ebdbn {ebdbNet}

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

Author(s)

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)