本文介绍DBSCAN算法的基本原理，包括密度可达与密度相连的概念，并详细解释了算法的步骤及实现过程。DBSCAN算法能有效识别出任意形状的簇类及噪声点。
[飞桨机器学习]DBSCAN算法
一、简介
DBSCAN（Density-Based Spatial Clustering of Applications with Noise），具有噪声的基于密度的聚类方法）是一种基于密度的空间聚类算法。 该算法将具有足够密度的区域划分为簇，并在具有噪声的空间数据库中发现任意形状的簇，它将簇定义为密度相连的点的最大集合。
二、相关概念
Ε邻域：给定对象半径为Ε内的区域称为该对象的Ε邻域；
核心对象：如果给定对象Ε邻域内的样本点数大于等于MinPts，则称该对象为核心对象；
直接密度可达：对于样本集合D，如果样本点q在p的Ε邻域内，并且p为核心对象，那么对象q从对象p直接密度可达。
密度可达：对于样本集合D，给定一串样本点p1,p2….pn，p= p1,q= pn,假如对象pi从pi-1直接密度可达，那么对象q从对象p密度可达。
密度相连：存在样本集合D中的一点o，如果对象o到对象p和对象q都是密度可达的，那么p和q密度相联。
可以发现，密度可达是直接密度可达的传递闭包，并且这种关系是非对称的。密度相连是对称关系。DBSCAN目的是找到密度相连对象的最大集合。
Eg: 假设半径Ε=3，MinPts=3，点p的E邻域中有点{m,p,p1,p2,o}, 点m的E邻域中有点{m,q,p,m1,m2},点q的E邻域中有点{q,m},点o的E邻域中有点{o,p,s},点s的E邻域中有点{o,s,s1}.
那么核心对象有p,m,o,s(q不是核心对象，因为它对应的E邻域中点数量等于2，小于MinPts=3)；
点m从点p直接密度可达，因为m在p的E邻域内，并且p为核心对象；
点q从点p密度可达，因为点q从点m直接密度可达，并且点m从点p直接密度可达；
点q到点s密度相连，因为点q从点p密度可达，并且s从点p密度可达。
三、DBSCAN算法描述:
输入: 包含n个对象的数据库，半径e，最少数目MinPts;
输出:所有生成的簇，达到密度要求。
(1)Repeat
(2)从数据库中抽出一个未处理的点；
(3)IF抽出的点是核心点 THEN 找出所有从该点密度可达的对象，形成一个簇；
(4)ELSE 抽出的点是边缘点(非核心对象)，跳出本次循环，寻找下一个点；
(5)UNTIL 所有的点都被处理。
DBSCAN对用户定义的参数很敏感，细微的不同都可能导致差别很大的结果，而参数的选择无规律可循，只能靠经验确定。
四、算法步骤
先说个人总结：先根据半径e，最少数目MinPts选择出核心对象，再根据核心对象划分簇
输入：样本集D=(x1,x2,…,xm)(x1,x2,…,xm)，邻域参数(ϵ,MinPts)(ϵ,MinPts), 样本距离度量方式
输出： 簇划分C.
1）初始化核心对象集合Ω=∅Ω=∅, 初始化聚类簇数k=0，初始化未访问样本集合ΓΓ = D, 簇划分C = ∅∅
2) 对于j=1,2,…m, 按下面的步骤找出所有的核心对象：
a) 通过距离度量方式，找到样本xjxj的ϵϵ-邻域子样本集Nϵ(xj)Nϵ(xj)
b) 如果子样本集样本个数满足|Nϵ(xj)|≥MinPts|Nϵ(xj)|≥MinPts， 将样本xjxj加入核心对象样本集合：Ω=Ω∪{xj}Ω=Ω∪{xj}
3）如果核心对象集合Ω=∅Ω=∅，则算法结束，否则转入步骤4.
4）在核心对象集合ΩΩ中，随机选择一个核心对象oo，初始化当前簇核心对象队列Ωcur={o}Ωcur={o}, 初始化类别序号k=k+1，初始化当前簇样本集合Ck={o}Ck={o}, 更新未访问样本集合Γ=Γ−{o}Γ=Γ−{o}
5）如果当前簇核心对象队列Ωcur=∅Ωcur=∅，则当前聚类簇CkCk生成完毕, 更新簇划分C={C1,C2,…,Ck}{C1,C2,…,Ck}, 更新核心对象集合Ω=Ω−CkΩ=Ω−Ck， 转入步骤3。否则更新核心对象集合Ω=Ω−CkΩ=Ω−Ck。
6）在当前簇核心对象队列ΩcurΩcur中取出一个核心对象o′o′,通过邻域距离阈值ϵϵ找出所有的ϵϵ-邻域子样本集Nϵ(o′)Nϵ(o′)，令Δ=Nϵ(o′)∩ΓΔ=Nϵ(o′)∩Γ, 更新当前簇样本集合Ck=Ck∪ΔCk=Ck∪Δ, 更新未访问样本集合Γ=Γ−ΔΓ=Γ−Δ, 更新Ωcur=Ωcur∪(Δ∩Ω)−o′Ωcur=Ωcur∪(Δ∩Ω)−o′，转入步骤5.
