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library(diffusionMap)
library(scatterplot3d)
library(svd)
library(cluster)
library(fpc)

Bacis Diffusion Mapping Algorithm for Dimensionality Reduction

Input: High dimensional data set phy_test

Define kernel k(x,y) , x,y in phy_test

Create kernel matrix K=k(x,y)

Create diffusion matrix by normalising the rows of the kernel matrix

Calculate eigenvectors of diffusion matrix

Map to the d-dimensional diffusion space at time t, using the d dominant eigenvectors and -values

Output: Lower dimensional data set Y (with same amount of records)

Data Preparation

select variables (5 out of 80) and 100 records

myvars<- phy_test[c(11,12,21,66,67)]
mydata <- myvars[1:100,]
head(mydata)
nrow(mydata) #100
ncol(mydata) #5

Matrix will be transported for distance measuring row-wise

B=t(mydata)
nrow(B)
#5

ncol(B)
#100

Check euclidean distance as a reference

D=as.matrix(dist(B))
D
nrow(D)
#5

ncol(D)
#5 - but transponed it would be 100 !

D=as.matrix(dist(mydata))
nrow(D) #100
ncol(D) #100

Pre- Preparation: PCA

PCA_mydata=prcomp(mydata)
print(PCA_mydata)
plot(PCA_mydata, type = "l")
summary(PCA_mydata)

taking the first three components, 96% of the models variance will be covered.

In high dimensions, while small distances are meaningful, large

distances are (almost) meaningless

Connectivity kernel (defines a local measure of similarity within a certain neighbourhood)

initial set-up: scale parameter a=1

kernel <- function(x,y){
exp(-(abs(sum((x-y))^2))/1)
}

kernel(c(1,2), c(3,4))

#does not work , although i do not know why
K=as.matrix(kernel(A[i,],A[j,]))

error subscript out ob bounds will be shown, but can be ignored since results seem to be correct

diffusion_matrix <- function(DifMat){
K=matrix(nrow=nrow(DifMat),ncol=nrow(DifMat))

for (i in 1:nrow(DifMat)) {
#go directly into the column loop
for (j in 1:nrow(DifMat)) {
if (i==j) {
K[i,j]= 1
}
else
K[i,j]=kernel(DifMat[i,],DifMat[j,])
}
}
return(K)
}
test=t(B)

(diffusion_matrix(test)[1:6,1:6])

Create normalized Matrix(by row)

library(som)
L= diffusion_matrix(A)
L
P= normalize(L,byrow=TRUE)
P
eigen(P)$values

[1] 6.825530e+00 4.613574e+00 1.964817e+00 9.890712e-01 9.083846e-02 1.083078e-02 3.758092e-04 1.022369e-06

[9] 6.625611e-09 -9.310607e-13

ev=cbind(eigen(P)$values)[c(1:4)]
vec=cbind(eigen(P)$vectors)
V=vec[1:10,1:4]
V

QC: A x= \lambda x

here (not symmetric matrix): (A - \lambda*I(nrow(A))) * x = 0

cbind(V[1,])
Q=P[,1:4]
Q

mATRIX multiplication is done by operator: %*%

Q%*%cbind(V[1,])

ev[1]* cbind(V[1,])

tbd

wie dim reduction ?

einfach nur die spalten weiter nutzen die zu den grössten eigenvalues gehören (siehe hierzu PCA)

oder gibts bestimmtes mapping-verfahren ?

geeigneten Datensatz finden und über t>1 laufen lassen.

Visualisierung