SMHS MixtureModeling - Revision history

Clgalla: /* Theory */

2014-10-20T18:57:09Z

‎Theory

Clgalla: /* Theory */

2014-10-20T18:51:40Z

‎Theory

Clgalla: /* Theory */

2014-10-20T18:49:20Z

‎Theory

Dinov: /* Theory */

2014-10-17T15:58:30Z

‎Theory

Dinov: /* Theory */

2014-10-17T15:53:28Z

‎Theory

Dinov: /* Theory */

2014-10-17T15:52:38Z

‎Theory

Clgalla at 20:42, 16 October 2014

2014-10-16T20:42:52Z

Clgalla at 20:41, 16 October 2014

2014-10-16T20:41:27Z

Clgalla at 20:39, 16 October 2014

2014-10-16T20:39:17Z

Show changes

Clgalla at 17:48, 16 October 2014

2014-10-16T17:48:22Z

Show changes

@@ Line 54: / Line 54: @@
   # Presumes the mixture object has the structure used by mixtools
   plot.normal.components <- function(mixture,component.number,...) {
-  curve(mixture$\$$lambda$[component.number]*
+  curve(mixture$\$$lambda[component.number]*
   dnorm(x,mean=mixture$\$$mu[component.number],
-  sd=mixture $\$$sigma[component.number]), add=TRUE, ...)
+  sd=mixture$\$$sigma[component.number]), add=TRUE, ...)
+  }
   plot(hist(snoq,breaks=101),col="grey",border="grey",freq=FALSE,

@@ Line 54: / Line 54: @@
   # Presumes the mixture object has the structure used by mixtools
   plot.normal.components <- function(mixture,component.number,...) {
-  curve(mixture $\lambda$ [component.number] *
+  curve(mixture$\$$lambda$[component.number]*
-  dnorm(x,mean=mixture $\mu$ [component.number],
+  dnorm(x,mean=mixture$\$$mu[component.number],
-  sd=mixture $\sigma$ [component.number]), add=TRUE, ...)
+  sd=mixture $\$$sigma[component.number]), add=TRUE, ...)
+  }
   plot(hist(snoq,breaks=101),col="grey",border="grey",freq=FALSE,
@@ Line 82: / Line 82: @@
   # returning the log probability, or the upper tail probability
   pnormmix <- function(x,mixture) {
-  lambda <- mixture$\$$lambda$
+  lambda <- mixture$\$$lambda
   k <- length(lambda)
   pnorm.from.mix <- function(x,component){

@@ Line 44: / Line 44: @@
   summary of normalmixEM object:
             comp 1    comp 2
-  $\lambda$  0.557524  0.442476
+  lambda  0.557524  0.442476
-  $\mu$  10.266172 60.009468
+  mu  10.266172 60.009468
-  $\sigma$  8.510244 44.997240
+  sigma  8.510244 44.997240
   loglik at estimate:  -32681.21
@@ Line 82: / Line 82: @@
   # returning the log probability, or the upper tail probability
   pnormmix <- function(x,mixture) {
-  $\lambda$ <- mixture $\lambda$
+  lambda <- mixture$\$$lambda$
-  k <- length($\lambda$)
+  k <- length(lambda)
   pnorm.from.mix <- function(x,component){
-  $\lambda$[component]*pnorm(x,mean=mixture$\mu$[component],
+  lambda[component]*pnorm(x,mean=mixture$\$$mu[component],
-  sd=mixture$\sigma$[component])
+  sd=mixture$\$$sigma[component])
+  }
   pnorms <- sapply(1:k,pnorm.from.mix,x=x)
@@ Line 118: / Line 118: @@
   # Presumes the mixture object has the structure used by mixtools
   dnormalmix <- function(x,mixture,log=FALSE) {
-  $\lambda$ <- mixture $\lambda$
+  lambda <- mixture$\$$\lambda
-  k <- length($\lambda$)
+  k <- length(lambda)
   # Calculate share of likelihood for all data for one component
   like.component <- function(x,component) {
-  $\lambda$ [component]*dnorm(x,mean=mixture$\mu$[component],
+  lambda[component]*dnorm(x,mean=mixture$\$$mu[component],
-  sd=mixture $\sigma$ [component])
+  sd=mixture$\$$sigma[component])
+  }
   # Create array with likelihood shares from all components over all data
@@ Line 152: / Line 152: @@
   loglikes <- vector(length=1+length(candidate.component.numbers))
   # k=1 needs special handling
-  $\mu$<-mean(snoq[train]) # MLE of mean
+  mu<-mean(snoq[train]) # MLE of mean
-  $\sigma$ <- sd(snoq[train])*sqrt((n-1)/n) # MLE of standard deviation
+  sigma<- sd(snoq[train])*sqrt((n-1)/n) # MLE of standard deviation
   loglikes[1] <- sum(dnorm(snoq[test],mu,sigma,log=TRUE))
   for (k in candidate.component.numbers) {
@@ Line 190: / Line 190: @@
 Calibration plot for the nine-component Gaussian mixture.
-  plot(0,xlim=range(snoq.k9$\$ \mu$),ylim=range(snoq.k9$\$ \sigma$),type="n",
+  plot(0,xlim=range(snoq.k9$\$$mu),ylim=range(snoq.k9$\$$sigma),type="n",
   xlab="Component mean", ylab="Component standard deviation")
-  points(x=snoq.k9$\$ \mu$,y=snoq.k9$\$ \sigma$,pch=as.character(1:9),
+  points(x=snoq.k9$\$$mu,y=snoq.k9$\$$sigma,pch=as.character(1:9),
-  cex=sqrt(0.5+5*snoq.k9$\$ \lambda$))
+  cex=sqrt(0.5+5*snoq.k9$\$$lambda))
 [[Image:SMHS_MixtureModel_Fig7.png|500px]]
@@ Line 212: / Line 212: @@
   # Do the classes of the Gaussian mixture make sense as annual weather patterns?
   # Most probable class for each day:
-  day.classes <- apply(snoq.k9$posterior,1,which.max)
+  day.classes <- apply(snoq.k9$\$$posterior,1,which.max)
   # Make a copy of the original, structured data set to edit
   snoqualmie.classes <- snoqualmie
@@ Line 222: / Line 222: @@
   # precipitation expected.  Solution: label by expected precipitation, not by
   # class number.
-  snoqualmie.classes[wet.days] <- snoq.k9$mu[day.classes]
+  snoqualmie.classes[wet.days] <- snoq.k9$\$$mu[day.classes]
-  plot(0,xlim=c(1,366),ylim=range(snoq.k9$mu),type="n",xaxt="n",
+  plot(0,xlim=c(1,366),ylim=range(snoq.k9$\$$mu),type="n",xaxt="n",
   xlab="Day of year",ylab="Expected precipiation (1/100 inch)")
   axis(1,at=1+(0:11)*30)
@@ Line 251: / Line 251: @@
   y.points <- x.points
   z <- matrix(0,nrow=100,ncol=100)
-  $\mu <- c(1,1)$
+  mu <- c(1,1)
-  $\sigma <- matrix(c(2,1,1,1),nrow=2)$
+  sigma <- matrix(c(2,1,1,1),nrow=2)
   for (i in 1:100) {
   for (j in 1:100) {

