of package SCUBA - University of Bathmasas/scuba/SCUBA_manual_1.0.pdf · 2011. 1. 21. · Common tools for overcoming this problem are Spectral Clustering, Multidimensional Scaling

R documentationof package SCUBA

January 21, 2011

Contents

Background 3

Information about SCUBA 6

SCUBA-package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Description of implemented functions 8

displayKMeans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

displaySpectClust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

plotEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

PlotOutData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

plotPairsEigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

saveAllFilesForRun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

scuba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

spectClust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

ZoomArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Index 23

2

Background

The analysis of data from a micro-array experiment is a major problem in countless biological

situations. For example, a common need is to cluster cancer samples using gene expression profiles (see,

for example, Chapter 9 of Wit and McClure [2004]). Typically, gene expression values are recorded in

a matrix (table) W whose elements are non-negative numbers and the entry in the i-th row and j-th

column of W , say Wij , represents the activity of gene i in sample j. Given the information in W ,

one might want to classify (or cluster) the samples, or the genes, according to how “similar” they are

to each other, where similarity is measured in terms of the activity of the genes across the samples.

There are many elements in W because of the number of genes usually involved in such analyses. As a

consequence, manipulating, visualising or clustering of micro-array data becomes a very hard problem.

Common tools for overcoming this problem are Spectral Clustering, Multidimensional Scaling (MDS)

and Principal Component Analysis (PCA), with the main idea of all these methods being the dimension

reduction of the data.

Our software is based on Spectral Clustering (c.f. Higham et al. [2005], von Luxburg [2007]) and

we shall give a few more details and references for this technique in the next paragraph. However, it is

well known that micro-array data is subject to noise, and hence it is obviously important to know how

this noise affects the classifications produced by Spectral Clustering. The program SCUBA (Spectral

Clustering Uncertainty via Bayesian Analysis) attempts to provide the user with a measure of how

the uncertainty in the micro-array data affects the Spectral Clustering using techniques from Bayesian

Statistics.

The algorithm of Spectral Clustering applied to a gene-sample data matrix, as described in Higham

et al. [2005] and Higham et al. [2007], is based on the Singular Value Decomposition (SVD) of a scaled

version of the gene-sample matrix W . Roughly speaking, the idea behind the algorithm is to represent

the data in a low-dimensional space in such a way that the Euclidean distances in low dimensions are

inversely related to the similarities derived from the data matrix W . This is achieved via the coordi-

nates of the singular vectors of the scaled version of W . In this respect, Spectral Clustering resembles

very closely MDS and PCA in that all three methods use the spectral decomposition (or the SVD) of

a distance or a similarity matrix associated with the data. Thus, all three methods provide low-rank

3

4 Background

approximations of such a distance (or similarity) matrix derived from W . (In fact, it can be proved that

Spectral Clustering is mathematically equivalent to a scaled version of MDS (c.f. Bavaud [2006]).)

In Oh and Raftery [2001] the idea of using Markov Chain Monte Carlo (MCMC) simulations

together with MDS was introduced. The aim of their approach was, given pairwise distances between

some data, to take into account possible uncertainties, or inaccuracies, in the measurement of these

distances and, based on this, to find an optimal low-dimensional configuration of the data. SCUBA

utilizes some of these ideas (c.f. Hurn et al. [2011]) and applies MCMC techniques to Spectral Clustering

in order to provide a tool for visualising the effects of the noise in gene-expression data. As an output,

SCUBA plots a low-dimensional configuration of points, obtained by applying Spectral Clustering to the

gene-expression data, as well as “clouds” around each point. These “clouds” are a visual representation

of the uncertainty in Spectral Clustering induced by the noise in micro-array data.

Bibliography

F. Bavaud. Spectral clustering and multidimensional scaling: a unified view. Proccedings of the IFCS

2006 Conference: “Data Science and Classification”, Ljubljana, Slovenia, July 25 - 29, 2006, 2006.

D. J. Higham, G. Kalna, and J. K. Vass. Analysis of the singular value decomposition as a tool for

processing microarray expression data. Algoritmy, pages 250–259, 2005.

D. J. Higham, G. Kalna, and M. Kibble. Spectral clustering and its use in bioinformatics. Journal of

Computational and Applied Mathematics, 204(1), 2007.

