SSANOVAwt {cosso} | R Documentation |

## Compute adaptive weights by fitting a SS-ANOVA model

### Description

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.

### Usage

```
SSANOVAwt(x,y,tau,family=c("Gaussian","Binomial","Cox","Quantile"),mscale=rep(1,ncol(x)),
gamma=1,scale=FALSE,nbasis,basis.id,cpus)
```

### Arguments

`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 |

### Details

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}`

.

### Value

`wt` |
The adaptive weights. |

### Author(s)

Hao Helen Zhang and Chen-Yen Lin

### References

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.

### Examples

```
## 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)
```

*cosso*version 2.1-2 Index]