Package 'PNAR'

Title: 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. <arXiv:2104.06296>. Armillotta, M. and K. Fokianos (2022b). Testing linearity for network autoregressive models. <arXiv:2202.03852>. Armillotta, M., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. <arXiv:2211.02582>.
Authors: Michail Tsagris [aut, cre], Mirko Armillotta [aut, cph], Konstantinos Fokianos [aut]
Maintainer: Michail Tsagris <mtsagris@uoc.gr>
License: GPL (>= 2)
Version: 1.6
Built: 2024-02-06 07:49:39 UTC
Source: CRAN

Help Index


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 KK 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/KN/K. KK should be chosen such that NN is divisible by KK.

alpha

The network density. A value in [0,1][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 NN and network density alphaalpha: Probability to draw an edge for a pair of nodes in the same block: αN0.3\alpha*N^{-0.3}. Probability to draw an edge for a pair of nodes in different blocks: αN1\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][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 αN0.3\alpha*N^{-0.3}, specified based on the number of nodes on the network NN 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 pp with qq covariates (ST-PNAR(pp)) with respect to the nuisance scale parameter γ\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 γ\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 γ\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 γ\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+q2p + 1 + q, where qq denotes the number of covariates.

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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 pp).

Z

An N×qN \times q matrix of covariates (one for each column), where qq 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 pp against the following ST-PNAR(pp) model, with respect to the unknown nuisance parameter (γ\gamma). For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0+h=1p(β1hXi,th+β2hYi,th+αheγXi,td2Xi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence λi,t\lambda_{i,t} is the expectation of Yi,tY_{i,t}, conditional to its past values.

The null hypothesis of the test is defined as H0:α1=...=αp=0H_{0}: \alpha_{1}=...=\alpha_{p}=0, versus the alternative that at least one among αh\alpha_{h} is not 0. The test statistic has the form

LM(γ)=S(θ^,γ)Σ1(θ^,γ)S(θ^,γ)LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma)

where

S(θ^,γ)=t=1TTi=1N(Yi,tλi,t(θ^,γ)1)λi,t(θ^,γ)αS(\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 α=(α1,...,αp)\alpha=(\alpha_{1},...,\alpha_{p}), evaluated at the estimated parameters θ^\hat{\theta} under the null assumption H0H_{0} (linear model) and Σ(θ^,γ)\Sigma(\hat{\theta},\gamma) is the variance of S(θ^,γ)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)/X2=-\log(0.9)/X^{2} and gama_U =log(0.1)/X2=-\log(0.1)/X^{2}, where XX is the overall mean of Xi,tX_{i,t} over the nodes i=1,...,Ni=1,...,N and times t=1,...,TTt=1,...,TT. Since the non-linear function eγXi,td2e^{-\gamma X_{i,t-d}^{2}} ranges between 0 and 1, by considering XX to be a representative value for the network mean, gama_U and gama_L would be the values of γ\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 γ\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 pp with qq covariates (T-PNAR(pp)) with respect to the nuisance threshold parameter γ\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 γ\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 γ\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 γ\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+q2p + 1 + q, where qq denotes the number of covariates.

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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 pp).

Z

An NN x qq matrix of covariates (one for each column), where qq 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 pp against the following T-PNAR(pp) model, with respect to the unknown nuisance parameter (γ\gamma). For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0+h=1p[β1hXi,th+β2hYi,th+(α0+α1hXi,th+α2hYi,th)I(Xi,tdγ)]+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N} W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW, and I()I() is the indicator function. The sequence λi,t\lambda_{i,t} is the expectation of Yi,tY_{i,t}, conditional to its past values.

The null hypothesis of the test is defined as H0:α0=α11=...=α2p=0H_{0}: \alpha_{0}=\alpha_{11}=...=\alpha_{2p}=0, versus the alternative that at least one among αs,h\alpha_{s,h} is not 00, for s=0,1,2s=0,1,2. The test statistic has the form

LM(γ)=S(θ^,γ)Σ1(θ^,γ)S(θ^,γ)LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma)

where

S(θ^,γ)=t=1TTi=1N(Yi,tλi,t(θ^,γ)1)λi,t(θ^,γ)αS(\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 α=(α0,...,α2p)\alpha=(\alpha_{0},...,\alpha_{2p}), evaluated at the estimated parameters θ^\hat{\theta} under the null assumption H0H_{0} (linear model) and Σ(θ^,γ)\Sigma(\hat{\theta},\gamma) is the variance of S(θ^,γ)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,...,Ni=1,...,N of 20%20\% and 80%80\% quantile of the empirical distribution of the network mean Xi,tX_{i,t} for t=1,...,TTt=1,...,TT. In this way the optimization is performed for values of γ\gamma such that the indicator function I(Xi,tdγ)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 γ\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 pp with qq covariates (PNAR(pp)).

