用TensorFlow实现戴明回归算法的示例_Python

如果最小二乘线性回归算法最小化到回归直线的竖直距离（即，平行于y轴方向），则戴明回归最小化到回归直线的总距离（即，垂直于回归直线）。其最小化x值和y值两个方向的误差，具体的对比图如下图。

用TensorFlow实现戴明回归算法的示例

线性回归算法和戴明回归算法的区别。左边的线性回归最小化到回归直线的竖直距离；右边的戴明回归最小化到回归直线的总距离。

线性回归算法的损失函数最小化竖直距离；而这里需要最小化总距离。给定直线的斜率和截距，则求解一个点到直线的垂直距离有已知的几何公式。代入几何公式并使tensorflow最小化距离。

损失函数是由分子和分母组成的几何公式。给定直线y=mx+b，点（x0，y0），则求两者间的距离的公式为：

用TensorFlow实现戴明回归算法的示例

									# 戴明回归

									#----------------------------------

									#

									# this function shows how to use tensorflow to

									# solve linear deming regression.

									# y = ax + b

									#

									# we will use the iris data, specifically:

									# y = sepal length

									# x = petal width

									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()

									# load the data

									# iris.data = [(sepal length, sepal width, petal length, petal width)]

									iris = datasets.load_iris()

									x_vals = np.array([x[3] for x in iris.data])

									y_vals = np.array([y[0] for y in iris.data])

									# declare batch size

									batch_size = 50

									# initialize placeholders

									x_data = tf.placeholder(shape=[none, 1], dtype=tf.float32)

									y_target = tf.placeholder(shape=[none, 1], dtype=tf.float32)

									# create variables for linear regression

									a = tf.variable(tf.random_normal(shape=[1,1]))

									b = tf.variable(tf.random_normal(shape=[1,1]))

									# declare model operations

									model_output = tf.add(tf.matmul(x_data, a), b)

									# declare demming loss function

									demming_numerator = tf.abs(tf.subtract(y_target, tf.add(tf.matmul(x_data, a), b)))

									demming_denominator = tf.sqrt(tf.add(tf.square(a),1))

									loss = tf.reduce_mean(tf.truediv(demming_numerator, demming_denominator))

									# declare optimizer

									my_opt = tf.train.gradientdescentoptimizer(0.1)

									train_step = my_opt.minimize(loss)

									# initialize variables

									init = tf.global_variables_initializer()

									sess.run(init)

									# training loop

									loss_vec = []

									for i in range(250):

									  rand_index = np.random.choice(len(x_vals), size=batch_size)

									  rand_x = np.transpose([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)

									  if (i+1)%50==0:

									    print('step #' + str(i+1) + ' a = ' + str(sess.run(a)) + ' b = ' + str(sess.run(b)))

									    print('loss = ' + str(temp_loss))

									# get the optimal coefficients

									[slope] = sess.run(a)

									[y_intercept] = sess.run(b)

									# get best fit line

									best_fit = []

									for i in x_vals:

									 best_fit.append(slope*i+y_intercept)

									# plot the result

									plt.plot(x_vals, y_vals, 'o', label='data points')

									plt.plot(x_vals, best_fit, 'r-', label='best fit line', linewidth=3)

									plt.legend(loc='upper left')

									plt.title('sepal length vs pedal width')

									plt.xlabel('pedal width')

									plt.ylabel('sepal length')

									plt.show()

									# plot loss over time

									plt.plot(loss_vec, 'k-')

									plt.title('l2 loss per generation')

									plt.xlabel('generation')

									plt.ylabel('l2 loss')

									plt.show()