacQueen J (1967). “Some methods for classification and analysis of multivariate observations.” In Proceedings of the fifth berkeley symposium on mathematical statistics and probability, volume 1: Statistics, 281–297.

See Also

riem.kmeanspp

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## K-MEANS WITH K=2,3,4
clust2 = riem.kmeans(myriem, k=2)
clust3 = riem.kmeans(myriem, k=3)
clust4 = riem.kmeans(myriem, k=4)

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
plot(mds2d, pch=19, main="true label", col=mylabs)
plot(mds2d, pch=19, main="K=2", col=clust2$cluster)
plot(mds2d, pch=19, main="K=3", col=clust3$cluster)
plot(mds2d, pch=19, main="K=4", col=clust4$cluster)
par(opar)

K-Means Clustering with Lightweight Coreset

Description

The modified version of lightweight coreset for scalable k-means computation is applied for manifold-valued data X_1,X_2,\ldots,X_N \in \mathcal{M}. The smaller the set is, the faster the execution becomes with potentially larger quantization errors.

Usage

riem.kmeans18B(
  riemobj,
  k = 2,
  M = length(riemobj$data)/2,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

a named list containing

cluster

a length-N vector of class labels (from 1:k).

means

a 3d array where each slice along 3rd dimension is a matrix representation of class mean.

score

within-cluster sum of squares (WCSS).

References

Bachem O, Lucic M, Krause A (2018). “Scalable k -Means Clustering via Lightweight Coresets.” In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery \& Data Mining, 1119–1127. ISBN 978-1-4503-5552-0.

See Also

riem.coreset18B

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## TRY DIFFERENT SIZES OF CORESET WITH K=4 FIXED
core1 = riem.kmeans18B(myriem, k=3, M=5)
core2 = riem.kmeans18B(myriem, k=3, M=10)
core3 = riem.kmeans18B(myriem, k=3, M=15)

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
plot(mds2d, pch=19, main="true label", col=mylabs)
plot(mds2d, pch=19, main="kmeans18B: M=5",  col=core1$cluster)
plot(mds2d, pch=19, main="kmeans18B: M=10", col=core2$cluster)
plot(mds2d, pch=19, main="kmeans18B: M=15", col=core3$cluster)
par(opar)

K-Means++ Clustering

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, perform k-means++ clustering algorithm using pairwise distances. The algorithm was originally designed as an efficient initialization method for k-means algorithm.

Usage

riem.kmeanspp(riemobj, k = 2, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

centers

a length-k vector of sampled centers' indices.

cluster

a length-N vector of class labels (from 1:k).

References

Arthur D, Vassilvitskii S (2007). “K-Means++: The advantages of careful seeding.” In Proceedings of the eighteenth annual ACM-SIAM symposium on discrete algorithms, SODA '07, 1027–1035. ISBN 978-0-89871-624-5, Number of pages: 9 Place: New Orleans, Louisiana.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## K-MEANS++ WITH K=2,3,4
clust2 = riem.kmeanspp(myriem, k=2)
clust3 = riem.kmeanspp(myriem, k=3)
clust4 = riem.kmeanspp(myriem, k=4)

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
plot(mds2d, pch=19, main="true label", col=mylabs)
plot(mds2d, pch=19, main="K=2", col=clust2$cluster)
plot(mds2d, pch=19, main="K=3", col=clust3$cluster)
plot(mds2d, pch=19, main="K=4", col=clust4$cluster)
par(opar)

K-Medoids Clustering

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, perform k-medoids clustering using pairwise distances.

Usage

riem.kmedoids(riemobj, k = 2, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

medoids

a length-k vector of medoids' indices.

cluster

a length-N vector of class labels (from 1:k).

See Also

pam

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## K-MEDOIDS WITH K=2,3,4
clust2 = riem.kmedoids(myriem, k=2)
clust3 = riem.kmedoids(myriem, k=3)
clust4 = riem.kmedoids(myriem, k=4)

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,2), pty="s")
plot(mds2d, pch=19, main="true label", col=mylabs)
plot(mds2d, pch=19, main="K=2", col=clust2$cluster)
plot(mds2d, pch=19, main="K=3", col=clust3$cluster)
plot(mds2d, pch=19, main="K=4", col=clust4$cluster)
par(opar)
 

Find K-Nearest Neighbors

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, riem.knn constructs k-nearest neighbors.

Usage

riem.knn(riemobj, k = 2, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

nn.idx

an (N \times k) neighborhood index matrix.

nn.dists

an (N\times k) distances from a point to its neighbors.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# * 10 perturbed data points near (1,0,0) on S^2 in R^3
# * 10 perturbed data points near (0,1,0) on S^2 in R^3
# * 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(2,3,4), each=10)

## K-NN CONSTRUCTION WITH K=5 & K=10
knn1 = riem.knn(myriem, k=5)
knn2 = riem.knn(myriem, k=10)

## MDS FOR VISUALIZATION
embed2 = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2, pch=19, main="knn with k=4", col=mylabs)
for (i in 1:30){
  for (j in 1:5){
    lines(embed2[c(i,knn1$nn.idx[i,j]),])
  }
}
plot(embed2, pch=19, main="knn with k=8", col=mylabs)
for (i in 1:30){
  for (j in 1:10){
    lines(embed2[c(i,knn2$nn.idx[i,j]),])
  }
}
par(opar)

Kernel Principal Component Analysis

Description

Although the method of Kernel Principal Component Analysis (KPCA) was originally developed to visualize non-linearly distributed data in Euclidean space, we graft this to the case for manifolds where extrinsic geometry is explicitly available. The algorithm uses Gaussian kernel with

K(X_i, X_j) = \exp\left( - \frac{d^2 (X_i, X_j)}{2 \sigma^2} \right )

where \sigma is a bandwidth parameter and d(\cdot, \cdot) is an extrinsic distance defined on a specific manifold.

Usage

riem.kpca(riemobj, ndim = 2, sigma = 1)

Arguments

Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

vars

a length-N vector of eigenvalues from kernelized covariance matrix.

References

Schölkopf B, Smola A, Müller K (1997). “Kernel principal component analysis.” In Goos G, Hartmanis J, van Leeuwen J, Gerstner W, Germond A, Hasler M, Nicoud J (eds.), Artificial Neural Networks — ICANN'97, volume 1327, 583–588. Springer Berlin Heidelberg, Berlin, Heidelberg. ISBN 978-3-540-63631-1 978-3-540-69620-9.

Examples

#-------------------------------------------------------------------
#          Example for Gorilla Skull Data : 'gorilla'
#-------------------------------------------------------------------
## PREPARE THE DATA
#  Aggregate two classes into one set
data(gorilla)

mygorilla = array(0,c(8,2,59))
for (i in 1:29){
  mygorilla[,,i] = gorilla$male[,,i]
}
for (i in 30:59){
  mygorilla[,,i] = gorilla$female[,,i-29]
}

gor.riem = wrap.landmark(mygorilla)
gor.labs = c(rep("red",29), rep("blue",30))

## APPLY KPCA WITH DIFFERENT KERNEL BANDWIDTHS
kpca1 = riem.kpca(gor.riem, sigma=0.01)
kpca2 = riem.kpca(gor.riem, sigma=1)
kpca3 = riem.kpca(gor.riem, sigma=100)
## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
plot(kpca1$embed, pch=19, col=gor.labs, main="sigma=1/100")
plot(kpca2$embed, pch=19, col=gor.labs, main="sigma=1")
plot(kpca3$embed, pch=19, col=gor.labs, main="sigma=100")
par(opar)

Manifold-to-Scalar Kernel Regression

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M} and scalars y_1, y_2, \ldots, y_N \in \mathbf{R}, perform the Nadaraya-Watson kernel regression by

\hat{m}_h (X) = \frac{\sum_{i=1}^n K \left( \frac{d(X,X_i)}{h} \right) y_i}{\sum_{i=1}^n K \left( \frac{d(X,X_i)}{h} \right)}

where the Gaussian kernel is defined as

K(x) := \frac{1}{\sqrt{2\pi}} \exp \left( - \frac{x^2}{2}\right)

with the bandwidth parameter h > 0 that controls the degree of smoothness.

Usage

riem.m2skreg(
  riemobj,
  y,
  bandwidth = 0.5,
  geometry = c("intrinsic", "extrinsic")
)

Arguments

Value

a named list of S3 class m2skreg containing

ypred

a length-N vector of smoothed responses.

bandwidth

the bandwidth value that was originally provided, which is saved for future use.

inputs

a list containing both riemobj and y for future use.

