An effective method for identifying clusters of robot strengths

In the analysis of qualification stage data from FIRST Robotics Competition (FRC) championships, the ratio (1.67–1.68) of the number of observations (110–114 matches) to the number of parameters (66–68 robots) in each division has been found to be quite small for the most commonly used winning margin power rating (WMPR) model. This usually leads to imprecise estimates and inaccurate predictions in such three-on-three matches that FRC tournaments are composed of. With the recognition of a clustering feature in estimated robot strengths, a more flexible model with latent clusters of robots was proposed to alleviate overparameterization of the WMPR model. Since its structure can be regarded as a dimension reduction of the parameter space in the WMPR model, the identification of clusters of robot strengths is naturally transformed into a model selection problem. Instead of comparing a huge number of competing models \((7.76\times 10^{67}\) to \(3.66\times 10^{70})\), we develop an effective method to estimate the number of clusters, clusters of robots and robot strengths in the format of qualification stage data from the FRC championships. The new method consists of two parts: (i) a combination of hierarchical and non-hierarchical classifications to determine candidate models; and (ii) variant goodness-of-fit criteria to select optimal models. In contrast to existing hierarchical classification, each step of our proposed non-hierarchical classification is based on estimated robot strengths from a candidate model in the preceding non-hierarchical classification step. A great advantage of the proposed methodology is its ability to consider the possibility of reassigning robots to other clusters. To reduce overestimation of the number of clusters by the mean squared prediction error criteria, corresponding Bayesian information criteria are further established as alternatives for model selection. With a coherent assembly of these essential elements, a systematic procedure is presented to perform the estimation of parameters. In addition, we propose two indices to measure the nested relation between clusters from any two models and monotonic association between robot strengths from any two models. Data from the 2018 and 2019 FRC championships and a simulation study are also used to illustrate the applicability and superiority of our proposed methodology.