In olobiolo/centroidr: Find Centroid of a Set of Points in n-dimensional Space

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(centroidr)

Estimate multidimensional variance of a sample.

This is the first package I am creating and as such it is sort of an excercise.

The statistics here is very much amateurish.

cat('\n\n\n')

The Purpose

Why calculate centroids? I was looking for an easy way to express two-dimensional spread of a three point sample. I have two variables for each observation that are plotted on an xy scatter plot and each sample has three separate measurements. This produces three points. Now, the variance of single variables can be easily expressed as standard deviation or, well, variance, but the spread of the three points in the plot is hard to estimate at an eye's glance.

a <- matrix(c(0.398, 0.2104, 1, 0.4865, 0.0389, 0), 3, 2)
plot(a, xaxt = 'n', yaxt = 'n', xlab = '', ylab = '', bty = 'n')
lines(rbind(a, a), lty = 1, type = 'c')

The Method

At first I focused on the triangle formed by the three points and considered:

• the triangle's circumference
• the tirangle's surface area

The methods fail if a sample has a point missing. Moreover, my case of a three-point sample is rather specific, if common. A more robust metric should work for any number of replicates. Calculating the circumference for a polygon of more than three vertices is a non-trivial matter.

On the other hand, one could calculate the sum of distances between each pair of points (for three points this is actually the circumference), making the method robust for any number of replicates.

The circumference (triangle-specific) and surface methods only worked well when the angles were relatively constant. The sum of distances approach is very susceptible to missing points.

Another approach was needed.

That is how I came to consider the centroid: the point in a triangle, where all medians intersect (the median being the line between a vertex and the middle point of its opposite side).

The coordinates of the centroid can be easily calculated by averaging the respective coordinates fo all vertices (this actually works for any polygon). Each vertex has a certain distance from the centroid and these distances can easily be used to express the spread of the points.

Scale <- function(x) return((x - min(x)) / max(x - min(x)))
plot(a, xaxt = 'n', yaxt = 'n', xlab = '', ylab = '', bty = 'n')
lines(rbind(a, a), lty = 1, type = 'c')
C <- centroid(a)
points(C[1], C[2], pch = 16)
lines(rbind(a[1,], C), lty = 2, type = 'c')
lines(rbind(a[2,], C), lty = 2, type = 'c')
lines(rbind(a[3,], C), lty = 2, type = 'c')

The basic problem was collapsing variances in two dimensions to a one-dimensional value that can be intuitively understood. Centroid distances can not only collapse any number of dimensions, they also work just as well with any number of replicates.

Once the centroid distances are computed, the issue then becomes how to translate them into a single numebr that will express the two-dimensional spread. Again, I tested two metrics:

• centroid distance mean
• centroid distance standard deviation

Let us consider some examples.

We will generate nine paris of random three-point samples, plot them together for visual inspection and have a look at how the four different spread metrics compare between them.

The samples are randomly generated. Feel free to reload the vignette to study more examples.

fun <- function (iterations = 9) {
  # prepare function that will test different metrics for a pair of triangles
  f <- function(iteration) {
    # generate and plot two triangles
      x1 <- Scale(matrix(rnorm(6, 0, 1.25), 3,2))
      x2 <- Scale(matrix(rnorm(6, 0, 1.25), 3,2))
      plot.new()
      plot.window(c(0, 1), c(0, 1))
      points(x1)
      lines(rbind(x1, x1), lty = 2, type = 'c')
      points(x2, pch = 16)
      lines(rbind(x2, x2))
      # add iteration number in upper-right corner
      text(x = 0.95, y = 0.95, labels = as.character(iteration))
      # function list to calculate various metrics
      fs <- list(distance_sum = function(x) sum(dist(x)),
                 surface_area = function(x) geometry::polyarea(x[,1], x[,2]),
                 cendist_sum = function(x) sum(cendist(x)),
                 cendist_mean = function(x) mean(cendist(x)),
                 cendist_var = function(x) var(cendist(x)))
      # get metrics for both triangles
      metrics <- sapply(list(x1, x2), function(X) sapply(fs, function(f) f(X)))
      # return the metrics' ratio
      round(metrics[, 1] / metrics[, 2], 2)
      }
 # prepare graphics layout
  panels = 1:(ceiling(sqrt(iterations))^2)
  layout(mat = matrix(panels, ceiling(sqrt(iterations)),ceiling(sqrt(iterations)), T))
  # run the desired numebr of simulations; triangle pairs are plotted and metric ratios returned
  sapply(1:iterations, f)
}

fun(9)

Major points:

• Surface area can vary greatly with slight sifts of a single point. For triangles this is easy to spot when two vertices are far apart and the third is only slightly removed from the opposite side.
• Centroid distance variance is completely unreliable as it stays constant in equilateral triangles of any size.
• Total vertex distance, total centroid distance and mean cendtoid distance are quite similar in efficiency. This holds for triangles but may fail for polygons with more vertices. Total distances will, however, certainly fail in the case of missing points, whereas mean centroid distance will be more robust.

Conclusion

Mean centroid distance seems to be a good measure of multidimensional spread. None of the function returns a summarized single value rather they return distances for all existing vertices so the user can easily go for the median rather than the mean as well as exclude specific points before summarizing.

olobiolo/centroidr documentation built on Dec. 3, 2019, 12:55 p.m.