Package 'nnlasso'

The car data

Description

Car data has 105 observations and 12 variables. All variables except the 12th are standardized such that mean of each of them is 0 and standard deviation is 1. First 10 variables are various characteristics of the cars. The 11th variable y is the price. The 12th variable is a binary variable.

Usage

data("car")

Format

A data frame with 105 observations on the following 12 variables.

Weight

weights of the cars

Length

overall length

Wheel.base

length of wheelbase

Width

width of car

Frt.Leg.Room

maximum front leg room

Front.Hd

distance between the car's head-liner and the head of a 5 ft. 9 in. front seat passenger

Turning

the radius of the turning circle

Disp

engine displacement

HP

net horsepower

Tank

fuel refill capacity

y

price

y1

High or low price

Details

The data is created from car90 data of rpart package with selected 11 variables. The selected variables are Weight,Length,Wheel.base,Width,Frt.Leg.Room,Front.Hd,Turning,Disp,HP,Tank,Price. All these variables are standardized such that each of them has mean 0 and standard deviation 1. Price variable has been renamed as y. The variable y1 is a dichotomous variable created from that the data such that if price >=25000, then y1=1 else y1=0. Only complete cases are considered, so the data has 105 observations in place of 111 observations in car90 data set.

Source

Terry Therneau, Beth Atkinson and Brian Ripley (2014). rpart: Recursive Partitioning and Regression Trees. R package version 4.1-8. http://CRAN.R-project.org/package=rpart

Examples

data(car)

---

Extract coefficients from a fitted nnlasso object

Description

The function returns the coefficients from a fitted nnlasso object

Usage

## S3 method for class 'nnlasso'
coef(object,...)

Arguments

object	A ‘nnlasso’ object obtained using ‘nnlasso’ function.
...	Not used

Value

Estimated coefficients for different lambdas starting from maximum value of lambda to minimum value of lambda

Author(s)

Baidya Nath Mandal and Jun Ma

References

Mandal, B.N. and Ma, J. (2016). L1 regularized multiplicative iterative path algorithm for non-negative generalized linear models

Examples

data(car)
attach(car)
x=car[,1:10]
g1=nnlasso(x,y1,family="binomial")
coef(g1)
g1=nnlasso(x,y,family="normal")
coef(g1)
detach(car)

---

k-fold cross validation for penalized generalized linear models for normal/binomial/poisson family

Description

The function does k-fold cross validation for selecting best value of regularization parameter.

Usage

cv.nnlasso(x,y,family=c("binomial","normal","poisson"),
k=5,nlambda=50,tau=1,plot=TRUE, errorbars=TRUE)

Arguments

x	x is matrix of order n x p where n is number of observations and p is number of predictor variables. Rows should represent observations and columns should represent predictor variables.
y	y is a vector of response variable of order n x 1.
family	family is either "normal" or "binomial" or "poisson".
k	Number of folds for cross validation. Default is k=5.
nlambda	Number of lambda values to be used for cross validation. Default is nlambda=50.
tau	Elastic net parameter, $0 \le \tau \le 1$ in elastic net penalty $\lambda{\tau|\beta|_1+(1-\tau)|\beta|_2^2}$. Default tau=1 corresponds to LASSO penalty.
plot	if TRUE, produces a plot of cross validated prediction mean squared errors/ deviances against lambda. Default is TRUE.
errorbars	If TRUE, error bars are drawn in the plot. Default is TRUE.

Value

Produces a plot and returns a list with following components:

lambda	Value of lambda for which average cross validation error is minimum
pmse	A vector of average cross validation errors for various lambda values
lambdas	A vector of lambda values used in cross validation
se	A vector containing standard errors of cross validation errors

Note

This function uses prediction means squared errors for normal family and deviance for binomial and poisson family.

Author(s)

Baidya Nath Mandal and Jun Ma

References

Mandal, B.N. and Ma, J. (2016). L1 regularized multiplicative iterative path algorithm for non-negative generalized linear models.

