原型聚類

原型聚類演算法假設聚類結構能通過一組原型刻畫，在現實聚類任務中極為常用。通常情形下，演算法先對原型進行初始化，然後對原型進行迭代更新求解。這裡的“原型”我認為實際上就是“原來的模型”，這類演算法企圖模擬出生成資料集的模型。

k均值演算法（k-means)

若存在一個樣本集$D=\begin{Bmatrix} x_{1},x_{2},...,x_{m} \end{Bmatrix}$,k均值演算法要求給出類別個數“k”，我們需要將這個樣本集D中的樣本劃分到類別集合$C=\begin{Bmatrix} C_{1},C_{2},...,C_{k} \end{Bmatrix}$

最小化E是一個NP難問題（NP是指非確定性多項式（non-deterministic polynomial，縮寫NP）），因此k均值演算法採用貪心策略，通過不斷迭代來近似求解上式

演算法缺點：

1. k值由使用者確定，不同的k值會獲得不同的結果
2. 對初始簇中心的選擇敏感
3. 不適合發現非凸形狀的簇或大小差別較大的簇
4. 特殊值（離群值）對模型的影響較大

演算法優點：

1. 容易理解，聚類效果不錯
2. 處理大資料集時，演算法可以保證較好的伸縮性（在處理各種規模的資料時都有很好的效能。隨著資料的增大，效率不會下降很快）和高效率
3. 當簇近似高斯分佈時，分類效果很好

python3.6實現：

# -*- coding: gbk -*-
 


import numpy as np
from sklearn.datasets import make_moons
from sklearn.datasets.samples_generator import make_blobs
import matplotlib.pyplot as plt
from collections import Counter
import copy


class KMeans():
    def __init__(self, k=3, max_iter=300):
        self.k = k
        self.max_iter = max_iter

    def dist(self, x1, x2):
        return np.linalg.norm(x1 - x2)

    def get_label(self, x):
        min_dist_with_mu = 999999
        label = -1

        for i in range(self.mus_array.shape[0]):
            dist_with_mu = self.dist(self.mus_array[i], x)
            if min_dist_with_mu > dist_with_mu:
                min_dist_with_mu = dist_with_mu
                label = i

        return label

    def get_mu(self, X):
        index = np.random.choice(X.shape[0], 1, replace=False)
        mus = []
        mus.append(X[index])
        for _ in range(self.k - 1):
            max_dist_index = 0
            max_distance = 0
            for j in range(X.shape[0]):
                min_dist_with_mu = 999999

                for mu in mus:
                    dist_with_mu = self.dist(mu, X[j])
                    if min_dist_with_mu > dist_with_mu:
                        min_dist_with_mu = dist_with_mu

                if max_distance < min_dist_with_mu:
                    max_distance = min_dist_with_mu
                    max_dist_index = j
            mus.append(X[max_dist_index])

        mus_array = np.array([])
        for i in range(self.k):
            if i == 0:
                mus_array = mus[i]
            else:
                mus[i] = mus[i].reshape(mus[0].shape)
                mus_array = np.append(mus_array, mus[i], axis=0)

        return mus_array

    def init_mus(self):
        for i in range(self.mus_array.shape[0]):
            self.mus_array[i] = np.array([0] * self.mus_array.shape[1])

    def fit(self, X):
        self.mus_array = self.get_mu(X)
        iter = 0

        while(iter < self.max_iter):

            old_mus_array = copy.deepcopy(self.mus_array)

            Y = []
            # 將X歸類
            for i in range(X.shape[0]):
                y = self.get_label(X[i])
                Y.append(y)

            self.init_mus()
            # 將同類的X累加
            for i in range(len(Y)):
                self.mus_array[Y[i]] += X[i]

            count = Counter(Y)
            # 計算新的mu
            for i in range(self.k):
                self.mus_array[i] = self.mus_array[i] / count[i]

            diff = 0
            for i in range(self.mus_array.shape[0]):
                diff += np.linalg.norm(self.mus_array[i] - old_mus_array[i])
            if diff == 0:
                break
            iter += 1

        self.E = 0
        for i in range(X.shape[0]):
            self.E += self.dist(X[i], self.mus_array[Y[i]])
        print('E = {}'.format(self.E))
        return np.array(Y)


if __name__ == '__main__':

    fig = plt.figure(1)

    plt.subplot(221)
    center = [[1, 1], [-1, -1], [1, -1]]
    cluster_std = 0.35
    X1, Y1 = make_blobs(n_samples=1000, centers=center,
                        n_features=3, cluster_std=cluster_std, random_state=1)
    plt.scatter(X1[:, 0], X1[:, 1], marker='o', c=Y1)

    plt.subplot(222)
    km1 = KMeans(k=3)
    km_Y1 = km1.fit(X1)
    mus = km1.mus_array
    plt.scatter(X1[:, 0], X1[:, 1], marker='o', c=km_Y1)
    plt.scatter(mus[:, 0], mus[:, 1], marker='^', c='r')

    plt.subplot(223)
    X2, Y2 = make_moons(n_samples=1000, noise=0.1)
    plt.scatter(X2[:, 0], X2[:, 1], marker='o', c=Y2)

    plt.subplot(224)
    km2 = KMeans(k=2)
    km_Y2 = km2.fit(X2)
    mus = km2.mus_array
    plt.scatter(X2[:, 0], X2[:, 1], marker='o', c=km_Y2)
    plt.scatter(mus[:, 0], mus[:, 1], marker='^', c='r')
    plt.show()

執行分類圖片如下：

圖中左側為原始生成的資料，右邊為k-means聚類效果，紅色三角為最終的均值向量

參考