Distribution-Free Exact High Dimensional Low Sample Size k-Sample Tests

Description

Testing homogeneity of k (\geq 2) multivariate distributions is a classical and challenging problem in statistics, and this becomes even more challenging when the dimension of the data exceeds the sample size. We construct some tests for this purpose which are exact level (size) \alpha tests based on clustering. These tests are easy to implement and distribution-free in finite sample situations. Under appropriate regularity conditions, these tests have the consistency property in HDLSS asymptotic regime, where the dimension of data d grows to \infty while the sample size remains fixed. We also consider a multiscale approach, where the results for the different number of partitions are aggregated judiciously. This package includes eight tests, namely (i) RI test, (ii) FS test, (iii) MRI test, (iv) MFS test, (v) MTRI test , (vi) MTFS test, (vii) ARI test and (viii) AFS test. In MRI and MFS test, we modified the RI and FS test, respectively, using an estimated clustering number. In the multiscale approach (MTRI and MTFS), we use Holm's step-down-procedure (1979) and Benjamini-Hochberg FDR controlling procedure (1995).

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Soham Sarkar and Anil K Ghosh (2019). On perfect clustering of high dimension, low sample size data, IEEE transactions on pattern analysis and machine intelligence, doi:10.1109/TPAMI.2019.2912599.

William M Rand (1971). Objective criteria for the evaluation of clustering methods, Journal of the American Statistical association, 66(336):846-850, doi:10.1080/01621459.1971.10482356.

Cyrus R Mehta and Nitin R Patel (1983). A network algorithm for performing Fisher's exact test in rxc contingency tables, Journal of the American Statistical Association, 78(382):427-434, doi:10.2307/2288652.

Joseph C Dunn (1973). A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters, doi:10.1080/01969727308546046.

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

k-Sample AFS Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime. This an aggregate test of the two sample versions of the FS test over \frac{k(k-1)}{2} numbers of two-sample comparisons, and the test statistic is the minimum of these two sample FS test statistics. Holm's step-down-procedure (1979) and Benjamini-Hochberg procedure (1995) are applied for multiple testing.

Usage

AFStest(M, sizes, randomization = TRUE, clust_alg = "knwClustNo", kmax = 4,
multTest = "Holm", s_psi = 1, s_h = 1, lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

Value

AFStest returns a list containing the following items:

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Cyrus R Mehta and Nitin R Patel (1983). A network algorithm for performing Fisher's exact test in rxc contingency tables, Journal of the American Statistical Association, 78(382):427-434, doi:10.2307/2288652.

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

Examples

  # muiltivariate normal distribution:
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d) 
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  #AFS test:
  results <- AFStest(M=X, sizes = c(n1,n2,n3,n4))
  
   ## outputs:
   results$AFSStat
   #[1] 5.412544e-06

   results$AFCutoff
   #[1] 0.0109604

   results$randomGamma
   #[1] 0

   results$decisionAFS
   #[1] 1

   results$multipleTest
   #  Population.1 Population.2 rejected pvalues
   #1            1            2     TRUE       0
   #2            1            3     TRUE       0
   #3            1            4     TRUE       0
   #4            2            3     TRUE       0
   #5            2            4     TRUE       0
   #6            3            4     TRUE       0

k-Sample ARI Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime. This an aggregate test of the two sample versions of the RI test over \frac{k(k-1)}{2} numbers of two-sample comparisons, and the test statistic is the minimum of these two sample RI test statistics. Holm's step-down-procedure (1979) and Benjamini-Hochberg procedure (1995) are applied for multiple testing.

Usage

ARItest(M, sizes, randomization = TRUE, clust_alg = "knwClustNo", kmax = 4, 
multTest = "Holm", s_psi = 1, s_h = 1, lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

Value

ARItest returns a list containing the following items:

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

William M Rand (1971). Objective criteria for the evaluation of clustering methods, Journal of the American Statistical association, 66(336):846-850, doi:10.1080/01621459.1971.10482356.

