This document summarizes a presentation on cluster stability estimation and determining the optimal number of clusters in a dataset. The presentation proposes a method that draws random samples from the dataset and compares the partitions obtained from each sample to estimate cluster stability. It quantifies the consistency between partitions using minimal spanning trees and the Friedman-Rafsky test statistic. Experiments on synthetic and real-world datasets show that the method can accurately determine the true number of clusters by finding the partition that maximizes cluster stability.
Polynomial matrices can help to elegantly formulate many broadband multi-sensor / multi-channel processing problems, and represent a direct extension of well-established narrowband techniques which typically involve eigen- (EVD) and singular value decompositions (SVD) for optimisation. Polynomial matrix decompositions extend the utility of the EVD to polynomial parahermitian matrices, and this talk presents a brief overview of such polynomial matrices, characteristics of the polynomial EVD (PEVD) and iterative algorithms for its solution. The presentation concludes with some surprising results when applying the PEVD to subband coding and broadband beamforming.
Covariance matrices are central to many adaptive filtering and optimisation problems. In practice, they have to be estimated from a finite number of samples; on this, I will review some known results from spectrum estimation and multiple-input multiple-output communications systems, and how properties that are assumed to be inherent in covariance and power spectral densities can easily be lost in the estimation process. I will discuss new results on space-time covariance estimation, and how the estimation from finite sample sets will impact on factorisations such as the eigenvalue decomposition, which is often key to solving the introductory optimisation problems. The purpose of the presentation is to give you some insight into estimating statistics as well as to provide a glimpse on classical signal processing challenges such as the separation of sources from a mixture of signals.
A discussion on sampling graphs to approximate network classification functionsLARCA UPC
The problem of network classification consists on assigning a finite set of labels to the nodes of the graphs; the underlying assumption is that nodes with the same label tend to be connected via strong paths in the graph. This is similar to the assumptions made by semi-supervised learning algorithms based on graphs, which build an artificial graph from vectorial data. Such semi-supervised algorithms are based on label propagation principles and their accuracy heavily relies on the structure (presence of edges) in the graph.
In this talk I will discuss ideas of how to perform sampling in the network graph, thus sparsifying the structure in order to apply semi-supervised algorithms and compute efficiently the classification function on the network. I will show very preliminary experiments indicating that the sampling technique has an important effect on the final results and discuss open theoretical and practical questions that are to be solved yet.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Bridging knowledge graphs_to_generate_scene_graphsWoen Yon Lai
The original paper link: https://arxiv.org/abs/2001.02314
* Disclaimer, I am not the author of this paper. I merely review this paper during a reading group discussion.
Polynomial matrices can help to elegantly formulate many broadband multi-sensor / multi-channel processing problems, and represent a direct extension of well-established narrowband techniques which typically involve eigen- (EVD) and singular value decompositions (SVD) for optimisation. Polynomial matrix decompositions extend the utility of the EVD to polynomial parahermitian matrices, and this talk presents a brief overview of such polynomial matrices, characteristics of the polynomial EVD (PEVD) and iterative algorithms for its solution. The presentation concludes with some surprising results when applying the PEVD to subband coding and broadband beamforming.
Covariance matrices are central to many adaptive filtering and optimisation problems. In practice, they have to be estimated from a finite number of samples; on this, I will review some known results from spectrum estimation and multiple-input multiple-output communications systems, and how properties that are assumed to be inherent in covariance and power spectral densities can easily be lost in the estimation process. I will discuss new results on space-time covariance estimation, and how the estimation from finite sample sets will impact on factorisations such as the eigenvalue decomposition, which is often key to solving the introductory optimisation problems. The purpose of the presentation is to give you some insight into estimating statistics as well as to provide a glimpse on classical signal processing challenges such as the separation of sources from a mixture of signals.
A discussion on sampling graphs to approximate network classification functionsLARCA UPC
The problem of network classification consists on assigning a finite set of labels to the nodes of the graphs; the underlying assumption is that nodes with the same label tend to be connected via strong paths in the graph. This is similar to the assumptions made by semi-supervised learning algorithms based on graphs, which build an artificial graph from vectorial data. Such semi-supervised algorithms are based on label propagation principles and their accuracy heavily relies on the structure (presence of edges) in the graph.
