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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Parameter selection, Validation, and Testing
Most models have parameters that influence how complex a model they can learn. Remember using KNeighborsRegressor.
If we change the number of neighbors we consider, we get a smoother and smoother prediction
Step1:  There is a function in scikit-learn, called validation_plot to reproduce the cartoon figure above. It plots one parameter, such as the number of neighbors, against training and validation error (using cross-validation)
Step2:  Note that many neighbors mean a "smooth" or "simple" model, so the plot is the mirror image of the diagram above.
If multiple parameters are important, like the parameters C and gamma in an SVM (more about that later), all possible combinations are tried
Step3:  As this is such a very common pattern, there is a built-in class for this in scikit-learn, GridSearchCV. GridSearchCV takes a dictionary that describes the parameters that should be tried and a model to train.
The grid of parameters is defined as a dictionary, where the keys are the parameters and the values are the settings to be tested.
Step4:  One of the great things about GridSearchCV is that it is a meta-estimator. It takes an estimator like SVR above, and creates a new estimator, that behaves exactly the same - in this case, like a regressor.
So we can call fit on it, to train it
Step5:  What fit does is a bit more involved then what we did above. First, it runs the same loop with cross-validation, to find the best parameter combination.
Once it has the best combination, it runs fit again on all data passed to fit (without cross-validation), to built a single new model using the best parameter setting.
Then, as with all models, we can use predict or score
Step6:  You can inspect the best parameters found by GridSearchCV in the best_params_ attribute, and the best score in the best_score_ attribute
Step7:  There is a problem with using this score for evaluation, however. You might be making what is called a multiple hypothesis testing error. If you try very many parameter settings, some of them will work better just by chance, and the score that you obtained might not reflect how your model would perform on new unseen data.
Therefore, it is good to split off a separate test-set before performing grid-search. This pattern can be seen as a training-validation-test split, and is common in machine learning
Step8:  We can also look at the parameters that were selected
Step9:  Some practitioners go for an easier scheme, splitting the data simply into three parts, training, validation and testing. This is a possible alternative if your training set is very large, or it is infeasible to train many models using cross-validation because training a model takes very long.
You can do this with scikit-learn for example by splitting of a test-set and then applying GridSearchCV with ShuffleSplit cross-validation with a single iteration
Step10:  This is much faster, but might result in worse hyperparameters and therefore worse results.
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Python Code:
from sklearn.model_selection import cross_val_score, KFold
from sklearn.neighbors import KNeighborsRegressor
# generate toy dataset:
x = np.linspace(-3, 3, 100)
rng = np.random.RandomState(42)
y = np.sin(4 * x) + x + rng.normal(size=len(x))
X = x[:, np.newaxis]
cv = KFold(shuffle=True)
# for each parameter setting do cross_validation:
for n_neighbors in [1, 3, 5, 10, 20]:
    scores = cross_val_score(KNeighborsRegressor(n_neighbors=n_neighbors), X, y, cv=cv)
    print("n_neighbors: %d, average score: %f" % (n_neighbors, np.mean(scores)))
Explanation: Parameter selection, Validation, and Testing
Most models have parameters that influence how complex a model they can learn. Remember using KNeighborsRegressor.
If we change the number of neighbors we consider, we get a smoother and smoother prediction:
<img src="figures/plot_kneigbors_regularization.png" width="100%">
In the above figure, we see fits for three different values of n_neighbors.
For n_neighbors=2, the data is overfit, the model is too flexible and can adjust too much to the noise in the training data. For n_neighbors=20, the model is not flexible enough, and can not model the variation in the data appropriately.
In the middle, for n_neighbors = 5, we have found a good mid-point. It fits
the data fairly well, and does not suffer from the overfit or underfit
problems seen in the figures on either side. What we would like is a
way to quantitatively identify overfit and underfit, and optimize the
hyperparameters (in this case, the polynomial degree d) in order to
determine the best algorithm.
We trade off remembering too much about the particularities and noise of the training data vs. not modeling enough of the variability. This is a trade-off that needs to be made in basically every machine learning application and is a central concept, called bias-variance-tradeoff or "overfitting vs underfitting".
<img src="figures/overfitting_underfitting_cartoon.svg" width="100%">
Hyperparameters, Over-fitting, and Under-fitting
Unfortunately, there is no general rule how to find the sweet spot, and so machine learning practitioners have to find the best trade-off of model-complexity and generalization by trying several hyperparameter settings. Hyperparameters are the internal knobs or tuning parameters of a machine learning algorithm (in contrast to model parameters that the algorithm learns from the training data -- for example, the weight coefficients of a linear regression model); the number of k in K-nearest neighbors is such a hyperparameter.
