Package 'PNAR'

Poisson Network Autoregressive Models

Description

Quasi likelihood-based methods for estimating linear and log-linear Poisson Network Autoregression models with p lags and covariates. Tools for testing the linearity versus several non-linear alternatives. Tools for simulation of multivariate count distributions, from linear and non-linear PNAR models, by using a specific copula construction. References include: 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. Armillotta, M., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. https://arxiv.org/abs/2211.02582.

Details

Package:	PNAR
Type:	Package
Version:	1.6
Date:	2023-10-09
License:	GPL(>=2)

Note

Disclaimer: Dr Mirko Armillotta and Konstantinos Fokianos wrote the initial functions. Dr Tsagris modified them, created the package and he is the maintainer.

We would to like to acknowledge Manos Papadakis for his help with the "htest" class object and S3 methods (print() and summary() functions).

Author(s)

Michail Tsagris, Mirko Armillotta 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

---

Generation of a network from the Stochastic Block Model

Description

This function generates a network from the Stochastic Block Model with $K$ blocks.

Usage

adja(N, K, alpha, directed = FALSE)

Arguments

N	The number of nodes on the network.
K	The number of blocks. Each block has dimension $N/K$. $K$ should be chosen such that $N$ is divisible by $K$.
alpha	The network density. A value in $[0,1]$ defining the frequency of connections in the network.
directed	Logical scalar, whether to generate a directed network or not. If TRUE a directed network is generated.

Details

For each pair of nodes it performs a Bernoulli trial with values 1 "draw an edge", 0 "otherwise". The probabilities of these trials are bigger if the two nodes are in the same block, lower otherwise, and they are specified based on the number of nodes on the network $N$ and network density $alpha$: Probability to draw an edge for a pair of nodes in the same block: $\alpha*N^{-0.3}$. Probability to draw an edge for a pair of nodes in different blocks: $\alpha*N^{-1}$.

Value

A row-normalized non-negative matrix describing the network. The main diagonal entries of the matrix are zeros, all the other entries are non-negative and the sum of elements over the rows equals one.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

Faust, K. and S. Wasserman (1992). Blockmodels: Interpretation and evaluation. Social Networks, 14, 5-61.

See Also

adja_gnp

Examples

W <- adja(N = 20, K = 5, alpha = 0.1)

---

Generation of a network from the Erdos-Renyi model

Description

This function generates a network from the Erdos-Renyi model.

Usage

adja_gnp(N, alpha, directed = FALSE)

Arguments

N	The number of nodes on the network.
alpha	The network density. A value in $[0,1]$ defining the frequency of connections in the network.
directed	Logical scalar, whether to generate a directed network. If TRUE a directed network is generated.

Details

For each pair of nodes it performs a Bernoulli trial with values 1 "draw an edge", 0 "otherwise". Each trial has the same probability of having an edge; this is equal to $\alpha*N^{-0.3}$, specified based on the number of nodes on the network $N$ and the network density alpha.

Value

A row-normalized non-negative matrix describing the network. The main diagonal entries of the matrix are zeros, all the other entries are non-negative and the maximum sum of elements over the rows equals one.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

Erdos, P. and A. Renyi (1959). On random graphs. Publicationes Mathematicae, 6, 290-297.

See Also

adja

Examples

W <- adja_gnp(N = 20, alpha= 0.1)

---

Chicago crime dataset

Description

Monthly number of burglaries on the south side of Chicago (552 blocks) during 2010-2015 (72 temporal observations).

Usage

crime

Format

A time series object ("ts" class) with multivariate time series, a matrix with 72 rows and 552 columns.

Source

Clark and Dixon (2021), available at https://github.com/nick3703/Chicago-Data.

References

Clark, N. J. and P. M. Dixon (2021). A class of spatially correlated self-exciting statistical models. Spatial Statistics, 43, 1-18.

See Also

crime_W, lin_estimnarpq, log_lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq( crime, crime_W, p = 1)
mod2 <- log_lin_estimnarpq( crime, crime_W, p = 1)

---

Network matrix for Chicago crime dataset

Description

Non-negative row-normized adjacency matrix describing the network structure between Chicago census blocks.

Usage

crime_W

Format

A matrix with 552 rows and 552 columns.

Source

Clark and Dixon (2021), available at https://github.com/nick3703/Chicago-Data.

