Posts Tagged: r

probability / r — 7 Comments
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?

art / r — 3 Comments
29
May 10

Weekend art in R (part 1?)


As usual click on the image for a full-size version. Code:

par(bg="black")
par(mar=c(0,0,0,0))
plot(c(0,1),c(0,1),col="white",pch=".",xlim=c(0,1),ylim=c(0,1))
iters = 500
for(i in 1:iters) {
	center = runif(2)
	size = rbeta(2,1,50)
 
	# Let's create random HTML-style colors
	color = sample(c(0:9,"A","B","C","D","E","F"),12,replace=T)
	fill = paste("#", paste(color[1:6],collapse=""),sep="")
	brdr = paste("#", paste(color[7:12],collapse=""),sep="")
 
	rect(center[1]-size[1], center[2]-size[2], center[1]+size[1], center[2]+size[2], col=fill, border=brdr, density=NA, lwd=1.5)
}
plot / r — 1 Comment
28
May 10

R: More plotting fun with Poission

Coded as follows:

x = seq(.001,50,.001)
par(bg="black")
par(mar=c(0,0,0,0)) 
plot(x,sin(1/x)*rpois(length(x),x),pch=20,col="blue")
games / r — 6 Comments
28
May 10

The guessing game in R (with a twist, of course)

Maybe you remember playing this one as a kid. If you are about my age, you may have even created a version of this game as one of your first computer programs. You guess a number, the computer tells you if you if you are too low or high. I’ve limited the number of maximum guesses, and randomized the computer’s choice based on the Poisson distribution (more on that later).

Here’s the code. This was part of my attempt to understand how R reads input from the command line. One of the things I learned: you may need to save this to a file and run it with “source()”, instead of running it directly from the console, line by line.

# Classic guessing game with twist
x = 0
gotRight = 0
failed = 0
 
# Initial lambda for our random var
correct = 2000
initial = correct
 
# How many guesses should we allow per number
maxGuesses = 7
 
while(x != Inf) {
	# The +1 part makes sure we never get zero, which would trigger 0's forever
	correct = rpois(1,correct) + 1
 
	# The advantage of using "cat" instead of "print" is that you remove those pesky quotation marks
	cat("I am thinking of a number between 1 and infinity. What is it? (Type Inf to quit)\n")
 
	# Solicit input from the user
	x = scan(n=1) # Just one item in this vector
 
	# Be nice and let the user quit. 
	if(x == Inf) {
		cat("The correct answer was", correct, "\n")
		cat("You got", gotRight, "right and failed", failed, "times. Maximum allowed guesses was", maxGuesses, "and initial lambda was", initial, ". Goodbye.\n")
		cat("Post your score to http://www.statisticsblog.com/2010/05/the-guessing-game-in-r-with-a-twist-of-course/#comments \n")
		break
	}
 
	for(i in 1:maxGuesses) {
		if(x == correct) {
			print("You rock!")
			gotRight = gotRight + 1
			break
		} else {		
			if(i == maxGuesses) {
				cat("You ran out of guesses. I will pick a new random number based on the last one.\n")
				failed = failed + 1
			} else {
				if(x < correct) {
					cat("You are too low. Guess again.\n")
				} else {
					cat("You are too high. Guess again.\n")
				}
 
				x = scan(n=1)
			}			
		}
	}
}

Note 1: My code makes a couple uses of the aparently controversial “break” function. I can still recall a heated debate I had with a CS professor who believed that calling “break” (in Python) was as bad as crossing the streams of your Proton Pack. That said, I have sucessfully used it on several occasions now without any appearance by Stay Puft Marshmallow Man or changing the natural order between dogs and cats. In R, the biggest problem with using constructs like “break” and “while” is that, for reasons clear only to readers of this blog but not myself, if you ask R for help about either of these tokens using

?term

you get an sent an error or to purgatory, respectively.

Hint: Because the random guesses are Poisson based, using a “half the distance” strategy for guessing may not be the best way to go. The hardcore amongst yourselves might want to calculate the median of the expected value conditional on having guessed too low or high.

Note 2: The Poisson isn’t a very good distribution for for this. Maybe you can find a better one, or at least jack up the dispersion like an overzealous offroader tweaking the suspension of his 4Runner.

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