Support Vector Machines (SVMs) are a powerful supervised learning algorithm used for classification or for regression. SVMs are a discriminative classifier: that is, they draw a boundary between clusters of data. In this post I will demonstrate hand-written digit recognition using the SVC classifier in scikit-learn. We'll make use of the online dataset available in the UCI machine learning repository.
Let's begin by importing required packages
import matplotlib.pylab as plt import numpy as np from scipy import sparse from sklearn.datasets import make_classification, make_blobs from sklearn.decomposition import PCA from sklearn.svm import SVC from sklearn.cross_validation import ShuffleSplit from sklearn import metrics from sklearn.learning_curve import learning_curve from pprint import pprint import urllib import seaborn %matplotlib inline
#Let's generate some simple blobs in sklearn, with 150 samples and 2 features X, y = make_blobs(n_samples=150, centers=2, random_state=0, cluster_std=0.4) #Let's plot the datasets to visualize plt.scatter(X[:,1], X[:,0], c=y, s=50, cmap=plt.cm.Paired) plt.show() #OK great the dataset looks fairly easily seperable (atleast to the naked eye). Let's get our hands dirty in SVC, then #move on to bigger and better things.
#Let's fit the SVC to our dataset. The linear kernel essentially means the decision boundary is a line. svc = SVC(kernel='linear').fit(X,y) #Great now, lets run predictions and see how well SVM has done on our dataset. y_pred=svc.predict(X) print metrics.classification_report(y, y_pred) #As expected it worked perfectly, basically seperated the data exactly as we generated it.
precision recall f1-score support 0 1.00 1.00 1.00 75 1 1.00 1.00 1.00 75 avg / total 1.00 1.00 1.00 150
#OK, let's now define a plot decision function that will help us visualize the decision boundary and hyperplane #We will be calling this routine multiple times from here on. #This will only work for 2D datasets def plot_decision_function(X,y,svc): #Plot the data points as a scatter plot first plt.scatter(X[:,0], X[:,1], c=y, s=30, cmap=plt.cm.Paired) #Form the Grid for the Plot x1=np.linspace(plt.xlim(), plt.xlim(), 30) y1=np.linspace(plt.ylim(), plt.ylim(), 30) y2,x2 = np.meshgrid(y1,x1) #OK now we will use the decision function parameter provided by SVC #to plot the hyperplane and decision boundary. Decision function has #the distance of the samples P = np.zeros_like(x2) for i,xi in enumerate(x1): for j,yj in enumerate(y1): P[i,j] = svc.decision_function([xi,yj]) return plt.contour(x2,y2,P,colors ='black', levels=[-1,0,1], linestyles=['--','-','--'])
#Let's plot the decision boundary for the SVC we just fit plt.title('SVC hyperplane with a well-seperated sample dataset', fontsize=14) plot_decision_function(X,y,svc) #Notice the couple of points that touch the dashed lines. These are the support vectors. They are actually the only #thing that influence the SVM. If another point on either side of the hyperplane move (without crossing the decision #boundary of course), the classification doesn't change. However if the points influencing the support vector moves #it directly influences the classification (In our case though, the datasets are extremely well seperated, doesn't #happen very often in real-time. #OK, let's highlight the support vectors in the plot. They are available in the support_vectors_ attribute of the #classifier. plt.scatter(svc.support_vectors_[:,0], svc.support_vectors_[:,1], s=200, facecolor='none', edgecolor='black', linewidths=1.5) plt.show()
#OK now let's make a slightly more complex sample dataset with 10K samples. X,y=make_classification(n_samples = 1000,n_features=2, n_redundant=0, n_informative=2, n_clusters_per_class=1, weights=[0.2], random_state=48) #Let's plot the data plt.scatter(X[:,0], X[:,1], c=y, s=40, cmap=plt.cm.Paired) #Alright this sure looks challenging. Let's use the rbf (radial basis function) Kernel since this data is definitely #not linearly seperable svc=SVC(kernel='rbf', C=1.0).fit(X,y) y_pred=svc.predict(X) #C is a parameter that is used for regularization (To address overfitting). For instance, if C is too small, the model #will overfit the training set, meaning it will try to map every single example correctly. But such a model will not #generalize (work on unknowns) very well since it is too finetuned for the training set. On the other hand if C is too large, #the model will underfit meaning it won't even fit the training set well.
