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第9章 逻辑回归 学习笔记 下

逻辑笔记学习 回归
2023-09-27 14:25:50 时间

目录

9-6 在逻辑回归中使用多项式特征

逻辑回归中添加多项式特征

使用逻辑回归

9-7 scikit-learn中的逻辑回归

使用scikit-learn中的逻辑回归

9-8 OvR与OvO

使用鸢尾花部分数据

使用所有的数据

kNN的决策边界

LogisticRegressionCV

使用逻辑回归处理 MNIST 数据集

PCA

Standardization

Logistic Regression

 

9-6 在逻辑回归中使用多项式特征

直线太简单了

x1方是一个特征，x2方是一个特征,相关于学习到了特征前面的系数为1，当然可系数也可以不为零，则是椭圆，可以是曲线也可添加x1x2则圆不一定在坐标原点，则可能在其它地方。

逻辑回归中添加多项式特征

y这样是一个布尔向量，np.array将其转化为0，1的向量

 

使用逻辑回归

def plot_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]

    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
plot_decision_boundary(log_reg, axis=[-4, 4, -4, 4])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

 

逻辑回归是用一条直线来分割

LogisticRegression.py

import numpy as np
from .metrics import accuracy_score

class LogisticRegression:

    def __init__(self):
        """初始化Logistic Regression模型"""
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def _sigmoid(self, t):
        return 1. / (1. + np.exp(-t))

    def fit(self, X_train, y_train, eta=0.01, n_iters=1e4):
        """根据训练数据集X_train, y_train, 使用梯度下降法训练Logistic Regression模型"""
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        def J(theta, X_b, y):
            y_hat = self._sigmoid(X_b.dot(theta))
            try:
                return - np.sum(y*np.log(y_hat) + (1-y)*np.log(1-y_hat)) / len(y)
            except:
                return float('inf')

        def dJ(theta, X_b, y):
            return X_b.T.dot(self._sigmoid(X_b.dot(theta)) - y) / len(y)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):

            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break

                cur_iter += 1

            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1])
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def predict_proba(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果概率向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return self._sigmoid(X_b.dot(self._theta))

    def predict(self, X_predict):
        """给定待预测数据集X_predict，返回表示X_predict的结果向量"""
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"

        proba = self.predict_proba(X_predict)
        return np.array(proba >= 0.5, dtype='int')

    def score(self, X_test, y_test):
        """根据测试数据集 X_test 和 y_test 确定当前模型的准确度"""

        y_predict = self.predict(X_test)
        return accuracy_score(y_test, y_predict)

    def __repr__(self):
        return "LogisticRegression()"

LogisticRegression()是自己写的模块

from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler

def PolynomialLogisticRegression(degree):
    return Pipeline([
        ('poly', PolynomialFeatures(degree=degree)),
        ('std_scaler', StandardScaler()),
        ('log_reg', LogisticRegression())
    ])
plot_decision_boundary(poly_log_reg, axis=[-4, 4, -4, 4])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

边界奇怪

 

 

9-7 scikit-learn中的逻辑回归

C太关注减小J，否则L正则项就要小

平衡这两项的关系

L前的系统不能为零ｓｃｉｋｉｔｌｅａｒｎ中

强制加入噪音２０个

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(666)
X = np.random.normal(0, 1, size=(200, 2))
y = np.array((X[:,0]**2+X[:,1])<1.5, dtype='int')
for _ in range(20):
    y[np.random.randint(200)] = 1

plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

使用scikit-learn中的逻辑回归

def plot_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]

    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)




plot_decision_boundary(log_reg, axis=[-4, 4, -4, 4])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import StandardScaler

def PolynomialLogisticRegression(degree):
    return Pipeline([
        ('poly', PolynomialFeatures(degree=degree)),
        ('std_scaler', StandardScaler()),
        ('log_reg', LogisticRegression())
    ])




plot_decision_boundary(poly_log_reg, axis=[-4, 4, -4, 4])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

def PolynomialLogisticRegression(degree, C):
    return Pipeline([
        ('poly', PolynomialFeatures(degree=degree)),
        ('std_scaler', StandardScaler()),
        ('log_reg', LogisticRegression(C=C))
    ])

poly_log_reg3 = PolynomialLogisticRegression(degree=20, C=0.1)
poly_log_reg3.fit(X_train, y_train)
plot_decision_boundary(poly_log_reg3, axis=[-4, 4, -4, 4])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

def PolynomialLogisticRegression(degree, C, penalty='l2'):
    return Pipeline([
        ('poly', PolynomialFeatures(degree=degree)),
        ('std_scaler', StandardScaler()),
        ('log_reg', LogisticRegression(C=C, penalty=penalty))
    ])

poly_log_reg4 = PolynomialLogisticRegression(degree=20, C=0.1, penalty='l1')
poly_log_reg4.fit(X_train, y_train)
plot_decision_boundary(poly_log_reg4, axis=[-4, 4, -4, 4])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

