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Clustering methods Course code: 175314. Part 1: Introduction. Pasi Fränti 10.3.2014 Speech & Image Processing Unit School of Computing University of Eastern Finland Joensuu, FINLAND. Sample data. Sources of R G B vectors. Red - Green plot of the vectors. Sample data.

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Clustering methods Course code: 175314

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Clustering methods course code 175314

Clustering methodsCourse code: 175314

Part 1: Introduction

Pasi Fränti


Speech & Image Processing Unit

School of Computing

University of Eastern Finland

Joensuu, FINLAND

Sample data

Sample data

Sources of RGB vectors

Red-Green plot of the vectors

Sample data1

Sample data

Employment statistics:

Application example 1 color reconstruction

Application example 1Color reconstruction

Image with original colors

Image with compression artifacts

Application example 2 speaker modeling for voice biometrics

Application example 2speaker modeling for voice biometrics


Feature extraction

and clustering





Training data


Feature extraction

Speaker models


Best match: Matti !

Speaker modeling

Speaker modeling

Speech data

Result of clustering

Application example 3 image segmentation

Application example 3Image segmentation

Image with 4 color clusters

Normalized color plots according to red and green components.



Application example 4 quantization

Application example 4Quantization

Approximation of continuous range values (or a very large set of possible discrete values) by a small set of discrete symbols or integer values

Quantized signal

Original signal

Color quantization of images

Color quantization of images

Color image

RGB samples


Application example 5 clustering of spatial data

Application example 5Clustering of spatial data

Clustered locations of users

Clustered locations of users

Clustered locations of users1

Timeline clustering

Clustering of photos

Clustered locations of users

Clustering gps trajectories mobile users taxi routes fleet management

Clustering GPS trajectoriesMobile users, taxi routes, fleet management

Conclusions from clusters

Conclusions from clusters

Cluster 2: Home

Cluster 1: Office

Part i clustering problem

Part I:Clustering problem

Subproblems of clustering

Subproblems of clustering

  • Where are the clusters?(Algorithmic problem)

  • How many clusters?(Methodological problem: which criterion?)

  • Selection of attributes (Application related problem)

  • Preprocessing the data(Practical problems: normalization, outliers)

Clustering result as partition

Clustering result as partition

Partition of data

Cluster prototypes

Illustrated by Voronoi diagram

Illustrated by Convex hulls

Clustering methods course code 175314

Duality of partition and centroids

Partition of data

Cluster prototypes

Partition by nearestprototype mapping

Centroids as prototypes

Clustering methods course code 175314

Challenges in clustering

Incorrect cluster allocation

Incorrect number of clusters

Too many clusters

Clusters missing

Cluster missing

How to solve

How to solve?

Algorithmic problem

Mathematical problem

Computer science problem

Solve the clustering:

  • Given input data (X) of N data vectors, and number of clusters (M), find the clusters.

  • Result given as a set of prototypes, or partition.

    Solve the number of clusters:

  • Define appropriate cluster validity function f.

  • Repeat the clustering algorithm for several M.

  • Select the best result according to f.

    Solve the problem efficiently.

Taxonomy of clustering jain murty flynn data clustering a review acm computing surveys 1999

Taxonomy of clustering[Jain, Murty, Flynn, Data clustering: A review, ACM Computing Surveys, 1999.]

  • One possible classification based on cost function.

  • MSE is well defined and most popular.

Definitions and data

Definitions and data

Set of N data points:

X={x1, x2, …, xN}

Partition of the data:

P={p1, p2, …, pM},

Set of M cluster prototypes (centroids):

C={c1, c2, …, cM},

Distance and cost function

Distance and cost function

Euclidean distance of data vectors:

Mean square error:

Dependency of data structures

Dependency of data structures

  • Centroid condition: for a given partition (P), optimal cluster centroids (C) for minimizing MSE are the average vectors of the clusters:

  • Optimal partition: for a given centroids (C), optimal partition is the one with nearest centroid :

Complexity of clustering

Complexity of clustering

  • Number of possible clusterings:

  • Clustering problem is NP complete [Garey et al., 1982]

  • Optimal solution by branch-and-bound in exponential time.

  • Practical solutions by heuristic algorithms.

Cluster software


Main area

Input area

Cluster software

  • Main area: working space for data

  • Input area: inputs to be processed

  • Output area:obtained results

  • Menu Process:selection of operation

Clustering methods course code 175314

Procedure to simulate k-means

Clustering image

Data set



Open data set (file *.ts), move it into Input area

Process – Random codebook, select number of clusters


Move obtained codebook from Output area into Input area

Process – Optimal partition, select Error function

Move codebook into Main area, partition into Input area

Process – Optimal codebook


Xlminer software

XLMiner software

Example of data in xlminer

Example of data in XLMiner

Distance matrix dendrogram

Distance matrix & dendrogram



  • Clustering is a fundamental tools needed in Speech and Image processing.

  • Failing to do clustering properly may defect the application analysis.

  • Good clustering tool needed so that researchers can focus on application requirements.



  • S. Theodoridis and K. Koutroumbas, Pattern Recognition, Academic Press, 3rd edition, 2006.

  • C. Bishop, Pattern Recognition and Machine Learning, Springer, 2006.

  • A.K. Jain, M.N. Murty and P.J. Flynn, Data clustering: A review, ACM Computing Surveys, 31(3): 264-323, September 1999.

  • M.R. Garey, D.S. Johnson and H.S. Witsenhausen, The complexity of the generalized Lloyd-Max problem, IEEE Transactions on Information Theory, 28(2): 255-256, March 1982.

  • F. Aurenhammer: Voronoi diagrams-a survey of a fundamental geometric data structure, ACM Computing Surveys, 23 (3), 345-405, September 1991.

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