r


11
Apr 13

High Obesity levels found among fat-tailed distributions

In my never ending quest to find the perfect measure of tail fatness, I ran across this recent paper by Cooke, Nieboer, and Misiewicz. They created a measure called the “Obesity index.” Here’s how it works:

  • Step 1: Sample four times from a distribution. The sample points should be independent and identically distributed (did your mind just say “IID”?)
  • Step 2: Sort the points from lowest to highest (that’s right, order statistics)
  • Step 3: Test whether the sum of the smallest and greatest number is larger than the sum of the two middle.

The Obesity index is the probability that the sum of these end points is larger than the sum of the middle numbers. In mathy symbols:

Ob(X) = P (X_1 + X_4 > X_2 + X_3 | X_1 \leq X_2 \leq X_3 \leq X_4), X_i~IID



The graph at the top of this post shows how the Obesity index converges for different distributions. As always, I’ve included my R code at the end of this article, so you can run this simulation for yourself (though, as usual, I forgot to set a random seed so that you can run it exactly like I did).

The dots in the graph represent the mean results from 8, 16, 32, and so on, up to 4096 trials from each of the distributions I tested. Note that each trial involves taking 4 sample points. Confused? Think of it this way: each sample of 4 points gives us one Bernoulli trial from a single distribution, which returns a 0 or 1. Find the average result after doing 4096 of these trials, and you get one of the colored dots at the far right of the graph. For example, the red dots are averages from a Uniform distribution. The more trials you do, the closer results from the Uniform will cluster around 0.5, which is the “true” Obesity value for this distribution. The Uniform distribution is, not coincidentally, symmetric. For symmetric distributions like the Normal, we only consider positive values.

The graph gives a feel for how many trials would be needed to distinguish between different distributions based on their Obesity index. I’ve done it this way as part of my Grand Master Plan to map every possible distribution based on how it performs in a variety of tail indices. Apparently the Obesity index can be used to estimate quantiles; I haven’t done this yet.

My initial impressions of this measure (and these are very initial!) are mixed. With a large enough number of trials, it does a good job of ordering distributions in a way that seems intuitively correct. On the other hand, I’d like to see a greater distance between the Uniform and Beta(0.01, 0.01) distribution, as the latter is an extreme case of small tails.

Note that Obesity is invariant to scaling:

Ob(x) = Ob(k*X)



but not to translations:

Ob(X) \neq Ob(X+c)



This could be a bug or a feature, depending on what you want to use the index for.

Extra special karma points to the first person who comes up with a distribution whose Obesity index is between the Uniform and Normal, and that isn’t a variant of one I already tested.

Here’s the code:

# Code by Matt Asher for StatisticsBlog.com
# Feel free to redistribute, but please keep this notice
 
# Create random varaibles from the function named in the string
generateFromList = function(n, dist, ...) {
	match.fun(paste('r', dist, sep=''))(n, ...)
}
 
# Powers of 2 for testAt
testAt = 3:12
testAtSeq = 2^testAt
testsPerLevel = 30
 
 
distros = c()
 
distros[1] = 'generateFromList(4,"norm")'
distros[2] = 'generateFromList(4,"unif")'
distros[3] = 'generateFromList(4,"cauchy")'
distros[4] = 'generateFromList(4,"exp")'
distros[5] = 'generateFromList(4,"chisq",1)'
distros[6] = 'generateFromList(4,"beta",.01,.01)'
distros[7] = 'generateFromList(4,"lnorm")'
distros[8] = 'generateFromList(4,"weibull",1,1)'
 
# Gotta be a better way to do this.
dWords = c("Normal", "Uniform", "Cauchy", "Exponential", "Chisquare", "Beta", "Lognormal", "Weibull")
 
par(mar=c(4,5,1.5,.5))
plot(0,0,col="white",xlim=c(min(testAt),max(testAt)), ylim=c(-.5,1), xlab="Sample size, expressed in powers of 2", ylab="Obesity index measure", main="Test of tail fatness using Obesity index")
 
abline(h=0)
 
colorList = list()
colorList[[1]]=rgb(0,0,1,.2)
colorList[[2]]=rgb(1,0,0,.2)
colorList[[3]]=rgb(0,1,0,.2)
colorList[[4]]=rgb(1,1,0,.2)
colorList[[5]]=rgb(1,0,1,.2)
colorList[[6]]=rgb(0,1,1,.2)
colorList[[7]]=rgb(0,0,0,.2)
colorList[[8]]=rgb(.5,.5,0,.2)
 