9-8 OvR与OvO

逻辑回归解决二分类问题，上面只能解决两分类，改造后可以实现多分类

ｏｎｅ　ｖｓ　Ｒｅｓｔ／Ａｌｌ

红色与其它区分开

逻辑回归就是概率最大的那个就是哪个类别

One vs One

每次只选两个类别

以次类推C42

投票后看哪个高，这种方法比ovr用的时间多，但其分类比较准确

使用鸢尾花部分数据

def plot_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]

    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)
plot_decision_boundary(log_reg, axis=[4, 8.5, 1.5, 4.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.scatter(X[y==2,0], X[y==2,1])
plt.show()

scklearn中默认支持多分类，采用ovr

 

正确调用OVO对应multinomial，同时solver要修改

log_reg2 = LogisticRegression(multi_class="multinomial", solver="newton-cg")
log_reg2.fit(X_train, y_train)
log_reg2.score(X_test, y_test)
plot_decision_boundary(log_reg2, axis=[4, 8.5, 1.5, 4.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.scatter(X[y==2,0], X[y==2,1])
plt.show()

使用所有的数据

 

X = iris.data
y = iris.target

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)
log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
log_reg.score(X_test, y_test)
log_reg2 = LogisticRegression(multi_class="multinomial", solver="newton-cg")
log_reg2.fit(X_train, y_train)
log_reg2.score(X_test, y_test)

 

有ovr,ovo这两类，可以于对应任意的二分类器都可以采用，从而支持多分类问题

 

from sklearn.multiclass import OneVsRestClassifier

ovr = OneVsRestClassifier(log_reg)
ovr.fit(X_train, y_train)
ovr.score(X_test, y_test)
from sklearn.multiclass import OneVsOneClassifier

ovo = OneVsOneClassifier(log_reg)
ovo.fit(X_train, y_train)
ovo.score(X_test, y_test)

kNN的决策边界

import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets

iris = datasets.load_iris()

X = iris.data[:,:2]
y = iris.target
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.scatter(X[y==2,0], X[y==2,1])
plt.show()
def plot_decision_boundary(model, axis):
    
    x0, x1 = np.meshgrid(
        np.linspace(axis[0], axis[1], int((axis[1]-axis[0])*100)).reshape(-1, 1),
        np.linspace(axis[2], axis[3], int((axis[3]-axis[2])*100)).reshape(-1, 1),
    )
    X_new = np.c_[x0.ravel(), x1.ravel()]

    y_predict = model.predict(X_new)
    zz = y_predict.reshape(x0.shape)

    from matplotlib.colors import ListedColormap
    custom_cmap = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])
    
    plt.contourf(x0, x1, zz, linewidth=5, cmap=custom_cmap)

from sklearn.neighbors import KNeighborsClassifier

knn_clf_1 = KNeighborsClassifier(n_neighbors=1)
knn_clf_1.fit(X, y)

plot_decision_boundary(knn_clf_1, axis=[4, 8, 1.5, 4.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.scatter(X[y==2,0], X[y==2,1])
plt.show()

 

knn_clf_5 = KNeighborsClassifier(n_neighbors=5)
knn_clf_5.fit(X, y)

plot_decision_boundary(knn_clf_5, axis=[4, 8, 1.5, 4.5])
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.scatter(X[y==2,0], X[y==2,1])
plt.show()

 

LogisticRegressionCV

使用逻辑回归处理 MNIST 数据集

PCA

 

Standardization

from sklearn.preprocessing import StandardScaler

std_scaler = StandardScaler()
std_scaler.fit(X_train_reduction, y_train)
X_train_std = std_scaler.transform(X_train_reduction)
X_test_std = std_scaler.transform(X_test_reduction)

 

Logistic Regression

from sklearn.linear_model import LogisticRegression

log_clf = LogisticRegression(solver="newton-cg")
%time log_clf.fit(X_train_std, y_train)

 

 

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