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%pylab inline
import pylab as pl
import numpy as np
Here is where we start diving into the field of machine learning.
By the end of this section you will
In addition, we will go over several basic tools within scikit-learn which can be used to accomplish the above tasks.
Machine Learning (ML) is about building programs with tunable parameters (typically an array of floating point values) that are adjusted automatically so as to improve their behavior by adapting to previously seen data.
In most ML applications, the data is in a 2D array of shape [n_samples x n_features],
where the number of features is the same for each object, and each feature column refers
to a related piece of information about each sample.
Machine learning can be broken into two broad regimes: supervised learning and unsupervised learning. We’ll introduce these concepts here, and discuss them in more detail below.
Every algorithm is exposed in scikit-learn via an ''Estimator'' object. For instance a linear regression is:
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from sklearn.linear_model import LinearRegression
Estimator parameters: All the parameters of an estimator can be set when it is instantiated:
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model = LinearRegression(normalize=True)
print model.normalize
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print model
Estimated parameters: When data is fitted with an estimator, parameters are estimated from the data at hand. All the estimated parameters are attributes of the estimator object ending by an underscore:
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x = np.array([0, 1, 2])
y = np.array([0, 1, 2])
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_ = pl.plot(x, y, marker='o')
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X = x[:, np.newaxis] # The input data for sklearn is 2D: (samples == 3 x features == 1)
X
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model.fit(X, y)
model.coef_
In Supervised Learning, we have a dataset consisting of both features and labels. The task is to construct an estimator which is able to predict the label of an object given the set of features. A relatively simple example is predicting the species of iris given a set of measurements of its flower. This is a relatively simple task. Some more complicated examples are:
What these tasks have in common is that there is one or more unknown quantities associated with the object which needs to be determined from other observed quantities.
Supervised learning is further broken down into two categories, classification and regression. In classification, the label is discrete, while in regression, the label is continuous. For example, in astronomy, the task of determining whether an object is a star, a galaxy, or a quasar is a classification problem: the label is from three distinct categories. On the other hand, we might wish to estimate the age of an object based on such observations: this would be a regression problem, because the label (age) is a continuous quantity.
Classification: K nearest neighbors (kNN) is one of the simplest learning strategies: given a new, unknown observation, look up in your reference database which ones have the closest features and assign the predominant class.
Let's try it out on our iris classification problem:
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from sklearn import neighbors, datasets
iris = datasets.load_iris()
X, y = iris.data, iris.target
knn = neighbors.KNeighborsClassifier(n_neighbors=1)
knn.fit(X, y)
# What kind of iris has 3cm x 5cm sepal and 4cm x 2cm petal?
print iris.target_names[knn.predict([[3, 5, 4, 2]])]
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# A plot of the sepal space and the prediction of the KNN
from helpers import plot_iris_knn
plot_iris_knn()
Exercise: Now use as an estimator on the same problem: sklearn.svm.SVC.
Note that you don't have to know what it is do use it.
If you finish early, do the same plot as above.
Regression: The simplest possible regression setting is the linear regression one:
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# Create some simple data
import numpy as np
np.random.seed(0)
X = np.random.random(size=(20, 1))
y = 3 * X.squeeze() + 2 + np.random.normal(size=20)
# Fit a linear regression to it
from sklearn.linear_model import LinearRegression
model = LinearRegression(fit_intercept=True)
model.fit(X, y)
print "Model coefficient: %.5f, and intercept: %.5f" % (model.coef_, model.intercept_)
# Plot the data and the model prediction
X_test = np.linspace(0, 1, 100)[:, np.newaxis]
y_test = model.predict(X_test)
import pylab as pl
pl.plot(X.squeeze(), y, 'o')
pl.plot(X_test.squeeze(), y_test)
Unsupervised Learning addresses a different sort of problem. Here the data has no labels, and we are interested in finding similarities between the objects in question. In a sense, you can think of unsupervised learning as a means of discovering labels from the data itself. Unsupervised learning comprises tasks such as dimensionality reduction, clustering, and density estimation. For example, in the iris data discussed above, we can used unsupervised methods to determine combinations of the measurements which best display the structure of the data. As we’ll see below, such a projection of the data can be used to visualize the four-dimensional dataset in two dimensions. Some more involved unsupervised learning problems are:
Sometimes the two may even be combined: e.g. Unsupervised learning can be used to find useful features in heterogeneous data, and then these features can be used within a supervised framework.
An example PCA for visualization Principle Component Analysis (PCA) is a dimension reduction technique that can find the combinations of variables that explain the most variance.