Examples

#-------------------------------------------------------------------
#                    Example on Sphere S^2
#
#  X : equi-spaced points from (0,0,1) to (0,1,0)
#  y : sin(x) with perturbation
#-------------------------------------------------------------------
# GENERATE DATA
npts = 100
nlev = 0.25
thetas = seq(from=0, to=pi/2, length.out=npts)
Xstack = cbind(rep(0,npts), sin(thetas), cos(thetas))

Xriem  = wrap.sphere(Xstack)
ytrue  = sin(seq(from=0, to=2*pi, length.out=npts))
ynoise = ytrue + rnorm(npts, sd=nlev)

# FIT WITH DIFFERENT BANDWIDTHS
fit1 = riem.m2skreg(Xriem, ynoise, bandwidth=0.001)
fit2 = riem.m2skreg(Xriem, ynoise, bandwidth=0.01)
fit3 = riem.m2skreg(Xriem, ynoise, bandwidth=0.1)

# VISUALIZE
xgrd <- 1:npts
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
plot(xgrd, fit1$ypred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="h=1e-3")
lines(xgrd, ytrue, col="red", lwd=1.5)
plot(xgrd, fit2$ypred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="h=1e-2")
lines(xgrd, ytrue, col="red", lwd=1.5)
plot(xgrd, fit3$ypred, pch=19, cex=0.5, "b", xlab="", ylim=c(-2,2), main="h=1e-1")
lines(xgrd, ytrue, col="red", lwd=1.5)
par(opar)

Manifold-to-Scalar Kernel Regression with K-Fold Cross Validation

Description

Manifold-to-Scalar Kernel Regression with K-Fold Cross Validation

Usage

riem.m2skregCV(
  riemobj,
  y,
  bandwidths = seq(from = 0.01, to = 1, length.out = 10),
  geometry = c("intrinsic", "extrinsic"),
  kfold = 5
)

Arguments

Value

a named list of S3 class m2skreg containing

ypred

a length-N vector of optimal smoothed responses.

bandwidth

the optimal bandwidth value.

inputs

a list containing both riemobj and y for future use.

errors

a matrix whose columns are bandwidths values and corresponding errors measure in SSE.

Examples

#-------------------------------------------------------------------
#                    Example on Sphere S^2
#
#  X : equi-spaced points from (0,0,1) to (0,1,0)
#  y : sin(x) with perturbation
#-------------------------------------------------------------------
# GENERATE DATA
set.seed(496) 
npts = 100
nlev = 0.25
thetas = seq(from=0, to=pi/2, length.out=npts)
Xstack = cbind(rep(0,npts), sin(thetas), cos(thetas))

Xriem  = wrap.sphere(Xstack)
ytrue  = sin(seq(from=0, to=2*pi, length.out=npts))
ynoise = ytrue + rnorm(npts, sd=nlev)

# FIT WITH 5-FOLD CV
cv_band = (10^seq(from=-4, to=-1, length.out=200))
cv_fit  = riem.m2skregCV(Xriem, ynoise, bandwidths=cv_band)
cv_err  = cv_fit$errors

# VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(1:npts, cv_fit$ypred, pch=19, cex=0.5, "b", xlab="", main="optimal prediction")
lines(1:npts, ytrue, col="red", lwd=1.5)
plot(cv_err[,1], cv_err[,2], "b", pch=19, cex=0.5, main="5-fold CV errors",
     xlab="bandwidth", ylab="SSE")
abline(v=cv_fit$bandwidth, col="blue", lwd=1.5)
par(opar)

Multidimensional Scaling

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, apply multidimensional scaling to get low-dimensional embedding in Euclidean space. Usually, ndim=2,3 are chosen for visualization.

Usage

riem.mds(riemobj, ndim = 2, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

stress

discrepancy between embedded and original distances as a measure of error.

References

Torgerson WS (1952). “Multidimensional scaling: I. Theory and method.” Psychometrika, 17(4), 401–419. ISSN 0033-3123, 1860-0980.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## MDS EMBEDDING WITH TWO GEOMETRIES
embed2int = riem.mds(myriem, geometry="intrinsic")$embed
embed2ext = riem.mds(myriem, geometry="extrinsic")$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2int, main="intrinsic MDS", ylim=c(-2,2), col=mylabs, pch=19)
plot(embed2ext, main="extrinsic MDS", ylim=c(-2,2), col=mylabs, pch=19)
par(opar)

Fréchet Mean and Variation

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, compute Fréchet mean and variation with respect to the geometry by minimizing

\textrm{min}_x \sum_{n=1}^N w_n \rho^2 (x, x_n),\quad x\in\mathcal{M}

where \rho (x, y) is a distance for two points x,y\in\mathcal{M}. If non-uniform weights are given, normalized version of the mean is computed and if weight=NULL, it automatically sets equal weights (w_i = 1/n) for all observations.

Usage

riem.mean(riemobj, weight = NULL, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a named list containing

mean

a mean matrix on \mathcal{M}.

variation

sum of (weighted) squared distances.

Examples

#-------------------------------------------------------------------
#        Example on Sphere : points near (0,1) on S^1 in R^2
#-------------------------------------------------------------------
## GENERATE DATA
ndata = 50
mydat = array(0,c(ndata,2))
for (i in 1:ndata){
  tgt = c(stats::rnorm(1, sd=2), 1)
  mydat[i,] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydat)

## COMPUTE TWO MEANS
mean.int = as.vector(riem.mean(myriem, geometry="intrinsic")$mean)
mean.ext = as.vector(riem.mean(myriem, geometry="extrinsic")$mean)

## VISUALIZE
opar <- par(no.readonly=TRUE)
plot(mydat[,1], mydat[,2], pch=19, xlim=c(-1.1,1.1), ylim=c(0,1.1),
     main="BLUE-extrinsic vs RED-intrinsic")
arrows(x0=0,y0=0,x1=mean.int[1],y1=mean.int[2],col="red")
arrows(x0=0,y0=0,x1=mean.ext[1],y1=mean.ext[2],col="blue")
par(opar)

Fréchet Median and Variation

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, compute Fréchet median and variation with respect to the geometry by minimizing

\textrm{min}_x \sum_{n=1}^N w_n \rho (x, x_n),\quad x\in\mathcal{M}

where \rho (x, y) is a distance for two points x,y\in\mathcal{M}. If non-uniform weights are given, normalized version of the mean is computed and if weight=NULL, it automatically sets equal weights for all observations.

Usage

riem.median(
  riemobj,
  weight = NULL,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

a named list containing

median

a median matrix on \mathcal{M}.

variation

sum of (weighted) distances.

Examples

#-------------------------------------------------------------------
#        Example on Sphere : points near (0,1) on S^1 in R^2
#-------------------------------------------------------------------
## GENERATE DATA
ndata = 50
mydat = array(0,c(ndata,2))
for (i in 1:ndata){
  tgt = c(stats::rnorm(1, sd=2), 1)
  mydat[i,] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydat)

## COMPUTE TWO MEANS
med.int = as.vector(riem.median(myriem, geometry="intrinsic")$median)
med.ext = as.vector(riem.median(myriem, geometry="extrinsic")$median)

## VISUALIZE
opar <- par(no.readonly=TRUE)
plot(mydat[,1], mydat[,2], pch=19, xlim=c(-1.1,1.1), ylim=c(0,1.1),
     main="BLUE-extrinsic vs RED-intrinsic")
arrows(x0=0,y0=0,x1=med.int[1],y1=med.int[2],col="red")
arrows(x0=0,y0=0,x1=med.ext[1],y1=med.ext[2],col="blue")
par(opar)

Nonlinear Mean Shift

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, perform clustering of the data based on the nonlinear mean shift algorithm. Gaussian kernel is used with the bandwidth h as of

G(x_i, x_j) \propto \exp \left( - \frac{\rho^2 (x_i,x_j)}{h^2} \right)

where \rho(x,y) is geodesic distance between two points x,y\in\mathcal{M}. Numerically, some of the limiting points that collapse into the same cluster are not exact. For such purpose, we require maxk parameter to search the optimal number of clusters based on k-medoids clustering algorithm in conjunction with silhouette criterion.

Usage

riem.nmshift(riemobj, h = 1, maxk = 5, maxiter = 50, eps = 1e-05)

Arguments

Value

a named list containing

distance

an (N\times N) distance between modes corresponding to each data point.

cluster

a length-N vector of class labels.

