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Thread: Jeff Russell: Mathematical Modeling of Attempt Selection

  1. #1
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    Default Jeff Russell: Mathematical Modeling of Attempt Selection

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  2. #2
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    I don't think P_1 and P_2 are calculated correctly.

    Looking at P1. Realistically you only have one way to succeed making the first attempt and fail the others (assuming that the 2nd and 3rd attempts both use w_2). You succeed with w_1 with a probability of p_1 and you fail 2x at w_2 with a probability (1-p_2)^2.

    P1 = Prob(w_1 && !w_2 && !w_2) = p_1 ( 1-p_2 )^2

    The way p_1 being calculated in the paper would be equivalent to
    succeeding the first time the selecting the same weight 2x more and failing OR
    succeeding the first time the selecting the same weight failing and then selecting the second weight and failing OR
    succeeding the first time then selecting the 2nd weight and failing 2x.

    Only the last one is needed because no one succeeds an attempt and then takes the next attempt at the same weight because it would be pointless.

    So

    P0 = (1-p_1)^3
    P1 = p_1 ( 1-p_2 )^2
    P2 = p_1 p_2 (1-p_3)
    p2 = p_1 p_2 p_3

    Also a note on common definitions

    w_1 P1 + w_2 P2 + w_3 P3 is the expected value of the weight lifted not the average value of the weight lifted

    (w_1 + w_2 + w_3 )/3 is the average value of weight lifted.

    Possible outcomes. Let S = success, F = Failure

    FFF : P = (1-p1)^2 : W = 0
    FFS : P = (1-p1)^2 : W = w_1
    FSF : P = (1-P1)p2(1-p2) : W = w_1
    SFF : P = p1(1-p2)^2 : W = w_1
    FSS : p = (1-p1)p1p2 : W = w_1 + w_2
    SFS : p = p1(1-p2)p2 : W = w_1 + w_2
    SSF : p = p1p2(1-p3) : W = w_1 + w_2
    SSS : p = p1p2p3 : W = w_1 + w_2 + w_3


    This should be the correct expected value

    Expected Value = (1-p1)^2 w_1 + (1-P1)p2(1-p2) w_1 + p1(1-p2)^2 w_1 + (1-p1)p1p2 ( w_1 + w_2 ) + p1(1-p2)p2 ( w_1 + w_2 ) + p1p2(1-p3) ( w_1 + w_2 ) + p1p2p3 (w_1 + w_2 + w_3 )

  3. #3
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    Quote Originally Posted by Raymondo View Post
    I don't think P_1 and P_2 are calculated correctly.
    I appreciate the spirit of the comment and the attention you've given my article, but I think there is some confusion about terms here which is leading to errors in your formulations.

    To clarify, w1, w2, and w3 are the three weights to attempt for a single lift (e.g. the squat). This probability exercise would be repeated three times for a full power meet, with three 'w' values chosen for each lift. In the table of equations in the article, these 'w' values are also (perhaps confusingly) referred to as the "first weight" (w1), "second weight" (w2) and "third weight" (w3). The w values are always chosen such that w1 < w2 < w3. Perhaps this was already clear, I'm not sure, but it's critical to what follows.

    Let's take the example then of P1, and why it is correct. P1 is the probability of ending the event with a score of w1 (that is, making only the "first planned weight"). There are three ways a person can do this:

    miss w1, miss w1, make w1, or
    miss w1, make w1, miss w2, or
    make w1, miss w2, miss w2

    which corresponds to:

    (1-p1)*(1-p1)*(p1) +
    (1-p1)*(p1)*(1-p2) +
    (p1)*(1-p2)*(1-p2)

    which is what's in the article. Your formulation of P1 fails to account for the first two terms (reading as only (p1)*(1-p2)*(1-p2)), and is for this reason incomplete. It seems like maybe you misunderstood the purpose of P1, and were instead formulating the probability of "making your first attempt only", which is different than "making w1 only, on any attempt", which is what we actually need to know.

    Similar reasoning applies to the P2 equation.

    Quote Originally Posted by Raymondo View Post
    w_1 P1 + w_2 P2 + w_3 P3 is the expected value of the weight lifted not the average value of the weight lifted

    (w_1 + w_2 + w_3 )/3 is the average value of weight lifted.
    The expected value and the average result (long-run average) are the same thing. See Expected value - Wikipedia. I use the term "average" here to mean "average resulting score", i.e. the expected value. The equation you've written is the "average of the planned attempts" which is quite different, and I think not useful for our purposes here.

