Package 'SparseLPM'

The Sparse Latent Position Model for Nonnegative Interaction Data

Description

Models the nonnegative entries of a rectangular adjacency matrix using a sparse latent position model, as illustrated in Rastelli, R. (2018) "The Sparse Latent Position Model for nonnegative weighted networks" <arXiv:1808.09262>.

Author(s)

Riccardo Rastelli

Mantainer: Riccardo Rastelli <[email protected]>

References

Rastelli, R. (2018) "The Sparse Latent Position Model for nonnegative weighted networks", https://arxiv.org/abs/1808.09262

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slpm_elbo

Description

Evaluates the evidence lower bound for a given configuration of variational parameters.

Usage

slpm_elbo(X, var_pars, hyper_pars, verbose = F)

Arguments

X	Rectangular adjacency matrix with non-negative entries.
var_pars	A list defining the variational parameters of the model. See Details for more specific indications.
hyper_pars	A list defining the hyperparameters of the model. The list should contain three vectors of length K denoted delta, a_gamma and b_gamma, where K is the number of latent dimensions.
verbose	TRUE or FALSE indicating whether a lengthy output should be printed out.

Details

The list var_pars must contain:

alpha_u_tilde

M*K matrix denoting the Gaussian means for senders.

alpha_v_tilde

N*K matrix denoting the Gaussian means for receivers.

beta_u_tilde

M*K matrix denoting the Gaussian variances for senders.

beta_v_tilde

N*K matrix denoting the Gaussian variances for receivers.

lambda_tilde

M*N*K array representing the soft clustering for the edges. This may be interpreted as the posterior probability that edge ij is determined by the k-th latent dimension.

delta_tilde

K dimensional vector containing the variational parameters for the mixing proportions. This may be interpreted as the importance of each latent dimension.

a_tilde

K dimensional vector containing the shapes of the variational Gamma distributions associated to the precisions.

b_tilde

K dimensional vector containing the rates of the variational Gamma distributions associated to the precisions.

Value

computing_time	Number of seconds required for the evaluation.
elbo	Value of the ELBO for the given variational parameters.

Examples

set.seed(12345)
M <- N <- 10
K <- 2
network <- slpm_gen(M = M, N = N, K = K)
var_pars <- slpm_init(X = network$adj, K = K)
hyper_pars <- list(delta = rep(1,K), a_gamma = rep(1,K), b_gamma = rep(1,K))
slpm_elbo(X = network$adj, var_pars = var_pars, hyper_pars = hyper_pars, verbose = FALSE)

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slpm_gen

Description

Generates the adjacency matrix adj of a SparseLPM by sampling both the data and model parameters from the posterior distribution.

Usage

slpm_gen(M, N, K, hyper_pars = NULL)

Arguments

M	Number of rows of adj.
N	Number of cols of adj.
K	Number of latent dimensions of the SparseLPM.
hyper_pars	A list defining the hyperparameters of the model. If left as NULL all the hyperparameters are initialised to 1. Otherwise, the list should contain three vectors of K positive values denoted delta, a_gamma and b_gamma, respectively.

Value

A list with components:

adj	Generated adjacency matrix.
U	Generated latent positions for senders.
V	Generated latent positions for receivers.
lambda	Latent variables attached to each of the edges, selecting which dimension determines the edge probability.
gamma	Vector of the Gaussian precisions associated to the latent dimensions.

Examples

set.seed(12345)
network <- slpm_gen(M = 10, N = 8, K = 2)

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slpm_gof

Description

Evaluates the expected adjacency matrix for a fitted SparseLPM.

Usage

slpm_gof(var_pars)

Arguments

var_pars

A list defining the variational parameters of the model. See Details for more specific indications.

Details

The list var_pars must contain:

alpha_u_tilde

M*K matrix denoting the Gaussian means for senders.

alpha_v_tilde

N*K matrix denoting the Gaussian means for receivers.

beta_u_tilde

M*K matrix denoting the Gaussian variances for senders.

beta_v_tilde

N*K matrix denoting the Gaussian variances for receivers.

lambda_tilde

M*N*K array representing the soft clustering for the edges. This may be interpreted as the posterior probability that edge ij is determined by the k-th latent dimension.

delta_tilde

K dimensional vector containing the variational parameters for the mixing proportions. This may be interpreted as the importance of each of the latent dimensions.

a_tilde

K dimensional vector containing the shapes of the variational Gamma distributions associated to the precisions.

b_tilde

K dimensional vector containing the rates of the variational Gamma distributions associated to the precisions.

Note that this function only uses the alphas and the lambdas. Also, to avoid numerical instability, the lambdas are automatically pre-transformed into a hard partitioning using a Maximum A Posterior method.

Value

An adjacency matrix with non-negative entries.