#Let's see how well we performed print metrics.classification_report(y,y_pred) #Plotting the results to visualize the hyperplane plt.title('SVC hyperplane with a well-seperated sample dataset', fontsize=14) plot_decision_function(X,y,svc) plt.scatter(svc.support_vectors_[:,0], svc.support_vectors_[:,1], s=200, facecolor='none', edgecolor='black', linewidths=1.5) plt.show() #There, the rbf kernel has done a very nice job of seperating out the data. Except for a few points all have been #classified correctly.
precision recall f1-score support 0 0.99 0.98 0.99 204 1 0.99 1.00 1.00 796 avg / total 0.99 0.99 0.99 1000
Recognizing Hand Written Digits (UCI ML Repo)
#OK let's actually get started with real data. We'll begin with the famous hand-written digit recognition. #For this we'll use the dataset provided by the UCI Machine Learning Repository. #The dataset basically has 8x8 pixel images of handwritten digits (stored as 64 feature dimensions) and corresponding #labels of the actual digits. Preprocessing has already been performed on the dataset and train/test datasets are #available for download seperately. Let's go get 'em. #Training dataset, last column is the labels y url="https://archive.ics.uci.edu/ml/machine-learning-databases/optdigits/optdigits.tra" raw_data = urllib.urlopen(url) digi=np.genfromtxt(raw_data, delimiter=',') X_train, y_train = digi[:,:-1], digi[:,-1:].squeeze() print X_train.shape, y_train.shape #Test dataset, last column is the labels y url="https://archive.ics.uci.edu/ml/machine-learning-databases/optdigits/optdigits.tes" raw_data = urllib.urlopen(url) digi=np.genfromtxt(raw_data, delimiter=',') X_test, y_test = digi[:,:-1], digi[:,-1:].squeeze() print X_test.shape, y_test.shape
(3823, 64) (3823,) (1797, 64) (1797,)
#Below is a plot_learning_curve module that's provided by scikit-learn. It allows us to quickly and easily visualize how #well the model is performing based on number of samples we're training on. It helps to understand situations such as #high variance or bias. #We'll call this module in the next segment. print(__doc__) import numpy as np import matplotlib.pyplot as plt from sklearn import cross_validation from sklearn.naive_bayes import GaussianNB from sklearn.svm import SVC from sklearn.datasets import load_digits from sklearn.learning_curve import learning_curve def plot_learning_curve(estimator, title, X, y, ylim=None, cv=None, n_jobs=1, train_sizes=np.linspace(.1, 1.0, 5)): """ Generate a simple plot of the test and traning learning curve. Parameters ---------- estimator : object type that implements the "fit" and "predict" methods An object of that type which is cloned for each validation. title : string Title for the chart. X : array-like, shape (n_samples, n_features) Training vector, where n_samples is the number of samples and n_features is the number of features. y : array-like, shape (n_samples) or (n_samples, n_features), optional Target relative to X for classification or regression; None for unsupervised learning. ylim : tuple, shape (ymin, ymax), optional Defines minimum and maximum yvalues plotted. cv : integer, cross-validation generator, optional If an integer is passed, it is the number of folds (defaults to 3). Specific cross-validation objects can be passed, see sklearn.cross_validation module for the list of possible objects n_jobs : integer, optional Number of jobs to run in parallel (default 1). """ plt.figure() plt.title(title) if ylim is not None: plt.ylim(*ylim) plt.xlabel("Training examples") plt.ylabel("Score") train_sizes, train_scores, test_scores = learning_curve( estimator, X, y, cv=cv, n_jobs=n_jobs, train_sizes=train_sizes) train_scores_mean = np.mean(train_scores, axis=1) train_scores_std = np.std(train_scores, axis=1) test_scores_mean = np.mean(test_scores, axis=1) test_scores_std = np.std(test_scores, axis=1) plt.grid() plt.fill_between(train_sizes, train_scores_mean - train_scores_std, train_scores_mean + train_scores_std, alpha=0.1, color="r") plt.fill_between(train_sizes, test_scores_mean - test_scores_std, test_scores_mean + test_scores_std, alpha=0.1, color="g") plt.plot(train_sizes, train_scores_mean, 'o-', color="r", label="Training score") plt.plot(train_sizes, test_scores_mean, 'o-', color="g", label="Cross-validation score") plt.legend(loc="best") return plt
