In this section, we'll look at model evaluation and the tuning of hyperparameters, which are parameters that define the model.
In [1]:
from __future__ import print_function, division
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
# Use seaborn for plotting defaults
import seaborn as sns; sns.set()
One of the most important pieces of machine learning is model validation: that is, checking how well your model fits a given dataset. But there are some pitfalls you need to watch out for.
Consider the digits example we've been looking at previously. How might we check how well our model fits the data?
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from sklearn.datasets import load_digits
digits = load_digits()
X = digits.data
y = digits.target
Let's fit a K-neighbors classifier
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from sklearn.neighbors import KNeighborsClassifier
knn = KNeighborsClassifier(n_neighbors=1)
knn.fit(X, y)
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Now we'll use this classifier to predict labels for the data
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y_pred = knn.predict(X)
Finally, we can check how well our prediction did:
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print("{0} / {1} correct".format(np.sum(y == y_pred), len(y)))
It seems we have a perfect classifier!
Question: what's wrong with this?
Above we made the mistake of testing our data on the same set of data that was used for training. This is not generally a good idea. If we optimize our estimator this way, we will tend to over-fit the data: that is, we learn the noise.
A better way to test a model is to use a hold-out set which doesn't enter the training. We've seen this before using scikit-learn's train/test split utility:
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from sklearn.cross_validation import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y)
X_train.shape, X_test.shape
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Now we train on the training data, and validate on the test data:
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knn = KNeighborsClassifier(n_neighbors=1)
knn.fit(X_train, y_train)
y_pred = knn.predict(X_test)
print("{0} / {1} correct".format(np.sum(y_test == y_pred), len(y_test)))
This gives us a more reliable estimate of how our model is doing.
The metric we're using here, comparing the number of matches to the total number of samples, is known as the accuracy score, and can be computed using the following routine:
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from sklearn.metrics import accuracy_score
accuracy_score(y_test, y_pred)
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This can also be computed directly from the model.score method:
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knn.score(X_test, y_test)
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Using this, we can ask how this changes as we change the model parameters, in this case the number of neighbors:
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for n_neighbors in [1, 5, 10, 20, 30]:
knn = KNeighborsClassifier(n_neighbors)
knn.fit(X_train, y_train)
print(n_neighbors, knn.score(X_test, y_test))
We see that in this case, a small number of neighbors seems to be the best option.
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X1, X2, y1, y2 = train_test_split(X, y, test_size=0.5, random_state=0)
X1.shape, X2.shape
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In [12]:
print(KNeighborsClassifier(1).fit(X2, y2).score(X1, y1))
print(KNeighborsClassifier(1).fit(X1, y1).score(X2, y2))
Thus a two-fold cross-validation gives us two estimates of the score for that parameter.
Because this is a bit of a pain to do by hand, scikit-learn has a utility routine to help:
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from sklearn.cross_validation import cross_val_score
cv = cross_val_score(KNeighborsClassifier(1), X, y, cv=10)
cv.mean()
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Here we've used 2-fold cross-validation. This is just one specialization of $K$-fold cross-validation, where we split the data into $K$ chunks and perform $K$ fits, where each chunk gets a turn as the validation set.
We can do this by changing the cv parameter above. Let's do 10-fold cross-validation:
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cross_val_score(KNeighborsClassifier(1), X, y, cv=10)
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This gives us an even better idea of how well our model is doing.
Now that we've gone over the basics of validation, and cross-validation, it's time to go into even more depth regarding model selection.
The issues associated with validation and cross-validation are some of the most important aspects of the practice of machine learning. Selecting the optimal model for your data is vital, and is a piece of the problem that is not often appreciated by machine learning practitioners.
Of core importance is the following question:
If our estimator is underperforming, how should we move forward?
The answer is often counter-intuitive. In particular, Sometimes using a more complicated model will give worse results. Also, Sometimes adding training data will not improve your results. The ability to determine what steps will improve your model is what separates the successful machine learning practitioners from the unsuccessful.
