DOTS支持的shader

Redis的基本使用方法和简介

  返回  

机器学习----逻辑回归

2021/8/21 14:09:11 浏览:

逻辑回归简介

logistic回归是一种广义的线性回归分析模型,以胃癌病情分析为例,选择两组人群,一组是胃癌组,一组是非胃癌组,两组人群必定具有不同的体征与生活方式等。因此因变量就为是否胃癌,值为“是”或“否”,自变量就可以包括很多了,如年龄、性别、饮食习惯、幽门螺杆菌感染等。

绘制sigmoid曲线
import numpy as np
import matplotlib.pyplot as plt
def sigmoid(t):
    return 1. / (1. + np.exp(-t))
x = np.linspace(-10, 10, 500)

plt.plot(x, sigmoid(x))
plt.show()

逻辑回归的损失函数

y=log(x)的函数图像如下图所示:

y=-log(x)的函数图像如下图所示:

由于p的取值只能是在0到1之间,故:

当p取值为0时,按之前的sigmoid函数定义,预测分类y应该为0,但此时真实分类y=1,而-log§趋于无穷大(损失函数大),很好的表达了预测得不准;当p取值为1时,按之前的sigmoid函数定义,预测分类y应该为1,此时真实分类y也等于1,-log§趋于0(损失函数小),很好的表达了预测得准确。

另外一种情况同理。

y=-log(-x)的函数图像如下图所示(与y=-log(x)关于y轴对称):

y=-log(1-x)的函数图像如下图所示(沿着x轴向右平移一个单位):

合成一个式子:

考虑多个样本:

损失函数求导

推导过程


向量化:

封装Logistics模型

在之前实现的线性回归的类的基础上改写:

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()"

实现逻辑回归

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

iris = datasets.load_iris()
X = iris.data
y = iris.target
X = X[y<2,:2]
y = y[y<2]
X.shape
(100, 2)
y.shape
(100,)
plt.scatter(X[y==0,0], X[y==0,1], color="red")
plt.scatter(X[y==1,0], X[y==1,1], color="blue")
plt.show()

使用逻辑回归

from playML.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, seed=666)
from playML.LogisticRegression import LogisticRegression

log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression()
log_reg.score(X_test, y_test)
1.0
log_reg.predict_proba(X_test)
array([ 0.92972035,  0.98664939,  0.14852024,  0.17601199,  0.0369836 ,
        0.0186637 ,  0.04936918,  0.99669244,  0.97993941,  0.74524655,
        0.04473194,  0.00339285,  0.26131273,  0.0369836 ,  0.84192923,
        0.79892262,  0.82890209,  0.32358166,  0.06535323,  0.20735334])
log_reg.predict(X_test)
array([1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0])
y_test
array([1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0])

决策边界

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

iris = datasets.load_iris()

X = iris.data
y = iris.target

X = X[y<2,:2]
y = y[y<2]
plt.scatter(X[y==0,0], X[y==0,1], color="red")
plt.scatter(X[y==1,0], X[y==1,1], color="blue")
plt.show()

from playML.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, seed=666)
from playML.LogisticRegression import LogisticRegression

log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression()
log_reg.coef_
array([ 3.01796521, -5.04447145])
log_reg.intercept_
-0.6937719272911228

在逻辑回归中,易得决策边界为 theta*X_b=0 的直线

如果只有两个特征值,则很容易通过公式画出逻辑回归的决策边界

def x2(x1):
    return (-log_reg.coef_[0] * x1 - log_reg.intercept_) / log_reg.coef_[1]
x1_plot = np.linspace(4, 8, 1000)
x2_plot = x2(x1_plot)
plt.scatter(X[y==0,0], X[y==0,1], color="red")
plt.scatter(X[y==1,0], X[y==1,1], color="blue")
plt.plot(x1_plot, x2_plot)
plt.show()

plt.scatter(X_test[y_test==0,0], X_test[y_test==0,1], color="red")
plt.scatter(X_test[y_test==1,0], X_test[y_test==1,1], color="blue")
plt.plot(x1_plot, x2_plot)
plt.show()

不规则的决策边界的绘制

def plot_decision_boundary(model, axis):
    # meshgrid网格矩阵
    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),
    )
  #  np.c_ 行拼接, ravel()高维矩阵平铺降成一维
    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, 7.5, 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.show()

kNN的决策边界

from sklearn.neighbors import KNeighborsClassifier

knn_clf = KNeighborsClassifier()
knn_clf.fit(X_train, y_train)
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
           metric_params=None, n_jobs=1, n_neighbors=5, p=2,
           weights='uniform')
knn_clf.score(X_test, y_test)
1.0
plot_decision_boundary(knn_clf, axis=[4, 7.5, 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.show()

knn_clf_all = KNeighborsClassifier()
knn_clf_all.fit(iris.data[:,:2], iris.target)
KNeighborsClassifier(algorithm='auto', leaf_size=30, metric='minkowski',
           metric_params=None, n_jobs=1, n_neighbors=5, p=2,
           weights='uniform')
plot_decision_boundary(knn_clf_all, axis=[4, 8, 1.5, 4.5])
plt.scatter(iris.data[iris.target==0,0], iris.data[iris.target==0,1])
plt.scatter(iris.data[iris.target==1,0], iris.data[iris.target==1,1])
plt.scatter(iris.data[iris.target==2,0], iris.data[iris.target==2,1])
plt.show()

image.png

knn_clf_all = KNeighborsClassifier(n_neighbors=50)
knn_clf_all.fit(iris.data[:,:2], iris.target)

plot_decision_boundary(knn_clf_all, axis=[4, 8, 1.5, 4.5])
plt.scatter(iris.data[iris.target==0,0], iris.data[iris.target==0,1])
plt.scatter(iris.data[iris.target==1,0], iris.data[iris.target==1,1])
plt.scatter(iris.data[iris.target==2,0], iris.data[iris.target==2,1])
plt.show()

