weights_matrix {GNAR}

Computes the weights matrix corresponding to the GNAR network object linked to the vector time series.

Description

Computes the weights matrix with normalised weights (i.e., add up to one) for the network time series with underlying network provided by the user. If the network is unweighted, then each r-stage neighbour is considered equally relevant, i.e., w_{ij} = \{ \mathcal{N}_r (i)\}^{-1} \mathbb{I} (d(i, j) = r) , where \mathbb{I} is the indicator function and the distance is the shortest path in the underlying network.

Usage

weights_matrix(network, max_r_stage)

Arguments

network

Network linked to the vector time series under study, must be a GNARnet object.

max_r_stage

Longest shortest path for which weights are non-zero. If not specified, then its set equal to the upper bound, which is the longest shortest path in the underlying network.

Value

Weight matrix \mathbf{W}, each entry is the weight w_{ij} between a pair of nodes. The matrix is not symmetric, and each row adds up to one when considering r-stage neighbours for a particular r.

Author(s)

Daniel Salnikov and Guy Nason.

References

Nason, G.P., Salnikov, D. and Cortina-Borja, M. (2023) New tools for network time series with an application to COVID-19 hospitalisations. https://arxiv.org/abs/2312.00530

Examples

#
# Weights matrix linked to the mechanical ventilation beds time series. 
# This network has a longest shortest path equal to six.
# 
#data(fiveNet)
W_norm = weights_matrix(fiveNet, 6)