For WP1, directly at the start of the project, I discussed with the supervisors the set-up of the IMARS project’s planning and ambitions, from realizing currently ongoing research to the planning of new research to the preparation of the job market. Deliverable D1 in the proposal has then been achieved.
For WP2 I developed a working paper investigating the properties of Tucker decomposition techniques in a tensor setting as a step towards developing a integer matrix autoregressive score-driven model. Contrarily to the original proposal, I was able to extend the modelling framework to integer tensors of any arbitrary order and not only for 2-order tensors (matrix time series) leading to a more general and powerful tensor formulation which is able to accommodate datasets of multidimensional arrays. Therefore deliverable D2 has been fulfilled and developed beyond the original proposal.
Given that the reduced form of a score-driven model can be written as an autoregressive moving average type model, I implemented a tensor count autoregressive model where the joint dependence between counts is described by copula functions. Finally, I have specified a suitable two-stage maximum likelihood estimator (2SMLE) that allows to estimate all the unknown parameters of the model in two separate optimization steps thereby reducing the computational complexity of the estimation task. This marks the achievement of milestone M2 of the proposal.
A complete theoretical treatment of the properties of such model together with an application to crime time series resulted in a publication.
For WP3 I proposed possible solutions to the parameters proliferation in the newly specified tensor count autoregression. For the general tensor model, I described in publication 1 from section 1.2.6 below suitable copula specifications and approximations of numerical optimizations which I showed to be effective in handling the curse of dimensionality problem in several simulation studies. Moreover, I have also specified a common temporal decay mechanism that allows to divide in half the number of parameters to be estimated. This method has been successfully applied in the crime data application (see section 1.2.4 below). Furthermore, several other dimension reduction techniques have been studied for the special case of 2-tensor data with spatiotemporal network structure (see section 1.2.5 below). These results define milestone M3.
Finally, I studied the asymptotic theory of the 2SMLE for integer-valued tensor models leading to new theoretical results in the literature (publication 1, section 1.2.6). This satisfies deliverable D3.
For WP4 I applied the copula tensor count autoregression developed in the previous work packages to an empirical application with a 2-order tensor of monthly number of police reports for different types of theft-related crimes in different cities of New South Wales, Australia (milestone M4). The tensor approach allowed to simultaneously model multiple temporal spillovers between different crimes in the same cities, same crimes in different cities and mixed effect of different crimes in different cities, all in one tensor modelling framework. Moreover, I showed that the tensor methodology was able to enhance both point and density forecasts of the crime time series with respect to a baseline approach which employs the same amount of parameters. This provides deliverable D4.
This part of the project was unified with the previous methodological and theoretical findings resulting in publication 1 from Section 1.2.6 below.
For WP5 it was clear that the planned cooperation with the SWOV would have been not implementable due to the shorter horizon of the project. Nevertheless, I have developed a set of other applications that I believe to be at least of equal interest of the application originally planned. For the special case of 2-tensor data with network structure (called network time series) I developed several methods of dimension reduction using spatial data as weighting matrices of coefficients. These methodologies have been applied to dynamic networks of hotel markets in Venice (publication 3 from section 1.2.6 below) and endemic-epidemic networks for influenza cases in Germany (publication 2, section 1.2.6). For the latter case an R package performing the analysis has been developed as well. These results substitute deliverable D5 and milestone M5 of the proposal.
Moreover, while investigating suitable estimation techniques for count time series I also obtained two new estimating methods enabling improved efficiency with respect to quasi-likelihood methods resulting in two publications listed as output of this search (publication 4 and 5, section 1.2.6). These methods are relevant as building blocks for future investigations on improved estimators for tensor time series.
