Why are LASSO in sklearn (python) and matlab statistical package different?

Question

I am using LaasoCV from sklearn to select the best model is selected by cross-validation. I found that the cross validation gives different result if I use sklearn or matlab statistical toolbox.

I used matlab and replicate the example given in http://www.mathworks.se/help/stats/lasso-and-elastic-net.html to get a figure like this

Then I saved the matlab data, and tried to replicate the figure with laaso_path from sklearn, I got

Although there are some similarity between these two figures, there are also certain differences. As far as I understand parameter lambda in matlab and alpha in sklearn are same, however in this figure it seems that there are some differences. Can somebody point out which is the correct one or am I missing something? Further the coefficient obtained are also different (which is my main concern).

Matlab Code:

rng(3,'twister') % for reproducibility
X = zeros(200,5);
for ii = 1:5
      X(:,ii) = exprnd(ii,200,1);
end
r = [0;2;0;-3;0];
Y = X*r + randn(200,1)*.1;

save randomData.mat % To be used in python code

[b fitinfo] = lasso(X,Y,'cv',10);
lassoPlot(b,fitinfo,'plottype','lambda','xscale','log');

disp('Lambda with min MSE')
fitinfo.LambdaMinMSE
disp('Lambda with 1SE')
fitinfo.Lambda1SE
disp('Quality of Fit')
lambdaindex = fitinfo.Index1SE;
fitinfo.MSE(lambdaindex)
disp('Number of non zero predictos')
fitinfo.DF(lambdaindex)
disp('Coefficient of fit at that lambda')
b(:,lambdaindex)

Python Code:

import scipy.io
import numpy as np
import pylab as pl
from sklearn.linear_model import lasso_path, LassoCV

data=scipy.io.loadmat('randomData.mat')
X=data['X']
Y=data['Y'].flatten()

model = LassoCV(cv=10,max_iter=1000).fit(X, Y)
print 'alpha', model.alpha_
print 'coef', model.coef_

eps = 1e-2 # the smaller it is the longer is the path
models = lasso_path(X, Y, eps=eps)
alphas_lasso = np.array([model.alpha for model in models])
coefs_lasso = np.array([model.coef_ for model in models])

pl.figure(1)
ax = pl.gca()
ax.set_color_cycle(2 * ['b', 'r', 'g', 'c', 'k'])
l1 = pl.semilogx(alphas_lasso,coefs_lasso)
pl.gca().invert_xaxis()
pl.xlabel('alpha')
pl.show()

Answer 1

I do not have matlab but be careful that the value obtained with the cross--validation can be unstable. This is because it influenced by the way you subdivide the samples.

Even if you run 2 times the cross-validation in python you can obtain 2 different results. consider this example :

kf=sklearn.cross_validation.KFold(len(y),n_folds=10,shuffle=True)
cv=sklearn.linear_model.LassoCV(cv=kf,normalize=True).fit(x,y)
print cv.alpha_
kf=sklearn.cross_validation.KFold(len(y),n_folds=10,shuffle=True)
cv=sklearn.linear_model.LassoCV(cv=kf,normalize=True).fit(x,y)
print cv.alpha_

0.00645093258722
0.00691712356467