Identification of the local productive systems in Spain: a new approach

Paper submitted to the 44th European Congress of the European Regional Science Association, Porto, 25-29 August 2004.

Motzkin decomposition of closed convex sets via truncation

A nonempty set F is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set C with a closed convex cone D. In that case, the sets C and D are called compact and conic components of F. This paper provides new characterizations of the Motzkin decomposable sets involving truncations of F (i.e., intersections of FF with closed halfspaces), when F contains no lines, and truncations of the intersection F̂ of F with the orthogonal complement of the lineality of F, otherwise. In particular, it is shown that a nonempty closed convex set F is Motzkin decomposable if and only if there exists a hyperplane H parallel to the lineality of F such that one of the truncations of F̂ induced by H is compact whereas the other one is a union of closed halflines emanating from H. Thus, any Motzkin decomposable set F can be expressed as F=C+D, where the compact component C is a truncation of F̂. These Motzkin decompositions are said to be of type T when F contains no lines, i.e., when C is a truncation of F. The minimality of this type of decompositions is also discussed.

Delineating zones to increase geographical detail in individual response data files: An application to the Spanish 2011 Census of population

Due to confidentiality considerations, the microdata available from the 2011 Spanish Census have been codified at a provincial (NUTS 3) level except when the municipal (LAU 2) population exceeds 20,000 inhabitants (a requirement that is met by less than 5% of all municipalities). For the remainder of the municipalities within a given province, information is only provided for their classification in wide population intervals. These limitations, hampering territorially-focused socio-economic analyses, and more specifically, those related to the labour market, are observed in many other countries. This article proposes and demonstrates an automatic procedure aimed at delineating a set of areas that meet such population requirements and that may be used to re-codify the geographic reference in these cases, thereby increasing the territorial detail at which individual information is available. The method aggregates municipalities into clusters based on the optimisation of a relevant objective function subject to a number of statistical constraints, and is implemented using evolutionary computation techniques. Clusters are defined to fit outer boundaries at the level of labour market areas.