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支持向量机(Support Vector Machine)

不适定问题不止一个决策边界

要找一个决策边界，不仅能将训练集很好的划分，而且提升模型的泛化能力
支持向量机直接将算法放在运行的内部，在不适定的问题中，使用svm去建模是好的
svm是统计学习中非常重要的方法
svm尝试寻找一个最优的决策边界，距离两个类别的最近的样本最远，距离决策边界最近的点称为支撑向量
svm算法要做的就是最大化margin，也就是要找到最大的d

margin=2d

解析几何，点到直线的距离

(x,y)到Ax+By+C=0的距离A2+B2​∣Ax+By+C∣​
扩展到n维空间θTxb​=0-->wTx+b=0-->∣∣w∣∣wT+b​
∣∣w∣∣=w12​+w22​+...+wn2​
​
∣∣w∣∣wTxi+b​≥d 其中∀yi=1
∣∣w∣∣wTxi+b​≤−d 其中∀yi=−1
∣∣w∣∣dwTxi+b​≥1 其中∀yi=1
∣∣w∣∣dwTxi+b​≤−1 其中∀yi=−1
wdT​xi+bd​≥1其中∀yi=1
wdT​xi+bd​≤−1其中∀yi=−1

yi(wTxi+b)≥1
对于任意支撑向量x有 max∣∣w∣∣∣wTx+b∣​-->max∣∣w∣∣1​-->min21​∣∣w∣∣2

Soft Margin和SVM的正则化

Soft Margin SVM yi(wTxi+b)≥1−ζ
ζi​≥0
min21​∣∣w∣∣2+C⋅Σi=1m​ζi​
C为超参数，平衡两部分的重要程度

使用SVM需要对数据进行标准化处理
对于SVM上，数据尺度不同，需要对数据进行标准化处理

什么是核函数

svm的本质：其实就是求解最优化问题，求解最优化过程中，需要变形为数学中最好解决的问题

核函数思想：K(xi,xk)=xi,xj
K(x,y)=(x⋅y+1)2

K(x,y)=(i=1∑n​x⋅y+1)2=i=1∑n​(xi2​)(yi2​)+i=2∑n​j=1∑i−1​(2​xi​xj​)(2​yi​yj​)+i=1∑n​(2​xi​)(2​yi​)+1

核函数会降低原算法的复杂度

K(x,y)=(x⋅y+c)d

d就是函数里的degree

高斯核函数

K(x,y)表示x和y的点乘

K(x,y)=e−γ∣∣x−y∣∣2

正态分布就是高斯函数 g(x)=σ2π
​1​e−21​(σx−μ​)2


RBF核 Radial Basis Function Kernel
将每一个样本点映射到一个无穷维的特征空间
多项式特征依靠升维是的原本线性不可分的数据线性可分

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 sklearn.preprocessing import StandardScaler

standardScaler = StandardScaler()

standardScaler.fit(X)

X_standard = standardScaler.transform(X)

from sklearn.svm import LinearSVC

#SVC使用向量做分类

svc = LinearSVC(C=1e9)

svc.fit(X_standard, y)

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_camp = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])

    plt.contourf(x0, x1, zz, linewith=5, camp=custom_camp)

    w = model.coef_[0]

    b = model.intercept_[0]

    plot_x = np.linspace(axis[0], axis[1], 200)

    up_y = -w[0]/w[1]*plot_x-b/w[1]+1/w[1]

    down_y = -w[0]/w[1]*plot_x-b/w[1]-1/w[1]

    up_index = (up_y>=axis[2])&(up_y<=axis[3])

    down_index = (down_y>=axis[2])&(down_y<=axis[3])

    plt.plot(plot_x[up_index],up_y[up_index],color='black')

    plt.plot(plot_x[down_index],down_y[down_index],col

plot_decision_boundary(svc, axis=[-3,3,-3,3])

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

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

plt.show()

               

svc2 = LinearSVC(C=0.01)

svc2.fit(X_standard, y)

plot_decision_boundary(svc2, axis=[-3,3,-3,3])

