http://www.stanford.edu/~boyd/papers/admm/lasso/lasso_example.html

http://www-stat.stanford.edu/~susan/courses/b494/index/node35.html

http://www-stat.stanford.edu/~susan/courses/s227/node5.html

http://www-stat.stanford.edu/~susan/courses/b494/index/node29.html

Syntax

Description

B = lasso(X,Y) returns fitted least-squares regression coefficients for a set of regularization coefficients Lambda.

[B,FitInfo] = lasso(X,Y) returns a structure containing information about the fits.

[B,FitInfo] = lasso(X,Y,Name,Value) fits regularized regressions with additional options specified by one or more Name,Value pair arguments.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

% File: Lasso.m

% Author: Jinzhu Jia

%

% One simple (Gradiant Descent) implementation of the Lasso

% The objective function is:

% ۱/۲*sum((Y – X * beta -beta0).^2) + lambda * sum(abs(beta))

% Comparison with CVX code is done

% Reference: To be uploaded

%

% Version 4.23.2010

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [beta0,beta] = Lasso(X,Y,lambda)

n = size(X,1);

p = size(X,2);

if(size(Y,1) ~= n)

fprintf( ‘Error: dim of X and dim of Y are not match!!\n’)

quit cancel

end

beta0 = 0;

beta= zeros(p,1);

crit = 1;

step = 0;

while(crit > 1E-5)

step = step + 1;

obj_ini = 1/2*sum((Y – X * beta -beta0).^2) + lambda * sum(abs(beta));

beta0 = mean(Y – X*beta);

for(j = 1:p)

a = sum(X(:,j).^2);

b = sum((Y – X*beta).* X(:,j)) + beta(j) * a;

if lambda >= abs(b)

beta(j) = 0;

else

if b>0

beta(j) = (b – lambda) / a;

else

beta(j) = (b+lambda) /a;

end

end

end

obj_now = 1/2*sum((Y – X * beta -beta0).^2) + lambda * sum(abs(beta));

crit = abs(obj_now – obj_ini) / abs(obj_ini);

end

————————————————————————-

%% Lasso regularization example

% Copyright (c) 2011, The MathWorks, Inc.

%% Introduction to using LASSO

using the lasso functionality introduced
% in R2011b. It is motivated by

% This demo explains how to star

tan example in Tibshiranis original paper

% on the lasso.

via the lasso.
% J. Royal. Statist. Soc B., Vol. 58, No. 1, pages 267-288)

% Tibshirani, R. (1996). Regression shrinkage and selection
 .

% The data set that were working with in this demo is a wide

ferent
% variables and only 20 observations. 5 out of the 8 variables h

% dataset with correlated variables. This data set includes 8 di
fave
% coefficients of zero. These variables have zero impact on the model. The

up workspace and set random seed

clear all
clc

% Set the random num

% other three variables have non negative values and impact the model

%% Clean
 ber stream
rng(1981);

%% Creating data set with specific characteristics

% Create eight X variables

with one another
% The covariance matrix is spec

% The mean of each variable will be equal to zero
mu = [0 0 0 0 0 0 0 0];

% The variable are correlate
dified as
i = 1:8;
matrix = abs(bsxfun(@minus,i',i));
covariance = repmat(.5,8,8).^matrix;

, covariance, 20);

% Create a hyperplane that describes Y = f(X)
Beta = [3; 1

% Use these parameters to generate a set of multivariate normal random numbers
X = mvnrnd(m
u.5; 0; 0; 2; 0; 0; 0];
ds = dataset(Beta);

% Add in a noise vector
Y = X * Beta + 3 * randn(20,1);

%% Use linear regression to fit the model 

b = regress(Y,X);

s, ‘PlotType’,

ds.Linear = b;

%% Use a lasso to fit the model

[B Stats] = lasso(X,Y, 'CV', 5);

disp(B)
disp(Stats)

%% Create a plot showing MSE versus lamba

lassoPlot(B, Sta
t 'CV')

%% Identify a reasonable set of lasso coefficients

% View the regression coefficients associated with Index1SE

ds.Lasso = B(:,Stats.Index1SE);
disp(ds)

(۱۰۰,۱);
Coeff_Num = zeros(100,1);
Betas = zeros(8,100);
cv

%%  Create a plot showing coefficient values versus L1 norm

lassoPlot(B, Stats)

%%  Run a Simulation

% Preallocate some variables
MSE = zeros(100,1);
mse = zero
s_Reg_MSE = zeros(1,100);

for i = 1 : 100

    X = mvnrnd(mu, covariance, 20);
    Y = X * Beta + randn(20,1);

    [B Stats] = lasso(X,Y, 'CV', 5);

E(i) = Stats.MSE(:, Shrink);

regf = @(XTRAIN, ytrain, XTEST)(XTEST*

    Shrink = Stats.Index1SE -  ceil((Stats.Index1SE - Stats.IndexMinMSE)/2);
    Betas(:,i) = B(:,Shrink) > 0;
    Coeff_Num(i) = sum(B(:,Shrink) > 0);
    M
Sregress(ytrain,XTRAIN));
    cv_Reg_MSE(i) = crossval('mse',X,Y,'predfun',regf, 'kfold', 5);

end

Number_Lasso_Coefficients = mean(Coeff_Num);
disp(Number_Lasso_Coefficients)

MSE_Ratio = median(cv_Reg_MSE)/median(MSE);

disp(MSE_Ratio)