Package 'BayesCVI'

BCVI-Correlation Cluster Validity (CCV) index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using the pearson correlation cluster validity (CCVP) and/or the spearman’s (rho) correlation cluster validity (CCVS) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_CCV.IDX(x, kmax, indexlist = "all", method = "FCM", fzm = 2,
        iter = 100, nstart = 20, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
indexlist	a character string indicating which The generalized C index be computed ("all","CCVP","CCVS"). More than one indexes can be selected.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
iter	a maximum number of iterations for method = "FCM". The default is 100.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-CCV is defined as follows. Let

$r_k(\bf x) = \dfrac{CVI(k)-\min_j CVI(j)}{\sum_{i=2}^K (CVI(i)-\min_j CVI(j))}$

where CVI is either CCVP or CCVS index.

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original CCVP$(k)$ or CCVS$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

M. Popescu, J. C. Bezdek, T. C. Havens and J. M. Keller (2013). "A Cluster Validity Framework Based on Induced Partition Dissimilarity." https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6246717&isnumber=6340245

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_XB.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(20,20,20,5,5,5,0.5,0.5,0.5)

B.CCV = B_CCV.IDX(x = scale(data), kmax=10, indexlist = "CCVP", method = "FCM", fzm = 2, iter = 100,
              nstart = 20, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI-CCVP

pplot = plot_BCVI(B.CCV$CCVP)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Calinski–Harabasz (CH) index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using Calinski–Harabasz (CH) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_CH.IDX(x, kmax, method = "kmeans", nstart = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("kmeans", "hclust_complete", "hclust_average", "hclust_single"). The default is "kmeans".
nstart	a maximum number of initial random sets for kmeans for method = "kmeans". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-CH is defined as follows. Let

$r_k(\bf x) = \dfrac{CH(k)-\min_j CH(j)}{\sum_{i=2}^K (CH(i)-\min_j CH(j))}$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original CH$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

T. Calinski, J. Harabasz, "A dendrite method for cluster analysis," Communications in Statistics, 3, 1-27 (1974).

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B2_data, B_TANG.IDX, B_XB.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B2_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.CH = B_CH.IDX(x = scale(data), kmax=10, method = "kmeans",
                nstart = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.CH)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Chou-Su-Lai (CSL) index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using Chou-Su-Lai (CSL) as the underlying cluster validity index (CVI) and Dirichlet prior parameters of the user's choice. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_CSL.IDX(x, kmax, method = "kmeans", nstart = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("kmeans", "hclust_complete", "hclust_average", "hclust_single"). The default is "kmeans".
nstart	a maximum number of initial random sets for kmeans for method = "kmeans". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-CSL is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j CSL(j)- CSL(k)}{\sum_{i=2}^K (\max_j CSL(j) - CSL(i))}.$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original CSL$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

C. H. Chou, M. C. Su, E. Lai, "A new cluster validity measure and its application to image compression," Pattern Anal Applic, 7, 205-220 (2004).

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B2_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B2_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.CSL = B_CSL.IDX(x = scale(data), kmax=10, method = "kmeans",
                nstart = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.CSL)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Davies–Bouldin (DB) and DB* (DBs) indexes

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using DB and/or DBs as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_DB.IDX(x, kmax, method = "kmeans", indexlist = "all", p = 2, q = 2,
        nstart = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("kmeans", "hclust_complete", "hclust_average", "hclust_single"). The default is "kmeans".
indexlist	a character string indicating which cluster validity indexes to be computed ("all", "DB", "DBs"). More than one indexes can be selected.
p	the power of the Minkowski distance between centroids of clusters. The default is 2.
q	the power of dispersion measure of a cluster. The default is 2.
nstart	a maximum number of initial random sets for kmeans for method = "kmeans". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-DB is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j CVI(j)- CVI(k)}{\sum_{i=2}^K (\max_j CVI(j) - CVI(i))}.$

where CVI indicates DB or DBs index.
Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original DB$(k)$ or DBs$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

D. L. Davies, D. W. Bouldin, "A cluster separation measure," IEEE Trans Pattern Anal Machine Intell, 1, 224-227 (1979).

M. Kim, R. S. Ramakrishna, "New indices for cluster validity assessment," Pattern Recognition Letters, 26, 2353-2363 (2005).