@@ Line 190: / Line 190: @@
 Calibration plot for the nine-component Gaussian mixture.
-  plot(0,xlim=range(snoq.k9$mu),ylim=range(snoq.k9$sigma),type="n",
+  plot(0,xlim=range(snoq.k9$\$ \mu$),ylim=range(snoq.k9$\$ \sigma$),type="n",
   xlab="Component mean", ylab="Component standard deviation")
-  points(x=snoq.k9$mu,y=snoq.k9$sigma,pch=as.character(1:9),
+  points(x=snoq.k9$\$ \mu$,y=snoq.k9$\$ \sigma$,pch=as.character(1:9),
-  cex=sqrt(0.5+5*snoq.k9$lambda))
+  cex=sqrt(0.5+5*snoq.k9$\$ \lambda$))
 [[Image:SMHS_MixtureModel_Fig7.png|500px]]

@@ Line 83: / Line 83: @@
   pnormmix <- function(x,mixture) {
   $\lambda$ <- mixture $\lambda$
-  k <- length(s\lambda$)
+  k <- length($\lambda$)
   pnorm.from.mix <- function(x,component){
   $\lambda$[component]*pnorm(x,mean=mixture$\mu$[component],

← Older revision		Revision as of 15:52, 17 October 2014
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−	We try the logarithm to turn multiplication into addition:$l(\theta) = \sum_{i=1}^{n} logf(x_{i};\theta) = \sum_{i=1}^{n} log \sum_{k=1}^{K} \lambda_{k} f(x_{i}; \theta_{k})$,	+	We try the logarithm to turn multiplication into addition: $l(\theta) = \sum_{i=1}^{n} logf(x_{i};\theta) = \sum_{i=1}^{n} log \sum_{k=1}^{K} \lambda_{k} f(x_{i}; \theta_{k})$,

← Older revision		Revision as of 20:42, 16 October 2014
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	[http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch20.pdf Chapter 20. Mixture Models]		[http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch20.pdf Chapter 20. Mixture Models]
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	<hr>		<hr>

@@ Line 262: / Line 262: @@
 [[Image:SMHS_MixtureModel_Fig11.png|500px]]
-Applications
+===Applications===
 ) [http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_2D_PointSegmentation_EM_Mixture This article] demonstrated activity of the mixture modeling and expectation maximization (EM) applied to the problem of 2D point cluster segmentation. It illustrated ways to use EM and mixture modeling to obtain cluster classification of points in 2D using SOCR charts activity and SOCR modeler with specific examples.
@@ Line 277: / Line 277: @@
 [http://www.stat.washington.edu/mclust/ mclust]
-[http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/mixture-examples.R  R code for examples in Lecture 20]
+[http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/mixture-examples.R  R code for examples in Chapter 20 (see references)]
 ===Problems===
@@ Line 289: / Line 289: @@
 ) Write a function to fit a mixture of exponential distributions using the EM algorithm.
-===References=====
+===References===
 [http://mirlyn.lib.umich.edu/Record/004199238 Statistical inference / George Casella, Roger L. Berger]