# Create the legend
for(d in 1:length(distros)) {
	x = abs(rnorm(20,min(testAt),.1))
	y = rep(-d/16,20)
	points(x, y, col=colorList[[d]], pch=20)
	text(min(testAt)+.25, y[1], dWords[d], cex=.7, pos=4)
 
}
 
 
dCounter = 1
for(d in 1:length(distros)) {
	for(l in testAtSeq) {
		for(i in 1:testsPerLevel) {
			count = 0
			for(m in 1:l) {
 
				# Get the estimate at that level, plot it testsPerLevel times	
				x = sort(abs(eval(parse( text=distros[dCounter] ))))
				if ( (x[4]+x[1])>(x[2]+x[3]) ) {
					count = count + 1
				}
			}
 
			# Tiny bit of scatter added
			ratio = count/l
			points(log(l, base=2), ( ratio+rnorm(1,0,ratio/100)), col=colorList[[dCounter]], pch=20)
		}
	}
 
	dCounter = dCounter + 1
}

18
Mar 13

Review of Mathematica 9 and R-link

VIDEO TRANSCRIPT: Hello, this is Matt Asher from StatisticsBlog.com. I’m going to be reviewing Mathematica 9, from Wolfram Research. In particular, I’ll be focusing on using it with R and to do Monte Carlo simulations and other statistical work. You can find a full transcript of this video at my blog, including the source code and links to all of the webpages I mention.

Before I begin I’d like to thank Jeff Hara and Andy Ross from Wolfram for their time. Also thanks to the folks at the Mathematica Stack Exchange, who helped with a couple of my questions.

I’m going to get started with a blank notebook. I’m going to clear out all of the variables that exist. I’ve found sometimes that if you have existing variables that can cause problems.

ClearAll["Global`*"]

After each line I’m hitting Shift+Enter to run the command, if you just hit enter Mathematica won’t run things yet.

So I’ve cleared my variables and I’m going to run

Needs["RLink`"]

which will bring in the link between Mathematica and R.

InstallR[]

I’m going to make sure it’s installed.

REvaluate["R.Version()"]

And then I’m going to run a test command here to make sure everything is up and running. As you can see this is the version of R I’m running and the connection succeeded.

Note that the free version of Mathematica, the evaluation version, doesn’t come with support for R, so if you want to test out Mathematica’s and its interactions with R you either have to have to buy the full version or maybe if you call up or contact Wolfram they’d be willing to let you have a free evaluation version that is full and allows you to test out R.

So how does the interface between R and Mathematica work?

Basically, you can run operations in R, then store the results to variables in R. You can also pass data types back and forth between R and Mathematica.

Here I’m setting a variable and this variable is set in R, not in Mathematica

RSet["hal", 9000]

So if I were to type just

hal

There is no response back. This is Mathematica’s way of saying that the variable is undefined or that it doesn’t know what to do with your input. So to get back information from R we have to use:

REvaluate["hal"]

We are putting “hal” in quotes so we are parsing this in R and not in Mathematica.

For example we can do things like grab a dataset from R

iris = REvaluate["iris"]

I’m grabbing the famous “iris” dataset in R and I am pulling it into Mathematica.

or I could do things like evaluate a command in R:

REvaluate["det(matrix(sample(-50:49),nrow=10))"]

and bring back the results. This grabs the determinant of a random matrix.

We can even do things like create our own functions in R, and this gets put into a variable in Mathematica.

perfectSample = RFunction["function(n, dist, ...) match.fun(paste('q', dist, sep=''))((1:n) / (n+1), ...)"]

This function creates a perfect sample of the length that you specify of a particular distribution. Then we can call that function directly in Mathematica.

perfectSample[100, "pois", 10]

and the results are returned.

Of course, if we just wanted to do things in R, we would be continuing to just use R, instead of having to learn this new interface between R and Mathematica. So then what can we do in Mathematica that goes beyond what we can easily do in R?

One of the biggest advantages to using Mathematica is that you get access to tools for creating interactive plots and simulations that can be changed on the fly.

I’m going to do an example using the Benini Distribution, which, according to Wolfram’s web page, can be used to model the weight of cats.