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

Examples

  # muiltivariate normal distribution:
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d) 
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  #ARI test:
  results <- ARItest(M=X, sizes = c(n1,n2,n3,n4))
  
   ## outputs:
   results$ARIStat
   #[1] 0

   results$ARICutoff
   #[1] 0.3368421

   results$randomGamma
   #[1] 0

   results$decisionARI
   #[1] 1

   results$multipleTest
   #  Population.1 Population.2 rejected pvalues
   #1            1            2     TRUE       0
   #2            1            3     TRUE       0
   #3            1            4     TRUE       0
   #4            2            3     TRUE       0
   #5            2            4     TRUE       0
   #6            3            4     TRUE       0

Benjamini-Hochbergs step-up-procedure (1995)

Description

Benjamini-Hochbergs step-up-procedure (1995) for multiple tests.

Usage

BenHoch(pvalues, alpha)

Arguments

Value

a vector of 0s and 1s. 0: fails to reject the corresponding hypothesis and 1: reject the corresponding hypothesis

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

Examples

   # Benjamini-Hochbergs step-up-procedure:
   pvalues <- c(0.50,0.01,0.001,0.69,0.02,0.05,0.0025)
   alpha <- 0.05
   BenHoch(pvalues, alpha)

   ## outputs:
   #[1] 0 1 1 0 1 0 1

k-Sample FS Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime.

Usage

FStest(M, labels, sizes, n_clust, randomization = TRUE, clust_alg = "knwClustNo", 
kmax = 2 * n_clust, s_psi = 1, s_h = 1, lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

Value

FStest returns a list containing the following items:

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Cyrus R Mehta and Nitin R Patel (1983). A network algorithm for performing Fisher's exact test in rxc contingency tables, Journal of the American Statistical Association, 78(382):427-434, doi:10.2307/2288652.

Examples

   # muiltivariate normal distribution:
   # generate data with dimension d = 500
   set.seed(151)
   n1=n2=n3=n4=10
   k = 4
   d = 500
   I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
   I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
   I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
   I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d) 
   levels <- c(rep(0,n1), rep(1,n2), rep(2,n3), rep(3,n4)) 
   X <- as.matrix(rbind(I1,I2,I3,I4)) 
   #FS test:
   results <- FStest(M=X, labels=levels, sizes = c(n1,n2,n3,n4), n_clust = k)
  
   ## outputs:
   results$estClustLabel
   #[1] 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3

   results$obsCtyTab
   #      [,1] [,2] [,3] [,4]
   #[1,]   10    0    0    0
   #[2,]    0   10    0    0
   #[3,]    0    0   10    0
   #[4,]    0    0    0   10

   results$ObservedProb
   #[1] 2.125236e-22

   results$FCutoff
   #[1] 1.115958e-07

   results$randomGamma
   #[1] 0

   results$estPvalue
   #[1] 0

   results$decisionF
   #[1] 1

Holm's step-down-procedure (1979)

Description

Holm's step-down-procedure (1979) for mutiple tests.

Usage

Holm(pvalues, alpha)

Arguments

Value

a vector of 0s and 1s. 0: fails to reject the corresponding hypothesis and 1: reject the corresponding hypothesis

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Examples

   # Holm's step down procedure:
   pvalues <- c(0.50,0.01,0.001,0.69,0.02,0.05,0.0025)
   alpha <- 0.05
   Holm(pvalues, alpha)

   ## outputs:
   #[1] 0 0 1 0 0 0 1
   

k-Sample MTFS Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime. This test is a multiscale approach based on FS test, where the results for different number of partitions are aggregated judiciously.

Usage

MTFStest(M, labels, sizes, k_max, multTest = "Holm", s_psi = 1, s_h = 1,
lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

Value

MTFStest returns a list containing the following items:

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

Examples

  # muiltivariate normal distribution:
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d)
  levels <- c(rep(0,n1), rep(1,n2), rep(2,n3), rep(3,n4)) 
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  #MTFS test:
  results <- MTFStest(X, levels, c(n1,n2,n3,n4), 8)
  
