Neural Network Modelling of Constrained Spatial Interaction Flows

Fundamental to regional science is the subject of spatial interaction. GeoComputation - a new research paradigm that represents the convergence of the disciplines of computer science, geographic information science, mathematics and statistics - has brought many scholars back to spatial interaction modeling. Neural spatial interaction modeling represents a clear break with traditional methods used for explicating spatial interaction. Neural spatial interaction models are termed neural in the sense that they are based on neurocomputing. They are clearly related to conventional unconstrained spatial interaction models of the gravity type, and under commonly met conditions they can be understood as a special class of general feedforward neural network models with a single hidden layer and sigmoidal transfer functions (Fischer 1998). These models have been used to model journey-to-work flows and telecommunications traffic (Fischer and Gopal 1994, Openshaw 1993). They appear to provide superior levels of performance when compared with unconstrained conventional models. In many practical situations, however, we have - in addition to the spatial interaction data itself - some information about various accounting constraints on the predicted flows. In principle, there are two ways to incorporate accounting constraints in neural spatial interaction modeling. The required constraint properties can be built into the post-processing stage, or they can be built directly into the model structure. While the first way is relatively straightforward, it suffers from the disadvantage of being inefficient. It will also result in a model which does not inherently respect the constraints. Thus we follow the second way. In this paper we present a novel class of neural spatial interaction models that incorporate origin-specific constraints into the model structure using product units rather than summation units at the hidden layer and softmax output units at the output layer. Product unit neural networks are powerful because of their ability to handle higher order combinations of inputs. But parameter estimation by standard techniques such as the gradient descent technique may be difficult. The performance of this novel class of spatial interaction models will be demonstrated by using the Austrian interregional traffic data and the conventional singly constrained spatial interaction model of the gravity type as benchmark. References Fischer M M (1998) Computational neural networks: A new paradigm for spatial analysis Environment and Planning A 30 (10): 1873-1891 Fischer M M, Gopal S (1994) Artificial neural networks: A new approach to modelling interregional telecommunciation flows, Journal of Regional Science 34(4): 503-527 Openshaw S (1993) Modelling spatial interaction using a neural net. In Fischer MM, Nijkamp P (eds) Geographical information systems, spatial modelling, and policy evaluation, pp. 147-164. Springer, Berlin

A methodology for neural spatial interaction modelling

This paper presents a methodology for neural spatial interaction modelling. Particular emphasis is laid on design, estimation and performance issues in both cases, unconstrained and singly constrained spatial interaction. Families of classical neural network models, but also less classical ones such as product unit neural network models are considered. Some novel classes of product unit and summation unit models are presented for the case of origin or destination constrained spatial interaction flows. The models are based on a modular connectionist architecture that may be viewed as a linked collection of functionally independent neural modules with identical feedforward topologies, operating under supervised learning algorithms. Parameter estimation is viewed as Maximum Likelihood (ML) learning. The nonconvex nature of the loss function makes the Alopex procedure, a global search procedure, an attractive and appropriate optimising scheme for ML learning. A benchmark comparison against the classical gravity models illustrates the superiority of both, the unconstrained and the origin constrained, neural network model versions in terms of generalization performance measured by Kullback and Leibler`s information criterion. Hereby, the authors make use of the bootstrapping pairs approach to overcome the largely neglected problem of sensitivity to the specific splitting of the data into training, internal validation and testing data sets, and to get a better statistical picture of prediction variability of the models. Keywords: Neural spatial interaction models, origin constrained or destination constrained spatial interaction, product unit network, Alopex procedure, boostrapping, benchmark performance tests.

Total factor productivity effects of interregional knowledge spillovers in manufacturing industries across Europe

The objective of this study is to identify knowledge spillovers that spread across regions in Europe and vary in magnitude for different industries. The study uses a panel of 203 NUTS-2 regions covering the 15 pre-2004 EU-member-states to estimate the impact over the period 1998-2003, and distinguish between five major industries. The study implements a fixed effects panel data regression model with spatial autocorrelation to estimate effects using patent applications as a measure of R&D output to capture the contribution of R&D (direct and spilled-over) to regional productivity at the industry level. The results suggest that interregional knowledge spillovers and their productivity effects are to a substantial degree geographically localised and this finding is consistent with the localisation hypothesis of knowledge spillovers. There is a substantial amount of heterogeneity across industries with evidence that two industries (electronics, and chemical industries) produce interregional knowledge spillovers that have positive and highly significant productivity effects. The study, moreover, confirms the importance of spatial autoregressive disturbance in the fixed effects model for measuring the TFP impact of interregional knowledge spillovers at the industry level.Total factor productivity, manufacturing industries, knowledge spillovers,patents, European regions, spatial econometrics