So to do that, what I’m going to do is use the Mathematica command “Manipulate”

Manipulate[Block[{data, dist, kmd},
  data = RandomVariate[dist = BeniniDistribution[\[Alpha],\[Beta], \[Sigma]], n];
  kmd = KernelMixtureDistribution[data, h, MaxMixtureKernels -> All];
  Plot[{PDF[dist, x], PDF[kmd, x]}, {x, 0, xRng}, 
   PlotRange -> Automatic]], {{\[Alpha], 1}, .01, 5}, {{\[Beta], 1}, 0.01, 
  5}, {{\[Sigma], 1}, .01, 2}, {{n, 100}, {10, 100, 1000, 10000}}, {{xRng, 5}, 1, 
  10}, {{h, .5}, 0.01, 1}]

And then I get back the results and what I’ve got here is a live Monte Carlo simulation where I am specifying the different parameters of the distribution and I’m also specifying how many variates that I’m creating. This is the smoothing, the kernel bandwidth that I am adjusting.

And I can adjust the size of it here. Make it bigger. And do all of these adjustments on the fly.

As you can see, you’ve got some good power here for generating interactive plots and simulations. You can do these natively in Mathematica, or you do live manipulation of results coming from R. This example comes from the Mathematica guys:

mathematicaRPlotWrapper = RFunction["function(filename, plotfun){
     pdf(filename)
     plotfun()
     dev.off()
     }"];
 
Clear[getRPlot];
getRPlot[plotFun_RFunction] := 
  With[{tempfile = FileNameJoin[{$TemporaryDirectory, "temp.pdf"}]}, 
   If[FileExistsQ[tempfile], DeleteFile[tempfile]];
   mathematicaRPlotWrapper[tempfile, plotFun];
   If[! FileExistsQ[tempfile], Return[$Failed]];
   Import[tempfile]];
 
myFun[t_] := 
 Show[#, ImageSize -> Medium, PlotRange -> All] &@
  getRPlot[RFunction["function(){
        x<- seq(1," <> ToString@t <> ",0.1)
        y<- sin(x)
        plot(y~x)
        }"]]

What’s going to happen here is I am calling an R function, doing all of my calculations, bringing them back into Mathematica.

I forgot the “Manipulate” part:

Manipulate[myFun[t], {t, 2, 10}]

So here we go. And what’s happening is everything is being sent to R for processing then coming all the way back to Mathematica. As you can see even though we are making that round trip the results are coming back at a good pace, it’s almost instantaneous this particular example.

What about speed though, more generally?

I tried creating some random variates like I did in my examination of JavaScript versus R. So I’m going to create 10 million random variates from a Normal distribution

Timing[data = RandomVariate[NormalDistribution[], 10^7];]

and that takes about a quarter of a second, a little bit more, which is about twice as fast as R for generating those.

But then let’s look at more of a worst-case scenario, a bunch of nested loops.

Timing[
 l = 0;
 For [i = 0, i < 10^2, i = i + 1,
  For[j = 0, j < 10^2, j = j + 1,
   For[k = 0, k < 10^2, k = k + 1,
   l = l + 1
   ]
  ]
  ]
 ]

Three of them, a total of a million loops it’s going through, and this takes about 1.2 seconds, and it will increase by an order of magnitude if I add an order of magnitude in here. That is slow, it’s about twice as slow as the same code would take to run in R, a language not known for it’s speed with loops. Of course, this is generally speaking not what you want to do if you want to run fast code. Mathematica itself on their website advises against using these kinds of procedural codes.

But at the same time I’ve found that there are times when there really is no other way to do things, especially if you are doing simulations with a bunch of objects that take place over time, that are iterative, in which case you do need a programming structure that is like this, so you’ll have to keep this in mind in terms of the speed.

Beside the live graphical manipulation of the results, another key benefit to using Mathematica is that you can do direct probability calculations, and sometimes even find closed form solutions for combining random variables.

I’ve got some code here that determines the probability that one standard normal variable will be greater than another.

Probability[
 Subscript[x, 1] > Subscript[x, 
  2], {Subscript[x, 1] \[Distributed] NormalDistribution[0, 1], 
  Subscript[x, 2] \[Distributed] NormalDistribution[0, 1]}]

And it is, of course, one-half. That’s a simple example. We can do things that are more complicated. I’m going to look here for a more general solution when you have two Normal variables with means μ1 and μ2 and we’re going to try and find the probability that one of them is greater than the other.

