Package 'NetworkChange'

Title:	Bayesian Package for Network Changepoint Analysis
Description:	Network changepoint analysis for undirected network data. The package implements a hidden Markov network change point model (Park and Sohn (2020)). Functions for break number detection using the approximate marginal likelihood and WAIC are also provided.
Authors:	Jong Hee Park [aut,cre], Yunkyu Sohn [aut]
Maintainer:	Jong Hee Park <[email protected]>
License:	GPL-3
Version:	0.8
Built:	2024-10-29 05:04:55 UTC
Source:	https://github.com/jongheepark/networkchange

Help Index

Detect a break number using different metrics
Compute the Average Loss of Hidden State Changes from Expected Break Points
Plot of latent node cluster
Plot of network by hidden regime
K-mean clustering of latent node positions
Major Power Alliance Network (1816 - 2012)
Build a synthetic block-structured temporal data with breaks
Compare Log Marginal Likelihood
Printing multiple ggplots in oone file
Changepoint analysis of a degree-corrected multilinear tensor model
Changepoint analysis of a degree-corrected multilinear tensor model with t-distributed error
Degree-corrected multilinear tensor model
Contour plot of latent node positions
Plot of network array data
Plot of latent node positions
Plot of layer-specific network generation rules.
Postwar Alliance Network (1846 - 2012)
Sample a starting value of hidden states
Starting values of U and V
Hidden State Sampler
Hidden State Sampler with precision
Update time-constant regression parameters
Update regime-changing regression parameters
Update transition matrix
Update latent states
Update regime-specific variance
Update time-constant latent node positions
Regime-specific latent node positions
Update layer specific network generation rules
Update V from a change-point network process
Compare WAIC

Detect a break number using different metrics

Description

Detect a break number using different metrics

Usage

BreakDiagnostic(
  Y,
  R = 2,
  mcmc = 100,
  burnin = 100,
  verbose = 100,
  thin = 1,
  UL.Normal = "Orthonormal",
  v0 = NULL,
  v1 = NULL,
  break.upper = 3,
  a = 1,
  b = 1
)
BreakDiagnostic(
  Y,
  R = 2,
  mcmc = 100,
  burnin = 100,
  verbose = 100,
  thin = 1,
  UL.Normal = "Orthonormal",
  v0 = NULL,
  v1 = NULL,
  break.upper = 3,
  a = 1,
  b = 1
)

Arguments

`Y`	Reponse tensor
`R`	Dimension of latent space. The default is 2.
`mcmc`	The number of MCMC iterations after burnin.
`burnin`	The number of burn-in iterations for the sampler.
`verbose`	A switch which determines whether or not the progress of the sampler is printed to the screen. If `verbose` is greater than 0 the iteration number, the $\beta$ vector, and the error variance are printed to the screen every `verbose`th iteration.
`thin`	The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
`UL.Normal`	Transformation of sampled U. Users can choose "NULL", "Normal" or "Orthonormal." "NULL" is no normalization. "Normal" is the standard normalization. "Orthonormal" is the Gram-Schmidt orthgonalization. Default is "NULL."
`v0`	$v_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for V. If `v0 = NULL`, a value is computed from a test run of `NetworkStatic`.
`v1`	$v_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for V. If `v1 = NULL`, a value is computed from a test run of `NetworkStatic`.
`break.upper`	Upper threshold for break number detection. The default is `break.upper = 3`.
`a`	$a$ is the shape1 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
`b`	$b$ is the shape2 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Examples

   ## Not run: 
   set.seed(19333)
   ## Generate an array (15 by 15 by 20) with a block merging transition
   Y <- MakeBlockNetworkChange(n=5, T=20, type ="merge")

   ## Fit 3 models (no break, one break, and two break) for break number detection 
   detect <- BreakDiagnostic(Y, R=2, break.upper = 2)
   
   ## Look at the graph
   detect[[1]]; print(detect[[2]])


## End(Not run)



## Not run: 
   set.seed(19333)
   ## Generate an array (15 by 15 by 20) with a block merging transition
   Y <- MakeBlockNetworkChange(n=5, T=20, type ="merge")

   ## Fit 3 models (no break, one break, and two break) for break number detection 
   detect <- BreakDiagnostic(Y, R=2, break.upper = 2)
   
   ## Look at the graph
   detect[[1]]; print(detect[[2]])


## End(Not run)

Compute the Average Loss of Hidden State Changes from Expected Break Points

Description

Compute the Average Loss of Hidden State Changes from Expected Break Points

Usage

BreakPointLoss(model.list, waic = FALSE, display = TRUE)
BreakPointLoss(model.list, waic = FALSE, display = TRUE)

Arguments

model.list

MCMC output objects. These have to be of class mcmc and have a logmarglike attribute. In what follows, we let M denote the total number of models to be compared.

waic

If waic is TRUE, waic(Watanabe information criterion) will be reported.

display

If display is TRUE, a plot of ave.loss will be produced.

BreakPointLoss. ave.loss, logmarglike, State, Tau, Tau.samp

Value

BreakPointLoss returns five objects. They are: ave.loss the expected loss for each model computed by the mean sqaured distance of hidden state changes from the expected break points. logmarglike the natural log of the marginal likelihood for each model; State sampled state vectors; Tau expected break points for each model; and Tau.samp sampled break points from hidden state draws.

