| Title: | A Bayesian Approach for Clustering Constant-Wise Change-Point Data |
|---|---|
| Description: | A Gibbs sampler algorithm was developed to estimate change points in constant-wise data sequences while performing clustering simultaneously. The algorithm is described in da Cruz, A. C. and de Souza, C. P. E "A Bayesian Approach for Clustering Constant-wise Change-point Data" <doi:10.48550/arXiv.2305.17631>. |
| Authors: | Ana Carolina da Cruz [aut, cre] |
| Maintainer: | Ana Carolina da Cruz <[email protected]> |
| License: | MIT + file LICENSE |
| Version: | 0.1.0 |
| Built: | 2025-01-29 20:31:21 UTC |
| Source: | CRAN |
A dataset generated for examplification of Gibbs sampler using the model proposed in the paper "BayesCPclust: A Bayesian Approach for Clustering Constant-Wise Change-Point Data". The generation process is described in the paper with N = 5, M = 50, w = 10, d = 2, K = 2.
datadata
A matrix with 50 rows and 5 columns
A.C. da Cruz, C.P.E. de Souza. "BayesCPclust: A Bayesian Approach for Clustering Constant-Wise Change-Point Data" arXiv, arXiv:2305.17631v3 .
A dataset generated for examplification of Gibbs sampler using the model proposed in the paper "BayesCPclust: A Bayesian Approach for Clustering Constant-Wise Change-Point Data". The generation process is described in the paper with N = 5, M = 50, w = 10, d = 2, K = 2.
data_adata_a
A list with three components: a matrix with 50 rows and 5 columns, a vector with the cluster assignments, a vector with variance components
A.C. da Cruz, C.P.E. de Souza. "BayesCPclust: A Bayesian Approach for Clustering Constant-Wise Change-Point Data" arXiv, arXiv:2305.17631v3 .
Full conditional for lambda
full_cond(kstar, lambda, cluster, al, bl, K, N)full_cond(kstar, lambda, cluster, al, bl, K, N)
kstar |
A scalar with the number maximum of change points in all clusters |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
al |
The hyperparameter value for the shape parameter in the gamma prior for lambda |
bl |
The hyperparameter value for the scale parameter in the gamma prior for lambda |
K |
A vector containing the number of change points for each cluster (or its initial values) |
N |
A scalar representing the number of data sequences |
'full_cond' returns a numerical value corresponding to a sample from the full conditional for lambda
This function is used within the Gibbs sampler, it is not expected to be used alone.
# Using hypothetical values to exemplification purposes clusters <- c(1,1,2,1,2) full_cond(kstar = 2, lambda = 3, cluster = clusters, al = 2, bl = 1000, K = c(2, 2), N = 5)# Using hypothetical values to exemplification purposes clusters <- c(1,1,2,1,2) full_cond(kstar = 2, lambda = 3, cluster = clusters, al = 2, bl = 1000, K = c(2, 2), N = 5)
Gibbs sampler algorithm for simulated scenarios or real datasets
gibbs_alg( N, w, M, K, Tl, cluster, alpha, sigma2, bs = 1000, as = 2, al = 2, bl = 1000, a = 2, b = 1000, alpha0 = 1/100, kstar, lambda, Y, d, maxIter = 10000 )gibbs_alg( N, w, M, K, Tl, cluster, alpha, sigma2, bs = 1000, as = 2, al = 2, bl = 1000, a = 2, b = 1000, alpha0 = 1/100, kstar, lambda, Y, d, maxIter = 10000 )
N |
A scalar representing the number of observations |
w |
A scalar representing the minimum number of points in each interval between two change points |
M |
A scalar representing the number of points available for each observation |
K |
A vector containing the number of change points for each cluster (or its initial values) |
Tl |
A list containing a vector for each cluster determining the change-point positions in each cluster (or its initial values) |
cluster |
A vector containing the cluster assignments for the observations (or its initial values) |
alpha |
A list containing a vector for each cluster determining the constant level values for each interval between change points in each cluster (or its initial values) |
sigma2 |
A vector with the variances of observations (or its initial values) |
bs |
The hyperparameter value for the scale parameter in the inverse-gamma prior for the variance component |
as |
The hyperparameter value for the shape parameter in the inverse-gamma prior for the variance component |
al |
The hyperparameter value for the shape parameter in the gamma prior for lambda |
