ebdbn {ebdbNet} R Documentation

## Empirical Bayes Dynamic Bayesian Network (EBDBN) Estimation

### Description

A function to infer the posterior mean and variance of network parameters using an empirical Bayes estimation procedure for a Dynamic Bayesian Network (DBN).

### Usage

ebdbn(y, K, input = "feedback", conv.1 = .15, conv.2 = .05, conv.3 = .01, verbose = TRUE,
max.iter = 100, max.subiter = 200)


### Arguments

 y A list of R (PxT) matrices of observed time course profiles (P genes, T time points) K Number of hidden states input "feedback" for feedback loop networks, or a list of R (MxT) matrices of input profiles conv.1 Value of convergence criterion 1 conv.2 Value of convergence criterion 2 conv.3 Value of convergence criterion 3 verbose Verbose output max.iter Maximum overall iterations (default value is 100) max.subiter Maximum iterations for hyperparameter updates (default value is 200)

### Details

An object of class ebdbNet.

This function infers the parameters of a network, based on the state space model

x_t = Ax_{t-1} + Bu_t + w_t

y_t = Cx_t + Du_t + z_t

where x_t represents the expression of K hidden states at time t, y_t represents the expression of P observed states (e.g., genes) at time t, u_t represents the values of M inputs at time t, w_t \sim MVN(0,I), and z_t \sim MVN(0,V^{-1}), with V = diag(v_1, \ldots, v_P). Note that the dimensions of the matrices A, B, C, and D are (KxK), (KxM), (PxK), and (PxM), respectively. When a network is estimated with feedback rather than inputs (input = "feedback"), the state space model is

x_t = Ax_{t-1} + By_{t-1} + w_t

y_t = Cx_t + Dy_{t-1} + z_t

The parameters of greatest interest are typically contained in the matrix D, which encodes the direct interactions among observed variables from one time to the next (in the case of feedback loops), or the direct interactions between inputs and observed variables at each time point (in the case of inputs).

The value of K is chosen prior to running the algorithm by using hankel. The hidden states are estimated using the classic Kalman filter. Posterior distributions of A, B, C, and D are estimated using an empirical Bayes procedure based on a hierarchical Bayesian structure defined over the parameter set. Namely, if a_{(j)}, b_{(j)}, c_{(j)}, d_{(j)}, denote vectors made up of the rows of matrices A, B, C, and D respectively, then

a_{(j)} \vert \alpha \sim N(0, diag(\alpha)^{-1})

b_{(j)} \vert \beta \sim N(0, diag(\beta)^{-1})

c_{(j)} \vert \gamma \sim N(0, diag(\gamma)^{-1})

d_{(j)} \vert \delta \sim N(0, diag(\delta)^{-1})

where \alpha = (\alpha_1, ..., \alpha_K), \beta = (\beta_1, ..., \beta_M), \gamma = (\gamma_1, ..., \gamma_K), and \delta = (\delta_1, ..., \delta_M). An EM-like algorithm is used to estimate the hyperparameters in an iterative procedure conditioned on current estimates of the hidden states.

conv.1, conv.2, and conv.3 correspond to convergence criteria \Delta_1, \Delta_2, and \Delta_3 in the reference below, respectively. After terminating the algorithm, the z-scores of the D matrix is calculated, which in turn determines the presence or absence of edges in the network.

See the reference below for additional details about the implementation of the algorithm.

### Value

 APost  Posterior mean of matrix A BPost  Posterior mean of matrix B CPost  Posterior mean of matrix C DPost  Posterior mean of matrix D CvarPost  Posterior variance of matrix C DvarPost  Posterior variance of matrix D xPost  Posterior mean of hidden states x alphaEst  Estimated value of \alpha betaEst  Estimated value of \beta gammaEst  Estimated value of \gamma deltaEst  Estimated value of \delta vEst  Estimated value of precisions v muEst  Estimated value of \mu sigmaEst  Estimated value of \Sigma alliterations  Total number of iterations run z  Z-statistics calculated from the posterior distribution of matrix D type  Either "input" or "feedback", as specified by the user

Andrea Rau

### References

Andrea Rau, Florence Jaffrezic, Jean-Louis Foulley, and R. W. Doerge (2010). An Empirical Bayesian Method for Estimating Biological Networks from Temporal Microarray Data. Statistical Applications in Genetics and Molecular Biology 9. Article 9.

### See Also

hankel, calcSensSpec, plot.ebdbNet

### Examples

library(ebdbNet)
tmp <- runif(1) ## Initialize random number generator
set.seed(125214) ## Save seed

## Simulate data
R <- 5
T <- 10
P <- 10
simData <- simulateVAR(R, T, P, v = rep(10, P), perc = 0.10)
Dtrue <- simData$Dtrue y <- simData$y

## Simulate 8 inputs
u <- vector("list", R)
M <- 8
for(r in 1:R) {
u[[r]] <- matrix(rnorm(M*T), nrow = M, ncol = T)
}

####################################################
## Run EB-DBN without hidden states
####################################################
## Choose alternative value of K using hankel if hidden states are to be estimated
## K <- hankel(y)\$dim

## Run algorithm
net <- ebdbn(y = y, K = 0, input = u, conv.1 = 0.15, conv.2 = 0.10, conv.3 = 0.10,
verbose = TRUE)

## Visualize results
## Note: no edges here, which is unsurprising as inputs were randomly simulated
## (in input networks, edges only go from inputs to genes)
## plot(net, sig.level = 0.95)



[Package ebdbNet version 1.2.6 Index]