For this section, we'll work with a simple 1D regression problem. This will help us to
easily visualize the data and the model, and the results generalize easily to higher-dimensional
datasets. We'll explore a simple linear regression problem.
This can be accomplished within scikit-learn with the sklearn.linear_model module.
We'll create a simple nonlinear function that we'd like to fit
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def test_func(x, err=0.5):
y = 10 - 1. / (x + 0.1)
if err > 0:
y = np.random.normal(y, err)
return y
Now let's create a realization of this dataset:
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def make_data(N=40, error=1.0, random_seed=1):
# randomly sample the data
np.random.seed(1)
X = np.random.random(N)[:, np.newaxis]
y = test_func(X.ravel(), error)
return X, y
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X, y = make_data(40, error=1)
plt.scatter(X.ravel(), y);
Now say we want to perform a regression on this data. Let's use the built-in linear regression function to compute a fit:
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X_test = np.linspace(-0.1, 1.1, 500)[:, None]
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error
model = LinearRegression()
model.fit(X, y)
y_test = model.predict(X_test)
plt.scatter(X.ravel(), y)
plt.plot(X_test.ravel(), y_test)
plt.title("mean squared error: {0:.3g}".format(mean_squared_error(model.predict(X), y)));
We have fit a straight line to the data, but clearly this model is not a good choice. We say that this model is biased, or that it under-fits the data.
Let's try to improve this by creating a more complicated model. We can do this by adding degrees of freedom, and computing a polynomial regression over the inputs. Scikit-learn makes this easy with the PolynomialFeatures preprocessor, which can be pipelined with a linear regression.
Let's make a convenience routine to do this:
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from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.pipeline import make_pipeline
def PolynomialRegression(degree=2, **kwargs):
return make_pipeline(PolynomialFeatures(degree),
LinearRegression(**kwargs))
Now we'll use this to fit a quadratic curve to the data.
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model = PolynomialRegression(2)
model.fit(X, y)
y_test = model.predict(X_test)
plt.scatter(X.ravel(), y)
plt.plot(X_test.ravel(), y_test)
plt.title("mean squared error: {0:.3g}".format(mean_squared_error(model.predict(X), y)));
This reduces the mean squared error, and makes a much better fit. What happens if we use an even higher-degree polynomial?
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model = PolynomialRegression(30)
model.fit(X, y)
y_test = model.predict(X_test)
plt.scatter(X.ravel(), y)
plt.plot(X_test.ravel(), y_test)
plt.title("mean squared error: {0:.3g}".format(mean_squared_error(model.predict(X), y)))
plt.ylim(-4, 14);
When we increase the degree to this extent, it's clear that the resulting fit is no longer reflecting the true underlying distribution, but is more sensitive to the noise in the training data. For this reason, we call it a high-variance model, and we say that it over-fits the data.
Just for fun, let's use IPython's interact capability (only in IPython 2.0+) to explore this interactively:
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from IPython.html.widgets import interact
def plot_fit(degree, Npts):
X, y = make_data(Npts, error=1)
X_test = np.linspace(-0.1, 1.1, 500)[:, None]
model = PolynomialRegression(degree=degree)
model.fit(X, y)
y_test = model.predict(X_test)
plt.scatter(X.ravel(), y)
plt.plot(X_test.ravel(), y_test)
plt.ylim(-4, 14)
plt.title("mean squared error: {0:.2f}".format(mean_squared_error(model.predict(X), y)))
interact(plot_fit, degree=[1, 30], Npts=[2, 100]);
Clearly, computing the error on the training data is not enough (we saw this previously). As above, we can use cross-validation to get a better handle on how the model fit is working.
