(This article was first published on bayesianbiologist » Rstats, and kindly contributed to R-bloggers)

In honour of π day (03.14 – can’t wait until 2015~) , I thought I’d share this little script I wrote a while back for an introductory lesson I gave on using Monte Carlo methods for integration.

The concept is simple – we can estimate the area of an object which is inside another object of known area by drawing many points at random in the larger area and counting how many of those land inside the smaller one. The ratio of this count to the total number of points drawn will approximate the ratio of the areas as the number of points grows large.

If we do this with a unit circle inside of a unit square, we can re-arrange our area estimate to yield an estimate of  π!

This R script lets us see this Monte Carlo routine in action:

##############################################
### Monte Carlo Simulation estimation of pi ##
## Author: Corey Chivers                    ##
##############################################

rm(list=ls())

options(digits=4)

## initialize ##
N=500 # Number of MC points
points <- data.frame(x=numeric(N),y=numeric(N))
pi_est <- numeric(N)
inner <-0
outer <-0

## BUILD Circle ##
circle <- data.frame(x=1:360,y=1:360)

for(i in 1:360)
{
circle$x[i] <-0.5+cos(i/180*pi)*0.5
circle$y[i] <-0.5+sin(i/180*pi)*0.5
}

## SIMULATE ##
pdf('MCpiT.pdf')

layout(matrix(c(2,3,1,1), 2, 2, byrow = TRUE))
for(i in 1:N)
{

# Draw a new point at random
points$x[i] <-runif(1)
points$y[i] <-runif(1)

# Check if the point is inside
# the circle
if( (points$x[i]-0.5)^2 + (points$y[i]-0.5)^2 > 0.25 )
{
outer=outer+1
}else
{
inner=inner+1
}

current_pi<-(inner/(outer+inner))/(0.25)
pi_est[i]= current_pi
print(current_pi)

par(mar = c(5, 4, 4, 2),pty='m')
plot(pi_est[1:i],type='l',
main=i,col="blue",ylim=c(0,5),
lwd=2,xlab="# of points drawn",ylab="estimate")
# Draw true pi for reference
abline(pi,0,col="red",lwd=2)

par(mar = c(1, 4, 4, 1),pty='s')
plot(points$x[1:i],points$y[1:i],
col="red",
main=c('Estimate of pi: ',formatC(current_pi, digits=4, format="g", flag="#")),
cex=0.5,pch=19,ylab='',xlab='',xlim=c(0,1),ylim=c(0,1))
lines(circle$x,circle$y,lw=4,col="blue")
frame() #blank

}
dev.off()
##############################################
##############################################

The resulting plot (multi-page pdf) lets us watch the estimate of π converge toward the true value.

At 500 sample points, I got an estimate of 3.122 – not super great. If you want to give your computer a workout, you can ramp up the number of iterations (N) and see how close your estimate can get. It should be noted that this is not an efficient way of estimating π, but rather a nice and simple example of how Monte Carlo can be used for integration.

In the lesson, before showing the simulation, I started by having students pair up and manually draw points, plot them, and calculate their own estimate.

If you use this in your classroom, drop me a note and let me know how it went!



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Frances Homans:

Maybe this could go into your "teaching" category.  I have students keep a log of externalities, both positive and negative, that they both generate and experience.  Winston Oakley sent me this, which I thought was env-econ worthy.  He gave me permission to send it to you.

I wake up in the morning to my alarm. I start flipping on lights. I turn on the radio to listen to the news. I cook my breakfast on my electric stove. Everything uses electricity; electricity generated by a coal-burning power plant. That burnt coal sends tons of greenhouse gases into the atmosphere, degrading air quality and contributing to global warming. But those costs aren’t included in my electric bill. The first externality of the day is on me.I get ready and head to the bus stop. I hop on the bus and ride to campus. It gets me there by burning diesel. Once again, I’m degrading air quality and contributing to global warming. These costs aren’t covered [in] the price of the bus fare. Another negative externality on my part.After attending a class I head to the student lounge to study. I sit down and start looking over my class materials. My concentration is broken when two other students start up a boisterous conversation. They’re having a great time, but my ability to study suffers. Negative externality on them. A few minutes later, a friend of mine comes into the lounge and it’s our turn create an externality. Now our conversation interferes with their work. Payback sucks.My classes get done and I wait at the bus stop to head home. Another student walks up and stands nearby, also waiting for the bus. He decides to take this opportunity to enjoy a cigarette. The smoke blows over to the air I’m breathing. He gets all the enjoyment from smoking, I get the second-hand externality.I get back to my apartment building and walk inside. It’s dinner time and the smell of each resident’s cooking seeps out into the hallway. It smells awful. They can’t really be eating something that smells like that, can they? Maybe it tastes better than it smells. Either way, they get the benefit of eating it; I have to suffer through the smell. I get in my apartment and make my own dinner. I wonder if it smells that bad out in the hallway…I strum the guitar for a while to unwind. I play hard and sing loud. The walls of my building aren’t that thick, so I know my neighbors can hear every note. They didn’t have to pay extra to live next to a talented musician. Nope, they get that all for free. At least I finish the day with a positive externality. If any of my neighbors say differently they’re just jealous.