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B2_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DI.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B2_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.DB = B_DB.IDX(x = scale(data), kmax=10, method = "kmeans", indexlist = "all",
              p = 2, q = 2, nstart = 100, alpha = "default", mult.alpha = 1/2)

# plot the BCVI-DB

pplot = plot_BCVI(B.DB$DB)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

# plot the BCVI-DBs

pplot = plot_BCVI(B.DB$DBs)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Dunn index (DI)

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using Dunn index (DI) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_DI.IDX(x, kmax, method = "kmeans", nstart = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("kmeans", "hclust_complete", "hclust_average", "hclust_single"). The default is "kmeans".
nstart	a maximum number of initial random sets for kmeans for method = "kmeans". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-DI is defined as follows. Let

$r_k(\bf x) = \dfrac{DI(k)-\min_j DI(j)}{\sum_{i=2}^K (DI(i)-\min_j DI(j))}$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original DI$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

J. C. Dunn, "A fuzzy relative of the ISODATA process and its use in detecting compact well-separated clusters," J Cybern, 3(3), 32-57 (1973).

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B2_data, B_TANG.IDX, B_XB.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B2_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.DI = B_DI.IDX(x = scale(data), kmax=10, method = "kmeans",
                nstart = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.DI)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-The generalized C (GC) index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using all or part of GC1 GC2 GC3 and GC4 as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_GC.IDX(x, kmax, indexlist = "all", method = "FCM", fzm = 2, iter = 100,
        nstart = 20, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
indexlist	a character string indicating which The generalized C index be computed ("all","GC1","GC2","GC3","GC4"). More than one indexes can be selected.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
iter	a maximum number of iterations for method = "FCM". The default is 100.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-GC is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j CVI(j)- CVI(k)}{\sum_{i=2}^K (\max_j CVI(j) - CVI(i))}.$

where CVI is one of the GC1 GC2 GC3 or GC4 index.

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original GC1$(k)$ GC2$(k)$ GC3$(k)$ GC4$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

J. C. Bezdek, M. Moshtaghi, T. Runkler, and C. Leckie, “The generalized c index for internal fuzzy cluster validity,” IEEE Transactions on Fuzzy Systems, vol. 24, no. 6, pp. 1500–1512, 2016. https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7429723&isnumber=7797168

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_XB.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.GC = B_GC.IDX(x = scale(data), kmax = 10, indexlist = "GC1",
              method = "FCM", fzm = 2, iter = 100,
                nstart = 20, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI-GC1

pplot = plot_BCVI(B.GC$GC1)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-HF index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using HF as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_HF.IDX(x, kmax, method = "FCM", fzm = 2, nstart = 20,
        iter = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
iter	a maximum number of iterations for method = "FCM". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-HF is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j HF(j)- HF(k)}{\sum_{i=2}^K (\max_j HF(j) - HF(i))}.$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original HF$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

F. Haouas, Z. Ben Dhiaf, A. Hammouda and B. Solaiman, "A new efficient fuzzy cluster validity index: Application to images clustering," 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), Naples, Italy, 2017, pp. 1-6. https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8015651&isnumber=8015374

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.HF = B_HF.IDX(x = scale(data), kmax =10, method = "FCM", fzm = 2,
              nstart = 20, iter = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.HF)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Modified Kernel form of Pakhira-Bandyopadhyay-Maulik (KPBM) index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using Modified Kernel form of Pakhira-Bandyopadhyay-Maulik (KPBM) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_KPBM.IDX(x, kmax, method = "FCM", fzm = 2, nstart = 20,
          iter = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
iter	a maximum number of iterations for method = "FCM". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-KPBM is defined as follows. Let

$r_k(\bf x) = \dfrac{KPBM(k)-\min_j KPBM(j)}{\sum_{i=2}^K (KPBM(i)-\min_j KPBM(j))}$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original KPBM$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