References

Clark, N. J. and P. M. Dixon (2021). A class of spatially correlated self-exciting statistical models. Spatial Statistics, 43, 1-18.

See Also

crime, lin_estimnarpq, log_lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 1)
mod2 <- log_lin_estimnarpq(crime, crime_W, p = 1)

---

Count the number of events within a specified time

Description

This function counts the number of events within a specified time.

Usage

getN(x, tt = 1)

Arguments

x	A matrix of (positive) inter-event times.
tt	A positive time.

Value

The number of events within time tt (possibly 0), for each column of x.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

See Also

rcopula, poisson.MODpq, poisson.MODpq.log

Examples

x <- rcopula(n = 100, N = 50, rho = 0.3)
getN(x)

---

Optimization of the score test statistic for the ST-PNAR(p) model

Description

Global optimization of the linearity test statistic for the Smooth Transition Poisson Network Autoregressive model of order $p$ with $q$ covariates (ST-PNAR($p$)) with respect to the nuisance scale parameter $\gamma$.

Usage

global_optimise_LM_stnarpq(gama_L = NULL, gama_U = NULL, len = 10, b, y, W,
p, d, Z = NULL, tol = 1e-9)

Arguments

gama_L	The lower value of the $\gamma$ values to consider. Use NULL if there is not information about its value. See the details for default computation.
gama_U	The upper value of the $\gamma$ values to consider. Use NULL if there is not information about its value. See the details for default computation.
len	The number of increments to consider for the $\gamma$ parameter.
b	The estimated parameters from the linear model, in the following order: (intercept, network parameters, autoregressive parameters, covariates). The length of the vector should be $2p + 1 + q$, where $q$ denotes the number of covariates.
y	A $TT \times N$ time series object or a $TT \times N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
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.
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.
tol	Tolerance level for the optimizer.

Details

The function optimizes the quasi score test statistic, under the null assumption of linearity, for testing linearity of Poisson Network Autoregressive model of order $p$ against the following ST-PNAR($p$) model, with respect to the unknown nuisance parameter ($\gamma$). For each node of the network $i=1,...,N$ over the time sample $t=1,...,TT$

$\lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h}+\alpha_{h}e^{-\gamma X_{i,t-d}^{2}}X_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l}$

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 sequence $\lambda_{i,t}$ is the expectation of $Y_{i,t}$, conditional to its past values.

The null hypothesis of the test is defined as $H_{0}: \alpha_{1}=...=\alpha_{p}=0$, versus the alternative that at least one among $\alpha_{h}$ is not 0. The test statistic has the form

$LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma)$

where

$S(\hat{\theta},\gamma)=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left(\frac{Y_{i,t}}{\lambda_{i,t}(\hat{\theta},\gamma)}-1\right) \frac{\partial\lambda_{i,t}(\hat{\theta},\gamma)}{\partial\alpha}$

is the partition of the quasi score related to the vector of non-linear parameters $\alpha=(\alpha_{1},...,\alpha_{p})$, evaluated at the estimated parameters $\hat{\theta}$ under the null assumption $H_{0}$ (linear model) and $\Sigma(\hat{\theta},\gamma)$ is the variance of $S(\hat{\theta},\gamma)$.

The optimization employes the Brent algorithm (Brent, 1973) applied in the interval from gama_L to gama_U. To be sure that the global optimum is found, the optimization is performed at (len-1) consecutive equidistant sub-intervals and then the maximum over them is taken as global optimum.

The values of gama_L and gama_U are computed internally as gama_L $=-\log(0.9)/X^{2}$ and gama_U $=-\log(0.1)/X^{2}$, where $X$ is the overall mean of $X_{i,t}$ over the nodes $i=1,...,N$ and times $t=1,...,TT$. Since the non-linear function $e^{-\gamma X_{i,t-d}^{2}}$ ranges between 0 and 1, by considering $X$ to be a representative value for the network mean, gama_U and gama_L would be the values of $\gamma$ leading the non-linear switching function to be 0.1 and 0.9, respectively, so that in the optimization procedure the extremes of the function domain are excluded. Alternatively, their value can be supplied by the user. For details see Armillotta and Fokianos (2022b, Sec. 4-5).