输出结果为： 簇划分C={C1,C2,…,Ck}
五、优缺点
优点：
-
与K-means方法相比，DBSCAN不需要事先知道要形成的簇类的数量。
-
与K-means方法相比，DBSCAN可以发现任意形状的簇类。
-
同时，DBSCAN能够识别出噪声点。
-
DBSCAN对于数据库中样本的顺序不敏感，即Pattern的输入顺序对结果的影响不大。但是，对于处于簇类之间边界样本，可能会根据哪个簇类优先被探测到而其归属有所摆动。
缺点：
-
DBScan不能很好反映高维数据。
-
DBScan不能很好反映数据集以变化的密度。
-
如果样本集的密度不均匀、聚类间距差相差很大时，聚类质量较差。
六、代码实现
1.引入库
import random
import numpy as np
import math
import copy
import matplotlib.pyplot as plt
2.计算欧式距离
# 计算欧式距离
def calculate_euclidean_distance(data1, data2, length):
distance = 0.0
for i in range(length):
distance += pow((data1[i]-data2[i]), 2)
return math.sqrt(distance)
3.密度聚类
def dbascn(data, min_distance, min_num):
q = []
for i in range(len(data)):
count = 0
for j in range(len(data)):
if calculate_euclidean_distance(data[i], data[j], 4) < min_distance:
count += 1
if count >= min_num:
q.append(data[i])
print(q)
c = []
for i in range(len(q)):
c.append([])
for i in range(len(q)):
for j in range(len(data)):
if data[j] != 0 and calculate_euclidean_distance(q[i], data[j], 4) < min_distance:
c[i].append(data[j])
data[j] = 0
print(data)
count = 0
for i in range(len(data)):
if data[i] == 0:
count += 1
for i in range(count):
data.remove(0)
return c, q, data
4.可视化
def veision(c, q, noise):
cc = []
for i in range(len(c)):
for j in range(len(c[i])):
cc.append(c[i][j])
data = np.array(cc)
data2 = np.array(noise)
data3 = np.array(q)
mean_vector = np.mean(data, axis=0)
mean_vector2 = np.mean(data2, axis=0)
mean_vector3 = np.mean(data3, axis=0)
cov_mat = np.cov(data - mean_vector, rowvar=0)
cov_mat2 = np.cov(data2 - mean_vector2, rowvar=0)
cov_mat3 = np.cov(data3 - mean_vector3, rowvar=0)
fvalue, fvector = np.linalg.eig(cov_mat)
fvalue2, fvector2 = np.linalg.eig(cov_mat2)
fvalue3, fvector3 = np.linalg.eig(cov_mat3)
fvaluesort = np.argsort(-fvalue)
fvaluesort2 = np.argsort(-fvalue2)
fvaluesort3 = np.argsort(-fvalue3)
fValueTopN = fvaluesort[:2]
fValueTopN2 = fvaluesort2[:2]
fValueTopN3 = fvaluesort3[:2]
newdata = fvector[:, fValueTopN]
newdata2 = fvector2[:, fValueTopN2]
newdata3 = fvector3[:, fValueTopN3]
new = np.dot(cc, newdata)
new2 = np.dot(noise, newdata2)
new3 = np.dot(q, newdata3)
c1 = []
c2 = []
for i in range(len(new)):
c1.append(new[i][0])
c2.append(new[i][1])
n1 = []
n2 = []
for i in range(len(new2)):
n1.append(new2[i][0])
n2.append(new2[i][1])
q1 = []
q2 = []
for i in range(len(new3)):
q1.append(new3[i][0])
q2.append(new3[i][1])
fig = plt.figure()
ax1 = fig.add_subplot(111)
ax1.scatter(c1, c2, marker='*')
ax1.scatter(q1, q2, marker='^')
ax1.scatter(n1, n2, marker='+')
plt.show()
5.主程序
if __name__ == "__main__":
# 生成随机数据
data_set = []
for i in range(100):
data_set.append([])
for j in range(4):
data_set[i].append(random.uniform(-50, 50))
print(data_set)
c, q, noise = dbascn(data_set, 25, 5)
print(c)
print(noise)
veision(c, q, noise)
6.完整代码
import random
import numpy as np
import math
import copy
import matplotlib.pyplot as plt
# 计算欧式距离
def calculate_euclidean_distance(data1, data2, length):
distance = 0.0
for i in range(length):
distance += pow((data1[i]-data2[i]), 2)
return math.sqrt(distance)
def dbascn(data, min_distance, min_num):
q = []
for i in range(len(data)):
count = 0
for j in range(len(data)):
if calculate_euclidean_distance(data[i], data[j], 4) < min_distance:
count += 1
if count >= min_num:
q.append(data[i])
print(q)
c = []
for i in range(len(q)):