Most commonly this "hyperparameter tuning" is done using a brute force search, for example over multiple values of n_neighbors:
End of explanation
from sklearn.model_selection import validation_curve
n_neighbors = [1, 3, 5, 10, 20, 50]
train_errors, test_errors = validation_curve(KNeighborsRegressor(), X, y, param_name="n_neighbors",
                                             param_range=n_neighbors, cv=cv)
plt.plot(n_neighbors, train_errors.mean(axis=1), label="train error")
plt.plot(n_neighbors, test_errors.mean(axis=1), label="test error")
plt.legend(loc="best")
Explanation: There is a function in scikit-learn, called validation_plot to reproduce the cartoon figure above. It plots one parameter, such as the number of neighbors, against training and validation error (using cross-validation):
End of explanation
from sklearn.model_selection import cross_val_score, KFold
from sklearn.svm import SVR
# each parameter setting do cross_validation:
for C in [0.001, 0.01, 0.1, 1, 10]:
    for gamma in [0.001, 0.01, 0.1, 1]:
        scores = cross_val_score(SVR(C=C, gamma=gamma), X, y, cv=cv)
        print("C: %f, gamma: %f, average score: %f" % (C, gamma, np.mean(scores)))
Explanation: Note that many neighbors mean a "smooth" or "simple" model, so the plot is the mirror image of the diagram above.
If multiple parameters are important, like the parameters C and gamma in an SVM (more about that later), all possible combinations are tried:
End of explanation
from sklearn.model_selection import GridSearchCV
param_grid = {'C': [0.001, 0.01, 0.1, 1, 10], 'gamma': [0.001, 0.01, 0.1, 1]}
grid = GridSearchCV(SVR(), param_grid=param_grid, cv=cv, verbose=3)
Explanation: As this is such a very common pattern, there is a built-in class for this in scikit-learn, GridSearchCV. GridSearchCV takes a dictionary that describes the parameters that should be tried and a model to train.
The grid of parameters is defined as a dictionary, where the keys are the parameters and the values are the settings to be tested.
End of explanation
grid.fit(X, y)
Explanation: One of the great things about GridSearchCV is that it is a meta-estimator. It takes an estimator like SVR above, and creates a new estimator, that behaves exactly the same - in this case, like a regressor.
So we can call fit on it, to train it:
End of explanation
grid.predict(X)
Explanation: What fit does is a bit more involved then what we did above. First, it runs the same loop with cross-validation, to find the best parameter combination.
Once it has the best combination, it runs fit again on all data passed to fit (without cross-validation), to built a single new model using the best parameter setting.
Then, as with all models, we can use predict or score:
End of explanation
print(grid.best_score_)
print(grid.best_params_)
Explanation: You can inspect the best parameters found by GridSearchCV in the best_params_ attribute, and the best score in the best_score_ attribute:
End of explanation
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
param_grid = {'C': [0.001, 0.01, 0.1, 1, 10], 'gamma': [0.001, 0.01, 0.1, 1]}
cv = KFold(n_splits=10, shuffle=True)
grid = GridSearchCV(SVR(), param_grid=param_grid, cv=cv)
grid.fit(X_train, y_train)
grid.score(X_test, y_test)
Explanation: There is a problem with using this score for evaluation, however. You might be making what is called a multiple hypothesis testing error. If you try very many parameter settings, some of them will work better just by chance, and the score that you obtained might not reflect how your model would perform on new unseen data.
Therefore, it is good to split off a separate test-set before performing grid-search. This pattern can be seen as a training-validation-test split, and is common in machine learning:
<img src="figures/grid_search_cross_validation.svg" width="100%">
We can do this very easily by splitting of some test data using train_test_split, training GridSearchCV on the training set, and applying the score method to the test set:
End of explanation
grid.best_params_
Explanation: We can also look at the parameters that were selected:
End of explanation
from sklearn.model_selection import train_test_split, ShuffleSplit
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)
param_grid = {'C': [0.001, 0.01, 0.1, 1, 10], 'gamma': [0.001, 0.01, 0.1, 1]}
single_split_cv = ShuffleSplit(n_splits=1)
grid = GridSearchCV(SVR(), param_grid=param_grid, cv=single_split_cv, verbose=3)
grid.fit(X_train, y_train)
grid.score(X_test, y_test)
Explanation: Some practitioners go for an easier scheme, splitting the data simply into three parts, training, validation and testing. This is a possible alternative if your training set is very large, or it is infeasible to train many models using cross-validation because training a model takes very long.
You can do this with scikit-learn for example by splitting of a test-set and then applying GridSearchCV with ShuffleSplit cross-validation with a single iteration:
<img src="figures/train_validation_test2.svg" width="100%">
End of explanation
clf = GridSearchCV(SVR(), param_grid=param_grid)
clf.fit(X_train, y_train)
clf.score(X_test, y_test)
Explanation: This is much faster, but might result in worse hyperparameters and therefore worse results.
End of explanation
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