Usage

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

Arguments

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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×qN \times q matrix of covariates (one for each column), where qq 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(pp) model with qq non-negative valued covariates, for each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT, defined as

λi,t=β0+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l,\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence λi,t\lambda_{i,t} is the expecation of Yi,tY_{i,t}, conditional to its past values. The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{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(θ)=t=1TTi=1N[Yi,tlogλi,t(θ)λi,t(θ)]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 θ\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 h=1p(β1h+β2h)<1\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 pp with qq covariates (PNAR(pp)).

Usage

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

Arguments

y

A TTTT x NN time series object or a TTTT x NN numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The NN x NN 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 NN x qq matrix of covariates (one for each column), where qq 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 pp with qq covariates (PNAR(pp)).

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 pp with qq covariates (PNAR(pp)).

Usage

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

Arguments

y

A TTTT x NN time series object or a TTTT x NN numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The NN x NN 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 NN x qq matrix of covariates (one for each column), where qq 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 pp with qq covariates (log-PNAR(pp)).

Usage

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

Arguments

y

A TTTT x NN time series object or a TTTT x NN numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The NN x NN 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 NN x qq matrix of covariates (one for each column), where qq 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(pp) model with qq non-negative valued covariates, for each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT, defined as

νi,t=β0+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l,\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence νi,t\nu_{i,t} is the log of the expectation of Yi,tY_{i,t}, conditional to its past values. The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{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(θ)=t=1TTi=1N[Yi,tνi,t(θ)eνi,t(θ)]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 θ\theta described above.

By default, the optimization is constrained in the stationary region where h=1p(β1h+β2h)<1\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 pp with qq covariates (log-PNAR(pp)).

Usage

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

Arguments

y

A TTTT x NN time series object or a TTTT x NN numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The NN x NN 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 NN x qq matrix of covariates (one for each column), where qq 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 pp with qq covariates (PNAR(pp)).

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 pp with qq covariates (log-PNAR(pp)).

Usage

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

Arguments

y

A TTTT x NN time series object or a TTTT x NN numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The NN x NN 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 NN x qq matrix of covariates (one for each column), where qq is the number of covariates in the model.

Details

This function computes initial values for the log-linear Poisson Network Autoregressive model of order pp with qq covariates (log-PNAR(pp)) 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 pp with qq covariates (PNAR(pp)).

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+q2p + 1 + q, where qq denotes the number of covariates.

W

The N×NN \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×qN \times q matrix of covariates (one for each column), where qq 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 (ρ\rho). A scalar in [1,1][-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 ρ\rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)(i,j)-element is ρij\rho^{|i-j|}.

dof

The degrees of freedom for Student's t copula.

Details

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

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

λi,t=β0+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{i,l}.

Value

A list including:

p2R

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

lambda

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

y

A TT×NTT \times N time series object matrix of simulated counts for NN time series over TTTT.

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 pp with qq covariates (log-PNAR(pp)).

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+q2p + 1 + q, where qq denotes the number of covariates.

W

The N×NN \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×qN \times q matrix of covariates (one for each column), where qq 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 (ρ\rho). A scalar in [1,1][-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 ρ\rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)(i,j)-element is ρij\rho^{|i-j|}.

dof

The degrees of freedom for Student's t copula.

Details

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

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

νi,t=β0+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{i,l}.

Value

A list including:

p2R

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

log_lambda

A TT×NTT \times N time series object matrix of simulated Poisson log-means for NN time series over TTTT.

y

A TT×NTT \times N time series object matrix of simulated counts for NN time series over TTTT.

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 pp with qq covariates (ID-PNAR(pp)).

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+q2p + 1 + q, where qq denotes the number of covariates.

W

The N×NN \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 pp).

