# Is there any prediction model that takes into account the regression towards the mean?

I am studying performance of football and basketball teams. The regression towards the mean is quite common in sports. One really great performance is usually followed by a less perfect one. So are there any prediction models that they take this phenomenon into account?

For example I have data for points scored by a basketball player. The average points scored in the last 100 games tells us about the quality of the player. The average points scored in the last 3 games tells us about the quality of the player as well, but in interaction the long term average it can tell us if the player has over/under achieved and will likely regress towards the mean in the future. What model could notice this interaction?

I will manually build a predictor which predicts points scored based on the long term average and the fact that somebody has over/under achieved in the short term.

• I read a blog series on using empirical bayes with baseball batting averages that might be of use: varianceexplained.org/r/simulation-bayes-baseball This is the final post, which links them all. "Empirical Bayes estimation" one may be appropriate. Idea is you include prior knowledge which would be your long term behaviour, and with your new data you would build a new distribution and can run tests to see how much it differs from the long term behaviour Aug 26, 2018 at 13:22

I think you misunderstand how the phenomenon arises.

It is the typical situation whenever there is random variation.

Let's look at the number of heads in 10 tosses of a fair coin. I do the set of 10 tosses and get 7 heads. Next I do another set of 10 tosses and get 4 heads, and so on. Let's imagine we like heads - 10 heads is a great performance for us and 0 heads is a terrible one, while 5 heads is average.

This will have exactly the property you mention -- a great performance will (nearly always) be followed by a less perfect one; similarly a terrible performance will nearly always be followed by a less terrible one.

But it's just a coin. It has no memory, it can't learn, it can't feel good or bad and it has no 'streaks' of performance (nor does it possess any 'anti-streakiness'). It simply knows nothing of its previous performance. There's no information in the previous good performance that will help us predict the number of heads next time.

It's just randomness.

If I get 10 heads, the chance that I get fewer than 10 heads is 1-P(10 heads) = 1023/1024 (so there's definitely a high chance I will perform less well the next time). But that was also the chance I'd get fewer than 10 heads if I previously got 5 heads -- nothing changed. The "10 heads" is of no value in prediction of the following outcome.

Now it may be that there's something more than this simple form of 'regression to the mean' going on in your favourite sport but you'd need to investigate to find out -- there's nothing inherent in the notion that good performances are followed by less good ones that suggest there's anything valuable in seeing it, it's a natural consequence of typical sorts of variation from one performance to the next and may be nothing more than the vicissitudes of chance.

Even in situations where good performances happen in streaks (above average performance tends to follow an above average performance), you would nevertheless still see that a great performance would nearly always be followed by a less outstanding one; taller than average fathers have taller than average sons, but on average the sons of the tall fathers are more often shorter than their fathers. This is regression to the mean.