Generating data sets via a factorial design, which has factors: degree of separation, number of clusters, number of non-noisy variables, number of noisy variables. The separation between any cluster and its nearest neighboring clusters can be set to a specified value. The covariance matrices of clusters can have arbitrary diameters, shapes and orientations.
simClustDesign(numClust = c(3,6,9),
sepVal = c(0.01, 0.21, 0.342),
sepLabels = c("L", "M", "H"),
numNonNoisy = c(4,8,20),
numNoisy = NULL,
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]J[j]G[g]v[p1]nv[p2]out[numOutlier]_[numReplicate]
(see details).
The function also returns three lists: datList
, memList
, and noisyList
.
a list of lists of data matrices for generated data sets.
a list of lists of cluster memberships for data points for generated data sets.
a list of lists of sets of noisy variables for generated data sets.
Vector of the number of clusters for data sets in the design.
Vector of desired values of the separation index between clusters
and their nearest neighboring clusters. Each element of sepVal
can
take values within the interval [-1, 1)
.
The closer to 1 an element of sepVal
is, the more separated the
pair of clusters are.
The values \(0.01, 0.21, 0.34\) are the values of the separation index for
two univariate clusters generated from \(N(0, 1)\) and \(N(0, A)\),
where \(A=4, 6, 8\), respectively. sepVal
\(=0.01 (A=4)\) 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.
Labels for "close", "separated", and "well-separated" cluster structures. By default, "L" (low) means "close", "M" (medium) means "separated", "H" (high) means "well-separated".
Vector of the number of non-noisy variables.
Vectors of the number of noisy variables. The default value of numNoisy
is NULL
so that the program can automatically assign the value of
numNoisy
as a vector with elements \(1, round(p1/2), p1\).
The 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
is rounded to an integer.
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 by 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 to 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 \(q\) as 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 simClustDesign
is an implementation of the design for
generating random clusters proposed in Qiu and Joe (2006a). In the design,
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 simClustDesign
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 unifcorrmat
, 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 simClustDesign
outputs
four files: data file, log file, membership file, and noisy set file.
All four files have the same format:
[fileName]J[j]G[g]v[p1]nv[p2]out[numOutlier]_[numReplicate].[extension]
where extension
can be dat
, log
, mem
, or
noisy
. ‘J’ indicates separation index, with ‘j’
indicating the level of the factor ‘separation index’;
‘G’ indicates number of clusters, with ‘g’ indicating the
level of the factor ‘number of clusters’; ‘v’ indicates
the number of non-noisy variables, with ‘p1’ indicating the level
of the factor ‘number of non-noisy variables’; ‘nv’ indicates
the number of noisy variables, with ‘p2’ indicating the level of
the factor ‘number of noisy variables’; ‘out’ indicates
number of outliers, with ‘numOutlier’ indicating the value of the
argument numOutlier
of the function simClustDesign
;
‘numReplicate’ indicates the value of the argument numReplicate
of the function simClustDesign
.
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.
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
if (FALSE) {
tmp <- simClustDesign(
numClust = 3,
sepVal = c(0.01, 0.21),
sepLabels = c("L", "M"),
numNonNoisy = 4,
numOutlier = 0,
numReplicate = 2,
clustszind = 2)}
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