    Quote Originally Posted by Raymondo View Post
    Possible outcomes. Let S = success, F = Failure
    ...
    Lastly, we have your S/F table, which is almost correct. The primary flaw is the summing of the 'w' values next to each term. Meet events aren't scored in this way: a lifter is credited with the heaviest weight, not the sum of weights made. So for example in the last term where the lifter goes 3 for 3, the score is not w1+w2+w3, but rather just w3, since that is the heaviest weight.

    Adopting your format, the full S/F table should read:

    FFF: P = (1-p1)^3 : W = 0
    FFS: P = (1-p1)^2*p1 : W = w1
    FSF: P = (1-p1)*p1*(1-p2) : W = w1
    SFF: P = p1*(1-p2)^2 : W = w1
    FSS: P = (1-p1)*p1*p2 : W = w2
    SFS: P = p1*(1-p2)*p2 : W = w2
    SSF: P = p1*p2*(1-p3) : W = w2
    SSS: P = p1*p2*p3 : W = w3

    With the expected value then being the sum of each 'P' multiplied by its corresponding 'W', in the same fashion as you've done. This result will be equivalent to the 'Ravg' equation in the article.

    Hope this makes sense, and thanks for reading.

  4. #4
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    Great read, thank you.
    I played around with the calculator a bit, interesting.

    I find this mathematical approach is also beneficial psychologically, it's kinda soothing: if I fail a weight, it doesn't necessarily mean I wasn't strong enough and the weight is too heavy for me, period. It's not the new law, set in stone. Rather, failing is just part of attempting maximum weights. Try again. Maybe I can lift this weight one out of three (two, four...) attempts. It reduces the psychological impact of a failed lift.
    (so far, I've only maxed out in my home gym, not in a competition. But I'm planning on doing just that)

  5. #5
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    This is interesting, but I'm not sure if it's really all that helpful in practice. Sample size and the control of multiple variables seems to be a monster in creating a model like this. I'm sure it could be done, but I would think it would take multiple years of collecting an individual lifter's competition results. A more generic/less granular model (which I think you're describing) could definitely be pulled together much more quickly with the available dates but I would still think unmentioned variables (time of the day the lift was performed, previous missed attempts, any lifter health issues, etc, etc) would need to be understood and modeled as well.

    This could be some smart/strong PhD student's dissertation.

    Thanks for the article!

  6. #6
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    Quote Originally Posted by Alexander Dargatz View Post
    It reduces the psychological impact of a failed lift.
    I feel the same way, but I get the impression that some people just hate to talk about misses, like they'll be jinxed or something if it's uttered in their presence. I don't know how you make a suitable plan without thinking on misses and where they come from, though.

  7. #7
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    Quote Originally Posted by atw_abn View Post
    Sample size and the control of multiple variables seems to be a monster in creating a model like this.
    It is. My goal was to avoid trying to correlate every factor that goes into a lift's success, and simply encompass it all with a probability approximation. Even doing that is hairy enough, and avenues for improvement on this simplified approach almost write themselves. One could, as you suggest, try to model how your strength its self changes with some factors, and through what range, and then run attempts against that distribution to find a different maximum. A model could become a blend of approaches like that, about as complicated as we cared to make it to be really.

    I did some searching to try to find previous work on this but came up empty handed. I was sure some Russian coach must have run numbers like this in the 70's or something.

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    Anyone aware of machine learning being trained with lifter’s physical specs, training logs, and competition numbers?

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    Quote Originally Posted by VNV View Post
    Anyone aware of machine learning being trained with lifter’s physical specs, training logs, and competition numbers?
    Not aware of any for attempt selection. I know there are a couple systems that purport to produce training programs with machine learning (or maybe it's something simpler and "AI" sounds good in marketing). The Juggernaut folks have one they're selling subscriptions to these days.

  10. #10
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    I'm not sure why a person would need to worry about machine learning.

    If you download the analysis add on to excel, you would be able to create a multi variable equation that could give a pretty good answer.

    I'm thinking something like..

    competition squat = max last training weight squat + lifters weight (or any other spec) + days since last training (or whatever data you think would be important.) + last competition bench

    competition bench = max last training weight bench + lifters weight (or any other spec) + days since last training (or whatever data you think would be important) + last competition bench

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