library(fastICA)

### See Westra et al. (2007, 2008)

##Monthly Colorado River flows at Lees Ferry
### 1906 - 2006

test=matrix(scan("http://civil.colorado.edu/~balajir/CVEN5454/r-project-data/LeesFerry-monflow-1906-2006.txt"),ncol=13,byrow=T)
allMonths = test[,2:13]/10^6   #MAF

## Scale Data - mean removed. 
### you can also divide by standard deviation

X = scale(allMonths,scale=FALSE)

### Get the monthly mean and monthly SD

allMonSd = apply(allMonths, 2, sd)
allMonMean = apply(allMonths, 2, mean)


### Perform ICA

K = dim(X)[2] # number of ICA components to obtain
zout = fastICA(X,K)


#### zout$S  ~ Independent Components
##### zout$S is equivalent to Principal Components

#### zout$A ~ Mixing Matrix  equivalent to Eigen Vectors
 
 ### X = AS
 
YY = zout$S %*% zout$A

Xback = t(t(YY) + allMonMean)


## XK  = Y  ## prewhitened matrix

Xpre = X %*% zout$K
ICS = Xpre %*% zout$W

### ICS and zout$S  should be the same..
## S is the ICA = YW

###################################################
### PCA

#get variance matrix..
Xf=scale(allMonths)
zs=var(Xf)

#do an Eigen decomposition..
zsvd=svd(zs)

#Principal Components...
pcs=Xf %*% zsvd$u
flowpcs = pcs

#Eigen Values.. - fraction variance 
lambdas=(zsvd$d/sum(zsvd$d))

############################## For Problem 8

### Perform ICA on the SST data
#### Perform ICA on the Rainfall data

### Using the exact same data from Problem 7
### Investigate the first four ICA 
## (i) & (ii) Plot the first four Mixing Vectors ~ similar to first four Eigen Vectors
## Compare with the first four Eigen vector patterns you saw from Problem 7
## Also plot the first four ICs as time series and do a histogram to see their distribution
### Compare with the first four PCS

### (iii) Relate the first three or four ICs of SST with that of rainfall
#### compare with the same from the PCs in problem 7

### Correlate the first three ICs of SST with rainfall and vice-versa and plot
### correlation maps. Compare with that of problem 7