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em-mixture-mv.Rmd
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em-mixture-mv.Rmd
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## Multivariate Mixture Model
The following code is based on algorithms noted in Murphy, 2012 Probabilistic Machine Learning.
Specifically, Chapter 11, section 4.
### Function
This estimating function will be used for both examples.
```{r em_mixture-mv}
em_mixture <- function(
params,
X,
clusters = 2,
tol = .00001,
maxits = 100,
showits = TRUE
) {
# Arguments are
# params: starting parameters (means, covariances, cluster probability)
# X: data
# clusters: number of clusters desired
# tol: tolerance
# maxits: maximum iterations
# showits: whether to show iterations
require(mvtnorm)
# Starting points
N = nrow(X)
mu = params$mu
var = params$var
probs = params$probs
# initializations
# cluster 'responsibilities', i.e. probability of cluster membership for each
# observation i
ri = matrix(0, ncol=clusters, nrow=N)
ll = 0 # log likelihood
it = 0 # iteration count
converged = FALSE # convergence
# Show iterations if showits == true
if (showits)
cat(paste("Iterations of EM:", "\n"))
while (!converged & it < maxits) {
probsOld = probs
# muOld = mu # Use direct values or loglike for convergence check
# varOld = var
llOld = ll
riOld = ri
### E
# Compute responsibilities
for (k in 1:clusters){
ri[,k] = probs[k] * dmvnorm(X, mu[k, ], sigma = var[[k]], log = FALSE)
}
ri = ri/rowSums(ri)
### M
rk = colSums(ri) # rk is weighted average cluster membership size
probs = rk/N
for (k in 1:clusters){
varmat = matrix(0, ncol = ncol(X), nrow = ncol(X)) # initialize to sum matrices
for (i in 1:N){
varmat = varmat + ri[i,k] * X[i,]%*%t(X[i,])
}
mu[k,] = (t(X) %*% ri[,k]) / rk[k]
var[[k]] = varmat/rk[k] - mu[k,]%*%t(mu[k,])
ll[k] = -.5*sum( ri[,k] * dmvnorm(X, mu[k,], sigma = var[[k]], log = TRUE) )
}
ll = sum(ll)
# compare old to current for convergence
parmlistold = c(llOld, probsOld) # c(muOld, unlist(varOld), probsOld)
parmlistcurrent = c(ll, probs) # c(mu, unlist(var), probs)
it = it + 1
# if showits true, & it =1 or modulo of 5 print message
if (showits & it == 1 | it%%5 == 0)
cat(paste(format(it), "...", "\n", sep = ""))
converged = min(abs(parmlistold - parmlistcurrent)) <= tol
}
clust = which(round(ri) == 1, arr.ind = TRUE) # create cluster membership
clust = clust[order(clust[,1]), 2] # order accoring to row rather than cluster
list(
probs = probs,
mu = mu,
var = var,
resp = ri,
cluster = clust,
ll = ll
)
}
```
### Example 1: Old Faithful
This example uses Old Faithful geyser eruptions as before. This is can be compared to the univariate code from the [other chapter][Mixture Model]. See also http://www.geyserstudy.org/geyser.aspx?pGeyserNo=OLDFAITHFUL
#### Data Setup
```{r mixmv-setup1}
library(tidyverse)
data("faithful")
```
#### Estimation
Create starting values and estimate.
```{r mixmv-starts1}
mustart = rbind(c(3, 60), c(3, 60.1)) # must be at least slightly different
covstart = list(cov(faithful), cov(faithful))
probs = c(.01, .99)
# params is a list of mu, var, and probs
starts = list(mu = mustart, var = covstart, probs = probs)
```
```{r mixmv-est1}
mix_faithful = em_mixture(
params = starts,
X = as.matrix(faithful),
clusters = 2,
tol = 1e-12,
maxits = 1500,
showits = TRUE
)
str(mix_faithful)
```
Visualize.
