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

## Description

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

## Usage

 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)

## Arguments

 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.ratio*lambda.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.

## Details

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

## Value

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.

## Author(s)

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

## References

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.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ## 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)

### Example output

Attaching package: 'igraph'

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

decompose, spectrum

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

union

Sparse Linear Regression with L1 Regularization.
SQRT Lasso regression via MFISTA.
Sparse Linear Regression with L1 Regularization.

camel documentation built on May 29, 2017, 10:32 p.m.