bl |
The hyperparameter value for the scale parameter in the gamma prior for lambda |
a |
The hyperparameter value for the shape parameter in the gamma prior for alpha0 |
b |
The hyperparameter value for the scale parameter in the gamma prior for alpha0 |
alpha0 |
A scalar defining the parameter for the Dirichlet process prior that controls the number of clusters (or its initial values) |
kstar |
A scalar with the number maximum of change points in all clusters |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
Y |
A matrix M x N with the data sequences |
d |
A scalar representing the number of clusters. |
maxIter |
A scalar for the number of iteration to run in the Gibbs sampler |
A list with each component representing the estimates for each iteration of the Gibbs sampler for each parameter
[run_gibbs()]
data(data) # initial values for each paramter and each cluster par.values <- list(K = c(0, 0), Tl = list(50, 50), alpha = list(5, 10)) #cluster assignment for each data sequence cluster <- kmeans(t(data), 2)$cluster # variance for each data sequence sigma2 <- apply(data, 2, var) res <- gibbs_alg(alpha0 = 1/100, N = 5, w = 10, M = 50, K = par.values$K, Tl = par.values$Tl, cluster = cluster, alpha = par.values$alpha, sigma2 = sigma2, bs = 1000, as = 2, al = 2, bl = 1000, a = 2, b = 1000, kstar = 2, lambda = 2, Y = data, d = 2, maxIter = 10)data(data) # initial values for each paramter and each cluster par.values <- list(K = c(0, 0), Tl = list(50, 50), alpha = list(5, 10)) #cluster assignment for each data sequence cluster <- kmeans(t(data), 2)$cluster # variance for each data sequence sigma2 <- apply(data, 2, var) res <- gibbs_alg(alpha0 = 1/100, N = 5, w = 10, M = 50, K = par.values$K, Tl = par.values$Tl, cluster = cluster, alpha = par.values$alpha, sigma2 = sigma2, bs = 1000, as = 2, al = 2, bl = 1000, a = 2, b = 1000, kstar = 2, lambda = 2, Y = data, d = 2, maxIter = 10)
Transfor a vector with over- or underflow
logsumexp(x, min_x = Inf)logsumexp(x, min_x = Inf)
x |
A vector with numbers |
min_x |
A numerical value to represent the minimum value to perform comparison with the actual minimum value of 'x' |
'logsumexp' returns each element of the vector 'x' transformed using the Log-Sum-Exp trick.
# Transforming all elements in a vector using the Log-Sum-Exp trick x <- c(1, 2, 3, 4, 5, 6) logsumexp(x)# Transforming all elements in a vector using the Log-Sum-Exp trick x <- c(1, 2, 3, 4, 5, 6) logsumexp(x)
Compute the mode of a numerical vector
Mode(x)Mode(x)
x |
A vector with numbers |
'Mode' returns a value representing the most frequent numerical value in the vector 'x'
# Finding the mode of a vector of numbers x <- c(1, 2, 2, 3, 5, 8, 10) Mode(x)# Finding the mode of a vector of numbers x <- c(1, 2, 2, 3, 5, 8, 10) Mode(x)
Probability mass function for truncated poisson
pk(k, kstar, lambda)pk(k, kstar, lambda)
k |
A scalar for the number of changes points in a cluster |
kstar |
A scalar with the number maximum of change points in all clusters |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
'pk' returns a numerical value representing the marginal probability for a given k
This function is used within the Gibbs sampler, it is not expected to be used alone.
[gibbs_alg()]
# Hypothetical values pk(k = 2, kstar = 3, lambda = 2)# Hypothetical values pk(k = 2, kstar = 3, lambda = 2)
Full conditional function for sigma2
possigma2n(as, bs, M, Yn, k, Tln, alphan)possigma2n(as, bs, M, Yn, k, Tln, alphan)
as |
The hyperparameter value for the shape parameter in the inverse-gamma prior for the variance component |
bs |
The hyperparameter value for the scale parameter in the inverse-gamma prior for the variance component |
M |
A scalar representing the number of points available for each data sequence |
Yn |
A vector or matrix with data sequences for a cluster |
k |
A scalar for the number of changes points in a cluster |
Tln |
A vector with the change-point positions for a cluster |
alphan |
A vector with the constant level values for each interval between change points for a cluster |
A numerical value corresponding to a sampled value from the full conditional of the variance component
This function is called within the Gibbs sampler, but it can be used separately as well.
