| Type: | Package |
| Title: | Estimate a Penalized Linear Model using the CDF Penalty Function |
| Version: | 0.1.0 |
| Author: | Daniele Cuntrera [aut, cre], Luigi Augugliaro [aut], Vito M.R. Muggeo [aut] |
| Maintainer: | Daniele Cuntrera <daniele.cuntrera@unipa.it> |
| Description: | Utilize the CDF penalty function to estimate a penalized linear model. It enables you to display some graphical representations and determine whether the Karush-Kuhn-Tucker conditions are met. For more details about the theory, please refer to Cuntrera, D., Augugliaro, L., & Muggeo, V. M. (2022) <doi:10.48550/arXiv.2212.08582>. |
| License: | GPL-2 | GPL-3 |
| Encoding: | UTF-8 |
| Imports: | plot.matrix |
| Suggests: | testthat (≥ 3.0.0) |
| Config/testthat/edition: | 3 |
| NeedsCompilation: | no |
| Packaged: | 2023-01-28 16:24:35 UTC; heghe |
| Repository: | CRAN |
| Date/Publication: | 2023-01-30 16:40:02 UTC |
BIC calculator function
Description
Function that takes the resulting values of the estimated model as input, to compute BIC
Usage
BIC_calc(X,
b.tld,
y,
n)
Arguments
X |
The covariates' matrix |
b.tld |
The estimated sparse-beta |
y |
The response variable |
n |
The number of observation |
Value
Returns the BIC value calculated for a single value of the tuning parameter.
Examples
p <- 10
n <- 100
X <- cbind(1, matrix(rnorm(n * p), n , p))
b.s <- c(1, rep(0, p))
b.s[sample(2:p, 3)] <- 1
y <- drop(crossprod(t(X), b.s))
out <- cdfPen(X = X, y = y)
(bic <- BIC_cdfpen(out))
plot(out$lmb, bic, "s")BIC computation from a "cdfpen" object
Description
Calculates the BIC for all estimated models in a "cdfpen" object
Usage
BIC_cdfpen(object)
Arguments
object |
Object containing the results. |
Value
Returns a vector containing the BIC values calculated over the entire estimated path
Threshold function for CDF penalty
Description
Applies the threshold rule to obtain the vector of sparse estimates
Usage
S(bm_gm,
db,
w)
Arguments
bm_gm |
Vector of pseudo-solution. |
db |
Lambda-rho ratio. |
w |
Weights obtained from the penalty function. |
Value
The estimated coefficient
Fit a Linear Model with with CDF regularization
Description
Uses the CDF penalty to estimate a linear model with the maximum penalized likelihood. The path of coefficients is computed for a grid of values for the lambda regularization parameter.
Usage
cdfPen(X,
y,
nu,
lmb,
nlmb = 100L,
e = 1E-3,
rho = 2,
algorithm = c("lla", "opt"),
nstep = 1E+5,
eps = 1E-6,
eps.lla = 1E-6,
nstep.lla = 1E+5)
Arguments
X |
Matrix of covariates, each row is a vector of observations. The matrix must not contain the intercept. |
y |
Vector of response variable. |
nu |
Shape parameter of the penalty. It affects the degree of the non-convexity of the penalty. If no value is specified, the smallest value that ensures a single solution will be used. |
lmb |
A user-supplied tuning parameter sequence. |
nlmb |
number of lambda values; 100 is the default value. |
e |
The smallest lambda value, expressed as a percentage of maximum lambda. Default value is .001. |
rho |
Parameter of the optimization algorithm. Default is 2. |
algorithm |
Approximation to be used to obtain the sparse solution. |
nstep |
Maximum number of iterations of the global algorithm. |
eps |
Convergence threshold of the global algorithm. |
eps.lla |
Convergence threshold of the LLA-algorithm (if used). |
nstep.lla |
Maximum number of iterations of the LLA-algorithm (if used). |
Details
We consider a local quadratic approximation of the likelihood to treat the problem as a weighted linear model.
The choice of value assigned to \nu is of fundamental importance: it affects both computational and estimation aspects. It affects the ”degree of non-convexity” of the penalty and determines which of the good and bad properties of convex and non-convex penalties are obtained. Using a high value of \nu ensures the uniqueness of solution, but the estimates will be biased. Conversely, a small value of \nu guarantees negligible bias in the estimates. The parameter \nu has the role of determining the convergence rate of non-null estimates$: the lower the value, the higher the convergence rate. Using lower values of \nu, the objective function will have local minima.
Value
coefficients |
The coefficients fit matrix. The number of columns is equal to nlmb, and the number of rows is equal to the number of coefficients. |
lmb |
The vector of lambda used. |
e |
The smallest lambda value, expressed as a percentage of maximum lambda. Default value is .001. |
rho |
The parameter of the optimization algorithm used |
nu |
The shape parameters of the penalty used. |
X |
The design matrix. |
y |
The response. |
algorithm |
Approximation used |
Author(s)
Daniele Cuntrera, Luigi Augugliaro, Vito Muggeo
References
Aggiungere Arxiv
Examples
p <- 10
n <- 100
X <- cbind(1, matrix(rnorm(n * p), n , p))
b.s <- c(1, rep(0, p))
b.s[sample(2:p, 3)] <- 1
y <- drop(crossprod(t(X), b.s))
out <- cdfPen(X = X, y = y)Fitter function for CDF penalty
Description
These are the fundamental computing algorithms that cdfPen invokes to estimate penalized linear models by varying lambda.