   ## outputs:
   results$fpmfvec
   #[1] 7.254445e-12 6.137740e-16 2.125236e-22 2.125236e-22 2.125236e-22 2.125236e-22 2.125236e-22

   results$Pvalues
   #[1] 0 0 0 0 0 0 0

   results$decisionMTFS
   #[1] 1

   results$contTabs
   #$contTabs[[1]]
   #     [,1] [,2]
   #[1,]   10    0
   #[2,]   10    0
   #[3,]    0   10
   #[4,]    0   10

   #$contTabs[[2]]
   #    [,1] [,2] [,3]
   #[1,]   10    0    0
   #[2,]    0   10    0
   #[3,]    0    8    2
   #[4,]    0    0   10

   #$contTabs[[3]]
   #     [,1] [,2] [,3] [,4]
   #[1,]   10    0    0    0
   #[2,]    0   10    0    0
   #[3,]    0    0   10    0
   #[4,]    0    0    0   10

   #$contTabs[[4]]
   #     [,1] [,2] [,3] [,4] [,5]
   #[1,]   10    0    0    0    0
   #[2,]    0   10    0    0    0
   #[3,]    0    0    4    6    0
   #[4,]    0    0    0    0   10

   #$contTabs[[5]]
   #    [,1] [,2] [,3] [,4] [,5] [,6]
   #[1,]   10    0    0    0    0    0
   #[2,]    0   10    0    0    0    0
   #[3,]    0    0    4    6    0    0
   #[4,]    0    0    0    0    8    2

   #$contTabs[[6]]
   #     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
   #[1,]   10    0    0    0    0    0    0
   #[2,]    0    5    5    0    0    0    0
   #[3,]    0    0    0    4    6    0    0
   #[4,]    0    0    0    0    0    8    2

   #$contTabs[[7]]
   #     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
   #[1,]    8    2    0    0    0    0    0    0
   #[2,]    0    0    5    5    0    0    0    0
   #[3,]    0    0    0    0    4    6    0    0
   #[4,]    0    0    0    0    0    0    8    2


   results$mulTestdec
   #[1] 1 1 1 1 1 1 1

k-Sample MTRI Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime. This test is a multiscale approach based on RI test, where the results for different number of partitions are aggregated judiciously.

Usage

MTRItest(M, labels, sizes, k_max, multTest = "Holm", s_psi = 1, s_h = 1, 
lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

Value

MTRItest returns a list containing the following items:

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Sture Holm (1979). A simple sequentially rejective multiple test procedure, Scandinavian journal of statistics, 65-70, doi:10.2307/4615733.

Yoav Benjamini and Yosef Hochberg (1995). Controlling the false discovery rate: a practical and powerful approach to multiple testing, Journal of the Royal statistical society: series B (Methodological) 57.1: 289-300, doi: 10.2307/2346101.

Examples

  # muiltivariate normal distribution:
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d)
  levels <- c(rep(0,n1), rep(1,n2), rep(2,n3), rep(3,n4)) 
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  #MTRI test:
  results <- MTRItest(X, levels, c(n1,n2,n3,n4), 8)
  
  ## outputs:
  results$RIvec
  #[1] 0.25641026 0.14871795 0.00000000 0.03076923 0.05128205 0.08333333 0.10384615

  results$Pvalues
  #[1] 0 0 0 0 0 0 0

  results$decisionMTRI
  #[1] 1

  results$contTabs
  #$contTabs[[1]]
  #     [,1] [,2]
  #[1,]   10    0
  #[2,]   10    0
  #[3,]    0   10
  #[4,]    0   10

  #$contTabs[[2]]
  #     [,1] [,2] [,3]
  #[1,]   10    0    0
  #[2,]    0   10    0
  #[3,]    0   10    0
  #[4,]    0    0   10

  #$contTabs[[3]]
  #     [,1] [,2] [,3] [,4]
  #[1,]   10    0    0    0
  #[2,]    0   10    0    0
  #[3,]    0    0   10    0
  #[4,]    0    0    0   10

  #$contTabs[[4]]
  #     [,1] [,2] [,3] [,4] [,5]
  #[1,]   10    0    0    0    0
  #[2,]    0   10    0    0    0
  #[3,]    0    0    4    6    0
  #[4,]    0    0    0    0   10