[gibbs_alg()]
data(data) possigma2n(as = 2, bs = 1000, M = 50, Yn = data[,1], k = 0, Tln = 50, alphan = 15)data(data) possigma2n(as = 2, bs = 1000, M = 50, Yn = data[,1], k = 0, Tln = 50, alphan = 15)
Posterior for alpha0
postalpha0(alpha0, a, b, N, cluster)postalpha0(alpha0, a, b, N, cluster)
alpha0 |
A scalar defining the parameter for the Dirichlet process prior that controls the number of clusters (or its initial values) |
a |
The hyperparameter value for the shape parameter in the gamma prior for alpha0 |
b |
The hyperparameter value for the scale parameter in the gamma prior for alpha0 |
N |
A scalar representing the number of data sequences |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
A numerical value corresponding to a sample from the posterior of alpha0
This function is called within the Gibbs sampler, but it can be called seperately.
postalpha0(alpha0 = 1/100, a = 2, b = 1000, N = 5, cluster = c(1,1,2,1,1))postalpha0(alpha0 = 1/100, a = 2, b = 1000, N = 5, cluster = c(1,1,2,1,1))
Full conditional for alphak
postalphak(M, Y, sigma2, K, Tl, cluster, clusteri)postalphak(M, Y, sigma2, K, Tl, cluster, clusteri)
M |
A scalar representing the number of points available for each data sequence |
Y |
A matrix M x N with the data sequences |
sigma2 |
A vector with the variances of the data sequences (or its initial values) |
K |
A vector containing the number of change points for each cluster (or its initial values) |
Tl |
A list containing a vector for each cluster determining the change-point positions in each cluster (or its initial values) |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
clusteri |
A scalar with the index of a cluster |
A numerical vector of size 'K' + 1 with sampled values from the full conditional of alphak for a given cluster 'clusteri'
This function is called within the Gibbs sampler, but it can be called separately as well.
[gibbs_alg()]
data(data) postalphak(M = 50, Y = data, sigma2 = 0.05, K = c(0, 0), Tl = c(50, 50), cluster = c(1,1,2,1,2), clusteri = 1)data(data) postalphak(M = 50, Y = data, sigma2 = 0.05, K = c(0, 0), Tl = c(50, 50), cluster = c(1,1,2,1,2), clusteri = 1)
Marginal probability of K
postK(kstar, w, M, Y, cluster, sigma2, lambda, clusteri)postK(kstar, w, M, Y, cluster, sigma2, lambda, clusteri)
kstar |
A scalar with the number maximum of change points in all clusters |
w |
A scalar representing the minimum number of points in each interval between two change points |
M |
A scalar representing the number of points available for each data sequence |
Y |
A matrix M x N with the data sequences |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
sigma2 |
A vector with the variances of the data sequences (or its initial values) |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
clusteri |
A scalar with the index of a cluster |
A numerical value corresponding to the sampled number of change points, k, for a given cluster
This function is called within the Gibbs sampler, but it can also de called separately.
[gibbs_alg()]
postK(kstar = 2, w = 10, M = 50, Y = data, cluster = c(1,1,2,1,2), sigma2 = apply(data, 2, var), lambda = 2, clusteri = 1)postK(kstar = 2, w = 10, M = 50, Y = data, cluster = c(1,1,2,1,2), sigma2 = apply(data, 2, var), lambda = 2, clusteri = 1)
Marginal probability of K per bin
postK_mk(k, m0, w, M, Yn, sigma2n, cellsn, mk, Cr)postK_mk(k, m0, w, M, Yn, sigma2n, cellsn, mk, Cr)
k |
A scalar for the number of changes points in a cluster |
m0 |
A scalar for the number of positions available to define change-points positions |
w |
A scalar representing the minimum number of points in each interval between two change points |
M |
A scalar representing the number of points available for each data sequence |
Yn |
A vector or matrix with data sequences for a cluster |
sigma2n |
A vector with the variance of the data sequences in a cluster |
cellsn |
A vector with the indices of the data sequences in a cluster |
mk |
A matrix with all possible values to distribute between change points |
Cr |
A scalar with the number of data sequences in a cluster |
'postK_mk' returns a numerical value representing the non-normalized probability for a given bin, given k, and a given cluster
This function is called within [postK()]. It should not be called alone.
