How to predict the future. How mathematical models work.
Have you ever wondered how to predict outcomes of events? Prediction of outcomes is incredibly important both to researchers and to practitioners, whether your cause is noble or you're just looking for a profit. There are many areas where prediction plays an important role: weather prediction, stock prices, or even the outcome of a sporting event. What one usually needs is a mathematical model into which one inputs certain data, called parameters, and then a computer cranks out a prediction for you according to your model. Weather prediction requires sophisticated models and supercomputers that are capable of performing on the order of one quadrillion tasks per second. Stock price prediction requires mathematical and statistical techniques of stochastic processes and time series, which one typically has to go to graduate school to learn. So we'll stick to a simpler example: modeling the flipping of a coin.
Suppose I flip a coin. Will it come up heads or tails? Well, you might have learned that the probability of either outcome is exactly 1/2. Sure in a theoretical world it is, but in a real world we can think about the parameters which might affect the outcome: how high the coin is from the ground, how much force you apply to the coin, where on the coin you apply this force, what material the floor is made of (tile, hardwood, carpeted?), how much wind there is at the time, and so on. Even the humidity could be a factor, which influences the outcome! A model would be kind of like an equation so that if you plug all these parameters in, it will tell you whether the coin will land heads or tails.
Suppose for simplicity, that we are in a weather-controlled room whose temperature, humidity, and barometric pressure are all controlled and constant, with no wind. We then install an automatic coin flipper at a fixed height, which will apply any force we choose to a certain spot on the coin, whose measurements and composition we've recorded carefully. We also know all about the floor, so we can collect data on how the coin bounces when it hits the ground a certain way. Using some basic physics, we can figure out how many times the coin will rotate before it hits the ground, and we can figure out when, where, and how hard it hits the ground. We then calculate the motion of its bounce and again compute when and how hard it will hit the ground next, and so on until the coin doesn't have enough force to bounce again. Whatever side is up on the coin then will be the outcome of that flip.
Of course, as intuition will tell you, the more bounces a coin makes, the more error will play a role. We can't measure these parameters perfectly; there will always be some round off. But if the floor is carpeted, the coin will bounce less and our predicted outcome will be more accurate. The point of all this is that a coin flips in the real world isn't really random. In fact, I remember reading a story about how some college students at a top university modeled the game of roulette. They had a device, which roughly clocked the speed and position of the roulette wheel, the speed and position of the ball, and tried to predict the outcome of where the ball would land. Since I couldn't find that story on the web, I'll link to this: www.rouletteprediction.com. In fact if you Google "roulette prediction" or "predicting roulette" you'll get a bunch of websites which claim to have software which models the game and predicts the outcome. I'm pretty sure that using such a device in a casino is illegal however, so don't say I didn't warn you.
Being a football fan myself, I've gotten interested in trying to find a model, which will predict the outcome of an NFL game. These models are usually very secret, and there are websites that charge a hefty fee in exchange for telling you which winner their model predicted. To give an example of a simple model of predicting the winner of an NFL match up, you can try the following. Let's consider the New England Patriots versus the Indianapolis Colts. For each team, start by giving a rating of 0-9 (0 worst, 9 best) for the quarterback, running back, and two wide receivers. For instance, let's give Peyton Manning a rating of 8, Joseph Addai a rating of 4, Reggie Wayne a rating of 7, and Anthony Gonzalez a rating of 5. Now give a rating of 0-19 (0 worst, 19 best) for each team's defense and special teams. Say we give the Colt's a rating of 11. Then we add up all the points and compare the two teams. Here the Colt's would have a total rating of 35. We then compare it to the rating we gave the Patriots, and we set the predicted winner to be the team with the most points.
Of course, this is an absurdly simple model, and we picked the ranges (0-9 and 0-19) randomly. We also didn't factor in things like each team's record, individual members of the defense, the offensive lines, third down conversion rates, home field advantage, etc. Also, you might disagree on the individual ratings themselves, claiming for example that the Colt's defense should be 13 instead of 11 and that Peyton should be ranked lower at 6. These discrepancies are what make the modeling fun! How do you assign ratings? How much weight should you give to each bit of information? And so on. Have fun coming up with your models!