Type: Package
Title: Longitudinal Graphical Lasso
Version: 0.1.0
Description: For high-dimensional correlated observations, this package carries out the L_1 penalized maximum likelihood estimation of the precision matrix (network) and the correlation parameters. The correlated data can be longitudinal data (may be irregularly spaced) with dampening correlation or clustered data with uniform correlation. For the details of the algorithms, please see the paper Jie Zhou et al. Identifying Microbial Interaction Networks Based on Irregularly Spaced Longitudinal 16S rRNA sequence data <doi:10.1101/2021.11.26.470159>.
License: GPL-3
Encoding: UTF-8
LazyData: true
RoxygenNote: 7.1.2
URL: https://github.com/jiezhou-2/lglasso
Suggests: knitr, rmarkdown
Imports: stats, glasso
Depends: R (≥ 2.10)
NeedsCompilation: no
Packaged: 2022-01-13 16:54:06 UTC; jie
Author: Jie Zhou [aut, cre, cph], Jiang Gui [aut], Weston Viles [aut], Anne Hoen [aut]
Maintainer: Jie Zhou <chowstat@gmail.com>
Repository: CRAN
Date/Publication: 2022-01-15 08:52:46 UTC

Estimates of correlation parameters and precision matrix

Description

Estimates of correlation parameters and precision matrix

Usage

heterlongraph(data, rho, type, tole, lower, upper)

Arguments

data

Data matrix in which the first column is subject id, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

rho

Tuning parameter used in graphical lasso

type

Type of correlation function, which can take either "abs" or "sqr".

tole

Error tolerance for determination of convergence of EM algorithm

lower

Lower bound for prediction of correlation parameter tau

upper

Upper bound for prediction of correlation parameter tau

Value

S list with three components which are the final estimate of alpha, tau and precision matrix omega

Author(s)

Jie Zhou

Estiamte of precision matrix and autocorrelaton parameter for homogeneous model

Description

Estiamte of precision matrix and autocorrelaton parameter for homogeneous model

Usage

homolongraph(data, rho, type, tole, lower, upper)

Arguments

data

Data matrix in which the first column is subject id, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

rho

Tuning parameter for graphical lasso

type

Type of correlation function, which can take either "abs" or "qua".

tole

Error tolerance for determination of convergence of EM algorithm

lower

Lower bound for prediction of correlation parameter tau

upper

Upper bound for prediction of correlation parameter tau

Value

A list for estimates of precision matrix and correlation parameter for given tuning parameter

Author(s)

Jie Zhou

Quasi covariance matrix for subject i

Description

Quasi covariance matrix for subject i

Usage

iss(idata, itau, type)

Arguments

idata

Data matrix for the subject i in which the first column is subject (cluster) id, the second column stands for the time points () of observation. Columns 2 to (p+2) is the observations for p variables respectively.

itau

Correlation parameter

type

Type of correlation function, which typically take either 0, 1 or 2.

Value

Empirical quasi covariance matrix

Author(s)

Jie Zhou

Graphical Lasso for Longitudinal Data

Description

This function implements the L_1 penalized maximum likelihood estimation for precision matrix (network) based on correlated data, e.g., irregularly spaced longitudinal data. It can be regarded as an extension of the package glasso (Friedman,Hastie and Tibshirani, 2008) which aims to find the sparse estimate of the network from independent continuous data.

Usage

lglasso(
  data,
  rho,
  heter = TRUE,
  type = 1,
  tole = 0.01,
  lower = 0.01,
  upper = 10
)

Arguments

data

Data matrix in which the first column is subject id, the second column is time points of observations for temporal data or site id for spatial data. Columns 3 to (p+2) is the observations for p variables.

rho

Tuning parameter used in L_1 penalty

heter

Binary variable TRUE or FALSE, indicating heterogeneous model or homogeneous model is fitted. In heterogeneous model, subjects are allowed to have his/her own temporal correlation parameter tau_i; while in homogeneous model, all the subjects are assumed to share the same temporal correlation parameter,i.e., tau_1=tau_2=...tau_m.

type

A positive number which specify the correlation function. The general form of correlation function is given by exp(tau|t_i-t_j|^type). in which type=0 can be used for spatial correlation while type>0 are used for temporal correlation. For latter, the default value is set to be type=1.

tole

Threshold for convergence. Default value is 1e-2. Iterations stop when maximum absolute difference between consecutive estimates of parameter change is less than tole.

lower

Lower bound for predicts of correlation parameter tau. Default value is 1e-2. The estimate of tau(alpha) will be searched in the interval [lower,upper], where parameter upper is explained in the following.

upper

Upper bound for predicts of correlation parameter tau.