References

Subbarao R, Meer P (2009). “Nonlinear Mean Shift over Riemannian Manifolds.” International Journal of Computer Vision, 84(1), 1–20. ISSN 0920-5691, 1573-1405.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
set.seed(496)
ndata  = 10
mydata = list()
for (i in 1:ndata){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in (ndata+1):(2*ndata)){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in ((2*ndata)+1):(3*ndata)){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=ndata)

## RUN NONLINEAR MEANSHIFT FOR DIFFERENT 'h' VALUES
run1 = riem.nmshift(myriem, maxk=10, h=0.1)
run2 = riem.nmshift(myriem, maxk=10, h=1)
run3 = riem.nmshift(myriem, maxk=10, h=10)

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,3), pty="s")
plot(mds2d, pch=19, main="label : h=0.1", col=run1$cluster)
plot(mds2d, pch=19, main="label : h=1",   col=run2$cluster)
plot(mds2d, pch=19, main="label : h=10",  col=run3$cluster)
image(run1$distance[,30:1], axes=FALSE, main="distance : h=0.1")
image(run2$distance[,30:1], axes=FALSE, main="distance : h=1")
image(run3$distance[,30:1], axes=FALSE, main="distance : h=10")
par(opar)

Compute Pairwise Distances for Data

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, compute pairwise distances.

Usage

riem.pdist(riemobj, geometry = c("intrinsic", "extrinsic"), as.dist = FALSE)

Arguments

Value

a S3 dist object or (N\times N) symmetric matrix of pairwise distances according to as.dist parameter.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with two types
#
#  group1 : perturbed data points near (0,0,1) on S^2 in R^3
#  group2 : perturbed data points near (1,0,0) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
sdval  = 0.1
for (i in 1:10){
  tgt = c(stats::rnorm(2, sd=sdval), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(1, stats::rnorm(2, sd=sdval))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)

## COMPARE TWO DISTANCES
dint = riem.pdist(myriem, geometry="intrinsic", as.dist=FALSE)
dext = riem.pdist(myriem, geometry="extrinsic", as.dist=FALSE)

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
image(dint[,nrow(dint):1], main="intrinsic", axes=FALSE)
image(dext[,nrow(dext):1], main="extrinsic", axes=FALSE)
par(opar)

Compute Pairwise Distances for Two Sets of Data

Description

Given M observations X_1, X_2, \ldots, X_M \in \mathcal{M} and N observations Y_1, Y_2, \ldots, Y_N \in \mathcal{M}, compute pairwise distances between two sets' elements.

Usage

riem.pdist2(riemobj1, riemobj2, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

an (M\times N) matrix of distances.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with two types
#
#  group1 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#  group2 : 10 perturbed data points near (1,0,0) on S^2 in R^3
#           10 perturbed data points near (0,1,0) on S^2 in R^3
#           10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata1 = list()
mydata2 = list()
for (i in 1:10){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata1[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem1 = wrap.sphere(mydata1)
myriem2 = wrap.sphere(mydata2)

## COMPARE TWO DISTANCES
dint = riem.pdist2(myriem1, myriem2, geometry="intrinsic")
dext = riem.pdist2(myriem1, myriem2, geometry="extrinsic")

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2))
image(dint[nrow(dint):1,], main="intrinsic", axes=FALSE)
image(dext[nrow(dext):1,], main="extrinsic", axes=FALSE)
par(opar)

Principal Geodesic Analysis

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, Principal Geodesic Analysis (PGA) finds a low-dimensional embedding by decomposing 2nd-order information in tangent space at an intrinsic mean of the data.

Usage

riem.pga(riemobj, ndim = 2)

Arguments

Value

a named list containing

center

an intrinsic mean in a matrix representation form.

embed

an (N\times ndim) matrix whose rows are embedded observations.

References

Fletcher PT, Lu C, Pizer SM, Joshi S (2004). “Principal Geodesic Analysis for the Study of Nonlinear Statistics of Shape.” IEEE Transactions on Medical Imaging, 23(8), 995–1005. ISSN 0278-0062.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## EMBEDDING WITH MDS AND PGA
embed2mds = riem.mds(myriem, ndim=2, geometry="intrinsic")$embed
embed2pga = riem.pga(myriem, ndim=2)$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2mds, main="Multidimensional Scaling",    col=mylabs, pch=19)
plot(embed2pga, main="Principal Geodesic Analysis", col=mylabs, pch=19)
par(opar)

PHATE

Description

PHATE is a nonlinear manifold learning method that is specifically targeted at improving diffusion maps by incorporating data-adaptive kernel construction, detection of optimal time scale, and information-theoretic metric measures.

Usage

riem.phate(riemobj, ndim = 2, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

References

Moon KR, van Dijk D, Wang Z, Gigante S, Burkhardt DB, Chen WS, Yim K, van den Elzen A, Hirn MJ, Coifman RR, Ivanova NB, Wolf G, Krishnaswamy S (2019). “Visualizing Structure and Transitions in High-Dimensional Biological Data.” Nature Biotechnology, 37(12), 1482–1492. ISSN 1087-0156, 1546-1696.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## PHATE EMBEDDING WITH LOG & SQRT POTENTIAL 
phate_log  = riem.phate(myriem, potential="log")$embed
phate_sqrt = riem.phate(myriem, potential="sqrt")$embed
embed_mds  = riem.mds(myriem)$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
plot(embed_mds,  col=mylabs, pch=19, main="MDS" )
plot(phate_log,  col=mylabs, pch=19, main="PHATE+Log")
plot(phate_sqrt, col=mylabs, pch=19, main="PHATE+Sqrt")
par(opar)

Riemannian Manifold Metric Learning

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M} and corresponding label information, riem.rmml computes pairwise distance of data under Riemannian Manifold Metric Learning (RMML) framework based on equivariant embedding. When the number of data points is not sufficient, an inverse of scatter matrix does not exist analytically so the small regularization parameter \lambda is recommended with default value of \lambda=0.1.

Usage

riem.rmml(riemobj, label, lambda = 0.1, as.dist = FALSE)

Arguments

Value

a S3 dist object or (N\times N) symmetric matrix of pairwise distances according to as.dist parameter.

References

Zhu P, Cheng H, Hu Q, Wang Q, Zhang C (2018). “Towards Generalized and Efficient Metric Learning on Riemannian Manifold.” In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, 3235–3241. ISBN 978-0-9992411-2-7.

Examples

#-------------------------------------------------------------------
#            Distance between Two Classes of SPD Matrices
#
#  Class 1 : Empirical Covariance from Standard Normal Distribution
#  Class 2 : Empirical Covariance from Perturbed 'iris' dataset
#-------------------------------------------------------------------
## DATA GENERATION
data(iris)
ndata  = 10
mydata = list()
for (i in 1:ndata){
  mydata[[i]] = stats::cov(matrix(rnorm(100*4),ncol=4))
}
for (i in (ndata+1):(2*ndata)){
  tmpdata = as.matrix(iris[,1:4]) + matrix(rnorm(150*4,sd=0.5),ncol=4)
  mydata[[i]] = stats::cov(tmpdata)
}
myriem = wrap.spd(mydata)
mylabs = rep(c(1,2), each=ndata)

## COMPUTE GEODESIC AND RMML PAIRWISE DISTANCE
pdgeo = riem.pdist(myriem)
pdmdl = riem.rmml(myriem, label=mylabs)

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
image(pdgeo[,(2*ndata):1], main="geodesic distance", axes=FALSE)
image(pdmdl[,(2*ndata):1], main="RMML distance", axes=FALSE)
par(opar)

Sammon Mapping

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, apply Sammon mapping, a non-linear dimensionality reduction method. Since the method depends only on the pairwise distances of the data, it can be adapted to the manifold-valued data.

Usage

riem.sammon(riemobj, ndim = 2, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

stress

discrepancy between embedded and original distances as a measure of error.

References

Sammon JW (1969). “A Nonlinear Mapping for Data Structure Analysis.” IEEE Transactions on Computers, C-18(5), 401–409. ISSN 0018-9340.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=10)

## COMPARE SAMMON WITH MDS
embed2mds = riem.mds(myriem, ndim=2)$embed
embed2sam = riem.sammon(myriem, ndim=2)$embed

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2mds, col=mylabs, pch=19, main="MDS")
plot(embed2sam, col=mylabs, pch=19, main="Sammon mapping")
par(opar)

Spectral Clustering by Zelnik-Manor and Perona (2005)

Description

Zelnik-Manor and Perona proposed a method to define a set of data-driven bandwidth parameters where \sigma_i is the distance from a point x_i to its nnbd-th nearest neighbor. Then the affinity matrix is defined as

A_{ij} = \exp(-d(x_i, d_j)^2 / \sigma_i \sigma_j)

and the standard spectral clustering of Ng, Jordan, and Weiss (riem.scNJW) is applied.

Usage

riem.sc05Z(riemobj, k = 2, nnbd = 7, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

cluster

a length-N vector of class labels (from 1:k).

eigval

eigenvalues of the graph laplacian's spectral decomposition.

embeds

an (N\times k) low-dimensional embedding.