M. A. Hurn, S. Shaw, A. Spence, and Z. Stoyanov. Spectral clustering uncertainty via bayesian analysis

(scuba): Report. University of Bath, 2011.

Man-Suk Oh and A. E. Raftery. Bayesian multidimensional scaling and choice of dimension. Journal

of the American Statistical Association, 96(455):1031–1044, 2001.

U. von Luxburg. A tutorial on spectral clustering. Stat. Comput., 17:395–416, 2007.

E. Wit and J. McClure. Statistics for Microarrays. Wiley, 2004.

5

Information about SCUBA

SCUBA-package Spectral Clustering Uncertainty via Bayesiand Analysis

Description

The package provides functionalities for performing an uncertainty analysis of Spectral Clustering

on a data matrix. It implements various plot functions for representing the results.

Details

Package: SCUBA

Type: Package

Version: 1.0

Date: 2011-01-20

License: GPL-3

The main function is scuba. This function applies MCMC techniques to Spectral Clustering in

the following way: it takes the low-dimensional configuration computed from Spectral Clustering

as an initial configuration for an MCMC algorithm, which updates that configuration (and other pa-

rameters) iteratively. The function scuba calls a Fortran subroutine, which performs the MCMC

uncertainty algorithm. This package also provides various functionalities for performing Spectral

Clustering on large data matrices (see, for example, the function spectClust). The package

was originally developed to analyse matrices with gene-sample data obtained from micro-array ex-

periments, but can easily be adapted for other data from other applications as well (as long as the

data can be represented as a rectangular matrix with non-negative elements). It also provides sev-

6

SCUBA-package 7

eral functions for visualisation (2D plots) of the results (see, for example, the following functions:

displaySpectClust, displayKMeans, scuba, PlotOutData). For more details about

the main function in the package, scuba, and about the package in general, please see our report,

Spectral Clustering Uncertainty via Bayesian Analysis (SCUBA): Report.

Author(s)

Merrilee Hurn, Simon Shaw, Alastair Spence, Zhivko Stoyanov, Ivelina Stoyanova

Maintainer: Zhivko Stoyanov , Ivelina Stoyanova

References

1. F. Bavaud Spectral clustering and multidimensional scaling: a unified view, Proccedings of

the IFCS 2006 Conference, 2006.

2. S. Butler Spectral Graph Theory: Three common spectra, Nankai University, Tianjin, 2006.

3. D. J. Higham, G. Kalna and J. K. Vass Analysis of the singular value decomposition as a tool

for processing microarray expression data, Algoritmy, pages 250–259, 2005.

4. D. J. Higham, G. Kalna and M. Kibble Spectral Clustering and its use in Bioinformatics,

Journal of Computational and Applied Mathematics, 204(1), 2007.

5. M. A. Hurn, S. Shaw, A. Spence and Z. V. Stoyanov Spectral Clustering Uncertainty via

Bayesian Analysis (SCUBA): Report, University of Bath, 2011.

6. P. Grindrod, D. J. Higham, G. Kalna, A. Spence, Z. V. Stoyanov and J. K. Vass DNA Meets

the SVD, Mathematics today, 2008.

7. Man-Suk Oh and A. E. Raftery Bayesian Multidimensional Scaling and Choice of dimension,

Journal of the American Statistical Association, 96(455), 2001.

8. R. Sibson Studies in the Robustness of Multidimensional Scaling: Procrustes Statistics, J. R.

Statist. Soc. B, 40, No. 2, pp. 234-238, 1978.

9. E. Wit and J. McClure Statistics for Microarrays: Design, Analysis and Inference, Wiley,

2004.

See Also

scuba, spectClust, displaySpectClust, displayKMeans, PlotOutData

Description of implemented functions

displayKMeans Displays clustering of data from K Means algorithm

Description

The function performs K-means clustering on the low-dimensional presentation of a data matrix

obtained by slectral clustering.

Usage

displayKMeans(X, q="samples", nclust = 2, d = 2)

Arguments

X A rectangular matrix of non-negative elements: rows of X represent genes and

its columns are samples.

q Either "samples" or "genes". Shows whether clustering is performed on

samples (columns) or genes (rows).

nclust Number of clusters. This specifies the number of points (centres) to be selected

on the plot.

d The dimensionality of the data, first d dimensions taken.

