Joonas Pääkkönen

Joonas Pääkkönen

In sports, individuals and teams are typically interested in final rankings. Final results, such as times or distances, dictate these rankings, also known as places. Places can be further associated with ordered random variables, commonly referred to as order statistics. In this work, we introduce a simple, yet accurate order statistical ordinal regression function that predicts relay race places with changeover-times. We call this function the Fenton-Wilkinson Order Statistics model. This model is built on the following educated assumption: individual leg-times follow log-normal distributions. Moreover, our key idea is to utilize Fenton-Wilkinson approximations of changeover-times alongside an estimator for the total number of teams as in the notorious German tank problem. This original place regression function is sigmoidal and thus correctly predicts the existence of a small number of elite teams that significantly outperform the rest of the teams. Our model also describes how place increases linearly with changeover-time at the inflection point of the log-normal distribution function. With real-world data from Jukola 2019, a massive orienteering relay race, the model is shown to be highly accurate even when the size of the training set is only 5% of the whole data set. Numerical results also show that our model exhibits smaller place prediction root-mean-square-errors than linear regression, mord regression and Gaussian process regression.

Joonas Pääkkönen

Ordinal regression falls between discrete-valued classification and continuous-valued regression. Ordinal target variables can be associated with ranked random variables. These random variables are known as order statistics and they are closely related to ordinal regression. However, the challenge of using order statistics for ordinal regression prediction is finding a suitable parent distribution. In this work, we provide a case study of a real-world orienteering relay race by viewing it as a random process. For this process, we show that accurate order statistical ordinal regression predictions of final team rankings, or places, can be obtained by assuming a lognormal distribution of individual leg times. Moreover, we apply Fenton-Wilkinson approximations to intermediate changeover times alongside an estimator for the total number of teams as in the notorious German tank problem. The purpose of this work is, in part, to spark interest in studying the applicability of order statistics in ordinal regression problems.