Dataframe initialization

Description

Dataframe initialization

Usage

BTdataframe(dataframe, home = TRUE)

Arguments

Details

Initial the raw dataframe and return an un-estimated ability vector and the worst team who loses most.

Note that even if the tournament does not have any home team or away team, you can still provide the match results according to the description above regardless of who is at home and who is away. By selecting the home = FALSE, We duplicate the dataset, switch the home, away teams and also the home, away match results. Then this dataset will be attached to the original dataset and all home and away win's number will be divided by 2. MLE estimation of home effect is proved to be an exact 0.

The elimination of home effect by duplicating the original dataset will be less efficient than eliminating the home parameter directly in iterations. Since most games such as football, basketball have home effect and this method provides an idea of handling the case where some games have home effect and some games are played on neutral place, this method is applied here.

Value

Bradley-Terry Model with Exponential Decayed weighted likelihood

Description

Exponential decay rate is applied to the likelihood function to achieve a better track of current abilities. When "decay.rate" is setting as 0, this is a standard Bradley-Terry Model whose estimated parameters are equivalent to package "BradleyTerry2". Further detailed description is attached in BTdecayLasso.

Usage

BTdecay(dataframe, ability, decay.rate = 0, fixed = 1, iter = 100)

Arguments

Details

The standard Bradley-Terry Model defines the winning probability of i against j,

P(Y_{ij}=1)=\frac{\exp(\tau h_{ij}^{t_{k}}+\mu_{i}-\mu_{j})}{1+\exp(\tau h_{ij}^{t_{k}}+\mu_{i}-\mu_{j})}

\tau is the home parameter and \mu_{i} is the team i's ability score. h_{ij} takes 1 if team i is at home, -1 otherwise. Given, a complete tournament's result. The objective likelihood function with an exponential decay rate is,

\sum_{k=1}^{n}\sum_{i<j}\exp(-\alpha t_{k})\cdot(y_{ij}(\tau h_{ij}^{t_{k}}+\mu_{i}-\mu_{j})-\log(1+\exp(\tau h_{ij}^{t_{k}}+\mu_{i}-\mu_{j})))

where n is the number of matches, \alpha is the exponential decay rate and y_{ij} takes 0 if i is defeated by j, 1 otherwise. t_{k} is the time lag (time until now). This likelihood function is optimized using L-BFGS-B method with package optimr and summary() function with S3 method can be applied to view the outputs.

Value

List with class "BT" contains estimated abilities and convergent code, 0 stands for convergence reaches, 1 stands for convergence not reaches. If 1 is returned, we suggest that decay rate should be set lower. Bradley-Terry model fails to model the situation when a team wins or loses in all matches. If a high decay rate is considered, a team who only loses or wins 1 matches long time ago will also causes the same problem.

Examples

##Initializing Dataframe
x <- BTdataframe(NFL2010)

##Standard Bradley-Terry Model optimization
y <- BTdecay(x$dataframe, x$ability, decay.rate = 0, fixed = x$worstTeam)
summary(y)

##Dynamic approximation of current ability scores using exponential decayed likelihood.
##If we take decay.rate = 0.005
##Match happens one month before will weight exp(-0.15)=0.86 on log-likelihood function
z <- BTdecay(x$dataframe, x$ability, decay.rate = 0.005, fixed = x$worstTeam)
summary(z)

Bradley-Terry Model with Exponential Decayed weighted likelihood and Adaptive Lasso

Description

Bradley-Terry model is applied for paired comparison data. Teams' ability score is estimated by maximizing log-likelihood function.

To achieve a better track of current abilities, we apply an exponential decay rate to weight the log-likelihood function. The most current matches will weight more than previous matches. Parameter "decay.rate" in most functions of this package is used to set the amount of exponential decay rate. decay.rate should be non-negative and the appropriate range of it depends on time scale in original dataframe. (see BTdataframe and parameter "dataframe"'s definition of fifth column) For example, a unit of week with a "decay.rate" 0.007 is equivalent to the unit of day with "decay.rate" 0.001. Usually, for sports matches, if we take the unit of day, it's ranging from 0 to 0.01. The higher choice of "decay.rate", the better track of current teams' ability with a side effect of higher variance.

If "decay.rate" is too large, for example "0.1" with a unit of day, \exp(-0.7) = 0.50. Only half weight will be add to the likelihood for matches played one week ago and \exp(-3.1) = 0.05 suggests that previous matches took place one month ago will have little effect. Therefore, Only a few matches are accounted for ability's estimation. It will lead to a very high variance and uncertainty. Since standard Bradley-Terry model can not handle the case where there is a team who wins or loses all matches, such estimation may not provide convergent results. Thus, if our estimation provides divergent result, an error will be returned and we suggest user to chose a smaller "decay.rate" or adding more match results into the same modeling period.

By default, the Adaptive Lasso is implemented for variance reduction and team's grouping. Adaptive Lasso is proved to have good grouping property. Apart from adaptive lasso, user can define own weight for different Lasso constraint \left|\mu_{i}-\mu_{j}\right| where \mu_{i} is team i's ability.