多项式特征应用于逻辑回归

逻辑回归中添加多项式特征

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]**2)<1.5, dtype='int')
plt.scatter(X[y==0,0], X[y==0,1])
plt.scatter(X[y==1,0], X[y==1,1])
plt.show()

使用逻辑回归

from playML.LogisticRegression import LogisticRegression
log_reg = LogisticRegression()
log_reg.fit(X, y)
LogisticRegression()
log_reg.score(X, y)
0.60499999999999998
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())
    ])
poly_log_reg = PolynomialLogisticRegression(degree=2)
poly_log_reg.fit(X, y)
Pipeline(steps=[('poly', PolynomialFeatures(degree=2, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression())])
poly_log_reg.score(X, y)
0.94999999999999996
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()

poly_log_reg2 = PolynomialLogisticRegression(degree=20)
poly_log_reg2.fit(X, y)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression())])
plot_decision_boundary(poly_log_reg2, 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()

逻辑回归的模型正则化

scikit-learn中的逻辑回归

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()

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)

使用scikit-learn中的逻辑回归

from sklearn.linear_model import LogisticRegression

log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
          penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
          verbose=0, warm_start=False)
log_reg.score(X_train, y_train)
0.79333333333333333
log_reg.score(X_test, y_test)
0.85999999999999999
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())
    ])
poly_log_reg = PolynomialLogisticRegression(degree=2)
poly_log_reg.fit(X_train, y_train)
Pipeline(steps=[('poly', PolynomialFeatures(degree=2, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
          penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
          verbose=0, warm_start=False))])
poly_log_reg.score(X_train, y_train)
0.91333333333333333
poly_log_reg.score(X_test, y_test)
0.93999999999999995
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()

poly_log_reg2 = PolynomialLogisticRegression(degree=20)
poly_log_reg2.fit(X_train, y_train)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
          penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
          verbose=0, warm_start=False))])
poly_log_reg2.score(X_train, y_train)
0.93999999999999995
poly_log_reg2.score(X_test, y_test)
0.92000000000000004
plot_decision_boundary(poly_log_reg2, 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)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=0.1, class_weight=None, dual=False, fit_intercept=True,
          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
          penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
          verbose=0, warm_start=False))])
poly_log_reg3.score(X_train, y_train)
0.85333333333333339
poly_log_reg3.score(X_test, y_test)
0.92000000000000004
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)
Pipeline(steps=[('poly', PolynomialFeatures(degree=20, include_bias=True, interaction_only=False)), ('std_scaler', StandardScaler(copy=True, with_mean=True, with_std=True)), ('log_reg', LogisticRegression(C=0.1, class_weight=None, dual=False, fit_intercept=True,
          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
          penalty='l1', random_state=None, solver='liblinear', tol=0.0001,
          verbose=0, warm_start=False))])
poly_log_reg4.score(X_train, y_train)
0.82666666666666666
poly_log_reg4.score(X_test, y_test)
0.90000000000000002
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()

使用逻辑回归解决多分类问题

OvR 和 OvO

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

iris = datasets.load_iris()
X = iris.data[:,:2]
y = iris.target
from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=666)
from sklearn.linear_model import LogisticRegression

log_reg = LogisticRegression()
log_reg.fit(X_train, y_train)
LogisticRegression(C=1.0, class_weight=None, dual=False, fit_intercept=True,
          intercept_scaling=1, max_iter=100, multi_class='ovr', n_jobs=1,
          penalty='l2', random_state=None, solver='liblinear', tol=0.0001,
          verbose=0, warm_start=False)
log_reg.score(X_test, y_test)
0.65789473684210531
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()

log_reg2 = LogisticRegression(multi_class="multinomial", solver="newton-cg")
log_reg2.fit(X_train, y_train)
log_reg2.score(X_test, y_test)
0.78947368421052633
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)
0.94736842105263153
log_reg2 = LogisticRegression(multi_class="multinomial", solver="newton-cg")
log_reg2.fit(X_train, y_train)
log_reg2.score(X_test, y_test)
1.0

OvO and OvR

from sklearn.multiclass import OneVsRestClassifier

ovr = OneVsRestClassifier(log_reg)
ovr.fit(X_train, y_train)
ovr.score(X_test, y_test)
0.94736842105263153
from sklearn.multiclass import OneVsOneClassifier

ovo = OneVsOneClassifier(log_reg)
ovo.fit(X_train, y_train)
ovo.score(X_test, y_test)
1.0

联系我们

如果您对我们的服务有兴趣,请及时和我们联系!

服务热线:18288888888
座机:18288888888
传真:
邮箱:888888@qq.com
地址:郑州市文化路红专路93号