Details

Given a data matrix X and the desired number of clusters nclust, the function plots the low-

dimensional representation obtained by spectral clustering and lets the user select (interactively) the

8

displaySpectClust 9

centres of the clusters. Then the function performes the K-means clustering algorithm implemented

in R’s package stats.

The function provides a graphical enhancement to the K-means clustering function (kmeans in

package stats).

Author(s)

Zhivko Stoyanov, Ivelina Stoyanova

See Also

spectClust

Examples

# using the preloaded data matrix Leukemia38

displayKMeans(Leukemia38)

# varying the number of clusters

displayKMeans(Leukemia38,nclust=3)

displaySpectClust Displays results of the spectral clustering algorithm

Description

The function plots the results of spectral clustering on either the "samples" or the "genes". In order

to do that it uses selected singular vectors of Xsc - a scaled version of the matrix X .

Usage

displaySpectClust(X, d=2, q="samples", keepFirst = FALSE)

10 displaySpectClust

Arguments


its columns are samples.

d Either 1 or 2. Present a plot of d-dimensions - i.e. first d eigenvectors. If 1 then

the first eigenvector is plotted against indices.

q Either "samples" or "genes". Shows whether clustering is performed on

samples (columns) or genes (rows).

keepFirst Either TRUE or FALSE. It is set to TRUE if the user wants to keep the first

eigenvalue (equal to 1) and its corresponding eigenvector. FALSE if the first

eugenvalue and eigenvector are to be ignored.

Details

The function plots the results of spectral clustering on either the "samples" (columns) or the "genes"

(rows) of the selected singular vectors of Xsc.

The function goes through the following stages:

1. Displays d-dimensional plot of the result of spectral clustering and next to it the plot with the

eigenvalues.

2. Allows the user to select interactively an eigenvalue from the plot and its index is obtained.

Displays 2-dimensional plots of all pairs of eigenvectors up to the index of the selected eigen-

value.

3. Allows the user to select a subplot of a particular pair of eigenvectors of interest.

4. Provides command line options for going back to (1) or (3) or exiting the program.

Author(s)


Examples

# testing on some random matrix

displaySpectClust(matrix(runif(8000),c(80,100)),2,"samples")

plotEigenvalues 11

#testing on the preloaded data matrix Leukemia38

displaySpectClust(Leukemia38,2,"genes")

# using the Zachary Karate Club set

displayKMeans(ZKC,nclust=2)

plotEigenvalues Plots eigenvalues against their indices

Description

The function displays the eigenvalues obtained by spectral clustering.

Usage

plotEigenvalues(data_matrix = NULL, Evals = NULL,

keepFirst = TRUE,

selecting = FALSE)

Arguments

data_matrix The data matrix to be analysed and its eigenvalues plotted. It can be NULL if

Evals is specified. In this case Evals contains the eigenvalues.

Evals The eigenvalues to be plotted. If this is provided, the eigenvalues are used di-

rectly, i.e. they are not computed.

keepFirst Either TRUE or FALSE (default TRUE). It is set to TRUE if the user wants to

keep the first eigenvalue (equal to 1) and its corresponding eigenvector. FALSE

if the first eigenvalue and eigenvector are to be ignored.

selecting Either TRUE or FALSE (default is FALSE). If TRUE the user is prompted to se-

lect an eigenvalue by clicking on the plot and the index of the selcted eigenvalue

is returned.

12 PlotOutData

Details

The function displays the eigenvalues obtained by spectral clustering. If data_matrix is speci-

fied the function calculates the eigenvalues using spectral clustering. If Evals is provided with the

pre-calculated eigenvalues, they are used directly.

Optionally, the first eigenvalue and eigenvector are omitted as they are not relevant for the clustering

problem. The function also can optionally return the index of the interactively selected eigenvalue.

Author(s)


See Also

spectClust, plotPairsEigenvectors

Examples

# the preloaded data matrix Leukemia38 is used

# first eigenvalue ignored

# allows selection of an eigenvalue from

the plot plotEigenvalues(data_matrix = Leukemia38, keepFirst =

FALSE, selecting = TRUE)

PlotOutData Plots the result of an MCMC uncertainty algorithm

Description

The main plotting function. It performs setup for plotting and calls the relevant plotting functions.