Consider the iris dataset. It cannot be visualized in a single 2D plot, as it has 4 features. We are going to extract 2 combinations of sepal and petal dimensions to visualize it:
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X, y = iris.data, iris.target
from sklearn.decomposition import PCA
pca = PCA(n_components=2)
pca.fit(X)
X_reduced = pca.transform(X)
print "Reduced dataset shape:", X_reduced.shape
import pylab as pl
pl.scatter(X_reduced[:, 0], X_reduced[:, 1], c=y)
print "Meaning of the 2 components:"
for component in pca.components_:
print " + ".join("%.3f x %s" % (value, name)
for value, name in zip(component, iris.feature_names))
Clustering: Clustering groups together observations that are homogeneous with respect to a given criterion, finding ''clusters'' in the data.
Note that these clusters will uncover relevent hidden structure of the data only if the criterion used highlights it.
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from sklearn.cluster import KMeans
k_means = KMeans(n_clusters=3, random_state=0) # Fixing the RNG in kmeans
k_means.fit(X_reduced)
y_pred = k_means.predict(X_reduced)
pl.scatter(X_reduced[:, 0], X_reduced[:, 1], c=y_pred)
Scikit-learn strives to have a uniform interface across all methods,
and we’ll see examples of these below. Given a scikit-learn estimator
object named model, the following methods are available:
model.fit() : fit training data. For supervised learning applications,
this accepts two arguments: the data X and the labels y (e.g. model.fit(X, y)).
For unsupervised learning applications, this accepts only a single argument,
the data X (e.g. model.fit(X)).model.predict() : given a trained model, predict the label of a new set of data.
This method accepts one argument, the new data X_new (e.g. model.predict(X_new)),
and returns the learned label for each object in the array.model.predict_proba() : For classification problems, some estimators also provide
this method, which returns the probability that a new observation has each categorical label.
In this case, the label with the highest probability is returned by model.predict().model.score() : for classification or regression problems, most (all?) estimators implement
a score method. Scores are between 0 and 1, with a larger score indicating a better fit.model.transform() : given an unsupervised model, transform new data into the new basis.
This also accepts one argument X_new, and returns the new representation of the data based
on the unsupervised model.model.fit_transform() : some estimators implement this method,
which more efficiently performs a fit and a transform on the same input data.Train errors Suppose you are using a 1-nearest neighbor estimator. How many errors do you expect on your train set?
This tells us that:
An example of regularization The core idea behind regularization is that we are going to prefer models that are simpler, for a certain definition of ''simpler'', even if they lead to more errors on the train set.
As an example, let's generate with a 9th order polynomial.
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rng = np.random.RandomState(0)
x = 2 * rng.rand(100) - 1
f = lambda t: 1.2 * t ** 2 + .1 * t ** 3 - .4 * t ** 5 - .5 * t ** 9
y = f(x) + .4 * rng.normal(size=100)
pl.figure()
pl.scatter(x, y, s=4)
And now, let's fit a 4th order and a 9th order polynomial to the data. For this we need to engineer features: the n_th powers of x:
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x_test = np.linspace(-1, 1, 100)
pl.figure()
pl.scatter(x, y, s=4)
X = np.array([x**i for i in range(5)]).T
X_test = np.array([x_test**i for i in range(5)]).T
order4 = LinearRegression()
order4.fit(X, y)
pl.plot(x_test, order4.predict(X_test), label='4th order')
X = np.array([x**i for i in range(10)]).T
X_test = np.array([x_test**i for i in range(10)]).T
order9 = LinearRegression()
order9.fit(X, y)
pl.plot(x_test, order9.predict(X_test), label='9th order')
pl.legend(loc='best')
pl.axis('tight')
pl.title('Fitting a 4th and a 9th order polynomial')
With your naked eyes, which model do you prefer, the 4th order one, or the 9th order one?
Let's look at the ground truth:
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pl.figure()
pl.scatter(x, y, s=4)
pl.plot(x_test, f(x_test), label="truth")
pl.axis('tight')
pl.title('Ground truth (9th order polynomial)')
Regularization is ubiquitous in machine learning. Most scikit-learn estimators have a parameter to tune the amount of regularization. For instance, with k-NN, it is 'k', the number of nearest neighbors used to make the decision. k=1 amounts to no regularization: 0 error on the training set, whereas large k will push toward smoother decision boundaries in the feature space.
Run the svm_gui.py file in the repository: https://github.com/jakevdp/sklearn_scipy2013