camel.slim: Calibrated Linear Regression
In camel: Calibrated Machine Learning

Description Usage Arguments Details Value Author(s) References See Also Examples

The function "camel.slime" implements LAD/L1 Lasso, SQRT/L2 Lasso, and carlibrated Dantizg selector using L1 regularization.

1
2
3

camel.slim(X, Y, lambda = NULL, nlambda = NULL, lambda.min.ratio = NULL,
           method="lq", q = 2, prec = 1e-4, max.ite = 1e4, mu = 0.01,
           intercept = TRUE, verbose = TRUE)

`Y`	The n dimensional response vector.
`X`	The n by d design matrix.
`lambda`	A sequence of decresing positive value to control the regularization. Typical usage is to leave the input `lambda = NULL` and have the program compute its own `lambda` sequence based on `nlambda` and `lambda.min.ratio`. Users can also specify a sequence to override this. Default value is from lambda.max to `lambda.min.ratiolambda.max`. For Lq regression, the default value of lambda.max* is π√{\log(d)/n}. For Dantzig selector, the default value of lambda.max is the minimum regularization parameter, which yields an all-zero estiamtes.
`nlambda`	The number of values used in `lambda`. Default value is 5.
`lambda.min.ratio`	The smallest value for `lambda`, as a fraction of the uppperbound (`MAX`) of the regularization parameter. The program can automatically generate `lambda` as a sequence of length = `nlambda` starting from `MAX` to `lambda.min.ratio*MAX` in log scale. The default value is `0.25` for Lq Lasso and `0.5` for Dantzig selector.
`method`	Dantzig selector is applied if `method = "dantzig"` and L_q Lasso is applied if `method = "lq"`. The default value is `"lq"`.
`q`	The loss function used in Lq Lasso. It is only applicable when `method = "lq"` and must be either 1 or 2. The default value is 2.
`prec`	Stopping criterion. The default value is 1e-4.
`max.ite`	The iteration limit. The default value is 1e4.
`mu`	The smoothing parameter. The default value is 0.01.
`intercept`	Whether the intercept is included in the model. The defulat value is `TRUE`.
`verbose`	Tracing information is disabled if `verbose = FALSE`. The default value is `TRUE`.

Calibrated Linear Regression adjust the regularization with respect to the noise level. Thus it achieves both improved finite sample performance and tuning insensitiveness.

An object with S3 class "camel.slim" is returned:

`beta`	A matrix of regression estimates whose columns correspond to regularization parameters.
`intercept`	The value of intercepts corresponding to regularization parameters.
`Y`	The value of `Y` used in the program.
`X`	The value of `X` used in the program.
`lambda`	The sequence of regularization parameters `lambda` used in the program.
`nlambda`	The number of values used in `lambda`.
`method`	The `method` from the input.
`sparsity`	The sparsity levels of the solution path.
`ite`	A list of vectors where ite[[1]] is the number of external iteration and ite[[2]] is the number of internal iteration with the i-th entry corresponding to the i-th regularization parameter.
`verbose`	The `verbose` from the input.

Xingguo Li, Tuo Zhao, and Han Liu
Maintainer: Xingguo Li <xingguo.leo@gmail.com>

1. A. Belloni, V. Chernozhukov and L. Wang. Pivotal recovery of sparse signals via conic programming. Biometrika, 2012.
2. L. Wang. L1 penalized LAD estimator for high dimensional linear regression. Journal of Multivariate Analysis, 2013.
3. E. Candes and T. Tao. The Dantzig selector: Statistical estimation when p is much larger than n. Annals of Statistics, 2007.

camel-package.

## Generate the design matrix and regression coefficient vector
n = 200
d = 400
X = matrix(rnorm(n*d), n, d)
beta = c(3,2,0,1.5,rep(0,d-4))

## Generate response using Gaussian noise, and fit a sparse linear model using SQRT Lasso
eps.sqrt = rnorm(n)
Y.sqrt = X%*%beta + eps.sqrt
out.sqrt = camel.slim(X = X, Y = Y.sqrt, lambda = seq(0.8,0.2,length.out=5))

## Generate response using Cauchy noise, and fit a sparse linear model using LAD Lasso
eps.lad = rt(n = n, df = 1)
Y.lad = X%*%beta + eps.lad
out.lad = camel.slim(X = X, Y = Y.lad, q = 1, lambda = seq(0.5,0.2,length.out=5))

## Visualize the solution path
plot(out.sqrt)
plot(out.lad)

Loading required package: lattice
Loading required package: igraph

Attaching package: 'igraph'

The following objects are masked from 'package:stats':

    decompose, spectrum

The following object is masked from 'package:base':

    union

Loading required package: MASS
Loading required package: Matrix
Sparse Linear Regression with L1 Regularization.
SQRT Lasso regression via MFISTA.
Sparse Linear Regression with L1 Regularization.
LAD Lasso regression via MFISTA.