Thanks Frances! Thanks Winston!

Environment economics classroom example of the day (I'm not even teaching the course, I do this only for you*):

Is the [Keystone XL] pipe in the national interest?

Addressing that question, though — especially in the sprawling sweep of six huge states through which the pipeline or its pump stations would run like a spine — takes in a universe of conflicting, interlocking issues, from short-term economics to global climate, from the discontent of a rural belt losing population to issues of national energy security, joblessness, corporate power and prices at the corner pump. ...

The State Department concluded last month that the project, Keystone XL, would cause minimal environmental impact if it was operated according to regulations, and the operator, TransCanada, has said the nearly 2,000-mile line would create 20,000 jobs in the United States. Opposition groups around the country, though, said the federal study did not consider the effects of a major spill, while supporters said the nation’s economy had continued to worsen, making Keystone XL all the more crucial. ...

Keystone XL’s opponents also point to a Enbridge Energy spill of 843,000 gallons of oil sands crude near Marshall, Mich., as an example of what can go wrong.

A little over a year after the spill, a 35-mile stretch of the Kalamazoo River remains closed. And just this week, Enbridge stated in federal filings that the cost associated with the spill, originally estimated at $585 million, might now increase by 20 percent.

via www.nytimes.com

Sometimes it helps to visualize the job gains. From Wikipedia, the U.S. section is 1379 miles long. At 5280 feet per mile, it is 7,281,120 feet long. That amounts to one job every 364 feet of pipe, or about the length of a football field including the endzones. So many jobs that those guys could wave to each other.

*And to solidify my microcelebrity status and a bit of Google Ad money.

I – and I believe many of my fellow economics professors – seek not to get our introductory students to learn specific facts or techniques but to begin to “think like an economist.”

I want them to abandon wishful thinking, good vs. evil analogies, just-so-stories and general ad hocery, in favor of treating human behavior as if it stemmed from some (perhaps unknown or even unknowable) set of systematic principles. In particular we are big on the notion that people respond to incentives.

Students it turns out are people. While “thinking like an economist” is often a bitter pill, this semester it went down much easier – and I think I see why. No less than half of the final papers were written on either immigration in general or the DREAM act in particular. A typical paragraph

Immigrants not only join the circular flow of the economy as labor, but also as consumers. They spend money on goods and services which results in firms having more revenue. Higher revenues tend to increase firm production to meet the higher demand of consumers. This increased production most likely would result in more jobs as firms expand to meet the needs of consumers. On a very basic level, this explains why immigration would only serve to expand the economy.

Unlike our discussions of taxes, rent control and the minimum wage “thinking like an economist” gives them in edge in an argument they already want to make: that the US should be more welcoming to immigrants.

While it is encouraging to see the tools taken up so easily, it is a warning that we cannot be sure students or the public generally has abandoned ad hocery when the systematic explanation does in fact suit their immediate needs.

Filed under: Bias and Rationality, Economics, Teaching

How an NFL quarterback taught complex analysis

Frank Ryan is not just a theorist, but is also a mathematician who specialized in complex analysis. He got his Ph.D from Rice University on “A Characterization of the Set of Asymptotic Values of a Function Holomorphic in the Unit Disc,” in 1965.

Today I would like to talk about how we learn, and how we teach.

I have a story to tell about Ryan—I just shared it the other day with Alan Kay—and he insisted I had to post on it. Right away. So here it is Alan.

Ryan was not only a professor at Case Western Reserve, where I was an undergraduate in the ’60’s, but he was at the same time the starting quarterback for the Cleveland Browns. The starting quarterback for a NFL football team, and a professor with a full teaching load. During the football season he taught his class early in the week, and then on Sundays he was behind the center taking the ball. Handing it off, throwing passes, and getting sacked—like any other quarterback in the league. He was one of the best quarterbacks of his day, and had great successes. For example, he appeared in three straight Pro Bowls.