In this talk I will discuss ideas of how to perform sampling in the network graph, thus sparsifying the structure in order to apply semi-supervised algorithms and compute efficiently the classification function on the network. I will show very preliminary experiments indicating that the sampling technique has an important effect on the final results and discuss open theoretical and practical questions that are to be solved yet.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Bridging knowledge graphs_to_generate_scene_graphsWoen Yon Lai
The original paper link: https://arxiv.org/abs/2001.02314
* Disclaimer, I am not the author of this paper. I merely review this paper during a reading group discussion.
ElectroencephalographySignalClassification based on Sub-Band Common Spatial P...IOSRJVSP
Brain-computer interface (BCI) is a communication pathway between brain and an external device. It translates human thought into commands to control the external devices.Electroencephalography (EEG) is cost effective and easier way to implement the BCI. This paper presents a novel method for classifying EEG during motor imagery by the combination of common spatial pattern (CSP) and linear discriminant analysis (LDA). In the proposed method, the EEG signal is bandpass-filtered into multiple frequency bands. The CSP features are then extracted from each of these bands. The LDA classifier is subsequently used to classify the CSP features. In this paper, experimental results are presented on a publicly available BCI competition dataset and the performance is compared with existing approaches. The experimental result shows that the proposed method yields comparatively superior cross validation accuracies compared to prevailing methods.
Machine learning in science and industry — day 2arogozhnikov
- decision trees
- random forest
- Boosting: adaboost
- reweighting with boosting
- gradient boosting
- learning to rank with gradient boosting
- multiclass classification
- trigger in LHCb
- boosting to uniformity and flatness loss
- particle identification
Machine learning in science and industry — day 1arogozhnikov
A course of machine learning in science and industry.
- notions and applications
- nearest neighbours: search and machine learning algorithms
- roc curve
- optimal classification and regression
- density estimation
- Gaussian mixtures and EM algorithm
- clustering, an example of clustering in the opera
Soft computing is likely to play aprogressively important role in many applications including image enhancement. The paradigm for soft computing is the human mind. The soft computing critique has been particularly strong with fuzzy logic. The fuzzy logic is facts representationas a
rule for management of uncertainty. Inthis paperthe Multi-Dimensional optimized problem is addressed by discussing the optimal thresholding usingfuzzyentropyfor Image enhancement. This technique is compared with bi-level and multi-level thresholding and obtained optimal
thresholding values for different levels of speckle noisy and low contrasted images. The fuzzy entropy method has produced better results compared to bi-level and multi-level thresholding techniques.
Steganographic Scheme Based on Message-Cover matchingIJECEIAES
Steganography is one of the techniques that enter into the field of information security, it is the art of dissimulating data into digital files in an imperceptible way that does not arise the suspicion. In this paper, a steganographic method based on the FaberSchauder discrete wavelet transform is proposed. The embedding of the secret data is performed in Least Significant Bit (LSB) of the integer part of the wavelet coefficients. The secret message is decomposed into pairs of bits, then each pair is transformed into another based on a permutation that allows to obtain the most matches possible between the message and the LSB of the coefficients. To assess the performance of the proposed method, experiments were carried out on a large set of images, and a comparison to prior works is accomplished. Results show a good level of imperceptibility and a good trade-off imperceptibility-capacity compared to literature.
Image Denoising Based On Sparse Representation In A Probabilistic FrameworkCSCJournals
Image denoising is an interesting inverse problem. By denoising we mean finding a clean image, given a noisy one. In this paper, we propose a novel image denoising technique based on the generalized k density model as an extension to the probabilistic framework for solving image denoising problem. The approach is based on using overcomplete basis dictionary for sparsely representing the image under interest. To learn the overcomplete basis, we used the generalized k density model based ICA. The learned dictionary used after that for denoising speech signals and other images. Experimental results confirm the effectiveness of the proposed method for image denoising. The comparison with other denoising methods is also made and it is shown that the proposed method produces the best denoising effect.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
ElectroencephalographySignalClassification based on Sub-Band Common Spatial P...IOSRJVSP
Brain-computer interface (BCI) is a communication pathway between brain and an external device. It translates human thought into commands to control the external devices.Electroencephalography (EEG) is cost effective and easier way to implement the BCI. This paper presents a novel method for classifying EEG during motor imagery by the combination of common spatial pattern (CSP) and linear discriminant analysis (LDA). In the proposed method, the EEG signal is bandpass-filtered into multiple frequency bands. The CSP features are then extracted from each of these bands. The LDA classifier is subsequently used to classify the CSP features. In this paper, experimental results are presented on a publicly available BCI competition dataset and the performance is compared with existing approaches. The experimental result shows that the proposed method yields comparatively superior cross validation accuracies compared to prevailing methods.
