The question posed for the pinball league was: can we say which of the players most improved over the season? I had data. I had the rankings of each of the players over the course of eight league nights. I had tools. I’ve taken statistics classes.
Could I say what a “most improved” pinball player looks like? Well, I can give a rough idea. A player’s improving if their rankings increase over the the season. The most-improved person would show the biggest improvement. This definition might go awry; maybe there’s some important factor I overlooked. But it was a place to start looking.
So here’s the first problem. It’s the plot of my own data, my league scores over the season. Yes, league night 2 is dismal. I’d had to miss the night and so got the lowest score possible.
Is this getting better? Or worse? The obvious thing to do is to look for a curve that goes through these points. Then look at what that curve is doing. The thing is, it’s always possible to draw a curve through a bunch of data points. As long as there’s not something crazy like there’s four data points for the same league night. As long as there’s one data point for each measurement you can always connect those points to some curve. Worse, you can always fit more than one curve through those points. We need to think harder.
Here’s the thing about pinball league night results. Or any other data that comes from the real world. It’s got noise in it. There’s some amount of it that’s just random. We don’t need to look for a curve that matches every data point. Or any data point particularly. What if the actual data is “some easy-to-understand curve, plus some random noise”?
It’s a good thought. It’s a dangerous thought. You need to have an idea of what the “real” curve should be. There’s infinitely many possibilities. You can bias your answer by choosing what curve you think the data ought to represent. Or by not thinking before you make a choice. As ever, the hard part is not in doing a calculation. It’s choosing what calculation to do.
That said there’s a couple safe bets. One of them is straight lines. Why? … Well, they’re easy to work with. But we have deeper reasons. Lots of stuff, when it changes, looks like it’s changing in a straight line. Take any curve that hasn’t got a corner or a jump or a break in it. There’s a straight line that looks close enough to it. Maybe not for long, but at least for some stretch. In the absence of a better idea of what ought to be right, a line is at least a starting point. You might learn something even if a line doesn’t fit well, and get ideas for why to look at particular other shapes.
So there’s good, steady mathematics business to be found in doing “linear regression”. That is, find the line that best fits a set of data points. What do we mean by “best fits”?
The mathematical community has an answer. I agree with it, surely to the comfort of the mathematical community. Here’s the premise. You have a bunch of data points, with a dependent variable ‘x’ and an independent variable ‘y’. So the data points are a bunch of points, for a couple values of j. You want the line that “best” matches that. Fine. In my pinball league case here, j is the whole numbers from 1 to 8. is … just j again. All right, as happens, this is more mechanism than we need for this problem. But there’s problems where it would be useful anyway. And for , well, here:
For the linear regression, propose a line described by the equation . No idea what ‘m’ and ‘b’ are just yet. But. Calculate for each of the values what the projection would be, that is, what . How far are those from the actual data?
Are there choices for ‘m’ and ‘b’ that make the difference smaller? It’s easy to convince yourself there are. Suppose we started out with ‘m’ equal to 0 and ‘b’ equal to 472. That’s an okay fit. Suppose we started out with ‘m’ equal to 100,000,000 and ‘b’ equal to -2,038. That’s a crazy bad fit. So there must be some ‘m’ and ‘b’ that make for better fits.
Is there a best fit? If you don’t think much about mathematics the answer is obvious: of course there’s a best fit. If there’s some poor, some decent, some good fits there must be a best. If you’re a bit better-learned and have thought more about mathematics you might grow suspicious. That term ‘best’ is dangerous. Maybe there’s several fits that are all different but equally good. Maybe there’s an endless series of ever-better fits but no one best. (If you’re not clear how this could work, ponder: what’s the largest negative real number?)
Good suspicions. If you learn a bit more mathematics you learn the calculus of variations. This is the study of how small changes in one quantity change something that depends on it; and it’s all about finding the maxima or minima of stuff. And that tells us that there is, indeed, a best choice for ‘m’ and ‘b’.
(Here I’m going to hedge. I’ve learned a bit more mathematics than that. I don’t think there’s some freaky set of data that will turn up multiple best-fit curves. But my gut won’t let me just declare that. There’s all kinds of crazy, intuition-busting stuff out there. But if there exists some data set that breaks linear regression you aren’t going to run into it by accident.)
So. How to find the best ‘m’ and ‘b’ for this? You’ve got choices. You can open up DuckDuckGo and search for ‘matlab linear regression’ and follow the instructions. Or ‘excel linear regression’, if you have an easier time entering data into spreadsheets. If you’re on the Mac, maybe ‘apple numbers linear regression’. Follow the directions on the second or third link returned. Oh, you can do the calculation yourself. It’s not hard. It’s just tedious. It’s a lot of multiplication and addition and you know what? We’ve already built tools that know how to do this. Use them. Not if your homework assignment is to do this by hand, but, for stuff you care about yes. (In Octave, an open-source clone of Matlab, you can do it by an admirably slick formula that might even be memorizable.)
If you suspect that some shape other than a line is best, okay. Then you’ll want to look up and understand the formulas for these linear regression coefficients. That’ll guide you to finding a best-fit for these other shapes. Or you can do a quick, dirty hack. Like, if you think it should be an exponential curve, then try fitting a line to x and the logarithm of y. And then don’t listen to those doubts about whether this would be the best-fit exponential curve. It’s a calculation, it’s done, isn’t that enough?
Back to lines, back to my data. I’ll spare you the calculations and show you the results.
Done. For me, this season, I ended up with a slope ‘m’ of about 2.48 and a ‘b’ of about 451.3. That is, the slightly diagonal black line here. The red circles are what my scores would have been if my performance exactly matched the line.
That seems like a claim that I’m improving over the season. Maybe not a compelling case. That missed night certainly dragged me down. But everybody had some outlier bad night, surely. Why not find the line that best fits everyone’s season, and declare the most-improved person to be the one with the largest positive slope?