pawls: Penalized adaptive weighted least squares regression
In pawls: Penalized Adaptive Weighted Least Squares Regression

Description Usage Arguments Value Author(s) Examples

Compute weighted least squares regression with L1 regularization on both the coefficients and weight vectors.

pawls(x, y, nlambda1 = 100, nlambda2 = 50, lambda1 = NULL,
  lambda2 = NULL, lambda1.min = 0.05, lambda2.min = 0.001, beta0 = NULL,
  w0 = NULL, initial = c("uniform", "PAWLS"), delta = 1e-06,
  maxIter = 1000, intercept = TRUE, standardize = TRUE,
  search = c("grid", "cross"))

`x`	a numeric matrix containing the predictor variables without an intercept. `pawls` standardizes the data and includes an intercept by default.
`y`	a numeric vector containing the response variable.
`nlambda1`	the number of lambda1 values (the default is 100).
`nlambda2`	the number of lambda2 values (the default is 50).
`lambda1`	a numeric vector of non-negative values to be used as tuning parameters of the penalty for coefficients. By default, a sequence of values of length `nlambda1` is computed, equally spaced on the log scale.
`lambda2`	a numeric vector of non-negative values to be used as tunning parameters of the penalty for weight vectors. By default, a sequence of values of length `nlambda2` is computed, equally spaced on the log scale.
`lambda1.min`	a numeric value giving the smallest value for `lambda1`, as a fraction of the `lambda1` maximum. The default is .05. Note that the `lambda1` maximum is an estimate of tuning parameter that set all the coefficientes to 0.
`lambda2.min`	a numeric value giving the smallest value for `lambda2`, as a fraction of the `lambda2` maximum. The default is .001. Note that the `lambda2` maximum is an estimate of tuning parameter that set all the weights to 1.
`beta0`	the initial estimates of coefficients to be used in the adaptive penalty for `beta`.
`w0`	the initial estimates of weight vector to be used in the adaptive penalty for `w`.
`initial`	a character string specifying the initial estimates of both coeffcients and weight vectors in the adaptive penalties. If "`uniform`", a non-adaptive `pawls` is performed. If "`pawls`", then the estimates are obtained by non-adaptive pawls (the default is "`uniform`").
`delta`	a small positive numeric value as a convergence threshold. The algorithm iterates until the RMSD for the change in both coefficents and weight vectors is less than delta (the default is 1e-06).
`maxIter`	a positive numeric value used to determin the maximum number of iterations (the default is 1000).
`intercept`	a logical indicating whether a constant term should be included in the model (the default is `TRUE`).
`standardize`	a logical indicating whether the predictor variables should be normalized to have unit L2 norm (the default is TRUE).
`search`	a character string specifying the algorithm to select tunning parameters for both coefficients and weight vectors. If "cross", the optimal tuning parameters are searched alternatively by minimizing `BIC`. If "grid", the optimal tuning parameters are selected as the pair that minimizs `BIC` over a fine grid. The default is "grid".

An object of class "pawls.cross"(if search=cross) or "pawls.grid" (if search=grid) contaning:

`beta`	a numeric vector containing the respective coefficient estimates with the optimal tuning parameters.
`w`	a numeric vector containing the respective weight estimates with the optimal tuning parameters.
`lambda1`	same as above.
`lambda2`	same as above.
`opt.lambda1`	a numeric value giving the optimal `lambda1` in the sense of minimzing `BIC`.
`opt.lambda2`	a numeric value giving the optimal `lambda2` in the sense of minimzing `BIC`.
`iter`	a numeric matrix with `nlambda2` rows and `nlambda1` columns giving the number of iteration until convergence at each pair of tuning parameters(`search=grid`) or a nueric value giving the number of iteration in corss search until convergence(`search=cross`).
`betas`	a 3-dimension numeric array containing the coefficient estimates. The dimensions are equal to `nlambda2`, `nlambda1` and the number of coefficients, respectively. It belongs to the object of class `"pawls.grid"` only.
`ws`	a 3-dimension numeric array containing the weight estimates. The dimensions are equal to `nlambda2`, `nlambda1` and the number of observations, respectively. It belongs to the object of class `"pawls.grid"` only.
`raw.bic`	a numeric matrix with `nlambda2` rows and `nlambda1` columns giving the raw `BIC` value for each pair of tuning prameters. It belongs to the object of class `"pawls.grid"` only.
`bic`	a numeric matrix with `nlambda2` rows and `nlambda1` columns giving the `BIC` value for each pair of tuning prameters. It belongs to the object of class `"pawls.grid"` only.

Bin Luo, Xiaoli Gao

## generate data
library("mvtnorm")
set.seed(123)  # for reproducibility
n = 100 # number of observations
p = 8 # number of variables
beta = c(1, 2, 3, 0, 0, 0, 0, 0) # coefficients
sigma <- 0.5      # controls signal-to-noise ratio
epsilon <- 0.1    # contamination level
Sigma <- 0.5^t(sapply(1:p, function(i, j) abs(i-j), 1:p))
x <- rmvnorm(n, sigma=Sigma)    # predictor matrix
e <- rnorm(n)                   # error terms
i <- 1:ceiling(epsilon*n)       # observations to be contaminated
e[i] <- e[i] + 5                # vertical outliers
y <- c(x %*% beta + sigma * e)  # response
x[i,] <- x[i,] + 5              # bad leverage points

## fit pawls model over a find grid of tuning parameters
pawls(x,y)

## fit adaptive pawls model over a find grid of tuning parameters
pawls(x,y,lambda1.min = 0.001, lambda2.min = 0.05, initial = "PAWLS")

## fit adaptive pawls model using corss search over a grid of tuning parameters
pawls(x,y,lambda1.min = 0.001, lambda2.min = 0.05, initial = "PAWLS", search="cross")