betway体育:《机械学习_07_03_svm_核函数与非线性支持向量机》

admin 3个月前 (12-18) 科技 67 1

一.简介

前两节划分实现了硬距离支持向量机与软距离支持向量机,它们本质上都是线性分类器,只是软距离对“异常点”加倍宽容,它们对形如如下的螺旋数据都没法举行优越分类,由于没法找到一个直线(超平面)能将其分离隔,必须使用曲线(超曲面)才气将其分开,而核技巧即是处置这类问题的一种常用手段。

import numpy as np
import matplotlib.pyplot as plt
import copy
import random
import os
os.chdir('../')
from ml_models import utils
from ml_models.svm import *
from sklearn import datasets
%matplotlib inline
data, target = datasets.make_moons(noise=0.01)
plt.scatter(data[:,0],data[:,1],c=target)
plt.show()

二.核技巧

核技巧简朴来说分为两步:
(1)将低维非线性可分数据\(x\),通过一个非线性映射函数\(\phi\),映射到一个新空间(高维度甚至是无限维空间);
(2)对新空间的数据\(\phi(x)\)训练线性分类器

好比如下的情形:

原始数据需要使用一个椭圆才气分离隔,但对原始数据施加一个非线性变换\(\phi:(x_1,x_2)->(x_1^2,x_2^2)\)变换后,在新空间中就可以线性分开了

行使核技巧后的SVM

以是,若是对原始数据施加一个映射,此时软距离SVM的对偶问题为:

\[\min_{\alpha} \frac{1}{2}\sum_{i=1}^N\sum_{j=1}^N\alpha_i\alpha_jy_iy_j\phi(x_i)^T\phi(x_j)-\sum_{i=1}^N\alpha_i\\ s.t.\sum_{i=1}^N\alpha_iy_i=0,\\ 0\leq\alpha_i\leq C,i=1,2,...,N \]

求解得最优\(\alpha_i^*\)后,SVM模子为:

\[f(x)=sign(\sum_{i=1}^N\alpha_iy_i\phi(x_i)^T\phi(x)+b^*) \]

三.核函数

考察一下上面公式,我们的目的其实是求解\(\phi(x_i)^T\phi(x_j)\),有没有一种函数让\((x_i,x_j)\)只在原始空间做盘算就到达\(\phi(x_i)^T\phi(x_j)\)的效果呢?有的,那就是核函数,即:

\[K(x_i,x_j)=\phi(x_i)^T\phi(x_j) \]

怎样的函数才气做核函数?

要成为核函数必须知足如下两点条件:

(1)对称性:\(K(x_i,x_j)=K(x_j,x_i)\)

(2)正定性:对随便的\(x_i,i=1,2,..,m\)\(K(x,z)\)对应的Gramm矩阵:

\[K=[K(x_i,x_j)]_{m\times m} \]

是半正定矩阵,这里的\(x_i\in\)可行域,并不要求一定要属于样本集

常见的核函数有哪些?

现在用的比较多的核函数有如下一些:

(1)多项式核函数:

\[K(x,z)=(x^Tz+1)^p \]

(2)高斯核函数:

\[K(x,z)=exp(-\frac{\mid\mid x-z\mid\mid^2}{2\sigma^2}) \]

显然,线性可分SVM中使用的是\(K(x,z)=x^Tz\)也是核函数

行使核函数后的SVM

行使核函数后,软距离SVM的对偶问题为:

\[\min_{\alpha} \frac{1}{2}\sum_{i=1}^N\sum_{j=1}^N\alpha_i\alpha_jy_iy_jK(x_i,x_j)-\sum_{i=1}^N\alpha_i\\ s.t.\sum_{i=1}^N\alpha_iy_i=0,\\ 0\leq\alpha_i\leq C,i=1,2,...,N \]

求解得最优\(\alpha_i^*\)后,SVM模子为:

\[f(x)=sign(\sum_{i=1}^N\alpha_iy_iK(x,x_i)+b^*) \]

四.代码实现

代码实现很简朴,就在软距离SVM的基础上将向量的内积盘算\(x^Tz\)替换为\(K(x,z)\)即可,首先界说一些核函数:

