poisson.MODpq.tnar {PNAR}

Generation of counts from a non-linear Threshold Poisson NAR(p) model with q covariates (T-PNAR(p))

Description

Generation of multivariate count time series from a non-linear Threshold Poisson network Autoregressive model of order p with q covariates (T-PNAR(p)).

Usage

poisson.MODpq.tnar(b, W, gama, a, p, d, Z = NULL, TT, N, copula = "gaussian",
corrtype = "equicorrelation", rho, dof = 1)

Arguments

b

The linear coefficients of the model, in the following order: (intercept, network parameters, autoregressive parameters, covariates). The dimension of the vector should be 2p + 1 + q, where q denotes the number of covariates.

W

The N \times N row-normalized non-negative adjacency matrix describing the network. The main diagonal entries of the matrix should be zeros, all the other entries should be non-negative and the maximum sum of elements over the rows should equal one. The function row-normalizes the matrix if a non-normalized adjacency matrix is provided.

gama

The scalar nuisance threshold parameter.

a

Vector of non-linear parameters. The dimension of the vector should be 2p + 1.

p

The number of lags in the model.

d

The lag parameter of non-linear variable (should be between 1 and p).

Z

An N \times q matrix of covariates (one for each column), where q is the number of covariates in the model. Note that they must be non-negative.

TT

The temporal sample size.

N

The number of nodes on the network.

copula

Which copula function to use? The "gaussian", "t", or "clayton".

rho

The value of the copula parameter (\rho). A scalar in [-1,1] for elliptical copulas (Gaussian, t), a value greater than or equal to -1 for Clayton copula.

corrtype

Used only for elliptical copulas. The type of correlation matrix employed for the copula; it will either be the "equicorrelation" or "toeplitz". The "equicorrelation" option generates a correlation matrix where all the off-diagonal entries equal \rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)-element is \rho^{|i-j|}.

dof

The degrees of freedom for Student's t copula.

Details

This function generates counts from a non-linear Threshold Poisson NAR(p) model, where q non time-varying covariates are allowed as well. The counts are simulated from Y_{t}=N_{t}(\lambda_{t}), where N_{t} is a sequence of N-dimensional IID Poisson count processes, with intensity 1, and whose structure of dependence is modelled through a copula construction C(\rho) on their associated exponential waiting times random variables. For details see Armillotta and Fokianos (2022a, Sec. 2.1-2.2).

The sequence \lambda_{i,t} is the expecation of Y_{i,t}, conditional to its past values and it is generated by means of the following T-PNAR(p) model. For each node of the network i=1,...,N over the time sample t=1,...,TT

\lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}\left[\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h}+(\alpha_{0}+\alpha_{1h}X_{i,t-h}+\alpha_{2h}Y_{i,t-h})I(X_{i,t-d}\leq\gamma)\right]+\sum_{l=1}^{q}\delta_{l}Z_{i,l}

where X_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node i connections, with the weights of the mean being W_{ij}, the single element of the network matrix W, and I() is the indicator function.

The parameter \beta_{0} is the intercept of the model, \beta_{1h} are the network coefficients, \beta_{2h} are the autoregressive parameters, the \alpha vector of non-linear parameters is divided as follows: \alpha_{0} is the intercept, \alpha_{1h} are the network coefficients, \alpha_{2h} are the autoregressive parameters; \gamma is the nuisance threshold parameter, and \delta_{l} are the coefficients assocciated to the covariates Z_{i,l}. The coefficient d is considered as an extra parameter defining the lag of the network effect in the non-linear part of the model and is left to be set by the user. For details on T-PNAR models see Armillotta and Fokianos (2022b, Sec. 2).

Value

A list including:

p2R

The Toeplitz correlation matrix, if employed in the copula or NULL else.

lambda

A TT \times N time series object matrix of simulated Poisson means for N time series over TT.

y

A TT \times N time series object matrix of simulated counts for N time series over TT.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

Armillotta, M. and K. Fokianos (2022a). Poisson network autoregression. https://arxiv.org/abs/2104.06296

Armillotta, M. and K. Fokianos (2022b). Testing linearity for network autoregressive models. https://arxiv.org/abs/2202.03852

See Also

poisson.MODpq, poisson.MODpq.log, poisson.MODpq.nonlin, poisson.MODpq.stnar

Examples

W <- adja( N = 20, K = 5, alpha= 0.5)
y <- poisson.MODpq.tnar( b = c(0.5, 0.3, 0.2), W = W, gama = 1,
a = c(0.2, 0.2, 0.2), p = 1, d = 1, Z = NULL, TT = 1000, N = 20,
copula = "gaussian", corrtype = "equicorrelation", rho = 0.5)$y