ECML PKDD September –, The Minimum Code Length for Clustering Using the Gray Code Mahito SUGIYAMA †,‡ ，Akihiro YAMAMOTO † † Kyoto University ‡ JSPS Research Fellow /
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
ECML PKDD September –,
The Minimum Code Length forClustering Using the Gray Code
Mahito SUGIYAMA†,‡，Akihiro YAMAMOTO†
†Kyoto University‡JSPS Research Fellow
/
Contributions
. TheMCL (MinimumCode Length)– A new measure to score clustering results– Needed to distinguish each cluster under some fixed encod-
ing scheme for real-valued variables
. COOL (COding-Oriented cLustering)– A general clustering approach– Always finds the best clusters (i.e., the global optimal solution)
which minimizes the MCL in O(nd)– Parameter tuning is not needed
. G-COOL (COOLwith the Gray code)– Achieves internal cohesion and external isolation– Finds arbitrary shaped clusters
/
Demonstration (Synthetic Dataset)
0
0.5
1
0 0.5 1
/
G-COOL
0
0.5
1
0 0.5 1
/
K-means
0
0.5
1
0 0.5 1
/
Results (Real datasets)Bi
nary
�lte
ring
Orig
inal
imag
e
Delta Dragon Europe Norway Ganges
/
Results (Real datasets)G-COOL
K-means
Delta Dragon Europe Norway Ganges
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
Clustering Focusing on Compression
• The MDL approach [Kontkanen et al., ]– Data encoding has to be optimized
∘ All encoding schemes are (implicitly) considered∘ The time complexity ⩾ O(n2)
• The Kolmogorov complexity approach [Cilibrasi, ]– Measures the distance between data points based on com-
pression of finite sequences∘ Difficult to apply multivariate data
– Actual clustering process is the traditional agglomerative hi-erarchical clustering∘ The time complexity ⩾ O(n2)
• Both approaches are not suitable for massive data
/
Our Strategy
• Requirements:. Fast, and linear in the data size. Robust to changes in input parameters. Can find arbitrary shaped clusters
/
Our Strategy
• Requirements:. Fast, and linear in the data size. Robust to changes in input parameters. Can find arbitrary shaped clusters
• Solutions:. Fix an encoding scheme for continuous variables
– Motivated by Computable Analysis [Weihrauch, ]. Clustering = Discretizing real-valued data
– Always finds the best results w.r.t. the MCL. Use the Gray code for real numbers [Tsuiki, ]
– Discretized data points are overlapped and adjacent clus-ters are merged
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
MCL (Minimum Code Length)
• The MCL is the code length of the maximally compressedclusters by using a fixed encoding scheme• The MCL is calculated in O(nd) by using radix sort– n and d are the number of data and dimension, resp.
/
MCL (Minimum Code Length)
• The MCL is the code length of the maximally compressedclusters by using a fixed encoding scheme• The MCL is calculated in O(nd) by using radix sort– n and d are the number of data and dimension, resp.
Example: X = {0.1, 0.2, 0.8, 0.9},𝒞1 = {{0.1, 0.2}, {0.8, 0.9}}𝒞2 = {{0.1}, {0.2, 0.8}, {0.9}}
– Use binary encoding– Which is preferred?
/
Binary Encoding
0 1
Positio
n
01234
0.50.1 0.2
00011...00110...
0.8 0.9
11001...11100...
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
id value
A .B .C .D .
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
id value
A .B .C .D .
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
id value
A .B .C .D .
MCL = 1 + 1 = 2
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
id value
A .B .C .D .
MCL = 1 + 1 = 2
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
id value
A .B .C .D .
MCL = 1 + 1 = 2
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
id value
A .B .C .D .
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
00 10 1101Lv. 2
Lv. 3000 001 110 111
id value
A .B .C .D .
MCL = 3 ⋅ 4 = 12
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
00 10 1101Lv. 2
Lv. 3000 001 110 111
id value
A .B .C .D .
MCL = 3 ⋅ 4 = 12
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
00 10 1101Lv. 2
Lv. 3000 001 110 111
id value
A .B .C .D .
