本文將詳細(xì)展示一個(gè)多類支持向量機(jī)分類器訓(xùn)練iris數(shù)據(jù)集來分類三種花。

SVM算法最初是為二值分類問題設(shè)計(jì)的，但是也可以通過一些策略使得其能進(jìn)行多類分類。主要的兩種策略是：一對(duì)多（one versus all）方法；一對(duì)一（one versus one）方法。

一對(duì)一方法是在任意兩類樣本之間設(shè)計(jì)創(chuàng)建一個(gè)二值分類器，然后得票最多的類別即為該未知樣本的預(yù)測(cè)類別。但是當(dāng)類別（k類）很多的時(shí)候，就必須創(chuàng)建k！/（k-2）！2！個(gè)分類器，計(jì)算的代價(jià)還是相當(dāng)大的。

另外一種實(shí)現(xiàn)多類分類器的方法是一對(duì)多，其為每類創(chuàng)建一個(gè)分類器。最后的預(yù)測(cè)類別是具有最大SVM間隔的類別。本文將實(shí)現(xiàn)該方法。

我們將加載iris數(shù)據(jù)集，使用高斯核函數(shù)的非線性多類SVM模型。iris數(shù)據(jù)集含有三個(gè)類別，山鳶尾、變色鳶尾和維吉尼亞鳶尾（I.setosa、I.virginica和I.versicolor），我們將為它們創(chuàng)建三個(gè)高斯核函數(shù)SVM來預(yù)測(cè)。

