Bradley-Terry Model with Exponential Time Decayed Log-Likelihood and Adaptive Lasso

We utilize the Bradley-Terry Model to estimate the abilities of teams using paired comparison data. For dynamic approximation of current rankings, we employ the Exponential Decayed Log-likelihood function, and we also apply the Lasso penalty for variance reduction and grouping. The main algorithm applies the Augmented Lagrangian Method described by Masarotto and Varin (2012) .

BTdecayLasso

Bradley-Terry model is used for ranking in sports tournament. Given the standard Bradley-Terry model, we use an exponential decay rate to weight its log-likelihood function and apply Lasso penalty to achieve a variance reduction and team grouping.

Installation

You can install BTdecayLasso from github with:

# install.packages("devtools")
devtools::install_github("heilokchow/BTdecayLasso")

Example

This is a basic example which shows you how to solve a common problem:

First, given raw datasets (five columns are home teams, away teams, home wins, away wins, time until now), we convert this dataset into a dataframe which can be used for other function’s input.

NFL <- BTdataframe(NFL2010)

Then, we comput the whole Lasso path for further analysis’s use. In this example, to track the dynamically changing abilities, we set ‘decay.rate’ to be 0.005. A higher decay rate will give more unbiased results for current abilites’ estimation with a side effect of higher variance.

BTM <- BTdecayLasso(NFL$dataframe, NFL$ability, decay.rate = 0.005, fixed = NFL$worstTeam)

We can use ‘plot’ function to view the whole Lasso path.

plot(BTM)

The optimal model is selected using AIC criteria on HYBRID Lasso’s run here.

BTO <- BTdecayLassoC(NFL$dataframe, NFL$ability, decay.rate = 0.005, fixed = NFL$worstTeam,
                     model = BTM, criteria = "AIC", type = "HYBRID")
summary(BTO)

Finally, we use bootstrapping to obtain the standard deviation of this choosen model with 100 times of simulation.

BT <- boot.BTdecayLasso(NFL$dataframe, NFL$ability, BTO$Optimal.lambda, decay.rate = 0.005, 
                        fixed = NFL$worstTeam, boot = 100)
summary(BT)