接上回书，这节我们介绍计算机如何找到合适的线性方程，即计算机如何学习？

我们来看一个简单的例子。如下图，有三个蓝点和三个红点，寻找区分这些点的直线。对于计算机来说，它可能随机从某个位置开始选择一个线性方程(如下方右图)。这条直线将整个样本空间化分为两个区域--蓝色区域与红色区域。可以看出，这条直线的分类效果比较差，所以我们需要移动它，即让这条直线更加靠近下图中两个分类错误的点(在蓝色区域的红点和在红色区域的蓝点)，这样才能使分类结果越来越好。

下面为大家讲授使直线更加靠近目标点的方法。如下图所示，假设直线的方程为，图中红色点为分类错误的点，其坐标为。要使直线靠近红点我们可以这么做：将直线方程的系数提取出来，同时为红点坐标增加偏置单元1，改为，然后对应相减(如下图所示)，将得到的值作为直线方程的系数即可。

          

但是！这样得到的直线方程移动较大，在数据点较多时可能会使分类结果更加不好，所以我们希望直线向红点做小幅移动。于是引入学习速率(Learning Rate)的概念。由上述分析可知，学习速率应是个较小的数字，此处假设学习速率为0.1，然后将0.1乘以增加了偏置单元的红点坐标值，然后相减(如下图)。得到新的直线方程，此时你会惊奇的发现，直线更加靠近分类错误的红点了！！！没错，就是这么简单和神奇。

同样，假如有个蓝色的点位于红色区域，也可以按照上述方法使直线更加靠近该点。但要注意，求新的直线方程参数时要做加法，而不是减法。请牢记此方法，后续会经常使用！

      

将上述知识点进行总结(伪代码如下图所示)。对于n维数据，起始先随机分配权重和偏置，然后计算所有点的结果。对于分类错误点我们执行以下操作：

1、预测为0的点，也就是被分到红色区域的蓝色点。我们将当前权重加上学习速率乘以该点的坐标作为新的权重值，将当前偏置加上学习速率作为新的偏置。

2、预测为1的点，也就是被分到蓝色区域的红色点。我们将当前权重减去学习速率乘以该点的坐标作为新的权重值，将当前偏置减去学习速率作为新的偏置。

下面进行练习。使用感知器算法分类下面的数据。

0.78051,-0.063669,1
0.28774,0.29139,1
0.40714,0.17878,1
0.2923,0.4217,1
0.50922,0.35256,1
0.27785,0.10802,1
0.27527,0.33223,1
0.43999,0.31245,1
0.33557,0.42984,1
0.23448,0.24986,1
0.0084492,0.13658,1
0.12419,0.33595,1
0.25644,0.42624,1
0.4591,0.40426,1
0.44547,0.45117,1
0.42218,0.20118,1
0.49563,0.21445,1
0.30848,0.24306,1
0.39707,0.44438,1
0.32945,0.39217,1
0.40739,0.40271,1
0.3106,0.50702,1
0.49638,0.45384,1
0.10073,0.32053,1
0.69907,0.37307,1
0.29767,0.69648,1
0.15099,0.57341,1
0.16427,0.27759,1
0.33259,0.055964,1
0.53741,0.28637,1
0.19503,0.36879,1
0.40278,0.035148,1
0.21296,0.55169,1
0.48447,0.56991,1
0.25476,0.34596,1
0.21726,0.28641,1
0.67078,0.46538,1
0.3815,0.4622,1
0.53838,0.32774,1
0.4849,0.26071,1
0.37095,0.38809,1
0.54527,0.63911,1
0.32149,0.12007,1
0.42216,0.61666,1
0.10194,0.060408,1
0.15254,0.2168,1
0.45558,0.43769,1
0.28488,0.52142,1
0.27633,0.21264,1
0.39748,0.31902,1
0.5533,1,0
0.44274,0.59205,0
0.85176,0.6612,0
0.60436,0.86605,0
0.68243,0.48301,0
1,0.76815,0
0.72989,0.8107,0
0.67377,0.77975,0
0.78761,0.58177,0
0.71442,0.7668,0
0.49379,0.54226,0
0.78974,0.74233,0
0.67905,0.60921,0
0.6642,0.72519,0
0.79396,0.56789,0
0.70758,0.76022,0
0.59421,0.61857,0
0.49364,0.56224,0
0.77707,0.35025,0
0.79785,0.76921,0
0.70876,0.96764,0
0.69176,0.60865,0
0.66408,0.92075,0
0.65973,0.66666,0
0.64574,0.56845,0
0.89639,0.7085,0
0.85476,0.63167,0
0.62091,0.80424,0
0.79057,0.56108,0
0.58935,0.71582,0
0.56846,0.7406,0
0.65912,0.71548,0
0.70938,0.74041,0
0.59154,0.62927,0
0.45829,0.4641,0
0.79982,0.74847,0
0.60974,0.54757,0
0.68127,0.86985,0
0.76694,0.64736,0
0.69048,0.83058,0
0.68122,0.96541,0
0.73229,0.64245,0
0.76145,0.60138,0
0.58985,0.86955,0
0.73145,0.74516,0
0.77029,0.7014,0
0.73156,0.71782,0
0.44556,0.57991,0
0.85275,0.85987,0
0.51912,0.62359,0

程序代码如下：

import numpy as np
# Setting the random seed, feel free to change it and see different solutions.
np.random.seed(42)

def stepFunction(t):
    if t >= 0:
        return 1
    return 0

def prediction(X, W, b):
    return stepFunction((np.matmul(X,W)+b)[0])

# TODO: Fill in the code below to implement the perceptron trick.
# The function should receive as inputs the data X, the labels y,
# the weights W (as an array), and the bias b,
# update the weights and bias W, b, according to the perceptron algorithm,
# and return W and b.
def perceptronStep(X, y, W, b, learn_rate = 0.01):
    # Fill in code
    for i in range(len(X)):
        y_hat = prediction(X[i],W,b)
        if y[i]-y_hat == 1:
            W[0] += X[i][0]*learn_rate
            W[1] += X[i][1]*learn_rate
            b += learn_rate
        elif y[i]-y_hat == -1:
            W[0] -= X[i][0]*learn_rate
            W[1] -= X[i][1]*learn_rate
            b -= learn_rate
    return W, b
    
# This function runs the perceptron algorithm repeatedly on the dataset,
# and returns a few of the boundary lines obtained in the iterations,
# for plotting purposes.
# Feel free to play with the learning rate and the num_epochs,
# and see your results plotted below.
def trainPerceptronAlgorithm(X, y, learn_rate = 0.01, num_epochs = 25):
    x_min, x_max = min(X.T[0]), max(X.T[0])
    y_min, y_max = min(X.T[1]), max(X.T[1])
    W = np.array(np.random.rand(2,1))
    b = np.random.rand(1)[0] + x_max
    # These are the solution lines that get plotted below.
    boundary_lines = []
    for i in range(num_epochs):
        # In each epoch, we apply the perceptron step.
        W, b = perceptronStep(X, y, W, b, learn_rate)
        boundary_lines.append((-W[0]/W[1], -b/W[1]))
    return boundary_lines

实验结果：

其中绿色虚线为每次更新的直线方程，黑线为最终的直线方程。

 

本篇主要介绍了线性数据的分类，那么对于非线性数据该怎么分类呢？咱们下节见。