MCL = 3 ⋅ 4 = 12
/
MCL with Binary Encoding
0 0.5 10.25 0.75
A B C D
0 1Lv. 1
00 10 1101Lv. 2
Lv. 3000 001 110 111
id value
A .B .C .D .
MCL = 3 ⋅ 4 = 12
/
Definition of MCL
• Fix an embedding γ :ℝd → Σω (Σ = {0,1} usually)• For p ∈ range(γ) and P ⊂ range(γ), define
Φ(p ∣P) ≔ {w ∈ Σ∗ p ∈ ↑w and P ∩ ↑v = ∅ for all vsuch that |v| = |w| and p ∈ ↑v }
– Each element in Φ(p ∣P) is a prefix that discriminates p from P
For a partition 𝒞 = {C1,… ,CK} of a data set X,MCL(𝒞) ≔ ∑ i∈{1,…,K} Li(𝒞), where
Li(𝒞) ≔ min{|W|γ(Ci) ⊆ ↑W and
W ⊆ x∈CiΦ(γ(x) ∣ γ(X ⧵ Ci)) }
/
Minimizing MCL and Clustering
Clusteringunder theMCL criterion is to find theglobal op-timal solution that minimizes the MCL
– Find 𝒞op such that𝒞op ∈ argmin
𝒞∈𝒞(X)⩾KMCL(𝒞),
where 𝒞(X)⩾K = {𝒞 is a partition of X ∣ #C ⩾ K }
• We give the lower bound of the number of clusters K as ainput parameter– 𝒞op becomes one set {X} without this assumption
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
Optimization by COOL
• COOL solves the optimization problem in O(nd)– n and d are the number of data and dimension, resp.
∘ The naïve approach takes exponential time and space– Computing process of theMCL becomes clustering process it-
self via discretization
• COOL is level-wise, and makes the level-k partition 𝒞k
from k = 1, 2,… , which holds the following condition:– For all x, y ∈ X, they are in the same cluster ⟺
v = w for some v ⊏ γ(x) andw ⊏ γ(y) with |v| = |w| = k∘ Level-k partitions form hierarchy∘ For C ∈ 𝒞k, there exists 𝒟 ⊆ 𝒞k+1 such that ⋃ 𝒟 = C
• For all C ∈ 𝒞op, there exists k such that C ∈ 𝒞k
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D Eid value
A .B .C .D .E .
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
id value
A .B .C .D .E .
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
id value
A 0B 0C 1D 1E 1
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
id value
A 0B 0C 1D 1E 1
MCL = 1 + 1 = 2
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
00 10 1101Lv. 2
id value
A 00B 01C 10D 10E 11
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
00 10 1101Lv. 2
id value
A 00B 01C 10D 10E 11
MCL = 2 ⋅ 4 = 8
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
00 10 1101Lv. 2
100 101Lv. 3
id value
A 00B 01C 100D 101E 11
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
00 10 1101Lv. 2
100 101Lv. 3
id value
A 00B 01C 100D 101E 11
MCL = 6 + 6 = 12
/
Noise Filtering by COOL
• Noise filtering is easily implemented in COOL• Define 𝒞⩾N ≔ {C ∈ 𝒞 ∣ #C ⩾ N} for a partition 𝒞– See a cluster C as noises if #C < N• Example: Given 𝒞 = {{0.1}, {0.4, 0.5, 0.6}, {0.9}}– 𝒞⩾2 = {{0.4, 0.5, 0.6}}, and 0.1 and 0.9 are noises• We input the lower bound N of the cluster size as a inputparameter