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Examples

   ## Not run: 
   set.seed(1973)
   ## Generate an array (30 by 30 by 40) with block transitions
   from 2 blocks to 3 blocks
   Y <- MakeBlockNetworkChange(n=10, T=40, type ="split")
   G <- 100 ## Small mcmc scans to save time

   ## Fit multiple models for break number detection using Bayesian model comparison
   out0 <- NetworkStatic(Y, R=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out1 <- NetworkChange(Y, R=2, m=1, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out2 <- NetworkChange(Y, R=2, m=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out3 <- NetworkChange(Y, R=2, m=3, mcmc=G, burnin=G, verbose=G, Waic=TRUE)

   ## The most probable model given break number 0 to 3 and data is out1 according to WAIC 
   out <- BreakPointLoss(out0, out1, out2, out3, waic=TRUE)

   print(out[["ave.loss"]])

## End(Not run)


## Not run: 
   set.seed(1973)
   ## Generate an array (30 by 30 by 40) with block transitions
   from 2 blocks to 3 blocks
   Y <- MakeBlockNetworkChange(n=10, T=40, type ="split")
   G <- 100 ## Small mcmc scans to save time

   ## Fit multiple models for break number detection using Bayesian model comparison
   out0 <- NetworkStatic(Y, R=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out1 <- NetworkChange(Y, R=2, m=1, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out2 <- NetworkChange(Y, R=2, m=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out3 <- NetworkChange(Y, R=2, m=3, mcmc=G, burnin=G, verbose=G, Waic=TRUE)

   ## The most probable model given break number 0 to 3 and data is out1 according to WAIC 
   out <- BreakPointLoss(out0, out1, out2, out3, waic=TRUE)

   print(out[["ave.loss"]])

## End(Not run)

Plot of latent node cluster

Description

Plot latent node cluster

Usage

drawPostAnalysis(
  mcmcout,
  Y,
  point.cex = 3,
  text.cex = 3,
  segment.size = 0.1,
  n.cluster = NULL,
  start = 1,
  frequency = 1
)
drawPostAnalysis(
  mcmcout,
  Y,
  point.cex = 3,
  text.cex = 3,
  segment.size = 0.1,
  n.cluster = NULL,
  start = 1,
  frequency = 1
)

Arguments

`mcmcout`	NetworkChange output
`Y`	Input raw data
`point.cex`	node point size. Default is 3.
`text.cex`	node label size. Default is 3.
`segment.size`	segment size. Default is 0.1.
`n.cluster`	number of cluster. Default is 3.
`start`	start of ts object
`frequency`	frequency of ts object

Value

A plot object

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Examples


   ## Not run: 
   set.seed(1973)
   ## generate an array with two constant blocks
   data(MajorAlly)
   Y <- MajorAlly
   fit <- NetworkChange(newY, R=2, m=2, mcmc=G, initial.s = initial.s,
          burnin=G, verbose=0, v0=v0, v1=v1)
   drawPostAnalysis(fit, Y, n.cluster=c(4, 4, 3))
   
## End(Not run)
## Not run: 
   set.seed(1973)
   ## generate an array with two constant blocks
   data(MajorAlly)
   Y <- MajorAlly
   fit <- NetworkChange(newY, R=2, m=2, mcmc=G, initial.s = initial.s,
          burnin=G, verbose=0, v0=v0, v1=v1)
   drawPostAnalysis(fit, Y, n.cluster=c(4, 4, 3))
   
## End(Not run)

Plot of network by hidden regime

Description

Plot latent node cluster

Usage

drawRegimeRaw(mcmcout, Y)
drawRegimeRaw(mcmcout, Y)

Arguments

`mcmcout`	NetworkChange output
`Y`	Input raw data

Value

A plot object

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Examples


   ## Not run: 
   set.seed(1973)
   ## generate an array with two constant blocks
   data(MajorAlly)
   Y <- MajorAlly
   fit <- NetworkChange(newY, R=2, m=2, mcmc=G, initial.s = initial.s,
          burnin=G, verbose=0, v0=v0, v1=v1)
   drawRegimeRaw(fit, newY)
   
## End(Not run)
## Not run: 
   set.seed(1973)
   ## generate an array with two constant blocks
   data(MajorAlly)
   Y <- MajorAlly
   fit <- NetworkChange(newY, R=2, m=2, mcmc=G, initial.s = initial.s,
          burnin=G, verbose=0, v0=v0, v1=v1)
   drawRegimeRaw(fit, newY)
   
## End(Not run)

K-mean clustering of latent node positions

Description

K-mean clustering of latent node positions

Usage

kmeansU(out, R = 2, n.cluster = 3, layer = 1, main = "")
kmeansU(out, R = 2, n.cluster = 3, layer = 1, main = "")

Arguments

`out`	Output of networkchange objects.
`R`	Number of latent space dimensions
`n.cluster`	Number of latent cluster
`layer`	Layer id for the cluster analysis
`main`	Title

Value

A plot object

Examples


   ## Not run: set.seed(1973)
   ## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=10, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   ## latent node positions
   kmeansU(out0)
   
## End(Not run)
## Not run: set.seed(1973)
   ## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=10, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   ## latent node positions
   kmeansU(out0)
   
## End(Not run)

Major Power Alliance Network (1816 - 2012)

Description

This dataframe contains major power alliance network data from 1816 to 2012 (2 year interval).

Format

The dataframe has contains data for major power alliance network data from 1816 to 2012. Major power definition is the COW data set, which incldues USA, UK, France, Germany (West Germany during 1954-1989), Austria, Italy, Russia, China, and Japan. In this data set, a defense pact (Type I), which is the highest level of military commitment, is coded as 1, and 0 otherwise.

Source

Correlates of War Project. 2017. "State System Membership List, v2016." Online, https://correlatesofwar.org/. Gibler, Douglas M. 2009. International military alliances, 1648-2008. CQ Press.