Evaluating Neural Spatial Interaction Modelling by Bootstrapping

This paper exposes problems of the commonly used technique of splitting the available data in neural spatial interaction modelling into training, validation, and test sets that are held fixed and warns about drawing too strong conclusions from such static splits. Using a bootstrapping procedure, we compare the uncertainty in the solution stemming from the data splitting with model specific uncertainties such as parameter initialization. Utilizing the Austrian interregional telecommunication traffic data and the differential evolution method for solving the parameter estimation task for a fixed topology of the network model [ i.e. J = 9] this paper illustrates that the variation due to different resamplings is significantly larger than the variation due to different parameter initializations. This result implies that it is important to not over-interpret a model, estimated on one specific static split of the data. (authors' abstract)Series: Discussion Papers of the Institute for Economic Geography and GIScienc

Cross region knowledge spillovers and total factor productivity. European evidence using a spatial panel data model

This paper concentrates on the central link between productivity and knowledge capital, and shifts attention from firms and industries to regions. The objective is to measure knowledge elasticity effects within a regional Cobb- Douglas production function framework, with an emphasis on knowledge spillovers. The analysis uses a panel of 203 European regions to estimate the effects over the period 1997-2002. The dependent variable is total factor productivity (TFP). We use a region-level relative TFP index as an approximation to the true TFP measure. This index describes how efficiently each region transforms physical capital and labour into outputs. The explanatory variables are internal and out-of-region stocks of knowledge, the latter capturing the contribution of interregional knowledge spillovers. We use patents to measure knowledge capital. Patent stocks are constructed such that patents applied at the European Patent Office in one year add to the stock in the following and then depreciate throughout the patents effective life according to a rate of knowledge obsolescence. A random effects panel data spatial error model is advocated and implemented for analyzing the productivity effects. The findings provide a fairly remarkable confirmation of the role of knowledge capital contributing to productivity differences among regions, and adding an important dimension to the discussion, showing that knowledge spillover effects increase with geographic proximity

Knowledge spillovers and total factor productivity. Evidence using a spatial panel data model

This paper investigates the impact of knowledge capital stocks on total factor productivity through the lens of the knowledge capital model proposed by Griliches (1979), augmented with a spatially discounted cross-region knowledge spillover pool variable. The objective is to shift attention from firms and industries to regions and to estimate the impact of cross-region knowledge spillovers on total factor productivity (TFP) in Europe. The dependent variable is the region-level TFP, measured in terms of the superlative TFP index suggested by Caves, Christensen and Diewert (1982). This index describes how efficiently each region transforms physical capital and labour into output. The explanatory variables are internal and out-of-region stocks of knowledge, the latter capturing the contribution of cross-region knowledge spillovers. We construct patent stocks to proxy regional knowledge capital stocks for N=203 regions over the 1997- 2002 time period. In estimating the effects we implement a spatial panel data model that controls for the spatial autocorrelation due to neighbouring regions and the individual heterogeneity across regions. The findings provide a fairly remarkable confirmation of the role of knowledge capital contributing to productivity differences among regions, and add an important spatial dimension to the discussion, by showing that productivity effects of knowledge spillovers increase with geographic proximity

A Methodology for Neural Spatial Interaction Modeling

This paper attempts to develop a mathematically rigid and unified framework for neural spatial interaction modeling. Families of classical neural network models, but also less classical ones such as product unit neural network ones are considered for the cases of unconstrained and singly constrained spatial interaction flows. Current practice appears to suffer from least squares and normality assumptions that ignore the true integer nature of the flows and approximate a discrete-valued process by an almost certainly misrepresentative continuous distribution. To overcome this deficiency we suggest a more suitable estimation approach, maximum likelihood estimation under more realistic distributional assumptions of Poisson processes, and utilize a global search procedure, called Alopex, to solve the maximum likelihood estimation problem. To identify the transition from underfitting to overfitting we split the data into training, internal validation and test sets. The bootstrapping pairs approach with replacement is adopted to combine the purity of data splitting with the power of a resampling procedure to overcome the generally neglected issue of fixed data splitting and the problem of scarce data. In addition, the approach has power to provide a better statistical picture of the prediction variability, Finally, a benchmark comparison against the classical gravity models illustrates the superiority of both, the unconstrained and the origin constrained neural network model versions in terms of generalization performance measured by Kullback and Leibler’s information criterion