[postK()], [gibbs_alg()]
data(data) M <- 50; k <- 0; w <- 10; m0 <- M - 1 -(k+1)*w for(k in 0:2){ mk <- RcppAlgos::permuteGeneral(0:m0, k + 1, constraintFun = "sum", comparisonFun = "==", limitConstraints = m0, repetition = TRUE)} out <- postK_mk(k = 0, m0 = m0, w = 10, M = 50, Yn = data[,c(1,2,4)], sigma2n = rep(0.05, 3), cellsn = c(1,2,4), mk = mk[1,], Cr = 3)data(data) M <- 50; k <- 0; w <- 10; m0 <- M - 1 -(k+1)*w for(k in 0:2){ mk <- RcppAlgos::permuteGeneral(0:m0, k + 1, constraintFun = "sum", comparisonFun = "==", limitConstraints = m0, repetition = TRUE)} out <- postK_mk(k = 0, m0 = m0, w = 10, M = 50, Yn = data[,c(1,2,4)], sigma2n = rep(0.05, 3), cellsn = c(1,2,4), mk = mk[1,], Cr = 3)
Marginal probability of m1,m2,m3,...,mk+1
postmk(w, M, Y, K, cluster, sigma2, clusteri)postmk(w, M, Y, K, cluster, sigma2, clusteri)
w |
A scalar representing the minimum number of points in each interval between two change points |
M |
A scalar representing the number of points available for each data sequence |
Y |
A matrix M x N with the data sequences |
K |
A vector containing the number of change points for each cluster (or its initial values) |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
sigma2 |
A vector with the variances of the data sequences (or its initial values) |
clusteri |
A scalar with the index of a cluster |
A numerical vector of size k + 1 with the sampled number of observations (or bin size, mk) between each change point for a given cluster
This function is called within the Gibbs sampler, but it can also be called separately.
data(data) postmk(w = 10, M = 50, Y = data, K = c(1, 1), cluster = c(2,1,1,1,1), sigma2 = apply(data, 2, var), clusteri = 1)data(data) postmk(w = 10, M = 50, Y = data, K = c(1, 1), cluster = c(2,1,1,1,1), sigma2 = apply(data, 2, var), clusteri = 1)
Mixing probability for creating new cluster
qn0(alpha0, w, N, M, bs, as, kstar, lambda, Yn)qn0(alpha0, w, N, M, bs, as, kstar, lambda, Yn)
alpha0 |
A scalar defining the parameter for the Dirichlet process prior that controls the number of clusters (or its initial values) |
w |
A scalar representing the minimum number of points in each interval between two change points |
N |
A scalar representing the number of data sequences |
M |
A scalar representing the number of points available for each data sequence |
bs |
The hyperparameter value for the scale parameter in the inverse-gamma prior for the variance component |
as |
The hyperparameter value for the shape parameter in the inverse-gamma prior for the variance component |
kstar |
A scalar with the number maximum of change points in all clusters |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
Yn |
A vector or matrix with data sequences for a cluster |
A numerical value representing the mixing value term used to compute the probability that the given data sequence should be a singleton cluster
This function is called within [gibbs_alg()]. It should not be called alone.