C. Alok. (2010). "An investigation of clustering algorithms and soft computing approaches for pattern recognition," Department of Computer Science, Assam University. http://hdl.handle.net/10603/93443

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.KPBM = B_KPBM.IDX(x = scale(data), kmax =10, method = "FCM", fzm = 2, nstart = 20,
                iter = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.KPBM)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-KWON index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using KWON as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_KWON.IDX(x, kmax, method = "FCM", fzm = 2, nstart = 20,
          iter = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
iter	a maximum number of iterations for method = "FCM". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-KWON is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j KWON(j)- KWON(k)}{\sum_{i=2}^K (\max_j KWON(j) - KWON(i))}.$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original KWON$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

S. H. Kwon, “Cluster validity index for fuzzy clustering,” Electronics letters, vol. 34, no. 22, pp. 2176–2177, 1998. doi:10.1049/el:19981523

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.KWON = B_KWON.IDX(x = scale(data), kmax =10, method = "FCM", fzm = 2, nstart = 20,
                    iter = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.KWON)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-KWON2 index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using KWON2 as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_KWON2.IDX(x, kmax, method = "FCM", fzm = 2, nstart = 20,
          iter = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
iter	a maximum number of iterations for method = "FCM". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-KWON2 is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j KWON2(j)- KWON2(k)}{\sum_{i=2}^K (\max_j KWON2(j) - KWON2(i))}.$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original KWON2$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

S. H. Kwon, J. Kim, and S. H. Son, “Improved cluster validity index for fuzzy clustering,” Electronics Letters, vol. 57, no. 21, pp. 792–794, 2021.

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.KWON2 = B_KWON2.IDX(x = scale(data), kmax =10, method = "FCM", fzm = 2,
                  nstart = 20, iter = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.KWON2)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Point biserial correlation (PB)

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using Point biserial correlation (PB) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_PB.IDX(x, kmax, method = "kmeans", corr = "pearson", nstart = 100,
        alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("kmeans", "hclust_complete", "hclust_average", "hclust_single"). The default is "kmeans".
corr	a character string indicating which correlation coefficient is to be computed ("pearson", "kendall" or "spearman"). The default is "pearson".
nstart	a maximum number of initial random sets for kmeans for method = "kmeans". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".)
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-PB is defined as follows. Let

$r_k(\bf x) = \dfrac{PB(k)-\min_j PB(j)}{\sum_{i=2}^K (PB(i)-\min_j PB(j))}$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original PB$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

G. W. Miligan, "An examination of the effect of six types of error perturbation on fifteen clustering algorithms," Psychometrika, 45, 325-342 (1980).

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B2_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B2_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.PB = B_PB.IDX(x = scale(data), kmax=10, method = "kmeans", corr = "pearson", nstart = 100,
                alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.PB)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-Pakhira-Bandyopadhyay-Maulik (PBM) index

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using Pakhira-Bandyopadhyay-Maulik (PBM) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_PBM.IDX(x, kmax, method = "FCM", fzm = 2, nstart = 20,
        iter = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("FCM" or "EM"). The default is "FCM".
fzm	a number greater than 1 giving the degree of fuzzification for method = "FCM". The default is 2.
nstart	a maximum number of initial random sets for FCM for method = "FCM". The default is 20.
iter	a maximum number of iterations for method = "FCM". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-PBM is defined as follows. Let

$r_k(\bf x) = \dfrac{PBM(k)-\min_j PBM(j)}{\sum_{i=2}^K (PBM(i)-\min_j PBM(j))}$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original PBM$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

M. K. Pakhira, S. Bandyopadhyay, and U. Maulik, “Validity index for crisp and fuzzy clusters,” Pattern recognition, vol. 37, no. 3, pp. 487–501, 2004.

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B7_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B7_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.PBM = B_PBM.IDX(x = scale(data), kmax =10, method = "FCM", fzm = 2, nstart = 20,
                iter = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.PBM)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

---

BCVI-The score function

Description

Compute Bayesian cluster validity index (BCVI) from two to kmax groups using the score function (SF) as the underling cluster validity index (CVI) with the user's selected Dirichlet prior parameters. The full detail of BCVI can be found in the paper Wiroonsri and Preedasawakul (2024).

