Generate cluster data sets with specified degree of separation. The separation between any cluster and its nearest neighboring cluster can be set to a specified value. The covariance matrices of clusters can have arbitrary diameters, shapes and orientations.
genRandomClust(numClust,
sepVal = 0.01,
numNonNoisy = 2,
numNoisy = 0,
numOutlier = 0,
numReplicate = 3,
fileName = "test",
clustszind = 2,
clustSizeEq = 50,
rangeN = c(50,200),
clustSizes = NULL,
covMethod = c("eigen", "onion", "c-vine", "unifcorrmat"),
eigenvalue = NULL,
rangeVar = c(1, 10),
lambdaLow = 1,
ratioLambda = 10,
alphad = 1,
eta = 1,
rotateind = TRUE,
iniProjDirMethod = c("SL", "naive"),
projDirMethod = c("newton", "fixedpoint"),
alpha = 0.05,
ITMAX = 20,
eps = 1.0e-10,
quiet = TRUE,
outputDatFlag = TRUE,
outputLogFlag = TRUE,
outputEmpirical = TRUE,
outputInfo = TRUE)The function outputs four data files for each data set (see details).
This function also returns separation information data frames
infoFrameTheory and infoFrameData based on population
and empirical mean vectors and covariance matrices of clusters for all
the data sets generated. Both infoFrameTheory and infoFrameData
contain the following seven columns:
Labels of clusters (\(1, 2, \ldots, numClust\)), where \(numClust\) is the number of clusters for the data set.
Labels of the corresponding nearest neighbors.
Separation indices of the clusters to their nearest neighboring clusters.
Labels of the corresponding farthest neighboring clusters.
Separation indices of the clusters to their farthest neighbors.
Median separation indices of the clusters to their neighbors.
Data file names with format [fileName]_[i], where \(i\) indicates
the replicate number.
The function also returns three lists: datList, memList, and noisyList.
a list of data matrices for generated data sets.
a list of luster memberships for data points for generated data sets.
a list of sets of noisy variables for generated data sets.
Number of clusters in a data set.
Desired value of the separation index between a cluster
and its nearest neighboring cluster. Theoretically, sepVal can take
values within the interval \([-1, 1)\)
(In practice, we set sepVal in \((-0.999, 0.999)\)).
The closer to \(1\) sepVal is, the more separated clusters are.
The default value is \(0.01\) which is the value of the separation index for
two univariate clusters generated from \(N(0, 1)\) and \(N(0, A)\),
respectively, where \(A=4\).
sepVal\(=0.01\) indicates a close cluster structure.
sepVal\(=0.21 (A=6)\) indicates a separated cluster structure.
sepVal\(=0.34 (A=8)\) indicates a well-separated cluster.
Number of non-noisy variables.
Number of noisy variables.
The default values of numNoisy and numOutlier are \(0\) so
that we get clean data sets.
Number or ratio of outliers. If numOutlier is a positive integer,
then numOutlier means the number of outliers.
If numOutlier is a real number between \((0, 1)\), then
numOutlier means the ratio of outliers, i.e. the number of outliers
is equal to round(numOutlier\(*n_1\)), where \(n_1\) is
the total number of non-outliers. If numOutlier is a real number
greater than \(1\), then numOutlier to rounded to an integer.
The default values of numNoisy and numOutlier are
\(0\) so that we get ‘clean’ data sets.
Number of data sets to be generated for the same cluster structure specified
by the other arguments of the function genRandomClust.
The default value \(3\) follows the design in Milligan (1985).
The first part of the names of data files that record the generated data sets
and associated information, such as cluster membership of data points, labels
of noisy variables, separation index matrix, projection directions, etc.
(see details). The default value of fileName is test.
Cluster size indicator.
clustszind\(=1\) indicates that all cluster have equal size.
The size is specified by the argument clustSizeEq.
clustszind\(=2\) indicates that the cluster sizes are randomly
generated from the range specified by the argument rangeN.
clustszind\(=3\) indicates that the cluster sizes are specified
via the vector clustSizes.
The default value is \(2\) so that the generated clusters are more
realistic.
Cluster size.
If the argument clustszind\(=1\), then all clusters will have the
equal number clustSizeEq of data points. The value of clustSizeEq
should be large enough to get non-singular cluster covariance matrices.
We recommend the clustSizeEq is at least \(10*p\), where \(p\)
is the total number of variables (including both non-noisy and noisy variables).
The default value \(100\) is a reasonable cluster size.
The range of cluster sizes.
If clustszind\(=2\), then cluster sizes will be randomly generated
from the range specified by rangeN. The lower bound of the number of
clusters should be large enough to get non-singular cluster covariance
matrices. We recommend the minimum cluster size is at least \(10*p\), where
\(p\) is the total number of variables (including both non-noisy and noisy
variables). The default range is \([50, 200]\) which
can produce reasonable variability of cluster sizes.
The sizes of clusters.
If clustszind\(=3\), then cluster sizes will be specified via the
vector clustSizes. We recommend the minimum cluster size is at least
\(10*p\), where \(p\) is the total number of variables (including both
non-noisy and noisy variables).
