Description Usage Arguments Details Value References Examples

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

1 2 | ```
corlatent(data, accuracy, n, R, p, lambda1, lambda2, lambda3, distribution = "Gaussian",
rule = "AND")
``` |

`data` |
data set. Can be a matrix, list, array, or data frame. If the data set is a matrix, it should have |

`accuracy` |
the threshhold where algorithm stops. The algorithm stops when the difference between estimaters of the |

`n` |
the number of observations. |

`R` |
the number of replicates for each observation. |

`p` |
the number of observed variables. |

`lambda1` |
tuning parameter that encourages estimated graph to be sparse. |

`lambda2` |
tuning parameter that models the effects of correlated replicates. Usually set to be equal to lambda1. |

`lambda3` |
tuning parameter that encourages the latent effect to be piecewise constants. |

`distribution` |
For a data set with Gaussian distribution, use "Gaussian"; For a data set with Ising distribution, use "Ising". Default is "Gaussian". |

`rule` |
rules to combine matrices that encode the conditional dependence relationships between sets of two observed variables. Options are "AND" and "OR". Default is "AND". |

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 ≤ j ≤ p*.

*
\min_{θ_{j,-j}, α_j, Δ_j} \{ -\frac{1}{nR}l(θ_{j,-j}, α_j, Δ_j) + λ\|θ_{j,-j}\|_1 + β\|α_j\|_1 + γ\|(I_n \otimes C)Δ_j\|_1 \},
*

where *l(θ_{j,-j}, α_j, Δ_j)* is the log likelihood function, *θ_{j,-j}* encodes the conditional dependence relationships between *j*th observed variable and the other observed variables, *α_j* models the correlation among replicates, *Δ_j* encodes the latent effect, *λ*, *β*, *γ* are the tuning parameters, *I_n* is an n-dimensional identity matrix and *C* is the discrete first derivative matrix where the *i*th 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.

`omega` |
a matrix that encodes the conditional dependence relationships between sets of two observed variables |

`theta` |
the adjacency matrix with 0 and 1 encoding conditional independence and dependence between sets of two observed variables, respectively |

`penalties` |
the penalty values |

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

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
# 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")
``` |

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