0 0.5 10.25 0.75
N = 2
/
Noise Filtering by COOL
• Noise filtering is easily implemented in COOL• Define 𝒞⩾N ≔ {C ∈ 𝒞 ∣ #C ⩾ N} for a partition 𝒞– See a cluster C as noises if #C < N• Example: Given 𝒞 = {{0.1}, {0.4, 0.5, 0.6}, {0.9}}– 𝒞⩾2 = {{0.4, 0.5, 0.6}}, and 0.1 and 0.9 are noises• We input the lower bound N of the cluster size as a inputparameter
0 0.5 10.25 0.75
N = 2
/
Algorithm of COOLInput: A data set X, two lower bounds K and NOutput: The optimal partition 𝒞op and noisesfunction COOL(X, K, N): Find partitions 𝒞1
⩾N,… ,𝒞m⩾N such that ‖𝒞m−1
⩾N ‖ < K ⩽ ‖𝒞m⩾N‖
: (𝒞op, MCL(𝒞op)) ← FINDCLUSTERS(X, K, {𝒞1⩾N,… ,𝒞m
⩾N}): return (𝒞op, X ⧵ ⋃ 𝒞op)function FINDCLUSTERS(X, K, {𝒞1,… ,𝒞m}): Find k such that ‖𝒞k−1‖ < K and ‖𝒞k‖ ⩾ K: 𝒞op ← 𝒞k
: if K = 2 then return (𝒞op, MCL(𝒞op)): for each C in 𝒞1 ∪ … ∪ 𝒞k−1
: (𝒞, L) ← FINDCLUSTERS(X ⧵ C, K − 1, {𝒞1,… ,𝒞k}): if MCL(𝒞 ∪ C) < MCL(𝒞op) then 𝒞op ← C ∪ 𝒞: return (𝒞op, MCL(𝒞op))
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
Gray Code
• Real numbers in [0, 1] are encoded with 0, 1, and ⊥Binary: 0.1 → 00011… , 0.25 → 00111…Gray: 0.1 → 00010… , 0.25 → 0⊥100…
• Originally, another binary encoding of natural numbers– Especially important in applications of conversion between
analog and digital information [Knuth, ]
The Gray code embedding is an injection γG that maps x ∈ [0, 1]to an infinite sequence p0p1p2 … , where– pi ≔ 1 if 2−im−2−(i+1) < x < 2−im+2−(i+1) for an oddm, pi ≔ 0
if the same holds for an evenm, and pi ≔ ⊥ if x = 2−im−2−(i+1)
for some integerm– For a vector x = (x1,… , xd), γG(x) = p1
1 … pd1p12 … pd2 …
/
Gray Code Embedding
0 0.5 1
0
1
235
Positio
n
0.8
10101...
0.25
0⊥100...
0.1
00010...
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D Eid value
A .B .C .D .E .
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥1
id value
A .B .C .D .E .
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥1
id value
A 0B 0, ⊥1C 1, ⊥1D 1, ⊥1E 1
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥1
id value
A 0B 0, ⊥1C 1, ⊥1D 1, ⊥1E 1
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥1
id value
A 0B 0, ⊥1C 1, ⊥1D 1, ⊥1E 1
MCL = 1 ⋅ 2 = 2
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥100 10 1101Lv. 2
0⊥1 ⊥10 1⊥1
0 1Lv. 1
0 1Lv. 1
id value
A 00B 01, ⊥10C 10, ⊥10D 10, 1⊥1E 11, 1⊥1
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥100 10 1101Lv. 2
0⊥1 ⊥10 1⊥1
0 1Lv. 1
0 1Lv. 1
id value
A 00B 01, ⊥10C 10, ⊥10D 10, 1⊥1E 11, 1⊥1
/
COOL with Gray Code (G-COOL)
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
⊥100 10 1101Lv. 2
0⊥1 ⊥10 1⊥1
0 1Lv. 1
0 1Lv. 1
id value
A 00B 01, ⊥10C 10, ⊥10D 10, 1⊥1E 11, 1⊥1
MCL = 2 ⋅ 3 = 6
/
