【ML從入門到入土系列08】EM

文章目錄

• • 1 理論
  • 2 代碼
  • 3 參考

1 理論

EM算法通過疊代求解觀測資料的對數似然函數 L ( θ ) = log ⁡ P ( Y ∣ θ ) {L}(\theta)=\log {P}(\mathrm{Y} | \theta) L(θ)=logP(Y∣θ)的極大化，實作極大似然估計。每次疊代包括兩步：

• E E E步：求期望

  Q ( θ , θ ( i ) ) = ∑ z log ⁡ P ( Y , Z ∣ θ ) P ( Z ∣ Y , θ ( i ) ) Q\left(\theta, \theta^{(i)}\right)=\sum_{z} \log P(Y, Z \mid \theta) P\left(Z \mid Y, \theta^{(i)}\right) Q(θ,θ(i))=z∑​logP(Y,Z∣θ)P(Z∣Y,θ(i))

• M M M步：求極大

  θ ( i + 1 ) = arg ⁡ max ⁡ θ Q ( θ , θ ( i ) ) \theta^{(i+1)}=\arg \max _{\theta} Q\left(\theta, \theta^{(i)}\right) θ(i+1)=argθmax​Q(θ,θ(i))

2 代碼

class EM:
    def __init__(self, prob):
        self.pro_A, self.pro_B, self.pro_C = prob

    # E步
    def pmf(self, i):
        pro_1 = self.pro_A * math.pow(self.pro_B, data[i]) * math.pow(
            (1 - self.pro_B), 1 - data[i])
        pro_2 = (1 - self.pro_A) * math.pow(self.pro_C, data[i]) * math.pow(
            (1 - self.pro_C), 1 - data[i])
        return pro_1 / (pro_1 + pro_2)

    # M步
    def fit(self, data):
        count = len(data)
        for d in range(count):
            _ = yield
            _pmf = [self.pmf(k) for k in range(count)]
            pro_A = 1 / count * sum(_pmf)
            pro_B = sum([_pmf[k] * data[k] for k in range(count)]) / sum(
                [_pmf[k] for k in range(count)])
            pro_C = sum([(1 - _pmf[k]) * data[k]
                         for k in range(count)]) / sum([(1 - _pmf[k])
                                                        for k in range(count)])
            self.pro_A = pro_A
            self.pro_B = pro_B
            self.pro_C = pro_C
           

3 參考

理論：周志華《機器學習》，李航《統計學習方法》

代碼：https://github.com/fengdu78/lihang-code