Examples

set.seed(12345)
M <- N <- 10
K <- 2
fitted_var_pars <- list()
fitted_var_pars$alpha_u_tilde = matrix(rnorm(M*K),M,K)
fitted_var_pars$alpha_v_tilde = matrix(rnorm(N*K),N,K)
fitted_var_pars$lambda_tilde = array(NA,c(M,N,K))
fitted_var_pars$lambda_tilde[,,1] = matrix(runif(M*N),M,N)
fitted_var_pars$lambda_tilde[,,2] = 1-fitted_var_pars$lambda_tilde[,,1]
expected_adj <- slpm_gof(fitted_var_pars)

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slpm_init

Description

Initialises the variational parameters of a SparseLPM.

Usage

slpm_init(X, K, method = "random", threshold = 0.1, stdev = NULL)

Arguments

X	Rectangular adjacency matrix with non-negative entries.
K	Number of latent dimensions of the SparseLPM.
method	The variational parameters are initialised at random. However, if method="distance", a distance-based method is used as described in Rastelli ... (2018).
threshold	A small number added to each of the entries of X to avoid numerical instability.
stdev	Standard deviation of the Gaussian used to generate the random latent positions.

Value

Returns a list of variational parameters that can be used as input for slpm_nga or slpm_elbo.

Examples

set.seed(12345)
M <- N <- 10
K <- 2
network <- slpm_gen(M = M, N = N, K = K)
var_pars_init <- slpm_init(X = network$adj, K = K)

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slpm_nga

Description

Runs a Natural Gradient Ascent algorithm to maximise the variational objective for a Sparse LPM.

Usage

slpm_nga(X, K, var_pars_init, hyper_pars = NULL, tol = 0.01, n_iter_max = 100000,
        natural_gradient = T, learning_rate_factor_up = 2, learning_rate_factor_down = 2,
        verbose = F)

Arguments

X	Rectangular adjacency matrix with non-negative entries.
K	The number of latent dimension of the model.
var_pars_init	List of variational parameters to be used as starting point for the optimisation. See Details for more specific indications.
hyper_pars	List defining the hyperparameters of the model. The list should contain three vectors of K positive values denoted delta, a_gamma and b_gamma, respectively, where K is the number of latent dimensions. If left as null, all delta parameters are set to 0.001, whereas the rest is set to 1.
tol	Positive number setting the stop condition: the algorithm stops if one entire iteration yields an increase in the objective function smaller than this value.
n_iter_max	Maximum number of iterations the algorithm should be run for.
natural_gradient	TRUE or FALSE indicating whether the natural gradient instead of the standard gradient should be used.
learning_rate_factor_up	Before any natural gradient ascent update, the current step size is multiplied by this number to ensure that the algorithms tries new solutions which are relatively far from the current one.
learning_rate_factor_down	During any natural gradient ascent update, if a certain step size leads to a decrease in the objective function, then the step is divided by this number repeatedly until an increase is ensured.
verbose	TRUE or FALSE indicating whether a lengthy output should be printed out.

Details

var_pars and var_pars_init are lists with components:

alpha_u_tilde

M*K matrix representing the Gaussian means for the latent positions of senders.

alpha_v_tilde

N*K matrix representing the Gaussian means for the latent positions of receivers.

beta_u_tilde

M*K matrix representing the Gaussian variances for the latent positions of senders.

beta_v_tilde

N*K matrix representing the Gaussian variances for the latent positions of receivers.

lambda_tilde

M*N*K array with entries corresponding to the posterior probabilities of assigning each edge to each latent dimension.

delta_tilde

Vector of K positive values representing the Dirichlet parameters generating the mixing proportions.

a_tilde

Vector of K positive values corresponding to the shapes of the variational Gamma distribution on the precisions.

b_tilde

Vector of K positive values corresponding to the rates of the variational Gamma distribution on the precisions.

Value

A list with components:

computing_time	Number of seconds required for the optimisation process.
var_pars	List containing the optimal values for the variational parameters.
learning_rates_u	Current step-size for the update of the variational parameters of each Gaussian distribution on the latent positions of senders.
learning_rates_v	Current step-size for the update of the variational parameters of each Gaussian distribution on the latent positions of receivers.
elbo_values	Values of the variational objective at the end of each of the iterations.
elbo_init	Value of the variational objective for the initial configuration.
elbo_final	Value of the variational objective for the optimal solution found.

References

Rastelli, R. (2018) "The Sparse Latent Position Model for nonnegative weighted networks", https://arxiv.org/abs/1808.09262

Examples

set.seed(12345)
network <- slpm_gen(M = 15, N = 10, K = 2)
K <- 6
var_pars_init <- slpm_init(X = network$adj, K = K)
res <- slpm_nga(X = network$adj, K = K, var_pars_init = var_pars_init)

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