COOL with Binary Encoding
0 0.5 10.25 0.75
A B C D E
0 1Lv. 1
id value
A 0B 0C 1D 1E 1
MCL = 1 + 1 = 2
/
Theoretical Analysis of G-COOL
• Use the Gray code as a fixed encoding in COOL– It achieves internal cohesion and external isolation
• Theorem: For the level-k partition 𝒞k, x, y ∈ X are in thesame cluster if d∞(x, y) < 2−(k+1)
– Thus x, y are in the different clusters only if d∞(x, y) ⩾ 2−(k+1)
– d∞(x, y) = maxi∈{1,…,d}|xi − yi| (L∞ metric)∘ Two adjacent intervals overlap and they are agglomerated
• Corollary: In the optimal partition 𝒞op, for all x ∈ C (C ∈𝒞op), its nearest neighbor y ∈ C– y is nearest neighbor of x ⟺ y ∈ argminy∈Xd∞(x, y)
/
Demonstration of G-COOL
0 0.5 10
0.5
1
G-COOL
0 0.5 10
0.5
1
COOL with the binary encoding
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
Experimental Methods
• Analyze G-COOL empirically with synthetic and realdatasets compared to DBSCAN and K-means– Synthetic datasets were generated by the R package cluster-
Generation [Qiu and Joe, ]∘ n = 1, 500 for each cluster and d = 3
– Real datasets were geospatial images from Earth-as-Art∘ reduced to × pixels, translated into binary images
– All data were normalized by min-max normalization
• G-COOL was implemented by R (version ..)• Internal and External measure were used– Internal: MCL, connectivity, Silhouette width– External: adjusted Rand index
/
Results (Synthetic datasets)
MCL
G-COOLDBSCANK-means
50000
5000
Number of clusters2 4 6
500Data show mean ± s.e.m.Each experiment was performed20 times
Bad
Good
/
Results (Synthetic datasets)
Connectivity Silhouette width1000
1
100
Number of clusters2 4 6
0.1
10
0.6
0.3
0.5
Number of clusters2 4 6
0.4
G-COOLDBSCANK-means
Good
Bad
Bad
Good
/
Results (Synthetic datasets)
Runtime (s)Adjusted Rand index20
5
15
Number of clusters2 4 6
0
10
1.0
0.3
0.9
Number of clusters2 4 6
0.6
G-COOLDBSCANK-means
Good
Bad
/
Results (Synthetic datasets)
MCL
G-COOL
Data show mean ± s.e.m.Each experiment was performed20 times
5000
3000
00 4 62 8 10
4000
2000
1000
The noise parameter N
Bad
Good
/
Results (Synthetic datasets)
Connectivity Silhouette width
80
20
60
0
40
0.6
0.2
0.4
0 4 62 8 10 0 4 62 8 10The noise parameter N The noise parameter N
0
Good
Bad
Bad
Good
/
Results (Synthetic datasets)
Runtime (s)Adjusted Rand index
2.0
0.5
1.5
0 4 60
1.0
1.0
0.4
0.8
0.6
2 8 10The noise parameter N
0 4 62 8 10The noise parameter N
0.2
0
Good
Bad
/
Results (Real datasets)Bi
nary
�lte
ring
Orig
inal
imag
e
Delta Dragon Europe Norway Ganges
/
Results (Real datasets)G-COOL
K-means
Delta Dragon Europe Norway Ganges
/
Results (Real datasets)
Name n K Running time (s) MCLGC KM GC KM
Delta . . Dragon . . Europe . . Norway . . Ganges . .
