Package {picasso}

picasso: Sparse Learning with Convex and Non-Convex Penalties

Description

Computationally efficient tools for fitting sparse generalized linear models with convex or non-convex penalties. Supported penalties include lasso, smoothly clipped absolute deviation (SCAD), and minimax concave penalty (MCP). Supported families are Gaussian, binomial, Poisson, square-root-lasso, and multinomial. Computation combines pathwise coordinate optimization with warm starts, active-set updates, screening rules, Proximal Newton iterations, and adaptive local linear approximation for applicable nonconvex models.

Details

Core optimization routines are implemented in C++. Gaussian models use active-set coordinate descent with an automatic residual/covariance backend; binomial and Poisson models use active-set Proximal Newton/IRLS, square-root-lasso uses a global quadratic majorizer with active-set coordinate updates, and multinomial uses class-coupled Proximal Newton/IRLS. The R interface returns coefficient paths as Matrix objects and provides prediction, assessment, confusion-matrix, and cross-validation helpers.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso.

Model Assessment for picasso Fits

Description

Compute prediction-quality metrics across the regularization path. assess.picasso evaluates deviance, MSE, MAE, or classification error on a test set; confusion.picasso returns confusion matrices for binary and multinomial classification. For scalar families, large lambda paths are evaluated in bounded link-predictor blocks; small paths still use one matrix multiplication.

Usage

assess.picasso(object, newx, newy, newoffset = NULL, ...)

## S3 method for class 'assess.picasso'
print(x, ...)

confusion.picasso(object, newx, newy, lambda.idx = NULL,
  newoffset = NULL, ...)

Arguments

Value

assess.picasso returns an object of class "assess.picasso" containing named numeric vectors (one value per lambda):

lambda

Lambda values.

deviance

Family-specific test loss: half mean squared error for Gaussian and square-root-lasso, Bernoulli or multinomial negative log-likelihood for categorical models, and conventional mean Poisson deviance for Poisson.

mse

Mean squared error (Gaussian, square-root-lasso, or Poisson).

mae

Mean absolute error (Gaussian or square-root-lasso).

class

Misclassification rate (binomial or multinomial).

confusion.picasso returns a list of table objects, one per selected lambda value, with predicted classes in rows and observed classes in columns. Multinomial tables retain all fitted class levels on both axes, including levels absent from the supplied test subset; binomial tables always retain levels 0 and 1.

print.assess.picasso prints the range of each metric across the fitted path and returns its input invisibly.

Scalar-family assessment targets about 8 MiB for each link-predictor and coefficient block, subject to a minimum of one lambda column. Metric calculation may use additional arrays of the same bounded block shape. This changes only working-memory use, not the returned metrics or their ordering.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso, cv.picasso

Examples

set.seed(1)
n <- 100; d <- 30
X <- matrix(rnorm(n * d), n, d)
Y <- rbinom(n, 1, 0.5)
fit <- picasso(X, Y, family = "binomial", nlambda = 20)
a <- assess.picasso(fit, X, Y)
print(a)

Extract Coefficients from an Object with S3 Class "gaussian"

Description

Extract estimated coefficient vectors from selected points on the regularization path.

Usage

## S3 method for class 'gaussian'
coef(object, lambda.idx = NULL, beta.idx = NULL, ...)

Arguments

Value

A numeric matrix of extracted coefficients. Columns correspond to lambda.idx. Rows contain the intercept ("(Intercept)") and the selected coefficients ("beta[i]" for i in beta.idx).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Extract Coefficients from an Object with S3 Class "logit"

Description

Extract estimated coefficient vectors from selected points on the regularization path.

Usage

## S3 method for class 'logit'
coef(object, lambda.idx = NULL, beta.idx = NULL, ...)

Arguments

Value

A numeric matrix of extracted coefficients. Columns correspond to lambda.idx. Rows contain the intercept ("(Intercept)") and the selected coefficients ("beta[i]" for i in beta.idx).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Extract Coefficients from an Object with S3 Class "poisson"

Description

Extract estimated coefficient vectors from selected points on the regularization path.

Usage

## S3 method for class 'poisson'
coef(object, lambda.idx = NULL, beta.idx = NULL, ...)

Arguments

Value

A numeric matrix of extracted coefficients. Columns correspond to lambda.idx. Rows contain the intercept ("(Intercept)") and the selected coefficients ("beta[i]" for i in beta.idx).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Extract Coefficients from an Object with S3 Class "sqrtlasso"

Description

Extract estimated coefficient vectors from selected points on the regularization path.