[gibbs_alg()]
qn0(alpha0 = 1/100, w = 10, N = 5, M = 50, bs = 1000, as = 2, kstar = 2, lambda = 2, Yn = data[,1])qn0(alpha0 = 1/100, w = 10, N = 5, M = 50, bs = 1000, as = 2, kstar = 2, lambda = 2, Yn = data[,1])
Mixing probability for creating new cluster per bin
qn0_mk(w, m0, bs, as, M, km, lambda, mk, Yn, kstar)qn0_mk(w, m0, bs, as, M, km, lambda, mk, Yn, kstar)
w |
A scalar representing the minimum number of points in each interval between two change points |
m0 |
A scalar for the number of positions available to define change-points positions |
bs |
The hyperparameter value for the scale parameter in the inverse-gamma prior for the variance component |
as |
The hyperparameter value for the shape parameter in the inverse-gamma prior for the variance component |
M |
A scalar representing the number of points available for each data sequence |
km |
A scalar for the number of changes points in a cluster |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
mk |
A matrix with all possible values to distribute between change points |
Yn |
A vector with a data sequence |
kstar |
A scalar with the number maximum of change points in all clusters |
A numerical value representing the mixing value term used to compute the probability that the given data sequence should be a singleton cluster for a given bin size.
This function is called within [qn0()]. It should not be called alone.
[qn0()], [gibbs_alg()]
data(data) M <- 50; k <- 0; w <- 10; m0 <- M - 1 -(k+1)*w for(k in 0:2){ mk <- RcppAlgos::permuteGeneral(0:m0, k + 1, constraintFun = "sum", comparisonFun = "==", limitConstraints = m0, repetition = TRUE)} out <- qn0_mk(w = 10, m0 = m0, bs = 1000, as = 2, M = 50, km = 1, lambda = 2, mk = mk[1,], Yn = data[,1], kstar = 2)data(data) M <- 50; k <- 0; w <- 10; m0 <- M - 1 -(k+1)*w for(k in 0:2){ mk <- RcppAlgos::permuteGeneral(0:m0, k + 1, constraintFun = "sum", comparisonFun = "==", limitConstraints = m0, repetition = TRUE)} out <- qn0_mk(w = 10, m0 = m0, bs = 1000, as = 2, M = 50, km = 1, lambda = 2, mk = mk[1,], Yn = data[,1], kstar = 2)
Mixing probability for getting assigned to an existing cluster
qnj(N, M, as, bs, Yn, alpha, cluster, Tl, K)qnj(N, M, as, bs, Yn, alpha, cluster, Tl, K)
N |
A scalar representing the number of data sequences |
M |
A scalar representing the number of points available for each data sequence |
as |
The hyperparameter value for the shape parameter in the inverse-gamma prior for the variance component |
bs |
The hyperparameter value for the scale parameter in the inverse-gamma prior for the variance component |
Yn |
A vector or matrix with data sequences for a cluster |
alpha |
A list containing a vector for each cluster determining the constant level values for each interval between change points in each cluster (or its initial values) |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
Tl |
A list containing a vector for each cluster determining the change-point positions in each cluster (or its initial values) |
K |
A vector containing the number of change points for each cluster (or its initial values) |
A vector of same size as the vector 'cluster' corresponding to the mixing term value used to compute the probability that the given data sequence 'Yn' should be part of each existing cluster
This function is called within the Gibbs sampler. It should not be called alone.
