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。
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:
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# 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() |
结果:
本文的戴明回归算法与线性回归算法得到的结果基本一致。两者之间的关键不同点在于预测值与数据点间的损失函数度量:线性回归算法的损失函数是竖直距离损失;而戴明回归算法是垂直距离损失(到x轴和y轴的总距离损失)。
注意,这里戴明回归算法的实现类型是总体回归(总的最小二乘法误差)。总体回归算法是假设x值和y值的误差是相似的。我们也可以根据不同的理念使用不同的误差来扩展x轴和y轴的距离计算。
以上就是本文的全部内容,希望对大家的学习有所帮助,也希望大家多多支持服务器之家。
原文链接:https://blog.csdn.net/lilongsy/article/details/79363189