二元高斯(Gauss)混合分布模型：γ1∼(π1,μ1,σ12)\gamma _1 \sim (\pi_1,\mu _1,\sigma _1^2)，γ2∼(π2,μ2,σ22)\gamma _2 \sim (\pi_2,\mu _2,\sigma _2^2) 其中π1\pi_1，π2\pi_2代表两个分布的比例 故π1+π2=1\pi_1+\pi_2=1
现有NN个数据x1,x2,⋯ ,xNx_1,x_2 ,\cdots ,x_N,试估计参数(π1,μ1,σ1;π2,μ2,σ2\pi_1,\mu _1,\sigma _1;\pi_2,\mu _2,\sigma _2)

1.公式推导

对于一维正态分布的概率公式为

P(x;μ,σ2)=12πσe−(x−μ)22σ2P(x;\mu ,\sigma ^2)=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^2}{2\sigma^2}}

  1. E步
    借助贝叶斯公式
    数据xix_i属于γ1\gamma _1分布的概率为

    γi1=P(γ1∣xi)=P(xi∣γ1)P(γ1)P(xi∣γ1)P(γ1)+P(xi∣γ2)P(γ2)\gamma _{i1}=P(\gamma _1|x_i )= \frac{P(x_i|\gamma _1)P(\gamma _1)}{P(x_i|\gamma _1)P(\gamma _1)+P(x_i|\gamma _2)P(\gamma _2)}

    数据xix_i属于γ2\gamma _2分布的概率为

    γi2=P(γ2∣xi)=P(xi∣γ2)P(γ2)P(xi∣γ1)P(γ1)+P(xi∣γ2)P(γ2)\gamma _{i2}=P(\gamma _2|x_i )= \frac{P(x_i|\gamma _2)P(\gamma _2)}{P(x_i|\gamma _1)P(\gamma _1)+P(x_i|\gamma _2)P(\gamma _2)}

    其中P(γ1)=π1P(\gamma _1)=\pi_1,P(γ2)=π2P(\gamma _2)=\pi_2​

    P(xi∣γ1)=12πσ1e−(x−μ1)22σ12P(xi∣γ2)=12πσ2e−(x−μ2)22σ22\begin{align*} &P(x_i|\gamma_1)=\frac{1}{\sqrt{2\pi}\sigma_1}e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}}\\ &P(x_i|\gamma_2)=\frac{1}{\sqrt{2\pi}\sigma_2}e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} \end{align*}

  2. M步
    估计μ1\mu_1,μ2\mu_2

    μ1=γ11x1+γ21x2+⋯+γN1xNγ11+γ21+⋯+γN1=∑i=1Nγi1xi∑i=1Nγi1μ2=γ12x1+γ22x2+⋯+γN2xNγ12+γ22+⋯+γN2=∑i=1Nγi2xi∑i=1Nγi2\begin{align*} &\mu_1=\frac{\gamma_{11}x_1+\gamma_{21}x_2+\cdots +\gamma_{N1}x_N}{\gamma_{11}+\gamma_{21}+\cdots +\gamma_{N1}}=\frac{\sum_{i=1}^{N}\gamma_{i1}x_i}{\sum_{i=1}^{N}\gamma_{i1}}\\ &\mu_2=\frac{\gamma_{12}x_1+\gamma_{22}x_2+\cdots +\gamma_{N2}x_N}{\gamma_{12}+\gamma_{22}+\cdots +\gamma_{N2}}=\frac{\sum_{i=1}^{N}\gamma_{i2}x_i}{\sum_{i=1}^{N}\gamma_{i2}} \end{align*}

    估计σ1\sigma _1，σ2\sigma _2

    σ12=γ11(x1−μ1)2+γ21(x2−μ1)2+⋯+γN1(xN−μ1)2γ11+γ21+⋯+γN1=∑i=1Nγi1(xi−μ1)2∑i=1Nγi1σ22=γ12(x1−μ2)2+γ22(x2−μ2)2+⋯+γN2(xN−μ2)2γ12+γ22+⋯+γN2=∑i=1Nγi2(xi−μ2)2∑i=1Nγi2\begin{align*} &\sigma _1^2=\frac{\gamma_{11}(x_1-\mu_1)^2+\gamma_{21}(x_2-\mu_1)^2+\cdots +\gamma_{N1}(x_N-\mu_1)^2}{\gamma_{11}+\gamma_{21}+\cdots +\gamma_{N1}}=\frac{\sum_{i=1}^{N}\gamma_{i1}(x_i-\mu_1)^2}{\sum_{i=1}^{N}\gamma_{i1}}\\ &\sigma _2^2=\frac{\gamma_{12}(x_1-\mu_2)^2+\gamma_{22}(x_2-\mu_2)^2+\cdots +\gamma_{N2}(x_N-\mu_2)^2}{\gamma_{12}+\gamma_{22}+\cdots +\gamma_{N2}}=\frac{\sum_{i=1}^{N}\gamma_{i2}(x_i-\mu_2)^2}{\sum_{i=1}^{N}\gamma_{i2}} \end{align*}

    估计π1\pi_1,π2\pi_2

    π1=P(γ1)=γ11+γ21+⋯+γN1N=∑i=1Nγi1Nπ2=P(γ2)=γ12+γ22+⋯+γN2N=∑i=1Nγi2N\begin{align*} &\pi_1=P(\gamma_1)=\frac{\gamma_{11}+\gamma_{21}+\cdots +\gamma_{N1}}{N}=\frac{\sum_{i=1}^{N}\gamma_{i1}}{N}\\ &\pi_2=P(\gamma_2)=\frac{\gamma_{12}+\gamma_{22}+\cdots +\gamma_{N2}}{N}=\frac{\sum_{i=1}^{N}\gamma_{i2}}{N} \end{align*}

