Model selection: choosing estimators and their parameters¶
Score, and cross-validated scores¶
As we have seen, every estimator exposes a score method that can judge the quality of the fit (or the prediction) on new data. Bigger is better.
>>> from sklearn import datasets, svm
>>> digits = datasets.load_digits()
>>> X_digits = digits.data
>>> y_digits = digits.target
>>> svc = svm.SVC(C=1, kernel='linear')
>>> svc.fit(X_digits[:-100], y_digits[:-100]).score(X_digits[-100:], y_digits[-100:])
0.97999999999999998
To get a better measure of prediction accuracy (which we can use as a proxy for goodness of fit of the model), we can successively split the data in folds that we use for training and testing:
>>> import numpy as np
>>> X_folds = np.array_split(X_digits, 3)
>>> y_folds = np.array_split(y_digits, 3)
>>> scores = list()
>>> for k in range(3):
... # We use 'list' to copy, in order to 'pop' later on
... X_train = list(X_folds)
... X_test = X_train.pop(k)
... X_train = np.concatenate(X_train)
... y_train = list(y_folds)
... y_test = y_train.pop(k)
... y_train = np.concatenate(y_train)
... scores.append(svc.fit(X_train, y_train).score(X_test, y_test))
>>> print(scores)
[0.93489148580968284, 0.95659432387312182, 0.93989983305509184]
This is called a KFold cross validation
Cross-validation generators¶
The code above to split data in train and test sets is tedious to write. The sklearn exposes cross-validation generators to generate list of indices for this purpose:
>>> from sklearn import cross_validation
>>> k_fold = cross_validation.KFold(n=6, n_folds=3, indices=True)
>>> for train_indices, test_indices in k_fold:
... print('Train: %s | test: %s' % (train_indices, test_indices))
Train: [2 3 4 5] | test: [0 1]
Train: [0 1 4 5] | test: [2 3]
Train: [0 1 2 3] | test: [4 5]
The cross-validation can then be implemented easily:
>>> kfold = cross_validation.KFold(len(X_digits), n_folds=3)
>>> [svc.fit(X_digits[train], y_digits[train]).score(X_digits[test], y_digits[test])
... for train, test in kfold]
[0.93489148580968284, 0.95659432387312182, 0.93989983305509184]
To compute the score method of an estimator, the sklearn exposes a helper function:
>>> cross_validation.cross_val_score(svc, X_digits, y_digits, cv=kfold, n_jobs=-1)
array([ 0.93489149, 0.95659432, 0.93989983])
n_jobs=-1 means that the computation will be dispatched on all the CPUs of the computer.
Cross-validation generators
| KFold (n, k) | StratifiedKFold (y, k) | LeaveOneOut (n) | LeaveOneLabelOut (labels) |
| Split it K folds, train on K-1 and then test on left-out | It preserves the class ratios / label distribution within each fold. | Leave one observation out | Takes a label array to group observations |
Exercise
On the digits dataset, plot the cross-validation score of a SVC estimator with an linear kernel as a function of parameter C (use a logarithmic grid of points, from 1 to 10).
import numpy as np
from sklearn import cross_validation, datasets, svm
digits = datasets.load_digits()
X = digits.data
y = digits.target
svc = svm.SVC(kernel='linear')
C_s = np.logspace(-10, 0, 10)
Grid-search and cross-validated estimators¶
Grid-search¶
The sklearn provides an object that, given data, computes the score during the fit of an estimator on a parameter grid and chooses the parameters to maximize the cross-validation score. This object takes an estimator during the construction and exposes an estimator API:
>>> from sklearn.grid_search import GridSearchCV
>>> gammas = np.logspace(-6, -1, 10)
>>> clf = GridSearchCV(estimator=svc, param_grid=dict(gamma=gammas),
... n_jobs=-1)
>>> clf.fit(X_digits[:1000], y_digits[:1000])
GridSearchCV(cv=None,...
>>> clf.best_score_
0.924...
>>> clf.best_estimator_.gamma == 1e-6
True
>>> # Prediction performance on test set is not as good as on train set
>>> clf.score(X_digits[1000:], y_digits[1000:])
0.94228356336260977
By default, the GridSearchCV uses a 3-fold cross-validation. However, if it detects that a classifier is passed, rather than a regressor, it uses a stratified 3-fold.
Nested cross-validation
>>> cross_validation.cross_val_score(clf, X_digits, y_digits)
... # doctest: +ELLIPSIS array([ 0.935..., 0.958..., 0.937...])
Two cross-validation loops are performed in parallel: one by the GridSearchCV estimator to set gamma and the other one by cross_val_score to measure the prediction performance of the estimator. The resulting scores are unbiased estimates of the prediction score on new data.
Warning
You cannot nest objects with parallel computing (n_jobs different than 1).
Cross-validated estimators¶
Cross-validation to set a parameter can be done more efficiently on an algorithm-by-algorithm basis. This is why for certain estimators the sklearn exposes Cross-validation: evaluating estimator performance estimators that set their parameter automatically by cross-validation:
>>> from sklearn import linear_model, datasets
>>> lasso = linear_model.LassoCV()
>>> diabetes = datasets.load_diabetes()
>>> X_diabetes = diabetes.data
>>> y_diabetes = diabetes.target
>>> lasso.fit(X_diabetes, y_diabetes)
LassoCV(alphas=None, copy_X=True, cv=None, eps=0.001, fit_intercept=True,
max_iter=1000, n_alphas=100, normalize=False, precompute='auto',
tol=0.0001, verbose=False)
>>> # The estimator chose automatically its lambda:
>>> lasso.alpha_
0.01229...
These estimators are called similarly to their counterparts, with ‘CV’ appended to their name.
Exercise
On the diabetes dataset, find the optimal regularization parameter alpha.
Bonus: How much can you trust the selection of alpha?
from __future__ import print_function
import numpy as np
import pylab as pl
from sklearn import cross_validation, datasets, linear_model
print(__doc__)
diabetes = datasets.load_diabetes()
X = diabetes.data[:150]
y = diabetes.target[:150]