Hengxu Liu’s contribution to the Discussion of ‘New tools for network time series with an application to COVID-19 hospitalisations’ by Nason et al.

Congratulations to the authors of the discussion paper.Their paper provides new tools for interpretable and effective modeling of network time series.A challenge of multivariate autoregression is the scaling with the number of variates.Numerous approaches have been proposed to circumvent this issue as the authors and others have discussed.An example is the Generalized Network Autoregressive (GNAR) model (Knight et al. (2016), Knight et al. (2020), Zhu et al. (2017)).The GNAR model is essentially a multivariate AR with restrictions on the transition matrices defined by the network.In this way, the number of parameters is significantly reduced.The GNAR model and its variants have shown great performance in various data fields.A limitation of the GNAR model is its linearity assumption, which might not be suitable for some data.In some cases, we can use a link function such as in GLM to transform the linearity.However, this is not always available, especially when the underlying data mechanism is unknown.We propose a more universal nonlinear version of GNAR, that uses P-splines (Eilers and Marx (1996)) and boosting to enhance the GNAR model.P-splines (Eilers and Marx (1996)) provide a flexible and interpretable way to model nonlinear relationships.Boosting is a common ensemble technique used in regression or classification to improve performance by combining multiple weak learners into a stronger learner.(Shafik and Tutz (2009)) propose a L 2 Boosting procedure with P-spline learners, which they use to fit univariate nonlinear additive autoregressive processes.(Taieb and Hyndman (2014)) use a similar boosting strategy, but focus on multi-step forecasts instead.Our proposed boosting strategy is similar to