Machine learning in science and industry — day 2arogozhnikov
- decision trees
- random forest
- Boosting: adaboost
- reweighting with boosting
- gradient boosting
- learning to rank with gradient boosting
- multiclass classification
- trigger in LHCb
- boosting to uniformity and flatness loss
- particle identification
Machine learning in science and industry — day 1arogozhnikov
A course of machine learning in science and industry.
- notions and applications
- nearest neighbours: search and machine learning algorithms
- roc curve
- optimal classification and regression
- density estimation
- Gaussian mixtures and EM algorithm
- clustering, an example of clustering in the opera
Soft computing is likely to play aprogressively important role in many applications including image enhancement. The paradigm for soft computing is the human mind. The soft computing critique has been particularly strong with fuzzy logic. The fuzzy logic is facts representationas a
rule for management of uncertainty. Inthis paperthe Multi-Dimensional optimized problem is addressed by discussing the optimal thresholding usingfuzzyentropyfor Image enhancement. This technique is compared with bi-level and multi-level thresholding and obtained optimal
thresholding values for different levels of speckle noisy and low contrasted images. The fuzzy entropy method has produced better results compared to bi-level and multi-level thresholding techniques.
Steganographic Scheme Based on Message-Cover matchingIJECEIAES
Steganography is one of the techniques that enter into the field of information security, it is the art of dissimulating data into digital files in an imperceptible way that does not arise the suspicion. In this paper, a steganographic method based on the FaberSchauder discrete wavelet transform is proposed. The embedding of the secret data is performed in Least Significant Bit (LSB) of the integer part of the wavelet coefficients. The secret message is decomposed into pairs of bits, then each pair is transformed into another based on a permutation that allows to obtain the most matches possible between the message and the LSB of the coefficients. To assess the performance of the proposed method, experiments were carried out on a large set of images, and a comparison to prior works is accomplished. Results show a good level of imperceptibility and a good trade-off imperceptibility-capacity compared to literature.
Image Denoising Based On Sparse Representation In A Probabilistic FrameworkCSCJournals
Image denoising is an interesting inverse problem. By denoising we mean finding a clean image, given a noisy one. In this paper, we propose a novel image denoising technique based on the generalized k density model as an extension to the probabilistic framework for solving image denoising problem. The approach is based on using overcomplete basis dictionary for sparsely representing the image under interest. To learn the overcomplete basis, we used the generalized k density model based ICA. The learned dictionary used after that for denoising speech signals and other images. Experimental results confirm the effectiveness of the proposed method for image denoising. The comparison with other denoising methods is also made and it is shown that the proposed method produces the best denoising effect.
Robust Image Denoising in RKHS via Orthogonal Matching PursuitPantelis Bouboulis
We present a robust method for the image denoising task based on kernel ridge regression and sparse modeling. Added noise is assumed to consist of two parts. One part is impulse noise assumed to be sparse (outliers), while the other part is bounded noise. The noisy image is divided into small regions of interest, whose pixels are regarded as points of a two-dimensional surface. A kernel based ridge regression method, whose parameters are selected adaptively, is employed to fit the data, whereas the outliers are detected via the use of the increasingly popular orthogonal matching pursuit (OMP) algorithm. To this end, a new variant of the OMP rationale is employed that has the additional advantage to automatically terminate, when all outliers have been selected.
Lesson 16: Inverse Trigonometric Functions (Section 041 slides)Mel Anthony Pepito
We cover the inverses to the trigonometric functions sine, cosine, tangent, cotangent, secant, cosecant, and their derivatives. The remarkable fact is that although these functions and their inverses are transcendental (complicated) functions, the derivatives are algebraic functions. Also, we meet my all-time favorite function: arctan.