Z

An N×qN \times q matrix of covariates (one for each column), where qq 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 (ρ\rho). A scalar in [1,1][-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 ρ\rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)(i,j)-element is ρij\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(pp) model, where qq non time-varying covariates are allowed as well. The counts are simulated from Yt=Nt(λt)Y_{t}=N_{t}(\lambda_{t}), where NtN_{t} is a sequence of NN-dimensional IID Poisson count processes, with intensity 1, and whose structure of dependence is modelled through a copula construction C(ρ)C(\rho) on their associated exponential waiting times random variables. For details see Armillotta and Fokianos (2022a, Sec. 2.1-2.2). The sequence λi,t\lambda_{i,t} is the expecation of Yi,tY_{i,t}, conditional to its past values and it is generated by means of the following ID-PNAR(pp) model. For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0(1+Xi,td)γ+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW.

The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, γ\gamma is the non-linear coefficient associated with the intercept drift, and δl\delta_{l} are the coefficients assocciated with the covariates Zi,lZ_{i,l}. The coefficient dd 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×NTT \times N time series object matrix of simulated Poisson means for NN time series over TTTT.

y

A TT×NTT \times N time series object matrix of simulated counts for NN time series over TTTT.

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 pp with qq covariates (ST-PNAR(pp)).

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+q2p + 1 + q, where qq denotes the number of covariates.

W

The N×NN \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 pp.

p

The number of lags in the model.

d

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

Z

An N×qN \times q matrix of covariates (one for each column), where qq 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 (ρ\rho). A scalar in [1,1][-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 ρ\rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)(i,j)-element is ρij\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(pp) model, where qq non time-varying covariates are allowed as well. The counts are simulated from Yt=Nt(λt)Y_{t}=N_{t}(\lambda_{t}), where NtN_{t} is a sequence of NN-dimensional IID Poisson count processes, with intensity 1, and whose structure of dependence is modelled through a copula construction C(ρ)C(\rho) on their associated exponential waiting times random variables. For details see Armillotta and Fokianos (2022a, Sec. 2.1-2.2).

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

λi,t=β0+h=1p(β1hXi,th+β2hYi,th+αheγXi,td2Xi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW.

The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, αh\alpha_{h} are the non-linear smooth transition parameters, γ\gamma is the nuisance smoothing parameter, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{i,l}. The coefficient dd 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×NTT \times N time series object matrix of simulated Poisson means for NN time series over TTTT.

y

A TT×NTT \times N time series object matrix of simulated counts for NN time series over TTTT.

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 pp with qq covariates (T-PNAR(pp)).

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+q2p + 1 + q, where qq denotes the number of covariates.

W

The N×NN \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+12p + 1.

p

The number of lags in the model.

d

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

Z

An N×qN \times q matrix of covariates (one for each column), where qq 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 (ρ\rho). A scalar in [1,1][-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 ρ\rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)(i,j)-element is ρij\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(pp) model, where qq non time-varying covariates are allowed as well. The counts are simulated from Yt=Nt(λt)Y_{t}=N_{t}(\lambda_{t}), where NtN_{t} is a sequence of NN-dimensional IID Poisson count processes, with intensity 1, and whose structure of dependence is modelled through a copula construction C(ρ)C(\rho) on their associated exponential waiting times random variables. For details see Armillotta and Fokianos (2022a, Sec. 2.1-2.2).

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

λi,t=β0+h=1p[β1hXi,th+β2hYi,th+(α0+α1hXi,th+α2hYi,th)I(Xi,tdγ)]+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW, and I()I() is the indicator function.

The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, the α\alpha vector of non-linear parameters is divided as follows: α0\alpha_{0} is the intercept, α1h\alpha_{1h} are the network coefficients, α2h\alpha_{2h} are the autoregressive parameters; γ\gamma is the nuisance threshold parameter, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{i,l}. The coefficient dd 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×NTT \times N time series object matrix of simulated Poisson means for NN time series over TTTT.

y

A TT×NTT \times N time series object matrix of simulated counts for NN time series over TTTT.

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

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

n

The number of random values to generate.

N

The number of variables for which random valeus will be generated.

copula

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

rho

The the value of the copula parameter (ρ\rho). A scalar in [1,1][-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 ρ\rho. The "toeplitz" option generates a correlation matrix whose generic off-diagonal (i,j)(i,j)-element is ρij\rho^{|i-j|}.

dof

The degrees of freedom for Student's t copula.

cholR

An alternative input for elliptic copulas, providing directly the Cholesky decomposition for a specific correlation matrix to be passed, otherwise leave it NULL.