  3. 将上式计算结果(π1,μ1,σ1;π2,μ2,σ2\pi_1,\mu _1,\sigma _1;\pi_2,\mu _2,\sigma _2)作为新的参数代入E步计算，重复上述步骤直到两个分布收敛不再更新。

Remark:传统EM算法寻求的是局部最优解，有较强的初值依赖性，初始值的准确设定与否对结果的估计误差有很大影响。

2.算法实现

先自行生成数据集，再进行参数估计。

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import numpy as np
import math
import random

class EM:
    def __init__(self, phi_1, phi_2, miu1, miu2, sigma1, sigma2, dataSize):
        """
        参数初始化
        :param phi_1: 隐变量取Gauss1的概率
        :param phi_2: 隐变量取Gauss2的概率
        :param miu1: Gauss1的伪均值
        :param miu2: Gauss2的伪均值
        :param sigma1: Gauss1的方差
        :param sigma2: Gauss2的方差
        :param dataSize: 样本数据长度
        """
        self.phi_1 = phi_1
        self.phi_2 = phi_2
        self.miu1 = miu1
        self.miu2 = miu2
        self.sigma1 = sigma1
        self.sigma2 = sigma2
        self.dataSize = dataSize

    def creat_gauss_dist(self):
        """
        构造一个高斯混合样本集
        :return:
        """
        data1 = np.random.normal(self.miu1, self.sigma1, int(self.dataSize * self.phi_1))
        data2 = np.random.normal(self.miu2, self.sigma2, int(self.dataSize * self.phi_2))
        dataset = []
        dataset.extend(data1)
        dataset.extend(data2)
        random.shuffle(dataset)  #打乱数据集顺序

        return dataset

    def calculate_gauss(self, dataset, miu, sigma):
        """
        计算高斯核函数
        :param miu: 高斯核伪均值
        :param sigma: 高斯核方差
        :return: 高斯分布概率值
        """
        gauss = (1 / (math.sqrt(2 * math.pi) * sigma)) * \
                 np.exp(-1 * (dataset - miu) * (dataset - miu) / (2 * sigma ** 2))

        return gauss

    def E_step(self, dataset, phi_1, phi_2, miu1, miu2, sigma1, sigma2):
        """
        E步：
        计算Q函数
        :return: Q_k(z), k=1, 2
        """

        q1_numerator = phi_1 * self.calculate_gauss(dataset, miu1, sigma1)
        q2_numerator = phi_2 * self.calculate_gauss(dataset, miu2, sigma2)

        q_denominator = q1_numerator + q2_numerator

        q1 = q1_numerator / q_denominator
        q2 = q2_numerator / q_denominator

        return q1, q2

    def M_step(self, dataset, miu1, miu2, q1, q2):
        """
        M步：
        计算参数的最大似然估计
        """

        nk1 = np.sum(q1)
        nk2 = np.sum(q2)

        phi_new_1 = np.sum(q1) / len(q1)
        phi_new_2 = np.sum(q2) / len(q2)

        miu_new_1 = np.dot(q1, dataset) / nk1
        miu_new_2 = np.dot(q2, dataset) / nk2  #np.dot() 做点积运算

        sigma_new_1 = math.sqrt(np.dot(q1, (dataset - miu1) ** 2) / nk1)
        sigma_new_2 = math.sqrt(np.dot(q2, (dataset - miu2) ** 2) / nk2)

        return miu_new_1, miu_new_2, sigma_new_1, sigma_new_2, phi_new_1, phi_new_2

    def train(self):
        dataset = self.creat_gauss_dist()
        dataset = np.array(dataset) #转换为数组
        max_iter = 10000  #迭代次数
        step = 0

        phi_1 = self.phi_1
        phi_2 = self.phi_2

        miu1 = self.miu1
        miu2 = self.miu2

        sigma1 = self.sigma1
        sigma2 = self.sigma2

        while step < max_iter:
            q1, q2 = self.E_step(dataset, phi_1=phi_1, phi_2=phi_2, miu1=miu1, miu2=miu2, sigma1=sigma1, sigma2=sigma2)
            miu1, miu2, sigma1, sigma2, phi_1, phi_2 = self.M_step(dataset, miu1, miu2, q1, q2)
            step += 1

        return miu1, miu2, sigma1, sigma2, phi_1, phi_2

if __name__ == '__main__':
    phi_1 = 0.75
    phi_2 = 0.25
    miu1 = 100
    miu2 = 200
    sigma1 = 5
    sigma2 = 8
    print('数据集参数:phi_1:%.2f, miu1:%.1f, sigma1:%.1f, phi_2:%.2f, miu2:%.1f, sigma2:%.1f' % (
        phi_1, miu1, sigma1, phi_2, miu2, sigma2
    ))
    em = EM(
        phi_1=0.75,
        phi_2=0.25,
        miu1=100,
        miu2=200,
        sigma1=8,
        sigma2=5,
        dataSize=10000
    )
    miu1, miu2, sigma1, sigma2, phi_1, phi_2 = em.train()
    print('拟合参数:phi_1:%.2f, miu1:%.1f, sigma1:%.1f, phi_2:%.2f, miu2:%.1f, sigma2:%.1f' % (
        phi_1, miu1, sigma1, phi_2, miu2, sigma2
    ))

拓展

机器学习-EM算法报告.pdf