The user needs to specify the value of clustSizes. Therefore, we
set the default value of clustSizes as NULL.
Method to generate covariance matrices for clusters (see details). The default method is 'eigen' so that the user can directly specify the range of the diameters of clusters.
numeric. user-specified eigenvalues when covMethod = "eigen". If eigenvalue = NULL and covMethod = "eigen", then eigenvalues will be automatically generated.
Range for variances of a covariance matrix (see details). The default range is \([1, 10]\) which can generate reasonable variability of variances.
Lower bound of the eigenvalues of cluster covariance matrices.
If the argument “covMethod="eigen"”, we need to generate eigenvalues for cluster covariance matrices.
The eigenvalues are randomly generated from the
interval [lambdaLow, lambdaLow\(*\)ratioLambda].
In our experience, lambdaLow\(=1\) and ratioLambda\(=10\)
can give reasonable variability of the diameters of clusters.
lambdaLow should be positive.
The ratio of the upper bound of the eigenvalues to the lower bound of the
eigenvalues of cluster covariance matrices.
If the argument covMethod="eigen", we need to generate eigenvalues for
cluster covariance matrices.
The eigenvalues are randomly generated from the
interval [lambdaLow, lambdaLow\(*\)ratioLambda].
In our experience, lambdaLow\(=1\) and ratioLambda\(=10\)
can give reasonable variability of the diameters of clusters.
ratioLambda should be larger than \(1\).
parameter for unifcorrmat method to generate random correlation matrix
alphad=1 for uniform. alphad should be positive.
parameter for “c-vine” and “onion” methods to generate random correlation matrix
eta=1 for uniform. eta should be positive.
Rotation indicator.
rotateind=TRUE indicates randomly rotating data in non-noisy
dimensions so that we may not detect the full cluster structure from
pair-wise scatter plots of the variables.
Indicating the method to get initial projection direction when calculating
the separation index between a pair of clusters (c.f. Qiu and Joe,
2006a, 2006b).
iniProjDirMethod=“SL”, the default, indicates the initial
projection direction is the sample version of the SL's projection direction
(Su and Liu, 1993, JASA)
\(\left(\boldsymbol{\Sigma}_1+\boldsymbol{\Sigma}_2\right)^{-1}\left(\boldsymbol{\mu}_2-\boldsymbol{\mu}_1\right)\)
iniProjDirMethod=“naive” indicates the initial projection
direction is \(\boldsymbol{\mu}_2-\boldsymbol{\mu}_1\)
Indicating the method to get the optimal projection direction when calculating
the separation index between a pair of clusters (c.f. Qiu and Joe,
2006a, 2006b).
projDirMethod=“newton” indicates we use the modified
Newton-Raphson method to search the optimal projection direction
(c.f. Qiu and Joe, 2006a). This requires the assumptions that both covariance
matrices of the pair of clusters are positive-definite. If this assumption
is violated, the “fixedpoint” method could be used. The
“fixedpoint” method iteratively searches the optimal projection
direction based on the first derivative of the separation index to the
projection direction (c.f. Qiu and Joe, 2006b).
Tuning parameter reflecting the percentage in the two
tails of a projected cluster that might be outlying.
We set alpha\(=0.05\) like we set
the significance level in hypothesis testing as \(0.05\).
Maximum iteration allowed when iteratively calculating the optimal projection direction. The actual number of iterations is usually much less than the default value 20.
Convergence threshold. A small positive number to check if a quantitiy \(q\)
is equal to zero. If \(|q|<\)eps, then we regard \(q\) is equal
to zero. eps is used to check if an algorithm converges.
The default value is \(1.0e-10\).
A flag to switch on/off the outputs of intermediate results and/or possible warning messages. The default value is TRUE.
Indicates if data set should be output to file.
Indicates if log info should be output to file.
Indicates if empirical separation indices and projection directions should be
calculated. This option is useful when generating clusters with sizes which
are not large enough so that the sample covariance matrices may be singular.
Hence, by default, outputEmpirical=TRUE.
Indicates if theoretical and empirical separation information data frames
should be output to a file with format [fileName]_info.log.
Weiliang Qiu weiliang.qiu@gmail.com
Harry Joe harry@stat.ubc.ca
The function genRandomClust is an implementation of the random cluster
generation method proposed in Qiu and Joe (2006a) which improve the cluster
generation method proposed in Milligan (1985) so that the degree of separation
between any cluster and its nearest neighboring cluster could be set to a
specified value while the cluster covariance matrices can be arbitrary positive definite matrices, and so that clusters generated might not be visualized
by pair-wise scatterplots of variables. The separation between a pair of
clusters is measured by the separation index proposed in Qiu and Joe (2006b).
The current version of the function genRandomClust implements two
methods to generate covariance matrices for clusters. The first method,
denoted by eigen, first randomly generates eigenvalues
(\(\lambda_1,\ldots>\lambda_p\)) for the covariance matrix
(\(\boldsymbol{\Sigma}\)), then uses columns of a randomly generated
orthogonal matrix
(\(\boldsymbol{Q}=(\boldsymbol{\alpha}_1,\ldots,\boldsymbol{\alpha}_p)\))
as eigenvectors. The covariance matrix
\(\boldsymbol{\Sigma}\) is then contructed as
\(\boldsymbol{Q}*diag(\lambda_1,\dots, \lambda_p)*\boldsymbol{Q}^T\).