Build a synthetic block-structured temporal data with breaks

Description

MakeBlockNetworkChange generates a block-structured temporal data with breaks.

Usage

MakeBlockNetworkChange(
  n = 10,
  break.point = 0.5,
  base.prob = 0.05,
  block.prob = 0.5,
  shape = 1,
  T = 40,
  break.point1 = 0.25,
  break.point2 = 0.75,
  type = "merge"
)
MakeBlockNetworkChange(
  n = 10,
  break.point = 0.5,
  base.prob = 0.05,
  block.prob = 0.5,
  shape = 1,
  T = 40,
  break.point1 = 0.25,
  break.point2 = 0.75,
  type = "merge"
)

Arguments

`n`	The number of nodes within a block. The total number of nodes is n*block.number.
`break.point`	The point of break. 0 indicates the beginning, 0.5 indicates the middle, and 1 indicates the end.
`base.prob`	The probability of link among non-block members.
`block.prob`	The probability of link among within-block members.
`shape`	The speed of breaks. The larger shape is, the faster the transition is. shape > 0 and shape < 8.
`T`	The length of time.
`break.point1`	The point of the first break in "merge-split" or "split-merge". Any number between 0 and 0.5 can be chosen. For example, 0 indicates #' the beginning, 0.25 indicates the 1/4th point, and 0.5 indicates the half point.
`break.point2`	The point of the second breakin "merge-split" or "split-merge". Any number between 0.5 and 1 can be chosen. For example, 0.5 indicates the beginning, 0.75 indicates the 3/4th point, and 1 indicates the end point.
`type`	The type of network changes. Options are "constant", "merge", "split", "merge-split", "split-merge." If "constant" is chosen, the number of breaks is zero. If "merge" or "split" is chosen, the number of breaks is one. If either "merge-split" or "split-merge" is chosen, the number of breaks is two.

Value

output An output of MakeBlockNetworkChange contains a symmetric block-structured temporal network data set with breaks.

Compare Log Marginal Likelihood

Description

Compare Log Marginal Likelihood

Usage

MarginalCompare(outlist)
MarginalCompare(outlist)

Arguments

outlist

List of NetworkChange objects

Value

A matrix of log marginal likelihoods.

References

Siddhartha Chib. 1995. “Marginal Likelihood from the Gibbs Output.” Journal of the American Statistical Association. 90: 1313-1321.

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Printing multiple ggplots in oone file

Description

Print multiple ggplots in one file. Slightly modified for packaging from the original version in the web.

Usage

multiplot(..., plotlist = NULL, cols = 1, layout = NULL)
multiplot(..., plotlist = NULL, cols = 1, layout = NULL)

Arguments

`...`	A list of ggplot objects separated by commas.
`plotlist`	A list of ggplot objects
`cols`	The number of columns.
`layout`	A matrix specifying the layout. If present, 'cols' is ignored.

Value

A plot object

Author(s)

http://www.cookbook-r.com/Graphs/Multiple_graphs_on_one_page_(ggplot2)/

Changepoint analysis of a degree-corrected multilinear tensor model

Description

NetworkChange implements Bayesian multiple changepoint models to network time series data using a degree-corrected multilinear tensor decomposition method

Usage

NetworkChange(
  Y,
  R = 2,
  m = 1,
  initial.s = NULL,
  mcmc = 100,
  burnin = 100,
  verbose = 0,
  thin = 1,
  reduce.mcmc = NULL,
  degree.normal = "eigen",
  UL.Normal = "Orthonormal",
  DIC = FALSE,
  Waic = FALSE,
  marginal = FALSE,
  plotUU = FALSE,
  plotZ = FALSE,
  constant = FALSE,
  b0 = 0,
  B0 = 1,
  c0 = NULL,
  d0 = NULL,
  u0 = NULL,
  u1 = NULL,
  v0 = NULL,
  v1 = NULL,
  a = NULL,
  b = NULL
)
NetworkChange(
  Y,
  R = 2,
  m = 1,
  initial.s = NULL,
  mcmc = 100,
  burnin = 100,
  verbose = 0,
  thin = 1,
  reduce.mcmc = NULL,
  degree.normal = "eigen",
  UL.Normal = "Orthonormal",
  DIC = FALSE,
  Waic = FALSE,
  marginal = FALSE,
  plotUU = FALSE,
  plotZ = FALSE,
  constant = FALSE,
  b0 = 0,
  B0 = 1,
  c0 = NULL,
  d0 = NULL,
  u0 = NULL,
  u1 = NULL,
  v0 = NULL,
  v1 = NULL,
  a = NULL,
  b = NULL
)

