smile.regression

Class LASSO

java.lang.Object

smile.regression.LASSO

All Implemented Interfaces:

Regression<double[]>

public class LASSO
extends Object
implements Regression<double[]>

Least absolute shrinkage and selection operator. 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.

Author:

Haifeng Li

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.

Method Summary

Methods

Modifier and Type	Method and Description
double[]	coefficients() Returns the linear coefficients.
double	intercept() Returns the intercept.
double	predict(double[] x) Predicts the dependent variable of an instance.
double	shrinkage() Returns the shrinkage parameter.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

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.

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.

y - the response values.

lambda - the shrinkage/regularization parameter.

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

maxIter - the maximum number of 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.