PlotOutData 13

Usage

PlotOutData(Xin, XinS, XoutR, XoutS, use_p,

cols=NULL,

first=2, second=NULL,

scaleOutData = FALSE, zoomOutData = FALSE)

Arguments

Xin Preprocessed input data matrix obtained from spectral clustering. This is the

initial configuration for the MCMC algorithm.

XinS Preprocessed input data matrix, scaled.

XoutR Postprocessed (rotated) output matrix from the MCMC algorithm.

XoutS Postprocessed (rotated and scaled) output matrix (a set of configurations).

use_p The dimension of the output configuration. This includes the first eigenvector

which is ignored in spectral clustering.

first The number of the eigenvector which is used for plotting. This defines the first

coordinate in the low-dimenional representation obtained from spectral cluster-

ing.

second The number of the other eigenvector used for plotting. If NULL, the first eigen-

value is plotted agains its indices.

scaleOutData

To be set to TRUE if the scaled result is required.

zoomOutData To be set to TRUE if it is required to zoom certain area(s) of the plot centred

at the selected point. The magnitude factor for zooming is user defined (on

command line).

Details

This function is called within the main function scuba but can be called with different data ob-

tained by independent method not in the package.

14 plotPairsEigenvectors

Author(s)


See Also

ZoomArea

plotPairsEigenvectors

Plots pairs of eigenvectors

Description

Displays all pairs of eigenvectors for a given matrix up to a specified index.

Usage

plotPairsEigenvectors(S, evalNumber, keepFirst = TRUE, selecting =

FALSE)

Arguments

S The matrix containing the eigenvectors as result from spectral clustering.

evalNumber The pairs of the first evalNumber eigenvectors will be plotted.

keepFirst Either TRUE or FALSE (default TRUE). It is set to TRUE if the user wants to

keep the first eigenvalue (equal to 1) and its corresponding eigenvector. FALSE

if the first eugenvalue and eigenvector are to be ignored.

selecting Either TRUE or FALSE. If TRUE the user is prompted to select a subplot from

the plot which is displayed enlarged.

saveAllFilesForRun 15

Details

This function displays all pairs of eigenvectors for a given matrix up to a specified index, evalNumber.

The evalNumber can be selected using the plotEigenvalues. Further, a subplot of a particular pair

can be selected to be displayed independently.

Author(s)


See Also

plotEigenvalues

Examples

# example for using the preloaded data matrix Leukemia38

# first, we get the matrix of the eigenvectors

# then plot the pairs of eigenvectors

sc

16 saveAllFilesForRun

Usage

saveAllFilesForRun(runName, outsideCall = TRUE)

Arguments

runName This argument can be set to a character string which is the name of the current

run of the program (with the current data matrix and parameters). All output

files with the result from the run will have their filenames prefixed by the name

of the run. This allows the user to distinguish between results from different

runs.

outsideCall TRUE if this is an independent call outside of the main function of the program.

If FALSE it is called from inside the main function and the working directory

need not be changed.

Details

The main MCMC algorithm uses Fortran and C subroutines which read and write data to/from

predefined files. Thus with each run of the program these files are rewritten and previous results are

lost.

This function provides the option of naming a run, i.e. with specific data and/or parameters, where

the filenames are prefixed with the name of the run. E.g. if we run the program with the ma-

trix Leukemia38 and save the result as run ’L38’ and then run the program with the matrix

ColonAlon and save it as run ’CA’ we can then compare the results from the two runs stored in

files L38-xs.out and CA-xs.out.

Author(s)


Examples

# after finishing a run which was not named

# deciding to save the results,

scuba 17

# e.g. they take considerable time to calculate again

saveAllFilesForRun(runName="someRun", outsideCall=TRUE)

scuba Display results of the MCMC uncertainty algorithm

Description

This is the main function of the SCUBA package. It performs Spectral Clustering of a data matrix

and uses that as in initial configuration of an MCMC algorithm (written in Fortran and C), which

updates that configuration (and other parameters of the model) iteratively. The output of the function

is a 2D plot, on which “clouds” serve as a visual representation of the noise in Spectral Clustering

indiced by uncertainty in the micro-array data.