Usage

## S3 method for class 'sqrtlasso'
coef(object, lambda.idx = NULL, beta.idx = NULL, ...)

Arguments

Value

A numeric matrix of extracted coefficients. Columns correspond to lambda.idx. Rows contain the intercept ("(Intercept)") and the selected coefficients ("beta[i]" for i in beta.idx).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Cross-Validation for picasso

Description

Performs K-fold cross-validation to select the regularization parameter for picasso models. Multinomial models use a fixed full-data class map and softmax negative log-likelihood or misclassification loss.

Usage

cv.picasso(X, Y, ..., nfolds = 10, foldid = NULL,
  type.measure = "default", fast.mode = FALSE)

## S3 method for class 'cv.picasso'
print(x, ...)

## S3 method for class 'cv.picasso'
plot(x, sign.lambda = 1, ...)

## S3 method for class 'cv.picasso'
coef(object, s = c("lambda.min", "lambda.1se"), ...)

## S3 method for class 'cv.picasso'
predict(object, newdata, s = c("lambda.min", "lambda.1se"),
  type = "response", ...)

Arguments

Details

Every binomial or multinomial training fold must contain every class observed in the full data; automatic stratification therefore requires at least two observations per class. A custom foldid that violates this rule or a failed, truncated, or otherwise unusable fold fit produces an explicit error rather than silently omitting the fold. Normal deviance-tail stopping may shorten the initial full-data generated path; that retained path is then passed explicitly to every fold and must be covered in full. An unusually deep scalar path can make a training fold reach normal saturation before completing that fixed sequence. This is reported as a truncated-fold error; request fewer lambdas or a larger lambda.min.ratio when it occurs.

Binomial factor responses use the full-data two-level map in every fold and are scored as zero/one outcomes. When predicting from an offset-fitted binomial or Poisson cv.picasso object, pass newoffset through ... to the underlying prediction method.

For Gaussian models, the full-data type.gaussian = "auto" decision is also reused in every fold, so changing training-fold sample sizes cannot switch the native backend.

Value

cv.picasso returns an object of class "cv.picasso" with components:

lambda

Ordered lambda sequence evaluated by every fold. A generated full-data fit may first stop normally and thereby establish a shorter path; every fold must then cover that fixed sequence completely.

cvm

Mean cross-validated error for each lambda.

cvsd

Standard error of the mean cross-validated error.

cvup, cvlo

Upper and lower error-bar bounds.

nzero

Number of nonzero coefficients at each lambda. For multinomial models this is the total across all classes.

lambda.min

Lambda that minimises cvm.

lambda.1se

Largest lambda within one standard error of the minimum.

name

Name of the loss measure used.

foldid

Fold assignment used for each observation.

family

Model family.

fast.mode

Logical precision preset used for all fits.

prec

Effective convergence tolerance used for all fits.

picasso.fit

Full-data picasso fit.

print.cv.picasso returns its input invisibly, and plot.cv.picasso draws the CV curve and returns NULL invisibly. coef.cv.picasso and predict.cv.picasso return the corresponding underlying fit-method result at lambda.min or lambda.1se.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso

Examples

set.seed(1)
n <- 100; d <- 50
X <- matrix(rnorm(n * d), n, d)
Y <- X[, 1:5] %*% rnorm(5) + rnorm(n)
cv <- cv.picasso(X, Y, family = "gaussian", nlambda = 20)
print(cv$lambda.min)

Bardet-Biedl Syndrome Gene Expression Data from Scheetz et al. (2006)

Description

Gene expression data (200 genes, 120 samples) from microarray experiments on mammalian eye tissue reported by Scheetz et al. (2006).

Usage

data(eyedata)

Format

A list with components:

x

A 120 by 200 matrix of gene probes.

y

A numeric response vector of length 120 representing the expression level of the TRIM32 gene.

Details

This data set contains 120 samples and 200 predictors.

