In the previous section, we learned how Rosenblatt's perceptron rule works; let us now go ahead and implement it in Python and apply it to the Iris dataset that we introduced in Chapter 1, Giving Computers the Ability to Learn from Data. We will take an objected-oriented approach to define the perceptron interface as a Python Class, which allows us to initialize new perceptron objects that can learn from data via a fit method, and make predictions via a separate predict method. As a convention, we add an underscore to attributes that are not being created upon the initialization of the object but by calling the object's other methods—for example, self.w_.
Note
If you are not yet familiar with Python's scientific libraries or need a refresher, please see the following resources:
NumPy: http://wiki.scipy.org/Tentative_NumPy_Tutorial
Pandas: http://pandas.pydata.org/pandas-docs/stable/tutorials.html
Matplotlib: http://matplotlib.org/users/beginner.html
Also, to better follow the code examples, I recommend you download the IPython notebooks from the Packt website. For a general introduction to IPython notebooks, please visit https://ipython.org/ipython-doc/3/notebook/index.html.
import numpy as np
class Perceptron(object):
"""Perceptron classifier.
Parameters
------------
eta : float
Learning rate (between 0.0 and 1.0)
n_iter : int
Passes over the training dataset.
Attributes
-----------
w_ : 1d-array
Weights after fitting.
errors_ : list
Number of misclassifications in every epoch.
"""
def __init__(self, eta=0.01, n_iter=10):
self.eta = eta
self.n_iter = n_iter
def fit(self, X, y):
"""Fit training data.
Parameters
----------
X : {array-like}, shape = [n_samples, n_features]
Training vectors, where n_samples
is the number of samples and
n_features is the number of features.
y : array-like, shape = [n_samples]
Target values.
Returns
-------
self : object
"""
self.w_ = np.zeros(1 + X.shape[1])
self.errors_ = []
for _ in range(self.n_iter):
errors = 0
for xi, target in zip(X, y):
update = self.eta * (target - self.predict(xi))
self.w_[1:] += update * xi
self.w_[0] += update
errors += int(update != 0.0)
self.errors_.append(errors)
return self
def net_input(self, X):
"""Calculate net input"""
return np.dot(X, self.w_[1:]) + self.w_[0]
def predict(self, X):
"""Return class label after unit step"""
return np.where(self.net_input(X) >= 0.0, 1, -1)Using this perceptron implementation, we can now initialize new Perceptron objects with a given learning rate eta and n_iter, which is the number of epochs (passes over the training set). Via the fit method we initialize the weights in self.w_ to a zero-vector where
stands for the number of dimensions (features) in the dataset where we add 1 for the zero-weight (that is, the threshold).
Note
NumPy indexing for one-dimensional arrays works similarly to Python lists using the square-bracket ([]) notation. For two-dimensional arrays, the first indexer refers to the row number, and the second indexer to the column number. For example, we would use X[2, 3] to select the third row and fourth column of a 2D array X.
After the weights have been initialized, the fit method loops over all individual samples in the training set and updates the weights according to the perceptron learning rule that we discussed in the previous section. The class labels are predicted by the predict method, which is also called in the fit method to predict the class label for the weight update, but predict can also be used to predict the class labels of new data after we have fitted our model. Furthermore, we also collect the number of misclassifications during each epoch in the list self.errors_ so that we can later analyze how well our perceptron performed during the training. The np.dot function that is used in the net_input method simply calculates the vector dot product .
Note
Instead of using NumPy to calculate the vector dot product between two arrays a and b via a.dot(b) or np.dot(a, b), we could also perform the calculation in pure Python via sum([i*j for i,j in zip(a, b)]. However, the advantage of using NumPy over classic Python for-loop structures is that its arithmetic operations are vectorized. Vectorization means that an elemental arithmetic operation is automatically applied to all elements in an array. By formulating our arithmetic operations as a sequence of instructions on an array rather than performing a set of operations for each element one at a time, we can make better use of our modern CPU architectures with Single Instruction, Multiple Data (SIMD) support. Furthermore, NumPy uses highly optimized linear algebra libraries, such as Basic Linear Algebra Subprograms (BLAS) and Linear Algebra Package (LAPACK) that have been written in C or Fortran. Lastly, NumPy also allows us to write our code in a more compact and intuitive way using the basics of linear algebra, such as vector and matrix dot products.
To test our perceptron implementation, we will load the two flower classes Setosa and Versicolor from the Iris dataset. Although, the perceptron rule is not restricted to two dimensions, we will only consider the two features sepal length and petal length for visualization purposes. Also, we only chose the two flower classes Setosa and Versicolor for practical reasons. However, the perceptron algorithm can be extended to multi-class classification—for example, through the One-vs.-All technique.
Note
One-vs.-All (OvA), or sometimes also called One-vs.-Rest (OvR), is a technique used to extend a binary classifier to multi-class problems. Using OvA, we can train one classifier per class, where the particular class is treated as the positive class and the samples from all other classes are considered as the negative class. If we were to classify a new data sample, we would use our classifiers, where
is the number of class labels, and assign the class label with the highest confidence to the particular sample. In the case of the perceptron, we would use OvA to choose the class label that is associated with the largest absolute net input value.
