Logistic迴歸（python實現）

一、邏輯迴歸（LogisticRegression）

       Logistic regression （邏輯迴歸）是當前業界比較常用的機器學習方法，用於估計某種事物的可能性。之前在經典之作《數學之美》中也看到了它用於廣告預測，也就是根據某廣告被用戶點擊的可能性，把最可能被用戶點擊的廣告擺在用戶能看到的地方，然後叫他“你點我啊！”用戶點了，你就有錢收了。這就是爲什麼我們的電腦現在廣告氾濫的原因了。

       還有類似的某用戶購買某商品的可能性，某病人患有某種疾病的可能性啊等等。這個世界是隨機的（當然了，人爲的確定性系統除外，但也有可能有噪聲或產生錯誤的結果，只是這個錯誤發生的可能性太小了，小到千萬年不遇，小到忽略不計而已），所以萬物的發生都可以用可能性或者機率（Odds）來表達。“機率”指的是某事物發生的可能性與不發生的可能性的比值。

       Logistic regression可以用來回歸，也可以用來分類，主要是二分類。還記得上幾節講的支持向量機SVM嗎？它就是個二分類的例如，它可以將兩個不同類別的樣本給分開，思想是找到最能區分它們的那個分類超平面。但當你給一個新的樣本給它，它能夠給你的只有一個答案，你這個樣本是正類還是負類。例如你問SVM，某個女生是否喜歡你，它只會回答你喜歡或者不喜歡。這對我們來說，顯得太粗魯了，要不希望，要不絕望，這都不利於身心健康。那如果它可以告訴我，她很喜歡、有一點喜歡、不怎麼喜歡或者一點都不喜歡，你想都不用想了等等，告訴你她有49%的機率喜歡你，總比直接說她不喜歡你，來得溫柔。而且還提供了額外的信息，她來到你的身邊你有多少希望，你得再努力多少倍，知己知彼百戰百勝，哈哈。Logistic regression就是這麼溫柔的，它給我們提供的就是你的這個樣本屬於正類的可能性是多少。

       還得來點數學。（更多的理解，請參閱參考文獻）假設我們的樣本是{x, y}，y是0或者1，表示正類或者負類，x是我們的m維的樣本特徵向量。那麼這個樣本x屬於正類，也就是y=1的“概率”可以通過下面的邏輯函數來表示：

反向傳播求梯度：

SGD求梯度：

 

代碼：

# _*_ coding:utf-8 _*_
#################################################
# logRegression: Logistic Regression
# Author : zouxy
# Date   : 2014-03-02
# HomePage : http://blog.csdn.net/zouxy09
# Email  : [email protected]
#################################################

from numpy import *
import matplotlib.pyplot as plt
import time


# calculate the sigmoid function
def sigmoid(inX):
    return 1.0 / (1 + exp(-inX))


# train a logistic regression model using some optional optimize algorithm
# input: train_x is a mat datatype, each row stands for one sample
#		 train_y is mat datatype too, each row is the corresponding label
#		 opts is optimize option include step and maximum number of iterations
def trainLogRegres(train_x, train_y, opts):
    # calculate training time
    startTime = time.time()

    numSamples, numFeatures = shape(train_x)
    alpha = opts['alpha']
    maxIter = opts['maxIter']
    weights = ones((numFeatures, 1))

    # optimize through gradient descent algorilthm
    for k in range(maxIter):
        if opts['optimizeType'] == 'gradDescent':  # gradient descent algorilthm
            output = sigmoid(train_x * weights)
            error = train_y - output  # 求導公式，推出
            weights = weights + alpha * train_x.transpose() * error
        elif opts['optimizeType'] == 'stocGradDescent':  # stochastic gradient descent
            for i in range(numSamples):
                output = sigmoid(train_x[i, :] * weights)
                error = train_y[i, 0] - output
                weights = weights + alpha * train_x[i, :].transpose() * error  # 求導公式，推出
        elif opts['optimizeType'] == 'smoothStocGradDescent':  # smooth stochastic gradient descent
            # randomly select samples to optimize for reducing cycle fluctuations
            dataIndex = range(numSamples)
            for i in range(numSamples):
                alpha = 4.0 / (1.0 + k + i) + 0.01
                randIndex = int(random.uniform(0, len(dataIndex)))
                output = sigmoid(train_x[randIndex, :] * weights)
                error = train_y[randIndex, 0] - output
                weights = weights + alpha * train_x[randIndex, :].transpose() * error
                del (dataIndex[randIndex])  # during one interation, delete the optimized sample
        else:
            raise NameError('Not support optimize method type!')

    print 'Congratulations, training complete! Took %fs!' % (time.time() - startTime)
    return weights


# test your trained Logistic Regression model given test set
def testLogRegres(weights, test_x, test_y):
    numSamples, numFeatures = shape(test_x)
    matchCount = 0
    for i in xrange(numSamples):
        predict = sigmoid(test_x[i, :] * weights)[0, 0] > 0.5  # 2-D array, so need [0,0] index
        if predict == bool(test_y[i, 0]):
            matchCount += 1
    accuracy = float(matchCount) / numSamples
    return accuracy


# show your trained logistic regression model only available with 2-D data
def showLogRegres(weights, train_x, train_y):
    # notice: train_x and train_y is mat datatype
    numSamples, numFeatures = shape(train_x)
    if numFeatures != 3:
        print
        "Sorry! I can not draw because the dimension of your data is not 2!"
        return 1

