简单易学的机器学习算法——Mean Shift聚类算法

felixzhao

发布于 2018-03-20 15:31:51

2.1K0

发布于 2018-03-20 15:31:51

文章被收录于专栏：null的专栏

一、Mean Shift算法概述

Mean Shift算法，又称为均值漂移算法，Mean Shift的概念最早是由Fukunage在1975年提出的，在后来由Yizong Cheng对其进行扩充，主要提出了两点的改进：

定义了核函数；
增加了权重系数。

核函数的定义使得偏移值对偏移向量的贡献随之样本与被偏移点的距离的不同而不同。权重系数使得不同样本的权重不同。Mean Shift算法在聚类，图像平滑、分割以及视频跟踪等方面有广泛的应用。

二、Mean Shift算法的核心原理

2.1、核函数

上图的画图脚本如下所示：

'''
Date:201604026
@author: zhaozhiyong
'''
import matplotlib.pyplot as plt
import math

def cal_Gaussian(x, h=1):
    molecule = x * x
    denominator = 2 * h * h
    left = 1 / (math.sqrt(2 * math.pi) * h)
    return left * math.exp(-molecule / denominator)

x = []

for i in xrange(-40,40):
    x.append(i * 0.5);

score_1 = []
score_2 = []
score_3 = []
score_4 = []

for i in x:
    score_1.append(cal_Gaussian(i,1))
    score_2.append(cal_Gaussian(i,2))
    score_3.append(cal_Gaussian(i,3))
    score_4.append(cal_Gaussian(i,4))

plt.plot(x, score_1, 'b--', label="h=1")
plt.plot(x, score_2, 'k--', label="h=2")
plt.plot(x, score_3, 'g--', label="h=3")
plt.plot(x, score_4, 'r--', label="h=4")

plt.legend(loc="upper right")
plt.xlabel("x")
plt.ylabel("N")
plt.show()

2.2、Mean Shift算法的核心思想

2.2.1、基本原理

对于Mean Shift算法，是一个迭代的步骤，即先算出当前点的偏移均值，将该点移动到此偏移均值，然后以此为新的起始点，继续移动，直到满足最终的条件。此过程可由下图的过程进行说明(图片来自参考文献3)：

步骤1：在指定的区域内计算偏移均值(如下图的黄色的圈)

步骤2：移动该点到偏移均值点处

步骤3：重复上述的过程(计算新的偏移均值，移动)

步骤4：满足了最终的条件，即退出

从上述过程可以看出，在Mean Shift算法中，最关键的就是计算每个点的偏移均值，然后根据新计算的偏移均值更新点的位置。

2.2.2、基本的Mean Shift向量形式

2.2.3、改进的Mean Shift向量形式

2.3、Mean Shift算法的解释

在Mean Shift算法中，实际上是利用了概率密度，求得概率密度的局部最优解。

2.3.1、概率密度梯度

2.3.2、Mean Shift向量的修正

2.4、Mean Shift算法流程

三、实验

3.1、实验数据

实验数据如下图所示(来自参考文献1)：

画图的代码如下：

'''
Date:20160426
@author: zhaozhiyong
'''
import matplotlib.pyplot as plt

f = open("data")
x = []
y = []
for line in f.readlines():
    lines = line.strip().split("\t")
    if len(lines) == 2:
        x.append(float(lines[0]))
        y.append(float(lines[1]))
f.close()  

plt.plot(x, y, 'b.', label="original data")
plt.title('Mean Shift')
plt.legend(loc="upper right")
plt.show()

3.2、实验的源码

#!/bin/python
#coding:UTF-8
'''
Date:20160426
@author: zhaozhiyong
'''

import math
import sys
import numpy as np

MIN_DISTANCE = 0.000001#mini error

def load_data(path, feature_num=2):
    f = open(path)
    data = []
    for line in f.readlines():
        lines = line.strip().split("\t")
        data_tmp = []
        if len(lines) != feature_num:
            continue
        for i in xrange(feature_num):
            data_tmp.append(float(lines[i]))

        data.append(data_tmp)
    f.close()
    return data

def gaussian_kernel(distance, bandwidth):
    m = np.shape(distance)[0]
    right = np.mat(np.zeros((m, 1)))
    for i in xrange(m):
        right[i, 0] = (-0.5 * distance[i] * distance[i].T) / (bandwidth * bandwidth)
        right[i, 0] = np.exp(right[i, 0])
    left = 1 / (bandwidth * math.sqrt(2 * math.pi))

    gaussian_val = left * right
    return gaussian_val

def shift_point(point, points, kernel_bandwidth):
    points = np.mat(points)
    m,n = np.shape(points)
    #计算距离
    point_distances = np.mat(np.zeros((m,1)))
    for i in xrange(m):
        point_distances[i, 0] = np.sqrt((point - points[i]) * (point - points[i]).T)