c.append([])
for i in range(len(q)):
for j in range(len(data)):
if data[j] != 0 and calculate_euclidean_distance(q[i], data[j], 4) < min_distance:
c[i].append(data[j])
data[j] = 0
print(data)
count = 0
for i in range(len(data)):
if data[i] == 0:
count += 1
for i in range(count):
data.remove(0)
return c, q, data
def veision(c, q, noise):
cc = []
for i in range(len(c)):
for j in range(len(c[i])):
cc.append(c[i][j])
data = np.array(cc)
data2 = np.array(noise)
data3 = np.array(q)
mean_vector = np.mean(data, axis=0)
mean_vector2 = np.mean(data2, axis=0)
mean_vector3 = np.mean(data3, axis=0)
cov_mat = np.cov(data - mean_vector, rowvar=0)
cov_mat2 = np.cov(data2 - mean_vector2, rowvar=0)
cov_mat3 = np.cov(data3 - mean_vector3, rowvar=0)
fvalue, fvector = np.linalg.eig(cov_mat)
fvalue2, fvector2 = np.linalg.eig(cov_mat2)
fvalue3, fvector3 = np.linalg.eig(cov_mat3)
fvaluesort = np.argsort(-fvalue)
fvaluesort2 = np.argsort(-fvalue2)
fvaluesort3 = np.argsort(-fvalue3)
fValueTopN = fvaluesort[:2]
fValueTopN2 = fvaluesort2[:2]
fValueTopN3 = fvaluesort3[:2]
newdata = fvector[:, fValueTopN]
newdata2 = fvector2[:, fValueTopN2]
newdata3 = fvector3[:, fValueTopN3]
new = np.dot(cc, newdata)
new2 = np.dot(noise, newdata2)
new3 = np.dot(q, newdata3)
c1 = []
c2 = []
for i in range(len(new)):
c1.append(new[i][0])
c2.append(new[i][1])
n1 = []
n2 = []
for i in range(len(new2)):
n1.append(new2[i][0])
n2.append(new2[i][1])
q1 = []
q2 = []
for i in range(len(new3)):
q1.append(new3[i][0])
q2.append(new3[i][1])
fig = plt.figure()
ax1 = fig.add_subplot(111)
ax1.scatter(c1, c2, marker='*')
ax1.scatter(q1, q2, marker='^')
ax1.scatter(n1, n2, marker='+')
plt.show()
if __name__ == "__main__":
# 生成随机数据
data_set = []
for i in range(100):
data_set.append([])
for j in range(4):
data_set[i].append(random.uniform(-50, 50))
print(data_set)
c, q, noise = dbascn(data_set, 25, 5)
print(c)
print(noise)
veision(c, q, noise)
7.结果
由于是随机生成数据，可以直观发现效果并不好
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# notebook显示plt
%matplotlib inline
import random
import numpy as np
import math
import copy
import matplotlib.pyplot as plt
# 计算欧式距离
def calculate_euclidean_distance(data1, data2, length):
distance = 0.0
for i in range(length):
distance += pow((data1[i]-data2[i]), 2)
return math.sqrt(distance)
def dbascn(data, min_distance, min_num):
q = []
for i in range(len(data)):
count = 0
for j in range(len(data)):
if calculate_euclidean_distance(data[i], data[j], 4) < min_distance:
count += 1
if count >= min_num:
q.append(data[i])
print(q)
c = []
for i in range(len(q)):
c.append([])
for i in range(len(q)):
for j in range(len(data)):
if data[j] != 0 and calculate_euclidean_distance(q[i], data[j], 4) < min_distance:
c[i].append(data[j])
data[j] = 0
print(data)
count = 0
for i in range(len(data)):
if data[i] == 0:
count += 1
for i in range(count):
data.remove(0)
return c, q, data
def veision(c, q, noise):
cc = []
for i in range(len(c)):
for j in range(len(c[i])):
cc.append(c[i][j])
data = np.array(cc)
data2 = np.array(noise)
data3 = np.array(q)
mean_vector = np.mean(data, axis=0)
mean_vector2 = np.mean(data2, axis=0)
mean_vector3 = np.mean(data3, axis=0)
cov_mat = np.cov(data - mean_vector, rowvar=0)
cov_mat2 = np.cov(data2 - mean_vector2, rowvar=0)
cov_mat3 = np.cov(data3 - mean_vector3, rowvar=0)
