Add visualization for linear vs. weighted interpolation

docs/t-digest-paper/weighted-vs-linear-interpolation.r

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+### Compare linear interpolation of centroids to weighted interpolation ###
+fade = rgb(0,0,0,alpha=0.5)
+dot.size = 0.7
+set.seed(5)
+
+pdf("weighted-vs-linear-interpolation.pdf", width=6, height=2.7, pointsize=10)
+layout(matrix(c(1,2),byrow=T, ncol=2), widths=c(1.1,1))
+x = sort(log(1-runif(10000))) # sorted exponential distribution
+F = ((0:(length(x)-1))+0.5)/length(x) # the y points for an x point to its percentile
+par(mar=c(2.5,3,1,1))
+plot(x, F, cex=dot.size, pch=21, bg=fade, col=NA, type='b', xlim=c(x[1], x[110]), ylim=c(0,0.01), xaxt='n', ylab=NA, mgp=c(1,0.5,0), xlab=NA)
+
+axis(side=1, at=-10:-1, labels=NA)
+title(xlab='x', line=0.8, cex.lab=1.5)
+title(ylab='q', line=1.5, cex.lab=1.5)
+
+left.end = min(x)
+
+lines(c(left.end, x[100]), c(0, 0.01), lwd=2)
+lines(c(left.end, left.end), c(-0.0005, 0.0005), lt=1, col='black', lwd=0.5)
+lines(c(x[100], x[100]), c(0.0085, 0.015), lt=1, col='black', lwd=0.5)
+text(-7, 0.006, "100")
+
+q.to.k = function(q) {
+    (asin(2*q-1)/pi + 1/2)
+}
+
+k.to.q = function(k,compression) {
+    sin(k/compression*pi - pi/2)/2 + 0.5
+}
+
+# This function makes a plot of the cdf of the sorted distribution in the global variable "x"
+# The graph limits are defined by x[rangeMin]/x[rangeMax], which are values from [1,n]. By default, it
+# with only graph ~ the first percentile.
+# It will then calculate the positions of the centroids based on the weights passed in.
+# Lastly, it will graph these positions with draw function which is passed in
+makeChart = function(weights, titleToDisp, drawFunc, rangeMin=1, rangeMax=round(1.1*length(x)/100)) {
+    xLimits = c(x[rangeMin], x[rangeMax])
+    yLimits = c((rangeMin-1)/length(x), rangeMax/length(x))
+    F = ((0:(length(x)-1))+0.5)/length(x) # the y points for an x point to its percentile
+
+    #plot the points of the distribution
+    plot(x, F, cex=dot.size, pch=21, bg=fade, col=NA, type='b', xlim=xLimits, ylim=yLimits, xaxt='n')
+    title(main=titleToDisp)
+    axis(side=1, at=-10:-1, labels=NA)
+    axis(side=2, at=(0:6)/10, labels=NA)
+    title(xlab='x', line=0.8, cex.lab=1.5)
+    title(ylab='q', line=2, cex.lab=1.5)
+
+    xCoordinatesEnds = c(1, cumsum(weights), length(x)) # x[coordinateEnds[i]] are the end points of each centroid
+
+    # xCoordinates[i] is the mean value of all of the points that in centroid[i]
+    xCoordinates = numeric(length(xCoordinatesEnds))
+    xCoordinates[1]=x[1] # first centroid is always the min
+    for(i in 2:length(xCoordinates)) {
+        xCoordinates[i] = mean(x[xCoordinatesEnds[i-1]:xCoordinatesEnds[i]])
+    }
+
+    # each centroid's height is all of the weight up to the centroid minus half the centroid weight.
+    yCoordinates = c(0, cumsum(weights)- weights / 2, length(x)) / sum(weights)
+
+    # note - weight of min and max is specified as 1. It could be k.to.q(1/compression) for better results.
+    weightsAlignedByPoints = c(1,weights,1)
+
+    # draw chart
+    drawFunc(xCoordinates, yCoordinates, weightsAlignedByPoints)
+    # write the weight of each centroid above it.
