Graphical Models with Latent Variables and Correlated Replicates

Description

Estimate graphical models with latent variables and correlated replicates using the method in Jin et al. (2020).

Usage

corlatent(data, accuracy, n, R, p, lambda1, lambda2, lambda3, distribution = "Gaussian",
rule = "AND")

Arguments

Details

The corlatent method has two assumptions. Assumption 1 states that the R replicates are assumed to follow a one-lag vector autoregressive model, conditioned on the latent variables. Assumption 2 states that the latent variables are piecewise constant across replicates. Based on these two assumptions, the method solve the following problem for 1 \le j \le p.

\min_{\theta_{j,-j}, \alpha_j, \Delta_j} \{ -\frac{1}{nR}l(\theta_{j,-j}, \alpha_j, \Delta_j) + \lambda\|\theta_{j,-j}\|_1 + \beta\|\alpha_j\|_1 + \gamma\|(I_n \otimes C)\Delta_j\|_1 \},

where l(\theta_{j,-j}, \alpha_j, \Delta_j) is the log likelihood function, \theta_{j,-j} encodes the conditional dependence relationships between jth observed variable and the other observed variables, \alpha_j models the correlation among replicates, \Delta_j encodes the latent effect, \lambda, \beta, \gamma are the tuning parameters, I_n is an n-dimensional identity matrix and C is the discrete first derivative matrix where the ith and (i+1)th column of every ith row are -1 and 1, respectively. This method aims at modeling exponential family graphical models with correlated replicates and latent variables.

Value

References

Jin, Y., Ning, Y., and Tan, K. M. (2020), ‘Exponential Family Graphical Models with Correlated Replicates and Unmeasured Confounders’, preprint available.

Examples

# Gaussian distribution with "AND" rule
n <- 20
R <- 10
p <- 5
l <- 2
s <- 2
seed <- 1

data <- generate_Gaussian(n, R, p, l, s, sparsityA = 0.95, sparsityobserved = 0.9,
sparsitylatent = 0.2, lwb = 0.3, upb = 0.3, seed)$X

result <- corlatent(data, accuracy = 1e-6, n, R, p,lambda1 = 0.1, lambda2 = 0.1,
lambda3 = 1e+5,distribution = "Gaussian", rule = "AND")

Generate a Gaussian distributed data set

Description

This function will generate a Gaussian distributed data set with latent variables and correlated replicates.

Usage

generate_Gaussian(n, R, p, l, s, sparsityA, sparsityobserved, sparsitylatent, lwb, 
upb, seed)

Arguments

Details

This function aims to generate a Gaussian distributed data set with latent variables and correlated replicates. For each observation, the latent variables are piecewise constant across replicates, and conditioned on the latent variables, the replicates follow a one-lag vector autoregressive model, where the transition matrix A is sparse with non-zero elements set equal to 0.3. The matrix \Sigma is the covariance matrix for the observed variables X and the latent variables U, and we partition \Sigma into matrices that quantify the relationships among the observed variables (\Sigma_{XX}), between the observed variables and latent variables (\Sigma_{XU} or \Sigma_{UX}), and of the latent variables (\Sigma_{UU}). In general, the data is generated with:

X_{i1} | U_{i1} \sim N_p(\Sigma_{XU}\Sigma^{-1}_{UU} U_{i1}, \Sigma_{XX} - \Sigma_{XU}\Sigma^{-1}_{UU}\Sigma_{UX}),

X_{it} | X_{i(t-1)},U_{it} \sim N_p(AX_{i(t-1)} + \Sigma_{XU}\Sigma^{-1}_{UU} U_{it}, \Sigma_{XX} - \Sigma_{XU}\Sigma^{-1}_{UU}\Sigma_{UX}),

where 1 \le i \le n and 1 \le t \le R.

Value

References

Jin, Y., Ning, Y., and Tan, K. M. (2020), ‘Exponential Family Graphical Models with Correlated Replicates and Unmeasured Confounders’, preprint available.

Examples

data <- generate_Gaussian(n = 50, R = 20, p = 30, l = 2, s = 2, sparsityA = 0.95,
sparsityobserved = 0.9, sparsitylatent = 0.2, lwb = 0.3, upb = 0.3, seed = 1)

Estimate Gaussian Graphical Models with Latent Variables

Description

Estimate Gaussian graphical models with latent variables using the method in Chandrasekaran et al. (2012).

Usage

lvglasso(data, n, p, lambda1, lambda2, rule = "AND")

Arguments

Details

The lvglasso method assumes that all the variables, both observed and latent, are jointly Gaussian, and specifies the conditional distribution of observed variables on the latent variables by a graphical model. Under the high-dimentional setting, this method provides consistent estimators for the conditional graphical model of observed variables conditioned on latent variables.

Value

References

Chandrasekaran, V., Parrilo, P. A. & Willsky, A. S. (2012), ‘Latent variable graphical model selection via convex optimization’, Ann. Statist. 40(4), 1935–1967.

Examples

#Gaussian distribution with "AND" rule
n <- 50
R <- 20
p <- 30
l <- 2
s <- 2
data <- generate_Gaussian(n, R, p, l, s, sparsityA = 0.95, sparsityobserved = 0.9,
sparsitylatent = 0.2, lwb = 0.3, upb = 0.3, seed = 1)$X

result <- lvglasso(data, n, p, lambda1 = 0.222, lambda2 = 0.1*0.222, rule = "AND")

Estimate Graphical Models with Latent Variables and Replicates

Description

Estimate graphical models with latent variables and replicates using the method in Tan et al. (2016).

Usage

semilatent(data, n, R, p, lambda, distribution = "Gaussian", rule = "AND")

Arguments

Details

The semilatent method has two assumptions. The first one states that the latent variables are constant across replicates. Assumption 2 states that given the latent variables, the replicates are mutually independent. With these two assumptions, the method seeks to solve the following problem for 1 \le j \le p.

\min_{\beta_{j,O / j}} \{l_j (\beta_{j,O / j}) + \lambda\|\beta_{j,O / j}\|_1 \},

where l_j (\beta_{j,O / j}) is a nuisance-free loss function, \beta_{j,O / j} is a parameter that represents the conditional dependence relationships between jth observed variable and the other observed variables, and \lambda is a tuning parameter. This method aims at modeling semiparametric exponential family graphical model with latent variables and replicates.

Value

References

Tan, K. M., Ning, Y., Witten, D. M. & Liu, H. (2016), ‘Replicates in high dimensions, with applications to latent variable graphical models’, Biometrika 103(4), 761–777.

Examples

#semilatent Gaussian with "AND" rule
n <- 50
R <- 20
p <- 30
seed <- 1
l <- 2
s <- 2
data <- generate_Gaussian(n, R, p, l, s, sparsityA = 0.95, sparsityobserved = 0.9,
sparsitylatent = 0.2, lwb = 0.3, upb = 0.3, seed)$X

result <- semilatent(data, n, R, p, lambda = 0.1,distribution = "Gaussian", 
rule = "AND")