Spatial Uncertainty

This paper presents a generic high dimensional model representation (HDMR) method for approximating the system response in terms of functions of lower dimensions. The proposed approach, which has been previously applied for problems dealing only with random variables, is extended in this paper for problems in which physical properties exhibit spatial random variation and may be modelled as random fields. The formulation of the extended HDMR is similar to the spectral stochastic finite element method in the sense that both of them utilize Karhunen–Loève expansion to represent the input, and lower-order expansion to represent the output. The method involves lower dimensional HDMR approximation of the system response, response surface generation of HDMR component functions, and Monte Carlo simulation. Each of the low order terms in HDMR is sub-dimensional, but they are not necessarily translating to low degree polynomials. It is an efficient formulation of the system response, if higher-order variable correlations are weak, allowing the physical model to be captured by the first few lower-order terms. Once the approximate form of the system response is defined, the failure probability can be obtained by statistical simulation. The proposed approach decouples the finite element computations and stochastic computations, and consecutively the finite element code can be treated as a black box, as in the case of a commercial software. Numerical examples are used to illustrate the features of the extended HDMR and to compare its performance with full scale simulation.

Geographic information systems have great potential as a tool for studying and teaching
historical geography. However, using traditional GIS data models, even spatio-temporal forms,
has been difficult due to the prevalence of uncertainty—both ambiguity and fuzziness—in source
information concerning space, time, and theme. Explicitly uncertain assertions of a geo-historical
datum can be modeled as an Evidentiary Set, a hybrid of a fuzzy set with probability and Dempster-
Shafer evidence theory. This set formalism is designed to represent continuous and discrete value
ambiguity (e.g., “about 10”), and fuzzy membership (e.g., “somewhat Central European”), including
ambiguous membership and other fuzzy-ambiguous combinations. The formal set structure can be
stored in GIS by representing continuous variation with a patch model, producing logical models for
object-oriented and relational GIS databases. The relational implementation was tested in two GIS
databases focused on human historical geography, showing the potential for the model to represent
explicit datum-level uncertainty in a wide variety of GIS applications.

Multiple criteria decision making usually means selection of the best objects or their parameters which best meet conditions or criteria. Human decision making often involves also uncertain and vague criteria. The terms as a short distance, a high building or long time are commonly used in human speech. Nevertheless, it is not necessary to describe them exactly. In case of decision making based on lot of data and multiple criteria, data are usually stored in databases and a query language is used for handling and querying them. In this case, the criteria should be defined in a formal computer language. The standard for querying data in relational databases is the Structured Query Language (SQL). All objects and phenomena, and everything people do, is experience in space and time. Space and time, therefore, should be used as a framework for querying and reasoning about information stored in database systems. Databases, in which spatial and temporal data types are included and functions for their handling are supported, are called the spatio-temporal databases. To spatio-temporal query a database, an extension of standard SQL to support spatial and temporal data is needed. Uncertain spatio-temporal queries are not yet standardized and they are a current topic of research. For uncertain queries creation and expression of uncertain decision criteria, development of new methods and techniques is required. One of the most widely used approach to model, analyse and process uncertain and vague data is fuzzy set theory. Therefore, in this paper, we propose a new way of application fuzzy set theory to spatio-temporal querying. For the case study realisation, the open source PostgreSQL database system extended by the PostGIS we have used. The implementation of the proposed principles of fuzzy set theory to spatio-temporal querying databases can bring an opportunity for the efficient decision making based on multiple uncertain criteria.