Also by default, the whole Lasso path will be run. Similar to package "glmnet", user can provide their own choice of Lasso penalty "lambda" and determine whether the whole Lasso path will be run (since such run is time-consuming). However, we suggest that if user is not familiar with the actual relationship among lambda, the amount of penalty, the amount of shrinkage and grouping effect, a whole Lasso path should be run and selection of an appropriate lambda is done by AIC or BIC criteria using BTdecayLassoC (since this model is time related, cross-validation method cannot be applied). Also, users can use BTdecayLassoF to run with a specific Lasso penalty ranging from 0 to 1 (1 penalty means all estimators will shrink to 0).

Two sets of estimated abilities will be given, the biased Lasso estimation and the HYBRID Lasso's estimation. HYBRID Lasso estimation solves the restricted Maximum Likelihood optimization based on the group determined by Lasso's estimation (Different team's ability will converges to the same value if Lasso penalty is added and these teams' ability is setting to be equal as a restriction).

In addition, summary() using S3 method can be applied to view the outputs.

Usage

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

Arguments

Details

According to BTdecay, the objective likelihood function to be optimized is,

\sum_{k=1}^{n}\sum_{i<j}\exp(-\alpha t_{k})\cdot(y_{ij}(\tau h_{ij}^{t_{k}}+\mu_{i}-\mu_{j})-\log(1+\exp(\tau h_{ij}^{t_{k}}+\mu_{i}-\mu_{j})))

The Lasso constraint is given as,

\sum_{i<j}w_{ij}\left|\mu_{i}-\mu_{j}\right|\leq s

where w_{ij} are predefined weight. For Adaptive Lasso, \left|w_{ij}=1/(\mu_{i}^{MLE}-\mu_{j}^{MLE})\right|.

Maximize this constraint objective function is equivalent to minimizing the following equation,

-l(\mu,\tau)+\lambda\sum_{i<j}w_{ij}|\mu_{i}-\mu_{j}|

Where -l(\mu,\tau) is taking negative value of objective function above. Increase "lambda" will decrease "s", their relationship is monotone. Here, we define "penalty" as 1-s/\max(s). Thus, "lambda" and "penalty" has a positive correlation.

Value

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, plot.swlasso, plot.wlasso are used for Lasso path plot if path = TRUE in this function's run

Examples

##Initializing Dataframe
x <- BTdataframe(NFL2010)

##The following code runs the main results
##Usually a single lambda's run will take 1-20 s
##The whole Adaptive Lasso run will take 5-20 min

##BTdecayLasso run with exponential decay rate 0.005 and 
##lambda 0.1, use path = TRUE if you want to run whole LASSO path
y1 <- BTdecayLasso(x$dataframe, x$ability, lambda = 0.1, path = FALSE,
                   decay.rate = 0.005, fixed = x$worstTeam)
summary(y1)

##Defining equal weight
##Note that comparing to Adaptive weight, the user defined weight may not be 
##efficient in groupiing. Therefore, to run the whole Lasso path 
##(evolving of distinct ability scores), it may take a much longer time. 
##We recommend the user to apply the default setting,
##where Adaptive Lasso will be run.

n <- nrow(x$ability) - 1
w2 <- matrix(1, nrow = n, ncol = n)
w2[lower.tri(w2, diag = TRUE)] <- 0

##BTdecayLasso run with exponential decay rate 0.005 and with a specific lambda 0.1
y2 <- BTdecayLasso(x$dataframe, x$ability, lambda = 0.1, weight = w2, 
                   path = FALSE, decay.rate = 0.005, fixed = x$worstTeam)

summary(y2)

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
)

Arguments

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.

Value

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 obtaining a whole Lasso path

Bradley-Terry Model with Exponential Decayed weighted likelihood and Adaptive Lasso with a given penalty rate

Description

This function provides a method to computed the estimated abilities and lambda given an intuitive fixed Lasso penalty rate. Since in Lasso method, the selection of lambda varies a lot with respect to different datasets. We can keep the consistency of amount of Lasso penalty induced in different datasets from different period by setting a fixed Lasso penalty rate "penalty". Please refer to BTdecayLasso for the definition of "penalty" and its relationship with "lambda".

Usage

BTdecayLassoF(
  dataframe,
  ability,
  penalty,
  decay.rate = 0,
  fixed = 1,
  thersh = 1e-05,
  max = 100,
  iter = 100
)

Arguments

Details

The estimated ability given fixed penalty p = 1- s/\max(s) where s is the sum of Lasso penalty part. When p = 0, this model is reduced to a standard Bradley-Terry Model. When p = 1, all ability scores are shrinking to 0.

The parameter "penalty" should be ranging from 0.01 to 0.99 due to the iteration's convergent error.

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

Value

The list with class "BTF" contains estimated abilities and other parameters.

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

The 2010 NFL Regular Season

Description

A dataframe containing all match results with 5 columns

Usage

NFL2010

Format

A dataframe containing all match results with 5 columns

home.team

Team who plays at home

away.team

Team who plays away

home.win

Take "1" if home team wins

away.win

Take "1" if away team wins

date

Number of days until now

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

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.

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

Plot the Lasso path

Description

Plot the whole lasso path run by BTdecayLasso() with given lambda and path = TRUE

Usage

##S3 method for class "swlasso"

Arguments

Plot the Lasso path

Description

Plot the whole lasso path run by BTdecayLasso() with lambda = NULL and path = TRUE

Usage

##S3 method for class "wlasso"

Arguments