  #$contTabs[[5]]
  #     [,1] [,2] [,3] [,4] [,5] [,6]
  #[1,]   10    0    0    0    0    0
  #[2,]    0   10    0    0    0    0
  #[3,]    0    0    4    6    0    0
  #[4,]    0    0    0    0    8    2

  #$contTabs[[6]]
  #     [,1] [,2] [,3] [,4] [,5] [,6] [,7]
  #[1,]   10    0    0    0    0    0    0
  #[2,]    0    5    5    0    0    0    0
  #[3,]    0    0    0    4    6    0    0
  #[4,]    0    0    0    0    0    8    2

  #$contTabs[[7]]
  #     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
  #[1,]    8    2    0    0    0    0    0    0
  #[2,]    0    0    5    5    0    0    0    0
  #[3,]    0    0    0    0    4    6    0    0
  #[4,]    0    0    0    0    0    0    8    2


  results$mulTestdec
  #[1] 1 1 1 1 1 1 1

k-Sample RI Test of Equal Distributions

Description

Performs the distribution free exact k-sample test for equality of multivariate distributions in the HDLSS regime.

Usage

RItest(M, labels, sizes, n_clust, randomization = TRUE, clust_alg = "knwClustNo", 
kmax = 2 * n_clust, s_psi = 1, s_h = 1, lb = 1, n_sts = 1000, alpha = 0.05)

Arguments

Value

RItest returns a list containing the following items:

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

William M Rand (1971). Objective criteria for the evaluation of clustering methods, Journal of the American Statistical association, 66(336):846-850, doi:10.1080/01621459.1971.10482356.

Examples

  # muiltivariate normal distribution:
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  k = 4
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d) 
  levels <- c(rep(0,n1), rep(1,n2), rep(2,n3), rep(3,n4)) 
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  # RI test:
  results <- RItest(M=X, labels=levels, sizes = c(n1,n2,n3,n4), n_clust = k)
  
   ## outputs:
   results$estClustLabel
   #[1] 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3

   results$obsCtyTab
   #      [,1] [,2] [,3] [,4]
   #[1,]   10    0    0    0
   #[2,]    0   10    0    0
   #[3,]    0    0   10    0
   #[4,]    0    0    0   10

   results$ObservedRI
   #[1] 0

   results$RICutoff
   #[1] 0.3307692

   results$randomGamma
   #[1] 0

   results$estPvalue
   #[1] 0

   results$decisionRI
   #[1] 1

Modified K-Means Algorithm by Using a New Dissimilarity Measure, MADD

Description

Performs modified K-means algorithm by using a new dissimilarity measure, called MADD, and provides estimated cluster (class) labels or memberships of observations.

Usage

gMADD(s_psi, s_h, n_clust, lb, M)

Arguments

Value

a vector of length n of estimated cluster (class) labels of observations

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Soham Sarkar and Anil K Ghosh (2019). On perfect clustering of high dimension, low sample size data, IEEE transactions on pattern analysis and machine intelligence, doi:10.1109/TPAMI.2019.2912599.

Examples

  # Modified K-means algorithm:
  # muiltivariate normal distribution
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d) 
  n_cl <- 4
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  gMADD(1,1,n_cl,1,X)
  
   ## outputs:
   #[1] 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3

Modified K-Means Algorithm by Using a New Dissimilarity Measure, MADD and DUNN Index

Description

Performs modified K-means algorithm by using a new dissimilarity measure, called MADD and DUNN index, and provides estimated cluster (class) labels or memberships and corresponding DUNN index of the observations.

Usage

gMADD_DI(s_psi, s_h, kmax, lb, M)

Arguments

Details

DUNN index is used for cluster validation, but here we use it to estimate total number of cluster k by \hat k = argmax_{2\le k' \le k^*}DI(k'). Here DI(k') represents the DUNN index and we use k^*=2*k.

Value

a kmax \times (n+1) matrix of the estimated cluster (class) labels and corresponding DUNN indexes of observations

Note

The result of this gMADD_DI function is a matrix. The 1st row of this matrix doesn't provide anything about estimated class labels or DUNN index of observations since the DUNN index is only defined for k\ge 2. The last column of this matrix represents the DUNN indexes. The estimated cluster labels of observations are calculated by finding out the corresponding row of maximum DUNN index.

