poisson.MODpq.nonlin: Generation of multivariate count time series from a non-linear Intercept Drift Poisson NAR(p) model with q covariates (ID-PNAR(p))

Description

Generation of counts from a non-linear Intercept Drift Poisson Network Autoregressive model of order $p$ with $q$ covariates (ID-PNAR($p$)).

Usage

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

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 $TT$.

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: A scalar non-linear intercept drift parameter.
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.

Author

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

Details

This function generates counts from a non-linear Intercept Drift 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 (2024, 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 ID-PNAR($p$) model. For each node of the network $i=1,...,N$ over the time sample $t=1,...,TT$ $$ \lambda_{i,t}=\frac{\beta_{0}}{(1+X_{i,t-d})^{\gamma}}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l} $$ 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 $W$.

The parameter $\beta_{0}$ is the intercept of the model, $\beta_{1h}$ are the network coefficients, $\beta_{2h}$ are the autoregressive parameters, $\gamma$ is the non-linear coefficient associated with the intercept drift, and $\delta_{l}$ are the coefficients assocciated with 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 ID-PNAR models see Armillotta and Fokianos (2023, Sec. 2).

References

Armillotta, M. and K. Fokianos (2023). Nonlinear network autoregression. Annals of Statistics, 51(6): 2526--2552.

Armillotta, M. and K. Fokianos (2024). Count network autoregression. Journal of Time Series Analysis, 45(4): 584--612.

Armillotta, M., Tsagris, M. and Fokianos, K. (2024). Inference for Network Count Time Series with the R Package PNAR. The R Journal, 15/4: 255--269.

Examples

Run this code

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

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