click through rate prediction

包括内容如下图：

 

使用直接估计法，置信区间置信率的估计：

1.使用二项分布直接估计

p(0.04<p^<0.06)=∑0.04n≤k≤0.06n(nk)0.05k0.95n−kp(0.04<p^<0.06)=∑0.04n≤k≤0.06n(nk)0.05k0.95n−k

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low=ceil(n*0.04);%上取整

high=floor(n*0.06);%下取整

prob = 0;

for i=low:1:high

    prob = prob+nchoosek(n,i)*(0.05^i)*(0.95^(n-i));

end

2.使用正态分布近似

μ=p=0.05,σ2=p(1−p)n=0.05∗0.95nμ=p=0.05,σ2=p(1−p)n=0.05∗0.95n

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normcdf(0.06,0.05,sigma/x(i)^0.5) - normcdf(0.04,0.05,sigma/x(i)^0.5)

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warning off all;

clear all;clc;close all;

x=500:1:1500;

y = zeros(1,size(x,2));

y2 = zeros(1,size(x,2));

sigma = sqrt(0.05*0.95);

for i =1:size(x,2)

    y(i) = adPredict(x(i));

    y2(i) = normcdf(0.06,0.05,sigma/x(i)^0.5) - normcdf(0.04,0.05,sigma/x(i)^0.5);

end

 

plot(x,y,'b-'); hold on;

plot(x,y2,'r-');

hold on;

x1=[500 1500];

y1=[0.85 0.85];

plot(x1,y1,'y-');

打印曲线：观测到，n=1000，差不多置信度会到达0.85

 

AUC概念及计算：

sklearn代码：sklearn中有现成方法，计算一组TPR,FPR，然后plot就可以；AUC也可以直接调用方法。

import numpy as npimport matplotlib.pyplot as pltfrom sklearn.linear_model import LogisticRegressionfrom sklearn import datasetsfrom sklearn.preprocessing import StandardScalerfrom sklearn.metrics import roc_auc_scorefrom sklearn.metrics import roc_curvedigits = datasets.load_digits()X, y = digits.data, digits.targetX = StandardScaler().fit_transform(X)# classify small against large digitsy = (y > 4).astype(np.int)X_train = X[:-400]y_train = y[:-400]X_test = X[-400:]y_test = y[-400:]lrg = LogisticRegression(penalty='l1')lrg.fit(X_train, y_train)y_test_prob=lrg.predict_proba(X_test)P = np.where(y_test==1)[0].shape[0];N  = np.where(y_test==0)[0].shape[0];dt = 10001TPR = np.zeros((dt,1))FPR = np.zeros((dt,1))for i in range(dt):    y_test_p = y_test_prob[:,1]>=i*(1.0/(dt-1))    TP = np.where((y_test==1)&(y_test_p==True))[0].shape[0];    FN = P-TP;    FP = np.where((y_test==0)&(y_test_p==True))[0].shape[0];    TN = N - FP;    TPR[i]=TP*1.0/P    FPR[i]=FP*1.0/Nplt.plot(FPR,TPR,color='black')plt.plot(np.array([[0],[1]]),np.array([[0],[1]]),color='red')plt.show()#use sklearn method# fpr, tpr, thresholds = roc_curve(y_test,y_test_prob[:,1],pos_label=1)# plt.plot(fpr,tpr,color='black')# plt.plot(np.array([[0],[1]]),np.array([[0],[1]]),color='red')# plt.show()rank = y_test_prob[:,1].argsort()rank = rank.argsort()+1auc = (sum(rank[np.where(y_test==1)[0]])-(P*1.0*(P+1)/2))/(P*N);print aucprint roc_auc_score(y_test, y_test_prob[:,1])