global_optimise_LM_stnarpq {PNAR}

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

Description

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

Usage

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

Arguments

gama_L

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

gama_U

The upper value of the \gamma values to consider. Use NULL if there is not information about its value. See the details for default computation.

len

The number of increments to consider for the \gamma parameter.

b

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

y

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

W

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

p

The number of lags in the model.

d

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

Z

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

tol

Tolerance level for the optimizer.

Details

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

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

where X_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node i connections, with the weights of the mean being W_{ij}, the single element of the network matrix W. The sequence \lambda_{i,t} is the expectation of Y_{i,t}, conditional to its past values.

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

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

where

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

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

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

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

Value

A list including:

gama

The optimum value of the \gamma parameter.

supLM

The value of the objective function at the optimum.

int

A vector with the extremes points of sub-intervals.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

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

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

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

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

See Also

score_test_stnarpq_j, global_optimise_LM_tnarpq, score_test_tnarpq_j

Examples

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