Arguments

`Y`	Reponse tensor
`R`	Dimension of latent space. The default is 2.
`m`	Number of change point. If `m = 0` is specified, the result should be the same as `NetworkStatic`.
`initial.s`	The starting value of latent state vector. The default is sampling from equal probabilities for all states.
`mcmc`	The number of MCMC iterations after burnin.
`burnin`	The number of burn-in iterations for the sampler.
`verbose`	A switch which determines whether or not the progress of the sampler is printed to the screen. If `verbose` is greater than 0 the iteration number, the $\beta$ vector, and the error variance are printed to the screen every `verbose`th iteration.
`thin`	The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
`reduce.mcmc`	The number of reduced MCMC iterations for marginal likelihood computations. If `reduce.mcmc = NULL`, `mcmc/thin` is used.
`degree.normal`	A null model for degree correction. Users can choose "NULL", "eigen" or "Lsym." "NULL" is no degree correction. "eigen" is a principal eigen-matrix consisting of the first eigenvalue and the corresponding eigenvector. " Lsym" is a modularity matrix. Default is "eigen."
`UL.Normal`	Transformation of sampled U. Users can choose "NULL", "Normal" or "Orthonormal." "NULL" is no normalization. "Normal" is the standard normalization. "Orthonormal" is the Gram-Schmidt orthgonalization. Default is "NULL."
`DIC`	If `DIC = TRUE`, the deviation information criterion is computed.
`Waic`	If `Waic = TRUE`, the Watanabe information criterion is computed.
`marginal`	If `marginal = TRUE`, the log marignal likelihood is computed using the method of Chib (1995).
`plotUU`	If `plotUU = TRUE` and `verbose > 0`, then the plot of the latent space will be printed to the screen at every `verbose`th iteration. The default is `plotUU = FALSE`.
`plotZ`	If `plotZ = TRUE` and `verbose > 0`, then the plot of the degree-corrected input matrix will be printed to the screen with the sampled mean values at every `verbose`th iteration. The default is `plotUU = FALSE`.
`constant`	If `constant = TRUE`, constant parameter is sampled and saved in the output as attribute `bmat`. Default is `constant = FALSE`.
`b0`	The prior mean of $\beta$ . This must be a scalar. The default value is 0.
`B0`	The prior variance of $\beta$ . This must be a scalar. The default value is 1.
`c0`	= 0.1
`d0`	= 0.1
`u0`	$u_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for U. The default is 10.
`u1`	$u_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for U. The default is 1.
`v0`	$v_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for V. The default is 10.
`v1`	$v_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for V. The default is the time length of Y.
`a`	$a$ is the shape1 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
`b`	$b$ is the shape2 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.

Value

An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package. The object contains an attribute Waic.out that contains results of WAIC and the log-marginal likelihood of the model (logmarglike). The object also contains an attribute prob.state storage matrix that contains the probability of $state_i$ for each period

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Peter D. Hoff 2011. "Hierarchical Multilinear Models for Multiway Data." Computational Statistics \& Data Analysis. 55: 530-543.

Siddhartha Chib. 1998. "Estimation and comparison of multiple change-point models." Journal of Econometrics. 86: 221-241.

Examples


   ## Not run: 
   set.seed(1973)
   \## Generate an array (30 by 30 by 40) with block transitions
   from 2 blocks to 3 blocks
   Y <- MakeBlockNetworkChange(n=10, T=40, type ="split")
   G <- 100 ## Small mcmc scans to save time

   \## Fit multiple models for break number detection using Bayesian model comparison
   out0 <- NetworkStatic(Y, R=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out1 <- NetworkChange(Y, R=2, m=1, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out2 <- NetworkChange(Y, R=2, m=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out3 <- NetworkChange(Y, R=2, m=3, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   outlist <- list(out0, out1, out2, out3)

   \## The most probable model given break number 0 to 3 and data is out1 according to WAIC 
   WaicCompare(outlist)

   plotU(out1)
 
   plotV(out1)
   
## End(Not run)
## Not run: 
   set.seed(1973)
   \## Generate an array (30 by 30 by 40) with block transitions
   from 2 blocks to 3 blocks
   Y <- MakeBlockNetworkChange(n=10, T=40, type ="split")
   G <- 100 ## Small mcmc scans to save time

   \## Fit multiple models for break number detection using Bayesian model comparison
   out0 <- NetworkStatic(Y, R=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out1 <- NetworkChange(Y, R=2, m=1, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out2 <- NetworkChange(Y, R=2, m=2, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   out3 <- NetworkChange(Y, R=2, m=3, mcmc=G, burnin=G, verbose=G, Waic=TRUE)
   outlist <- list(out0, out1, out2, out3)

   \## The most probable model given break number 0 to 3 and data is out1 according to WAIC 
   WaicCompare(outlist)

   plotU(out1)
 
   plotV(out1)
   
## End(Not run)

Changepoint analysis of a degree-corrected multilinear tensor model with t-distributed error

Description

NetworkChangeRobust implements Bayesian multiple changepoint models to network time series data using a degree-corrected multilinear tensor decomposition method with t-distributed error

Usage

NetworkChangeRobust(
  Y,
  R = 2,
  m = 1,
  initial.s = NULL,
  mcmc = 100,
  burnin = 100,
  verbose = 0,
  thin = 1,
  degree.normal = "eigen",
  UL.Normal = "Orthonormal",
  plotUU = FALSE,
  plotZ = FALSE,
  b0 = 0,
  B0 = 1,
  c0 = NULL,
  d0 = NULL,
  n0 = 2,
  m0 = 2,
  u0 = NULL,
  u1 = NULL,
  v0 = NULL,
  v1 = NULL,
  a = NULL,
  b = NULL
)
NetworkChangeRobust(
  Y,
  R = 2,
  m = 1,
  initial.s = NULL,
  mcmc = 100,
  burnin = 100,
  verbose = 0,
  thin = 1,
  degree.normal = "eigen",
  UL.Normal = "Orthonormal",
  plotUU = FALSE,
  plotZ = FALSE,
  b0 = 0,
  B0 = 1,
  c0 = NULL,
  d0 = NULL,
  n0 = 2,
  m0 = 2,
  u0 = NULL,
  u1 = NULL,
  v0 = NULL,
  v1 = NULL,
  a = NULL,
  b = NULL
)