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Soham Sarkar and Anil K Ghosh (2019). On perfect clustering of high dimension, low sample size data, IEEE transactions on pattern analysis and machine intelligence, doi:10.1109/TPAMI.2019.2912599.

Joseph C Dunn (1973). A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters, doi:10.1080/01969727308546046.

Examples

  # Modified K-means algorithm:
  # muiltivariate normal distribution
  # generate data with dimension d = 500
  set.seed(151)
  n1=n2=n3=n4=10
  d = 500
  I1 <- matrix(rnorm(n1*d,mean=0,sd=1),n1,d)
  I2 <- matrix(rnorm(n2*d,mean=0.5,sd=1),n2,d) 
  I3 <- matrix(rnorm(n3*d,mean=1,sd=1),n3,d) 
  I4 <- matrix(rnorm(n4*d,mean=1.5,sd=1),n4,d) 
  n_cl <- 4
  N <- n1+n2+n3+n4
  X <- as.matrix(rbind(I1,I2,I3,I4)) 
  dvec_di_mat <-  gMADD_DI(1,1,2*n_cl,1,X)
  est_no_cl <- which.max(dvec_di_mat[ ,(N+1)])
  dvec_di_mat[est_no_cl,1:N]

   ## outputs:
   #[1] 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3
   

Generalized Hypergeometric Probability

Description

A function that provides the probability of observing an r\times c contingency table using generalized hypergeometric probability.

Usage

pmf(M)

Arguments

Value

a single value between 0 and 1

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Biplab Paul, Shyamal K De and Anil K Ghosh (2021). Some clustering based exact distribution-free k-sample tests applicable to high dimension, low sample size data, Journal of Multivariate Analysis, doi:10.1016/j.jmva.2021.104897.

Cyrus R Mehta and Nitin R Patel (1983). A network algorithm for performing Fisher's exact test in rxc contingency tables, Journal of the American Statistical Association, 78(382):427-434, doi:10.2307/2288652.

Examples

   # Generalized hypergeometric probability of rxc Contingency Table:
   mat <- matrix(1:20,5,4, byrow = TRUE)
   pmf(mat)

   ## outputs:
   #[1] 4.556478e-09

Rand Index

Description

Measures to compare the dissimilarity of exact cluster labels (memberships) and estimated cluster labels (memberships) of the observations.

Usage

randfun(lvel, dv)

Arguments

Value

a single value between 0 and 1

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

William M Rand (1971). Objective criteria for the evaluation of clustering methods, Journal of the American Statistical association, 66(336):846-850, doi:10.1080/01621459.1971.10482356.

Examples

   # Measures of dissimilarity:
   exl <- c(rep(0,5), rep(1,5), rep(2,5), rep(3,5))
   el <- c(0,0,1,0,0,1,2,1,0,1,2,2,3,2,2,3,2,3,1,3)
   randfun(exl,el)

   ## outputs:
   #[1] 0.2368421

Generates an r\times c Contingency Table

Description

A function that generates an r\times c contingency table with the same marginal totals as given r\times c contingency table.

Usage

rctab(M)

Arguments

Value

generated r\times c contingency table

Author(s)

Biplab Paul, Shyamal K. De and Anil K. Ghosh

Maintainer: Biplab Paul<paul.biplab497@gmail.com>

References

Cyrus R Mehta and Nitin R Patel (1983). A network algorithm for performing Fisher's exact test in rxc contingency tables, Journal of the American Statistical Association, 78(382):427-434, doi:10.2307/2288652.

Examples

# Generation of rxc Contingency Table:
   set.seed(151)
   mat <- matrix(1:20,5,4, byrow = TRUE)
   rctab(mat)

   ## outputs:
   #      [,1] [,2] [,3] [,4]
   #[1,]    3    4    0    3
   #[2,]    4    5   10    7
   #[3,]    8    7   12   15
   #[4,]   18   16   13   11
   #[5,]   12   18   20   24