Implicit differentiation allows us to find slopes of lines tangent to curves that are not graphs of functions. Almost all of the time (yes, that is a mathematical term!) we can assume the curve comprises the graph of a function and differentiate using the chain rule.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
Mine Blood Donors Information through Improved K-Means Clusteringijcsity
The number of accidents and health diseases which are increasing at an alarming rate are resulting in a huge increase in the demand for blood. There is a necessity for the organized analysis of the blood donor database or blood banks repositories. Clustering analysis is one of the data mining applications and K-means clustering algorithm is the fundamental algorithm for modern clustering techniques. K-means clustering algorithm is traditional approach and iterative algorithm. At every iteration, it attempts to find the distance from the centroid of each cluster to each and every data point. This paper gives the improvement to the original k-means algorithm by improving the initial centroids with distribution of data. Results and discussions show that improved K-means algorithm produces accurate clusters in less computation time to find the donors information
Jiawei Han, Micheline Kamber and Jian Pei
Data Mining: Concepts and Techniques, 3rd ed.
The Morgan Kaufmann Series in Data Management Systems
Morgan Kaufmann Publishers, July 2011. ISBN 978-0123814791
K-Means, its Variants and its ApplicationsVarad Meru
This presentation was given by our project group at the Lead College competition at Shivaji University. Our project got the 1st Prize. We focused mainly on Rough K-Means and build a Social-Network-Recommender System based on Rough K-Means.
The Members of the Project group were -
Mansi Kulkarni,
Nikhil Ingole,
Prasad Mohite,
Varad Meru
Vishal Bhavsar.
Wonderful Experience !!!
This research paper demonstrates the invention of the kinetic bands, based on Romanian mathematician and statistician Octav Onicescu’s kinetic energy, also known as “informational energy”, where we use historical data of foreign exchange currencies or indexes to predict the trend displayed by a stock or an index and whether it will go up or down in the future. Here, we explore the imperfections of the Bollinger Bands to determine a more sophisticated triplet of indicators that predict the future movement of prices in the Stock Market. An Extreme Gradient Boosting Modelling was conducted in Python using historical data set from Kaggle, the historical data set spanning all current 500 companies listed. An invariable importance feature was plotted. The results displayed that Kinetic Bands, derived from (KE) are very influential as features or technical indicators of stock market trends. Furthermore, experiments done through this invention provide tangible evidence of the empirical aspects of it. The machine learning code has low chances of error if all the proper procedures and coding are in play. The experiment samples are attached to this study for future references or scrutiny.
Chapter 10. Cluster Analysis Basic Concepts and Methods.pptSubrata Kumer Paul
Jiawei Han, Micheline Kamber and Jian Pei
Data Mining: Concepts and Techniques, 3rd ed.
The Morgan Kaufmann Series in Data Management Systems
Morgan Kaufmann Publishers, July 2011. ISBN 978-0123814791
A COMPREHENSIVE ANALYSIS OF QUANTUM CLUSTERING : FINDING ALL THE POTENTIAL MI...IJDKP
Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is
accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a
σ value, a hyper-parameter which can be manually defined and manipulated to suit the application.
Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster
centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the
exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an
outstanding task because normally such expressions are impossible to solve analytically. However, we
prove that if the points are all included in a square region of size σ, there is only one minimum. This bound
is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new
numerical approach “per block”. This technique decreases the number of particles by approximating some
groups of particles to weighted particles. These findings are not only useful to the quantum clustering
problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics
and other applications.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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Methods from Mathematical Data Mining (Supported by Optimization)
1. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Methods from Mathematical Data Mining
(Supported by Optimization)
Gerhard-Wilhelm Weber * and Başak Akteke-Öztürk
Gerhard- Akteke-
Institute of Applied Mathematics
Middle East Technical University, Ankara, Turkey
* Faculty of Economics, Management and Law, University of Siegen, Germany
Center for Research on Optimization and Control, University of Aveiro, Portugal
1
EURO CBBM
EURO EURO ORD
EURO CE*OC August 8, 2009
2. 4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
National University of Technology of the Ukraine
Kiev, Ukraine, August 5-16, 2009
Clustering Theory
Cluster Number and Cluster Stability Estimation
Z. Volkovich
Software Engineering Department, ORT Braude College of Engineering, Karmiel 21982, Israel
Z. Barzily
Software Engineering Department, ORT Braude College of Engineering, Karmiel 21982, Israel
G.-W. Weber
Departments of Scientific Computing, Financial Mathematics and Actuarial Sciences,
Institute of Applied Mathematics, Middle East Technical University, 06531, Ankara, Turkey
D. Toledano-Kitai
Software Engineering Department, ORT Braude College of Engineering, Karmiel 21982, Israel
2
August 8, 2009
3. Clustering
• An essential tool for “unsupervised” learning is
cluster analysis which suggests categorizing data
(objects, instances) into groups such that the
likeness within a group is much higher than the one
between the groups.