Arguments

`Y`	Reponse tensor
`R`	Dimension of latent space. The default is 2.
`m`	Number of change point. If `m = 0` is specified, the result should be the same as `NetworkStatic`.
`initial.s`	The starting value of latent state vector. The default is sampling from equal probabilities for all states.
`mcmc`	The number of MCMC iterations after burnin.
`burnin`	The number of burn-in iterations for the sampler.
`verbose`	A switch which determines whether or not the progress of the sampler is printed to the screen. If `verbose` is greater than 0 the iteration number, the $\beta$ vector, and the error variance are printed to the screen every `verbose`th iteration.
`thin`	The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
`degree.normal`	A null model for degree correction. Users can choose "NULL", "eigen" or "Lsym." "NULL" is no degree correction. "eigen" is a principal eigen-matrix consisting of the first eigenvalue and the corresponding eigenvector. " Lsym" is a modularity matrix. Default is "eigen."
`UL.Normal`	Transformation of sampled U. Users can choose "NULL", "Normal" or "Orthonormal." "NULL" is no normalization. "Normal" is the standard normalization. "Orthonormal" is the Gram-Schmidt orthgonalization. Default is "NULL."
`plotUU`	If `plotUU = TRUE` and `verbose > 0`, then the plot of the latent space will be printed to the screen at every `verbose`th iteration. The default is `plotUU = FALSE`.
`plotZ`	If `plotZ = TRUE` and `verbose > 0`, then the plot of the degree-corrected input matrix will be printed to the screen with the sampled mean values at every `verbose`th iteration. The default is `plotUU = FALSE`.
`b0`	The prior mean of $\beta$ . This must be a scalar. The default value is 0.
`B0`	The prior variance of $\beta$ . This must be a scalar. The default value is 1.
`c0`	= 0.1 The shape parameter of inverse gamma prior for $\sigma^2$ .
`d0`	= 0.1 The rate parameter of inverse gamma prior for $\sigma^2$ .
`n0`	= 0.1 The shape parameter of inverse gamma prior for $\gamma$ of Student-t distribution.
`m0`	= 0.1 The rate parameter of inverse gamma prior for $\gamma$ of Student-t distribution.
`u0`	$u_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for U. The default is 10.
`u1`	$u_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for U. The default is 1.
`v0`	$v_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for V. The default is 10.
`v1`	$v_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for V. The default is the time length of Y.
`a`	$a$ is the shape1 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.
`b`	$b$ is the shape2 beta prior for transition probabilities. By default, the expected duration is computed and corresponding a and b values are assigned. The expected duration is the sample period divided by the number of states.

Value

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Peter D. Hoff 2011. "Hierarchical Multilinear Models for Multiway Data." Computational Statistics \& Data Analysis. 55: 530-543.

Siddhartha Chib. 1998. "Estimation and comparison of multiple change-point models." Journal of Econometrics. 86: 221-241.

Sumio Watanabe. 2010. "Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory." Journal of Machine Learning Research. 11: 3571-3594. Siddhartha Chib. 1995. “Marginal Likelihood from the Gibbs Output.” Journal of the American Statistical Association. 90: 1313-1321.

Examples


   ## Not run: 
   set.seed(1973)
   ## Generate an array (30 by 30 by 40) with block transitions
   from 2 blocks to 3 blocks
   Y <- MakeBlockNetworkChange(n=10, T=40, type ="split")
   G <- 100 ## only 100 mcmc scans to save time
   ## Fit models
   out1 <- NetworkChangeRobust(Y, R=2, m=1, mcmc=G, burnin=G, verbose=G)
   ## plot latent node positions
   plotU(out1)
   ## plot layer-specific network generation rules
   plotV(out1)
   
## End(Not run)
## Not run: 
   set.seed(1973)
   ## Generate an array (30 by 30 by 40) with block transitions
   from 2 blocks to 3 blocks
   Y <- MakeBlockNetworkChange(n=10, T=40, type ="split")
   G <- 100 ## only 100 mcmc scans to save time
   ## Fit models
   out1 <- NetworkChangeRobust(Y, R=2, m=1, mcmc=G, burnin=G, verbose=G)
   ## plot latent node positions
   plotU(out1)
   ## plot layer-specific network generation rules
   plotV(out1)
   
## End(Not run)

Degree-corrected multilinear tensor model

Description

NetworkStatic implements a degree-corrected Bayesian multilinear tensor decomposition method

Usage

NetworkStatic(
  Y,
  R = 2,
  mcmc = 100,
  burnin = 100,
  verbose = 0,
  thin = 1,
  reduce.mcmc = NULL,
  degree.normal = "eigen",
  UL.Normal = "Orthonormal",
  plotUU = FALSE,
  plotZ = FALSE,
  constant = FALSE,
  b0 = 0,
  B0 = 1,
  c0 = NULL,
  d0 = NULL,
  u0 = NULL,
  u1 = NULL,
  v0 = NULL,
  v1 = NULL,
  marginal = FALSE,
  DIC = FALSE,
  Waic = FALSE
)
NetworkStatic(
  Y,
  R = 2,
  mcmc = 100,
  burnin = 100,
  verbose = 0,
  thin = 1,
  reduce.mcmc = NULL,
  degree.normal = "eigen",
  UL.Normal = "Orthonormal",
  plotUU = FALSE,
  plotZ = FALSE,
  constant = FALSE,
  b0 = 0,
  B0 = 1,
  c0 = NULL,
  d0 = NULL,
  u0 = NULL,
  u1 = NULL,
  v0 = NULL,
  v1 = NULL,
  marginal = FALSE,
  DIC = FALSE,
  Waic = FALSE
)

