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 (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.

Details

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 (2024). Count network autoregression. Journal of Time Series Analysis, 45(4): 584–612.

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

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.

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

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

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

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

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 (2023, Sec. 4-5).

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

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 (2023, Sec. 4-5).

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

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:

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 (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..

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

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 (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.

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

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 (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.

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

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:

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 (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.

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

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 (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.

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

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 (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.

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

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 (2024, 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:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

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 (2024, 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:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

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).

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

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 (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 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 (2023, Sec. 2).

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

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 (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 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 (2023, Sec. 2).

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

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

Random number generation of copula functions

Description

Random number generation of copula functions.

Usage

rcopula(n, N, copula = "gaussian", corrtype = "equicorrelation",
rho, dof = 1, cholR = NULL)

Arguments

Details

This function generates random copula values from Gaussian, Student's t, or Clayton copulas based on a single copula paremeter and different correlation structures.

Value

An n \times N matrix with the simulated copula values.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

Nelsen, Roger B. (1999). An Introduction to Copulas, Springer.

See Also

getN, poisson.MODpq, poisson.MODpq.log

Examples

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

Linearity test against non-linear ID-PNAR(p) model

Description

Quasi score test for testing linearity of Poisson Network Autoregressive model of order p against the non-linear Intercep Drift (ID) version (ID-PNAR(p)).

Usage

score_test_nonlinpq_h0(b, y, W, p, d, Z = NULL)

Arguments

Details

The function computes the quasi score test for testing linearity of Poisson Network Autoregressive model of order p against 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 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}: \gamma=0, versus the alternative H_{1}: \gamma >0. The test statistic has the form

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

where

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

is the partition of the quasi score related to the non-linear parameter \gamma, evaluated at the estimated parameters \hat{\theta} under the null assumption H_{0} (linear model), and \Sigma(\hat{\theta}) is the variance of S(\hat{\theta}). Under H_{0}, the test asymptotically follows the \chi^2 distribution with 1 degree of freedom. For details see Armillotta and Fokianos (2023, Sec. 4).

Value

A list with attribute class "htest" including:

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

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

See Also

score_test_stnarpq_j, score_test_tnarpq_j, lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
ca <- mod1$coefs[, 1]
score_test_nonlinpq_h0(ca, crime, crime_W, p = 2, d = 1)

Bound p-value for testing for smooth transition effects on PNAR(p) model

Description

Computation of Davies bound p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the non-linear Smooth Transition alternative (ST-PNAR(p)).

Usage

score_test_stnarpq_DV(b, y, W, p, d, Z = NULL, gama_L = NULL,
gama_U = NULL, len = 100)

Arguments

Details

The function computes an upper-bound for the p-value of the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the following Smooth Transition alternative (ST-PNAR(p)). 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). Since the test statistic depends on an unknown nuisance parameter (\gamma), the supremum of the statistic is considered in the test, \sup_{\gamma}LM(\gamma). The function computes the bound of the p-value, suggested by Davies (1987), for the test statistic \sup_{\gamma}LM(\gamma), with scalar nuisance parameter \gamma, as follows.

P(\chi^{2}_{k} \geq M)+V M^{1/2(k-1)}\frac{e^{-M/2}2^{-k/2}}{\Gamma(k/2)}

where M is the maximum of the test statistic LM(\gamma), computed by the available sample, over a grid of values for the nuisance parameter \gamma_{F}=(\gamma_{L},\gamma_{1},...,\gamma_{l},\gamma_{U}); k is the number of non-linear parameters tested. So the first summand of the bound is just the p-value of a chi-square test with k degrees of freedom. The second summand is a correction term depending on V, which is the approximated total variation computed as

V=|LM^{1/2}(\gamma_{1})-LM^{1/2}(\gamma_{L})|+|LM^{1/2}(\gamma_{2})-LM^{1/2}(\gamma_{1})|+...+|LM^{1/2}(\gamma_{U})-LM^{1/2}(\gamma_{l})|.

The feasible bound allows to approximate the p-values of the sup-type test in a straightforward way, by adding to the tail probability of a chi-square distribution a correction term which depends on the total variation of the process. For details see Armillotta and Fokianos (2023, Sec. 5).

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 values can be supplied by the user.

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.

See Also

score_test_stnarpq_j, global_optimise_LM_stnarpq

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 1)
ca <- mod1$coefs[, 1]
score_test_stnarpq_DV(ca, crime, crime_W, p = 1, d = 1)

Bootstrap test for smooth transition effects on PNAR(p) model

Description

Computation of bootstrap p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the non-linear Smooth Transition alternative (ST-PNAR(p)).

Usage

score_test_stnarpq_j(supLM, b, y, W, p, d, Z = NULL, J = 499,
gama_L = NULL, gama_U = NULL, tol = 1e-9, ncores = 1, seed = NULL)

Arguments

Details

The function computes a bootstrap p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the following Smooth Transition alternative (ST-PNAR(p)). 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).