GC: G-COOL, KM: K-means
/
Outline
. Overview
. Background and Our Strategy
. MCL and Clustering
. COOL Algorithm
. G-COOL: COOL with the Gray Code
. Experiments
. Conclusion
/
Conclusion
• Integrate clustering and its evaluation in the coding-oriented manner– An effective solution for two essential problems, how to mea-
sure goodness of results and how to find good clusters∘ No distance calculation and no data distribution
• Key ideas:. Fix of an encoding scheme for real-valued variables
– Introduced the MCL focusing on compression of clusters– Formulated clusteringwith theMCL, and constructedCOOL
that finds the global optimal solution linearly. The Gray code
– We showed efficiency and effectiveness of G-COOL by the-oretically and experimentally
/
Appendix
A-/A-
Notation (/)• A datum x ∈ ℝd, a data set X = {x1,… , xn}– #X is the number of elements in X– X ⧵ Y is the relative complement of Y in X• Clustering is partition of X into K subsets (clusters) C1, … , CK
– Ci ≠ ∅ and Ci ∩ Cj = ∅– We call 𝒞 = {C1,… ,CK} a partition of X– 𝒞(X) = {𝒞 ∣ 𝒞 is a partition of X}• The set of finite and infinite sequences over an alphabet Σ aredenoted by Σ∗ and Σω, resp.– The length |w| is the number of symbols other than ⊥
∘ Ifw = 11⊥100⊥⊥ … , then |w| = 5– For a set of sequencesW, |W| = ∑w∈W|w|
A-/A-
Notation (/)• An embedding of ℝd is an injective function γ from ℝd to Σω
• For p, q ∈ Σω, define p ⩽ q if pi = qi for all iwith pi ≠ ⊥– Intuitively, q is more concrete than p• For w ∈ Σ∗, we writew ⊏ p ifw⊥ω ⩽ p– ↑w = {p ∈ range(γ) ∣ w ⊏ p} forw ∈ Σ∗
– ↑W = {p ∈ range(γ) ∣ w ⊏ p for some w ∈ W} forW ⊆ Σ∗
• The following monotonicity holds– γ−1(↑v) ⊆ γ−1(↑w) iff v⊥ω ⩾ w⊥ω
A-/A-
Optimization by COOL
• The optimal partition𝒞op can be constructed by the level-k partitions
For all C ∈ 𝒞op, there exists k such that C ∈ 𝒞k
• The level-k partitions have the hierarchical structure– For each C ∈ 𝒞k we have ⋃ 𝒟 = C for some D ⊆ 𝒞k+1
– COOL is similar to divisive hierarchical clustering
COOL always outputs the global optimal partition 𝒞op
• The time complexity is O(nd) (best) and O(nd + K! ) (worst)– Usually K ≪ n holds, hence O(nd)
A-/A-
Clustering Process of COOL
0 10
1
A-/A-
Clustering Process of COOL
0 10
1 K = 2
A-/A-
Clustering Process of COOL
0 10
1 K = 2
A-/A-
Clustering Process of COOL
0 10
1 K = 2
Lv. 1
Cluster
1 2 31
2 3
A-/A-
Clustering Process of COOL
0 10
1 K = 2
Lv. 1
Cluster
1 2 31
2 3
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
Cluster
1 2 31
2 3K = 5
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
Cluster
1 2 31
2 3K = 5
Lv. 2
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
ClusterK = 5
Lv. 21 2
3
5
4 6
21 3 4 5 6
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
ClusterK = 5
Lv. 2
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
ClusterK = 5
Lv. 2
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
ClusterK = 5
Lv. 2
A-/A-
Clustering Process of COOL
0 10
1
Lv. 1
ClusterK = 5
Lv. 2
A-/A-
The Multi-Dimensional Gray Code
• Use thewrapping functionφ(p1,… , pd) ≔ p11 … pd1p
12 … pd2 …
Define the d-dimensional Gray code embedding γdG:ℐ →Σω
⊥,d by γdG(x1,… , xd) ≔ φ(γG(x1),… , γG(xd))
• We abbreviate d of γdG if it is understood from the context
A-/A-
Internal Measures
• Connectivity [Handl et al., ]– Conn(𝒞) = ∑x∈X ∑Mi=1 f(x,nn(x, i))/i
∘ nn(x, j) is the i-th neighbor of x, f(x, y) is 0 if x and y be-long to the same cluster, and 1 otherwise
∘ M is an input parameter (we set as )– Takes values from 0 to ∞, should be minimized• Silhouette width– The average of Silhouette value S(x) for each xS(x) = (b(x) − a(x)/max(b(x), a(x)))∘ a(x) = ‖C‖−1 ∑y∈C d(x, y) (x ∈ C)∘ b(x) = minD∈𝒞⧵C ‖D‖−1 ∑y∈D d(x, y)
– Takes values from −1 to 1, should be maximizedA-/A-