Value

A list including:

gama	The optimum value of the $\gamma$ parameter.
supLM	The value of the objective function at the optimum.
int	A vector with the extremes points of sub-intervals.

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

Armillotta, M., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. https://arxiv.org/abs/2211.02582

Brent, R. (1973) Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs N.J.

See Also

score_test_stnarpq_j, global_optimise_LM_tnarpq, score_test_tnarpq_j

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 1)
b <- mod1$coefs[, 1]
global_optimise_LM_stnarpq(b = b, y = crime, W = crime_W, p = 1, d = 1)

---

Optimization of the score test statistic for the T-PNAR(p) model

Description

Global optimization of the linearity test statistic for the Threshold Poisson Network Autoregressive model of order $p$ with $q$ covariates (T-PNAR($p$)) with respect to the nuisance threshold parameter $\gamma$.

Usage

global_optimise_LM_tnarpq(gama_L = NULL, gama_U = NULL, len = 10, b, y, W,
p, d, Z = NULL, tol = 1e-9)

Arguments

gama_L	The lower value of the $\gamma$ values to consider. Use NULL if there is not information about its value.. See the details for default computation.
gama_U	The upper value of the $\gamma$ values to consider. Use NULL if there is not information about its value.. See the details for default computation.
len	The number of increments to consider for the $\gamma$ parameter.
b	The estimated parameters from the linear 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.
y	A $TT \times N$ time series object or a $TT \times N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
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.
p	The number of lags in the model.
d	The lag parameter of non-linear variable (should be between 1 and $p$).
Z	An $N$ x $q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model. Note that they must be non-negative.
tol	Tolerance level for the optimizer.

Details

The function optimizes the quasi score test statistic, under the null assumption of linearity, for testing linearity of Poisson Network Autoregressive model of order $p$ against the following T-PNAR($p$) model, with respect to the unknown nuisance parameter ($\gamma$). 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 sequence $\lambda_{i,t}$ is the expectation of $Y_{i,t}$, conditional to its past values.

The null hypothesis of the test is defined as $H_{0}: \alpha_{0}=\alpha_{11}=...=\alpha_{2p}=0$, versus the alternative that at least one among $\alpha_{s,h}$ is not $0$, for $s=0,1,2$. The test statistic has the form

$LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma)$

where

$S(\hat{\theta},\gamma)=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left(\frac{Y_{i,t}}{\lambda_{i,t}(\hat{\theta},\gamma)}-1\right)\frac{\partial\lambda_{i,t}(\hat{\theta},\gamma)}{\partial\alpha}$

is the partition of the quasi score related to the vector of non-linear parameters $\alpha=(\alpha_{0},...,\alpha_{2p})$, evaluated at the estimated parameters $\hat{\theta}$ under the null assumption $H_{0}$ (linear model) and $\Sigma(\hat{\theta},\gamma)$ is the variance of $S(\hat{\theta},\gamma)$.

The optimization employes the Brent algorithm (Brent, 1973) applied in the interval from gama_L to gama_U. To be sure that the global optimum is found, the optimization is performed at (len-1) consecutive equidistant sub-intervals and then the maximum over them is taken as global optimum.

The values of gama_L and gama_U are computed internally as the mean over $i=1,...,N$ of $20\%$ and $80\%$ quantile of the empirical distribution of the network mean $X_{i,t}$ for $t=1,...,TT$. In this way the optimization is performed for values of $\gamma$ such that the indicator function $I(X_{i,t-d}\leq\gamma)$ is not always close to 0 or 1. Alternatively, their value can be supplied by the user. For details see Armillotta and Fokianos (2022b, Sec. 4-5).

Value

A list including:

gama	The optimum value of the $\gamma$ parameter.
supLM	The value of the objective function at the optimum.
int	A vector with the extremes points of sub-intervals.

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

Armillotta, M., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. https://arxiv.org/abs/2211.02582

Brent, R. (1973) Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs N.J.

See Also

score_test_tnarpq_j, global_optimise_LM_stnarpq, score_test_stnarpq_j

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
b <- mod1$coefs[, 1]
global_optimise_LM_tnarpq(b = b, y = crime, W = crime_W, p = 2, d = 1)

---

Estimation of the linear Poisson NAR(p) model model with p lags and q covariates (PNAR(p))

Description

Estimation of the linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (PNAR($p$)).