References

Zelnik-manor L, Perona P (2005). "Self-Tuning Spectral Clustering." In Saul LK, Weiss Y, Bottou L (eds.), Advances in Neural Information Processing Systems 17, 1601–1608. MIT Press.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
lab    = rep(c(1,2,3), each=10)

## CLUSTERING WITH DIFFERENT K VALUES
cl2 = riem.sc05Z(myriem, k=2)$cluster
cl3 = riem.sc05Z(myriem, k=3)$cluster
cl4 = riem.sc05Z(myriem, k=4)$cluster

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,4), pty="s")
plot(mds2d, col=lab, pch=19, main="true label")
plot(mds2d, col=cl2, pch=19, main="riem.sc05Z: k=2")
plot(mds2d, col=cl3, pch=19, main="riem.sc05Z: k=3")
plot(mds2d, col=cl4, pch=19, main="riem.sc05Z: k=4")
par(opar)

Spectral Clustering by Ng, Jordan, and Weiss (2002)

Description

The version of Ng, Jordan, and Weiss first constructs the affinity matrix

A_{ij} = \exp(-d(x_i, d_j)^2 / \sigma^2)

where \sigma is a common bandwidth parameter and performs k-means clustering on the row-space of eigenvectors for the symmetric graph laplacian matrix

L=D^{-1/2}(D-A)D^{-1/2}

.

Usage

riem.scNJW(riemobj, k = 2, sigma = 1, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

cluster

a length-N vector of class labels (from 1:k).

eigval

eigenvalues of the graph laplacian's spectral decomposition.

embeds

an (N\times k) low-dimensional embedding.

References

Ng AY, Jordan MI, Weiss Y (2002). "On Spectral Clustering: Analysis and an Algorithm." In Dietterich TG, Becker S, Ghahramani Z (eds.), Advances in Neural Information Processing Systems 14, 849–856. MIT Press.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
lab    = rep(c(1,2,3), each=10)

## CLUSTERING WITH DIFFERENT K VALUES
cl2 = riem.scNJW(myriem, k=2)$cluster
cl3 = riem.scNJW(myriem, k=3)$cluster
cl4 = riem.scNJW(myriem, k=4)$cluster

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,4), pty="s")
plot(mds2d, col=lab, pch=19, main="true label")
plot(mds2d, col=cl2, pch=19, main="riem.scNJW: k=2")
plot(mds2d, col=cl3, pch=19, main="riem.scNJW: k=3")
plot(mds2d, col=cl4, pch=19, main="riem.scNJW: k=4")
par(opar)

Spectral Clustering by Shi and Malik (2000)

Description

The version of Shi and Malik first constructs the affinity matrix

A_{ij} = \exp(-d(x_i, d_j)^2 / \sigma^2)

where \sigma is a common bandwidth parameter and performs k-means clustering on the row-space of eigenvectors for the random-walk graph laplacian matrix

L=D^{-1}(D-A)

.

Usage

riem.scSM(riemobj, k = 2, sigma = 1, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

cluster

a length-N vector of class labels (from 1:k).

eigval

eigenvalues of the graph laplacian's spectral decomposition.

embeds

an (N\times k) low-dimensional embedding.

References

Shi J, Malik J (2000). “Normalized Cuts and Image Segmentation." IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888–905.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
lab    = rep(c(1,2,3), each=10)

## CLUSTERING WITH DIFFERENT K VALUES
cl2 = riem.scSM(myriem, k=2)$cluster
cl3 = riem.scSM(myriem, k=3)$cluster
cl4 = riem.scSM(myriem, k=4)$cluster

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,4), pty="s")
plot(mds2d, col=lab, pch=19, main="true label")
plot(mds2d, col=cl2, pch=19, main="riem.scSM: k=2")
plot(mds2d, col=cl3, pch=19, main="riem.scSM: k=3")
plot(mds2d, col=cl4, pch=19, main="riem.scSM: k=4")
par(opar)

Spectral Clustering with Unnormalized Laplacian

Description

The version of Shi and Malik first constructs the affinity matrix

A_{ij} = \exp(-d(x_i, d_j)^2 / \sigma^2)

where \sigma is a common bandwidth parameter and performs k-means clustering on the row-space of eigenvectors for the unnormalized graph laplacian matrix

L=D-A

.

Usage

riem.scUL(riemobj, k = 2, sigma = 1, geometry = c("intrinsic", "extrinsic"))

Arguments

Value

a named list containing

cluster

a length-N vector of class labels (from 1:k).

eigval

eigenvalues of the graph laplacian's spectral decomposition.

embeds

an (N\times k) low-dimensional embedding.

References

von Luxburg U (2007). “A Tutorial on Spectral Clustering.” Statistics and Computing, 17(4):395–416.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# class 1 : 10 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 10 perturbed data points near (0,1,0) on S^2 in R^3
# class 3 : 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:10){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 11:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:30){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
lab    = rep(c(1,2,3), each=10)

## CLUSTERING WITH DIFFERENT K VALUES
cl2 = riem.scUL(myriem, k=2)$cluster
cl3 = riem.scUL(myriem, k=3)$cluster
cl4 = riem.scUL(myriem, k=4)$cluster

## MDS FOR VISUALIZATION
mds2d = riem.mds(myriem, ndim=2)$embed

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,4), pty="s")
plot(mds2d, col=lab, pch=19, main="true label")
plot(mds2d, col=cl2, pch=19, main="riem.scUL: k=2")
plot(mds2d, col=cl3, pch=19, main="riem.scUL: k=3")
plot(mds2d, col=cl4, pch=19, main="riem.scUL: k=4")
par(opar)

Find the Smallest Enclosing Ball

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, find the smallest enclosing ball.

Usage

riem.seb(riemobj, method = c("aa2013"), ...)

Arguments

Value

a named list containing

center

a matrix on \mathcal{M} that minimizes the radius.

radius

the minimal radius with respect to the center.

References

Bâdoiu M, Clarkson KL (2003). “Smaller core-sets for balls.” In Proceedings of the fourteenth annual ACM-SIAM symposium on discrete algorithms, SODA '03, 801–802. ISBN 0-89871-538-5.

Arnaudon M, Nielsen F (2013). “On approximating the Riemannian 1-center.” Computational Geometry, 46(1), 93–104. ISSN 09257721.

Examples

#-------------------------------------------------------------------
#       Euclidean Example : samples from Standard Normal in R^2
#-------------------------------------------------------------------
## GENERATE 25 OBSERVATIONS FROM N(0,I)
ndata  = 25
mymats = array(0,c(ndata, 2))
mydata = list()
for (i in 1:ndata){
  mydata[[i]] = stats::rnorm(2)
  mymats[i,]  = mydata[[i]]
}
myriem = wrap.euclidean(mydata)

## COMPUTE
sebobj = riem.seb(myriem)
center = as.vector(sebobj$center) 
radius = sebobj$radius

## VISUALIZE
#  1. prepare the circle for drawing
theta  = seq(from=0, to=2*pi, length.out=100)
coords = radius*cbind(cos(theta), sin(theta))
coords = coords + matrix(rep(center, each=100), ncol=2)

#  2. draw
opar <- par(no.readonly=TRUE)
par(pty="s")
plot(coords, type="l", lwd=2, col="red",
     main="Euclidean SEB", xlab="x", ylab="y")
points(mymats, pch=19)                           # data
points(center[1], center[2], pch=19, col="blue") # center 
par(opar)

Two-Sample Test modified from Biswas and Ghosh (2014)

Description

Given M observations X_1, X_2, \ldots, X_M \in \mathcal{M} and N observations Y_1, Y_2, \ldots, Y_N \in \mathcal{M}, perform the permutation test of equal distribution

H_0~:~\mathcal{P}_X = \mathcal{P}_Y

by the method from Biswas and Ghosh (2014). The method, originally proposed for Euclidean-valued data, is adapted to the general Riemannian manifold with intrinsic/extrinsic distance.

Usage

riem.test2bg14(riemobj1, riemobj2, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

p-value under H_0.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Biswas M, Ghosh AK (2014). “A nonparametric two-sample test applicable to high dimensional data.” Journal of Multivariate Analysis, 123, 160–171. ISSN 0047259X.