External Measures
• Adjusted Rand index– Let the result be 𝒞 = {C1,… ,CK} and the correct parti-
tion be 𝒟 = {D1,… ,DM}– Suppose nij ≔ ‖{x ∈ X ∣ x ∈ Ci, x ∈ Dj}‖. Then
∑i, j nijC2 − (∑i ‖Ci‖C2 ∑h ‖Dj‖C2)/nC2
2−1(∑i ‖Ci‖C2 + ∑h ‖Dj‖C2) − (∑i ‖Ci‖C2 ∑h ‖Dj‖C2)/nC2
A-/A-
Discussion
• Results for synthetic datasets– Best performance under the internal measures– (nearly) Best performance under the internal measures– G-COOL is efficient and effective
∘ DBSCAN is sensitive to input parameters– The MCL works well as an internal measure• Results for real datasets– not good, and not bad
∘ There are no clear clusters originally• G-COOL is a good clustering method
A-/A-
RelatedWork
• Partitional methods [Chaoji et al., ]
• Mass-based methods [Ting and Wells, ]
• Density-based methods (DBSCAN [Ester et al., ]）• Hierarchical clustering methods(CURE [Guha et al., ], CHAMELEON [Karypis et al.,])• Grid-based methods(STING [Wang et al., ], WaveCluster [Sheikholeslami etal., ])
A-/A-
Future Works
• Speeding up by using tree-structures such as BDD• Apply to anomaly detection• Theoretical analysis, in particular relation with Com-putable Analysis– Admissibility is a key property
A-/A-
References
[Berkhin, ] P. Berkhin. A survey of clustering data mining techniques.GroupingMultidimensional Data, pages –, .
[Chaoji et al., ] V. Chaoji, M. A. Hasan, S. Salem, andM. J. Zaki. SPARCL:An effective and efficient algorithm formining arbitrary shape-basedclusters. Knowledge and Information Systems, ():–, .
[Cilibrasi and Vitányi, ] R. Cilibrasi and P. M. B. Vitányi. Cluster-ing by compression. IEEE Transactions on Information Theory,():–, .
[Ester et al., ] M. Ester, H. P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databaseswith noise. In Proceedings of KDD, , –, .
[Guha et al., ] S. Guha, R. Rastogi, and K. Shim. CURE: An effi-cient clustering algorithm for large databases. Information Systems,():–, .
[Handl et al., ] J. Handl, J. Knowles, and D. B. Kell. Computational
A-/A-
cluster validation in post-genomic data analysis. Bioinformatics,():, .
[Karypis et al., ] G. Karypis, H. Eui-Hong, and V. Kumar. CHAMELEON:Hierarchical clustering using dynamic modeling. Computer,():–, .
[Kontkanen and Myllymäki, ] P. Kontkanen and P. Myllymäki. An em-pirical comparison of NML clustering algorithms. In Proceedings ofInformation Theory and Statistical Learning, .
[Kontkanen et al., ] P. Kontkanen, P.Myllymäki,W. Buntine, J. Rissanen,andH. Tirri. AnMDL framework for data clustering. In P. Grünwald, I. J.Myung, andM. Pitt, editors, Advances inMinimumDescription Length:Theory and Applications. MIT Press, .
[Knuth, ] D. E. Knuth. TheArt of Computer Programming, Volume, Fas-cicle : Generating All Tuples and Permutations. Addison-Wesley Pro-fessional, .
[Qiu and Joe, ] W. Qiu and H. Joe. Generation of random clusters withspecified degree of separation. Journal of Classification, :–,.
A-/A-
[Sheikholeslami et al., ] G. Sheikholeslami, S. Chatterjee, andA. Zhang. WaveCluster: A multi-resolution clustering approach forvery large spatial databases. In Proceedings of the th InternationalConference on Very Large Data Bases, pages –, .
[Ting and Wells, ] K. M. Ting and J. R. Wells. Multi-dimensional massestimation and mass-based clustering. In Proceedings of th IEEE In-ternational Conference on DataMining, pages – , .
[Tsuiki, ] Hideki Tsuiki. Real number computation through Gray codeembedding. Theoretical Computer Science, ():–, .
[Wang et al., ] W. Wang, J. Yang, and R. Muntz. STING: A statisticalinformation grid approach to spatial data mining. In Proceedingsof the rd International Conference on Very Large Data Bases, pages–, .
[Weihrauch, ] K. Weihrauch. Computable Analysis. Springer, .
A-/A-
Related Documents