I still remember listening to him explain a fine point in complex analysis on Tuesday, and then watching him on TV getting tackled, on Sunday. It was hard to believe, even though I knew it was the same person, the player being taken hard to the ground was my professor. The player being tackled knew how to throw a perfect spiral 40 yards, and also could go to the board and prove Picard’s Little Theorem. Amazing.

Today, I believe there is no chance a quarterback—or any player—on an NFL team would want to or be allowed to be a full time professor during the season. The game has gotten very technical, the pay is too great, and the stakes are too high for any team to allow this to happen. But, back when I was taking complex analysis it happened. Really.

I still recall wincing when he got sacked during one tough game. Then, a few days later I saw him in class, with his arm in a sling and his speech slightly slurred. I assumed the slurring came from pain killers he was taking for the shoulder injury. Sling or not we pushed on, deeper into the beautiful structure of complex analysis.

Ryan’s Seminar

As an undergraduate I took a seminar with Ryan on complex analysis. This was one of strangest classes I ever had in mathematics, and probably one of the best. It was a small group, about eight of us, taking his class on advanced topics in complex analysis. The course was based on a thin monograph, but Ryan did zero lecturing. Instead, at the beginning of each class he ran the following protocol:

He would shuffle up a deck of playing cards, and we all would gather around a table.
He then would deal out the cards one at a time face up on the table: we each got the cards landing in front of us.
There were two bad cards: the Queen of Spades and the Queen of Diamonds.

Once these two cards were dealt, the class really began. The player who was “lucky” enough to get the Queen of Spades went to the blackboard and was expected to start explaining from exactly where we stopped at the end of the last class. You were allowed to use the book or notes and you typically explained the proof of some theorem.

After half the class was over, it was the other “lucky” person—who got the Queen of Diamonds—to take over from the first student.

Sounds not too hard, but it was a killer. The main problem was the thin book’s concept of a proof was not a detailed proof, but at best a high level sketch. Proofs were filled with phrases like: “it is easy to see that is continuous,” or “it clearly follows that is never in the unit disk.” The person at the board would say these phrases, but Ryan would usually jump in with a simple “why?” Why indeed was continuous? Why indeed was never ?

Sometimes the student at the board could answer and we moved on to the next point, but often they got stuck. The rest of us could help and make suggestions—of course we were usually lost too. The class might stay on a single question for the entire first half of class. If this happened, then the next student would have to get up and try to convince Ryan why it was true.

The student who was up second had an interesting prediction problem. They had 45 minutes to prepare for their turn, but they had no idea where the first student would get to in their 45 minutes. I remember being in this position—half listening to the class while trying to guess where I would have to start explaining.

The cards, the Queen of Spades and the Queen of Diamonds, were picked as the “bad” cards for a reason. The first is, of course, the worst card to get in the game of hearts: the player who is stuck with this card gets 13 points. The second is based on the original movie “The Manchurian Candidate”. In the movie this card is used to trigger Laurence Harvey to follow orders without question. One of the great movies, in my opinion

Learning Methods

I sometimes wonder how we learn and how we should teach. A while ago I posted on EEE and still think about this—the Educational Extinct Event.

I hated Ryan’s class. One consequence of the way the class was organized was I learned relatively little material. In a more conventional class I think I would have learned more theorems, more proofs, more concrete facts from complex analysis.

I loved Ryan’s class. The class taught me to think on my feet—literally. I learned how to read a proof and find the “gaps” I needed to fill in myself. I may have learned relatively little complex analysis, but I learned a great deal about mathematics in general.

By the way Picard’s Little Theorem, named for Charles Picard, is:

Theorem: Suppose is an entire function. Then, the range of is either the whole complex plane or the plane minus a single point.

The function shows the theorem is best. A very beautiful theorem.

Open Problems

The main open problem is this: what is the best way to present mathematical material? I am especially interested in hearing what you think about Ryan’s method. Did you have some similar experience? Should we teach more classes in this way? Or is it better to cover lots of material?

I am currently at the major meeting of the ASL—the Association of Symbolic Logic. I will give you an update on what it is like at a logic meeting. The reason I am here is to chair a session held in honor of the Gödel lecturer, who this year is our own Sasha Razborov.