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Binary segmentation procedures using the bivariate binomial distribution for detecting streakiness in sports data

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Abstract

Streakiness is an important measure in many sports data for individual players or teams in which the success rate is not a constant over time. That is, there are many successes/failures during some periods and few or no successes/failures during other periods. In this paper we propose a Bayesian binary segmentation procedure using a bivariate binomial distribution to locate the changepoints and estimate the associated success rates. The proposed method consists of a series of nested hypothesis tests based on the Bayes factors or posterior probabilities. At each stage, we compare three different changepoint models to the constant success rate model using the bivariate binary data. The proposed method is applied to analyze real sports datasets on baseball and basketball players as illustration. Extensive simulation studies are performed to demonstrate the usefulness of the proposed methodologies.

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Acknowledgements

The first author’s research was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07045804).

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Hanyang University, Ansan, 15588, South Korea

    Seong W. Kim

  2. Department of Mathematics and Statistics, Karakoram International University, Gilgit, 15100, Pakistan

    Sabina Shahin

  3. Department of Statistical Science, Southern Methodist University, Dallas, TX, 75275-0332, USA

    Hon Keung Tony Ng

  4. Department of Applied Statistics, University of Suwon, 17 Wauan-gil, Bongdam-eup, Hwaseong, 18323, South Korea

    Jinheum Kim

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  1. Seong W. Kim
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  2. Sabina Shahin
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  3. Hon Keung Tony Ng
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  4. Jinheum Kim
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Correspondence to Jinheum Kim.

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Appendix

Appendix

Proof of Proposition 3.1

We only prove parts (b) and (c). The derivations for \(\pi _0\) and \(\pi _3\) can be done using similar arguments in the proofs of (b) and (c). Note that the Jeffreys prior for a multi-parameter case with \({\varvec{\theta }}=(\theta _1,\dots ,\theta _k)\) is \(\pi ({\varvec{\theta }}) \propto |I({\varvec{\theta }})|^{1/2}\), where \(I({\varvec{\theta }})\) is the Fisher information matrix for \({\varvec{\theta }}\) and the i’s row and j’s column of \(I({\varvec{\theta }})\) is

$$\begin{aligned} I_{ij}=-\text{ E } \biggl [\frac{\partial ^2 \log L({\varvec{\theta }})}{\partial \theta _i \partial \theta _j}\biggl ] \quad \text { for } 1 \le i, j \le k. \end{aligned}$$

(b) For a fixed value of c, the log-likelihood of model \({{{\mathcal {M}}}}_1\) is

$$\begin{aligned} \lambda _1= & {} \log {L_{1}(\theta _{11},\theta _{12},\theta _{20})}\\= & {} \sum _{i=1}^{c}x_{1i} \log {\theta _{11}} + \sum _{i=1}^{c} (m-x_{1i})\log {(1-\theta _{11})}+\sum _{i=c+1}^{n}x_{1i} \log {\theta _{12}} \\&+\sum _{i=c+1}^{n} (m-x_{1i})\log {(1-\theta _{12})}+\sum _{i=1}^{n}x_{2i} \log {\theta _{20}}+\sum _{i=1}^{n}(x_{1i}-x_{2i})\log (1-\theta _{20}). \end{aligned}$$

The partial derivatives can be obtained as

$$\begin{aligned} \frac{\partial \lambda _1}{\partial \theta _{11}}= & {} \frac{\sum _{i=1}^{c}x_{1i}}{\theta _{11}}- \frac{\sum _{i=1}^{c}(m-x_{1i})}{1-\theta _{11}},\\ \frac{\partial ^{2}\lambda _1}{\partial \theta _{11}^{2}}= & {} -\frac{\sum _{i=1}^{c}x_{1i}}{\theta _{11}^{2}}- \frac{\sum _{i=1}^{c}(m-x_{1i})}{(1-\theta _{11})^{2}},\\ \frac{\partial \lambda _1}{\partial \theta _{12}}= & {} \frac{\sum _{i=c+1}^{n}x_{1i}}{\theta _{12}}- \frac{\sum _{i=c+1}^{n}(m-x_{1i})}{1-\theta _{12}},\\ \frac{\partial ^{2}\lambda _1}{\partial \theta _{12}^{2}}= & {} -\frac{\sum _{i=c+1}^{n}x_{1i}}{\theta _{12}^{2}}- \frac{\sum _{i=c+1}^{n}(m-x_{1i})}{(1-\theta _{12})^{2}},\\ \frac{\partial \lambda _1}{\partial \theta _{20}}= & {} \frac{\sum _{i=1}^{n}x_{2i}}{\theta _{20}}- \frac{\sum _{i=1}^{n}(x_{1i}-x_{2i})}{1-\theta _{20}},\\ \frac{\partial ^{2} \lambda }{\partial \theta _{20}^{2}}= & {} -\frac{\sum _{i=1}^{n}x_{2i}}{\theta _{20}^{2}}- \frac{\sum _{i=1}^{n}(x_{1i}-x_{2i})}{(1-\theta _{20})^{2}}. \end{aligned}$$