Examples

#normal family
data(car)
attach(car)
x=as.matrix(car[,1:10])
## Not run: cv.nnlasso(x,y,family="normal",k=5)
#binomial family with same data
## Not run: cv.nnlasso(x,y1,family="binomial",k=5)
detach(car)

#poisson family
x=matrix(rnorm(100*30),100,30)
y=sample(c(1:5),100,replace=TRUE)
## Not run: cv.nnlasso(x,y,family="poisson",k=5)

---

Entire regularization path of non-negative penalized generalized linear model for normal/binomial/poisson family using multiplicative iterative algorithm

Description

The function computes coefficients of a penalized generalized linear model subject to non-negativity constraints for normal/binomial/poisson family using multiplicative iterative algorithm for a sequence of lambda values. Currently lasso and elastic net penalty are supported.

Usage

nnlasso(x,y,family=c("normal","binomial","poisson"),lambda=NULL,
intercept=TRUE,normalize=TRUE,tau=1,tol=1e-6,maxiter=1e5,nstep=100,min.lambda=1e-4,
eps=1e-6,path=TRUE,SE=FALSE)

Arguments

x	x is matrix of order n x p where n is number of observations and p is number of predictor variables. Rows should represent observations and columns should represent predictor variables.
y	y is a vector of response variable of order n x 1. y should follow either normal/binomial/poisson distribution.
family	family should be one of these: "normal","binomial","poisson"
lambda	The value of lambda for which coefficients are desired. The value of path must be FALSE in this case.
intercept	If TRUE, model includes intercept, else the model does not have intercept.
normalize	If TRUE, columns of x matrix are normalized with mean 0 and norm 1 prior to fitting the model. The coefficients at end are returned on the original scale. Default is normalize = TRUE.
tau	Elastic net parameter, $0 \le \tau \le 1$ in elastic net penalty $\lambda{\tau||\beta||_1+(1-\tau)||\beta||_2^2}$. Default tau = 1 corresponds to LASSO penalty.
tol	Tolerance criteria for convergence of solutions. Default is tol = 1e-6.
maxiter	Maximum number of iterations permissible for solving optimization problem for a particular lambda. Default is 10000. Rarely you need to change this to higher value.
nstep	Number of steps from maximum value of lambda to minimum value of lambda. Default is nstep = 100.
min.lambda	Minimum value of lambda. Default is min.lambda=1e-4.
eps	A small value below which a coefficient would be considered as zero.
path	Logical. If path=TRUE, entire regularization path will be obtained for a sequence of lambda values which are calculated automatically. To get coefficient estimates for a single lambda value, set path=FALSE with lambda=value. Default is path=TRUE.
SE	logical. If SE=TRUE, standard errors are produced for estimated coefficient at a given lambda. Standard errors are not produced if path=TRUE. Default is SE=FALSE.

Value

An object of class ‘nnlasso’ with following components:

beta0	A vector of order nstep of intercept estimates. Each value denote an estimate for a particular lambda. Corresponding lambda values are available in ‘lambdas’ element of the ‘nnlasso’ object.
coef	A matrix of order nstep x p of slope estimates. Each row denotes solution for a particular lambda. Corresponding lambda values are available in ‘lambdas’ element of the ‘nnlasso’ object. Here p is number of predictor variables.
lambdas	Sequence of lambda values for which coefficients are obtained
L1norm	L1norm of the coefficients
norm.frac	Fractions of norm computed as L1 norm at current lambda divided by maximum L1 norm
lambda.iter	Number of iterations used for different lambdas
of.value	Objective function values
normx	Norm of x variables
se	The standard errors of coefficient estimates

Author(s)

Baidya Nath Mandal and Jun Ma

References

Mandal, B.N. and Ma, J. (2016). L1 regularized multiplicative iterative path algorithm for non-negative generalized linear models.

