boot.BTdecayLasso {BTdecayLasso}R Documentation

Compute the standard deviation of Bradley-Terry decay Lasso model by bootstrapping

Description

Bootstrapping is done assuming that Maximum Likelihood's estimation reflects the true abilities. Same level of Lasso penalty "lambda" should be applied in different simulation models for Lasso induced estimation.

Usage

boot.BTdecayLasso(dataframe, ability, lambda, boot = 100, weight = NULL,
  decay.rate = 0, fixed = 1, thersh = 1e-05, max = 100, iter = 100)

Arguments

dataframe

Generated using BTdataframe given raw data.

ability

A column vector of teams ability, the last row is the home parameter. The row number is consistent with the team's index shown in dataframe. It can be generated using BTdataframe given raw data.

lambda

The amount of Lasso penalty induced, only a single scalar is accepted in bootstrapping.

boot

Amount of simulations.

weight

Weight for Lasso penalty on different abilities.

decay.rate

The exponential decay rate. Usually ranging from (0, 0.01), A larger decay rate weights more importance to most recent matches and the estimated parameters reflect more on recent behaviour.

fixed

A teams index whose ability will be fixed as 0. The worstTeam's index can be generated using BTdataframe given raw data.

thersh

Threshold for convergence

max

Maximum weight for w_{ij} (weight used for Adaptive Lasso).

iter

Number of iterations used in L-BFGS-B algorithm.

Details

100 times of simulation will be done by default, user can adjust the numbers of simulation by input of boot. However, bootstrapping process is time consuming and usually 1000 time of simulations is enough to provide a stable result.

More detailed description of "lambda", "penalty" and "weight" are documented in BTdecayLasso.

summary() function follows S3 method can be applied to view the outputs.

Value

A list with class "boot" contain Lasso and Hybrid Lasso's bootstrapping's mean and standard deviation.

Lasso

Lasso bootstrapping's result. A three column matrix where first column is the original estimation, the second column is bootstrapping mean and the last column is the bootstrapping standard deviation

HYBRID.Lasso

HYBRID Lasso bootstrapping's result. A three column matrix where the first column is the original estimation, the second column is bootstrapping mean and the last column is the bootstrapping standard deviation

References

Masarotto, G. and Varin, C.(2012) The Ranking Lasso and its Application to Sport Tournaments. *The Annals of Applied Statistics* **6** 1949–1970.

Zou, H. (2006) The adaptive lasso and its oracle properties. *J.Amer.Statist.Assoc* **101** 1418–1429.

See Also

BTdataframe for dataframe initialization, BTdecayLasso for detailed description

Examples

##Initialize the dataframe and ability
x <- BTdataframe(NFL2010)

##The following code runs the main results

##Run Lasso estimate for whole Lasso path
z <- BTdecayLasso(x$dataframe, x$ability, fixed = x$worstTeam)

##Model selection using AIC with Lasso's likelihood
z1 <- BTdecayLassoC(x$dataframe, x$ability, model = z, 
                    criteria = "AIC", type = "LASSO", fixed = x$worstTeam)

##Bootstrapping for model with lowest AIC score for 100 times.
##Note that the decay.rate used in model selection should be consistent with
##the one which is used in whole Lasso path's run (keep the same model)
z2 <- boot.BTdecayLasso(x$dataframe, x$ability, lambda = z1$Optimal.lambda, 
                        boot = 100, fixed = x$worstTeam)



[Package BTdecayLasso version 0.1.0 Index]