Usage

lin_estimnarpq(y, W, p, Z = NULL, uncons = FALSE, init = NULL,
xtol_rel = 1e-8, maxeval = 100)

Arguments

y	A $TT \times N$ time series object or a $TT \times N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
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.
p	The number of lags in the model.
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.
uncons	logical, if TRUE an unconstrained optimization is run (default is FALSE).
init	A vector of starting values for the optimization algorithm. If this is NULL, the function computes them internally.
xtol_rel	The stopping tolerance of the optimization algorithm.
maxeval	The maximum number of evalutions the optimization algorithm will perform.

Details

This function performs constrained estimation of the linear Poisson NAR($p$) model with $q$ non-negative valued covariates, for each node of the network $i=1,...,N$ over the time sample $t=1,...,TT$, defined as

$\lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l},$

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 sequence $\lambda_{i,t}$ is the expecation of $Y_{i,t}$, conditional to its past values. The parameter $\beta_{0}$ is the intercept of the model, $\beta_{1h}$ are the network coefficients, $\beta_{2h}$ are the autoregressive parameters, and $\delta_{l}$ are the coefficients assocciated to the covariates $Z_{i,l}$.

The estimation of the parameters of the model is performed by Quasi Maximum Likelihood Estimation (QMLE), maximizing the following quasi log-likelihood

$l(\theta)=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left[Y_{i,t}\log\lambda_{i,t}(\theta)-\lambda_{i,t}(\theta)\right]$

with respect to the vector of unknown parameters $\theta$ described above. The coefficients are defined only in the non-negative real line.

By default, the optimization is constrained in the stationary region where $\sum_{h=1}^{p}(\beta_{1h}+\beta_{2h})<1$; this can be removed by setting uncons = TRUE. However, the model estimates might be inconsistent if the estimated parameters lie outside the stationary region.

The ordinary least squares estimates are employed as starting values of the optimization procedure. Robust standard errors and z-tests are also returned.

Value

A list with attribute class "PNAR" including:

coefs	A matrix with the estimated QMLE coefficients, their standard errors their Z-test statistics and the relevant p-values computed via the standard normal approximation.
score	The value of the quasi score function at the optimization point. It should be close to 0 if the optimization is successful.
loglik	The value of the maximized quasi log-likelihood.
ic	A vector with the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Quasi information criterion (QIC).

Alternatively, these can be printed via the function summary.PNAR.

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., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. https://arxiv.org/abs/2211.02582

See Also

log_lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
summary(mod1)

---

Scatter plot of information criteria versus the number of lags in the linear Poisson NAR(p) model model with p lags and q covariates (PNAR(p))

Description

Scatter plot of information criteria versus the number of lags in the linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (PNAR($p$)).

Usage

lin_ic_plot(y, W, p = 1:10, Z = NULL, uncons = FALSE, ic = "QIC")

Arguments

y	A $TT$ x $N$ time series object or a $TT$ x $N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
W	The $N$ x $N$ 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.
p	A vector with integer numbers, the range of lags in the model, for which the AIC, BIC and QIC will be computed.
Z	An $N$ x $q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model. Note that they must be non-negative.
uncons	Logical, if TRUE an unconstrained optimization without stationarity constraints is performed (default is FALSE).
ic	The information criterion you want to plot, "QIC" (default value), "AIC" or "BIC".

Details

The function computes the AIC, BIC or QIC for a range of lag orders of the linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (PNAR($p$)).

Value

A scatter plot with the lag order versus either QIC (default), AIC or BIC, and a vector with their values, for each lag order.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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

See Also

lin_estimnarpq, log_lin_ic_plot

Examples

data(crime)
data(crime_W)
lin_ic_plot(crime, crime_W, p = 1:3)

---

Starting values for the linear Poisson NAR(p) model model with p lags and q covariates (PNAR(p))

Description

Starting values for the linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (PNAR($p$)).

Usage

lin_narpq_init(y, W, p, Z = NULL)

Arguments

y	A $TT$ x $N$ time series object or a $TT$ x $N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
W	The $N$ x $N$ 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.
p	The number of lags in the model.
Z	An $N$ x $q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model. Note that they must be non-negative.