Probability[
 Subscript[x, 1] > Subscript[x, 
  2], {Subscript[x, 1] \[Distributed] 
   NormalDistribution[Subscript[\[Mu], 1], 1], 
  Subscript[x, 2] \[Distributed] 
   NormalDistribution[Subscript[\[Mu], 2], 1]}]

As you can see it’s taking more time to run this calculation. And eventually we get back the response. And we do have a closed form solution.

“Erfc” stands for the complementary error function.

Unfortunately, not all of the problems that I tried to do work and sometimes they just freeze up completely.

Probability[
 Subscript[x, 3] > Subscript[x, 
  4], {Subscript[x, 3] \[Distributed] PoissonDistribution[5], 
  Subscript[x, 4] \[Distributed] PoissonDistribution[10]}]

Here I’m running a query to Mathematica to try and find the probability that a Poisson(5) will be greater than a Poission(10) random variable. I found that this just freezes up and will not return a response. Though I only waited a certain number of minutes. Actually, one time it locked my compter up entirely, the other time I gave up after a few minutes. I’m going to hit Alt-comma to abort the command and back out of that.

So, by comparison to R, you can do the same calculation of two Poissions. I’m going to make sure that’s run in R:

x = rpois(10^6,5); y=rpois(10^6,10); z = x<y; length(z[z==TRUE])/length(x)

(NOTE: This code actually finds the probability that a Poission(5) is LESS than a Poission(10))

As you can see I’ve run that a couple times already, and it takes about .9 seconds to run this. Of course, this is not trying to find a closed form solution, but for me anyway, these are perfectly good solutions, numerical solutions, to the problems.

So, going back in to Mathematica. Besides getting closed form solutions for probabilities, you can combine distributions, do convolutions and see what kind of formula you get back out.

I found this worked fairly well for simple, well-know distributions:

dist = TransformedDistribution[u + v, 
 {u \[Distributed] NormalDistribution[μ1, σ1], 
  v \[Distributed] NormalDistribution[μ2, σ2]}];
PDF[dist, x]

I do it here for the Normal. I’m adding two Normally distributed variables together and we get back out very quickly the formula for that.

But what happens if we try to work with a less well known distribution. In fact, lets go ahead and see what happens if we want to add together cats and find out the final weight distribution.

dist = TransformedDistribution[u + v, 
 {u \[Distributed] BeniniDistribution[\[Alpha]1,\[Beta]1,\[Sigma]1], 
  v \[Distributed] BeniniDistribution[\[Alpha]2, \[Beta]2,\[Sigma]2]}];
PDF[dist, x]

And, apparently, cats don’t add so well. I tried this one as well a couple times and wasn’t able to get results returned from Mathematica unfortunately.

So, besides these bugs and issues, there are a couple other significant downsides to Mathematica. Sharing results over the Internet can be done: you can export notebooks to HTML, but people if they want to use the interactive graphs they’ll need to have a special browser plugin installed. I asked the guys from Wolfram if they know what percent of web users already have it installed. They didn’t know, I suspect the number is very low, much lower than, say, the number who have Flash installed. Of course R by itself doesn’t provide much support for sharing results over the web, though Rstudio makes something called Shiny that can do some exporting over the web. I haven’t checked that out very much yet. I plan to do that soon.

So, beyond sharing, the biggest impediment to using Mathematica on a daily basis is the interface. The GUI. I’ll bring in some of the buttons here, the palettes of buttons. Overall the look is extremely crude. Things are disordered. The floating palettes have buttons of various sizes different places, and it looks essentially like something you might have hacked together in Visual Basic 15 years ago and then hadn’t been touched since. Clearly, the modern era of interface design has passed Mathematica by. At this point even open source programs that began with horrific interfaces, like the Gimp, have now begun to focus on making their interfaces look good. And looks may seem superficial, but if you are going to be working with an interface like this, if you are going to be paying $1000 for a license of Mathematica, I don’t think it’s too much to expect that the design be easy to use, that it be easy to find things, and that there are cues as to where to find things. Color would certainly go a long way to help with that, as would other cues to help you.

I’m going to bring back in Mathematica here.

Based on the GUI, I wonder… One more thing about the GUI, if you’re moving the palettes along it doesn’t do ghosting, so it pops back in.

So, despite these issues, I can see using Mathematica as an occasional tool to find exact results for probabilities and distributions, or quickly testing out how changing parameters affects a distribution’s shape. For now though, I’m going to continue experimenting with JavaScript as an alternative to R that runs code quickly and also I’ll be looking some more into Shiny.

