BTdecayLassoC: Bradley-Terry Model with Exponential Decayed weighted likelihood and weighted Lasso with AIC or BIC criteria

Description

Model selection via AIC or BIC criteria. For Lasso estimators, the degree of freedom is the number of distinct groups of estimated abilities.

Usage

BTdecayLassoC(
  dataframe,
  ability,
  weight = NULL,
  criteria = "AIC",
  type = "HYBRID",
  model = NULL,
  decay.rate = 0,
  fixed = 1,
  thersh = 1e-05,
  iter = 100,
  max = 100
)

Value

Score: Lowest AIC or BIC score
Optimal.degree: The degree of freedom where lowest AIC or BIC score is achieved
Optimal.ability: The ability where lowest AIC or BIC score is achieved
ability: Matrix contains all abilities computed in this algorithm
Optimal.lambda: The lambda where lowest score is attained
Optimal.penalty: The penalty (1- s/\(\max(s)\)) where lowest score is attained
type: Type of model selection method
decay.rate: Decay rate of this model

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.
weight: Weight for Lasso penalty on different abilities
criteria: "AIC" or "BIC"
type: "HYBRID" or "LASSO"
model: An Lasso path object with class wlasso or swlasso. If NULL, the whole lasso path will be run.
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
iter: Number of iterations used in L-BFGS-B algorithm.
max: Maximum weight for \(w_{ij}\) (weight used for Adaptive Lasso)

Details

This function is usually run after the run of whole Lasso path. "model" parameter is obtained by whole Lasso pass's run using BTdecayLasso. If no model is provided, this function will run Lasso path first (time-consuming).

Users can select the information score added to HYBRID Lasso's likelihood or original Lasso's likelihood. ("HYBRID" is recommended)

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

References

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