You K, Park H (2020). “Re-visiting Riemannian geometry of symmetric positive definite matrices for the analysis of functional connectivity.” NeuroImage, 117464. ISSN 10538119.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with two types
#
# class 1 : 20 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 30 perturbed data points near (0,1,0) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata1 = list()
mydata2 = list()
for (i in 1:20){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata1[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 1:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem1 = wrap.sphere(mydata1)
myriem2 = wrap.sphere(mydata2)

## PERFORM PERMUTATION TEST
#  it is expected to return a very small number.

riem.test2bg14(myriem1, myriem2, nperm=999)


## Not run: 
## CHECK WITH EMPIRICAL TYPE-1 ERROR
set.seed(777)
ntest = 1000
pvals = rep(0,ntest)

for (i in 1:ntest){
  X = cbind(matrix(rnorm(30*2, sd=0.1),ncol=2), rep(1,30))
  Y = cbind(matrix(rnorm(30*2, sd=0.1),ncol=2), rep(1,30))
  Xnorm = X/sqrt(rowSums(X^2))
  Ynorm = Y/sqrt(rowSums(Y^2))
  
  Xriem = wrap.sphere(Xnorm)
  Yriem = wrap.sphere(Ynorm)
  pvals[i] = riem.test2bg14(Xriem, Yriem, nperm=999)$p.value
}

emperr = round(sum((pvals <= 0.05))/ntest, 5)
print(paste0("* EMPIRICAL TYPE-1 ERROR=", emperr))

## End(Not run)

Two-Sample Test with Wasserstein Metric

Description

Given M observations X_1, X_2, \ldots, X_M \in \mathcal{M} and N observations Y_1, Y_2, \ldots, Y_N \in \mathcal{M}, permutation test based on the Wasserstein metric (see riem.wasserstein for more details) is applied to test whether two distributions are same or not, i.e.,

H_0~:~\mathcal{P}_X = \mathcal{P}_Y

with Wasserstein metric \mathcal{W}_p being the measure of discrepancy between two samples.

Usage

riem.test2wass(
  riemobj1,
  riemobj2,
  p = 2,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

p-value under H_0.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with two types
#
# class 1 : 20 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 30 perturbed data points near (0,1,0) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata1 = list()
mydata2 = list()
for (i in 1:20){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata1[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 1:20){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem1 = wrap.sphere(mydata1)
myriem2 = wrap.sphere(mydata2)

## PERFORM PERMUTATION TEST
#  it is expected to return a very small number, but 
#  small number of 'nperm' may not give a reasonable p-value.

riem.test2wass(myriem1, myriem2, nperm=99, use.smooth=FALSE)


## Not run: 
## CHECK WITH EMPIRICAL TYPE-1 ERROR
set.seed(777)
ntest = 1000
pvals = rep(0,ntest)

for (i in 1:ntest){
  X = cbind(matrix(rnorm(30*2, sd=0.1),ncol=2), rep(1,30))
  Y = cbind(matrix(rnorm(30*2, sd=0.1),ncol=2), rep(1,30))
  Xnorm = X/sqrt(rowSums(X^2))
  Ynorm = Y/sqrt(rowSums(Y^2))
  
  Xriem = wrap.sphere(Xnorm)
  Yriem = wrap.sphere(Ynorm)
  pvals[i] = riem.test2wass(Xriem, Yriem, nperm=999)$p.value
  print(paste0("iteration ",i,"/",ntest," complete.."))
}

emperr = round(sum((pvals <= 0.05))/ntest, 5)
print(paste0("* EMPIRICAL TYPE-1 ERROR=", emperr))

## End(Not run)

t-distributed Stochastic Neighbor Embedding

Description

Given N observations X_1, X_2, \ldots, X_N \in \mathcal{M}, t-SNE mimicks the pattern of probability distributions over pairs of manifold-valued objects on low-dimensional target embedding space by minimizing Kullback-Leibler divergence.

Usage

riem.tsne(riemobj, ndim = 2, geometry = c("intrinsic", "extrinsic"), ...)

Arguments

Value

a named list containing

embed

an (N\times ndim) matrix whose rows are embedded observations.

stress

discrepancy between embedded and original distances as a measure of error.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with three types
#
# 10 perturbed data points near (1,0,0) on S^2 in R^3
# 10 perturbed data points near (0,1,0) on S^2 in R^3
# 10 perturbed data points near (0,0,1) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata = list()
for (i in 1:20){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 21:40){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 41:60){
  tgt = c(stats::rnorm(2, sd=0.1), 1)
  mydata[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem = wrap.sphere(mydata)
mylabs = rep(c(1,2,3), each=20)

## RUN THE ALGORITHM IN TWO GEOMETRIES
mypx = 5
embed2int = riem.tsne(myriem, ndim=2, geometry="intrinsic", perplexity=mypx)
embed2ext = riem.tsne(myriem, ndim=2, geometry="extrinsic", perplexity=mypx)

## VISUALIZE
opar = par(no.readonly=TRUE)
par(mfrow=c(1,2), pty="s")
plot(embed2int$embed, main="intrinsic t-SNE", col=mylabs, pch=19)
plot(embed2ext$embed, main="extrinsic t-SNE", col=mylabs, pch=19)
par(opar)

Wasserstein Distance between Empirical Measures

Description

Given two empirical measures \mu, \nu consisting of M and N observations, p-Wasserstein distance for p\geq 1 between two empirical measures is defined as

\mathcal{W}_p (\mu, \nu) = \left( \inf_{\gamma \in \Gamma(\mu, \nu)} \int_{\mathcal{M}\times \mathcal{M}} d(x,y)^p d \gamma(x,y) \right)^{1/p}

where \Gamma(\mu, \nu) denotes the collection of all measures/couplings on \mathcal{M}\times \mathcal{M} whose marginals are \mu and \nu on the first and second factors, respectively.

Usage

riem.wasserstein(
  riemobj1,
  riemobj2,
  p = 2,
  geometry = c("intrinsic", "extrinsic"),
  ...
)

Arguments

Value

a named list containing

distance

\mathcal{W_p} distance between two empirical measures.

plan

an (M\times N) matrix whose rowSums and columnSums are weight1 and weight2 respectively.

Examples

#-------------------------------------------------------------------
#          Example on Sphere : a dataset with two types
#
# class 1 : 20 perturbed data points near (1,0,0) on S^2 in R^3
# class 2 : 30 perturbed data points near (0,1,0) on S^2 in R^3
#-------------------------------------------------------------------
## GENERATE DATA
mydata1 = list()
mydata2 = list()
for (i in 1:20){
  tgt = c(1, stats::rnorm(2, sd=0.1))
  mydata1[[i]] = tgt/sqrt(sum(tgt^2))
}
for (i in 1:30){
  tgt = c(rnorm(1,sd=0.1),1,rnorm(1,sd=0.1))
  mydata2[[i]] = tgt/sqrt(sum(tgt^2))
}
myriem1 = wrap.sphere(mydata1)
myriem2 = wrap.sphere(mydata2)

## COMPUTE p-WASSERSTEIN DISTANCES
dist1 = riem.wasserstein(myriem1, myriem2, p=1)
dist2 = riem.wasserstein(myriem1, myriem2, p=2)
dist5 = riem.wasserstein(myriem1, myriem2, p=5)

pm1 = paste0("p=1: dist=",round(dist1$distance,3))
pm2 = paste0("p=2: dist=",round(dist2$distance,3))
pm5 = paste0("p=5: dist=",round(dist5$distance,3))

## VISUALIZE TRANSPORT PLAN AND DISTANCE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3))
image(dist1$plan, axes=FALSE, main=pm1)
image(dist2$plan, axes=FALSE, main=pm2)
image(dist5$plan, axes=FALSE, main=pm5)
par(opar)

Generate Random Samples from Multivariate Normal Distribution

Description

In \mathbf{R}^p, random samples are drawn

X_1,X_2,\ldots,X_n~ \sim ~ \mathcal{N}(\mu, \Sigma)

where \mu \in \mathbf{R}^p is a mean vector and \Sigma \in \textrm{SPD}(p) is a positive definite covariance matrix.

Usage

rmvnorm(n = 1, mu, sigma)

Arguments

Value

either (1) a length-p vector (n=1) or (2) an (n\times p) matrix where rows are random samples.

Examples

#-------------------------------------------------------------------
#   Generate Random Data and Compare with Empirical Covariances
#
# In R^5 with zero mean and diagonal covariance, 
# generate 100 and 200 observations and compute MLE covariance.
#-------------------------------------------------------------------
## GENERATE DATA
mymu  = rep(0,5)
mysig = diag(5)

## MLE FOR COVARIANCE
smat1 = stats::cov(rmvnorm(n=100, mymu, mysig))
smat2 = stats::cov(rmvnorm(n=200, mymu, mysig))

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(mysig[,5:1], axes=FALSE, main="true covariance")
image(smat1[,5:1], axes=FALSE, main="empirical cov with n=100")
image(smat2[,5:1], axes=FALSE, main="empirical cov with n=200")
par(opar)

Supported Geometries on SPD Manifold

Description

SPD manifold is a well-studied space in that there have been many geometries proposed on the space. For special functions on under SPD category, this function finds whether there exists a matching name that is currently supported in Riemann. If there is none, it will return an error message.

Usage

spd.geometry(geometry)

Arguments

Value

a matching name in lower-case.