Usage

B_SF.IDX(x, kmax, method = "kmeans", nstart = 100, alpha = "default", mult.alpha = 1/2)

Arguments

x	a numeric data frame or matrix where each column is a variable to be used for cluster analysis and each row is a data point.
kmax	a maximum number of clusters to be considered.
method	a character string indicating which clustering method to be used ("kmeans", "hclust_complete", "hclust_average", "hclust_single"). The default is "kmeans".
nstart	a maximum number of initial random sets for kmeans for method = "kmeans". The default is 100.
alpha	Dirichlet prior parameters $\alpha_2,...,\alpha_k$ where $\alpha_k$ is the parameter corresponding to "the probability of having k groups" (selecting each $\alpha_k$ between 0 to 30 is recommended and using the other parameter mult.alpha to be its multiplier. The default is "default".
mult.alpha	the power $s$ from $n^s$ to be multiplied to the Dirichlet prior parameters alpha (selecting mult.alpha in [0,1) is recommended). The default is $\frac{1}{2}$.

Details

BCVI-SF is defined as follows.

Let

$r_k(\bf x) = \dfrac{\max_j SF(j)- SF(k)}{\sum_{i=2}^K (\max_j SF(j) - SF(i))}.$

Assume that

$f({\bf x}|{\bf p}) = C({\bf p}) \prod_{k=2}^Kp_k^{nr_k(x)}$

represents the conditional probability density function of the dataset given $\bf p$, where $C({\bf p})$ is the normalizing constant. Assume further that ${\bf p}$ follows a Dirichlet prior distribution with parameters ${\bm \alpha} = (\alpha_2,\ldots,\alpha_K)$. The posterior distribution of $\bf p$ still remains a Dirichlet distribution with parameters $(\alpha_2+nr_2({\bf x}),\ldots,\alpha_K+nr_K({\bf x}))$.

The BCVI is then defined as

$BCVI(k) = E[p_k|{\bf x}] = \frac{\alpha_k + nr_k({\bf x})}{\alpha_0+n}$

where $\alpha_0 = \sum_{k=2}^K \alpha_k.$

The variance of $p_k$ can be computed as

$Var(p_k|{\bf x}) = \dfrac{(\alpha_k + nr_k(x))(\alpha_0 + n -\alpha_k-nr_k(x))}{(\alpha_0 + n)^2(\alpha_0 + n +1 )}.$

Value

BCVI	the dataframe where the first and the second columns are the number of groups k and BCVI$(k)$, respectively, for k from 2 to kmax.
VAR	the data frame where the first and the second columns are the number of groups k and the variance of $p_k$, respectively, for k from 2 to kmax.
CVI	the data frame where the first and the second columns are the number of groups k and the original SF$(k)$, respectively, for k from 2 to kmax.

Author(s)

Nathakhun Wiroonsri and Onthada Preedasawakul

References

S. Saitta, B. Raphael, I. Smith, "A bounded index for cluster validity," In Perner, P.: Machine Learning and Data Mining in Pattern Recognition, Lecture Notes in Computer Science, 4571, Springer (2007).

O. Preedasawakul, and N. Wiroonsri, A Bayesian Cluster Validity Index, Computational Statistics & Data Analysis, 202, 108053, 2025. doi:10.1016/j.csda.2024.108053

See Also

B2_data, B_TANG.IDX, B_WP.IDX, B_Wvalid, B_DB.IDX

Examples

library(BayesCVI)

# The data included in this package.
data = B2_data[,1:2]

# alpha
aalpha = c(5,5,5,20,20,20,0.5,0.5,0.5)

B.SF = B_SF.IDX(x = scale(data), kmax=10, method = "kmeans",
                nstart = 100, alpha = aalpha, mult.alpha = 1/2)

# plot the BCVI

pplot = plot_BCVI(B.SF)
pplot$plot_index
pplot$plot_BCVI
pplot$error_bar_plot

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