cv.smog: Cross-valiation for smog

Description

cv.smog conducts a greedy-search for optimal lambda's and yields a sparse model given a provided model selection criterion. When type is ''nloglike'', the method allows the nfolds to be processed in parallel for speeding up the cross-validation.

Usage

cv.smog(x, y, g, v, label, type = "nloglike", family = "gaussian",
  lambda.max = NULL, nlambda.max = 20, delta = 0.9, nfolds = 10,
  parallel = FALSE, ncores = NULL, ...)
# S3 method for cv.smog
print(x, ...)

Arguments

a model matrix, or a data frame of dimensions n by p, in which the columns represents the predictor variables.

response variable, corresponds to the family description. When family is ''gaussian'' or ''binomial'', y ought to be a numeric vector of observations of length n; when family is ''coxph'', y represents the survival objects, containing the survival time and the censoring status. See Surv.

a vector of group labels for the predictor variables.

a vector of binary values, represents whether or not the predictor variables are penalized. Note that 1 indicates penalization and 0 for not penalization.

label

a character vector, represents the type of predictors in terms of treatment, prognostic, and predictive effects by using ''t'', ''prog'', and ''pred'', respectively.

type

model selction criterion, should choose from ''nloglike'', ''cAIC'', ''AIC'', ''BIC'', and ''GCV'', respectively.

family

a description of the distribution family for the response variable variable. For continuous response variable, family is ''gaussian''; for multinomial or binary response variable, family is ''binomial''; for survival response variable, family is ''coxph'', respectively.

lambda.max

the maximum value for lambda's. If NULL, the default lambda.max is $1 / λ_{m i n} (x^{'} x)$ .

nlambda.max

the maximum number of lambdas' shrunk down from the maximum lambda lambda.max. Default is 20.

delta

the damping rate for lambda's such that $λ_{k} = δ^{k} λ_{0}$ . Default is 0.9.

nfolds

number of folds. One fold of the observations in the data are used as the testing, and the remaining are fitted for model training. Default is 5.

parallel

Whether or not process the nfolds cross-validations in parallel. If TRUE, use foreach to do each cross-validation in parallel. Default is FALSE.

ncores

number of cpu's for parallel computing. See makeCluster and registerDoParallel. Default is NULL.

...

other arguments that can be supplied to smog.

Value

Includes the profile containing a path of lambda's and the corresponding model selectio criterion value, the optimal lambda's, and the optimal model, respectively. The type comes from a list of model selection criteria values, includes the average of the negative log-likelihood values and the correction AIC for each fold of the data.

cvfit: the fitted model based on the optimal lambda's.
lhat: the optimal lambda's which has the minimum model selection criterion.
profile: a data frame contains the path of lambda's and the corresponding model selection criterion, which is determined by the type.

Details

When the type is ''nloglike'', it requires doing nfolds cross-validations. For each cross-validation, evenly split the whole data into nfolds, and one fold of the observations are used as the testing data, and the remaining are used for model training. After calculating the (deviance) residuals for each fold of testing data, return the average of the (deviance) residuals. Note that we keep lambda2 $= 0$ during the greedy search for lambda's.

Model selection criteria

Besides the n-fold cross-validation, cv.smog provides several AIC based model selection criteria.

cAIC: $\frac{n}{2}$ log(|2*log-likelihood|) + $\frac{n}{2} (\frac{1 + k / n}{1 - k + 2 / n})$
AIC: log(|2*log-likelihood|/n) + $2 \frac{k}{n}$
BIC: log(|2*log-likelihood |/n) + $2 \frac{k}{n}$ log(n)
GCV: |2*log-likelihood| / $(n (1 - k / n)^{2})$

Where k is the degrees of freedom DF, which is related to the penalty parameters $λ$ 's.

References

ma2019structuralsmog

Examples

Run this code

# NOT RUN {
# generate design matrix x
set.seed(2018)
n=100;p=20
s=10
x=matrix(0,n,1+2*p)
x[,1]=sample(c(0,1),n,replace = TRUE)
x[,seq(2,1+2*p,2)]=matrix(rnorm(n*p),n,p)
x[,seq(3,1+2*p,2)]=x[,seq(2,1+2*p,2)]*x[,1]

g=c(p+1,rep(1:p,rep(2,p)))  # groups 
v=c(0,rep(1,2*p))           # penalization status
label=c("t",rep(c("prog","pred"),p))  # type of predictor variables

# generate beta
beta=c(rnorm(13,0,2),rep(0,ncol(x)-13))
beta[c(2,4,7,9)]=0

# generate y
data=x%*%beta
noise=rnorm(n)
snr=as.numeric(sqrt(var(data)/(s*var(noise))))
y=data+snr*noise

cvfit=cv.smog(x,y,g,v,label,type = "GCV", family="gaussian")

# }

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