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.

You can install BTdecayLasso from github with:

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

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.

`<- BTdataframe(NFL2010) NFL `

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.

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

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.

```
<- BTdecayLassoC(NFL$dataframe, NFL$ability, decay.rate = 0.005, fixed = NFL$worstTeam,
BTO 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.

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