Author(s)

Xingguo Li, Tuo Zhao, Tong Zhang and Han Liu
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

References

1. T. Scheetz, K. Kim, R. Swiderski, A. Philp, T. Braun, K. Knudtson, A. Dorrance, G. DiBona, J. Huang, T. Casavant, V. Sheffield, and E. Stone. Regulation of gene expression in the mammalian eye and its relevance to eye disease. Proceedings of the National Academy of Sciences of the United States of America, 2006.

See Also

picasso-package.

Examples

data(eyedata)
image(eyedata$x)

Methods for Multinomial picasso Fits

Description

Fit multinomial models with picasso using family = "multinomial", then print, plot, extract coefficients, or predict from the fitted class-coupled regularization path.

Usage

## S3 method for class 'multinomial'
print(x, ...)

## S3 method for class 'multinomial'
plot(x, which.class = 1, ...)

## S3 method for class 'multinomial'
coef(object, lambda.idx = NULL, beta.idx = NULL, ...)

## S3 method for class 'multinomial'
predict(object, newdata, lambda.idx = NULL,
  type = "response", s = NULL, ...)

Arguments

Details

The public fitting call is picasso(X, Y, family = "multinomial", ...). Responses must contain at least three observed classes; numeric, character, and factor labels are accepted, and unused factor levels are dropped.

The solver first identifies a strong working set and then solves the class-coupled restricted problem with Proximal Newton/IRLS iterations until the weighted-L1 subproblem converges. Full KKT checks update the active set and certify every retained subproblem. Pathwise warm starts, sequential strong screening, adaptive inexact Newton tolerances, vectorized coordinate kernels, and accepted-probability reuse reduce repeated work.

MCP and SCAD use adaptive local linear approximation. Every fit performs the minimum three total stages (one L1 master and two weighted-L1 updates). Raising lla.max.stages permits further stages, which stop once target stationarity is at most prec; exhausting the budget yields the usable status lla_stationarity_limit.

After at least five lambdas, an automatically generated path stops normally when explained deviance exceeds 0.999 or its improvement is below 1e-5. Explicit lambda sequences disable only this saturation rule; dfmax or a hard failure can still truncate them. A hard failure after one or more completed lambdas returns the committed prefix with a warning and failure metadata; failure before the first completed lambda is an error.

With no intercept, standardized predictors are scaled about the origin rather than centered. The zero model then uses uniform class probabilities for the automatic lambda path and null deviance.

Value

print.multinomial returns x invisibly. plot.multinomial draws one class's coefficient paths and returns NULL invisibly.

coef.multinomial returns a list of K numeric matrices, one per class. Rows contain the intercept followed by selected coefficients, and columns correspond to selected lambdas.

For one selected lambda, predict.multinomial returns an n_{\mathrm{new}} \times K matrix for "response" or "link", a factor of length n_{\mathrm{new}} for "class", or a list of K integer vectors for "nonzero". Multiple lambda.idx or s values return a list of those objects.

The fitted object returned by picasso contains class-specific beta and intercept lists, lambda, df, levels, K, dev.ratio, status/failure metadata, per-lambda diagnostics, path.early.stopped, and requested.nlambda; see picasso.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso, cv.picasso, and assess.picasso.

Examples

set.seed(1)
n <- 120; d <- 12
X <- matrix(rnorm(n * d), n, d)
score <- cbind(X[, 1], -X[, 1] + X[, 2], -X[, 2])
Y <- factor(c("red", "green", "blue")[max.col(score)])
fit <- picasso(X, Y, family = "multinomial", nlambda = 12)
prob <- predict(fit, X[1:5, ], lambda.idx = fit$nlambda,
                type = "response")
pred <- predict(fit, X[1:5, ], lambda.idx = fit$nlambda,
                type = "class")

Fit Sparse Regression Models with Convex and Non-Convex Penalties

Description

picasso fits regularization paths for sparse Gaussian, logistic (family = "binomial"), Poisson, sqrt-lasso, and multinomial models using lasso, MCP, or SCAD penalties.

Usage

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

Arguments

Details

When lambda is not supplied, picasso constructs a logarithmically spaced path between \lambda_{\max} and \lambda_{\min} = \mathrm{lambda.min.ratio} \cdot \lambda_{\max}.

The method solves a penalized optimization problem:

\min_{\beta_0, \beta}\; \mathcal{L}(\beta_0,\beta) + \sum_{j=1}^d p_{\lambda,\gamma}(|\beta_j|),

where p_{\lambda,\gamma} is lasso, MCP, or SCAD. For multinomial models, the penalty term is instead \sum_{k=1}^K\sum_{j=1}^d p_{\lambda,\gamma}(|B_{jk}|); intercepts are never penalized.