• This resemblance is often described by a
distance function.
3
August 8, 2009
4. Clustering
For a given set S ⊂ IR d a clustering algorithm CL
constructs a clustered set:
CL(S, int-part, k) = Π(S) = (π1(S) ,…, πk (S)),
such that CL(x) = CL(y) = i, if x and y are similar:
x, y ∈ πi(S), for some i=1,…,k;
and CL(x) ≠ CL(y), if x and y are dissimilar.
4
August 8, 2009
5. Clustering
The disjoint subsets πi (S), i=1,…,k, are named
clusters:
k
U π (S )
i =1
i = S , and π i ∩ π j = ∅ for i ≠ j.
5
August 8, 2009
7. Clustering
The iterative clustering process is usually carried out in two phases:
a partitioning phase and a quality assessment phase.
In the partitioning phase, a label is assigned to each element
in view of the assumption that, in addition to the observed features,
for each data item, there is a hidden, unobserved feature
representing cluster membership.
The quality assessment phase measures the grouping quality.
The outcome of the clustering process is a partition that acquires
the highest quality score.
Except for the data itself, two essential input parameters are
typically required: an initial partition and a suggested number of
clusters. Here, the parameters are denoted as
• int-part ;
• k. 7
August 8, 2009
8. The Problem
Partitions generated by the iterative algorithms are commonly
sensitive to initial partitions fed in as an input parameter.
Selection of “good” initial partitions is an essential
clustering problem.
Another problem arising here is choosing the right number of the
clusters. It is well known that this key task of the cluster analysis
is ill posed. For instance, the “correct” number of clusters in a
data set can depend on the scale in which the data are measured.
In this talk, we address to the last problem concerning
determination of the number of clusters.
8
August 8, 2009
9. The Problem
Partitions generated by the iterative algorithms are commonly
sensitive to initial partitions fed in as an input parameter.
Selection of “good” initial partitions is an essential
clustering problem.
Another problem arising here is choosing the right number of the
clusters. It is well known that this key task of the cluster analysis
is ill posed. For instance, the “correct” number of clusters in a
data set can depend on the scale in which the data are measured.
9
August 8, 2009
10. The Problem
Many approaches to this problem exploit the within-cluster
dispersion matrix (defined according to the pattern of a
covariance matrix). The span of this matrix (column space)
usually decreases as the number of groups rises, and may have
a point in which it “falls”. Such an “elbow” on the graph locates,
in several known methods, the “true” number of clusters.
Stability based approaches, for the cluster validation problem,
evaluate the partitions’ variability under repeated applications
of a clustering algorithm. Low variability is understood as
high consistency in the result obtained, and the number of clusters
that maximizes cluster stability is accepted as an estimate for the
“true” number of clusters.
10
August 8, 2009
11. The Concept
In the current talk, the problem of determining the
true number of clusters is addressed by the cluster
stability approach.
We propose a method for the study of cluster stability.
This method suggests a geometrical stability of a
partition.
• We draw samples from the source data and estimate
the clusters by means of each of the drawn samples.
• We compare pairs of the partitions obtained.
• A pair is considered to be consistent if the obtained
division is close.
11
August 8, 2009
12. The Concept
• We quantify this closeness by the number of edges
connecting points from different samples in a
minimal spanning tree (MST) constructed for each one
of the clusters.
• We use the Friedman and Rafsky two sample test
statistic which measures these quantities. Under the
null hypothesis on the homogeneity of the source data,
this statistic is approximately normally distributed.
So, the case of well mingled samples within the clusters
leads to normal distribution of the considered statistic.
12
August 8, 2009
14. The Concept
The left-side picture is an example of “a good cluster”
where the quantity of edges connecting points from
different samples (marked by solid red lines) is
relatively big.
The right-side picture images a “poor situation” when
only one (and long) edge connects the (sub-) clusters.
