Finite Mixture of Spherical Laplace Distributions

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

data

data vectors in form of either an $(n\times p)$ matrix or a length-$n$ list. See wrap.sphere for descriptions on supported input types.

k

the number of clusters (default: 2).

same.sigma

a logical; TRUE to use same scale parameter across all components, or FALSE otherwise.

variants

type of the class assignment methods, one of "soft","hard", and "stochastic".

...

extra parameters including

maxiter: the maximum number of iterations (default: 50).
eps: stopping criterion for the EM algorithm (default: 1e-6).
printer: a logical; TRUE to show history of the algorithm, FALSE otherwise.

object

a fitted moSL model from the moSL function.

newdata

data vectors in form of either an $(m\times p)$ matrix or a length-$m$ list. See wrap.sphere for descriptions on supported input types.

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.

Validation status

This retained legacy interface is experimental. Its full numerical and statistical contract has not been independently verified across supported inputs. See riem-method-contracts and the installed contract table for method-specific assumptions, restrictions, and evidence scope.

Examples

# \donttest{
# ---------------------------------------------------- #
#                 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))
#> [1] "Log-likelihood for K=3 fitted    : 89.23077"
print(paste0("Log-likelihood for K=3 predicted : ", newloglkd))
#> [1] "Log-likelihood for K=3 predicted : 89.23077"

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

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