Arguments

`Y`	Reponse tensor
`R`	Dimension of latent space. The default is 2.
`mcmc`	The number of MCMC iterations after burnin.
`burnin`	The number of burn-in iterations for the sampler.
`verbose`	A switch which determines whether or not the progress of the sampler is printed to the screen. If `verbose` is greater than 0 the iteration number, the $\beta$ vector, and the error variance are printed to the screen every `verbose`th iteration.
`thin`	The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value.
`reduce.mcmc`	The number of reduced MCMC iterations for marginal likelihood computations. If `reduce.mcmc = NULL`, `mcmc/thin` is used.
`degree.normal`	A null model for degree correction. Users can choose "NULL", "eigen" or "Lsym." "NULL" is no degree correction. "eigen" is a principal eigen-matrix consisting of the first eigenvalue and the corresponding eigenvector. " Lsym" is a modularity matrix. Default is "eigen."
`UL.Normal`	Transformation of sampled U. Users can choose "NULL", "Normal" or "Orthonormal." "NULL" is no normalization. "Normal" is the standard normalization. "Orthonormal" is the Gram-Schmidt orthgonalization. Default is "NULL."
`plotUU`	If `plotUU = TRUE` and `verbose > 0`, then the plot of the latent space will be printed to the screen at every `verbose`th iteration. The default is `plotUU = FALSE`.
`plotZ`	If `plotZ = TRUE` and `verbose > 0`, then the plot of the degree-corrected input matrix will be printed to the screen with the sampled mean values at every `verbose`th iteration. The default is `plotUU = FALSE`.
`constant`	If `constant = TRUE`, constant parameter is sampled and saved in the output as attribute `bmat`. Default is `constant = FALSE`.
`b0`	The prior mean of $\beta$ . This must be a scalar. The default value is 0.
`B0`	The prior variance of $\beta$ . This must be a scalar. The default value is 1.
`c0`	= 0.1
`d0`	= 0.1
`u0`	$u_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for U. The default is 10.
`u1`	$u_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for U. The default is 1.
`v0`	$v_0/2$ is the shape parameter for the inverse Gamma prior on variance parameters for V. The default is 10.
`v1`	$v_1/2$ is the scale parameter for the inverse Gamma prior on variance parameters for V. The default is the time length of Y.
`marginal`	If `marginal = TRUE`, the log marignal likelihood is computed using the method of Chib (1995).
`DIC`	If `DIC = TRUE`, the deviation information criterion is computed.
`Waic`	If `Waic = TRUE`, the Watanabe information criterion is computed.

Value

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Peter D. Hoff 2011. "Hierarchical Multilinear Models for Multiway Data." Computational Statistics \& Data Analysis. 55: 530-543.

Examples


   ## Not run: 
   set.seed(1973)

   \## generate an array with three constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=10, type ="constant")
   G <- 100 ## Small mcmc scans to save time
   out0 <- NetworkStatic(Y, R=2, mcmc=G, burnin=G, verbose=G)

   \## recovered latent blocks
   Kmeans(out0, n.cluster=3, main="Recovered Blocks")

   \## contour plot of latent node positions
   plotContour(out0)

   \## plot latent node positions
   plotU(out0)

   \## plot layer-specific network connection rules
   plotV(out0)
   
## End(Not run)

## Not run: 
   set.seed(1973)

   \## generate an array with three constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=10, type ="constant")
   G <- 100 ## Small mcmc scans to save time
   out0 <- NetworkStatic(Y, R=2, mcmc=G, burnin=G, verbose=G)

   \## recovered latent blocks
   Kmeans(out0, n.cluster=3, main="Recovered Blocks")

   \## contour plot of latent node positions
   plotContour(out0)

   \## plot latent node positions
   plotU(out0)

   \## plot layer-specific network connection rules
   plotV(out0)
   
## End(Not run)

Contour plot of latent node positions

Description

Draw a contour plot of latent node positions

Usage

plotContour(OUT, main = "", k = 8, my.cols = brewer.pal(k, "Spectral"))
plotContour(OUT, main = "", k = 8, my.cols = brewer.pal(k, "Spectral"))

Arguments

`OUT`	Output of networkchange objects.
`main`	The title of plot
`k`	The number of levels (nlevels in contour ()).
`my.cols`	Color scale. Use brewer.pal() from RColorBrewer.

Value

A plot object

Examples


   ## Not run: set.seed(1973)
   \## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=40, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   \## contour plot of latent node positions
   plotContour(out0)
   
## End(Not run)
## Not run: set.seed(1973)
   \## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=40, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   \## contour plot of latent node positions
   plotContour(out0)
   
## End(Not run)

Plot of network array data

Description

Plot network array data

Usage

plotnetarray(
  Y,
  n.graph = 4,
  node.size = 2,
  node.color = "brown",
  edge.alpha = 0.5,
  edge.size = 0.2,
  edge.color = "grey"
)
plotnetarray(
  Y,
  n.graph = 4,
  node.size = 2,
  node.color = "brown",
  edge.alpha = 0.5,
  edge.size = 0.2,
  edge.color = "grey"
)

Arguments

`Y`	network array data
`n.graph`	number of subgraphs. Default is 4.
`node.size`	node size. Default is 2.
`node.color`	node color. Default is "brown."
`edge.alpha`	transparency of edge. Default is 0.5.
`edge.size`	edge size. Default is 0.2.
`edge.color`	edge color. Default is "grey."