14
August 8, 2009
15. The Two-Sample MST-Test
Henze and Penrose (1979) considered the asymptotic behavior of
Rmn :
the number of edges of V which connect a point of S to a point of T .
Suppose that |S|=m → ∞ and |T|=n → ∞ such that
m /(m+n) → p∈ (0, 1).
∈
Introducing q = 1 − p and r = 2pq, they obtained:
1
Rmn −
2mn
m+n
(
→ N 0, σ d
2
),
m+n
2
where the convergence is in distribution and N(0, σ d ) denotes
the normal distribution with a 0 expectation and a variance
2
σ d := r (r + Cd (1 − 2r)), for some constant Cd
depending only on the space’s dimension d.
15
August 8, 2009
16. Concept
• Resting upon this fact, the standard score
2K m
Y j := Rj −
m K
of the mentioned edges quantity is calculated
for each cluster j=1,…, K ,
where m is the sample size and
K denotes the number of clusters.
%
• The partition quality Y is represented by the
worst cluster corresponding to the
minimal standard score value obtained.
16
August 8, 2009
17. Concept
• It is natural to expect that the true number of
clusters can be characterized by the empirical
distribution of the partition standard score
having the shortest left tail.
• The proposed methodology is expressed as a
sequential creation of the described distribution
with its left-asymmetry estimation.
17
August 8, 2009
18. Concept
One of important problems appearing here is the
so-called clusters coordination problem.
Actually, the same cluster can be differently tagged
within repeated rerunning of the algorithm.
This fact results from the inherent symmetry of
the partitions according to their clusters labels.
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19. Concept
We solve this problem by the following way:
Let S = S1 ∪ S 2 . Consider three categorizations:
Π K := Cl ( S , K ) ,
Π K ,1 := Cl ( S1, K ) ,
Π K ,2 := Cl ( S2 , K ) .
Thus, we get two partitions for each of the samples
Si, i=1,2. The first one is induced by ΠK and the
second one is Π K ,i , i = 1, 2 .
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20. Concept
For each one of the samples i =1,2, our purpose is
to find the permutation ψ of the set {1,…,K} which
minimizes the quantities of the misclassified items:
( i ) x , i = 1, 2 ,
ψ i* ψ α
= arg min ∑ I ( )
K ,i ( x ) ≠ α K ( )
ψ x∈ X
where I(z) is the indicator function of the event z and
α K ,i , α Ki ) are assignments defined by ∏ K , ∏ K ,i ,
(
correspondingly.
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21. Concept
The well-known Hungarian method for solving
this problem has computational complexity of O(K3).
After changing the cluster labels of the partitions
∏ K ,i , i = 1, 2 , consistent with ψ i , i = 1, 2 ,
*
we can assume that these partitions are coordinated,
i.e., the clusters are consistently designated.
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22. Algorithm
1. Choose the parameters: K*, J, m, Cl .
2. For K = 2 to K*
3. For j = 1 to J
4. Sj,1= sample (X, m) , Sj,2= sample (X Sj,1, m)
5. Calculate
ΠK , j =Cl( S(j), K) ,
ΠK , j,1 =Cl( Sj ,1, K) ,
ΠK , j,2 =Cl( Sj ,2, K) .
6. Solve the coordination problem.
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23. Algorithm
7. Calculate Yj(k), k=1,…,K, % (jK ) .
Y
8. end if j
9. Calculate an asymmetry index (percentile) IK
% (jK ) | j = 1,...,J }.
for {Y
10. end if K
11. The “true” K* is selected as the one which yields
the maximal value of the index.
Here, sample(S,m) is a procedure which selects a
random sample of size m from the set S, without
replacement. 23
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24. Numerical Experiments
We have carried out various numerical experiments on synthetic
and real data sets. We choose K*=7 in all tests, and we provide
10 trials for each experiment.
The results are presented via the error-bar plots of the sample
percentiles’ mean within the trials. The sizes of the error bars
equal two standard deviations, found inside the trials of the results.
The standard version of the Partitioning Around Medoids (PAM)
algorithm has been used for clustering.
The empirical percentiles of 25%, 75% and 90% have been used
as the asymmetry indexes.
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25. Numerical Experiments – Synthetic Data
The synthesized data are mixtures of 2-dimensional
Gaussian distributions with independent coordinates
owning the same standard deviation σ.