"""
该部门放到ml_model.kernel_functions中
"""

def linear():
    """
    线性核函数
    :return:linear function
    """

    def _linear(x, y):
        return np.dot(x, y)

    return _linear


def poly(p=2):
    """
    多项式核函数
    :param p:
    :return: poly function
    """

    def _poly(x, y):
        return np.power(np.dot(x, y) + 1, p)

    return _poly


def rbf(sigma=0.1):
    """
    径向基/高斯核函数
    :param sigma:
    :return:
    """

    def _rbf(x, y):
        np_x = np.asarray(x)
        if np_x.ndim <= 1:
            return np.exp((-1 * np.dot(x - y, x - y) / (2 * sigma * sigma)))
        else:
            return np.exp((-1 * np.multiply(x - y, x - y).sum(axis=1) / (2 * sigma * sigma)))

    return _rbf
from ml_models import kernel_functions


class SVC(object):
    def __init__(self, epochs=100, C=1.0, tol=1e-3, kernel=None, degree=3, gamma=0.1):
        """
        :param epochs: 迭代次数上限
        :param C: C越小,对于误分类的责罚越小
        :param tol:提前中止训练时的误差值上限,制止迭代太久
        :param kernel:核函数
        :param degree:kernel='poly'时生效
        :param gamma:kernel='rbf'时生效
        """
        self.b = None
        self.alpha = None
        self.E = None
        self.epochs = epochs
        self.C = C
        self.tol = tol
        # 界说核函数
        if kernel is None:
            self.kernel_function = kernel_functions.linear()
        elif kernel == 'poly':
            self.kernel_function = kernel_functions.poly(degree)
        elif kernel == 'rbf':
            self.kernel_function = kernel_functions.rbf(gamma)
        else:
            self.kernel_function = kernel_functions.linear()
        # 纪录支持向量
        self.support_vectors = None
        # 纪录支持向量的x
        self.support_vector_x = []
        # 纪录支持向量的y
        self.support_vector_y = []
        # 纪录支持向量的alpha
        self.support_vector_alpha = []

    def f(self, x):
        """
        :param x:
        :return: wx+b
        """
        x_np = np.asarray(x)
        if len(self.support_vector_x) == 0:
            if x_np.ndim <= 1:
                return 0
            else:
                return np.zeros((x_np.shape[:-1]))
        else:
            if x_np.ndim <= 1:
                wx = 0
            else:
                wx = np.zeros((x_np.shape[:-1]))
            for i in range(0, len(self.support_vector_x)):
                wx += self.kernel_function(x, self.support_vector_x[i]) * self.support_vector_alpha[i] * \
                      self.support_vector_y[i]
            return wx + self.b

    def init_params(self, X, y):
        """
        :param X: (n_samples,n_features)
        :param y: (n_samples,) y_i\in\{0,1\}
        :return:
        """
        n_samples, n_features = X.shape
        self.b = .0
        self.alpha = np.zeros(n_samples)
        self.E = np.zeros(n_samples)
        # 初始化E
        for i in range(0, n_samples):
            self.E[i] = self.f(X[i, :]) - y[i]

    def _select_j(self, best_i):
        """
        选择j
        :param best_i:
        :return:
        """
        valid_j_list = [i for i in range(0, len(self.alpha)) if self.alpha[i] > 0 and i != best_i]
        best_j = -1
        # 优先选择使得|E_i-E_j|最大的j
        if len(valid_j_list) > 0:
            max_e = 0
            for j in valid_j_list:
                current_e = np.abs(self.E[best_i] - self.E[j])
                if current_e > max_e:
                    best_j = j
                    max_e = current_e
        else:
            # 随机选择
            l = list(range(len(self.alpha)))
            seq = l[: best_i] + l[best_i + 1:]
            best_j = random.choice(seq)
        return best_j

    def _meet_kkt(self, x_i, y_i, alpha_i):
        """
        判断是否知足KKT条件

        :param w:
        :param b:
        :param x_i:
        :param y_i:
        :return:
        """
        if alpha_i < self.C:
            return y_i * self.f(x_i) >= 1 - self.tol
        else:
            return y_i * self.f(x_i) <= 1 + self.tol

    def fit(self, X, y2, show_train_process=False):
        """

        :param X:
        :param y2:
        :param show_train_process: 显示训练历程
        :return:
        """
        y = copy.deepcopy(y2)
        y[y == 0] = -1
        # 初始化参数
        self.init_params(X, y)
        for _ in range(0, self.epochs):
            if_all_match_kkt = True
            for i in range(0, len(self.alpha)):
                x_i = X[i, :]
                y_i = y[i]
                alpha_i_old = self.alpha[i]
                E_i_old = self.E[i]
                # 外层循环:选择违反KKT条件的点i
                if not self._meet_kkt(x_i, y_i, alpha_i_old):
                    if_all_match_kkt = False
                    # 内层循环,选择使|Ei-Ej|最大的点j
                    best_j = self._select_j(i)

                    alpha_j_old = self.alpha[best_j]
                    x_j = X[best_j, :]
                    y_j = y[best_j]
                    E_j_old = self.E[best_j]