+    text(xCoordinates, yCoordinates + (rangeMax-rangeMin)*.06/length(x), round(c(1, weights, 1)))
+}
+
+# Draw the centroids linearly interpolated
+drawLinear = function(xCoordinates, yCoordinates, weights) {
+    lines(xCoordinates, yCoordinates, type='o', lwd=1, col='blue')
+}
+# Draw the centroids interpolated by weight
+drawWeighted = function(xCoordinates, yCoordinates, weights) {
+    points(xCoordinates, yCoordinates, col='blue')
+    for(i in 1:(length(xCoordinates)-1)) {
+        w1 = weights[i]
+        w2 = weights[i+1]
+        x1 = xCoordinates[i]
+        x2 = xCoordinates[i+1]
+        mean1 = yCoordinates[i]
+        mean2 = yCoordinates[i+1]
+
+        # Weighted average with centroid weight as 
+        weightFunc = function(q) {
+            (mean1*w1*(x2-q)+mean2*w2*(q-x1)) / (w1*(x2-q)+w2*(q-x1))
+        }
+        curve(weightFunc, x1, x2, add=TRUE, col="blue")
+    }
+}
+
+# This function calculates the ideal centroid weights for the specified compression / dataset size
+getIdealBounds = function(n, compression) {
+    leftBounds = c(0, k.to.q(1:(compression-1), compression))
+    rightBounds = k.to.q(1:compression, compression)
+    (rightBounds - leftBounds) * length(x) 
+}
+
+# these weights were taken from linear-interpolation.r
+weights = c(2, 8, 19, 35, 56, 81, 111)
+
+n=length(x)
+makeChart(c(weights, n-sum(weights)), "Preset Weights", drawLinear)
+
+# Show comparison for these preset weights
+makeChart(c(weights, n-sum(weights)), "Preset Weights", drawLinear)
+makeChart(c(weights, n-sum(weights)), "Preset Weights", drawWeighted)
+
+drawExampleCharts = function(distName) {
+    n = length(x)
+
+    # make comparison charts for ideal centroid placements
+    for (i in c(50,100,200)) {
+        titleToDisp = paste(distName," c=", i)
+        makeChart(getIdealBounds(n, i), titleToDisp, drawLinear)
+        makeChart(getIdealBounds(n, i), titleToDisp, drawWeighted)
+    }
+
+    midLen = n/20 # 5%
+    titleToDisp = paste(distName," c=", 100)
+
+    # make chart of top 0.025-0.05 quantiles
+    makeChart(getIdealBounds(n, 100), titleToDisp, drawLinear, rangeMin=round(midLen/2), rangeMax=midLen)
+    makeChart(getIdealBounds(n, 100), titleToDisp, drawWeighted, rangeMin=round(midLen/2), rangeMax=midLen)
+
+    # make chart of top 0.45-0.55 quantiles
+    makeChart(getIdealBounds(n, 100), titleToDisp, drawLinear, rangeMin=round(n/2-midLen), rangeMax=round(n/2+midLen))
+    makeChart(getIdealBounds(n, 100), titleToDisp, drawWeighted, rangeMin=round(n/2-midLen), rangeMax=round(n/2+midLen))
+
+
+    # make chart of top 0.01 quantile
+    bottomPercent = (n*1.1)/100 # 1.1% of elements
+    makeChart(getIdealBounds(n, 100), titleToDisp, drawLinear, rangeMin=n-bottomPercent, rangeMax=n)
+    makeChart(getIdealBounds(n, 100), titleToDisp, drawWeighted, rangeMin=n-bottomPercent, rangeMax=n)
+}
+
+drawExampleCharts(paste("Exp n=", length(x)))
+
+x = sort(log(1-runif(50000))) # sorted exponential distribution
+drawExampleCharts(paste("Exp n=", length(x)))
+
+x = sort(rnorm(50000)) # sorted normal distribution
+drawExampleCharts(paste("Normal n=", length(x)))
+
+x = sort(rnorm(50000)) # sorted normal distribution
+drawExampleCharts(paste("Normal n=", length(x)))
+
+x = sort(runif(50000)) # sorted uniform distribution
+drawExampleCharts(paste("Uniform n=", length(x)))
+
+dev.off()