Details

The function computes starting values to be used in the function lin_estimnarpq. These are simply the ordinary least squares estimators with a correction. If any of the the resulting coefficients is negative they become equal to 0.001

Value

A vector with the initial values.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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

See Also

lin_estimnarpq

Examples

data(crime)
data(crime_W)
x0 <- lin_narpq_init(crime, crime_W, p = 2)

---

Estimation of the log-linear Poisson NAR(p) model with p lags and q covariates (log-PNAR(p))

Description

Estimation of the log-linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (log-PNAR($p$)).

Usage

log_lin_estimnarpq(y, W, p, Z = NULL, uncons = FALSE, init = NULL,
xtol_rel = 1e-8, maxeval = 100)

Arguments

y	A $TT$ x $N$ time series object or a $TT$ x $N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
W	The $N$ x $N$ 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.
p	The number of lags in the model.
Z	An $N$ x $q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model.
uncons	logical, if TRUE an unconstrained optimization is performed (default is FALSE).
init	A vector of starting values for the optimization algorithm. If this is NULL, the function computes them internally.
xtol_rel	The stopping tolerance of the optimization algorithm.
maxeval	The maximum number of evalutions the optimization algorithm will perform.

Details

This function performs a constrained estimation of the linear Poisson NAR($p$) model with $q$ non-negative valued covariates, for each node of the network $i=1,...,N$ over the time sample $t=1,...,TT$, defined as

$\nu_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l},$

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 sequence $\nu_{i,t}$ is the log of the expectation of $Y_{i,t}$, conditional to its past values. The parameter $\beta_{0}$ is the intercept of the model, $\beta_{1h}$ are the network coefficients, $\beta_{2h}$ are the autoregressive parameters, and $\delta_{l}$ are the coefficients assocciated to the covariates $Z_{i,l}$.

The estimation of the parameters of the model is performed by Quasi Maximum Likelihood Estimation (QMLE), maximizing the following quasi log-likelihood

$l(\theta)=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left[Y_{i,t}\nu_{i,t}(\theta)-e^{\nu_{i,t}(\theta)}\right]$

with respect to the vector of unknown parameters $\theta$ described above.

By default, the optimization is constrained in the stationary region where $\sum_{h=1}^{p}(|\beta_{1h}|+|\beta_{2h}|)<1$; this can be removed by setting uncons = TRUE. However, the model estimates might be inconsistent if the estimated parameters lie outside the stationary region.

The ordinary least squares estimates are employed as starting values of the optimization procedure. Robust standard errors and z-tests are also returned.

Value

A list with attribute class "PNAR" including:

coefs	A matrix with the estimated QMLE coefficients, their standard errors, their Z-test statistics and the relevant p-values computed via the standard normal approximation.
score	The value of the quasi score function at the optimization point. It should be close to 0 if the optimization is successful.
loglik	The value of the maximized quasi log-likelihood.
ic	A vector with the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Quasi information criterion (QIC).

Alternatively, these can be printed via the function summary.PNAR.

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., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. https://arxiv.org/abs/2211.02582

See Also

lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- log_lin_estimnarpq(crime, crime_W, p = 2)
summary(mod1)

---

Scatter plot of information criteria versus the number of lags in the log-linear Poisson NAR(p) model with p lags and q covariates (log-PNAR(p))

Description

Scatter plot of information criteria versus the number of lags in log-linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (log-PNAR($p$)).

Usage

log_lin_ic_plot(y, W, p = 1:10, Z = NULL, uncons = FALSE, ic = "QIC")

Arguments

y	A $TT$ x $N$ time series object or a $TT$ x $N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
W	The $N$ x $N$ 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.
p	A vector with integer numbers, the range of lags in the model, for which the AIC, BIC and QIC will be computed.
Z	An $N$ x $q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model. Note that they must be non-negative.
uncons	Logical, if TRUE an unconstrained optimization without stationarity constraints is performed (default is FALSE).
ic	The information criterion you want to plot, "QIC" (default value), "AIC" or "BIC".

Details

The function computes the AIC, BIC or QIC for a range of lag orders of the log-linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (PNAR($p$)).

Value

A scatter plot with the lag order versus either QIC (default), AIC or BIC, and a vector with their values, for each lag order.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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

See Also

log_lin_estimnarpq, lin_ic_plot

Examples

data(crime)
data(crime_W)
log_lin_ic_plot(crime, crime_W, p = 1:3)

---

Starting values for the log-linear Poisson NAR(p) model with p lags and q covariates (log-PNAR(p))

Description

Starting values for the log-linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (log-PNAR($p$)).

