I could not find where the Manhattan distance of weights is calculated and multiplied with alpha (L1 reg. coefficient) in the Lasso Regression and the Quantile Regression source code of scikit-learn.
I was trying to implement Lasso Regression and Quantile Regression w/ NumPy and compare results w/ scikit-learn models.
I don't believe the loss function (including the regularization penalty) is ever explicitly calculated, no.
Instead, the loss function is optimized by coordinate descent, and so we only ever need to actually calculate derivatives of the loss function. That happens in the enet_coordinate_descent function (or relatives), and I think the relevant bit is here.
I'm using an sklearn implementation of support vector machine for a binary classification problem. I wanted to store loss function values for each iteration and plot it to visualize model performance. Is this possible?
The Linear regression model from sklearn uses a closed or normal equation to find the parameters. However with large datasets Gradient Descent is said to be more efficient. Is there any way to use the LinearRegression from sklearn using gradient descent.
The function you are looking for is: sklearn.linear_model.SGDRegressor
You can modify the loss hyperparameter which will define the loss function to be used.
Be aware that the SGD of SGDRegressor stands for Stochastic Gradient Descent. Which means that the gradient of the loss is estimated each sample at a time and the model is updated along the way with a decreasing strength schedule (aka learning rate).
I am confused by this example here: https://scikit-learn.org/stable/visualizations.html
If we plot the ROC curve for a Logistic Regression Classifier the ROC curve is parametrized by the threshold parameter. But a usual SVM spits out binary values instead of probabilities.
Consequently there should not be a threshold which can be varied to obtain an ROC curve.
But which parameter is then varied in the example above?
SVMs have a measure of confidence in their predictions using the distance from the separating hyperplane (before the kernel, if you're not doing a linear SVM). These are obviously not probabilities, but they do rank-order the data points, and so you can get an ROC curve. In sklearn, this is done via the decision_function method. (You can also set probability=True in the SVC to calibrate the decision function values into probability estimates.)
See this section of the User Guide for some of the details on the decision function.
One option of the SVM classifier (SVC) is probability which is false by default. The documentation does not say what it does. Looking at libsvm source code, it seems to do some sort of cross-validation.
This option does not exist for LinearSVC nor OneSVM.
I need to calculate AUC scores for several SVM models, including these last two. Should I calculate the AUC score using decision_function(X) as the thresholds?
Answering my own question.
Firstly, it is a common "myth" that you need probabilities to draw the ROC curve. No, you need some kind of threshold in your model that you can change. The ROC curve is then drawn by changing this threshold. The point of the ROC curve being, of course, to see how well your model is reproducing the hypothesis by seeing how well it is ordering the observations.
In the case of SVM, there are two ways I see people drawing ROC curves for them:
using distance to the decision bondary, as I mentioned in my own question
using the bias term as your threshold in the SVM: http://researchgate.net/post/How_can_I_plot_determine_ROC_AUC_for_SVM. In fact, if you use SVC(probabilities=True) then probabilities will be calculated for you in this manner, by using CV, which you can then use to draw the ROC curve. But as mentioned in the link I provide, it is much faster if you draw the ROC curve directly by varying the bias.
I think #2 is the same as #1 if we are using a linear kernel, as in my own case, because varying the bias is varying the distance in this particular case.
In order to calculate AUC, using sklearn, you need a predict_proba method on your classifier; this is what the probability parameter on SVC does (you are correct that it's calculated using cross-validation). From the docs:
probability : boolean, optional (default=False)
Whether to enable probability estimates. This must be enabled prior to calling fit, and will slow down that method.
You can't use the decision function directly to compute AUC, since it's not a probability. I suppose you could scale the decision function to take values in the range [0,1], and compute AUC, however I'm not sure what statistical properties this will have; you certainly won't be able to use it to compare with ROC calculated using probabilities.