Value

If heter=TRUE, then a list with three components is returned which are respectively the estimate of parameter alpha in exponent distribution, correlation parameter tau and precision matrix omega. If heter=FALSE, then a list with two components is returned which are respectively the estimate of correlation parameter tau and precision matrix omega.

Author(s)

Jie Zhou

References

Jie Zhou, Jiang Gui, Weston D.Viles, Anne G.Hoen Identifying Microbial Interaction Networks Based on Irregularly Spaced Longitudinal 16S rRNA sequence data. bioRxiv 2021.11.26.470159; doi: https://doi.org/10.1101/2021.11.26.470159

Friedman J, Tibshirani TH and R. Glasso: Graphical Lasso: Estimation of Gaussian Graphical Models.; 2019. Accessed November 28, 2021. https://CRAN.R-project.org/package=glasso

Friedman J, Hastie T, Tibshirani TH, Sparse inverse covariance estimation with the graphical lasso, Biostatistics, Volume 9, Issue 3, July 2008, Pages 432–441, https://doi.org/10.1093/biostatistics/kxm045

Examples

sample_data[1:5,1:5]
dim(sample_data)
## Heterogeneous model with dampening correlation rate using the first three clusters
a=lglasso(data = sample_data[1:11,], rho = 0.7,heter=TRUE, type=1)
### Estimates of correlation parameters
a$tau
### Sub-network for the first five variables
a$omega[1:5,1:5]
### Total number of the edges in the estimated network
(length(which(a$omega!=0))-ncol(a$omega))/2
## Homogeneous model with dampening correlation rate using the first three clusters
b=lglasso(data = sample_data[1:11,], rho = 0.7,heter=FALSE,type=1)
### Estimates of correlation parameters
b$tau
### Sub-network for the first five  variables
b$omega[1:5,1:5]
### Total number of the edges in the estimated network
(length(which(b$omega!=0))-ncol(b$omega))/2
## Heterogeneous model with uniform correlation rate using the first three clusters
c=lglasso(data = sample_data[1:11,], rho = 0.7,heter=TRUE,type=0)
### Estimates of correlation parameters
c$tau
### Sub-network for the first five  variables
c$omega[1:5,1:5]
### Total number of the edges in the estimated network
(length(which(c$omega!=0))-ncol(c$omega))/2
## Homogeneous model with uniform correlation rate using the first three clusters
d=lglasso(data = sample_data[1:11,], rho = 0.7,heter=FALSE,type=0)
### Estimates of correlation parameters
d$tau
### Sub-network for the first five  variables
d$omega[1:5,1:5]
### Total number of the edges in the estimated network
(length(which(d$omega!=0))-ncol(d$omega))/2

Value of likelihood function at given parameter

Description

Value of likelihood function at given parameter

Usage

ll_homo(data, omega, tau, type)

Arguments

data

Data matrix in which the first column is subject id, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

omega

Precision matrix

tau

Correlation parameter

type

Type of correlation function, which can take either "abs" or "qua".

Value

Value of likelihood function at given omega and tau

Author(s)

Jie Zhou

full log likelihood used in EBIC computation

Description

full log likelihood used in EBIC computation

Usage

lli_homo(idata, omega, tau, type)

Arguments

idata

Data matrix for the subject i in which the first column is id for subject, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

omega

Precision matrix

tau

Correlation parameter

type

Type of correlation function, which can take either "abs" or "qua".

Value

Value of likelihood function for subject i at given omega and tau

Author(s)

Jie Zhou

Complete likelihood function used in EM algorithm of heterogeneous marginal graphical lasso model

Description

Complete likelihood function used in EM algorithm of heterogeneous marginal graphical lasso model

Usage

logdensity(idata, omega, tau, alpha, type)

Arguments

idata

Data matrix for the subject i in which the first column is id for subject, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

omega

Precision matrix

tau

Correlation parameter

alpha

Parameter in exponential distribution

type

Type of correlation function, which can take either "abs" or "qua".