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×Nn \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 pp against the non-linear Intercep Drift (ID) version (ID-PNAR(pp)).

Usage

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

Arguments

b

The estimated parameters from the linear PNAR model, in the following order: (intercept, network parameters, autoregressive parameters, covariates). The dimension of the vector should be 2p+1+q2p + 1 + q, where qq denotes the number of covariates.

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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 pp).

Z

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

Details

The function computes the quasi score test for testing linearity of Poisson Network Autoregressive model of order pp against the following ID-PNAR(pp) model. For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0(1+Xi,td)γ+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence λi,t\lambda_{i,t} is the expectation of Yi,tY_{i,t} conditional to its past values.

The null hypothesis of the test is defined as H0:γ=0H_{0}: \gamma=0, versus the alternative H1:γ>0H_{1}: \gamma >0. The test statistic has the form

LM=S(θ^)Σ1(θ^)S(θ^),LM=S^{'}(\hat{\theta})\Sigma^{-1}(\hat{\theta})S(\hat{\theta}),

where

S(θ^)=t=1TTi=1N(Yi,tλi,t(θ^)1)λi,t(θ^)γS(\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 γ\gamma, evaluated at the estimated parameters θ^\hat{\theta} under the null assumption H0H_{0} (linear model), and Σ(θ^)\Sigma(\hat{\theta}) is the variance of S(θ^)S(\hat{\theta}). Under H0H_{0}, the test asymptotically follows the χ2\chi^2 distribution with 1 degree of freedom. For details see Armillotta and Fokianos (2022b, Sec. 4).

Value

A list with attribute class "htest" including:

statistic

The value of the χ2\chi^2 test statistic.

parameter

The degrees of freedom of the χ2\chi^2 distribution. This is always 1.

p.value

The p-value of the χ2\chi^2 test statistic.

null.value

The value of the γ\gamma parameter, which is equal to 0 under the null hypothesis.

alternative

The alternative hypothesis, γ\gamma has to be greater than 0.

method

The name of the test.

data.name

Information on the arguments used.

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 (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

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 pp (PNAR(pp)) versus the non-linear Smooth Transition alternative (ST-PNAR(pp)).

Usage

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

Arguments

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+q2p + 1 + q, where qq denotes the number of covariates.

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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 pp).

Z

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

gama_L

The lower value of the nuisance parameter γ\gamma 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 nuisance parameter γ\gamma to consider. Use NULL if there is not information about its value. See the details for default computation.

len

The length of the grid of values of γ\gamma values to consider.

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 pp (PNAR(pp)) versus the following Smooth Transition alternative (ST-PNAR(pp)). For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0+h=1p(β1hXi,th+β2hYi,th+αheγXi,td2Xi,th)+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence λi,t\lambda_{i,t} is the expectation of Yi,tY_{i,t}, conditional to its past values.

The null hypothesis of the test is defined as H0:α1=...=αp=0H_{0}: \alpha_{1}=...=\alpha_{p}=0, versus the alternative that at least one among αh\alpha_{h} is not 0. The test statistic has the form

LM(γ)=S(θ^,γ)Σ1(θ^,γ)S(θ^,γ),LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma),

where

S(θ^,γ)=t=1TTi=1N(Yi,tλi,t(θ^,γ)1)λi,t(θ^,γ)αS(\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 α=(α1,...,αp)\alpha=(\alpha_{1},...,\alpha_{p}), evaluated at the estimated parameters θ^\hat{\theta} under the null assumption H0H_{0} (linear model), and Σ(θ^,γ)\Sigma(\hat{\theta},\gamma) is the variance of S(θ^,γ)S(\hat{\theta},\gamma). Since the test statistic depends on an unknown nuisance parameter (γ\gamma), the supremum of the statistic is considered in the test, supγLM(γ)\sup_{\gamma}LM(\gamma). The function computes the bound of the p-value, suggested by Davies (1987), for the test statistic supγLM(γ)\sup_{\gamma}LM(\gamma), with scalar nuisance parameter γ\gamma, as follows.

