# Estimate the graphical VAR model.

### Description

Estimates the graphical VAR (Wild et al., 2010) model through LASSO estimation coupled with extended Bayesian information criterion for choosing the optimal tuning parameters. The estimation procedure is outlined by Rothman, Levina and Zhu (2010) and is further described by Abegaz and Wit (2013). The procedure here is based on the work done in the R package SparseTSCGM (Abegaz and Wit, 2014).

### Usage

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### Arguments

`data` |
A matrix or data frame containing repeated measures (rows) on a set of variables (columns). Must not contain missing data. |

`nLambda` |
The number of both lambda parameters to test. Defaults to 50, which results in 2500 models to evaluate. |

`verbose` |
Logical, should a progress bar be printed to the console? |

`gamma` |
The EBIC hyper-parameter. Set to 0 to use regular BIC. |

`scale` |
Logical, should responses be standardized before estimation? |

`lambda_beta` |
An optional vector of lambda_beta values to test. Set |

`lambda_kappa` |
An optional vector of lambda_kappa values to test. Set |

`maxit.in` |
Maximum number of iterations in the inner loop (computing beta) |

`maxit.out` |
Maximum number of iterations in the outer loop |

`deleteMissings` |
Logical, should missing responses be deleted? |

`penalize.diagonal` |
Logical, should the diagonal of beta be penalized (i.e., penalize auto-regressions)? |

`lambda_min_kappa` |
Multiplier of maximal tuning parameter for kappa |

`lambda_min_beta` |
Multiplier of maximal tuning parameter for beta |

### Details

Let y_t denote the vector of centered responses of a subject on a set of items on time point t. The graphical VAR model, using only one lag, is defined as follows:

y_t = Beta y_y-1 + epsilon_t

In which epsilon_t is a vector of error and is independent between time points but not within time points. Within time points, the error is normally distributed with mean vector 0 and precision matrix (inverse covariance matrix) Kappa. The Beta matrix encodes the between time point interactions and the Kappa matrix encodes the within time point interactions. We aim to find a sparse solution for both Beta and Kappa, and do so by applying the LASSO algorithm as detailed by Rothman, Levina and Zhu (2010). The LASSO algorithm uses two tuning parameters, lambda_beta controlling the sparsity in Beta and lambda_kappa controlling the sparsity in Kappa. We estimate the model under a (by default) 50 by 50 grid of tuning parameters and choose the tuning parameters that optimize the extended Bayesian Information Criterion (EBIC; Chen and Chen,2008).

After estimation, the Beta and Kappa matrices can be standardized as described by Wild et al. (2010). The Kappa matrix can be standardized to partial contemporaneous correlations (PCC) as follows:

PCC(y_i,t, y_j,t) = - kappa_ij / ( sqrtkappa_ii kappa_jj)

Similarly, the beta matrix can be standardized to partial directed correlations (PDC):

PDC(y_i,t-1, y_j,t) = beta_ji / sqrtsigma_jj kappa_ii + beta_ji^2

In which sigma is the inverse of kappa. Note that this process transposes the beta matrix. This is done because in representing a directed network it is typical to let rows indicate the node of origin and columns the node of destination.

Set `lambda_beta = 0`

argument and `lambda_kappa = 0`

for unregularized estimation.

### Value

A `graphicalVAR`

object, which is a list containing:

`PCC` |
The partial contemporaneous correlation network |

`PDC` |
The partial directed correlation network |

`beta` |
The estimated beta matrix |

`kappa` |
The estimated kappa matrix |

`EBIC` |
The optimal EBIC |

`path` |
Results of all tested tuning parameters |

`labels` |
A vector containing the node labels |

### Author(s)

Sacha Epskamp <mail@sachaepskamp.com>

### References

Chen, J., & Chen, Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95(3), 759-771.

Fentaw Abegaz and Ernst Wit (2013). Sparse time series chain graphical models for reconstructing genetic networks. Biostatistics. 14, 3: 586-599.

Fentaw Abegaz and Ernst Wit (2014). SparseTSCGM: Sparse time series chain graphical models. R package version 2.1.1. http://CRAN.R-project.org/package=SparseTSCGM

Rothman, A.J., Levina, E., and Zhu, J. (2010). Sparse multivariate regression with covariance estimation. Journal of Computational and Graphical Statistics. 19: 947-962.

Wild, B., Eichler, M., Friederich, H. C., Hartmann, M., Zipfel, S., & Herzog, W. (2010). A graphical vector autoregressive modelling approach to the analysis of electronic diary data. BMC medical research methodology, 10(1), 28.

### Examples

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# Simulate model:
Mod <- randomGVARmodel(4,probKappaEdge = 0.8,probBetaEdge = 0.8)
# Simulate data:
Data <- graphicalVARsim(100,Mod$beta,Mod$kappa)
# Estimate model:
Res <- graphicalVAR(Data, gamma = 0, nLambda = 10)
# Plot results:
layout(t(1:2))
plot(Mod, "PCC", layout = "circle")
plot(Res, "PCC", layout = "circle")
plot(Mod, "PDC", layout = "circle")
plot(Res, "PDC", layout = "circle")
``` |