picasso: PathwIse CAlibrated Sparse Shooting algOrithm (PICASSO)
In picasso: Pathwise Calibrated Sparse Shooting Algorithm

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

The function "picasso" implements the user interface.

picasso(X, Y, lambda = NULL, nlambda = 100, lambda.min.ratio =
                 0.05, family = "gaussian", method = "l1",
                 type.gaussian = "naive", gamma = 3, df = NULL,
                 standardize = TRUE, intercept = TRUE, prec = 1e-07,
                 max.ite = 1000, verbose = FALSE)

`X`	`X` is an n by d design matrix where `n` is the sample size and `d` is the data dimension.
`Y`	`Y` is the n dimensional response vector. Y is numeric vector for family=``gaussian'' and family=``sqrtlasso'', or a two-level factor for family=``binomial'', or a non-negative integer vector representing counts for family = ``gaussian''.
`lambda`	A sequence of decresing positive values 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`. The default value of lambda.max* is the minimum regularization parameter which yields an all-zero estimates.
`nlambda`	The number of values used in `lambda`. Default value is 100.
`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 `0.05`.

Caution: logistic and poisson regression can be ill-conditioned if lambda is too small for nonconvex penalty. We suggest the user to avoid using any lambda.min.raito smaller than 0.05 for logistic/poisson regression under nonconvex penalty.

`family`	Options for model. Sparse linear regression and sparse multivariate regression is applied if `family = "gaussian"`, sqrt lasso is applied if `family = "sqrtlasso"`, sparse logistic regression is applied if `family = "binomial"` and sparse poisson regression is applied if `family = "poisson"`. The default value is `"gaussian"`.
`method`	Options for regularization. Lasso is applied if `method = "l1"`, MCP is applied if `method = "mcp"` and SCAD Lasso is applied if `method = "scad"`. The default value is `"l1"`.
`type.gaussian`	Options for updating residuals in sparse linear regression. The naive update rule is applied if `opt = "naive"`, and the covariance update rule is applied if `opt = "covariance"`. The default value is `"naive"`.
`gamma`	The concavity parameter for MCP and SCAD. The default value is `3`.
`df`	Maximum degree of freedom for the covariance update. The default value is `2*n`.
`standardize`	Design matrix X will be standardized to have mean zero and unit standard deviation if `standardize = TRUE`. The default value is `TRUE`.
`intercept`	Does the model has intercept term or not. Default value is `TRUE`.
`prec`	Stopping precision. The default value is 1e-7.
`max.ite`	Max number of iterations for the algorithm. The default value is 1000.
`verbose`	Tracing information is disabled if `verbose = FALSE`. The default value is `FALSE`.

For sparse linear regression,

\min_{β} {\frac{1}{2n}}|| Y - X β - β_0||_2^2 + λ R(β),

where R(β) can be \ell_1 norm, MCP, SCAD regularizers.

For sparse logistic regression,

\min_{β} {\frac{1}{n}}∑_{i=1}^n (\log(1+e^{x_i^T β+ β_0}) - y_i x_i^T β) + λ R(β),

where R(β) can be \ell_1 norm, MCP, and SCAD regularizers.

For sparse poisson regression,

\min_{β} {\frac{1}{n}}∑_{i=1}^n (e^{x_i^T β + β_0} - y_i (x_i^T β+β_0) + λ R(β),

where R(β) can be \ell_1 norm, MCP or SCAD regularizers.

An object with S3 classes "gaussian", "binomial", and "poisson" corresponding to sparse linear regression, sparse logistic regression, and sparse poisson regression respectively is returned:

`beta`	A matrix of regression estimates whose columns correspond to regularization parameters for sparse linear regression and sparse logistic regression. A list of matrices of regression estimation corresponding to regularization parameters for sparse column inverse operator.
`intercept`	The value of intercepts corresponding to regularization parameters for sparse linear regression, and sparse logistic regression.
`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`.
`family`	The `family` from the input.
`method`	The `method` from the input.
`path`	A list of `d` by `d` adjacency matrices of estimated graphs as a graph path corresponding to `lambda`.
`sparsity`	The sparsity levels of the graph path for sparse inverse column operator.
`standardize`	The `standardize` from the input.
`df`	The degree of freecom (number of nonzero coefficients) along the solution path for sparse linear regression, nd sparse logistic regression.
`ite`	A list of vectors where the i-th entries of ite[[1]] and ite[[2]] correspond to the outer iteration and inner iteration of i-th regularization parameter respectively.
`verbose`	The `verbose` from the input.