Loss functions by family are:

• "gaussian": \mathcal{L}(\beta_0,\beta)=\frac{1}{2n}\lVert Y-\beta_0\mathbf{1}-X\beta \rVert_2^2.

• "binomial": \mathcal{L}(\beta_0,\beta)=\frac{1}{n}\sum_{i=1}^n[\log(1+\exp(\eta_i))-y_i\eta_i].

• "poisson": \mathcal{L}(\beta_0,\beta)=\frac{1}{n}\sum_{i=1}^n[\exp(\eta_i)-y_i\eta_i].

• "sqrtlasso": \mathcal{L}(\beta_0,\beta)=\frac{1}{\sqrt{n}}\lVert Y-\beta_0\mathbf{1}-X\beta \rVert_2.

• "multinomial": \mathcal{L}(b,B)=-\frac{1}{n}\sum_{i=1}^n\log\left(\frac{\exp(b_{y_i}+x_i^\top B_{y_i})}{\sum_{k=1}^K\exp(b_k+x_i^\top B_k)}\right).

For binomial and Poisson models, \eta_i=\beta_0+x_i^\top\beta+o_i, where o_i is the supplied offset or zero.

Gaussian paths use active-set coordinate descent with residual or lazy covariance updates. Binomial and Poisson weighted-L1 subproblems use active-set Proximal Newton/IRLS, square-root-lasso uses a global quadratic majorizer with active-set coordinate updates, and multinomial uses class-coupled active-set Proximal Newton/IRLS. MCP and SCAD use adaptive LLA around those weighted-L1 solvers except for the direct Gaussian nonconvex coordinate-descent implementation.

Value

A fitted object of class "gaussian", "logit", "poisson", "sqrtlasso", or "multinomial" depending on family.

Common components include:

beta

A Matrix object of dimension d \times \mathrm{nlambda} containing fitted coefficients. Its concrete representation may be sparse or dense. Multinomial fits instead return a list of K such objects, one per class.

intercept

Numeric vector of intercepts. Multinomial fits return a list of K numeric vectors, one per class.

lambda

Regularization sequence used for fitting.

nlambda

Number of regularization values.

df

Degrees of freedom (number of nonzero coefficients) along the path.

nulldev

Null-model loss or deviance under the fitted intercept convention and family-specific scale.

dev.ratio

Fraction of null deviance explained at each lambda, analogous to R^2 for Gaussian fits. Values are clipped to zero through one.

method

Penalty type used for fitting.

alg

Identifier for the resolved native algorithm.

ite

Iteration information returned by the solver.

verbose

Input verbose value.

runtime

Elapsed fitting time.

family

Input family value.

fast.mode

Logical precision preset used for fitting.

prec

Effective convergence tolerance used by the native solver.

type.gaussian.requested

For Gaussian fits, the requested solver mode.

type.gaussian

For Gaussian fits, the resolved native solver mode.

status

Native termination label. Gaussian iteration-limit failures retain only the previously converged lambda prefix.

status.code

Integer form of status.

failure

Details for the first uncommitted hard failure, otherwise NULL.

Binomial, Poisson, sqrt-lasso, and multinomial fits additionally include:

runt

Native per-lambda runtime.

lla.max.stages

Requested adaptive-LLA stage budget.

diagnostics

Per-lambda runtime, penalized objective, weighted-L1 KKT residual, and target stationarity. Scalar-family diagnostics also report solver iterations, active-set size, and completed LLA stages; multinomial diagnostics instead report outer iterations, inner sweeps, coordinate updates, and active-set size.

For non-Gaussian MCP/SCAD fits, status lla_stationarity_limit means that every returned model is usable, but at least one point reached its LLA-stage budget before meeting prec.

Native path fitting polls for a pending user interrupt (Ctrl-C) at every lambda boundary, so a long path can be stopped without waiting for completion; the interrupt is then raised as a standard R interrupt condition. A single very expensive lambda still runs to completion before the poll.

