Abstract: 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.
Abstract: The purpose of this article is to find a method
of comparing designs for ordinal regression models using
quantile dispersion graphs in the presence of linear predictor
misspecification. The true relationship between response variable
and the corresponding control variables are usually unknown.
Experimenter assumes certain form of the linear predictor of the
ordinal regression models. The assumed form of the linear predictor
may not be correct always. Thus, the maximum likelihood estimates
(MLE) of the unknown parameters of the model may be biased due to
misspecification of the linear predictor. In this article, the uncertainty
in the linear predictor is represented by an unknown function. An
algorithm is provided to estimate the unknown function at the
design points where observations are available. The unknown function
is estimated at all points in the design region using multivariate
parametric kriging. The comparison of the designs are based on
a scalar valued function of the mean squared error of prediction
(MSEP) matrix, which incorporates both variance and bias of the
prediction caused by the misspecification in the linear predictor. The
designs are compared using quantile dispersion graphs approach.
The graphs also visually depict the robustness of the designs on the
changes in the parameter values. Numerical examples are presented
to illustrate the proposed methodology.
Abstract: Instead of traditional (nominal) classification we investigate
the subject of ordinal classification or ranking. An enhanced
method based on an ensemble of Support Vector Machines (SVM-s)
is proposed. Each binary classifier is trained with specific weights
for each object in the training data set. Experiments on benchmark
datasets and synthetic data indicate that the performance of our
approach is comparable to state of the art kernel methods for
ordinal regression. The ensemble method, which is straightforward
to implement, provides a very good sensitivity-specificity trade-off
for the highest and lowest rank.