```{r mixmv-vis1, echo=FALSE}
library(ggplot2)
faithful %>%
mutate(cluster = factor(mix_faithful$cluster)) %>%
ggplot(aes(x = eruptions, y = waiting) ) +
geom_density2d(color = 'gray92') +
geom_point(aes(color = cluster)) +
scico::scale_color_scico_d(begin = .25, end = .75, alpha = .5) +
guides(color = guide_legend('Cluster'))
faithful %>%
mutate(prob_clus_1 = mix_faithful$resp[, 1]) %>%
ggplot(aes(x = eruptions, y = waiting)) +
geom_density2d(color = 'gray92') +
geom_point(aes(color = prob_clus_1)) +
scico::scale_color_scico(palette = 'bilbao', begin = .25) +
guides(color = guide_legend('Prob. Cluster 1'))
# relatively speaking, these are extremely well-separated clusters
worst = apply(mix_faithful$resp, 1, function(x) max(x) < .99)
ggplot(aes(x = eruptions, y = waiting), data = faithful) +
geom_point(aes(color = worst)) +
scico::scale_color_scico_d(palette = 'bilbao', begin = .25)
```
#### Comparison
Compare to <span class="pack" style = "">mclust</span> results. Options are set to be more similar to the settings demonstrated.
```{r mixmv-compare1}
library(mclust)
mix_mclust = mclust::Mclust(
faithful[, 1:2],
2,
modelNames = 'VVV',
control = emControl(tol = 1e-12)
)
detach(package:mclust)
# str(mix_mclust, 1)
```
Compare means.
```{r mixmv-compare1-means}
t(mix_faithful$mu)
mix_mclust$parameters$mean
```
Compare variances.
```{r mixmv-compare1-vars}
mix_faithful$var
mix_mclust$parameters$variance$sigma
```
Compare classifications. Reverse in case arbitrary labeling of one of the clusters is opposite.
```{r mixmv-compare1-class}
table(mix_faithful$cluster, mix_mclust$classification)
table(ifelse(mix_faithful$cluster == 2, 1, 2),
mix_mclust$classification)
# compare responsibilities; reverse one if arbitrary numbering of one of them is opposite
# cbind(round(mix_faithful$resp[,1], 2), round(mix_mclust$z[,2], 2)) # cluster '1'
# cbind(round(mix_faithful$resp[,2], 2), round(mix_mclust$z[,1], 2)) # cluster '2'
```
### Example 2: Iris
#### Data Setup
Set up the data.
```{r mixmv-setup2}
iris2 = iris %>% select(-Species)
```
#### Estimation
Run and examine. We add noise to our starting value, and the function is notably sensitive to starts, but we don't want to cheat too badly.
```{r mixmv-starts2}
mustart = iris %>%
group_by(Species) %>%
summarise(across(.fns = function(x) mean(x) + runif(1, 0, .5))) %>%
select(-Species) %>%
as.matrix()
# use purrr::map due to mclust::map masking
covstart = iris %>%
split(.$Species) %>%
purrr::map(select, -Species) %>%
purrr::map(function(x) cov(x) + diag(runif(4, 0, .5)))
probs = c(.1, .2, .7)
starts = list(mu = mustart, var = covstart, probs = probs)
```
```{r mixmv-est2}
mix_mclust_iris = em_mixture(
params = starts,
X = as.matrix(iris2),
clusters = 3,
tol = 1e-8,
maxits = 1500,
showits = T
)
table(mix_mclust_iris$cluster, iris$Species)
```
#### Comparison
Compare to <span class="pack" style = "">mclust</span> results.
```{r mixmv-compare2}
library(mclust)
mclust_iris = mclust::Mclust(iris[,1:4], 3)
table(mclust_iris$classification, iris$Species)
detach(package:mclust)
```
### Source
Original code available at
https://github.com/m-clark/Miscellaneous-R-Code/blob/master/ModelFitting/EM%20Examples/EM%20Mixture%20MV.R