    # draw all samples
    for i in xrange(numSamples):
        if int(train_y[i, 0]) == 0:
            plt.plot(train_x[i, 1], train_x[i, 2], 'or')
        elif int(train_y[i, 0]) == 1:
            plt.plot(train_x[i, 1], train_x[i, 2], 'ob')

    # draw the classify line
    min_x = min(train_x[:, 1])[0, 0]
    max_x = max(train_x[:, 1])[0, 0]
    weights = weights.getA()  # convert mat to array
    y_min_x = float(-weights[0] - weights[1] * min_x) / weights[2]
    y_max_x = float(-weights[0] - weights[1] * max_x) / weights[2]
    plt.plot([min_x, max_x], [y_min_x, y_max_x], '-g')
    plt.xlabel('X1');
    plt.ylabel('X2')
    plt.show()


#################################################
# logRegression: Logistic Regression
# Author : zouxy
# Date   : 2014-03-02
# HomePage : http://blog.csdn.net/zouxy09
# Email  : [email protected]
#################################################




def loadData():
    train_x = []
    train_y = []
    fileIn = open('D:/MLCode/logesticRegression/testSet.txt')
    for line in fileIn.readlines():
        lineArr = line.strip().split()
        train_x.append([1.0, float(lineArr[0]), float(lineArr[1])])
        train_y.append(float(lineArr[2]))
    return mat(train_x), mat(train_y).transpose()


## step 1: load data
print "step 1: load data..."
train_x, train_y = loadData()
test_x = train_x
test_y = train_y

## step 2: training...
print "step 2: training..."
opts = {'alpha': 0.01, 'maxIter': 20, 'optimizeType': 'smoothStocGradDescent'}
optimalWeights = trainLogRegres(train_x, train_y, opts)

## step 3: testing
print "step 3: testing..."
accuracy = testLogRegres(optimalWeights, test_x, test_y)

## step 4: show the result
print "step 4: show the result..."
print 'The classify accuracy is: %.3f%%' % (accuracy * 100)
showLogRegres(optimalWeights, train_x, train_y)

testSet.txt

-0.017612   14.053064  0
-1.395634  4.662541   1
-0.752157  6.538620   0
-1.322371  7.152853   0
0.423363   11.054677  0
0.406704   7.067335   1
0.667394   12.741452  0
-2.460150  6.866805   1
0.569411   9.548755   0
-0.026632  10.427743  0
0.850433   6.920334   1
1.347183   13.175500  0
1.176813   3.167020   1
-1.781871  9.097953   0
-0.566606  5.749003   1
0.931635   1.589505   1
-0.024205  6.151823   1
-0.036453  2.690988   1
-0.196949  0.444165   1
1.014459   5.754399   1
1.985298   3.230619   1
-1.693453  -0.557540  1
-0.576525  11.778922  0
-0.346811  -1.678730  1
-2.124484  2.672471   1
1.217916   9.597015   0
-0.733928  9.098687   0
-3.642001  -1.618087  1
0.315985   3.523953   1
1.416614   9.619232   0
-0.386323  3.989286   1
0.556921   8.294984   1
1.224863   11.587360  0
-1.347803  -2.406051  1
1.196604   4.951851   1
0.275221   9.543647   0
0.470575   9.332488   0
-1.889567  9.542662   0
-1.527893  12.150579  0
-1.185247  11.309318  0
-0.445678  3.297303   1
1.042222   6.105155   1
-0.618787  10.320986  0
1.152083   0.548467   1
0.828534   2.676045   1
-1.237728  10.549033  0
-0.683565  -2.166125  1
0.229456   5.921938   1
-0.959885  11.555336  0
0.492911   10.993324  0
0.184992   8.721488   0
-0.355715  10.325976  0
-0.397822  8.058397   0
0.824839   13.730343  0
1.507278   5.027866   1
0.099671   6.835839   1
-0.344008  10.717485  0
1.785928   7.718645   1
-0.918801  11.560217  0
-0.364009  4.747300   1
-0.841722  4.119083   1
0.490426   1.960539   1
-0.007194  9.075792   0
0.356107   12.447863  0
0.342578   12.281162  0
-0.810823  -1.466018  1
2.530777   6.476801   1
1.296683   11.607559  0
0.475487   12.040035  0
-0.783277  11.009725  0
0.074798   11.023650  0
-1.337472  0.468339   1
-0.102781  13.763651  0
-0.147324  2.874846   1
0.518389   9.887035   0
1.015399   7.571882   0
-1.658086  -0.027255  1
1.319944   2.171228   1
2.056216   5.019981   1
-0.851633  4.375691   1
-1.510047  6.061992   0
-1.076637  -3.181888  1
1.821096   10.283990  0
3.010150   8.401766   1
-1.099458  1.688274   1
-0.834872  -1.733869  1
-0.846637  3.849075   1
1.400102   12.628781  0
1.752842   5.468166   1
0.078557   0.059736   1
0.089392   -0.715300  1
1.825662   12.693808  0
0.197445   9.744638   0
0.126117   0.922311   1
-0.679797  1.220530   1
0.677983   2.556666   1
0.761349   10.693862  0
-2.168791  0.143632   1
1.388610   9.341997   0
0.317029   14.739025  0

 

運行結果：

準確率爲：95% （由於使用的是隨機梯度下降SGD，因此每次訓練的準確率可能會不一樣，93、95、97都有）

 

注：

• 反向傳播求導公式，簡單推理需要掌握
• 簡單實用的模型，在廣告預測上很實用
• 簡單的優化法則需要掌握

 

參考資料：

https://blog.csdn.net/zouxy09/article/details/20319673

https://blog.csdn.net/panxiaoxie/article/details/79567426