1 – Generate a test set y=x+err

#"noisy" x=y
x = runif(n = 1000,min=-100,max=100) # x has a uniform distribution
y = x + rnorm(n=1000,mean=0,sd=20) # y has a normal distribution

d <- data.frame(x=x,y=y)
#ver primeros puntos
head(d)

##           x            y
## 1 -77.25932 -57.56371837
## 2  24.45988  -0.03487656
## 3  21.85495  36.04947094
## 4  24.67589  22.49148848
## 5  72.18308 107.83523462
## 6  28.06212  23.19322747

plot(d)

2 – Perform PCA

pca <- prcomp(d,center = F,scale. = F)
summary(pca)

## Importance of components:
##                           PC1      PC2
## Standard deviation     83.040 13.30478
## Proportion of Variance  0.975  0.02503
## Cumulative Proportion   0.975  1.00000

pca$rotation

##        PC1       PC2
## x 0.691987 -0.721910
## y 0.721910  0.691987

dt <- pca$x #d transformed
head(dt)

##            PC1        PC2
## [1,] -95.01826  15.940928
## [2,]  16.90074 -17.681966
## [3,]  41.14781   9.168461
## [4,]  33.31222  -2.249953
## [5,] 127.79708  22.510896
## [6,]  36.16204  -4.208914

plot(dt)

3 – For a given vector, calculate it’s relation (cosine similarity) to all the others in both spaces

a <- d[2,] #choose a point on the original space
s<-apply(d,1,function(x) cos.sim(a,x)) #calculate the angles to all the other vectors
at <- dt[2,] #get the same point in the new space
st<-apply(dt,1,function(x) cos.sim(at,x)) #calculate the angles in the new space

head(data.frame(s=s,st=st))

##            s         st
## 1 -0.8010403 -0.8010403
## 2  1.0000000  1.0000000
## 3  0.5171996  0.5171996
## 4  0.7381007  0.7381007
## 5  0.5550765  0.5550765
## 6  0.7698978  0.7698978

head(data.frame(s=order(s,decreasing = T),st=order(st,decreasing = T)))

##     s  st
## 1   2   2
## 2 898 898
## 3 387 387
## 4  47  47
## 5 571 571
## 6 299 299

4 – Bonus: magnitudes are also the same

magd <- sqrt(rowSums(d*d))
magdt <- sqrt(rowSums(dt*dt))

head(data.frame(magd=magd,magdt=magdt))

##        magd     magdt
## 1  96.34617  96.34617
## 2  24.45991  24.45991
## 3  42.15689  42.15689
## 4  33.38812  33.38812
## 5 129.76453 129.76453
## 6  36.40616  36.40616

5 – Bonus: you can retreive the distributions

hist(as.data.frame(dt)$PC1) #this is close to uniform

hist(as.data.frame(dt)$PC2) #this is close to normal

Looks closer than the original x,y

hist(d$x)

hist(d$y)