# Multi-class (Nonlinear) SVM Example#----------------------------------## This function wll illustrate how to# implement the gaussian kernel with# multiple classes on the iris dataset.## Gaussian Kernel:# K(x1, x2) = exp(-gamma * abs(x1 - x2)^2)## X : (Sepal Length, Petal Width)# Y: (I. setosa, I. virginica, I. versicolor) (3 classes)## Basic idea: introduce an extra dimension to do# one vs all classification.## The prediction of a point will be the category with# the largest margin or distance to boundary.import matplotlib.pyplot as pltimport numpy as npimport tensorflow as tffrom sklearn import datasetsfrom tensorflow.python.framework import opsops.reset_default_graph()# Create graphsess = tf.Session()# Load the data# 加載iris數(shù)據(jù)集并為每類分離目標(biāo)值。# 因?yàn)槲覀兿肜L制結(jié)果圖，所以只使用花萼長度和花瓣寬度兩個(gè)特征。# 為了便于繪圖，也會(huì)分離x值和y值# iris.data = [(Sepal Length, Sepal Width, Petal Length, Petal Width)]iris = datasets.load_iris()x_vals = np.array([[x[0], x[3]] for x in iris.data])y_vals1 = np.array([1 if y==0 else -1 for y in iris.target])y_vals2 = np.array([1 if y==1 else -1 for y in iris.target])y_vals3 = np.array([1 if y==2 else -1 for y in iris.target])y_vals = np.array([y_vals1, y_vals2, y_vals3])class1_x = [x[0] for i,x in enumerate(x_vals) if iris.target[i]==0]class1_y = [x[1] for i,x in enumerate(x_vals) if iris.target[i]==0]class2_x = [x[0] for i,x in enumerate(x_vals) if iris.target[i]==1]class2_y = [x[1] for i,x in enumerate(x_vals) if iris.target[i]==1]class3_x = [x[0] for i,x in enumerate(x_vals) if iris.target[i]==2]class3_y = [x[1] for i,x in enumerate(x_vals) if iris.target[i]==2]# Declare batch sizebatch_size = 50# Initialize placeholders# 數(shù)據(jù)集的維度在變化，從單類目標(biāo)分類到三類目標(biāo)分類。# 我們將利用矩陣傳播和reshape技術(shù)一次性計(jì)算所有的三類SVM。# 注意，由于一次性計(jì)算所有分類，# y_target占位符的維度是[3，None]，模型變量b初始化大小為[3，batch_size]x_data = tf.placeholder(shape=[None, 2], dtype=tf.float32)y_target = tf.placeholder(shape=[3, None], dtype=tf.float32)prediction_grid = tf.placeholder(shape=[None, 2], dtype=tf.float32)# Create variables for svmb = tf.Variable(tf.random_normal(shape=[3,batch_size]))# Gaussian (RBF) kernel 核函數(shù)只依賴x_datagamma = tf.constant(-10.0)dist = tf.reduce_sum(tf.square(x_data), 1)dist = tf.reshape(dist, [-1,1])sq_dists = tf.multiply(2., tf.matmul(x_data, tf.transpose(x_data)))my_kernel = tf.exp(tf.multiply(gamma, tf.abs(sq_dists)))# Declare function to do reshape/batch multiplication# 最大的變化是批量矩陣乘法。# 最終的結(jié)果是三維矩陣，并且需要傳播矩陣乘法。# 所以數(shù)據(jù)矩陣和目標(biāo)矩陣需要預(yù)處理，比如xT·x操作需額外增加一個(gè)維度。# 這里創(chuàng)建一個(gè)函數(shù)來擴(kuò)展矩陣維度，然后進(jìn)行矩陣轉(zhuǎn)置，# 接著調(diào)用TensorFlow的tf.batch_matmul（）函數(shù)def reshape_matmul(mat):  v1 = tf.expand_dims(mat, 1)  v2 = tf.reshape(v1, [3, batch_size, 1])  return(tf.matmul(v2, v1))# Compute SVM Model 計(jì)算對(duì)偶損失函數(shù)first_term = tf.reduce_sum(b)b_vec_cross = tf.matmul(tf.transpose(b), b)y_target_cross = reshape_matmul(y_target)second_term = tf.reduce_sum(tf.multiply(my_kernel, tf.multiply(b_vec_cross, y_target_cross)),[1,2])loss = tf.reduce_sum(tf.negative(tf.subtract(first_term, second_term)))# Gaussian (RBF) prediction kernel# 現(xiàn)在創(chuàng)建預(yù)測(cè)核函數(shù)。# 要當(dāng)心reduce_sum（）函數(shù)，這里我們并不想聚合三個(gè)SVM預(yù)測(cè)，# 所以需要通過第二個(gè)參數(shù)告訴TensorFlow求和哪幾個(gè)rA = tf.reshape(tf.reduce_sum(tf.square(x_data), 1),[-1,1])rB = tf.reshape(tf.reduce_sum(tf.square(prediction_grid), 1),[-1,1])pred_sq_dist = tf.add(tf.subtract(rA, tf.multiply(2., tf.matmul(x_data, tf.transpose(prediction_grid)))), tf.transpose(rB))pred_kernel = tf.exp(tf.multiply(gamma, tf.abs(pred_sq_dist)))# 實(shí)現(xiàn)預(yù)測(cè)核函數(shù)后，我們創(chuàng)建預(yù)測(cè)函數(shù)。# 與二類不同的是，不再對(duì)模型輸出進(jìn)行sign（）運(yùn)算。# 因?yàn)檫@里實(shí)現(xiàn)的是一對(duì)多方法，所以預(yù)測(cè)值是分類器有最大返回值的類別。# 使用TensorFlow的內(nèi)建函數(shù)argmax（）來實(shí)現(xiàn)該功能prediction_output = tf.matmul(tf.multiply(y_target,b), pred_kernel)prediction = tf.arg_max(prediction_output-tf.expand_dims(tf.reduce_mean(prediction_output,1), 1), 0)accuracy = tf.reduce_mean(tf.cast(tf.equal(prediction, tf.argmax(y_target,0)), tf.float32))# Declare optimizermy_opt = tf.train.GradientDescentOptimizer(0.01)train_step = my_opt.minimize(loss)# Initialize variablesinit = tf.global_variables_initializer()sess.run(init)# Training looploss_vec = []batch_accuracy = []for i in range(100):  rand_index = np.random.choice(len(x_vals), size=batch_size)  rand_x = x_vals[rand_index]  rand_y = y_vals[:,rand_index]  sess.run(train_step, feed_dict={x_data: rand_x, y_target: rand_y})  temp_loss = sess.run(loss, feed_dict={x_data: rand_x, y_target: rand_y})  loss_vec.append(temp_loss)  acc_temp = sess.run(accuracy, feed_dict={x_data: rand_x,                       y_target: rand_y,                       prediction_grid:rand_x})  batch_accuracy.append(acc_temp)  if (i+1)%25==0:    print('Step #' + str(i+1))    print('Loss = ' + str(temp_loss))# 創(chuàng)建數(shù)據(jù)點(diǎn)的預(yù)測(cè)網(wǎng)格，運(yùn)行預(yù)測(cè)函數(shù)x_min, x_max = x_vals[:, 0].min() - 1, x_vals[:, 0].max() + 1y_min, y_max = x_vals[:, 1].min() - 1, x_vals[:, 1].max() + 1xx, yy = np.meshgrid(np.arange(x_min, x_max, 0.02),           np.arange(y_min, y_max, 0.02))grid_points = np.c_[xx.ravel(), yy.ravel()]grid_predictions = sess.run(prediction, feed_dict={x_data: rand_x,                          y_target: rand_y,                          prediction_grid: grid_points})grid_predictions = grid_predictions.reshape(xx.shape)# Plot points and gridplt.contourf(xx, yy, grid_predictions, cmap=plt.cm.Paired, alpha=0.8)plt.plot(class1_x, class1_y, 'ro', label='I. setosa')plt.plot(class2_x, class2_y, 'kx', label='I. versicolor')plt.plot(class3_x, class3_y, 'gv', label='I. virginica')plt.title('Gaussian SVM Results on Iris Data')plt.xlabel('Pedal Length')plt.ylabel('Sepal Width')plt.legend(loc='lower right')plt.ylim([-0.5, 3.0])plt.xlim([3.5, 8.5])plt.show()# Plot batch accuracyplt.plot(batch_accuracy, 'k-', label='Accuracy')plt.title('Batch Accuracy')plt.xlabel('Generation')plt.ylabel('Accuracy')plt.legend(loc='lower right')plt.show()# Plot loss over timeplt.plot(loss_vec, 'k-')plt.title('Loss per Generation')plt.xlabel('Generation')plt.ylabel('Loss')plt.show()