Automatically created module for IPython interactive environment
#All right let's do this the right way. We'll use a cross-validation generator to select train and CV datasets to finetune #parameters such as C (Regularization parameter we saw earlier). These hyperparameters are extremely critical to the model. #Now, if we tune parameters against the Test dataset, we will end up biasing towards the test set and will once again #not generalize very well. We will also have no good way to find out since we have essentially trained on all our data. #Luckily scikit-learn has builit-in packages that can help with this. We'll use a crossvalidation generator that #can train the model by tuning the parameters based on a cross-validation subset (cv) that is picked from within the #training set. A different cv subset will be picked for each iteration, we control the number of iterations. Then we will #use these cv/train splits and run a gridsearch function that will evaluate the model with each split and tune parameters #to give us the best parameter that gives the optimal result. #Defining this as a function so we can call it anytime we want def fit_SVM(kernel, n_jobs, C, gamma): #Choose Estimator as SVM estimator = SVC(kernel=kernel) #Choose cross-validation generator - let's choose ShuffleSplit which randomly shuffles and selects Train and CV sets #for each iteration. There are other methods like the KFold split. cv = ShuffleSplit(X_train.shape, n_iter=10, test_size=0.2, random_state=0) #Apply the cross-validation iterator on the Training set using GridSearchCV. This will run the classifier on the #different train/cv splits using parameters specified and return the model that has the best results #Note that we are tuning based on the F1 score 2PR/P+R where P is Precision and R is Recall. This may not always be #the best score to tune our model on. I will explore this area further in a seperate exercise. For now, we'll use F1. from sklearn.grid_search import GridSearchCV classifier = GridSearchCV(estimator=estimator, cv=cv, param_grid=dict(C=C, gamma=gamma), n_jobs=n_jobs, scoring='f1') #Also note that we're feeding multiple neighbors to the GridSearch to try out. #We'll now fit the training dataset to this classifier classifier.fit(X_train, y_train) #Let's look at the best estimator that was found by GridSearchCV print "Best Estimator learned through GridSearch" print classifier.best_estimator_ return classifier.best_estimator_.C, classifier.best_estimator_.gamma, cv
#WARNING - THIS WILL TAKE A WHILE TO RUN. TRY ADJUSTING parameters such as n_jobs (jobs to run in parallel, before #increasing this make sure your system can handle it), n_iter for ShuffleSplit (in the function definition) and reducing #number of values being tried for C and gamma. #SELECT INTERRUPT IN THE MENU AND PRESS INTERRUPT KERNEL IF YOU NEEDD TO STOP EXECUTION #C_vals=[.01, .03, .1, .3, 1, 3, 10, 30] #gamma_vals=[0.001, .003, .01, .03, .1, 1, 3] #C_vals=[.3, 1, 3] #gamma_vals=[0.001, .003, .01] C_vals= gamma_vals=[.001] #Let's fit SVM to the digits training dataset by calling the function we just created. C, gamma,cv=fit_SVM('rbf', 10, C_vals, gamma_vals)
Best Estimator learned through GridSearch SVC(C=3, cache_size=200, class_weight=None, coef0=0.0, degree=3, gamma=0.001, kernel='rbf', max_iter=-1, probability=False, random_state=None, shrinking=True, tol=0.001, verbose=False)