Usage

scuba(filename = NULL, data_matrix = NULL,

nburn = 1000, nsamp = 1000,

usePreviousResultFilename = NULL,

scaleOutData = FALSE, zoomOutData = FALSE,

runName = NULL)

Arguments

filename Name of file containing a data matrix. It can be NULL if data_matrix is

specified instead.

data_matrix Preloaded data matrix. It can be NULL if data are loaded from a file.

use_p Dimension of result.

seed This is the seed used by the MCMC algorithm.

nsweep The number of iterations performed by the MCMC algorithm.

18 scuba

nburn The number of burn-in iterations used in the MCMC algorithm.

nsamp This is the number of samples to be output from the MCMC algorithm and

should therefore be a positive integer, which is less than nsweep-nburn. This

parameter becomes necessary when nsweep, the total number of iterations per-

formed by the MCMC algorithm, is “too large.” The latter would make the size

of the configuration, output from the MCMC algorithm, too big for R to handle.

Thus, choosing nsamp < nsweep-nburn selects (uniformly, with respect to the

number of the iteration) fewer samples to plot.usePreviousResultFilename

If this is specified a file with results from a previous run will be used and the

call to the MCMC algorithm will be omitted. This offers the opportunity to save

time for repeated anayses of a large data matrix.

scaleOutData To be set to TRUE if the scaled result needs to be plotted.

zoomOutData To be set to TRUE if it is required to perform some zooming on the plot with the

result.

runName This argument can be set to a character string which is the name of the current

run of the program (with the current data matrix and parameters). All output

files with the result from the run will have their filenames prefixed by the name

of the run. This allows the user to distinguish between results from different

runs.

Details

This is the main function of the SCUBA package. This function applies MCMC uncertainty algo-

rithm to the output from Spectral Clustering in the following way: it takes the low-dimensional con-

figuration, computed by Spectral Clustering, as an initial configuration for an MCMC algorithm,

which updates that configuration (and other parameters of the Bayesian model) iteratively. The

function scuba calls a Fortran subroutine (via CallMCMCFortran), which performs the MCMC

uncertainty algorithm. The name of the Fortran file, which contains the MCMC uncertainty algo-

rithm is positive.f. Experienced Fortran users may edit positive.f, and in this way alter

the MCMC algorithm, according to their own needs. The random number generators, used by the

MCMC algorithm, are the random generators used in R and are called in the C file rng.c. The

output of the function is a 2D plot, on which “clouds” serve as a visual representation of the noise

in Spectral Clustering indiced by uncertainty in the micro-array data. There are variaous options

for the plots (for more details, see the function PlotOutData). The function also provides the option

spectClust 19

of zooming particular regions of the 2D plot (see description of the function ZoomArea and the

variable zoomOutData).

Author(s)


References





3. M. A. Hurn, S. Shaw, A. Spence and Z. V. Stoyanov Spectral Clustering Uncertainty via

Bayesian Analysis (SCUBA): Report, University of Bath, 2011.

4. P. Grindrod, D. J. Higham, G. Kalna, A. Spence, Z. V. Stoyanov and J. K. Vass DNA Meets

the SVD, Mathematics today, 2008.

5. Man-Suk Oh and A. E. Raftery Bayesian Multidimensional Scaling and Choice of dimension,

Journal of the American Statistical Association, 96(455), 2001.

6. R. Sibson Studies in the Robustness of Multidimensional Scaling: Procrustes Statistics, J. R.

Statist. Soc. B, 40, No. 2, pp. 234-238, 1978.

7. E. Wit and J. McClure Statistics for Microarrays: Design, Analysis and Inference, Wiley,

2004.

Examples

scuba(data_matrix = Leukemia38, zoomOutData = TRUE)

scuba(data_matrix = Leukemia38, runName = "TestRun")

spectClust Spectral clustering algorithm

20 spectClust

Description

The function returns the singular values and singular vectors of a scaled version of the input matrix.

Usage

spectClust(X, econ = 0, keepFirst=FALSE)

Arguments


its columns are samples

econ Either 0, 1 or 2. Default value is 0. Shows whether economy of calcula-

tions/output is applied. Suppose Xsc = USV T is the result of SVD. If econ

is 0, then both matrices U and V are output, if 1 - only matrix U , if 2 - only

matrix V (this is the first to be calculated so for greater econ value less calcu-

lations are performed).

keepFirst Either TRUE or FALSE. Shows whether the first eigenvalue and corresponding

eigenvector need to be kept or not.

