How to drop a perpendicular line from each point in a scatterplot to an (Eigen)vector?

Question

I'm creating a visualization to illustrate how Principal Components Analysis works, by plotting Eigenvalues for some actual data (for the purposes of the illustration, I'm subsetting to 2 dimensions).

I'm want a combination of these two plots from this fantastic PCA tutorial, only for my real data.

I can plot the vectors and all ok:

Person1 <- c(-3,1,1,-3,0,-1,-1,0,-1,-1,3,4,5,-2,1,2,-2,-1,1,-2,1,-3,4,-6,1,-3,-4,3,3,-5,0,3,0,-3,1,-2,-1,0,-3,3,-4,-4,-7,-5,-2,-2,-1,1,1,2,0,0,2,-2,4,2,1,2,2,7,0,3,2,5,2,6,0,4,0,-2,-1,2,0,-1,-2,-4,-1)
Person2 <- c(-4,-3,4,-5,-1,-1,-2,2,1,0,3,2,3,-4,2,-1,2,-1,4,-2,6,-2,-1,-2,-1,-1,-3,5,2,-1,3,3,1,-3,1,3,-3,2,-2,4,-4,-6,-4,-7,0,-3,1,-2,0,2,-5,2,-2,-1,4,1,1,0,1,5,1,0,1,1,0,2,0,7,-2,3,-1,-2,-3,0,0,0,0)
df <- data.frame(cbind(Person1, Person2))
g <- ggplot(data = df, mapping = aes(x = Person1, y = Person2))
g <- g + geom_point(alpha = 1/3)  # alpha b/c of overplotting
g <- g + geom_smooth(method = "lm")  # just for comparsion
g <- g + coord_fixed()  # otherwise, the angles of vectors are off
corre <- cor(x = df$Person1, y = df$Person2, method = "spearman")  # calculate correlation, must be spearman b/c of measurement
matrix <- matrix(c(1, corre, corre, 1), nrow = 2)  # make this into a matrix
eigen <- eigen(matrix)  # calculate eigenvectors and values
eigen$vectors.scaled <- eigen$vectors %*% diag(sqrt(eigen$values))  
  # scale eigenvectors to length = square-root
  # as per http://stats.stackexchange.com/questions/9898/how-to-plot-an-ellipse-from-eigenvalues-and-eigenvectors-in-r
g <- g + stat_ellipse(type = "norm")
g <- g + stat_ellipse(type = "t")
  # add ellipse, though I am not sure which is the adequate type
  # as per https://github.com/hadley/ggplot2/blob/master/R/stat-ellipse.R
g <- g + geom_abline(intercept = 0, slope = eigen$vectors.scaled[1,1], colour = "green")  # add slope for pc1
g <- g + geom_abline(intercept = 0, slope = eigen$vectors.scaled[1,2], colour = "red")  # add slope for pc2
g <- g + geom_segment(aes(x = 0, y = 0, xend = max(df), yend = eigen$vectors.scaled[1,1] * max(df)), colour = "green", arrow = arrow(length = unit(0.2, "cm")))  # add arrow for pc1
g <- g + geom_segment(aes(x = 0, y = 0, xend = max(df), yend = eigen$vectors.scaled[1,2] * max(df)), colour = "red", arrow = arrow(length = unit(0.2, "cm")))  # add arrow for pc1
g

So far so good (well). How do I know use geom_segment to drop a perpendicular from every datapoint to, say, the green first principal component?

Ramiah

I know the visualization has some more (statistical/conceptual) problems – I asked about those on Cross Validated - > stats.stackexchange.com/questions/153564/… – Ramiah

Your example seems overly complex for just finding segments perpendicular to a line. I would start with this question as a guide: #2639930 – Bindweed

Answer 1

Adapting a previous answer, you can do

perp.segment.coord <- function(x0, y0, a=0,b=1){
 #finds endpoint for a perpendicular segment from the point (x0,y0) to the line
 # defined by lm.mod as y=a+b*x
  x1 <- (x0+b*y0-a*b)/(1+b^2)
  y1 <- a + b*x1
  list(x0=x0, y0=y0, x1=x1, y1=y1)
}


ss<-perp.segment.coord(df$Person1, df$Person2,0,eigen$vectors.scaled[1,1])

g + geom_segment(data=as.data.frame(ss), aes(x = x0, y = y0, xend = x1, yend = y1), colour = "blue")

Bindweed