SSANOVAwt | R Documentation |
A preliminary estimate \tilde{\eta}
is first obtained by fitting a smoothing spline ANOVA model,
and then use the inverse L_2
-norm, ||\tilde{\eta}_j||^{-\gamma}
, as the initial weight for the j
-th functional component.
SSANOVAwt(x,y,tau,family=c("Gaussian","Binomial","Cox","Quantile"),mscale=rep(1,ncol(x)),
gamma=1,scale=FALSE,nbasis,basis.id,cpus)
x |
input matrix; the number of rows is sample size, the number of columns is the data dimension. The range of input variables is scaled to [0,1] for continuous variables. |
y |
response vector. Quantitative for |
tau |
the quantile to be estimated, a number strictly between 0 and 1. Argument required when |
family |
response type. Abbreviations are allowed. |
mscale |
scale parameter for the Gram matrix associated with each function component. Default is |
gamma |
power of inverse |
scale |
if |
nbasis |
number of "knots" to be selected. Ignored when |
basis.id |
index designating selected "knots". Argument is not valid if |
cpus |
number of available processor units. Default is |
The initial mean function is estimated via a smooothing spline objective function. In the SS-ANOVA model framework, the regression function is assumed to have an additive form
\eta(x)=b+\sum_{j=1}^p\eta_j(x^{(j)}),
where b
denotes intercept and \eta_j
denotes the main effect of the j
-th covariate.
For "Gaussian"
response, the mean regression function is estimated by minimizing the objective function:
\sum_i(y_i-\eta(x_i))^2/nobs+\lambda_0\sum_{j=1}^p\alpha_j||\eta_j||^2.
where RSS is residual sum of squares.
For "Binomial"
response, the regression function is estimated by minimizing the objective function:
-log-likelihood/nobs+\lambda_0\sum_{j=1}^p\alpha_j||\eta_j||^2
For "Quantile"
regression model, the quantile function, is estimated by minimizing the objective function:
\sum_i\rho(y_i-\eta(x_i))/nobs+\lambda_0\sum_{j=1}^p\alpha_j||\eta_j||^2.
For "Cox"
regression model, the log-hazard function, is estimated by minimizing the objective function:
-log-Partial Likelihood/nobs+\lambda_0\sum_{j=1}^p\alpha_j||\eta_j||^2.
The smoothing parameter \lambda_0
is tuned by 5-fold Cross-Validation, if family="Gaussian"
, "Binomial"
or "Quantile"
,
and Approximate Cross-Validation, if family="Cox"
. But the smoothing parameters \alpha_j
are given in the argument mscale
.
The adaptive weights are then fiven by ||\tilde{\eta}_j||^{-\gamma}
.
wt |
The adaptive weights. |
Hao Helen Zhang and Chen-Yen Lin
Storlie, C. B., Bondell, H. D., Reich, B. J. and Zhang, H. H. (2011) "Surface Estimation, Variable Selection, and the Nonparametric Oracle Property", Statistica Sinica, 21, 679–705.
## Adaptive COSSO Model
## Binomial
set.seed(20130310)
x=cbind(rbinom(200,1,.7),matrix(runif(200*7,0,1),nc=7))
trueProb=1/(1+exp(-x[,1]-sin(2*pi*x[,2])-5*(x[,4]-0.4)^2))
y=rbinom(200,1,trueProb)
Binomial.wt=SSANOVAwt(x,y,family="Bin")
ada.B.Obj=cosso(x,y,wt=Binomial.wt,family="Bin")
## Not run:
## Gaussian
set.seed(20130310)
x=cbind(rbinom(200,1,.7),matrix(runif(200*7,0,1),nc=7))
y=x[,1]+sin(2*pi*x[,2])+5*(x[,4]-0.4)^2+rnorm(200,0,1)
Gaussian.wt=SSANOVAwt(designx,response,family="Gau")
ada.G.Obj=cosso(x,y,wt=Gaussian.wt,family="Gaussian")
## End(Not run)
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