                    # 举行更新
                    # 1.首先获取无裁剪的最优alpha_2
                    eta = self.kernel_function(x_i, x_i) + self.kernel_function(x_j, x_j) - 2.0 * self.kernel_function(
                        x_i, x_j)
                    # 若是x_i和x_j很靠近,则跳过
                    if eta < 1e-3:
                        continue
                    alpha_j_unc = alpha_j_old + y_j * (E_i_old - E_j_old) / eta
                    # 2.裁剪并获得new alpha_2
                    if y_i == y_j:
                        L = max(0., alpha_i_old + alpha_j_old - self.C)
                        H = min(self.C, alpha_i_old + alpha_j_old)
                    else:
                        L = max(0, alpha_j_old - alpha_i_old)
                        H = min(self.C, self.C + alpha_j_old - alpha_i_old)

                    if alpha_j_unc < L:
                        alpha_j_new = L
                    elif alpha_j_unc > H:
                        alpha_j_new = H
                    else:
                        alpha_j_new = alpha_j_unc

                    # 若是转变不够大则跳过
                    if np.abs(alpha_j_new - alpha_j_old) < 1e-5:
                        continue
                    # 3.获得alpha_1_new
                    alpha_i_new = alpha_i_old + y_i * y_j * (alpha_j_old - alpha_j_new)
                    # 5.更新alpha_1,alpha_2
                    self.alpha[i] = alpha_i_new
                    self.alpha[best_j] = alpha_j_new
                    # 6.更新b
                    b_i_new = y_i - self.f(x_i) + self.b
                    b_j_new = y_j - self.f(x_j) + self.b
                    if self.C > alpha_i_new > 0:
                        self.b = b_i_new
                    elif self.C > alpha_j_new > 0:
                        self.b = b_j_new
                    else:
                        self.b = (b_i_new + b_j_new) / 2.0
                    # 7.更新E
                    for k in range(0, len(self.E)):
                        self.E[k] = self.f(X[k, :]) - y[k]

                    # 8.更新支持向量相关的信息
                    self.support_vectors = np.where(self.alpha > 1e-3)[0]
                    self.support_vector_x = [X[i, :] for i in self.support_vectors]
                    self.support_vector_y = [y[i] for i in self.support_vectors]
                    self.support_vector_alpha = [self.alpha[i] for i in self.support_vectors]

                    # 显示训练历程
                    if show_train_process is True:
                        utils.plot_decision_function(X, y2, self, [i, best_j])
                        utils.plt.pause(0.1)
                        utils.plt.clf()

            # 若是所有的点都知足KKT条件,则中止
            if if_all_match_kkt is True:
                break

        # 显示最终效果
        if show_train_process is True:
            utils.plot_decision_function(X, y2, self, self.support_vectors)
            utils.plt.show()

    def get_params(self):
        """
        输出原始的系数
        :return: w
        """

        return self.w, self.b

    def predict_proba(self, x):
        """
        :param x:ndarray花样数据: m x n
        :return: m x 1
        """
        return utils.sigmoid(self.f(x))

    def predict(self, x):
        """
        :param x:ndarray花样数据: m x n
        :return: m x 1
        """
        proba = self.predict_proba(x)
        return (proba >= 0.5).astype(int)

五.查看效果

#查看rbf的效果
svm = SVC(C=3.0, kernel='rbf',gamma=0.1, epochs=10, tol=0.2)
svm.fit(data, target)
utils.plot_decision_function(data, target, svm, svm.support_vectors)

#查看poly的效果
svm = SVC(C=3.0, kernel='poly',degree=3, epochs=10, tol=0.2)
svm.fit(data, target)
utils.plot_decision_function(data, target, svm, svm.support_vectors)

betway体育:《机械学习_07_03_svm_核函数与非线性支持向量机》 第1张

六.问题讨论

1.RBF函数中\(\sigma\)的差别取值对训练的影响

为了探索该问题,我们对\(\sigma\)从小到大取一组数,在另外一个伪数据上查看效果

from sklearn.datasets import make_classification
data, target = make_classification(n_samples=100, n_features=2, n_classes=2, n_informative=1, n_redundant=0,
                                   n_repeated=0, n_clusters_per_class=1, class_sep=.5,random_state=21)
c1 = SVC(C=3.0, kernel='rbf',gamma=0.1, epochs=10, tol=0.01)
c1.fit(data, target)
c2 = SVC(C=3.0, kernel='rbf',gamma=0.5, epochs=10, tol=0.01)
c2.fit(data, target)
c3 = SVC(C=3.0, kernel='rbf',gamma=2, epochs=10, tol=0.01)
c3.fit(data, target)
plt.figure(figsize=(16,4))
plt.subplot(1,3,1)
utils.plot_decision_function(data,target,c1)
plt.subplot(1,3,2)
utils.plot_decision_function(data,target,c2)
plt.subplot(1,3,3)
utils.plot_decision_function(data,target,c3)

betway体育:《机械学习_07_03_svm_核函数与非线性支持向量机》 第2张

上面\(\sigma\)划分取值\([0.1,0.5,2]\),通过效果可以简朴总结如下:

