用python根据考生成绩对学生预测是否被高校录取

机器学习AI算法工程

发布于 2018-03-14 18:21:00

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文章被收录于专栏：机器学习AI算法工程机器学习AI算法工程

Dataset

每年高中生和大学生都会申请进入到各种各样的高校中去。每个学生都有一组唯一的考试分数，成绩和背景数据。录取委员会根据这个数据决定是否接受这些申请者。在这种情况下一个二元分类算法可用于接受或拒绝申请，逻辑回归是个不错的方法。

数据集admissions.csv包含了1000个申请者的信息，特征如下：

gre -（入学考试成绩） gpa - （累积平均绩点） admit - 适合被录取 0或1

Use Linear Regression To Predict Admission

这是原本的数据，admit的值是0或者1。可以发现”gpa”和”admit”并没有线性关系，因为”admit”只取两个值 import pandas import matplotlib.pyplot as plt admissions = pandas.read_csv("admissions.csv") plt.scatter(admissions["gpa"], admissions["admit"]) plt.show()

这是通过线性回归模型预测的admit的值，发现admit_prediction 取值范围较大，有负值，不是我们想要的。

# The admissions DataFrame is in memory

# Import linear regression class

from sklearn.linear_model import LinearRegression

# Initialize a linear regression model

model = LinearRegression()

# Fit model

model.fit(admissions[['gre', 'gpa']], admissions["admit"])

# Prediction of admission

admit_prediction = model.predict(admissions[['gre', 'gpa']])

# Plot Estimated Function

plt.scatter(admissions["gpa"], admit_prediction)

因此我们期望构造一个模型，能够给我们一个接受（admission）的概率，并且这个概率取值在[0~1]，然后选择合适的阈值进行分类。

The Logit Function

逻辑回归是一个流行的分类方法,它将输出限制在0和1之间。这个输出可以被视为一个给定一组输入某个事件的概率,就像任何其他分类方法。

logit function是逻辑回归的基础，这个函数的形式如下：

观察一下logit function的样子：

# Logistic Function def logit(x): # np.exp(x) raises x to the exponential power, ie e^x. e ~= 2.71828 return np.exp(x) / (1 + np.exp(x)) # Linspace is as numpy function to produced evenly spaced numbers over a specified interval. # Create an array with 50 values between -6 and 6 as t t = np.linspace(-6,6,50, dtype=float) # Get logistic fits ylogit = logit(t) # plot the logistic function plt.plot(t, ylogit, label="logistic") plt.ylabel("Probability") plt.xlabel("t") plt.title("Logistic Function") plt.show() a = logit(-10) b = logit(10) ''' a:4.5397868702434395e-05 b:0.99995460213129761 '''

The Logistic Regression

逻辑回归就是将线性回归的输出当做Logit Function的输入然后产生一个输出当做最终的概率。其中β0是截距，其他的βi是斜率，也是特征的系数。

与线性模型一样，我们想要找到最优的βi的值使得预测值与真实值之间的误差最小。通常用来最小化误差的方法是最大似然法和梯度下降法。

Model Data

下面进行逻辑回归实验，每次进行训练测试集划分之前，需要将样本数据进行洗牌，这样抽样具有随机性。看到最后的gre和预测值的关系发现，当gre越大时，被接受的概率越大，这是符合实际情况的。

from sklearn.linear_model import LogisticRegression # Randomly shuffle our data for the training and test set admissions = admissions.loc[np.random.permutation(admissions.index)] # train with 700 and test with the following 300, split dataset num_train = 700 data_train = admissions[:num_train] data_test = admissions[num_train:] # Fit Logistic regression to admit with gpa and gre as features using the training set logistic_model = LogisticRegression() logistic_model.fit(data_train[['gpa', 'gre']], data_train['admit']) # Print the Models Coefficients print(logistic_model.coef_) ''' [[ 0.38004023 0.00791207]] ''' # Predict the chance of admission from those in the training set fitted_vals = logistic_model.predict_proba(data_train[['gpa', 'gre']])[:,1] fitted_test = logistic_model.predict_proba(data_test[['gpa', 'gre']])[:,1] plt.scatter(data_test["gre"], fitted_test) plt.show()

Predictive Power

这里有个用法需要提一下，accuracy_train = (predicted == data_train[‘admit’]).mean()中predicted == data_train[‘admit’]得到是一个布尔型array,在计算mean()时，会将True记作1，False记作0，然后求均值。但是在list中是不行的，list对象的布尔型数据没有mean()这个函数。

# .predict() using a threshold of 0.50 by default predicted = logistic_model.predict(data_train[['gpa','gre']]) # The average of the binary array will give us the accuracy accuracy_train = (predicted == data_train['admit']).mean() # Print the accuracy print("Accuracy in Training Set = {s}".format(s=accuracy_train)) ''' # 这种输出方式也很好 Accuracy in Training Set = 0.7785714285714286 ''' # Percentage of those admitted percent_admitted = data_test["admit"].mean() * 100 # Predicted to be admitted predicted = logistic_model.predict(data_test[['gpa','gre']]) # What proportion of our predictions were true accuracy_test = (predicted == data_test['admit']).mean()

sklearn中的逻辑回归的阈值默认设置为0.5

Admissions ROC Curve

逻辑回归中的predict_proba这个函数返回的不是类标签，而是接受的概率，这可以允许我们自己修改阈值。首先我们需要作出它的ROC曲线来观察合适阈值：

from sklearn.metrics import roc_curve, roc_auc_score # Compute the probabilities predicted by the training and test set # predict_proba returns probabilies for each class. We want the second column train_probs = logistic_model.predict_proba(data_train[['gpa', 'gre']])[:,1] test_probs = logistic_model.predict_proba(data_test[['gpa', 'gre']])[:,1] # Compute auc for training set auc_train = roc_auc_score(data_train["admit"], train_probs) # Compute auc for test set auc_test = roc_auc_score(data_test["admit"], test_probs) # Difference in auc values auc_diff = auc_train - auc_test # Compute ROC Curves roc_train = roc_curve(data_train["admit"], train_probs) roc_test = roc_curve(data_test["admit"], test_probs) # Plot false positives by true positives plt.plot(roc_train[0], roc_train[1]) plt.plot(roc_test[0], roc_test[1])