Examples

# it just returns a small-letter string.
mygeom = spd.geometry("stein")

Pairwise Distance on SPD Manifold

Description

Given N observations X_1, X_2, \ldots, X_N in SPD manifold, compute pairwise distances among observations.

Usage

spd.pdist(spdobj, geometry, as.dist = FALSE)

Arguments

Value

a S3 dist object or (N\times N) symmetric matrix of pairwise distances according to as.dist parameter.

Examples

#-------------------------------------------------------------------
#                   Two Types of Covariances
#
#  group1 : perturbed from data by N(0,1) in R^3
#  group2 : perturbed from data by [sin(x); cos(x); sin(x)*cos(x)]
#-------------------------------------------------------------------
## GENERATE DATA
spd_mats = array(0,c(3,3,20))
for (i in 1:10){
  spd_mats[,,i] = stats::cov(matrix(rnorm(50*3), ncol=3))
}
for (j in 11:20){
  randvec = stats::rnorm(50, sd=3)
  randmat = cbind(sin(randvec), cos(randvec), sin(randvec)*cos(randvec))
  spd_mats[,,j] = stats::cov(randmat + matrix(rnorm(50*3, sd=0.1), ncol=3))
}

## WRAP IT AS SPD OBJECT
spd_obj = wrap.spd(spd_mats)

## COMPUTE PAIRWISE DISTANCES
#  Geometries are case-insensitive.
pdA = spd.pdist(spd_obj, "airM")
pdL = spd.pdist(spd_obj, "lErm")
pdJ = spd.pdist(spd_obj, "Jeffrey")
pdS = spd.pdist(spd_obj, "stEin")
pdW = spd.pdist(spd_obj, "wasserstein")

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(2,3), pty="s")
image(pdA, axes=FALSE, main="AIRM")
image(pdL, axes=FALSE, main="LERM")
image(pdJ, axes=FALSE, main="Jeffrey")
image(pdS, axes=FALSE, main="Stein")
image(pdW, axes=FALSE, main="Wasserstein")
par(opar)

Wasserstein Barycenter of SPD Matrices

Description

Given N observations X_1, X_2, \ldots, X_N in SPD manifold, compute the L_2-Wasserstein barycenter that minimizes

\sum_{n=1}^N \lambda_i \mathcal{W}_2 (N(X), N(X_i))^2

where N(X) denotes the zero-mean Gaussian measure with covariance X.

Usage

spd.wassbary(spdobj, weight = NULL, method = c("RU02", "AE16"), ...)

Arguments

Value

a (p\times p) Wasserstein barycenter matrix.

Examples

#-------------------------------------------------------------------
#        Covariances from standard multivariate Gaussians.
#-------------------------------------------------------------------
## GENERATE DATA
ndata = 20
pdim  = 10
mydat = array(0,c(pdim,pdim,ndata))
for (i in 1:ndata){
  mydat[,,i] = stats::cov(matrix(rnorm(100*pdim), ncol=pdim))
}
myriem = wrap.spd(mydat)

## COMPUTE BY DIFFERENT ALGORITHMS
baryRU <- spd.wassbary(myriem, method="RU02")
baryAE <- spd.wassbary(myriem, method="AE16")

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,3), pty="s")
image(diag(pdim), axes=FALSE, main="True Covariance")
image(baryRU, axes=FALSE, main="by RU02")
image(baryAE, axes=FALSE, main="by AE16")
par(opar)

Convert between Cartesian Coordinates and Geographic Coordinates

Description

In geospatial data analysis, it is common to consider locations on the Earth as data. These locations, usually provided by latitude and longitude, are not directly applicable for spherical data analysis. We provide two functions - sphere.geo2xyz and sphere.xyz2geo - that convert geographic coordinates in longitude/latitude into a unit-norm vector on \mathcal{S}^2, and vice versa. As a convention, latitude and longitude are represented as decimal degrees.

Usage

sphere.geo2xyz(lat, lon)

sphere.xyz2geo(xyz)

Arguments

Value

transformed data.

Examples

## EXAMPLE DATA WITH POPULATED US CITIES
data(cities)

## SELECT ALBUQUERQUE
geo = cities$coord[1,]
xyz = cities$cartesian[1,]

## CHECK TWO INPUT TYPES AND THEIR CONVERSIONS
sphere.geo2xyz(geo[1], geo[2])
sphere.xyz2geo(xyz)

Generate Uniform Samples on Sphere

Description

It generates n random samples from \mathcal{S}^{p-1}. For convenient usage of users, we provide a number of options in terms of the return type.

Usage

sphere.runif(n, p, type = c("list", "matrix", "riemdata"))

Arguments

Value

an object from one of the above by type option.

References

Chikuse Y (2003). Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-00160-9 978-0-387-21540-2.

See Also

wrap.sphere

Examples

#-------------------------------------------------------------------
#                       Draw Samples on Sphere
#
# Multiple return types on S^4 in R^5
#-------------------------------------------------------------------
dat.list = sphere.runif(n=10, p=5, type="list")
dat.matx = sphere.runif(n=10, p=5, type="matrix")
dat.riem = sphere.runif(n=10, p=5, type="riemdata")

Test of Uniformity on Sphere

Description

Given N observations \lbrace X_1, X_2, \ldots, X_M \brace on \mathcal{S}^{p-1}, it tests whether the data is distributed uniformly on the sphere.

Usage

sphere.utest(spobj, method = c("Rayleigh", "RayleighM"))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

p-value under H_0.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Chikuse Y (2003). Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-00160-9 978-0-387-21540-2.

Mardia KV, Jupp PE (eds.) (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley \& Sons, Inc., Hoboken, NJ, USA. ISBN 978-0-470-31697-9 978-0-471-95333-3.

See Also

wrap.sphere

Examples

#-------------------------------------------------------------------
#   Compare Rayleigh's original and modified versions of the test
#-------------------------------------------------------------------
#  Data Generation
myobj = sphere.runif(n=100, p=5, type="riemdata")

#  Compare 2 versions : Original vs Modified Rayleigh
sphere.utest(myobj, method="rayleigh")
sphere.utest(myobj, method="rayleighm")

Spherical Laplace Distribution

Description

This is a collection of tools for learning with spherical Laplace (SL) distribution on a (p-1)-dimensional sphere in \mathbf{R}^p including sampling, density evaluation, and maximum likelihood estimation of the parameters. The SL distribution is characterized by the following density function,

f_{SL}(x; \mu, \sigma) = \frac{1}{C(\sigma)} \exp \left( -\frac{d(x,\mu)}{\sigma} \right)

for location and scale parameters \mu and \sigma respectively.

Usage

dsplaplace(data, mu, sigma, log = FALSE)

rsplaplace(n, mu, sigma)

mle.splaplace(data, method = c("DE", "Optimize", "Newton"), ...)

Arguments

Value

dsplaplace gives a vector of evaluated densities given samples. rsplaplace generates unit-norm vectors in \mathbf{R}^p wrapped in a list. mle.splaplace computes MLEs and returns a list containing estimates of location (mu) and scale (sigma) parameters.

Examples

# -------------------------------------------------------------------
#          Example with Spherical Laplace Distribution
#
# Given a fixed set of parameters, generate samples and acquire MLEs.
# Especially, we will see the evolution of estimation accuracy.
# -------------------------------------------------------------------
## DEFAULT PARAMETERS
true.mu  = c(1,0,0,0,0)
true.sig = 1

## GENERATE A RANDOM SAMPLE OF SIZE N=1000
big.data = rsplaplace(1000, true.mu, true.sig)

## ITERATE FROM 50 TO 1000 by 10
idseq = seq(from=50, to=1000, by=10)
nseq  = length(idseq)

hist.mu  = rep(0, nseq)
hist.sig = rep(0, nseq)

for (i in 1:nseq){
  small.data = big.data[1:idseq[i]]             # data subsetting
  small.MLE  = mle.splaplace(small.data)        # compute MLE
  
  hist.mu[i]  = acos(sum(small.MLE$mu*true.mu)) # difference in mu
  hist.sig[i] = small.MLE$sigma
}

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(idseq, hist.mu,  "b", pch=19, cex=0.5, 
     main="difference in location", xlab="sample size")
plot(idseq, hist.sig, "b", pch=19, cex=0.5, 
     main="scale parameter", xlab="sample size")
abline(h=true.sig, lwd=2, col="red")
par(opar)

Spherical Normal Distribution

Description

We provide tools for an isotropic spherical normal (SN) distributions on a (p-1)-sphere in \mathbf{R}^p for sampling, density evaluation, and maximum likelihood estimation of the parameters where the density is defined as

f_{SN}(x; \mu, \lambda) = \frac{1}{Z(\lambda)} \exp \left( -\frac{\lambda}{2} d^2(x,\mu) \right)

for location and concentration parameters \mu and \lambda respectively and the normalizing constant Z(\lambda).