Let's do this here, again using the validation_curve utility. To make things more clear, we'll use a slightly larger dataset:
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X, y = make_data(120, error=1.0)
plt.scatter(X, y);
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from sklearn.learning_curve import validation_curve
def rms_error(model, X, y):
y_pred = model.predict(X)
return np.sqrt(np.mean((y - y_pred) ** 2))
degree = np.arange(0, 18)
val_train, val_test = validation_curve(PolynomialRegression(), X, y,
'polynomialfeatures__degree', degree, cv=7,
scoring=rms_error)
Now let's plot the validation curves:
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def plot_with_err(x, data, color, **kwargs):
mu, std = data.mean(1), data.std(1)
plt.plot(x, mu, '-', c=color, **kwargs)
plt.fill_between(x, mu - std, mu + std, edgecolor='none', facecolor=color, alpha=0.2)
plot_with_err(degree, val_train, 'r', label='training scores')
plot_with_err(degree, val_test, 'b', label='validation scores')
plt.xlabel('degree'); plt.ylabel('rms error')
plt.legend();
Notice the trend here, which is common for this type of plot.
For a small model complexity, the training error and validation error are very similar. This indicates that the model is under-fitting the data: it doesn't have enough complexity to represent the data. Another way of putting it is that this is a high-bias model.
As the model complexity grows, the training and validation scores diverge. This indicates that the model is over-fitting the data: it has so much flexibility, that it fits the noise rather than the underlying trend. Another way of putting it is that this is a high-variance model.
Note that the training score (nearly) always improves with model complexity. This is because a more complicated model can fit the noise better, so the model improves. The validation data generally has a sweet spot, which here is around 5 terms.
Here's our best-fit model according to the cross-validation:
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model = PolynomialRegression(4).fit(X, y)
plt.scatter(X, y)
plt.plot(X_test, model.predict(X_test));
As you might guess, the exact turning-point of the tradeoff between bias and variance is highly dependent on the number of training points used. Here we'll illustrate the use of learning curves, which display this property.
The idea is to plot the mean-squared-error for the training and test set as a function of Number of Training Points
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from sklearn.learning_curve import learning_curve
def plot_learning_curve(degree=3):
train_sizes = np.linspace(0.05, 1, 20)
N_train, val_train, val_test = learning_curve(PolynomialRegression(degree),
X, y, train_sizes, cv=5,
scoring=rms_error)
plot_with_err(N_train, val_train, 'r', label='training scores')
plot_with_err(N_train, val_test, 'b', label='validation scores')
plt.xlabel('Training Set Size'); plt.ylabel('rms error')
plt.ylim(0, 3)
plt.xlim(5, 80)
plt.legend()
Let's see what the learning curves look like for a linear model:
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plot_learning_curve(1)
This shows a typical learning curve: for very few training points, there is a large separation between the training and test error, which indicates over-fitting. Given the same model, for a large number of training points, the training and testing errors converge, which indicates potential under-fitting.
As you add more data points, the training error will never increase, and the testing error will never decrease (why do you think this is?)
It is easy to see that, in this plot, if you'd like to reduce the MSE down to the nominal value of 1.0 (which is the magnitude of the scatter we put in when constructing the data), then adding more samples will never get you there. For $d=1$, the two curves have converged and cannot move lower. What about for a larger value of $d$?
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plot_learning_curve(3)
Here we see that by adding more model complexity, we've managed to lower the level of convergence to an rms error of 1.0!
What if we get even more complex?
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plot_learning_curve(10)
For an even more complex model, we still converge, but the convergence only happens for large amounts of training data.
So we see the following:
Thus these curves can give you hints about how you might improve a sub-optimal model. If the curves are already close together, you need more model complexity. If the curves are far apart, you might also improve the model by adding more data.
To make this more concrete, imagine some telescope data in which the results are not robust enough. You must think about whether to spend your valuable telescope time observing more objects to get a larger training set, or more attributes of each object in order to improve the model. The answer to this question has real consequences, and can be addressed using these metrics.
We've gone over several useful tools for model validation
These tools are powerful means of evaluating your model on your data.