Multinomial fits also include path.early.stopped and requested.nlambda. A successful automatically generated path may contain fewer lambdas than requested after its deviance has saturated. Explicit paths do not apply saturation stopping, but may still be truncated by dfmax or a hard failure. Every retained weighted-L1 subproblem meets its KKT tolerance; for MCP/SCAD, inspect target stationarity when status.code == 10.

Binomial and Poisson fits include offset.used, a logical flag recording whether offset was explicitly supplied. Binomial fits include the two response levels used for zero/one encoding; multinomial fits include K and the retained class levels.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

References

1. J. Friedman, T. Hastie, 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. The 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. J. Ge, X. Li, H. Jiang, H. Liu, T. Zhang, M. Wang, and T. Zhao. PICASSO: A sparse learning library for high-dimensional data analysis in R and Python. Journal of Machine Learning Research, 20(44):1–5, 2019. https://www.jmlr.org/papers/v20/17-722.html.

See Also

picasso-package.

Examples

################################################################
## 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

## automatic Gaussian backend selection (the default)
fitted.l1.auto = picasso(X, Y, nlambda=100)

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

## covariance updates can be faster when n >> d
fitted.l1.covariance  = picasso(X, Y, nlambda=100, type.gaussian="covariance")

## early stopping: stop when more than 10 nonzero coefficients
fitted.l1.dfmax = picasso(X, Y, nlambda=100, dfmax=10)

## 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)

## Predicted probabilities for the first 5 observations at lambda[30]
param.l1 = predict(fitted.l1, X[1:5, ], lambda.idx = 30,
                   type = "response")


################################################################
## 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)


################################################################
## Square-root-lasso
set.seed(2)
X <- matrix(rnorm(80 * 12), 80, 12)
Y <- X[, 1] - X[, 2] + rnorm(80)
fitted.sqrt <- picasso(
  X, Y, family = "sqrtlasso", nlambda = 12, method = "l1"
)
predict(fitted.sqrt, X[1:3, ], lambda.idx = fitted.sqrt$nlambda)


################################################################
## Multinomial regression with retained class labels
score <- cbind(X[, 1], -X[, 1] + X[, 2], -X[, 2])
Y <- factor(c("red", "green", "blue")[max.col(score)])
fitted.multi <- picasso(
  X, Y, family = "multinomial", nlambda = 12, method = "l1"
)
predict(fitted.multi, X[1:3, ], lambda.idx = fitted.multi$nlambda,
        type = "class")

Plot Method for an Object with S3 Class "gaussian"

Description

Plot coefficient paths across regularization parameters.

Usage

## S3 method for class 'gaussian'
plot(x, ...)

Arguments

Value

No return value, called for side effects (drawing the coefficient-path plot).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Plot Method for an Object with S3 Class "logit"

Description

Plot coefficient paths across regularization parameters.

Usage

## S3 method for class 'logit'
plot(x, ...)

Arguments

Value

No return value, called for side effects (drawing the coefficient-path plot).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Plot Method for an Object with S3 Class "poisson"

Description

Plot coefficient paths across regularization parameters.

Usage

## S3 method for class 'poisson'
plot(x, ...)

Arguments

Value

No return value, called for side effects (drawing the coefficient-path plot).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Plot Method for an Object with S3 Class "sqrtlasso"

Description

Plot coefficient paths across regularization parameters.

Usage

## S3 method for class 'sqrtlasso'
plot(x, ...)

Arguments

Value

No return value, called for side effects (drawing the coefficient-path plot).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Prediction Method for an Object with S3 Class "gaussian"

Description

Predict responses for new data using fitted models.

Usage

## S3 method for class 'gaussian'
predict(object, newdata, lambda.idx = NULL, Y.pred.idx = NULL,
  type = "response", s = NULL, ...)

Arguments

Details

predict.gaussian returns predicted responses for newdata using fitted coefficients from object:

\hat{Y} = \hat{\beta}_0 + X_{new} \hat{\beta}.

Value

Return type depends on type:

• "response" or "link" (default): numeric matrix of predicted values \hat{\beta}_0 + X \hat{\beta}.

• "nonzero": list of integer vectors of nonzero coefficient indices, one element per selected lambda.

Rows correspond to observations; columns correspond to lambda.idx (or s values when s is specified).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Prediction Method for an Object with S3 Class "logit"

Description

Predict responses for new data using fitted models.