Jason Ge, Xingguo Li, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Jason Ge <jiange@princeton.edu>

1. J. Friedman, T. Hastie and H. Hofling and R. Tibshirani. Pathwise coordinate optimization. The Annals of Applied Statistics, 2007.
2. C.H. Zhang. Nearly unbiased variable selection under minimax concave penalty. Annals of Statistics, 2010.
3. J. Fan and R. Li. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 2001.
4. R. Tibshirani, J. Bien, J. Friedman, T. Hastie, N. Simon, J. Taylor and R. Tibshirani. Strong rules for discarding predictors in lasso-type problems. Journal of the Royal Statistical Society: Series B, 2012.
5. T. Zhao, H. Liu, and T. Zhang. A General Theory of Pathwise Coordinate Optimization. Techinical Report, Princeton Univeristy.

picasso-package.

################################################################
## Sparse linear regression
## Generate the design matrix and regression coefficient vector
n = 100 # sample number 
d = 80 # sample dimension
c = 0.5 # correlation parameter
s = 20  # support size of coefficient
set.seed(2016)
X = scale(matrix(rnorm(n*d),n,d)+c*rnorm(n))/sqrt(n-1)*sqrt(n)
beta = c(runif(s), rep(0, d-s))

## Generate response using Gaussian noise, and fit sparse linear models
noise = rnorm(n)
Y = X%*%beta + noise

## l1 regularization solved with naive update
fitted.l1.naive = picasso(X, Y, nlambda=100, type.gaussian="naive")

## l1 regularization solved with covariance update
fitted.l1.covariance  = picasso(X, Y, nlambda=100, type.gaussian="covariance")

## mcp regularization
fitted.mcp = picasso(X, Y, nlambda=100, method="mcp")

## scad regularization
fitted.scad = picasso(X, Y, nlambda=100, method="scad")

## lambdas used 
print(fitted.l1.naive$lambda)

## number of nonzero coefficients for each lambda
print(fitted.l1.naive$df)

## coefficients and intercept for the i-th lambda
i = 30
print(fitted.l1.naive$lambda[i])
print(fitted.l1.naive$beta[,i])
print(fitted.l1.naive$intercept[i])

## Visualize the solution path
plot(fitted.l1.naive)
plot(fitted.l1.covariance)
plot(fitted.mcp)
plot(fitted.scad)


################################################################
## Sparse logistic regression
## Generate the design matrix and regression coefficient vector
n <- 100  # sample number 
d <- 80   # sample dimension
c <- 0.5   # parameter controlling the correlation between columns of X
s <- 20    # support size of coefficient
set.seed(2016)
X <- scale(matrix(rnorm(n*d),n,d)+c*rnorm(n))/sqrt(n-1)*sqrt(n)
beta <- c(runif(s), rep(0, d-s))

## Generate response and fit sparse logistic models
p = 1/(1+exp(-X%*%beta))
Y = rbinom(n, rep(1,n), p)

## l1 regularization
fitted.l1 = picasso(X, Y, nlambda=100, family="binomial", method="l1")

## mcp regularization
fitted.mcp = picasso(X, Y, nlambda=100, family="binomial", method="mcp")

## scad regularization
fitted.scad = picasso(X, Y, nlambda=100, family="binomial", method="scad")

## lambdas used 
print(fitted.l1$lambda)

## number of nonzero coefficients for each lambda
print(fitted.l1$df)

## coefficients and intercept for the i-th lambda
i = 30
print(fitted.l1$lambda[i])
print(fitted.l1$beta[,i])
print(fitted.l1$intercept[i])

## Visualize the solution path
plot(fitted.l1)

## Estimate of Bernoulli parameters
param.l1 = fitted.l1$p


################################################################
## Sparse poisson regression
## Generate the design matrix and regression coefficient vector
n <- 100  # sample number 
d <- 80   # sample dimension
c <- 0.5   # parameter controlling the correlation between columns of X
s <- 20    # support size of coefficient
set.seed(2016)
X <- scale(matrix(rnorm(n*d),n,d)+c*rnorm(n))/sqrt(n-1)*sqrt(n)
beta <- c(runif(s), rep(0, d-s))/sqrt(s)

## Generate response and fit sparse poisson models
p = X%*%beta+rnorm(n)
Y = rpois(n, exp(p))

## l1 regularization
fitted.l1 = picasso(X, Y, nlambda=100, family="poisson", method="l1")

## mcp regularization
fitted.mcp = picasso(X, Y, nlambda=100, family="poisson", method="mcp")

## scad regularization
fitted.scad = picasso(X, Y, nlambda=100, family="poisson", method="scad")

## lambdas used 
print(fitted.l1$lambda)

## number of nonzero coefficients for each lambda
print(fitted.l1$df)

## coefficients and intercept for the i-th lambda
i = 30
print(fitted.l1$lambda[i])
print(fitted.l1$beta[,i])
print(fitted.l1$intercept[i])

## Visualize the solution path
plot(fitted.l1)

Loading required package: MASS
Loading required package: Matrix
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