Make sure to check out the links related to this video at StatisticsBlog.com, and if you like these videos click on the subscribe button.


28
Feb 13

Statistical computation in JavaScript — am I nuts?

Over the past couple weeks, I’ve been considering alternatives to R. I’d heard Python was much faster, so I translated a piece of R code with several nested loops into Python (it ran an order of magnitude faster). To find out more about Mathematica 9, I had an extended conversation with some representatives from Wolfram Research (Mathematica can run R code, I’ll post a detailed review soon). And I’ve been experimenting with JavaScript and HTML5′s “canvas” feature.

JavaScript may seem like an unlikely competitor for R, and in may ways it is. It has no repository of statistical analysis packages, doesn’t support vectorization, and requires the additional layer of a web browser to run. This last drawback, though, could be it’s killer feature. Once a piece of code is written in JavaScript, it can be instantly shared with anyone in the world directly on a web page. No additional software needed to install, no images to upload separately. And unlike Adobe’s (very slowly dying) Flash, the output renders perfectly on your smartphone. R has dozens of packages and hundreds of options for charts, but the interactivity of these is highly limited. JavaScript has fewer charting libraries, but it does have some which produce nice output.

Nice output? What matters is the content; the rest is just window dressing, right? Not so fast. Visually pleasing, interactive information display is more than window dressing, and it’s more in demand than ever. As statisticians have stepped up their game, consumers of data analysis have come to expect more from their graphics. In my experience, users spend more time looking at graphs that are pleasing, and get more out of charts with (useful) interactive elements. Beyond that, there’s a whole world of simulations which only provide insight if they are visual and interactive.

Pretty legs, but can she type?
Alright, so there are some advantages to using JavaScript when it comes to creating and sharing output, but what about speed? The last time I used JavaScript for a computationally intensive project, I was frustrated by its slow speed and browser (usually IE!) lockups. I’d heard, though, that improvements had been made, that a new “V8″ engine made quick work of even the nastiest js code. Could it be true?

If there’s one thing I rely on R for, it’s creating random variables. To see if JavaScript could keep up on R’s home court, I ran the following code in R:

start = proc.time()[3]
x = rnorm(10^7,0,1)
end = proc.time()[3]
cat(start-end)

Time needed to create 10 million standard Normal variates in R? About half-a-second on my desktop computer. JavaScript has no native function to generate Normals, and while I know very little about how these are created in R, it seemed like cheating to use a simple inverse CDF method (I’ve heard bad things about these, especially when it comes to tails, can anyone confirm or deny?). After some googling, I found this function by Yu-Jie Lin for generating JS Normals via a “polar” method:

function normal_random(mean, variance) {
  if (mean == undefined)
    mean = 0.0;
  if (variance == undefined)
    variance = 1.0;
  var V1, V2, S;
  do {
    var U1 = Math.random();
    var U2 = Math.random();
    V1 = 2 * U1 - 1;
    V2 = 2 * U2 - 1;
    S = V1 * V1 + V2 * V2;
  } while (S > 1);
 
  X = Math.sqrt(-2 * Math.log(S) / S) * V1;
//  Y = Math.sqrt(-2 * Math.log(S) / S) * V2;
  X = mean + Math.sqrt(variance) * X;
//  Y = mean + Math.sqrt(variance) * Y ;
  return X;
  }

So how long did it take Yu-Jie’s function to run 10 million times and store the results into an array? In Chrome, it took about half-a-second, same as in R (in Firefox it took about 3 times as long). Got that? No speed difference between R and JS running in Chrome. For loops, JS seems blazing fast (compared to R). Take another look at the demo simulation I created. Each iteration of the code requires on the order of N-squared operations, and the entire display area is re-rendered from scratch. Try adding new balls using the “+” button and see if your browser keeps up.

It’s only a flesh wound!
So have I found the Holy Grail of computer languages for statistical computation? That’s much too strong a statement, especially given the crude state of JS libraries for even basic scientific needs like matrix operations. For now, R is safe. In the long term, though, I suspect the pressures to create easily shared, interactive interfaces, combined with improvements in speed, will push more people to JS/HTML5. Bridges like The Omega Project (has anyone used this?) might speed up the outflow, until people pour out of R and into JavaScript like blood from a butchered knight.


