putting some math to the problem of shot selection

[schtevie]

No, I thank you.

But first let me add a few more words. What (RPPP t) purports to represent is for each second of shot clock time the average number of points per possession that ensue from that point on.

The numerator of RPPP for t=n would include for t = n, n+1, ...24, all points scored prior to possession reverting to the opposing team. These include:

(1) points scored on all field goal attempts within the specified period of the initial shot clock,
(2) points scored on all foul shots within the specified period of the initial shot clock,
(3) and then the same again for all points scored on initial and any subsequent offensive rebounds.

The denominator of RPPP for t=n would include for t = n, n+1, ...24 all possessions within the interval. These are defined as the sum of:

(1) all field goal attempts within the specified interval of the initial shot clock.
(2) all shooting fouls within the specified interval of the initial shot clock (that is two or three shot, non-technical fouls only - that is no technicals and no "and 1s", the former not being possession based and the latter already being counted in the corresponding field goal attempt)
(3) all turnovers within the specified interval of the initial shot clock

I think that gets it right.

With this, you get a (Th t+x) that can be compared to a given scoring opportunity at time t (also to include, of course, the augment from expected foul shots and subsequent offensive rebounds).

Make sense? (Please let me know if I have misspecified something.)

[mcoughlin]

Also, "As suggested above, for each second of shot clock time, the probability of a turnover occurring prior to the next "look" (TO t+x, where x = 3 or 4)."

Is this just essentially taking the plot of turnover probability as a function of time and for each point, summing it with the probabilities of the next 3 or 4 seconds, so an average turnover probability weighted by the number of possessions?

Michael

[EvanZ]

If I understand correctly, the idea is that for time>=t seconds, the denominator (possessions) should not contain the possessions that occurred before time<t, because clearly, if you turn the ball over at 4 seconds, you can't turn it over again at 8 seconds. Same for shooting, rebounding, etc. Am I right?

[schtevie]

Hmmm. Thinking out loud here...

The goal is to get a "realistic" sense of the likelihood of a turnover between t and t+x.

I am not exactly sure how the turnover probability is defined in the graph you provide. I suspect it is the sum of turnovers within the shot clock remainder divided by all possessions, similarly defined.

This said, I don't know if that quotient best relates to the current concept, as all scoring opportunities are being assumed away in the 3 or 4 second interval before the next "look", and many realized turnovers are directly "caused" by scoring attempts.

I suppose the preferred approach would be to tabulate all TOs realized between t and t+x and have this be the numerator of the TO probability. Then the denominator would be defined as this TO sum added to the same possession count used in the denominator of RPPP t+x.

This, it seems to me, would bias the expected TO probability high (for the aforementioned effect) but it would be in the spirit of the exercise.

Make sense?

[schtevie]

If I understand correctly, the idea is that for time>=t seconds, the denominator (possessions) should not contain the possessions that occurred before time<t, because clearly, if you turn the ball over at 4 seconds, you can't turn it over again at 8 seconds. Same for shooting, rebounding, etc. Am I right?

Right. That's why you only count scoring opportunities (fga and foul shots) and turnovers occurring subsequent to t (and at or before 24 seconds).

[EvanZ]

The cumulative probability of a turnover should be a curve that starts at 0 at t=0 and goes to 14% (you'd have to normalize it or whatever to be 1) at t=24 s.

At time t (3 s, 6s, whatever), the probability of a *future* turnover is the integral of the cumulative probability for time>t. Correct? The probability of a turnover up until that point is the integral for time<t. That is a "mathy" way to state the problem.

[schtevie]

Hmmm. Thinking out loud here...

The goal is to get a "realistic" sense of the likelihood of a turnover between t and t+x.

I am not exactly sure how the turnover probability is defined in the graph you provide. I suspect it is the sum of turnovers within the shot clock remainder divided by all possessions, similarly defined.

This said, I don't know if that quotient best relates to the current concept, as all scoring opportunities are being assumed away in the 3 or 4 second interval before the next "look", and many realized turnovers are directly "caused" by scoring attempts.

I suppose the preferred approach would be to tabulate all TOs realized between t and t+x and have this be the numerator of the TO probability. Then the denominator would be defined as this TO sum added to the same possession count used in the denominator of RPPP t+x.

This, it seems to me, would bias the expected TO probability high (for the aforementioned effect) but it would be in the spirit of the exercise.

Make sense?

Actually, I don't think it makes sense. To not count the non-turnover, "dispensed" possessions in the interval "x" in the denominator would impart an absurdly high bias to the TO%. The actual TO count in the numerator alone imparts an upward bias.

Accordingly, the numerator should be the actual turnovers between t and t+x, and then the denominator should include:

(1) these same turnovers
(2) field goal attempts between t and t+x
(3) all shooting fouls between t and t+x (again, two or three-shot, non-technical fouls)
(4) and then all possessions as defined for RPPP t+x

I think this makes much better sense.

[mcoughlin]

Hi again,

I believe these to be drafts of the eventual plots, but I have not yet correctly accounted for putbacks from offensive rebounds though. The first is x = 3 and the second x = 4. You have to look closely to see that they differ.




I will note that for any t that was too close to the end of the shotclock to have a full t + x count, I just had it count as much as was available (i.e. for t = 23, t + x included 23 and 24 for both of them).

Thank you,
Michael

[Crow]

Michael's charts for fouls and turnovers vs elapsed time are interesting.

[schtevie]

Something(s) aren't right. The likelihood of a turnover within 3 or 4 second intervals shouldn't be the same (the latter should be about 1.33 larger, right?) and shouldn't be higher than the overall probability of turning the ball over. Perhaps my suggested definition is incorrect. Also, the RPPP at the start of the shot clock should very closely approximate league average offensive efficiency, last year's being 1.07.