P(χk2M)+VM1/2(k1)eM/22k/2Γ(k/2)P(\chi^{2}_{k} \geq M)+V M^{1/2(k-1)}\frac{e^{-M/2}2^{-k/2}}{\Gamma(k/2)}

where MM is the maximum of the test statistic LM(γ)LM(\gamma), computed by the available sample, over a grid of values for the nuisance parameter γF=(γL,γ1,...,γl,γU)\gamma_{F}=(\gamma_{L},\gamma_{1},...,\gamma_{l},\gamma_{U}); kk 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 kk degrees of freedom. The second summand is a correction term depending on VV, which is the approximated total variation computed as

V=LM1/2(γ1)LM1/2(γL)+LM1/2(γ2)LM1/2(γ1)+...+LM1/2(γU)LM1/2(γl).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 (2022b, Sec. 5).

The values of gama_L and gama_U are computed internally as gama_L =log(0.9)/X2=-\log(0.9)/X^{2} and gama_U =log(0.1)/X2=-\log(0.1)/X^{2}, where XX is the overall mean of Xi,tX_{i,t} over the nodes i=1,...,Ni=1,...,N and times t=1,...,TTt=1,...,TT. Since the non-linear function eγXi,td2e^{-\gamma X_{i,t-d}^{2}} ranges between 0 and 1, by considering XX to be a representative value for the network mean, gama_U and gama_L would be the values of γ\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:

DV

The Davies bound of p-values for sup test.

supLM

The value of the sup test statistic in the sample yy.

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

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

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

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 pp (PNAR(pp)) versus the non-linear Smooth Transition alternative (ST-PNAR(pp)).

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

supLM

The optimized value of the test statistic. See the function global_optimise_LM_stnarpq.

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+q2p + 1 + q, where qq denotes the number of covariates.

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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 pp).

Z

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

J

The number of bootstrap samples to draw.

gama_L

The lower value of the nuisance parameter γ\gamma 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 nuisance parameter γ\gamma to consider. Use NULL if there is not information about its value. See the details for default computation.

tol

Tolerance level for the optimizer.

ncores

Number of cores to use for parallel computing. By default the number of cores is set to 1 (no parallel computing). Note: If for some reason the parallel does not work then load the doParallel package yourseleves.

seed

To replicate the results use a seed for the generator, an integer number.

Details

The function computes a bootstrap p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order pp (PNAR(pp)) versus the following Smooth Transition alternative (ST-PNAR(pp)). For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0+h=1p(β1hXi,th+β2hYi,th+αheγXi,td2Xi,th)+l=1qδlZi,l,\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence λi,t\lambda_{i,t} is the expectation of Yi,tY_{i,t}, conditional to its past values.

The null hypothesis of the test is defined as H0:α1=...=αp=0H_{0}: \alpha_{1}=...=\alpha_{p}=0, versus the alternative that at least one among αh\alpha_{h} is not 00. The test statistic has the form

LM(γ)=S(θ^,γ)Σ1(θ^,γ)S(θ^,γ)LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma)

where

S(θ^,γ)=t=1TTi=1N(Yi,tλi,t(θ^,γ)1)λi,t(θ^,γ)αS(\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 α=(α1,...,αp)\alpha=(\alpha_{1},...,\alpha_{p}), evaluated at the estimated parameters θ^\hat{\theta} under the null assumption H0H_{0} (linear model), and Σ(θ^,γ)\Sigma(\hat{\theta},\gamma) is the variance of S(θ^,γ)S(\hat{\theta},\gamma).

Since the test statistic depends on an unknown nuisance parameter (γ\gamma), the supremum of the statistic is considered in the test, supγLM(γ)\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 supγLM(γ)\sup_{\gamma}LM(\gamma) by employing Gaussian perturbations of the score S(θ^,γ)S(\hat{\theta},\gamma). For details see Armillotta and Fokianos (2022b, Sec. 5).

The values of gama_L and gama_U are computed internally as gama_L =log(0.9)/X2=-\log(0.9)/X^{2} and gama_U =log(0.1)/X2=-\log(0.1)/X^{2}, where XX is the overall mean of Xi,tX_{i,t} over the nodes i=1,...,Ni=1,...,N and times t=1,...,TTt=1,...,TT. Since the non-linear function eγXi,td2e^{-\gamma X_{i,t-d}^{2}} ranges between 0 and 1, by considering XX to be a representative value for the network mean, gama_U and gama_L would be the values of γ\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:

pJ

The bootstrap p-value of the sup test.

cpJ

The adjusted version of bootstrap p-value of the sup test.

gamaj

The optimal values of the γ\gamma parameter for score test boostrap replications.

supLMj

The values of perturbed test statistic at the optimum point gamaj.

