今天介绍 logistic regression,虽然里面有 regression 这个词,但是这其实是一种分类的方法,这个分类方法输出的也是 0-1 之间的一个数,可以看成是一种概率输出,这个分类器利用一种 BP 迭代和随机梯度下降的方法来训练求得参数和建立分类模型。

首先来看看这个分类器用到的主要函数,即 sigmoid 函数:

y=σ(x)=11+ex

这个函数有一个很好的特性,就是它的导数,

yx=σ(x)(1σ(x))

下面看看,如何利用这个函数来做分类,假设样本为向量 x, 经过权重系数 w 以及 bias 的转换,变成 u=wTx+b,再经过 sigmoid 函数的转换,最终输出一个预测概率 y=σ(u) , 样本的 ground truth 为 t, 则预测值与真实 label 之间的误差可以用最小均方误差表示:

e=12(yt)2

我们可以通过不断的调整 wb 让预测值和真实 label 之间逐渐接近,根据链式法则,我们可以得到:

ew=eyyuuw

而每一部分的偏导数都可以求得:

ey=yt
yu=σ(u)(1σ(u))
uw=x

根据求得的偏导数,可以对权重系数进行更新:

w:=w+αew

下面给出一个用 logistic regression 做分类的例子:

import numpy as np
from sklearn import datasets

def Sigmoid(x):
    return 1.0/(1 + np.exp(-x))

def Generate_label(y, N_class):
    N_sample = len(y)
    label = np.zeros((N_sample, N_class))
    for ii in range(N_sample):
        label[ii, int(y[ii])]=1     
    return label

# load the iris data
iris = datasets.load_iris()
x_data = iris.data
y_label = iris.target
class_name = iris.target_names

n_sample = len(x_data)
n_class = len(set(y_label))

np.random.seed(0)
index = np.random.permutation(n_sample)
x_data = x_data[index]
y_label = y_label[index].astype(np.float)

train_x = x_data[: int(.8 * n_sample)]
train_y = y_label[: int( .8 * n_sample)]
test_x = x_data[int(.8 * n_sample) :]
test_y = y_label[int(.8 * n_sample) :]

train_label = Generate_label(train_y, n_class)
test_label = Generate_label(test_y, n_class)

# training process
D = train_x.shape[1]
W = 0.01 * np.random.rand(D, n_class)
b = np.zeros((1, n_class))    

step_size = 1e-1
reg = 1e-3
train_sample = train_x.shape[0]    
batch_size = 10
num_batch = train_sample / batch_size
train_epoch = 1000

for ii in range (train_epoch):

    for batch_ii in range(num_batch):

        batch_x = train_x[batch_ii * batch_size:
            (batch_ii+1) * batch_size, :]
        batch_y = train_label[batch_ii * batch_size:
            (batch_ii+1) * batch_size, :]

        scores = np.dot(batch_x, W) + b
        y_out = Sigmoid(scores)

        e = y_out - batch_y

        dataloss = 0.5 * np.sum(e*e) / batch_size
        regloss = 0.5 * reg *  np.sum(W*W)

        L = dataloss + regloss

        dscores = e * y_out * (1 - y_out) / batch_size
        dw = np.dot(batch_x.T, dscores)
        db = np.sum(dscores, axis=0, keepdims=True)

        dw += reg*W

        W = W - step_size * dw
        b = b - step_size * db

    if (ii % 10 == 0):
        print 'the training loss is: %.4f' % L

# test process
scores = np.dot(test_x, W) + b
y_out = Sigmoid(scores)

predict_out = np.argmax(y_out, axis=1)

print 'test accuracy: %.2f' % (np.mean(predict_out == test_y))
posted on 2017-11-18 15:24  未雨愁眸  阅读(564)  评论(0编辑  收藏  举报