smile.regression

Class LASSO

java.lang.Object

smile.regression.LASSO

All Implemented Interfaces:

java.io.Serializable, Regression<double[]>

public class LASSO
extends java.lang.Object
implements Regression<double[]>, java.io.Serializable

Lasso (least absolute shrinkage and selection operator) regression. The Lasso is a shrinkage and selection method for linear regression. It minimizes the usual sum of squared errors, with a bound on the sum of the absolute values of the coefficients (i.e. L₁-regularized). It has connections to soft-thresholding of wavelet coefficients, forward stage-wise regression, and boosting methods.

The Lasso typically yields a sparse solution, of which the parameter vector β has relatively few nonzero coefficients. In contrast, the solution of L₂-regularized least squares (i.e. ridge regression) typically has all coefficients nonzero. Because it effectively reduces the number of variables, the Lasso is useful in some contexts.

For over-determined systems (more instances than variables, commonly in machine learning), we normalize variables with mean 0 and standard deviation 1. For under-determined systems (less instances than variables, e.g. compressed sensing), we assume white noise (i.e. no intercept in the linear model) and do not perform normalization. Note that the solution is not unique in this case.

There is no analytic formula or expression for the optimal solution to the L₁-regularized least squares problems. Therefore, its solution must be computed numerically. The objective function in the L₁-regularized least squares is convex but not differentiable, so solving it is more of a computational challenge than solving the L₂-regularized least squares. The Lasso may be solved using quadratic programming or more general convex optimization methods, as well as by specific algorithms such as the least angle regression algorithm.

References

1. R. Tibshirani. Regression shrinkage and selection via the lasso. J. Royal. Statist. Soc B., 58(1):267-288, 1996.
2. B. Efron, I. Johnstone, T. Hastie, and R. Tibshirani. Least angle regression. Annals of Statistics, 2003
3. Seung-Jean Kim, K. Koh, M. Lustig, Stephen Boyd, and Dimitry Gorinevsky. An Interior-Point Method for Large-Scale L1-Regularized Least Squares. IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 1, NO. 4, 2007.

See Also:

Serialized Form

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	LASSO.Trainer Trainer for LASSO regression.

Constructor Summary

Constructors

Constructor and Description
LASSO(double[][] x, double[] y, double lambda) Constructor.
LASSO(double[][] x, double[] y, double lambda, double tol, int maxIter) Constructor.
LASSO(Matrix x, double[] y, double lambda) Constructor.
LASSO(Matrix x, double[] y, double lambda, double tol, int maxIter) Constructor.

Method Summary

Modifier and Type	Method and Description
double	adjustedRSquared() Returns adjusted R2 statistic.
double[]	coefficients() Returns the linear coefficients.
int	df() Returns the degree-of-freedom of residual standard error.
double	error() Returns the residual standard error.
double	ftest() Returns the F-statistic of goodness-of-fit.
double	intercept() Returns the intercept.
double	predict(double[] x) Predicts the dependent variable of an instance.
double	pvalue() Returns the p-value of goodness-of-fit test.
double[]	residuals() Returns the residuals, that is response minus fitted values.
double	RSquared() Returns R2 statistic.
double	RSS() Returns the residual sum of squares.
double	shrinkage() Returns the shrinkage parameter.
java.lang.String	toString()

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Methods inherited from interface smile.regression.Regression

predict

Constructor Detail

LASSO

public LASSO(double[][] x,
             double[] y,
             double lambda)

Constructor. Learn the L1-regularized least squares model.

Parameters:

x - a matrix containing the explanatory variables. NO NEED to include a constant column of 1s for bias.

y - the response values.

lambda - the shrinkage/regularization parameter.

LASSO

public LASSO(double[][] x,
             double[] y,
             double lambda,
             double tol,
             int maxIter)

Constructor. Learn the L1-regularized least squares model.

Parameters:

x - a matrix containing the explanatory variables. NO NEED to include a constant column of 1s for bias.

y - the response values.

lambda - the shrinkage/regularization parameter.

tol - the tolerance for stopping iterations (relative target duality gap).

maxIter - the maximum number of IPM (Newton) iterations.

LASSO

public LASSO(Matrix x,
             double[] y,
             double lambda)

Constructor. Learn the L1-regularized least squares model.

Parameters:

x - a matrix containing the explanatory variables. The variables should be centered and standardized. NO NEED to include a constant column of 1s for bias.

y - the response values.

lambda - the shrinkage/regularization parameter.

LASSO

public LASSO(Matrix x,
             double[] y,
             double lambda,
             double tol,
             int maxIter)

Constructor. Learn the L1-regularized least squares model.

Parameters:

x - a matrix containing the explanatory variables. The variables should be centered and standardized. NO NEED to include a constant column of 1s for bias.

y - the response values.

lambda - the shrinkage/regularization parameter.

tol - the tolerance for stopping iterations (relative target duality gap).

maxIter - the maximum number of IPM (Newton) iterations.

Method Detail

coefficients

public double[] coefficients()

Returns the linear coefficients.

intercept

public double intercept()

Returns the intercept.

shrinkage

public double shrinkage()

Returns the shrinkage parameter.

predict

public double predict(double[] x)

Description copied from interface: Regression

Predicts the dependent variable of an instance.

Specified by:

predict in interface Regression<double[]>

Parameters:

x - the instance.

Returns:

the predicted value of dependent variable.

residuals

public double[] residuals()

Returns the residuals, that is response minus fitted values.

RSS

public double RSS()

Returns the residual sum of squares.

error

public double error()

Returns the residual standard error.

df

public int df()

Returns the degree-of-freedom of residual standard error.

RSquared

public double RSquared()

Returns R² statistic. In regression, the R² coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R² of 1.0 indicates that the regression line perfectly fits the data.

In the case of ordinary least-squares regression, R² increases as we increase the number of variables in the model (R² will not decrease). This illustrates a drawback to one possible use of R², where one might try to include more variables in the model until "there is no more improvement". This leads to the alternative approach of looking at the adjusted R².

adjustedRSquared

public double adjustedRSquared()

Returns adjusted R² statistic. The adjusted R² has almost same explanation as R² but it penalizes the statistic as extra variables are included in the model.

ftest

public double ftest()

Returns the F-statistic of goodness-of-fit.

pvalue

public double pvalue()

Returns the p-value of goodness-of-fit test.

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object