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

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 pp (PNAR(pp)) versus the non-linear Threshold alternative (T-PNAR(pp)).

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

supLM

The optimized value of the test statistic. See the function global_optimise_LM_tnarpq.

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+q2p + 1 + q, where qq denotes the number of covariates.

y

A TT×NTT \times N time series object or a TT×NTT \times N numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The N×NN \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 pp).

Z

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

J

The number of bootstrap samples to draw.

gama_L

The lower value of the nuisance parameter γ\gamma 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 nuisance parameter γ\gamma to consider. Use NULL if there is not information about its value. See the details for default computation.

tol

Tolerance level for the optimizer.

ncores

Number of cores to use for parallel computing. By default the number of cores is set to 1 (no parallel computing). Note: If for some reason the parallel does not work then load the doParallel package yourseleves.

seed

To replicate the results use a seed for the generator, an integer number.

Details

The function computes a bootstrap p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order pp (PNAR(pp)) versus the following Threshold alternative (T-PNAR(pp)). For each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT

λi,t=β0+h=1p[β1hXi,th+β2hYi,th+(α0+α1hXi,th+α2hYi,th)I(Xi,tdγ)]+l=1qδlZi,l\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 Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence λi,t\lambda_{i,t} is the expectation of Yi,tY_{i,t}, conditional to its past values.

The null hypothesis of the test is defined as H0:α0=α11=...=α2p=0H_{0}: \alpha_{0}=\alpha_{11}=...=\alpha_{2p}=0, versus the alternative that at least one among αs,h\alpha_{s,h} is not 00, for s=0,1,2s=0,1,2. The test statistic has the form

LM(γ)=S(θ^,γ)Σ1(θ^,γ)S(θ^,γ)LM(\gamma)=S^{'}(\hat{\theta},\gamma)\Sigma^{-1}(\hat{\theta},\gamma)S(\hat{\theta},\gamma)

where

S(θ^,γ)=t=1TTi=1N(Yi,tλi,t(θ^,γ)1)λi,t(θ^,γ)αS(\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 α=(α0,...,α2p)\alpha=(\alpha_{0},...,\alpha_{2p}), evaluated at the estimated parameters θ^\hat{\theta} under the null assumption H0H_{0} (linear model), and Σ(θ^,γ)\Sigma(\hat{\theta},\gamma) is the variance of S(θ^,γ)S(\hat{\theta},\gamma).

Since the test statistic depends on an unknown nuisance parameter (γ\gamma), the supremum of the statistic is considered in the test, supγLM(γ)\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 supγLM(γ)\sup_{\gamma}LM(\gamma) by employing Gaussian perturbations of the score S(θ^,γ)S(\hat{\theta},\gamma). For details see Armillotta and Fokianos (2022b, Sec. 5).

The values of gama_L and gama_U are computed internally as the mean over i=1,...,Ni=1,...,N of 20%20\% and 80%80\% quantiles of the empirical distribution of the network mean Xi,tX_{i,t} for t=1,...,TTt=1,...,TT. In this way the optimization is performed for values of γ\gamma such that the indicator function I(Xi,tdγ)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).

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:

pJ

The bootstrap p-value of the sup test.

cpJ

The adjusted version of bootstrap p-value of the sup test.

gamaj

The optimal values of the γ\gamma parameter for score test boostrap replications.

supLMj

The values of perturbed test statistic at the optimum point gamaj.

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

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(pp) 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

object

An object containing the results of the function score_test_stnarpq_DV.

x

An object containing the results of the function score_test_stnarpq_DV.

...

Extra arguments the user can pass.

Details

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

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

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

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 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

object

An object containing the results of the function score_test_nonlinpq_h0.

x

An object containing the results of the function score_test_nonlinpq_h0.

...

Extra arguments the user can pass.

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. 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

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)

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

object

An object containing the results of the estimation function lin_estimnarpq or log_lin_estimnarpq.

x

An object containing the results of the estimation function lin_estimnarpq or log_lin_estimnarpq.

...

Extra arguments the user can pass.

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. 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)
mod1
print(mod1)
summary(mod1)