    #计算高斯核      
    point_weights = gaussian_kernel(point_distances, kernel_bandwidth)

    #计算分母
    all = 0.0
    for i in xrange(m):
        all += point_weights[i, 0]

    #均值偏移
    point_shifted = point_weights.T * points / all
    return point_shifted

def euclidean_dist(pointA, pointB):
    #计算pointA和pointB之间的欧式距离
    total = (pointA - pointB) * (pointA - pointB).T
    return math.sqrt(total)

def distance_to_group(point, group):
    min_distance = 10000.0
    for pt in group:
        dist = euclidean_dist(point, pt)
        if dist < min_distance:
            min_distance = dist
    return min_distance

def group_points(mean_shift_points):
    group_assignment = []
    m,n = np.shape(mean_shift_points)
    index = 0
    index_dict = {}
    for i in xrange(m):
        item = []
        for j in xrange(n):
            item.append(str(("%5.2f" % mean_shift_points[i, j])))

        item_1 = "_".join(item)
        print item_1
        if item_1 not in index_dict:
            index_dict[item_1] = index
            index += 1

    for i in xrange(m):
        item = []
                for j in xrange(n):
                        item.append(str(("%5.2f" % mean_shift_points[i, j])))

                item_1 = "_".join(item)
        group_assignment.append(index_dict[item_1])

    return group_assignment

def train_mean_shift(points, kenel_bandwidth=2):
    #shift_points = np.array(points)
    mean_shift_points = np.mat(points)
    max_min_dist = 1
    iter = 0
    m, n = np.shape(mean_shift_points)
    need_shift = [True] * m

    #cal the mean shift vector
    while max_min_dist > MIN_DISTANCE:
        max_min_dist = 0
        iter += 1
        print "iter : " + str(iter)
        for i in range(0, m):
            #判断每一个样本点是否需要计算偏置均值
            if not need_shift[i]:
                continue
            p_new = mean_shift_points[i]
            p_new_start = p_new
            p_new = shift_point(p_new, points, kenel_bandwidth)
            dist = euclidean_dist(p_new, p_new_start)

            if dist > max_min_dist:#record the max in all points
                max_min_dist = dist
            if dist < MIN_DISTANCE:#no need to move
                need_shift[i] = False

            mean_shift_points[i] = p_new
    #计算最终的group
    group = group_points(mean_shift_points)

    return np.mat(points), mean_shift_points, group

if __name__ == "__main__":
    #导入数据集
    path = "./data"
    data = load_data(path, 2)

    #训练，h=2
    points, shift_points, cluster = train_mean_shift(data, 2)

    for i in xrange(len(cluster)):
        print "%5.2f,%5.2f\t%5.2f,%5.2f\t%i" % (points[i,0], points[i, 1], shift_points[i, 0], shift_points[i, 1], cluster[i])

3.3、实验的结果

经过Mean Shift算法聚类后的数据如下所示：

'''
Date:20160426
@author: zhaozhiyong
'''
import matplotlib.pyplot as plt

f = open("data_mean")
cluster_x_0 = []
cluster_x_1 = []
cluster_x_2 = []
cluster_y_0 = []
cluster_y_1 = []
cluster_y_2 = []
center_x = []
center_y = []
center_dict = {}

for line in f.readlines():
    lines = line.strip().split("\t")
    if len(lines) == 3:
        label = int(lines[2])
        if label == 0:
            data_1 = lines[0].strip().split(",")
            cluster_x_0.append(float(data_1[0]))
            cluster_y_0.append(float(data_1[1]))
            if label not in center_dict:
                center_dict[label] = 1
                data_2 = lines[1].strip().split(",")
                center_x.append(float(data_2[0]))
                center_y.append(float(data_2[1]))
        elif label == 1:
            data_1 = lines[0].strip().split(",")
            cluster_x_1.append(float(data_1[0]))
            cluster_y_1.append(float(data_1[1]))
            if label not in center_dict:
                center_dict[label] = 1
                data_2 = lines[1].strip().split(",")
                center_x.append(float(data_2[0]))
                center_y.append(float(data_2[1]))
        else:
            data_1 = lines[0].strip().split(",")
            cluster_x_2.append(float(data_1[0]))
            cluster_y_2.append(float(data_1[1]))
            if label not in center_dict:
                center_dict[label] = 1
                data_2 = lines[1].strip().split(",")
                center_x.append(float(data_2[0]))
                center_y.append(float(data_2[1]))    
f.close()


plt.plot(cluster_x_0, cluster_y_0, 'b.', label="cluster_0")
plt.plot(cluster_x_1, cluster_y_1, 'g.', label="cluster_1")
plt.plot(cluster_x_2, cluster_y_2, 'k.', label="cluster_2")
plt.plot(center_x, center_y, 'r+', label="mean point")
plt.title('Mean Shift 2')
#plt.legend(loc="best")
plt.show()

参考文献

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