[gibbs_alg()]
qnj(N = 5, M = 50, as = 2, bs = 1000, Yn = data[,1], alpha = c(10, 10), cluster = c(1,1,2,1,2), Tl = c(50,50), K = c(0,0))qnj(N = 5, M = 50, as = 2, bs = 1000, Yn = data[,1], alpha = c(10, 10), cluster = c(1,1,2,1,2), Tl = c(50,50), K = c(0,0))
Runs the Gibbs sampler algorithm using using initial values for the parameters
run_gibbs( M, N, w, d, as = 2, bs = 100, al = 2, bl = 1000, a = 2, b = 1000, alpha0 = 1/100, lambda = 2, maxIter = 10000, par.values, data, cluster, sigma2 )run_gibbs( M, N, w, d, as = 2, bs = 100, al = 2, bl = 1000, a = 2, b = 1000, alpha0 = 1/100, lambda = 2, maxIter = 10000, par.values, data, cluster, sigma2 )
M |
A scalar representing the number of points available for each observation |
N |
A scalar representing the number of observations |
w |
A scalar representing the minimum number of points in each interval between two change points |
d |
A scalar representing the number of clusters. |
as |
The hyperparameter value for the shape parameter in the inverse-gamma prior for the variance component |
bs |
The hyperparameter value for the scale parameter in the inverse-gamma prior for the variance component |
al |
The hyperparameter value for the shape parameter in the gamma prior for lambda |
bl |
The hyperparameter value for the scale parameter in the gamma prior for lambda |
a |
The hyperparameter value for the shape parameter in the gamma prior for alpha0 |
b |
The hyperparameter value for the scale parameter in the gamma prior for alpha0 |
alpha0 |
A scalar defining the parameter for the Dirichlet process prior that controls the number of clusters (or its initial values) |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
maxIter |
A scalar for the number of iteration to run in the Gibbs sampler |
par.values |
A list with lists with parameters for each cluster. The first argument in each list is the number of change points, then the positions for the change points, where T_1 = 1, T_last = M + 1, and for each interval between change points you need to specify a value for the constant level. If running the Gibbs sampler for a dataset with unknown number of change points, we suggest setting the number of change points for each cluster to be zero. Check example in README file. |
data |
a matrix of size M x N with data sequences in the columns |
cluster |
a vector with cluster assignments for each data sequence |
sigma2 |
a vector with variance components for each data sequence |
A list with estimates for each iteration of the Gibbs sampler for each parameter
d = 2 # two clusters N = 5 # 5 data sequences M = 50 # 50 observations for each data sequence maxIter = 10 # number of Gibbs sampler iterations data(data) # initial values for each paramter and each cluster par.values <- list(K = c(0, 0), Tl = list(50, 50), alpha = list(5, 10)) #cluster assignment for each data sequence cluster <- kmeans(t(data), 2)$cluster # variance for each data sequence sigma2 <- apply(data, 2, var) res <- run_gibbs(M, N, w = 10, d, as = 2, bs = 100, al = 2, bl = 1000, a = 2, b = 1000, alpha0 = 1/100, lambda = 2, maxIter = 10, par.values, data, cluster, sigma2)d = 2 # two clusters N = 5 # 5 data sequences M = 50 # 50 observations for each data sequence maxIter = 10 # number of Gibbs sampler iterations data(data) # initial values for each paramter and each cluster par.values <- list(K = c(0, 0), Tl = list(50, 50), alpha = list(5, 10)) #cluster assignment for each data sequence cluster <- kmeans(t(data), 2)$cluster # variance for each data sequence sigma2 <- apply(data, 2, var) res <- run_gibbs(M, N, w = 10, d, as = 2, bs = 100, al = 2, bl = 1000, a = 2, b = 1000, alpha0 = 1/100, lambda = 2, maxIter = 10, par.values, data, cluster, sigma2)
Update equation for lambda
update_lambda(a = 4, b = 2, kstar, lambda, cluster, al, bl, K, N)update_lambda(a = 4, b = 2, kstar, lambda, cluster, al, bl, K, N)
a |
The hyperparameter value for the shape parameter in the gamma prior for alpha0 |
b |
The hyperparameter value for the scale parameter in the gamma prior for alpha0 |
kstar |
A scalar with the number maximum of change points in all clusters |
lambda |
A scalar defining the parameter for the Truncate Poisson distribution that controls the number of change points (or its initial values) |
cluster |
A vector containing the cluster assignments for the data sequences (or its initial values) |
al |
The hyperparameter value for the shape parameter in the gamma prior for lambda |
bl |
The hyperparameter value for the scale parameter in the gamma prior for lambda |
K |
A vector containing the number of change points for each cluster (or its initial values) |
N |
A scalar representing the number of data sequences |
A numerical value corresponding to a sample from the posterior of the parameter lambda
This function is called within the Gibbs sampler, but it can also be called separately.
[gibbs_alg()]
update_lambda(a = 4, b = 2, kstar = 2, lambda = 2, cluster = c(1,1,2,1,2), al = 2, bl = 1000, K = c(2,2), N = 5)update_lambda(a = 4, b = 2, kstar = 2, lambda = 2, cluster = c(1,1,2,1,2), al = 2, bl = 1000, K = c(2,2), N = 5)