Usage

dspnorm(data, mu, lambda, log = FALSE)

rspnorm(n, mu, lambda)

mle.spnorm(data, method = c("Newton", "Halley", "Optimize", "DE"), ...)

Arguments

Value

dspnorm gives a vector of evaluated densities given samples. rspnorm generates unit-norm vectors in \mathbf{R}^p wrapped in a list. mle.spnorm computes MLEs and returns a list containing estimates of location (mu) and concentration (lambda) parameters.

References

Hauberg S (2018). “Directional Statistics with the Spherical Normal Distribution.” In 2018 21st International Conference on Information Fusion (FUSION), 704–711. ISBN 978-0-9964527-6-2.

You K, Suh C (2022). “Parameter Estimation and Model-Based Clustering with Spherical Normal Distribution on the Unit Hypersphere.” Computational Statistics \& Data Analysis, 107457. ISSN 01679473.

Examples

# -------------------------------------------------------------------
#          Example with Spherical Normal Distribution
#
# Given a fixed set of parameters, generate samples and acquire MLEs.
# Especially, we will see the evolution of estimation accuracy.
# -------------------------------------------------------------------
## DEFAULT PARAMETERS
true.mu  = c(1,0,0,0,0)
true.lbd = 5

## GENERATE DATA N=1000
big.data = rspnorm(1000, true.mu, true.lbd)

## ITERATE FROM 50 TO 1000 by 10
idseq = seq(from=50, to=1000, by=10)
nseq  = length(idseq)

hist.mu  = rep(0, nseq)
hist.lbd = rep(0, nseq)

for (i in 1:nseq){
  small.data = big.data[1:idseq[i]]          # data subsetting
  small.MLE  = mle.spnorm(small.data) # compute MLE
  
  hist.mu[i]  = acos(sum(small.MLE$mu*true.mu)) # difference in mu
  hist.lbd[i] = small.MLE$lambda
}

## VISUALIZE
opar <- par(no.readonly=TRUE)
par(mfrow=c(1,2))
plot(idseq, hist.mu,  "b", pch=19, cex=0.5, main="difference in location")
plot(idseq, hist.lbd, "b", pch=19, cex=0.5, main="concentration param")
abline(h=true.lbd, lwd=2, col="red")
par(opar)

Simulated Annealing on Stiefel Manifold

Description

Simulated Annealing is a black-box, derivative-free optimization algorithm that iterates via stochastic search in the neighborhood of current position. stiefel.optSA solves the following problem

\min_X f(X),\quad X \in St(p,k)

without any other auxiliary information such as gradient or hessian involved.

Usage

stiefel.optSA(func, p, k, ...)

Arguments

Value

a named list containing:

cost

minimized function value.

solution

a (p\times k) matrix that attains the cost.

accfreq

frequency of acceptance moves.

Examples

#-------------------------------------------------------------------
#               Optimization for Eigen-Decomposition
#
# Given (5x5) covariance matrix S, eigendecomposition is indeed 
# an optimization problem cast on the stiefel manifold. Here, 
# we are trying to find top 3 eigenvalues and compare.
#-------------------------------------------------------------------
## PREPARE
set.seed(121)                         # set seed
A = cov(matrix(rnorm(100*5), ncol=5)) # define covariance
myfunc <- function(p){                # cost function to minimize
  return(sum(-diag(t(p)%*%A%*%p)))
} 

## SOLVE THE OPTIMIZATION PROBLEM
Aout = stiefel.optSA(myfunc, p=5, k=3, n.start=40, maxiter=200)

## COMPUTE EIGENVALUES
#  1. USE SOLUTIONS TO THE ABOVE OPTIMIZATION 
abase   = Aout$solution
eig3sol = sort(diag(t(abase)%*%A%*%abase), decreasing=TRUE)

#  2. USE BASIC 'EIGEN' FUNCTION
eig3dec = sort(eigen(A)$values, decreasing=TRUE)[1:3]

## VISUALIZE
opar <- par(no.readonly=TRUE)
yran = c(min(min(eig3sol),min(eig3dec))*0.95,
         max(max(eig3sol),max(eig3dec))*1.05)
plot(1:3, eig3sol, type="b", col="red",  pch=19, ylim=yran,
     xlab="index", ylab="eigenvalue", main="compare top 3 eigenvalues")
lines(1:3, eig3dec, type="b", col="blue", pch=19)
legend(1, 1, legend=c("optimization","decomposition"), col=c("red","blue"),
       lty=rep(1,2), pch=19)
par(opar)

Generate Uniform Samples on Stiefel Manifold

Description

It generates n random samples from Stiefel manifold St(k,p).

Usage

stiefel.runif(n, k, p, type = c("list", "array", "riemdata"))

Arguments

Value

an object from one of the above by type option.

References

Chikuse Y (2003). Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-00160-9 978-0-387-21540-2.

See Also

wrap.stiefel

Examples

#-------------------------------------------------------------------
#                 Draw Samples on Stiefel Manifold 
#
# Try Different Return Types with 3 Observations of 5-frames in R^10
#-------------------------------------------------------------------
#  GENERATION
dat.list = stiefel.runif(n=3, k=5, p=10, type="list")
dat.arr3 = stiefel.runif(n=3, k=5, p=10, type="array")
dat.riem = stiefel.runif(n=3, k=5, p=10, type="riemdata")

Test of Uniformity on Stiefel Manifold

Description

Given the data on Stiefel manifold St(k,p), it tests whether the data is distributed uniformly.

Usage

stiefel.utest(stobj, method = c("Rayleigh", "RayleighM"))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

p-value under H_0.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Chikuse Y (2003). Statistics on Special Manifolds, volume 174 of Lecture Notes in Statistics. Springer New York, New York, NY. ISBN 978-0-387-00160-9 978-0-387-21540-2.

Mardia KV, Jupp PE (eds.) (1999). Directional Statistics, Wiley Series in Probability and Statistics. John Wiley \& Sons, Inc., Hoboken, NJ, USA. ISBN 978-0-470-31697-9 978-0-471-95333-3.

See Also

wrap.stiefel

Examples

#-------------------------------------------------------------------
#   Compare Rayleigh's original and modified versions of the test
# 
# Test 1. sample uniformly from St(2,4)
# Test 2. use perturbed principal components from 'iris' data in R^4
#         which is concentrated around a point to reject H0.
#-------------------------------------------------------------------
## DATA GENERATION
#  1. uniform data
myobj1 = stiefel.runif(n=100, k=2, p=4)

#  2. perturbed principal components
data(iris)
irdat = list()
for (n in 1:100){
   tmpdata    = iris[1:50,1:4] + matrix(rnorm(50*4,sd=0.5),ncol=4)
   irdat[[n]] = eigen(cov(tmpdata))$vectors[,1:2]
}
myobj2 = wrap.stiefel(irdat)

## TEST
#  1. uniform data
stiefel.utest(myobj1, method="Rayleigh")
stiefel.utest(myobj1, method="RayleighM")

#  2. concentrated data
stiefel.utest(myobj2, method="rayleIgh")   # method names are 
stiefel.utest(myobj2, method="raYleiGhM")  # CASE - INSENSITIVE !

Prepare Data on Correlation Manifold

Description

The collection of correlation matrices is considered as a subset (and quotient) of the well-known SPD manifold. In our package, it is defined as

\mathcal{C}_{++}^p = \lbrace X \in \mathbf{R}^{p\times p} ~\vert~ X^\top = X,~ \textrm{rank}(X)=p,~ \textrm{diag}(X) = 1 \rbrace

where the rank condition means it is strictly positive definite. Please note that the geometry involving semi-definite correlation matrices is not the objective here.

Usage

wrap.correlation(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times p) correlation matrices.

size

size of each correlation matrix.

name

name of the manifold of interests, "correlation"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  5 observations; empirical correlation of normal observations.
#-------------------------------------------------------------------
#  Data Generation
d1 = array(0,c(3,3,5))
d2 = list()
for (i in 1:5){
  dat = matrix(rnorm(10*3),ncol=3)
  d1[,,i] = stats::cor(dat)
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.correlation(d1)
test2 = wrap.correlation(d2)

Prepare Data on Euclidean Space

Description

Euclidean space \mathbf{R}^p is the most common space for data analysis, which can be considered as a Riemannian manifold with flat metric. Since the space of matrices is isomorphic to Euclidean space after vectorization, we consider the inputs as p-dimensional vectors.