The second method, denoted as “unifcorrmax”, first generates a random
correlation matrix (\(\boldsymbol{R}\)) via the method proposed in Joe (2006),
then randomly generates variances (\(\sigma_1^2,\ldots, \sigma_p^2\)) from
an interval specified by the argument rangeVar. The covariance matrix
\(\boldsymbol{\Sigma}\) is then constructed as
\(diag(\sigma_1,\ldots,\sigma_p)*\boldsymbol{R}*diag(\sigma_1,\ldots,\sigma_p)\).
For each data set generated, the function genRandomClust outputs
four files: data file, log file, membership file, and noisy set file.
All four files have the same format: [fileName]_[i].[extension],
where \(i\) indicates the replicate number, and extension can be
dat, log, mem, and noisy.
The data file with file extension dat contains \(n+1\) rows and
\(p\) columns, where \(n\) is the number of data points and \(p\)
is the number of variables. The first row is the variable names.
The log file with file extension log contains information such
as cluster sizes, mean vectors, covariance matrices, projection directions,
separation index matrices, etc. The membership file with file extension
mem contains \(n\) rows and one column of cluster memberships for
data points. The noisy set file with file extension noisy contains
a row of labels of noisy variables.
When generating clusters, population covariance matrices are all
positive-definite. However sample covariance matrices might be
semi-positive-definite due to small cluster sizes. In this case, the
function genRandomClust will automatically use the
“fixedpoint” method to search the optimal projection direction.
The current version of the function genPositiveDefMat implements four
methods to generate random covariance matrices. The first method, denoted by
“eigen”, first randomly generates eigenvalues
(\(\lambda_1,\ldots,\lambda_p\)) for the covariance matrix
(\(\boldsymbol{\Sigma}\)), then
uses columns of a randomly generated orthogonal matrix
(\(\boldsymbol{Q}=(\boldsymbol{\alpha}_1,\ldots,\boldsymbol{\alpha}_p)\))
as eigenvectors. The covariance matrix \(\boldsymbol{\Sigma}\) is then
contructed as
\(\boldsymbol{Q}*diag(\lambda_1,\ldots,\lambda_p)*\boldsymbol{Q}^T\).
The remaining methods, denoted as “onion”, “c-vine”, and “unifcorrmat”
respectively, first generates a random
correlation matrix (\(\boldsymbol{R}\)) via the method mentioned and proposed in Joe (2006),
then randomly generates variances (\(\sigma_1^2,\ldots,\sigma_p^2\)) from
an interval specified by the argument rangeVar. The covariance matrix
\(\boldsymbol{\Sigma}\) is then constructed as
\(diag(\sigma_1,\ldots,\sigma_p)*\boldsymbol{R}*diag(\sigma_1,\ldots,\sigma_p)\).
Joe, H. (2006) Generating Random Correlation Matrices Based on Partial Correlations. Journal of Multivariate Analysis, 97, 2177--2189.
Milligan G. W. (1985) An Algorithm for Generating Artificial Test Clusters. Psychometrika 50, 123--127.
Qiu, W.-L. and Joe, H. (2006a) Generation of Random Clusters with Specified Degree of Separaion. Journal of Classification, 23(2), 315-334.
Qiu, W.-L. and Joe, H. (2006b) Separation Index and Partial Membership for Clustering. Computational Statistics and Data Analysis, 50, 585--603.
Su, J. Q. and Liu, J. S. (1993) Linear Combinations of Multiple Diagnostic Markers. Journal of the American Statistical Association, 88, 1350--1355.
Ghosh, S., Henderson, S. G. (2003). Behavior of the NORTA method for correlated random vector generation as the dimension increases. ACM Transactions on Modeling and Computer Simulation (TOMACS), 13(3), 276--294.
Kurowicka and Cooke, 2006. Uncertainty Analysis with High Dimensional Dependence Modelling, Wiley, 2006.
if (FALSE) {
tmp1 <- genRandomClust(
numClust = 7,
sepVal = 0.3,
numNonNoisy = 5,
numNoisy = 3,
numOutlier = 5,
numReplicate = 2,
fileName = "chk1")
}
if (FALSE) {
tmp2 <- genRandomClust(
numClust = 7,
sepVal = 0.3,
numNonNoisy = 5,
numNoisy = 3,
numOutlier = 5,
numReplicate = 2,
covMethod = "unifcorrmat",
fileName = "chk2")
}
if (FALSE) {
tmp3 <- genRandomClust(
numClust = 2,
sepVal = -0.1,
numNonNoisy = 2,
numNoisy = 6,
numOutlier = 30,
numReplicate = 1,
clustszind = 1,
clustSizeEq = 80,
rangeVar = c(10, 20),
covMethod = "unifcorrmat",
iniProjDirMethod = "naive",
projDirMethod = "fixedpoint",
fileName = "chk3")
}
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