Since the test statistic depends on an unknown nuisance parameter (\gamma), the supremum of the statistic is considered in the test, \sup_{\gamma}LM(\gamma). This value can be computed for the available sample by using the function global_optimise_LM_stnarpq and should be supplied here as an input supLM.

The function performs the bootstrap resampling of the test statistic \sup_{\gamma}LM(\gamma) by employing Gaussian perturbations of the score S(\hat{\theta},\gamma). For details see Armillotta and Fokianos (2023, Sec. 5).

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.

Note: For large datasets the function may require few minutes to run. Parallel computing is suggested to speed up the computations.

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

See Also

score_test_stnarpq_DV, global_optimise_LM_stnarpq, score_test_tnarpq_j

Examples

# load data
data(crime)
data(crime_W)
#estimate linear PNAR model
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
b <- mod1$coefs[, 1]

g <- global_optimise_LM_stnarpq(b = b, y = crime, W = crime_W, p = 2, d = 1)
supg <- g$supLM
score_test_stnarpq_j(supLM = supg, b = b, y = crime, W = crime_W, p = 2, d = 1, J = 5)

Bootstrap test for threshold effects on PNAR(p) model

Description

Computation of bootstrap p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the non-linear Threshold alternative (T-PNAR(p)).

Usage

score_test_tnarpq_j(supLM, b, y, W, p, d, Z = NULL, J = 499,
gama_L = NULL, gama_U = NULL, tol = 1e-9, ncores = 1, seed = NULL)

Arguments

Details

The function computes a bootstrap p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the following Threshold alternative (T-PNAR(p)). 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. 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).

Since the test statistic depends on an unknown nuisance parameter (\gamma), the supremum of the statistic is considered in the test, \sup_{\gamma}LM(\gamma). This value can be computed for the available sample by using the function global_optimise_LM_tnarpq and should be supplied here as an input supLM.

The function performs the bootstrap resampling of the test statistic \sup_{\gamma}LM(\gamma) by employing Gaussian perturbations of the score S(\hat{\theta},\gamma). For details see Armillotta and Fokianos (2023, Sec. 5).

The values of gama_L and gama_U are computed internally as the mean over i=1,...,N of 20\% and 80\% quantiles 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 (2023, Sec. 4-5).

Note: For large datasets the function may require few minutes to run. Parallel computing is suggested to speed up the computations.

Value

A list including:

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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.

See Also

global_optimise_LM_tnarpq, global_optimise_LM_stnarpq, score_test_stnarpq_j

Examples

# load data
data(crime)
data(crime_W)
#estimate linear PNAR model
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
b <- mod1$coefs[, 1]

g <- global_optimise_LM_tnarpq(b = b, y = crime, W = crime_W, p = 2, d = 1)
supg <- g$supLM
score_test_tnarpq_j(supLM = supg, b = b, y = crime, W = crime_W, p = 2, d = 1, J = 5)

S3 methods for extracting the results of the bound p-value for testing for smooth transition effects on PNAR(p) model

Description

S3 methods for extracting the results of the bound p-value for testing for smooth transition effects on PNAR(p) model.

Usage

## S3 method for class 'DV'
summary(object, ...)
## S3 method for class 'summary.DV'
print(x, ...)
## S3 method for class 'DV'
print(x, ...)

Arguments

Details

The functions print the output of the bound p-value for testing for smooth transition effects on PNAR(p) model.

Value

The functions print the results of the function score_test_stnarpq_DV.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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.

Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.

See Also

score_test_stnarpq_DV

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 1)
ca <- mod1$coefs[, 1]
a <- score_test_stnarpq_DV(ca, crime, crime_W, p = 1, d = 1)
print(a)
summary(a)

S3 methods for extracting the results of the estimation functions

Description

S3 methods for extracting the results of the estimation functions.

Usage

## S3 method for class 'PNAR'
summary(object, ...)
## S3 method for class 'summary.PNAR'
print(x, ...)
## S3 method for class 'PNAR'
print(x, ...)

Arguments

Details

These functions print the output of the estimation functions.

Value

The print.PNAR() function prints the coefficients of the model. The summary.PNAR() function prints the output in the lm() style.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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.

See Also

log_lin_estimnarpq

Examples

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

S3 methods for extracting the results of the non-linear hypothesis test

Description

S3 methods for extracting the results of the non-linear hypothesis test.

Usage

## S3 method for class 'nonlin'
summary(object, ...)
## S3 method for class 'summary.nonlin'
print(x, ...)
## S3 method for class 'nonlin'
print(x, ...)

Arguments

Details

The functions print the output of the non-linear hypothesis test.

Value

The functions print the results of the function score_test_nonlinpq_h0.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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.

See Also

score_test_nonlinpq_h0

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
ca <- mod1$coefs[, 1]
a <- score_test_nonlinpq_h0(ca, crime, crime_W, p = 2, d = 1)
print(a)
summary(a)