First, we will use the pandas library to load the Iris dataset directly from the UCI Machine Learning Repository into a DataFrame object and print the last five lines via the tail method to check that the data was loaded correctly:
>>> import pandas as pd >>> df = pd.read_csv('https://archive.ics.uci.edu/ml/' ... 'machine-learning-databases/iris/iris.data', header=None) >>> df.tail()
Next, we extract the first 100 class labels that correspond to the 50 Iris-Setosa and 50 Iris-Versicolor flowers, respectively, and convert the class labels into the two integer class labels 1 (Versicolor) and -1 (Setosa) that we assign to a vector y where the values method of a pandas DataFrame yields the corresponding NumPy representation. Similarly, we extract the first feature column (sepal length) and the third feature column (petal length) of those 100 training samples and assign them to a feature matrix X, which we can visualize via a two-dimensional scatter plot:
>>> import matplotlib.pyplot as plt >>> import numpy as np >>> y = df.iloc[0:100, 4].values >>> y = np.where(y == 'Iris-setosa', -1, 1) >>> X = df.iloc[0:100, [0, 2]].values >>> plt.scatter(X[:50, 0], X[:50, 1], ... color='red', marker='o', label='setosa') >>> plt.scatter(X[50:100, 0], X[50:100, 1], ... color='blue', marker='x', label='versicolor') >>> plt.xlabel('sepal length') >>> plt.ylabel('petal length') >>> plt.legend(loc='upper left') >>> plt.show()
After executing the preceding code example we should now see the following scatterplot:
Now it's time to train our perceptron algorithm on the Iris data subset that we just extracted. Also, we will plot the misclassification error for each epoch to check if the algorithm converged and found a decision boundary that separates the two Iris flower classes:
>>> ppn = Perceptron(eta=0.1, n_iter=10) >>> ppn.fit(X, y) >>> plt.plot(range(1, len(ppn.errors_) + 1), ppn.errors_, ... marker='o') >>> plt.xlabel('Epochs') >>> plt.ylabel('Number of misclassifications') >>> plt.show()
After executing the preceding code, we should see the plot of the misclassification errors versus the number of epochs, as shown next:
As we can see in the preceding plot, our perceptron already converged after the sixth epoch and should now be able to classify the training samples perfectly. Let us implement a small convenience function to visualize the decision boundaries for 2D datasets:
from matplotlib.colors import ListedColormap def plot_decision_regions(X, y, classifier, resolution=0.02): # setup marker generator and color map markers = ('s', 'x', 'o', '^', 'v') colors = ('red', 'blue', 'lightgreen', 'gray', 'cyan') cmap = ListedColormap(colors[:len(np.unique(y))]) # plot the decision surface x1_min, x1_max = X[:, 0].min() - 1, X[:, 0].max() + 1 x2_min, x2_max = X[:, 1].min() - 1, X[:, 1].max() + 1 xx1, xx2 = np.meshgrid(np.arange(x1_min, x1_max, resolution), np.arange(x2_min, x2_max, resolution)) Z = classifier.predict(np.array([xx1.ravel(), xx2.ravel()]).T) Z = Z.reshape(xx1.shape) plt.contourf(xx1, xx2, Z, alpha=0.4, cmap=cmap) plt.xlim(xx1.min(), xx1.max()) plt.ylim(xx2.min(), xx2.max()) # plot class samples for idx, cl in enumerate(np.unique(y)): plt.scatter(x=X[y == cl, 0], y=X[y == cl, 1], alpha=0.8, c=cmap(idx), marker=markers[idx], label=cl)
First, we define a number of colors and markers and create a color map from the list of colors via ListedColormap. Then, we determine the minimum and maximum values for the two features and use those feature vectors to create a pair of grid arrays xx1 and xx2 via the NumPy meshgrid function. Since we trained our perceptron classifier on two feature dimensions, we need to flatten the grid arrays and create a matrix that has the same number of columns as the Iris training subset so that we can use the predict method to predict the class labels Z of the corresponding grid points. After reshaping the predicted class labels Z into a grid with the same dimensions as xx1 and xx2, we can now draw a contour plot via matplotlib's contourf function that maps the different decision regions to different colors for each predicted class in the grid array:
>>> plot_decision_regions(X, y, classifier=ppn) >>> plt.xlabel('sepal length [cm]') >>> plt.ylabel('petal length [cm]') >>> plt.legend(loc='upper left') >>> plt.show()
After executing the preceding code example, we should now see a plot of the decision regions, as shown in the following figure:
As we can see in the preceding plot, the perceptron learned a decision boundary that was able to classify all flower samples in the Iris training subset perfectly.
Note
Although the perceptron classified the two Iris flower classes perfectly, convergence is one of the biggest problems of the perceptron. Frank Rosenblatt proved mathematically that the perceptron learning rule converges if the two classes can be separated by a linear hyperplane. However, if classes cannot be separated perfectly by such a linear decision boundary, the weights will never stop updating unless we set a maximum number of epochs.