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In competitive climbing, it is well known that there are many different styles, differentiated by e.g. the steepness of the climbing wall. Some athletes excel at certain styles more than others. I would like to see if these styles can be discovered empirically from competition data, such as that publicly available for climbing world cups (example).

My question is how to best infer styles from the available data. The nature of the data is as follows:

  • Each competition round consists of four or five climbs, and each athlete has a few minutes to try each climb, with unlimited attempts.
  • For each climb and each athlete, the scoring system records (roughly) whether the athlete made it halfway to the top or all the way to the top of the climb, along with how many attempts it took them to reach those two points.
  • Many athletes compete in many rounds of competition, but most do not participate in all rounds over a season. There are probably about 70-120 climbs total per season, and about 200 athletes who try them, but not every athlete tries every climb.

Because some athletes are better at certain styles, we expect to see certain groups of athletes perform better on certain climbs, and this clustering is how I'd expect styles to emerge from the data. The question is how to finesse this out.

My most basic thought would be to assign a scalar value to each athlete's performance on a given climb and then do principal component analysis. Two issues I see with this are:

  1. Dealing with the missing data, where athletes didn't attempt a given climb. There seem to be ways of dealing with this, e.g. as described on this stats.se post.
  2. There is substantial arbitrariness in assigning a scalar value to an athlete's performance on a given climb.

Are there better approaches to this problem? Are there principled ways of converting a discrete performance measure (like "reached the top" or "didn't reach the top") into a numeric value for PCA?

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  • $\begingroup$ How many styles are there, and are they distinct and unrelated? You mentioned steepness, and I wondered if that is a scalar (e.g. it could be anything from 5.0 deg to 10.0 deg), or if there are just e.g. steep, very steep, and this-is-silly ? What other factors contribute to style? (The link you gave didn't describe any of the style factors, as far as I could see, only the list of competitors.) $\endgroup$
    – Darren Cook
    Commented Jan 3, 2023 at 17:16
  • $\begingroup$ @DarrenCook There are several dimensions to style--perhaps 5-10. Wall steepness is one dimension, and is indeed a scalar, roughly ranging from -10 deg (less than overhanging, known as "slab climbing") to 90 deg (completely horizontal, known as "roof climbing"). However, the techniques involved can change almost discontinuously at certain angles--e.g. once the wall is less than vertical balance often becomes much more important than strength. So one could probably model the steepness as discrete if that were easier. (1/2) $\endgroup$
    – Yly
    Commented Jan 3, 2023 at 21:51
  • $\begingroup$ @DarrenCook Many other dimensions of style correspond to the presence or absence of certain types of movement, for instance pinches, jumps, coordination moves, presses, etc. These in some ways are discrete (either present or not), but one could also quantify the difficulty of the movement in each of these categories, which is scalar. For example, some jumps are bigger than others, which makes them harder, and the climb is consequently "jumpier". So I think one could model them as either discrete or continuous, whichever is convenient for analysis. (2/2) $\endgroup$
    – Yly
    Commented Jan 3, 2023 at 21:56
  • $\begingroup$ @DarrenCook As for whether styles are unrelated, I would hazard that after you normalize by overall ability, most dimensions of style that humans would identify are uncorrelated, but not all. For instance, an athlete's skill at coordination moves is likely uncorrelated with their pressing strength. However, I wouldn't be surprised if some types of strength that humans consider distinct are actually closely related (e.g. pinch strength and crimp strength). $\endgroup$
    – Yly
    Commented Jan 3, 2023 at 22:09

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I think the bigger challenge here is to create the dataset. Once you can show the real data, and the columns you have, your questions will become answerable.

I'd suggest looking into ELO as a possible way to turn a climber's performance into a scalar number. The "Use outside of chess" section there shows it has been applied to team sports, puzzles, video games, etc.

TrueSkill is another one that comes to mind, for supporting when more than two players; though Microsoft killed off the adoption of that by patenting it. Other ideas are at https://en.wikipedia.org/wiki/Sports_rating_system.

I don't see the missing data as much of a problem; if you must have a entry for each climber for each climb you could use their Elo rating to predict a reasonable value. Once you have a baseline clustering model and how each cluster affects certain climber, you could use that to refine that value, then repeat.

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