DBSCAN(Density-Based Spatial Clustering of Applications with Noise)是一个比较有代表性的基于密度的聚类算法。与划分和层次聚类方法不同，它将簇定义为密度相连的点的最大集合，能够把具有足够高密度的区域划分为簇，并可在噪声的空间数据库中发现任意形状的聚类。
DBSCAN算法最重要的两个参数是ε \varepsilon ε M i n P t s MinPts M i n P t s ε \varepsilon ε M i n P t s MinPts M i n P t s ε \varepsilon ε
ε \varepsilon ε D D D x x x ε \varepsilon ε 核心对象： 对于任意一个样本x x x ε \varepsilon ε M i n P t s MinPts M i n P t s x x x 密度直达： 若x i x_i x i x j x_j x j ε \varepsilon ε x j x_j x j x i x_i x i x j x_j x j 密度可达： 对于x i x_i x i x j x_j x j { P 1 , P 2 , . . . , P t } \begin{Bmatrix}P_1,P_2,...,P_t\end{Bmatrix} { P 1 , P 2 , . . . , P t } P 1 = x i , P t = x j P_1=x_i,P_t=x_j P 1 = x i , P t = x j P t + 1 P_{t+1} P t + 1 P t P_{t} P t x i x_i x i x j x_j x j P 1 , P 2 , . . . , P t − 1 P_1,P_2,...,P_{t-1} P 1 , P 2 , . . . , P t − 1 密度相连： 对于x i x_i x i x j x_j x j x k x_k x k x i x_i x i x j x_j x j x k x_k x k x i x_i x i x j x_j x j
输入样本集D = { x 1 , x 2 , . . . , x m } D=\begin{Bmatrix}x_1,x_2,...,x_m\end{Bmatrix} D = { x 1 , x 2 , . . . , x m } ε \varepsilon ε M i n P t s MinPts M i n P t s
1.初始化核心对象集合Ω = ∅ \Omega= \varnothing Ω = ∅ Γ = D \Gamma=D Γ = D C = ∅ C= \varnothing C = ∅
2.对于j = 1 , 2 , 3... m j=1,2,3...m j = 1 , 2 , 3 . . . m x j x_j x j ε \varepsilon ε N ε ( x j ) N_\varepsilon(x_j) N ε ( x j ) N ε ( x j ) ≥ M i n P t s N_\varepsilon(x_j)\geq MinPts N ε ( x j ) ≥ M i n P t s x j x_j x j Ω = Ω ∪ { x j } \Omega= \Omega\cup\begin{Bmatrix}x_j\end{Bmatrix} Ω = Ω ∪ { x j }
3.如果Ω = ∅ \Omega= \varnothing Ω = ∅
4.在核心对象集合中，随机选择一个核心对象o，初始化当前簇核心对象队列Ω c u t = { o } \Omega_{cut}=\begin{Bmatrix}o\end{Bmatrix} Ω c u t = { o } K = K + 1 K=K+1 K = K + 1 C K = { o } C_K=\begin{Bmatrix}o\end{Bmatrix} C K = { o } Γ = Γ − { o } \Gamma=\Gamma-\begin{Bmatrix}o\end{Bmatrix} Γ = Γ − { o }
5.若当前簇核心对象队列Ω c u t = ∅ \Omega_{cut}=\varnothing Ω c u t = ∅ C K C_K C K C = { C 1 , C 2 , C 3 . . . C K } C= \begin{Bmatrix}C_1,C_2,C_3...C_K\end{Bmatrix} C = { C 1 , C 2 , C 3 . . . C K } Ω = Ω − C K \Omega= \Omega-C_K Ω = Ω − C K
6.在当前簇核心对象队列Ω c u t \Omega_{cut} Ω c u t o ′ o' o ′ ε \varepsilon ε ε \varepsilon ε N ε ( o ′ ) N_\varepsilon(o') N ε ( o ′ ) Δ = N ε ( o ′ ) ∩ Γ \Delta =N_\varepsilon (o')\cap \Gamma Δ = N ε ( o ′ ) ∩ Γ C K = C K ∪ Δ C_K=C_K\cup\Delta C K = C K ∪ Δ Γ = Γ − Δ \Gamma=\Gamma-\Delta Γ = Γ − Δ Ω c u t = Ω c u t ∪ ( N ε ( o ′ ) ∩ Ω ) \Omega_{cut}=\Omega_{cut}\cup(N_\varepsilon(o')\cap\Omega) Ω c u t = Ω c u t ∪ ( N ε ( o ′ ) ∩ Ω )
输出结果：簇划分C = { C 1 , C 2 , C 3 . . . C K } C= \begin{Bmatrix}C_1,C_2,C_3...C_K\end{Bmatrix} C = { C 1 , C 2 , C 3 . . . C K }
优点：
缺点： ε \varepsilon ε M i n P t s MinPts M i n P t s
