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Ch13_DataAssim_Analytical.Rmd
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Ch13_DataAssim_Analytical.Rmd
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---
title: "Chapter 12 - Data Assimilation"
author: "Mike Dietze"
date: "June 15, 2015"
output: html_document
---
## Kalman Filter
```{r}
##### define parameters
mu = 2 # prior mean
tau = 3 # prior sd
Y = 0 # observation
sigma = 2 # observation sd
```
### Helper functions
```{r}
filled.dnorm <- function(x,mu,tau,vertical=TRUE,offset=0,...){
if(vertical){
polygon(c(0,dnorm(x,mu,tau),0)+offset,c(x[1],x,x[length(x)]),...)
}else{
polygon(c(x[1],x,x[length(x)]),c(0,dnorm(x,mu,tau),0)+offset,...)
}
}
filled.density <- function(x,adjust,vertical=TRUE,offset=0,...){
den = density(x,adjust=adjust)
if(vertical){
polygon(c(0,den$y,0)+offset,c(den$x[1],den$x,den$x[length(den$x)]),...)
}else{
polygon(c(den$x[1],den$x,den$x[length(x)]),c(0,den$y,0)+offset,...)
}
}
col.alpha <- function(col,alpha=1){
rgb = col2rgb(col)
rgb(rgb[1],rgb[2],rgb[3],alpha*255,maxColorValue=255)
}
plot.KFupdate <- function(mu,tau,Y,sigma){
xlim = range(qnorm(c(0.01,0.99),mu,tau),qnorm(c(0.01,0.99),Y,sigma))
prec.new = (1/tau^2+1/sigma^2)
mu.new = (mu/tau^2+Y/sigma^2)/prec.new
sigma.new = sqrt(1/prec.new)
x = seq(xlim[1],xlim[2],length=1000)
ylim=c(0,max(dnorm(x,mu.new,sigma.new)))
alpha=0.4
plot(x,dnorm(x,mu,tau),xlim=xlim,ylim=ylim,ylab="Density",type='n',bty='n',
xaxt="n",yaxt="n",xlab=" ",mgp=c(1,1,0)*1.7)
axis(1, pos=0)
axis(2, pos=x[1])
filled.dnorm(x,mu,tau,FALSE,col=col.alpha(4,alpha),lwd=2)
filled.dnorm(x,Y,sigma,FALSE,col=col.alpha(3,alpha),lwd=2)
filled.dnorm(x,mu.new,sigma.new,FALSE,col=col.alpha(2,alpha),lwd=2)
abline(v=Y,lty=2,lwd=3)
legend("topright",c("Prior","Data","Posterior"),lwd=4,col=4:2)
}
```
```{r}
par(mfrow=c(1,1))
plot.KFupdate(mu,tau,Y,sigma)
```
```{r}
par(mfrow=c(2,1))
par(mar=c(2,3,2,2))
plot.KFupdate(mu,tau/2.5,Y,sigma)
plot.KFupdate(mu,tau,Y,sigma/2.5)
```
## Analysis step: comparison of Normal updates
```{r}
## KF one-step
mu = 5 # prior mean
tau = 0.4 # prior sd
Y = 4 # observation
sigma = 0.4 # observation sd
m = 1.2 # process slope
v = 0.25 # process error
offset=ceiling(1.3*max(dnorm(x,mu,tau)))
mu.a = m*mu
tau.a = m*m*tau + v
prec.new = (1/tau.a^2+1/sigma^2)
mu.f = (mu.a/tau.a^2+Y/sigma^2)/prec.new
tau.f = sqrt(1/prec.new)
offset2 = offset + ceiling(1.2*max(dnorm(x,mu.f,tau.f)))
xlim=c(0,offset2)
ylim=c(min(qnorm(0.01,c(mu,Y,mu.a,mu.f),c(tau,sigma,tau.a,tau.f))),
max(qnorm(0.99,c(mu,Y,mu.a,mu.f),c(tau,sigma,tau.a,tau.f))))
par(mfrow=c(1,1))
par(mar=c(4,3,2,2))
x = seq(ylim[1],ylim[2],length=1000)
plot(0,0,xlim=xlim,ylim=ylim,xlab="Time",ylab="X",type='n',bty='n',
xaxt="n",yaxt="n",mgp=c(1,1,0)*1.7)
axis(1, pos=ylim[1])
axis(2, pos=0)
##prior
filled.dnorm(x,mu,tau,vertical=TRUE,col=col.alpha(2,alpha))
text(0.4,mu-0.2,"1: Initial State",cex=1.5)
##Forecast
text(1,7.5,"Forecast Step",cex=2.5)
filled.dnorm(x,mu.a,tau.a,vertical=TRUE,col=col.alpha(4,alpha),offset=offset)
lines(c(0,offset),c(mu,mu.a),lty=2)
lines(c(0,offset),qnorm(0.025,c(mu,mu.a),c(tau,tau.a)),lty=2)
lines(c(0,offset),qnorm(0.975,c(mu,mu.a),c(tau,tau.a)),lty=2)
text(2.8,6,"2: Forecast",cex=1.5)
##Observation
filled.dnorm(x,Y,sigma,col=col.alpha(3,alpha),vertical=TRUE,offset=offset)
lines(c(offset,offset2),c(Y,Y),lty=3,lwd=2)
text(3.5,Y-0.2,"3: New Observation", cex = 1.5)
##Analysis
text(3,7.5,"Analysis Step",cex=2.5)
filled.dnorm(x,mu.f,tau.f,col=col.alpha(2,alpha),vertical=TRUE,offset=offset)
text(3.6,mu.f,"4: Updated State",cex=1.5)
```
## Logistic EKF
```{r}
par(mfrow=c(1,1),lwd=4)
N0 = 3
r = 0.3
p1 = 1
p2 = 1
q1 = 0
q2 = 1
K = 10
g = function(N){
N+r*N*(1-N/K)
}
dg = function(N){
1+r-2*r*N/K
}
#for(r in c(0.2,1)){
nt = 20
N = N0
EN = N0
for(t in 2:nt){
N[t]=g(N[t-1])
m = dg(N[t-1])
p1[t] = m^2*p1[t-1]+q1
p2[t] = m^2*p2[t-1]+q2
}
UC = N+1.96*sqrt(p2)
LC = N-1.96*sqrt(p2)
plot(N,ylim=c(min(LC),max(UC)),type='l',cex.lab=1.5,cex.axis=1.5)
lines(UC,lty=2)
lines(LC,lty=2)
lines(N+1.96*sqrt(p1),lty=3)
lines(N-1.96*sqrt(p1),lty=3)
legend("bottomright",legend=c("mean",paste("q =",c(q2,q1))),lty=1:3,cex=2)
#}
plot(UC-LC)
plot(sqrt(p1)/sqrt(p2))
abline(h=0.5)
```