I don't know who you mean by "their".
How? You don't compute percentage of balls won, games won and matches won.
I was able to point out two things.
The quadratic regression was correct given the data and for that range, and does what it is supposed to do (give you an expected value for that range). Thus the confusion when you called it "wrong". But you are right, logistic regression is the proper regression to use here, as it accurately predicts beyond the range of supplied data, whereas the quadratic falls short of that task.Neither is anybody else. What is worse is that no one tried different rating differences. I thought that OldUser would figure it out since his graphs went up to a ratings difference well beyond 250.
Sure, math class begins.
First, what is wrong.
Have you heard of even and odd polynomials? In algebra class did you ever need to plot them out? If you plotted out your function beyond the range 0 to 250 you would see the problem.
Your equation is
The problem is the the -8.4E-6*x^2 is an even and looks like an upside down U with the peak at about 250. The percentages start to decrease. No one on the forum even bothered to try a ratings difference of 300, 400, 500 even though I hinted at it. The x^2 term should have been enough but then you should have tried numbers outside of the range from 0 to 250.
Look up odd and even functions. Khan academy has lessons on this topic.
Anyway, it is obvious that the -x^2 term would be a problem.
The correct formula is not a simple polynomial. It is sigmoid function. The theoretical percent vs ratings difference in OldUser's link is a sigmoid function. Notice that when the ratings difference is positive, the curve looks much like yours but the percentage will approach 1 as the rating difference increases, unlike your equation. A sigmoid function will also handle the case where the ratings difference is negative so the probabilities for the weaker player winning is less than 0.5. Your equation doesn't work when the ratings difference is negative.
Sigmoid function - Wikipedia
en.wikipedia.org
Sigmoid Function -- from Wolfram MathWorld
The sigmoid function, also called the sigmoidal curve (von Seggern 2007, p. 148) or logistic function, is the function y=1/(1+e^(-x)). (1) It has derivative (dy)/(dx) = [1-y(x)]y(x) (2) = (e^(-x))/((1+e^(-x))^2) (3) = (e^x)/((1+e^x)^2) (4) and indefinite integral intydx =...mathworld.wolfram.com
Now use the sigmoid function as your equation. It should be in the form of y=1/(1+exp(-k*x)). k determines how rapidly the sigmoid function reaches +/- 1 and the slope at a 0 ratings difference.
Note, a tanh() function will also yield a sigmoid.
Sigmoid functions are using in other ratings systems like rating central and in neural nets, anywhere where a number must be converted to a range of percentage from 0 to 1. I have used them for motion control. One more thing. In this application a ratings difference of 0 means the probability for winning is 0.5. Another application a x might need to be offset to get a probability of 0.5. In this case, offset x like this y=1/(1+exp(-K(x-b)) where b is the offset. Your curve fitting code should include an offset if trying to fit the sigmoid to raw data.
Your curve fitting is good but you can't assume a simple polynomial will match the data. Now apply your curve fitting program to ratings difference from -500 to +500 and plot it.
I have doubts about the so called "experts" in OldUser's link. The predicted graph doesn't match the actual data at all.
If you don't read the WHOLE ARTICLE written by Jeff Sonas, you won't get the point : it's always subjective, because it depends on what algorithm you choose to push your theory.The first graph with the picture doesn't make sense. Later we find it out is a graph of the predicted vs actual. The predicted line is not good.
The second graph, the one labeled "Theoretical Relationship:" makes sense.
The next graph titled "Ratings difference vs %Score(predicted) does not. A ratings difference of 900 points should not mean the stronger player wins only 92% of the time. This is non-sense.
The graph titled "Ratings difference vs %Score (actual) is real data that must have been gathered from many tournaments. I wonder which tournaments match players with a 900 point difference.
Yes, the USATT table is derived from the ELO system but the graph you showed doesn't make sense. The predicted data does not match the actual data at all. A real expert would be able to do this.
Actually, I bet if the theoretical and actual data were overlaid, they would match very closely.
I have no idea why they wanted that lame "predicted" graph.
This still doesn't address the problem with hcInnkhg's equation. This was taught in high school algebra. There are lots of videos and websites that cover this topic. That is why it should be obvious! Don't be so woke and butt hurt if you don't remember your high school algebra! I am 69 and I remember.
About ELO, I even remembered his first name
ELO is good but I like David Marcus's Ratings Central better.
Arpad Elo - Wikipedia
So what is f(x)? The reason I ask that y shouldn't be 1 if x>=238. Did you read OldUser's link?
www.protocols.io
What point? Fitting an equation to data isn't subjective. Yes, multiple forms of equations may need to be tried but one equation will result in minimizing the sum of squared errors. The goal is to have an equation that matches the data as closely as possible by minimizing the sum of squared errors between the data and the equation that that estimates the data.
f(x) is the original quadratic fit. I'm using a piecewise fit. y should be 1 if x >=238 because if when x>=238, the higher-rated player gains 0 rating point i.e. they are expected to win 100%. If this is wrong then you'd need to explain a bit more coz I don't get what you are getting at.So what is f(x)? The reason I ask that y shouldn't be 1 if x>=238. Did you read OldUser's link?
You are not using a sigmoid function.
OldUser's article show the actual data looks like a sigmoid function.
All you need to find is the k value in f(x)=1/(1+exp(-k*x)) that minimize the sum of squared errors between the table and f(x) . I suspect you are using Excel. In this case you must use and Add-In
Sigmoid fitting in Excel (Excel Solver Add-In)
This protocol covers how to fit sigmoidal curve to data within Excel, and allows rapid estimation of EC50/IC50 values from experimental dose-response data. Although R or other s...
BTW, when not playing TT you should look up the functions that Excel uses to minimize the sum of squared errors between your equation and the data. Gradient descent is a good start.
I am trying to show hcInnkng how to fit a curve to the data correctly.
How do you know? How do you measure that it is a close to reality as possible?
Reality is the actual win percentage vs ratings difference. We don't have the USATT data so what hcInnkng was trying to do is match an equation to the table but we really don't know how accurate the table is but it is all we have to go on.
It is not subjective. There is a best fit with the minimum sum of squared errors.
Is that all you care about? No one is ever wrong?
Glad to be able to lighten up your mood (until I get banned again). At this rate, it won't be long before I go dark again.
So is that my problem?
Rude? It was a clue that hcInnkngs equations should be verified. No one did. Is that my problem?
Yep, that’s your problem because you have no common sense of what is being a decent human being. But if you’re what you say you are, a broken ball, then you are not wrong. And yes, obvious to you, only, and if you don’t say what it is then it’ll still be you, only. So who’s the problem here?
Sure, I’m the instigator, but I believe everyone else knows what happened and who’s to blame. At times, someone has to speak up.