Value

Value of complete likelihood function at given value of omega, tau and alpha

Author(s)

Jie Zhou

Maximum Likelihood Estimate of Precision Matrix and Correlation Parameters for Given Network

Description

Maximum Likelihood Estimate of Precision Matrix and Correlation Parameters for Given Network

Usage

mle(
  data,
  network,
  heter = TRUE,
  type = 1,
  tole = 0.01,
  lower = 0.01,
  upper = 10
)

Arguments

data

Data matrix in which the first column is subject id, the second column is time points of observations for temporal data or site id for spatial data. Columns 3 to (p+2) is the observations for p variables.

network

The network selected by function lglasso

heter

Binary variable TRUE or FALSE, indicating heterogeneous model or homogeneous model is fitted. In heterogeneous model, subjects are allowed to have his/her own temporal correlation parameter tau_i; while in homogeneous model, all the subjects are assumed to share the same temporal correlation parameter,i.e., tau_1=tau_2=...tau_m.

type

A positive number which specify the correlation function. The general form of correlation function is given by exp(tau|t_i-t_j|^type). in which type=0 can be used for spatial correlation while type>0 are used for temporal correlation. For latter, the default value is set to be type=1.

tole

Threshold for convergence. Default value is 1e-2. Iterations stop when maximum absolute difference between consecutive estimates of parameter change is less than tole.

lower

Lower bound for predicts of correlation parameter tau. Default value is 1e-2. The estimate of tau(alpha) will be searched in the interval [lower,upper], where parameter upper is explained in the following.

upper

Upper bound for predicts of correlation parameter tau.

Value

A list which include the maximum likelihood estimate of precision matrix, correlation parameter tau. If heter=TRUE, the output also include the estimate of alpha where tau~exp(alpha)

Author(s)

Jie Zhou

Maximum likelihood estimate of correlation parameter for given structure of precision matrix

Description

Maximum likelihood estimate of correlation parameter for given structure of precision matrix

Usage

mle_alpha(data, alpha0, omega, type, tole, lower, upper)

Arguments

data

Data matrix in which the first column is subject id, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

alpha0

Initial value for the parameter in exponential distribution

omega

Fixed value for precision matrix

type

Type of correlation function, which can take either "abs" or "qua".

tole

Error tolerance for determination of convergence of EM algorithm

lower

Lower bound for prediction of correlation parameter tau

upper

Upper bound for prediction of correlation parameter tau

Author(s)

Jie Zhou

Title

Description

Title

Usage

mle_net(data, priori)

Arguments

data

A Longitudinal data set

priori

Given structure of precision matrix

Value

The maximum likelihood estimation

Author(s)

Jie Zhou

Estiamte of precision matrix and autocorrelaton parameter for homogeneous model

Description

Estiamte of precision matrix and autocorrelaton parameter for homogeneous model

Usage

mle_tau(data, omega, type, lower, upper)

Arguments

data

Data matrix in which the first column is subject id, the second column is the time points of observation. Columns 2 to (p+2) is the observations for p variables.

omega

The maximum likelihood estiamte of precision matrix

type

Type of correlation function, which can take either "abs" or "qua".

lower

Lower bound for prediction of correlation parameter tau

upper

Upper bound for prediction of correlation parameter tau

Value

A list for estimates of precision matrix and correlation parameter for given tuning parameter

Author(s)

Jie Zhou

Construct the temporal component fo correlation function

Description

Construct the temporal component fo correlation function

Usage

phifunction(t, tau, type = 1)

Arguments

t

Time points of observations

tau

correlation parameter

type

The type of correlation function, which typically take either 0,1 or 2.

Value

A square matrix with dimension equal to the length of vector t

Author(s)

Jie Zhou

Sample Data

Description

The sample data are subset of a larger longitudinal data set from an ongoing large-scale prospective project. There are 13 cluster are involved in the sample data.

Usage

sample_data

Format

A 100-by-22 matrix

Column 1

Cluster id

;

Column 2

Time points of observations

;

Columns 3-22

Observations for 20 microbes

.