(1)若是\(\sigma\)取值越小,SVM越能捉住个体样本的信息,越容易过拟合;

(2)\(\sigma\)取值越大SVM的泛化能力越强

若何对该效果举行明白呢?可以通过样本点在映射空间的距离来看,对随便两个样本点\(x,z\),它们在映射空间中的距离的平方可以示意如下:

\[||\phi(x)-\phi(z)||^2=(\phi(x)-\phi(z))^T(\phi(x)-\phi(z))\\ =\phi(x)^T\phi(x)+\phi(z)^T\phi(z)-2\phi(x)^T\phi(z)\\ =K(x,x)+K(z,z)-2K(x,z)\\ =2-2\cdot exp(-\frac{\mid\mid x-z\mid\mid^2}{2\sigma^2})(将K(x,z)替换为RBF函数) \]

以是:

(1)若是\(\sigma\rightarrow 0\),那么\(-\frac{\mid\mid x-z\mid\mid^2}{2\sigma^2}\rightarrow -\infty\),那么\(exp(-\frac{\mid\mid x-z\mid\mid^2}{2\sigma^2})\rightarrow 0\),那么\(||\phi(x)-\phi(z)||\rightarrow \sqrt 2\)

(2)若是\(\sigma\rightarrow \infty\),那么\(-\frac{\mid\mid x-z\mid\mid^2}{2\sigma^2}\rightarrow 0\),那么\(exp(-\frac{\mid\mid x-z\mid\mid^2}{2\sigma^2})\rightarrow 1\),那么\(||\phi(x)-\phi(z)||\rightarrow 0\)

我们可以验证上面的总结,若\(\sigma\)取值越小,样本点在映射空间越涣散,则在高维空间越容易线性可分,表现在低维空间则越容易过拟合;\(\sigma\)取值越大,样本点在映射空间越集中,越不易线性可分,表现在低维空间也是不易线性可分

2.若何明白RBF可将数据映射到无限维空间

原谅自己,这部门公式不想码了,具体内容参考大神的知乎帖子>>>,其中主要需要用到两个等式变换:

(1)指数函数的泰勒级数:\(e^x=\sum_{n=1}^{\infty}\frac{x^n}{n!}\),将RBF函数举行睁开;

(2)行使多项式睁开定理,将样本\(x\)\(z\)在原始空间的内积的\(n\)次方举行睁开,如果\(x,z\in R^k\),那么:

\[(x^Tz)^n=(\sum_{i=1}^kx_iz_i)^n\\ =\sum_{l=1}^L\frac{n!}{n_{l_1}!n_{l_2}!\cdots n_{l_k}!}(x_1z_1)^{n_{l_1}}(x_2z_2)^{n_{l_2}}\cdots (x_kz_k)^{n_{l_k}} \]

这里,\(\sum_{i=1}^kn_{l_i}=n\)\(L=\frac{(n+k-1)!}{n!(k-1)!}\),进一步的,上面等式可以化简为形如这样的表达式:\(\Phi(x)^T\Phi(z)\)\(\Phi(x)=[\Phi_1(x),\Phi_2(x),\cdots ,\Phi_L(x)]\)

,

Sunbet

Sunbet www.6358917.com Sunbet(www.sunbet.in)是进入Sunbet的主力站点。Sunbet开放Sunbet会员开户网址、Sunbet代理开户、Sunbet手机版下载、Sunbet电脑客户端下载等业务。

allbet声明:该文看法仅代表作者自己,与本平台无关。转载请注明:betway体育:《机械学习_07_03_svm_核函数与非线性支持向量机》

网友评论

  • (*)

最新评论

  • Allbet 2020-12-18 00:03:01 回复

    Allbetwww.allbet6.com欢迎进入欧搏平台(Allbet Game),欧搏平台开放欧搏Allbet开户、欧搏Allbet代理开户、欧搏Allbet电脑客户端、欧搏AllbetAPP下载等业务。不错的,能写成这样

    1

标签列表

    文章归档

      站点信息

      • 文章总数:981
      • 页面总数:0
      • 分类总数:8
      • 标签总数:1244
      • 评论总数:853
      • 浏览总数:59609