Tensorflow用SVM(高斯核函数)分类非线性数据
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如果想分割非线性数据集,该如何改变线性分类器映射到数据集?答案是,改变SVM损失函数中的核函数。
# Illustration of Various Kernels
#----------------------------------
#
# This function wll illustrate how to
# implement various kernels in TensorFlow.
#
# Linear Kernel:
# K(x1, x2) = t(x1) * x2
#
# Gaussian Kernel (RBF):
# K(x1, x2) = exp(-gamma * abs(x1 - x2)^2)
import matplotlib.pyplot as plt
import numpy as np
import tensorflow as tf
from sklearn import datasets
from tensorflow.python.framework import ops
ops.reset_default_graph()
# Create graph
sess = tf.Session()
# Generate non-lnear data
# 生成的数据是两个同心圆数据,每个不同的环代表不同的类,确保只有类-1或者
# 为了让绘图方便,这里将每类数据分成x值和y值
(x_vals, y_vals) = datasets.make_circles(n_samples=350, factor=.5, noise=.1)
y_vals = np.array([1 if y==1 else -1 for y in y_vals])
class1_x = [x[0] for i,x in enumerate(x_vals) if y_vals[i]==1]
class1_y = [x[1] for i,x in enumerate(x_vals) if y_vals[i]==1]
class2_x = [x[0] for i,x in enumerate(x_vals) if y_vals[i]==-1]
class2_y = [x[1] for i,x in enumerate(x_vals) if y_vals[i]==-1]
# Declare batch size
# 对于SVM算法,为了让每次迭代训练不波动,得到一个稳定的训练模型,
# 这时批量大小得取值更大
# 注意,本例为预测数据点声明有额外的占位符。
# 最后创建彩色的网格来可视化不同的区域代表不同的类别
batch_size = 350
# Initialize placeholders
x_data = tf.placeholder(shape=[None, 2], dtype=tf.float32)
y_target = tf.placeholder(shape=[None, 1], dtype=tf.float32)
prediction_grid = tf.placeholder(shape=[None, 2], dtype=tf.float32)
# Create variables for svm
b = tf.Variable(tf.random_normal(shape=[1,batch_size]))
# Apply kernel
# Linear Kernel
# my_kernel = tf.matmul(x_data, tf.transpose(x_data))
# Gaussian (RBF) kernel
# 该核函数用矩阵操作来表示
# 在sq_dists中应用广播加法和减法操作
# 线性核函数可以表示为:my_kernel=tf.matmul(x_data,tf.transpose(x_data)。
gamma = tf.constant(-50.0)
dist = tf.reduce_sum(tf.square(x_data), 1)
dist = tf.reshape(dist, [-1,1])
sq_dists = tf.add(tf.subtract(dist, tf.multiply(2., tf.matmul(x_data, tf.transpose(x_data)))), tf.transpose(dist))
my_kernel = tf.exp(tf.multiply(gamma, tf.abs(sq_dists)))
# Compute SVM Model
# 对偶问题。为了最大化,这里采用最小化损失函数的负数tf.negative
first_term = tf.reduce_sum(b)
b_vec_cross = tf.matmul(tf.transpose(b), b)
y_target_cross = tf.matmul(y_target, tf.transpose(y_target))
second_term = tf.reduce_sum(tf.multiply(my_kernel, tf.multiply(b_vec_cross, y_target_cross)))
loss = tf.negative(tf.subtract(first_term, second_term))
# Create Prediction Kernel
# Linear prediction kernel
# my_kernel = tf.matmul(x_data, tf.transpose(prediction_grid))
# Gaussian (RBF) prediction kernel
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)))
prediction_output = tf.matmul(tf.multiply(tf.transpose(y_target),b), pred_kernel)
prediction = tf.sign(prediction_output-tf.reduce_mean(prediction_output))
accuracy = tf.reduce_mean(tf.cast(tf.equal(tf.squeeze(prediction), tf.squeeze(y_target)), tf.float32))
# Declare optimizer
my_opt = tf.train.GradientDescentOptimizer(0.002)
train_step = my_opt.minimize(loss)
# Initialize variables
init = tf.global_variables_initializer()
sess.run(init)
# Training loop
loss_vec = []
batch_accuracy = []
for i in range(1000):
rand_index = np.random.choice(len(x_vals), size=batch_size)
rand_x = x_vals[rand_index]
rand_y = np.transpose([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)%250==0:
print('Step #' + str(i+1))
print('Loss = ' + str(temp_loss))
# Create a mesh to plot points in
x_min, x_max = x_vals[:, 0].min() - 1, x_vals[:, 0].max() + 1
y_min, y_max = x_vals[:, 1].min() - 1, x_vals[:, 1].max() + 1
xx, 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 grid
plt.contourf(xx, yy, grid_predictions, cmap=plt.cm.Paired, alpha=0.8)
plt.plot(class1_x, class1_y, 'ro', label='Class 1')
plt.plot(class2_x, class2_y, 'kx', label='Class -1')
plt.title('Gaussian SVM Results')
plt.xlabel('x')
plt.ylabel('y')
plt.legend(loc='lower right')
plt.ylim([-1.5, 1.5])
plt.xlim([-1.5, 1.5])
plt.show()
# Plot batch accuracy
plt.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 time
plt.plot(loss_vec, 'k-')
plt.title('Loss per Generation')
plt.xlabel('Generation')
plt.ylabel('Loss')
plt.show()
线性核函数下输出:
Step #250
Loss = -74.7393
Step #500
Loss = -190.832
Step #750
Loss = -220.781
Step #1000
Loss = -221.795
使用线性支持向量机在非线性可分的数据集上进行分割
高斯核函数下输出:
Step #250
Loss = 35.2813
Step #500
Loss = -7.39628
Step #750
Loss = -10.6262
Step #1000
Loss = -12.2222
使用非线性的高斯核函数SVM在非线性可分的数据集上进行分割
上述的代码里有两个重要的部分:如何为SVM对偶优化问题完成核函数和损失函数。我们已经实现了线性核函数和高斯核函数,其中高斯核函数能够分割非线性数据集。
我们也应该注意到高斯核函数中有一个参数——gamma。该参数控制数据集分割的弯曲部分的影响程度,一般情况下选择较小值,但是也严重依赖于数据集。在理想情况下,gamma值是通过统计技术(比如,交叉验证)来确定的。
本文是《TensorFlow机器学习实战指南》的读书笔记和动手实践结果。
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