Archive for radial speed

fast and curious

Posted in Running, Statistics with tags , , , , , , , , , on April 28, 2024 by xi'an

A paper in Frontiers attempted modelling the optimization of world-class 400 m and 1,500 m running performances using high-resolution data.

“In the present study, rather than using statistical analyses of 100-m split times or big data, we choose to analyze a select sample of World-class athletes individually using high-resolution data and fit a mathematical model to their pacing profiles. This mathematical model allows us to perform further predictive simulations and analyze the effect  of different physiological variables on performance. “

The model used in this analysis is a deterministic system of equations depending on half-a-dozen parameters like (of course!) the VO² max. Most of them “are not measured for any individual athlete (e.g., using experimental methods), but are estimated through a computation to fit the data for a specific race and athlete.”

The authors then turn cranks to see how each parameter increases the speed curve, as in the graph below for Femke Bol winning the 2022 European Championship 400m in München and the impact of increasing final VO², anaerobic energy e⁰, or propulsive force. And the same for Jakob Ingebrigtsen’s 1,500m win [above graph]—in a time shorter than my current 1,000m best! What is unclear to me is how the model repercusses the toll of such increases on the athlete’s other parameters—since Ingebrigtsen already has an amazing VO² max!. Some of the changes leads to a 5sec improvement, which is quite large at record levels—the World record for men moved by 1s between 1998 and 2022. I’d thus be interested in a field experiment where one of these top runners would attempt to implement the suggested changes in their pace to see how much this impacts the other factors… Another experiment with the fitted model is to check the impact of the lane allocated to the runner for a 400m race, the fastest lane being the first one despite higher centrifugal forces on the bends. Which seem to have a lesser if still noticeable effect on the 1,500m race [above graph].

 

thermodynamic integration plus temperings

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on July 30, 2019 by xi'an

Biljana Stojkova and David Campbel recently arXived a paper on the used of parallel simulated tempering for thermodynamic integration towards producing estimates of marginal likelihoods. Resulting into a rather unwieldy acronym of PT-STWNC for “Parallel Tempering – Simulated Tempering Without Normalizing Constants”. Remember that parallel tempering runs T chains in parallel for T different powers of the likelihood (from 0 to 1), potentially swapping chain values at each iteration. Simulated tempering monitors a single chain that explores both the parameter space and the temperature range. Requiring a prior on the temperature. Whose optimal if unrealistic choice was found by Geyer and Thomson (1995) to be proportional to the inverse (and unknown) normalising constant (albeit over a finite set of temperatures). Proposing the new temperature instead via a random walk, the Metropolis within Gibbs update of the temperature τ then involves normalising constants.

“This approach is explored as proof of concept and not in a general sense because the precision of the approximation depends on the quality of the interpolator which in turn will be impacted by smoothness and continuity of the manifold, properties which are difficult to characterize or guarantee given the multi-modal nature of the likelihoods.”

To bypass this issue, the authors pick for their (formal) prior on the temperature τ, a prior such that the profile posterior distribution on τ is constant, i.e. the joint distribution at τ and at the mode [of the conditional posterior distribution of the parameter] is constant. This choice makes for a closed form prior, provided this mode of the tempered posterior can de facto be computed for each value of τ. (However it is unclear to me why the exact mode would need to be used.) The resulting Metropolis ratio becomes independent of the normalising constants. The final version of the algorithm runs an extra exchange step on both this simulated tempering version and the untempered version, i.e., the original unnormalised posterior. For the marginal likelihood, thermodynamic integration is invoked, following Friel and Pettitt (2008), using simulated tempering samples of (θ,τ) pairs (associated instead with the above constant profile posterior) and simple Riemann integration of the expected log posterior. The paper stresses the gain due to a continuous temperature scale, as it “removes the need for optimal temperature discretization schedule.” The method is applied to the Glaxy (mixture) dataset in order to compare it with the earlier approach of Friel and Pettitt (2008), resulting in (a) a selection of the mixture with five components and (b) much more variability between the estimated marginal  likelihoods for different numbers of components than in the earlier approach (where the estimates hardly move with k). And (c) a trimodal distribution on the means [and unimodal on the variances]. This example is however hard to interpret, since there are many contradicting interpretations for the various numbers of components in the model. (I recall Radford Neal giving an impromptu talks at an ICMS workshop in Edinburgh in 2001 to warn us we should not use the dataset without a clear(er) understanding of the astrophysics behind. If I remember well he was excluded all low values for the number of components as being inappropriate…. I also remember taking two days off with Peter Green to go climbing Craigh Meagaidh, as the only authorised climbing place around during the foot-and-mouth epidemics.) In conclusion, after presumably too light a read (I did not referee the paper!), it remains unclear to me why the combination of the various tempering schemes is bringing a noticeable improvement over the existing. At a given computational cost. As the temperature distribution does not seem to favour spending time in the regions where the target is most quickly changing. As such the algorithm rather appears as a special form of exchange algorithm.