Since all the cross partial derivatives are zero, we have the following Fisher information

$$\begin{aligned} I(\theta _{11},\theta _{12},\theta _{20})= \left( \begin{array}{ccc}{\frac{cm}{\theta _{11}(1-\theta _{11})}} &{} {0} &{} {0} \\ {0} &{} {\frac{(n-c)m}{\theta _{12}(1-\theta _{12})}}&{}{0} \\ {0}&{}{0}&{}{\frac{cm\theta _{11}+(n-c)m\theta _{12}}{\theta _{20}(1-\theta _{20})}} \end{array}\right) . \end{aligned}$$
(A1)

The Jeffreys prior for model \({{{\mathcal {M}}}}_1\) readily follows from (A1), since the off-diagonal elements are all zero. \(\square \)

(c) Again, for a fixed value of c the log-likelihood of model \({{{\mathcal {M}}}}_2\) is

$$\begin{aligned} \lambda _2= & {} \log {L_{2}(\theta _{10},\theta _{21},\theta _{22})}\\= & {} \sum _{i=1}^{n}x_{1i} \log {\theta _{10}} + \sum _{i=1}^{n} (m-x_{1i})\log {(1-\theta _{10})}+\sum _{i=1}^{c}x_{1i} \log {\theta _{21}} \nonumber \\&+\sum _{i=1}^{c} (x_{1i}-x_{2i})\log {(1-\theta _{21})}+ \sum _{i=c+1}^{n}x_{2i}\log {\theta _{22}}\\&+\sum _{i=c+1}^{n}(x_{1i}-x_{2i})\log (1-\theta _{22}). \end{aligned}$$

Differentiate \(\lambda _2\) with respect to \(\theta _{10}\), \(\theta _{21}\), and \(\theta _{22}\) to obtain the partial derivatives as

$$\begin{aligned} \frac{\partial \lambda _2}{\partial \theta _{10}}= & {} \frac{\sum _{i=1}^{n}x_{1i}}{\theta _{10}}- \frac{\sum _{i=1}^{n}(m-x_{1i})}{1-\theta _{10}},\\ \frac{\partial ^{2}\lambda _2}{\partial \theta _{10}^{2}}= & {} -\frac{\sum _{i=1}^{n}x_{1i}}{\theta _{10}^{2}}- \frac{\sum _{i=1}^{n}(m-x_{1i})}{(1-\theta _{10})^{2}},\\ \frac{\partial \lambda _2}{\partial \theta _{21}}= & {} \frac{\sum _{i=1}^{c}x_{2i}}{\theta _{21}}- \frac{\sum _{i=1}^{c}(x_{1i}-x_{2i})}{1-\theta _{21}},\\ \frac{\partial ^{2}\lambda _2}{\partial \theta _{21}^{2}}= & {} -\frac{\sum _{i=1}^{c}x_{2i}}{\theta _{21}^{2}}- \frac{\sum _{i=1}^{c}(x_{1i}-x_{2i})}{(1-\theta _{21})^{2}},\\ \frac{\partial \lambda _2}{\partial \theta _{22}}= & {} \frac{\sum _{i=c+1}^{n}x_{2i}}{\theta _{22}}- \frac{\sum _{i=c+1}^{n}(x_{1i}-x_{2i})}{1-\theta _{22}},\\ \frac{\partial ^{2} \lambda _2}{\partial \theta _{22}^{2}}= & {} -\frac{\sum _{i=c+1}^{n}x_{2i}}{\theta _{22}^{2}}-\frac{\sum _{i=c+1}^{n}(x_{1i}-x_{2i})}{(1-\theta _{22})^{2}}. \end{aligned}$$

Similar to the proof of part (b), the off-diagonal elements of the Fisher information are all zero. Thus, the result follows from some routine algebra. \(\square \)

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Kim, S.W., Shahin, S., Ng, H.K.T. et al. Binary segmentation procedures using the bivariate binomial distribution for detecting streakiness in sports data. Comput Stat 36, 1821–1843 (2021). https://doi.org/10.1007/s00180-020-00992-2

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