fvalue, fvector = np.linalg.eig(cov_mat)
fvalue2, fvector2 = np.linalg.eig(cov_mat2)
fvalue3, fvector3 = np.linalg.eig(cov_mat3)
fvaluesort = np.argsort(-fvalue)
fvaluesort2 = np.argsort(-fvalue2)
fvaluesort3 = np.argsort(-fvalue3)
fValueTopN = fvaluesort[:2]
fValueTopN2 = fvaluesort2[:2]
fValueTopN3 = fvaluesort3[:2]
newdata = fvector[:, fValueTopN]
newdata2 = fvector2[:, fValueTopN2]
newdata3 = fvector3[:, fValueTopN3]
new = np.dot(cc, newdata)
new2 = np.dot(noise, newdata2)
new3 = np.dot(q, newdata3)
c1 = []
c2 = []
for i in range(len(new)):
c1.append(new[i][0])
c2.append(new[i][1])
n1 = []
n2 = []
for i in range(len(new2)):
n1.append(new2[i][0])
n2.append(new2[i][1])
q1 = []
q2 = []
for i in range(len(new3)):
q1.append(new3[i][0])
q2.append(new3[i][1])
fig = plt.figure()
ax1 = fig.add_subplot(111)
ax1.scatter(c1, c2, marker='*')
ax1.scatter(q1, q2, marker='^')
ax1.scatter(n1, n2, marker='+')
plt.show()
if __name__ == "__main__":
# 生成随机数据
data_set = []
for i in range(100):
data_set.append([])
for j in range(4):
data_set[i].append(random.uniform(-50, 50))
print(data_set)
c, q, noise = dbascn(data_set, 25, 5)
print(c)
print(noise)
veision(c, q, noise)
[[-15.379777735134503, -37.844364923635425, -41.755347510432294, -10.597228127051729], [39.15011395769959, -43.99579779457326, 21.258601551506914, -19.398430154651734], [38.69286619758692, -7.407664915859783, -33.393081198067776, 28.431060800624323], [-49.57831907291702, 2.428308178244812, 34.11223152089018, 23.767658448062008], [47.905456781794655, 28.35294800298925, 26.0236278292613, 9.680039107809655], [-41.22836632698631, 35.27063771889078, -20.75265385750784, -1.3882004384521238], [-9.076114874574102, -24.762374364538587, -23.891614323168685, 3.116076153897808], [3.002277730328615, 17.76470015424914, -17.257817272240516, -34.74188180866875], [13.141800930567562, 38.85081627137474, -34.8639917921499, -48.98593730117731], [-9.614791249147245, -14.453714733289367, 38.84847043962519, -26.65360151552264], [-43.247112024583735, -27.03238299385957, -31.673935498235227, 30.552227510157223], [-8.092232719135993, -34.21791475446994, -11.834498734508237, 2.6325560661688883], [-17.954133312676603, -20.37850885307594, -31.047864477182298, 21.717087185442168], [-38.150583048822526, 19.177770786802355, -24.680536820571174, 24.249103163657054], [-29.154545613170836, 0.46739324781639624, -49.13555020475455, -19.111223678809885], [-24.5897208817225, -46.004064571806815, -11.113514672100145, -9.453213405231878], [-15.285558935576773, 9.666631902126475, -7.7727823000732315, 41.7872917272836], [29.48533595789496, 10.231566863883515, 45.749740424917746, 20.65191194829154], [-46.87194196061512, 45.52495764002913, -29.60028066676933, -44.063796677195135], [12.53143664033405, 23.32352310525704, 20.31910127193281, -30.702325256345773], [37.62636397954449, 48.901876742791444, -41.94844330551318, -25.16637641814965], [-20.06874806820885, 21.039361298859916, 5.18714403709334, 5.639089289221097], [-48.991514838126726, -8.33888473451568, -6.4325119914144935, -26.730740413008668], [6.608798840745337, 30.885274067753528, 2.3690081318134872, 40.31795644381933], 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运行代码请点击：https://aistudio.baidu.com/aistudio/projectdetail/619613?shared=1
欢迎三连！
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