Usage
cdfPen.fit(b,
b.tld,
g,
b.rho,
H.rho,
lmb.rho,
nu,
algorithm,
nstep = 1E+5,
eps = 1E-5,
eps.lla = 1E-6,
nstep.lla = 1E+5)
Arguments
b |
Starting values of beta-vector. |
b.tld |
Starting values of sparse beta-vector. |
g |
Starting values of pseudo-variable. |
b.rho |
Ridge solution. |
H.rho |
Second part of ridge solution. |
lmb.rho |
Lambda-rho ratio. |
nu |
Shape parameter of the penalty. It affects the degree of the non-convexity of the penalty. |
algorithm |
Approximation to be used to obtain the sparse solution. |
nstep |
Maximum number of iterations of the global algorithm. |
eps |
Convergence threshold of the global algorithm. |
eps.lla |
Convergence threshold of the LLA-algorithm (if used). |
nstep.lla |
Maximum number of iterations of the LLA-algorithm (if used). |
Value
b |
Estimated beta-vector. |
b.tld |
Estimated sparse beta-vector. |
g |
Final values of pseudo-variable. |
i |
Number of iterations. |
conv |
Convergence check status (0 if converged). |
Author(s)
Daniele Cuntrera, Luigi Augugliaro, Vito Muggeo
References
Aggiungere Arxiv
Check on the condition of Karush-Kuhn-Tucker
Description
Control over Karush-Kuhn-Tucker (Karush, 1939) conditions for the estimates obtained.
Usage
check_KKT(obj,
intercept = TRUE)
Arguments
obj |
Object to be checked. |
intercept |
Is the intercept used in the model? |
Value
grd |
The value of gradient. |
hx |
The value of equality constraint. |
glob |
The global value of derivative (grd + hx). |
test |
Is the condition verified? |
lmb |
The values of lambda used in the model |
Author(s)
Daniele Cuntrera, Luigi Augugliaro, Vito Muggeo
References
Karush, W. (1939). Minima of functions of several variables with inequalities as side constraints. M. Sc. Dissertation. Dept. of Mathematics, Univ. of Chicago.
Examples
p <- 10
n <- 100
X <- cbind(1, matrix(rnorm(n * p), n , p))
b.s <- c(1, rep(0, p))
b.s[sample(2:p, 3)] <- 1
y <- drop(crossprod(t(X), b.s))
out <- cdfPen(X = X, y = y)
KKT <- check_KKT(out)
plot(KKT$test)
LLA approximation for CDF penalty
Description
Linearly approximate a part of the objective function to greatly speed up computations.
Usage
lla(b.o,
lmb.rho,
bm_gm,
nu,
nstep.lla = 100L,
eps.lla = 1E-6)
Arguments
b.o |
Vector of sparse-solution. |
lmb.rho |
Lambda-rho ratio. |
bm_gm |
Vector of pseudo-solution |
nu |
Shape parameter of the penalty. |
nstep.lla |
Maximum number of iterations of the LLA-algorithm (if used). |
eps.lla |
Convergence threshhold of the LLA-algorithm (if used). |
Details
The LLA approximation allows the computationally intensive part to be treated as a weighted LASSO (Tibshirani, 1996) problem. In this way the computational effort is significantly less while maintaining satisfactory accuracy of the results. See Zou and Li (2008).
Value
b |
Vector of the estimated sparse-solution. |
Conv |
Convergence check (0 if converged). |
nstep.lla |
Number of iterations done. |
References
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B (Methodological), 58(1):267–288.
Zou, H. and Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. Annals of statistics, 36(4):1509
Plot coefficients or BIC from a "cdfpen" object
Description
Plot coefficient profile plot or BIC trend
Usage
plot_cdfpen(object,
...)
Arguments
object |
Object to be plotted. |
... |
Other graphical parameters to plot. |
Details
A graph showing the BIC trend or profile of coefficients is displayed.
Value
No return value
Author(s)
Daniele Cuntrera, Luigi Augugliaro, Vito Muggeo
Examples
p <- 10
n <- 100
X <- cbind(1, matrix(rnorm(n * p), n , p))
b.s <- c(1, rep(0, p))
b.s[sample(2:p, 3)] <- 1
y <- drop(crossprod(t(X), b.s))
out <- cdfPen(X = X, y = y)
plot_cdfpen(out) #Coefficients' path ~ lambda
plot_cdfpen(out, "l1") #Coefficients' path ~ L1 norm
plot_cdfpen(out, "BIC") #BIC ~ lambda Plotter function for cdfpen class
Description
Function that takes user requests as input, to show the requested graph
Usage
plot_path(obj,
lmb,
coeff,
type = c("path", "l1", "BIC"),
...)
Arguments
obj |
Object to be plotted |
lmb |
lambda values used in the model |
coeff |
the coefficients' matrix |
type |
type of graph to be ploted |
... |
Other characteristics to be added |
Value
No return value