大学生录取预测——逻辑回归

Dataset

每年高中生和大学生都会申请进入到各种各样的高校和事业单位中去。每个学生都有一组独一无二的考试分数,成绩,和背景。录取委员会根据这个决定接受或拒绝这些申请者。在这种情况下一个二进制分类算法可用于接受或拒绝申请。逻辑回归是一个合适的方法,我们将在这个任务中解决这个问题

• 数据集admissions.csv包含了1000个申请者的信息，特征如下：

gre - Graduate Record Exam（研究生入学考试）, a generalized test for prospective graduate students（一个通用的测试未来的研究生）, continuous between 200 and 800.
gpa - Cumulative grade point average（累积平均绩点）, continuous between 0.0 and 4.0.
admit - Binary variable, 0 or 1, where 1 means the applicant was admitted to the program.

Use Linear Regression To Predict Admission

• 这是原本的数据，admit的值是0或者1

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，发现取值范围较大，甚至有负值，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]，然后我们根据银行信用卡批准——模型评估ROC&AUC这篇文章的方法来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])

可以看到ROC曲线开始非常的陡峭，慢慢地变得平缓。测试集的AUC值是0.79小于训练集的AUC值0.82.这些迹象表明我们的模型可以根据gre和gpa来预测是否录取了。