August, 2010


30
Aug 10

The Chosen One

Toss one hundred different balls into your basket. Shuffle them up and select one with equal probability amongst the balls. That ball you just selected, it’s special. Before you put it back, increase its weight by 1/100th. Then put it back, mix up the balls and pick again. If you do this enough, at some point there will be a consistent winner which begins to stand out.

The graph above shows the results of 1000 iterations with 20 balls (each victory increases the weight of the winner by 5%). The more balls you have, the longer it takes before a clear winner appears. Here’s the graph for 200 balls (0.5% weight boost for each victory).

As you can see, in this simulation it took about 85,000 iterations before a clear winner appeared.

I contend that as the number of iterations grows, the probability of seeing a Chosen One approaches unity, no matter how many balls you use. In other words, for any number of balls, a single one of them will eventually see its relative weight, compared to the others, diverge. Can you prove this is true?

BTW this is a good Monte Carlo simulation of the Matthew Effect (no relation).

Here is the code in R to replicate:

numbItems = 200
items = 1:numbItems
itemWeights = rep(1/numbItems,numbItems) # Start out uniform
iterations = 100000
itemHistory = rep(0,iterations)
 
for(i in 1:iterations) {
	chosen = sample(items, 1, prob=itemWeights)
	itemWeights[chosen] = itemWeights[chosen] + (itemWeights[chosen] * (1/numbItems))
	itemWeights = itemWeights / sum(itemWeights) # re-Normalze
	itemHistory[i] = chosen
}
 
plot(itemHistory, 1:iterations, pch=".", col="blue")

After many trials using a fixed large number of balls and iterations, I found that the moment of divergence was amazingly consistent. Do you get the same results?


21
Aug 10

Weekend art in R (Part 3)

I have a few posts nearing completion, but meanwhile a weekend break for art. Big thanks to Simon Urbanek and Jeffrey Horner, creators of Cairo, a library for the programming language R. Have you noticed how R can’t anti-alias (fancy way for saying smooth out lines and curves when creating a bit-mapped image)? Cairo can.

Make sure to click the image above for the full version. Here’s my code:

# The Cairo library produces nice, smooth graphics
Cairo(1200, 1200, file="D:/Your/Path/Here/Dots.png", type="png", bg="#FF6A00")
 
# How big should the grid for placing dots be?
myWidth=40
myHeight=40
 
dotsPlaced = myWidth*myHeight
 
# Optional default colors and sizes for dots
myColors = rep(c("#0000F0","#00F000"),dotsPlaced)
myCex = rep(3.2,dotsPlaced)
 
for(i in 1:dotsPlaced) {
	# Change this to allow more of the default color dots to survive
	if(runif(1)<1) {
		myColors[i] = paste("#",paste(sample(c(3:9,"A","B","C","D","E","F"),6,replace=T),collapse=""),collapse="",sep="")
	}
	myCex[i] = runif(1,3,6)
}
 
# Keeping this is marginal
par(oma=c(0,0,0,0))
par(mar=c(0,0,0,0))
 
# Start off with a blank plot. The white dot helps with cropping later
plot(0,0,pch=".",xlim=c(0,40),ylim=c(0,40),col="white", xaxt = "n", yaxt = "n")
 
for(m in 1:myWidth) {
	for(n in 1:myHeight) {
		if(runif(1) < .93) {
			points(n,m,pch=20,col=myColors[((m*n)+n)],cex=myCex[((m*n)+n)])
		}
	}
}
 
dev.off() # Tell Cairo to burn the plot to disk

8
Aug 10

Seeing angels in the architecture

Sorry for the long delay between posts; I was temporarily sucked in to the infinite. While doing some reading about set theory (foundational stuff for probability and, in fact, all of mathematics), I veered off into the infinite and had a hard time climbing back out. I’m guessing you already know most of the basics about sets: compliments and unions and intersections. You may even know some of the stranger parts, like G. Cantor’s cascading crescendo of cardinalities. But knowing those in a cursory way (and really, that’s all a work-a-day statistician or even probabilist needs) isn’t the same as really exploring them.

Looking up again now after several weeks, I feel like I’ve traveled three levels deep in a dream, lost in a purgatory I could only escape by answering questions likeĀ  “Is a line made up of points, or does it have points?”, “Is it possible to count what you cannot fully name”, and “In an unbounded universe, is the compliment of the compliment of an object the same as the original object?”. I know, I know. I should have taken that left back at Albuquerque, I shouldn’t have swallowed the red pill. Still, it’s been an interesting trip to say the least, and I feel like I may now be coming back up the the surface, a little bit wiser and a lot more confused than when I began.

Meanwhile, I’ve added a couple items to the “Manifesto” and, The Architect permitting, will be posting a theory on Types of Randomness soon. Post should take between 1 and 10 days to complete, with 95% confidence. Hum… better make that an 80% confidence interval, I still haven’t wrapped my head around the whole idea of forcing.