22
Feb 13

What’s my daughter listening to? HTML chart gen in R

 

My daughter, who turns 10 in April, has discovered pop music. She’s been listing to Virgin Radio 99.9, one of our local stations. Virgin provides an online playlist that goes back four days, so I scraped the data and brought it into R. The chart shown at top shows all of the songs played from February 17th through the 20th, listed by frequency.

Broadly speaking, the data follows a power law. But only broadly speaking. Instead of a smoothly shaped curve from the single most frequently played song to a tail of single plays, Virgin Toronto has four songs that all share the heaviest level of rotation, then a drop-off of almost 50% to the next level. There was one big surprise in the data, at least for me. Listening to the station, it seems like they are playing the same 10 songs over and over. This impression is true to some extent, as the top 10 songs represented about one-third of all plays. But in just four days there were 57 single plays, and 44 songs played just twice. In all, 173 unique songs were played, with a much longer tail than I had expected.

That said, it would be interesting to compare Virgin’s playlist distribution with the widely eclectic (at least to my ears) Radio Paradise. Anyone want to give it a try? Here’s my code after I scraped the four pages of data by hand and put them into a text file.

To get the link to the Youtube videos, I used Google’s “I feel lucky” option paired with a search for the song name. If you get an unexpected result, take it up with Google. In the past I’ve used R’s “brew” library to generate HTML code from a template, this time I just hand coded the snippets. To make the red bars I found out the maximum number of plays for any song, then stretched each bar relative to this maximum.


14
Feb 13

Population simulation leads to Valentine’s Day a[R]t

Working on a quick-and-dirty simulation of people wandering around until they find neighbors, then settling down. After playing with the coloring a bit I arrived at the above image, which I quite like. Code below:

# Code by Matt Asher for statisticsblog.com
# Feel free to modify and redistribute, but please keep this notice 
 
maxSettlers = 150000
 
# Size of the area
areaW = 300
areaH = 300
 
# How many small movements will they make to find a neighbor
maxSteps = 200
 
# Homesteaders, they don't care about finding a neighbor
numbHomesteaders = 10
 
areaMatrix = matrix(0, nrow=areaW, ncol=areaH)
 
# For the walk part
adjacents = array(c(1,0,1,1,0,1,-1,1,-1,0,-1,-1,0,-1,1,-1), dim=c(2,8))
 
# Is an adjacent cell occupied?
hasNeighbor <- function(m,n,theMatrix) {
	toReturn = FALSE
	for(k in 1:8) {
		yCheck = m + adjacents[,k][1]
		xCheck = n + adjacents[,k][2]
		if( !((xCheck > areaW) | (xCheck < 1) | (yCheck > areaH) | (yCheck < 1)) ) {
			if(theMatrix[yCheck,xCheck]>0) {
				toReturn = TRUE
			}
		}
	}
	return(toReturn)
}
 
 
# Main loop
for(i in 1:maxSettlers) {
	steps = 1
	xPos = sample(1:areaW, 1)
	yPos = sample(1:areaH, 1)
 
	if(i <= numbHomesteaders) {
		# Seed it with homesteaders
		areaMatrix[xPos,yPos] = 1
	} else {
		if(areaMatrix[xPos,yPos]==0 & hasNeighbor(xPos,yPos,areaMatrix)) {
			areaMatrix[xPos,yPos] = 1
		} else {
			spotFound = FALSE
			outOfBounds = FALSE
 
			while(!spotFound & !outOfBounds & (steps<maxSteps)) {
 
				# Look for a new location in one of adjacent 9 cells, while still in area
				steps = steps + 1
				movement = adjacents[,sample(1:8,1)]
				xPos = xPos + movement[1]
				yPos = yPos + movement[2]
 
				if( (xPos > areaW) | (xPos < 1) | (yPos > areaH) | (yPos < 1)) {
					outOfBounds = TRUE
				} else if(hasNeighbor(xPos,yPos,areaMatrix) ) {
					areaMatrix[xPos,yPos] = steps
					spotFound = TRUE
				}
			}
		}
 
	}
 
}
 
image(areaMatrix, col=rev(rgb(seq(0.01,1,0.01),seq(0.01,1,0.01),seq(0.01,1,0.01))))
 
# I think this version looks nicer!
# areaMatrix[areaMatrix !=0] = 1
# image(areaMatrix, col=rev(rgb(.5,0,seq(0.2,1,0.2))))