Mean values of the components are placed on the
unit circle on the angular neighboring distance 2π / k .
ˆ
Each data set contains 4000 items.
Here, we took J=100 (J: number of samples) and
m=200 (m: size of samples).
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26. Synthetic Data - Example 1
The first data set has the parameters k = 4 and σ = 0.3.
ˆ
As we see, all of the three indexes clearly indicate
four clusters. 26
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27. Synthetic Data - Example 2
The second synthetic data set has the parameters k = 5
ˆ
and σ = 0.3.
The components are obviously overlapping in this case.
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28. Synthetic Data - Example 2
As it can be seen, the true number of clusters has been
successfully found by all indexes.
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29. Numerical Experiments – Real-World Data
First Data Sets
The first real data set was chosen from the text collection
http://ftp.cs.cornell.edu/pub/smart/ .
This set consists of the following three sub-collections
DC0: Medlars Collection (1033 medical abstracts),
DC1: CISI Collection (1460 information science abstracts),
DC2: Cranfield Collection (1400 aerodynamics abstracts).
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30. Numerical Experiments – Real-World Data
First Data Sets
We picked the 600 “best” terms, following the common
bag of words method.
It is known that this collection is well separated
by means of its first two leading principal components.
Here, we also took J=100 and m=200.
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31. Real-World Data - First Data Sets
All the indexes receive their maximal values at K=3,
i.e., the number of clusters is properly determined.
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32. Numerical Experiments – Real-World Data
Second Data Set
Another considered data set is the famous
Iris Flower Data Set, available, for example, at
http://archive.ics.uci.edu/ml/datasets/Iris .
This dataset is composed from 150 4-dimensional
feature vectors of three equally sized sets of iris flowers.
We choose J=200 and the sample size equals 70.
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33. Real-World Data – Iris Flower Data Set
Our method turns out a three clusters structure.
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34. Conclusions -
The Rationale of Our Approach
• In this paper, we propose a novel approach, based on
the Minimal Spanning Tree two sample test, for the
cluster stability assessment.
• The method offers to quantify the partitions’ features
through the test statistic computed within the clusters
built by means of sample pairs.
• The worst cluster, determined by the lowest
standardized statistic value, characterizes the
partition quality.
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35. Conclusions -
The Rationale of Our Approach
• The departure from the theoretical model, which
suggests well-mingled samples within the clusters,
is described by the left tail of the score distribution.
• The shortest tail corresponds to the “true” number
of clusters.
• All presented experiments detect the true number
of clusters.
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36. Conclusions
• In the case of the five components Gaussian data set,
the true number of clusters was found even though
a certain overlapping of the clusters exists.
• The four Gaussian components data set contains
sufficiently separated components. Therefore,
it is of no revelation that the true number of clusters
is attained here.
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37. Conclusions
• The analysis of the abstracts data set is carried out
with 600 terms and the true number of clusters
was also detected.
• The Iris Flower dataset is sufficiently difficult to
analyze due to the fact that two clusters are not
linearly separable. However, the true number
of clusters was found here as well.
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38. References
Barzily, Z., Volkovich, Z.V., Akteke-Öztürk, B., and Weber, G.-W., Cluster stability using minimal spanning trees,
ISI Proceedings of 20th Mini-EURO Conference Continuous Optimization and Knowledge-Based Technologies
(Neringa, Lithuania, May 20-23, 2008) 248-252.
Barzily, Z., Volkovich, Z.V., Akteke-Öztürk, B., and Weber, G.-W., On a minimal spanning tree approach in the
cluster validation problem, to appear in the special issue of INFORMATICA at the occasion of 20th Mini-EURO
Conference Continuous Optimization and Knowledge Based Technologies (Neringa, Lithuania, May 20-23, 2008),
Dzemyda, G., Miettinen, K., and Sakalauskas, L., guest editors.
Volkovich, V., Barzily, Z., Weber, G.-W., and Toledano-Kitai, D., Cluster stability estimation based on a minimal
spanning trees approach, Proceedings of the Second Global Conference on Power Control and Optimization, AIP
Conference Proceedings 1159, Bali, Indonesia, 1-3 June 2009, Subseries: Mathematical and Statistical Physics; ISBN
978-0-7354-0696-4 (August 2009) 299-305; Hakim, A.H., Vasant, P., and Barsoum, N., guest eds..
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