Value

A plot object

References

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Examples


   ## Not run: 
   set.seed(1973)
   ## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=1, T=20, type ="split")
   plotnetarray(Y)
   
## End(Not run)
## Not run: 
   set.seed(1973)
   ## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=1, T=20, type ="split")
   plotnetarray(Y)
   
## End(Not run)

Plot of latent node positions

Description

Plot latent node positions

Usage

plotU(OUT, Time = NULL, names = NULL, main = NULL, label.prob = 0.9)
plotU(OUT, Time = NULL, names = NULL, main = NULL, label.prob = 0.9)

Arguments

`OUT`	Output of networkchange objects.
`Time`	Starting of the time period. If NULL, 1.
`names`	Node names. If NULL, use natural numbers.
`main`	The title of plot
`label.prob`	Label print threshold. 0.9 is the default.

Value

A plot object

Examples


   ## Not run: 
   set.seed(1973)
   \## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=40, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   \## latent node positions
   plotU(out0)
   
## End(Not run)
## Not run: 
   set.seed(1973)
   \## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=40, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   \## latent node positions
   plotU(out0)
   
## End(Not run)

Plot of layer-specific network generation rules.

Description

Plot layer-specific network generation rules.

Usage

plotV(OUT, main = "", cex = 2)
plotV(OUT, main = "", cex = 2)

Arguments

`OUT`	Output of networkchange objects.
`main`	The title of plot
`cex`	point size

Value

A plot object

Examples


   ## Not run: set.seed(1973)
   \## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=40, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   \## latent node positions
   plotV(out0)
   
## End(Not run)
## Not run: set.seed(1973)
   \## generate an array with two constant blocks
   Y <- MakeBlockNetworkChange(n=10, shape=10, T=40, type ="constant")
   out0 <- NetworkStatic(Y, R=2, mcmc=10, burnin=10,
   verbose=10, UL.Normal = "Orthonormal")
   \## latent node positions
   plotV(out0)
   
## End(Not run)

Postwar Alliance Network (1846 - 2012)

Description

This dataframe contains postwar alliance network data from 1946 to 2012 (2 year interval).

Format

The dataframe has contains data for postwar alliance network data from 1946 to 2012 with 2 year interval. After removing disconnected components, 104 countries are included. In this data set, a defense pact (Type I), which is the highest level of military commitment, is coded as 1, and 0 otherwise.

Source

Correlates of War Project. 2017. "State System Membership List, v2016." Online, https://correlatesofwar.org/. Gibler, Douglas M. 2009. International military alliances, 1648-2008. CQ Press.

Sample a starting value of hidden states

Description

Sample a starting value of hidden states

Usage

startS(Z, Time, m, initial.U, V, s2, R)
startS(Z, Time, m, initial.U, V, s2, R)

Arguments

`Z`	Degree-corrected network array data
`Time`	The length of time.
`m`	The number of breaks
`initial.U`	Initialized U matrix.
`V`	Initialized V matrix.
`s2`	Initialized error variance
`R`	The dimensionality of latent space

Value

A state vector

Starting values of U and V

Description

Initialize starting values of U and V

Usage

startUV(Z, R, K)
startUV(Z, R, K)

Arguments

`Z`	Degree-corrected network array data.
`R`	The dimensionality of latent space.
`K`	The dimensionality of Z.

Value

A list of U and V

Hidden State Sampler

Description

Sample hidden states from hidden Markov multilinear model

Usage

ULUstateSample(m, s, ZMUt, s2, P, SOS.random)
ULUstateSample(m, s, ZMUt, s2, P, SOS.random)

Arguments

`m`	The number of break
`s`	Latent state vector
`ZMUt`	Z - MU
`s2`	error variance
`P`	Transition matrix
`SOS.random`	single observation state random perturbation

Value

A list of a state vector, state probabilities, and SOS.random.

Hidden State Sampler with precision

Description

Sample hidden states from hidden Markov multilinear model with precision using Rmpfr package

Usage

ULUstateSample.mpfr(m, s, ZMUt, s2, P, SOS.random)
ULUstateSample.mpfr(m, s, ZMUt, s2, P, SOS.random)

Arguments

`m`	The number of break
`s`	Latent state vector
`ZMUt`	Z - MU
`s2`	error variance
`P`	Transition matrix
`SOS.random`	single observation state random perturbation

Value

A list of a state vector, state probabilities, and SOS.random.

Update time-constant regression parameters

Description

Update time-constant regression parameters

Usage

updateb(Z, MU, s2, XtX, b0, B0)
updateb(Z, MU, s2, XtX, b0, B0)

Arguments

`Z`	Degree corrected response tensor
`MU`	Mean array
`s2`	Error variance
`XtX`	X'X
`b0`	Prior mean of beta
`B0`	Prior variance of beta

Value

A vector of regression parameters

Update regime-changing regression parameters

Description

Update regime-changing beta

Usage

updatebm(ns, K, s, s2, B0, p, ZU)
updatebm(ns, K, s, s2, B0, p, ZU)

Arguments

`ns`	The number of hidden states
`K`	The dimensionality of Z
`s`	Latent state vector
`s2`	The variance of error
`B0`	The prior variance of beta
`p`	The rank of X
`ZU`	Z - ULU

Value

A vector of regime-changing regression parameters

Update transition matrix

Description

Update transition matrix

Usage

updateP(s, ns, P, A0)
updateP(s, ns, P, A0)

Arguments

`s`	Latent state vector
`ns`	The number of hidden states
`P`	Transition matrix
`A0`	Prior of transition matrix

Value

A transtion matrix

Update latent states

Description

Update latent states

Usage

updateS(
  iter,
  s,
  V,
  m,
  Zb,
  Zt,
  Time,
  MU.state,
  P,
  s2,
  N.upper.tri,
  random.perturb
)
updateS(
  iter,
  s,
  V,
  m,
  Zb,
  Zt,
  Time,
  MU.state,
  P,
  s2,
  N.upper.tri,
  random.perturb
)

Arguments

`iter`	iteration number
`s`	the most recent latent states
`V`	Network generation rules
`m`	The number of breaks
`Zb`	Z - b
`Zt`	Z stacked by time
`Time`	The length of time
`MU.state`	UVU for each state
`P`	Transition matrix
`s2`	error variance
`N.upper.tri`	The number of upper triangular elements
`random.perturb`	If `random.perturb` = TRUE and a single state observation is found, the latent state is randomly selected by equal weights.