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

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

plt.show()


             

#多项式处理非线性问题

import numpy as npimport matplotlib.pyplot as plt

from sklearn import datasets

X, y = datasets.make_moons()

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



#构建数据集

X,y = datasets.make_moons(noise=0.15, random_state=666)

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, StandardScaler

from sklearn.svm import LinearSVC

from sklearn.pipeline import Pipeline

def PolynomialSVC(degree,C=1.0):

    return Pipeline([

        ("poly", PolynomialFeatures(degree=degree)),

        ("std_scaler", StandardScaler()),

        ("linearSVC", LinearSVC(C=C)),

    ])

poly_svc = PolynomialSVC(degree=3)

poly_svc.fit(X,y)

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_camp = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])

plt.contourf(x0, x1, zz, linewith=5, camp=custom_camp)

plot_decision_boundary(poly_svc,axis=[-1.5,2.5,-1.0,1.5])

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

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

plt.show()



#多项式核函数

from sklearn.svm import SVC

def PolynomialKernelSVC(degree, C=1.0):

    return Pipeline([

        ("std_scaler", StandardScaler()),

        ("kernelSVC", SVC(kernel='poly',degree=degree,C=C)),

    ])

poly_kernel_svc = PolynomialKernelSVC(degree=3)

poly_kernel_svc.fit(X,y)

plot_decision_boundary(poly_kernel_svc,axis=[-1.5,2.5,-1.0,1.5])

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

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

plt.show()

scikit-learn的高斯核函数

K(x,y)=e^{-\gamma||x-y||}^2

gamma越大，高斯分布越窄
gamma越小，高斯分布越宽

import numpy as np

import matplotlib.pyplot as plt

from sklearn import datasets

X, y = datasets.make_moons(noise=0.15, random_state=666)

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 StandardScaler

from sklearn.svm import SVC

from sklearn.pipeline import Pipeline

def RBFkernelSVC(gamma=1.0):

    return Pipeline([

        ("std_scaler", StandardScaler()),

        ("svc", SVC(kernel='rbf', gamma=gamma)),

    ])

svc = RBFkernelSVC(gamma=1.0)

svc.fit(X, y)

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_camp = ListedColormap(['#EF9A9A','#FFF59D','#90CAF9'])

    plt.contourf(x0, x1, zz, linewith=5,camp=custom_camp)

plot_decision_boundary(svc, axis=[-1.5, 2.5, -1.0, 1.5])

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

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

plt.show()



svc_gamma100 = RBFkernelSVC(gamma=100)

svc_gamma100.fit(X,y)

plot_decision_boundary(svc_gamma100, axis=[-1.5, 2.5, -1.0, 1.5])

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

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

plt.show()

#决策边界的这一类，其中在周围都形成了中型的图案



svc_gamma10 = RBFkernelSVC(gamma=10)

svc_gamma10.fit(X,y)

plot_decision_boundary(svc_gamma10, axis=[-1.5, 2.5, -1.0, 1.5])

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

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

plt.show()



svc_gamma03 = RBFkernelSVC(gamma=0.3)

svc_gamma03.fit(X,y)

plot_decision_boundary(svc_gamma03, axis=[-1.5, 2.5, -1.0, 1.5])

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

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

plt.show()

SVM思路解决回归问题

使用margin的方式，类似于soft margin svm方式
就是找两个距离余量，需要留着距离之间的距离

import numpy as np

import matplotlib.pyplot as plt

from sklearn import datasets

boston = datasets.load_boston()

X = boston.data

y = boston.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.svm import LinearSVR

from sklearn.svm import SVR

from sklearn.preprocessing import StandardScaler

from sklearn.pipeline import Pipeline

def SrandardlinearSVR(epsilon=0.1):

    return Pipeline([

        ("std_scaler", StandardScaler()),

        ("linearSVR", LinearSVR(epsilon=epsilon))

    ])

svr = SrandardlinearSVR()

svr.fit(X_train, y_train)

svr.score(X_test, y_test)

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