Usage

log_lin_narpq_init(y, W, p, Z = NULL)

Arguments

y	A $TT$ x $N$ time series object or a $TT$ x $N$ numerical matrix with the $N$ multivariate count time series over $TT$ time periods.
W	The $N$ x $N$ 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.
p	The number of lags in the model.
Z	An $N$ x $q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model.

Details

This function computes initial values for the log-linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (log-PNAR($p$)) with stationarity conditions. These initial values are simply the ordinary least squares estimators with a correction.

Value

A vector with the initial values.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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

See Also

log_lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- log_lin_narpq_init(crime, crime_W, p = 2)

---

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

Description

Generation of multivariate count time series from a linear Poisson Network Autoregressive model of order $p$ with $q$ covariates (PNAR($p$)).

Usage

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

Arguments

b	The 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.
p	The number of lags in the model.
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 choices are "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 linear 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 (2022, Sec. 2.1-2.2).

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

$\lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l}$

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, and $\delta_{l}$ are the coefficients assocciated to the covariates $Z_{i,l}$.

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 (2022). Poisson network autoregression. https://arxiv.org/abs/2104.06296

Fokianos, K., Stove, B., Tjostheim, D., and P. Doukhan (2020). Multivariate count autoregression. Bernoulli, 26(1), 471-499.

See Also

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

Examples

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

---

Generation of multivariate count time series from a log-linear Poisson NAR(p) model with q covariates (log-PNAR(p))

Description

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

Usage

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

Arguments

b	The 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.
p	The number of lags in the model.
Z	An $N \times q$ matrix of covariates (one for each column), where $q$ is the number of covariates in the model.
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 the value of the copula parameter ($\rho$). A scalar in $[-1,1]$ for elliptical copulas (Gaussian, t), a value greater 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 log-linear Poisson NAR($p$) model, where $q$ non time-varying covariates are allowed as well. The counts are simulated from $Y_{t}=N_{t}(e^{\nu_{t}})$, 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 (2022, Sec. 2.1-2.2).

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

$\nu_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l}$

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, and $\delta_{l}$ are the coefficients assocciated to the covariates $Z_{i,l}$.

Value

A list including:

p2R	The Toeplitz correlation matrix, if employed in the copula or NULL else.
log_lambda	A $TT \times N$ time series object matrix of simulated Poisson log-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 (2022). Poisson network autoregression. https://arxiv.org/abs/2104.06296

Fokianos, K., Stove, B., Tjostheim, D., and P. Doukhan (2020). Multivariate count autoregression. Bernoulli, 26(1), 471-499.

See Also

poisson.MODpq, poisson.MODpq.nonlin, poisson.MODpq.stnar, poisson.MODpq.tnar

Examples

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

---

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)

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.

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 (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 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 (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.stnar, poisson.MODpq.tnar

Examples

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

---

Generation of counts from a non-linear Smooth Transition Poisson NAR(p) model with q covariates (ST-PNAR(p))

Description

Generation of multivariate count time series from a non-linear Smooth Transition Poisson Network Autoregressive model of order $p$ with $q$ covariates (ST-PNAR($p$)).

Usage

poisson.MODpq.stnar(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 smoothing parameter.
a	Vector of non-linear parameters. The dimension of the vector should be $p$.
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 choices are "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 Smooth Transition 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 ST-PNAR($p$) model. For each node of the network $i=1,...,N$ over the time sample $t=1,...,TT$

$\lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h}+\alpha_{h}e^{-\gamma X_{i,t-d}^{2}}X_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l}$

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, $\alpha_{h}$ are the non-linear smooth transition parameters, $\gamma$ is the nuisance smoothing 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 ST-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.tnar

Examples

W <- adja( N = 20, K = 5, alpha= 0.5)
y <- poisson.MODpq.stnar( b = c(0.5, 0.3, 0.2), W = W, gama = 0.2, a = 0.4,
p = 1, d = 1, Z = NULL, TT = 1000, N = 20, copula = "gaussian",
corrtype = "equicorrelation", rho = 0.5)$y

---

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

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