Value

A list of vectors containing latent states and their probabilities

Update regime-specific variance

Description

Update regime-specific variance parameter

Usage

updates2m(ns, Zm, MU, c0, d0, Km)
updates2m(ns, Zm, MU, c0, d0, Km)

Arguments

`ns`	The number of hidden states
`Zm`	The regime-specific holder of Z - beta
`MU`	The mean array.
`c0`	Scalar shape parameter
`d0`	Scalar scale parameter
`Km`	Regime-specific dimensions

Value

A scalar for a regime-specific variance

Update time-constant latent node positions

Description

Update time-constant latent node positions

Usage

updateU(K, U, V, R, Zb, s2, eU, iVU)
updateU(K, U, V, R, Zb, s2, eU, iVU)

Arguments

`K`	The dimensionality of Z
`U`	The most recent draw of latent node positions
`V`	Layer-specific network generation rule
`R`	The dimensionality of latent space
`Zb`	Z - beta
`s2`	error variance
`eU`	The mean of U
`iVU`	The variance of U

Value

A matrix of time-constant latent node positions

Regime-specific latent node positions

Description

Update regime-specific latent node positions.

Usage

updateUm(ns, U, V, R, Zm, Km, ej, s2, eU, iVU, UL.Normal)
updateUm(ns, U, V, R, Zm, Km, ej, s2, eU, iVU, UL.Normal)

Arguments

`ns`	The number of latent states
`U`	THe latent node positions
`V`	Layer-specific network generation rule.
`R`	The dimensionality of latent space
`Zm`	Regim-specific Z - beta
`Km`	The dimension of regime-specific Z.
`ej`	Regime indicator.
`s2`	The variance of error.
`eU`	The regim-specific mean of U.
`iVU`	The regim-specific variance of U.
`UL.Normal`	Normalization method for U. "Normal" or "Orthonormal" are supported.

Value

A matrix of regime-specific latent node positions

Update layer specific network generation rules

Description

Update layer specific network generation rules

Usage

updateV(Zb, U, R, K, s2, eV, iVV, UTA)
updateV(Zb, U, R, K, s2, eV, iVV, UTA)

Arguments

`Zb`	Z - beta.
`U`	The latent node positions.
`R`	The dimension of latent space.
`K`	The dimension of Z.
`s2`	The variance of error.
`eV`	The mean of V.
`iVV`	The variance of V.
`UTA`	Indicator of upper triangular array

Value

A matrix of layer specific network generation rules

Update V from a change-point network process

Description

Update layer specific network generation rules from a change-point network process

Usage

updateVm(ns, U, V, Zm, Km, R, s2, eV, iVV, UTA)
updateVm(ns, U, V, Zm, Km, R, s2, eV, iVV, UTA)

Arguments

`ns`	The number of hidden regimes.
`U`	The latent node positions.
`V`	The layer-specific network generation rule.
`Zm`	The holder of Z - beta.
`Km`	The dimension of regime-specific Z.
`R`	The dimension of latent space.
`s2`	The variance of error.
`eV`	The mean of V
`iVV`	The variance of V
`UTA`	Indicator of upper triangular array

Value

A matrix of regime-specific layer specific network generation rules

Compare WAIC

Description

Compare Widely Applicable Information Criterion

Usage

WaicCompare(outlist)
WaicCompare(outlist)

Arguments

outlist

List of NetworkChange objects

Value

Results of WAIC computation

A matrix of log marginal likelihoods.

References

Andrew Gelman, Jessica Hwang, and Aki Vehtari. 2014. "Understanding predictive information criteria for Bayesian models." Statistics and Computing. 24(6):997-1016.

Jong Hee Park and Yunkyun Sohn. 2020. "Detecting Structural Change in Longitudinal Network Data." Bayesian Analysis. Vol.15, No.1, pp.133-157.

Package 'NetworkChange'

Help Index

Detect a break number using different metrics

Description

Usage

Arguments

References

Examples

Compute the Average Loss of Hidden State Changes from Expected Break Points

Description

Usage

Arguments

Value

References

Examples

Plot of latent node cluster

Description

Usage

Arguments

Value

References

Examples

Plot of network by hidden regime

Description

Usage

Arguments

Value

References

Examples

K-mean clustering of latent node positions

Description

Usage

Arguments

Value

Examples

Major Power Alliance Network (1816 - 2012)

Description

Format

Source

Build a synthetic block-structured temporal data with breaks

Description

Usage

Arguments

Value

Compare Log Marginal Likelihood

Description

Usage

Arguments

Value

References

See Also

Printing multiple ggplots in oone file

Description

Usage

Arguments

Value

Author(s)

Changepoint analysis of a degree-corrected multilinear tensor model

Description

Usage

Arguments

Value

References

See Also

Examples

Changepoint analysis of a degree-corrected multilinear tensor model with t-distributed error

Description

Usage

Arguments

Value

References

See Also

Examples

Degree-corrected multilinear tensor model

Description

Usage

Arguments

Value

References

See Also