Usage

## S3 method for class 'logit'
predict(object, newdata, lambda.idx = NULL, p.pred.idx = NULL,
  type = "response", s = NULL, newoffset = NULL, ...)

Arguments

Details

predict.logit returns predicted Bernoulli probabilities for newdata using fitted coefficients from object:

\hat{p} = \frac{e^{o_{new} + \hat{\beta}_0 + X_{new} \hat{\beta}}}{1+e^{o_{new} + \hat{\beta}_0 + X_{new} \hat{\beta}}}.

Value

Return type depends on type:

• "response" (default): numeric matrix of predicted probabilities \hat{p} = \sigma(o_{new} + \hat{\beta}_0 + X \hat{\beta}).

• "link": numeric matrix of log-odds o_{new} + \hat{\beta}_0 + X \hat{\beta}.

• "class": integer matrix of predicted class labels (0 or 1).

• "nonzero": list of integer vectors of nonzero coefficient indices, one element per selected lambda.

Rows correspond to observations; columns correspond to lambda.idx (or s values when s is specified).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Prediction Method for an Object with S3 Class "poisson"

Description

Predict responses for new data using fitted models.

Usage

## S3 method for class 'poisson'
predict(object, newdata, lambda.idx = NULL, p.pred.idx = NULL,
  type = "response", s = NULL, newoffset = NULL, ...)

Arguments

Details

predict.poisson returns predicted Poisson means for newdata using fitted coefficients from object:

\hat{\mu} = e^{o_{new} + \hat{\beta}_0 + X_{new} \hat{\beta}}.

Value

Return type depends on type:

• "response" (default): numeric matrix of predicted Poisson means \hat{\mu} = \exp(o_{new} + \hat{\beta}_0 + X \hat{\beta}).

• "link": numeric matrix of log-means o_{new} + \hat{\beta}_0 + X \hat{\beta}.

• "nonzero": list of integer vectors of nonzero coefficient indices, one element per selected lambda.

Rows correspond to observations; columns correspond to lambda.idx (or s values when s is specified).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Prediction Method for an Object with S3 Class "sqrtlasso"

Description

Predict responses for new data using fitted models.

Usage

## S3 method for class 'sqrtlasso'
predict(object, newdata, lambda.idx = NULL, Y.pred.idx = NULL,
  type = "response", s = NULL, ...)

Arguments

Details

predict.sqrtlasso returns predicted responses for newdata using fitted coefficients from object:

\hat{Y} = \hat{\beta}_0 + X_{new} \hat{\beta}.

Value

Return type depends on type:

• "response" or "link" (default): numeric matrix of predicted values \hat{\beta}_0 + X \hat{\beta}.

• "nonzero": list of integer vectors of nonzero coefficient indices, one element per selected lambda.

Rows correspond to observations; columns correspond to lambda.idx (or s values when s is specified).

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Print Method for an Object with S3 Class "gaussian"

Description

Print a concise summary for an object with S3 class "gaussian".

Usage

## S3 method for class 'gaussian'
print(x, ...)

Arguments

Details

This method prints tuning parameters and fit statistics for a fitted model object.

Value

The input object x, returned invisibly.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Print Method for an Object with S3 Class "logit"

Description

Print a concise summary for an object with S3 class "logit".

Usage

## S3 method for class 'logit'
print(x, ...)

Arguments

Details

This method prints tuning parameters and fit statistics for a fitted logistic model object.

Value

The input object x, returned invisibly.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Print Method for an Object with S3 Class "poisson"

Description

Print a concise summary for an object with S3 class "poisson".

Usage

## S3 method for class 'poisson'
print(x, ...)

Arguments

Details

This method prints tuning parameters and fit statistics for a fitted Poisson model object.

Value

The input object x, returned invisibly.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.

Print Method for an Object with S3 Class "sqrtlasso"

Description

Print a concise summary for an object with S3 class "sqrtlasso".

Usage

## S3 method for class 'sqrtlasso'
print(x, ...)

Arguments

Details

This method prints tuning parameters and fit statistics for a fitted model object.

Value

The input object x, returned invisibly.

Author(s)

Jason Ge, Xingguo Li, Haoming Jiang, Mengdi Wang, Tong Zhang, Han Liu and Tuo Zhao
Maintainer: Tuo Zhao <tourzhao@gatech.edu>

See Also

picasso and picasso-package.