python与R语言手推logistic回归（梯度下降法/牛顿法）

logistic回归主要用于分类问题中，遇到k分类问题时则转化为k个二分类问题即可。
logistic回归是将logit曲线套用在解释变量线性组合上，利用极大似然法进行参数估计，将似然函数（二项分布交叉熵）作为目标函数，利用最优化方法（牛顿法、梯度下降法）进行求解。

from sklearn import datasets
from sklearn.model_selection import train_test_split
iris = datasets.load_iris()
X = iris.data
y = iris.target
X = X[y != 2]
y = y[y != 2]
xtrain,xtest,ytrain,ytest=train_test_split(
    X, y, test_size=0.3, random_state=42)

tip:由于exp(x)呈现指数级增长，易导致float溢出，可以对x范围进行控制防止溢出。

def sigmoid(z):
  # #防止溢出在RuntimeWarning: overflow encountered in exp
    return 1 / (1.0 + np.exp(-np.clip(z,-100,10000)))
def f(x,w):#x为n*k w为k*1
    return sigmoid(x@w )
def predict(x,w):
    return np.round(f(x, w))

#损失函数为两个伯努利分布的交叉熵由极大似然估计进行推导
def cross_entropy_loss(y_pred, y_label):
    cross_loss=-np.dot(y_label,np.log(y_pred))-np.dot(np.log(1-y_label),1-y_pred)
    return cross_loss

def gradient(x, y, w):
    y_pred=predict(x,w)
    w_grad=np.matmul(x.T,y_pred-y_label)
    return w_grad
#随机梯度下降进行迭代
def training(x,y_label,alpha):
#     x=x.reset_index(drop=True)
#     y_label=y_label.reset_index(drop=True)
    dim=x.shape[1]
    w = np.random.rand(dim, 1)
    for i in range(1000):
        index=np.random.randint(0,len(y_label)-1)
#         y_pred=f(np.array(x.iloc[index,:],ndmin=2),w)
#         gradient=np.array(x.iloc[index,:],ndmin=2).T@(y_pred-y_label.iloc[index])
        y_pred=f(np.array(x[index,:],ndmin=2),w)
        gradient=np.array(x[index,:],ndmin=2).T@(y_pred-y_label[index])
        w-=alpha*gradient
    return w

w=training(xtrain,ytrain,0.001)
y_train_pred=predict(xtrain,w)
y_test_pred=predict(xtest,w)

from sklearn.metrics import classification_report,confusion_matrix
print(classification_report(ytrain, y_train_pred)) 
print(classification_report(ytest, y_test_pred)) 
print(confusion_matrix(ytrain, y_train_pred)) 
print(confusion_matrix(ytest, y_test_pred)) 

输出结果：
precision recall f1-score support

       0       1.00      1.00      1.00        33
       1       1.00      1.00      1.00        37

accuracy                           1.00        70

macro avg 1.00 1.00 1.00 70
weighted avg 1.00 1.00 1.00 70

          precision    recall  f1-score   support

       0       1.00      1.00      1.00        17
       1       1.00      1.00      1.00        13

accuracy                           1.00        30

macro avg 1.00 1.00 1.00 30
weighted avg 1.00 1.00 1.00 30

[[33 0]
[ 0 37]]
[[17 0]
[ 0 13]]