Details

The input matrix X is a gene-sample matrix, whose entries are non-negative and represent the

activity of a given gene in a particular sample. We assume that the i-th row of X represents the

activity of the i-th gene, and the j-th column of X gives the activity of all the genes in the j-th

sample.

The algorithm finds the spectrum (SVD) of a scaled version of the matrix X , that is Xsc = USV T ,

and returns a list of:

1. the array of eigenvalues;

2. the matrix U - which contains the eigenvectors of the matrix XT X; and/or

3. the matrix V - which contains the eigenvectors of the matrix XXT .

For more details see the references below.

ZoomArea 21

Author(s)


References





3. Peter Grindrod, Desmond J. Higham, Gabriela Kalna, Alastair Spence, Zhivko Stoyanov and

Keith Vass DNA Meets the SVD, Mathematics today, 2008.

Examples

spectClust(matrix(runif(8000),c(80,100)),0)

spectClust(Leukemia38)

ZoomArea Zooms the plot centered at a selected point

Description

Allows the user to zoom a particular region of a given plot by a user-defined magnitude factor and

centre point.

Usage

ZoomArea(Xin, Xout, use_p=3,

cols=NULL,

first = 2, second = NULL,

resultType = "rotated", zoom = NULL)

22 ZoomArea

Arguments

Xin Preprocessed input data matrix obtained from spectral clustering. This is the

initial configuration for the MCMC uncertainty algorithm.

Xout Postprocessed (rotated and/or scaled) output matrix (a set of configurations).

use_p The dimension of the output configuration. This includes the first eigenvector

which is ignored in spectral clustering.

first The number of the eigenvector which is used for plotting. This defines the first

coordinate in the low-dimensional representation obtained from spectral cluster-

ing.

second The number of the other eigenvector used for plotting. If NULL, the first eigen-

value is plotted against its indices.

resultType The type of the result to be plotted. It is important for correct labelling of axes.

The default is ’rotated’.

zoom The magnitude factor for zooming. If NULL the user is prompted to input a

magnitude factor on the command line.

Details

Allows the user to zoom a particular region of a plot centered at selected point. The magnitude

factor is user defined (on command line).

A click on the zoomed plot returns to the original (not zoomed) plot.

Author(s)


Index

∗Topic Gene-expression dataplotEigenvalues, 12

PlotOutData, 13

plotPairsEigenvectors, 15

saveAllFilesForRun, 16

ZoomArea, 22

∗Topic MCMCplotEigenvalues, 12

PlotOutData, 13



scuba, 18

SCUBA-package, 6

ZoomArea, 22

∗Topic MDSdisplaySpectClust, 10

spectClust, 21

∗Topic PCAdisplaySpectClust, 10

spectClust, 21

∗Topic SVDdisplayKMeans, 9

displaySpectClust, 10

spectClust, 21

∗Topic Spectral ClusteringdisplayKMeans, 9


plotEigenvalues, 12

PlotOutData, 13



scuba, 18

SCUBA-package, 6

spectClust, 21

ZoomArea, 22

∗Topic gene-expression datadisplayKMeans, 9


scuba, 18

SCUBA-package, 6

spectClust, 21

∗Topic k-meansdisplayKMeans, 9

Background, 3

CallMCMCFortran, 19

Description of implemented

functions, 9

displayKMeans, 7, 8, 9

displaySpectClust, 7, 8, 10

Information about SCUBA, 6

plotEigenvalues, 12, 16

PlotOutData, 7, 8, 13, 20

plotPairsEigenvectors, 13, 15


SCUBA (SCUBA-package), 6

23

24 INDEX

scuba, 6–8, 18

SCUBA-package, 6

spectClust, 6, 8, 10, 13, 21

ZoomArea, 15, 20, 22

ContentsBackgroundInformation about SCUBASCUBA-package

Description of implemented functionsdisplayKMeansdisplaySpectClustplotEigenvaluesPlotOutDataplotPairsEigenvectorssaveAllFilesForRunscubaspectClustZoomArea

Index

Documents

of package SCUBA - University of Bathmasas/scuba/SCUBA_manual_1.0.pdf · 2011. 1. 21. · Common tools for overcoming this problem are Spectral Clustering, Multidimensional Scaling