Usage

wrap.euclidean(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times 1) matrices in \mathbf{R}^p.

size

dimension of the ambient space.

name

name of the manifold of interests, "euclidean"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations in R^3 in Matrix and List.
#-------------------------------------------------------------------
## DATA GENERATION
d1 = array(0,c(5,3))
d2 = list()
for (i in 1:5){
  single  = stats::rnorm(3)
  d1[i,]  = single
  d2[[i]] = single
}

## RUN
test1 = wrap.euclidean(d1)
test2 = wrap.euclidean(d2)

Prepare Data on Grassmann Manifold

Description

Grassmann manifold Gr(k,p) is the set of k-planes, or k-dimensional subspaces in R^p, which means that for a given matrix Y \in \mathbf{R}{p\times k}, the column space SPAN(Y) is an element in Grassmann manifold. We use a convention that each element in Gr(k,p) is represented as an orthonormal basis (ONB) X \in \mathbf{R}^{p\times k} where

X^\top X = I_k.

If not provided in such a form, this wrapper takes a QR decomposition of the given data to recover a corresponding ONB.

Usage

wrap.grassmann(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of k-subspace basis matrices.

size

size of each k-subspace basis matrix.

name

name of the manifold of interests, "grassmann"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations in Gr(2,4)
#-------------------------------------------------------------------
#  Generation
d1 = array(0,c(4,2,5))
d2 = list()
for (i in 1:5){
  d1[,,i] = matrix(rnorm(4*2), ncol=2)
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.grassmann(d1)
test2 = wrap.grassmann(d2)

Wrap Landmark Data on Shape Space

Description

One of the frameworks used in shape space is to represent the data as landmarks. Each shape is a point set of k points in \mathbf{R}^p where each point is a labeled object. We consider general landmarks in p=2,3,\ldots. Note that when p > 2, it is stratified space but we assume singularities do not exist or are omitted. The wrapper takes translation and scaling out from the data to make it preshape (centered, unit-norm). Also, for convenience, orthogonal Procrustes analysis is applied with the first observation being the reference so that all the other data are rotated to match the shape of the first.

Usage

wrap.landmark(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of preshapes in \mathbf{R}^p.

size

size of each preshape.

name

name of the manifold of interests, "landmark"

References

Dryden IL, Mardia KV (2016). Statistical shape analysis with applications in R, Wiley series in probability and statistics, Second edition edition. John Wiley \& Sons, Chichester, UK ; Hoboken, NJ. ISBN 978-1-119-07251-5 978-1-119-07250-8.

Examples

## USE 'GORILLA' DATA
data(gorilla)
riemobj = wrap.landmark(gorilla$male)

Prepare Data on Multinomial Manifold

Description

Multinomial manifold is referred to the data that is nonnegative and sums to 1. Also known as probability simplex or positive orthant, we denote (p-1) simplex in \mathbf{R}^p by

\Delta^{p-1} = \lbrace x \in \mathbf{R}^p~\vert~ \sum_{i=1}^p x_i = 1, x_i > 0 \rbrace

in that data are positive L_1 unit-norm vectors. In wrap.multinomial, normalization is applied when each data point is not on the simplex, but if vectors contain values not in (0,1), it returns errors.

Usage

wrap.multinomial(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times 1) matrices in \Delta^{p-1}.

size

dimension of the ambient space.

name

name of the manifold of interests, "multinomial"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#-------------------------------------------------------------------
## DATA GENERATION
d1 = array(0,c(5,3))
d2 = list()
for (i in 1:5){
  single  = abs(stats::rnorm(3))
  d1[i,]  = single
  d2[[i]] = single
}

## RUN
test1 = wrap.multinomial(d1)
test2 = wrap.multinomial(d2)

Prepare Data on Rotation Group

Description

Rotation group, also known as special orthogonal group, is a Riemannian manifold

SO(p) = \lbrace Q \in \mathbf{R}^{p\times p}~\vert~ Q^\top Q = I, \textrm{det}(Q)=1 \rbrace

where the name originates from an observation that when p=2,3 these matrices are rotation of shapes/configurations.

Usage

wrap.rotation(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times p) rotation matrices.

size

size of each rotation matrix.

name

name of the manifold of interests, "rotation"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#-------------------------------------------------------------------
## DATA GENERATION
d1 = array(0,c(3,3,5))
d2 = list()
for (i in 1:5){
  single  = qr.Q(qr(matrix(rnorm(9),nrow=3)))
  d1[,,i] = single
  d2[[i]] = single
}

## RUN
test1 = wrap.rotation(d1)
test2 = wrap.rotation(d2)

Prepare Data on Symmetric Positive-Definite (SPD) Manifold

Description

The collection of symmetric positive-definite matrices is a well-known example of matrix manifold. It is defined as

\mathcal{S}_{++}^p = \lbrace X \in \mathbf{R}^{p\times p} ~\vert~ X^\top = X,~ \textrm{rank}(X)=p \rbrace

where the rank condition means it is strictly positive definite. Please note that the geometry involving semi-definite matrices is considered in wrap.spdk.

Usage

wrap.spd(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times p) SPD matrices.

size

size of each SPD matrix.

name

name of the manifold of interests, "spd"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations; empirical covariance of normal observations.
#-------------------------------------------------------------------
#  Data Generation
d1 = array(0,c(3,3,5))
d2 = list()
for (i in 1:5){
  dat = matrix(rnorm(10*3),ncol=3)
  d1[,,i] = stats::cov(dat)
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.spd(d1)
test2 = wrap.spd(d2)

Prepare Data on SPD Manifold of Fixed-Rank

Description

When (p\times p) SPD matrices are of fixed-rank k < p, they form a geometric structure represented by (p\times k) matrices,

SPD(k,p) = \lbrace X \in \mathbf{R}^{(p\times p)}~\vert~ Y Y^\top = X, \textrm{rank}(X) = k \rbrace

It's key difference from \mathcal{S}_{++}^p is that all matrices should be of fixed rank k where k is usually smaller than p. Inputs are given as (p\times p) matrices with specified k and wrap.spdk automatically decomposes input square matrices into rank-k representation matrices.

Usage

wrap.spdk(input, k)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times k) representation of the corresponding rank-k SPSD matrices.

size

size of each representation matrix.

name

name of the manifold of interests, "spdk"

References

Journée M, Bach F, Absil P, Sepulchre R (2010). “Low-rank optimization on the cone of positive semidefinite matrices.” SIAM Journal on Optimization, 20(5), 2327–2351.

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#-------------------------------------------------------------------
#  Data Generation
d1 = array(0,c(10,10,3))
d2 = list()
for (i in 1:3){
  dat = matrix(rnorm(10*10),ncol=10)
  d1[,,i] = stats::cov(dat)
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.spdk(d1, k=2)
test2 = wrap.spdk(d2, k=2)

Prepare Data on Sphere

Description

The unit hypersphere (sphere, for short) is one of the most fundamental curved space in studying geometry. Precisely, we denote (p-1) sphere in \mathbf{R}^p by

\mathcal{S}^{p-1} = \lbrace x \in \mathbf{R}^p ~ \vert ~ x^\top x = \|x\|^2 = 1 \rbrace

where vectors are of unit norm. In wrap.sphere, normalization is applied when each data point is not on the unit sphere.

Usage

wrap.sphere(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of (p\times 1) matrices in \mathcal{S}^{p-1}.

size

dimension of the ambient space.

name

name of the manifold of interests, "sphere"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations in S^2 embedded in R^3.
#-------------------------------------------------------------------
## DATA GENERATION
d1 = array(0,c(5,3))
d2 = list()
for (i in 1:5){
  single  = stats::rnorm(3)
  d1[i,]  = single
  d2[[i]] = single
}

## RUN
test1 = wrap.sphere(d1)
test2 = wrap.sphere(d2)

Prepare Data on (Compact) Stiefel Manifold

Description

Stiefel manifold St(k,p) is the set of k-frames in \mathbf{R}^p, which is indeed a Riemannian manifold. For usage in Riemann package, each data point is represented as a matrix by the convention

St(k,p) = \lbrace X \in \mathbf{R}^{p\times k} ~\vert~ X^\top X = I_k \rbrace

which means that columns are orthonormal. When the provided matrix is not an orthonormal basis as above, wrap.stiefel applies orthogonalization to extract valid basis information.

Usage

wrap.stiefel(input)

Arguments

Value

a named riemdata S3 object containing

data

a list of k-frame orthonormal matrices.

size

size of each k-frame basis matrix.

name

name of the manifold of interests, "stiefel"

Examples

#-------------------------------------------------------------------
#                 Checker for Two Types of Inputs
#
#  Generate 5 observations in St(2,4)
#-------------------------------------------------------------------
#  Data Generation by QR Decomposition
d1 = array(0,c(4,2,5))
d2 = list()
for (i in 1:5){
  d1[,,i] = qr.Q(qr(matrix(rnorm(4*2),ncol=2)))
  d2[[i]] = d1[,,i]
}

#  Run
test1 = wrap.stiefel(d1)
test2 = wrap.stiefel(d2)