Examples

#Non-negative LASSO
data(car)
attach(car)
x=as.matrix(car[,1:10])
g1=nnlasso(x,y,family="normal")
plot(g1)
plot(g1,xvar="lambda")


#Non-negative Elastic net with same data
## Not run: 
g2=nnlasso(x,y,family="normal",tau=0.6)
plot(g2)
plot(g2,xvar="lambda")

## End(Not run)

#Non-negative Ridge regression with same data
## Not run: 
g3=nnlasso(x,y,family="normal",tau=0)
plot(g3)
plot(g3,xvar="lambda")

## End(Not run)

#Non-negative L1 penalized GLM for binomial family
## Not run: 
g1=nnlasso(x,y1,family="binomial")
plot(g1)
plot(g1,xvar="lambda")

## End(Not run)

#Non-negative Elastic net with GLM with binomial family
## Not run: 
g2=nnlasso(x,y1,family="binomial",tau=0.8)
plot(g2)
plot(g2,xvar="lambda")

## End(Not run)

#coefficient estimates for a particular lambda for normal family
g1=nnlasso(x,y,lambda=0.01,family="normal",path=FALSE,SE=TRUE)
coef(g1)
round(g1$se,3)


#coefficient estimates for a particular lambda for binomial family
## Not run: 
g2=nnlasso(x,y1,lambda=0.01,family="binomial",path=FALSE,SE=TRUE)
coef(g2)
round(g2$se,3)
detach(car)

## End(Not run)

---

Plot of regularization path

Description

Produces a plot of entire regularization path from a 'nnlasso' object obtained using ‘nnlasso’ function.

Usage

## S3 method for class 'nnlasso'
plot(x,xvar=c("lambda","L1norm","fraction of norm"),...)

Arguments

x	A ‘nnlasso’ object obtained using ‘nnlasso’ function.
xvar	What should be on x-axis? xvar="lambda" produces a plot of regularization path with respect to lambda, xvar="L1norm" produces a plot of regularization path with respect to L1 norm of coefficients and xvar="fraction of norm" produces a plot of regularization path with respect to fraction of norm of coefficients. Default is xvar="L1norm".
...	Optional graphical parameters to matplot() function

Value

A plot of regularization path is produced.

Author(s)

Baidya Nath Mandal and Jun Ma

References

Mandal, B.N. and Ma, J. (2016). L1 regularized multiplicative iterative path algorithm for non-negative generalized linear models.

Examples

data(car)
attach(car)
x=as.matrix(car[,1:10])
g1=nnlasso(x,y,family="normal")
plot(g1)
plot(g1,xvar="lambda")
g1=nnlasso(x,y1,family="binomial")
plot(g1)
plot(g1,xvar="lambda")
detach(car)

---

Prediction of coefficients of a penalized linear regression or generalized linear models

Description

The function computes estimated coefficients value at a given lambda or L1 norm or fraction of norm using a ‘nnlasso’ object obtained using ‘nnlasso’ function.

Usage

## S3 method for class 'nnlasso'
predict(object,mode=c("fraction","norm","lambda"),at=0,...)

Arguments

object	A ‘nnlasso’ object obtained using ‘nnlasso’ function.
mode	If mode="lambda", prediction is made for a given lambda, if mode="norm", prediction is made for a given L1 norm and if mode="fraction", prediction is made for a fraction of norm value. Default is mode="lambda"
at	A value at which prediction is to be made. Default is at = 0.
...	Not used. Other arguments to predict.

Value

A vector of estimated coefficients of length p or p+1 at the given value of lambda or L1 norm or fraction of norm, depending on intercept=TRUE or FALSE in ‘nnlasso’ object. Here p is number of predictor variables.

Author(s)

Baidya Nath Mandal and Jun Ma

References

Mandal, B.N. and Ma, J. (2016). L1 regularized multiplicative iterative path algorithm for non-negative generalized linear models.

Examples

data(car)
attach(car)
x=as.matrix(car[,1:10])
g1=nnlasso(x,y1,family="binomial")
predict(g1,mode="lambda",at=0.1)
predict(g1,mode="L1norm",at=1)
predict(g1,mode="fraction",at=0.5)
g1=nnlasso(x,y,family="normal")
predict(g1,mode="lambda",at=0.09)
predict(g1,mode="L1norm",at=0.6)
predict(g1,mode="fraction",at=0.8)
detach(car)

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