#OK, let's look at the learning curve for our model based on number of training examples. We'll call the plot_learning_curve #module by feeding it the estimator (best estimator returned from GS) and train/cv sets. #The module simply runs the estimator multiple times on subsets of the data provided and plots the train and cv scores. #Note that we're feeding the best parameters we've learned from GridSearchCV to the estimator now. #We may need to adjust the hyperparameters further if there is overfitting (or underfitting, though unlikely) title = "Learning Curves (SVM, C=%.6f, gamma=%.6f)" %(C,gamma) estimator = SVC(kernel='rbf', C=C, gamma=gamma) plot_learning_curve(estimator, title, X_train, y_train, cv=cv) plt.show()
#Looks like there is overfitting, look at the gulf between the training score and cv. Considering we don't have #any more examples to train on let's make some adjustments to the regularization parameter C. Bringing this value down #(from the best learned through GridSearchCV) will address the problem. C=.5 #Run the learning curve again title = "Learning Curves (SVM, C=%.6f, gamma=%.6f)" %(C,gamma) estimator = SVC(kernel='rbf', C=C, gamma=gamma) plot_learning_curve(estimator, title, X_train, y_train, cv=cv) plt.show()
#Great, we see that as the number of training samples increases, both training and cv scores are on the rise and they both #are pretty high as well as close to each other at 3000 examples. This is a good sign. If training score rose but cv score #fell that would indicate high variance (overfitting). If both train and cv scores are low that would indicate high bias #(underfitting). #Let's see how our trained model performs on the test set. We are not going to train on this set merely looking at how well #our model can generalize. #Calling Fit on the estimator object so we can predict. We're NOT retraining the classifier here. estimator.fit(X_train, y_train) y_pred=estimator.predict(X_test) print metrics.classification_report(y_test,y_pred) print "Final Generalization Accuracy: %.6f" %metrics.accuracy_score(y_test,y_pred) #Excellent, we have a 97.7% accuracy on the Test Set, which is inline with what is to be expected for this dataset.
precision recall f1-score support 0.0 1.00 0.99 1.00 178 1.0 0.96 0.99 0.98 182 2.0 1.00 0.98 0.99 177 3.0 0.99 0.97 0.98 183 4.0 0.99 0.99 0.99 181 5.0 0.98 0.99 0.99 182 6.0 1.00 0.99 1.00 181 7.0 0.99 0.94 0.96 179 8.0 0.97 0.94 0.96 174 9.0 0.90 0.98 0.94 180 avg / total 0.98 0.98 0.98 1797 Final Generalization Accuracy: 0.977741
#Just for kickers, let's visualize images of some digits and label them with what our model is predicting. fig=plt.figure(figsize=(6,6)) fig.suptitle('Sample Digit predictions by SVM', fontsize=15, alpha=2) for i in range(20): ax=fig.add_subplot(5,4,i+1,xticks=, yticks=) ax.imshow(X_test[i,:].reshape(8,8), cmap=plt.cm.binary) ax.text(-1, 7, y_pred[i].astype(int), fontsize=15, color='black', alpha=2) plt.show()
#scikit-learn provides something called a confusion matrix that helps us easily visualize the differences b/w #actuals and predictions. It's especially helpful with digits data as the results are easier to comprehend. print metrics.confusion_matrix(y_test,y_pred) #Now, interestingly this is much better to visualize if we plotted it. seaborn.heatmap(metrics.confusion_matrix(y_test,y_pred)) plt.xlabel('True Labels') plt.ylabel('Predicted Labels') plt.show() #The confusion matrix is basically a utility to compare predicted with Actuals. The ideal matrix will be solid diagonally #across the board and plain everywhere else. We can see below that we're almost there but for a few patches here and there.
[[177 0 0 0 1 0 0 0 0 0] [ 0 181 0 0 0 0 0 0 1 0] [ 0 3 174 0 0 0 0 0 0 0] [ 0 0 0 177 0 2 0 2 1 1] [ 0 0 0 0 180 0 0 0 1 0] [ 0 0 0 0 0 180 0 0 0 2] [ 0 0 0 0 0 0 180 0 1 0] [ 0 0 0 0 0 0 0 168 0 11] [ 0 5 0 0 0 0 0 0 164 5] [ 0 0 0 2 0 1 0 0 1 176]]
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