A Multidimensional Model for Exploratory Spatiotemporal Analysis

Zarine Kemp and Howard Lee
Computing Laboratory
University of Kent
Canterbury, Kent CT2 7RY
E-mail: Z.Kemp@ukc.ac.uk

Abstract

Geographic phenomena often exhibit different characteristics depending on the scale of the observations. Hierarchical reasoning enables the understanding of scale and categorization effects in data analysis. This research focuses on the computational support that is required for reasoning about data at various levels and at multiple dimensions. The model proposed within this framework, enables researchers to obtain insights into information in data repositories by enabling access to a wide range of views of the data along dimensions relevant to the application domain. The framework is characterized by its focus on the multidimensional data cube as the logical model for spatiotemporal analysis. This logical data structure supports functionality that is crucial to exploratory analysis such as calculations and modeling across dimensions, through hierarchies, over temporal intervals and derivation of relevant subsets of the data. Data subsets are extracted by flexible operations for ‘slicing’ and ‘dicing’ through the multidimensional cube, ‘roll-up’ and ‘drill-down’ to enable aggregation at required levels of consolidation and ‘pivoting’ to view the data from different perspectives. The research challenges inherent in the analysis of spatiotemporal data have also been recognized in other application domains such as scientific and statistical databases where similar requirements arise for advanced classification structures, dynamic hierarchies and the need for dimension reduction of data through high level abstractions.

1. Introduction

Geographic phenomena are characterized predominantly by their spatial and temporal characteristics but also by related properties, often referred to in the geographic information systems (GIS) literature as aspatial attributes. Due to the unique characteristics of space and time, capturing geographic phenomena in spatiotemporal databases has required special representations. Consequently, much attention has focused on object-based versus field-based models and related vector/raster representations. Similarly, research effort has also concentrated on the evolutionary, temporal aspects of these phenomena and various space-time composite models have been proposed. However, it has been clear for some time that complex phenomena such as plankton blooms or oil spills need to be represented by heterogeneous, multi-source data comprising diverse spatial and temporal representations as well as procedures to model their dynamic behaviours.

A parallel development in geographic information science has been the recognition of the central role played by hierarchy theory. Hierarchy theory has implications for spatiotemporal information systems and analyses from a variety of perspectives: as an organizing principle for the understanding of complex systems (Pattee 1973, Simon 1973), to explain the differences that exist at different levels of analysis (Ahl and Allen 1996) and in the cognition of space and spatial information (Golledge and Montello 1998).

Current trends in GIS and spatiotemporal information systems have been moving towards the integration of flexible data representation and manipulation capabilities of database management systems and the spatial analysis functionality of GIS (Kafatos 1999, Kineman 1993, Kucera 1999). This paper highlights the following consequences of this trend:

it has become easier to explore the inherently multidimensional aspects of geographic phenomena using computational tools

it is now possible to represent the hierarchical nature of spatiotemporal phenomena and to consider them at different levels of abstraction depending on the requirements of the analysis.

The rest of the paper is organized as follows. Section 2 explores the relevance of hierarchy theory for the representation and modeling of spatiotemporal analysis. Section 3 presents an overview of the system design, section 4 describes a small motivating example, section 5 discusses the concepts underlying multidimensional analysis and the architecture of the framework to enable hierarchical reasoning in multiple dimensions and section 6 concludes the paper.

2. Hierarchy theory and spatiotemporal systems

We discuss the relevance of hierarchy theory from two perspectives. First, from the ‘holistic’ point of view considering complex systems at a high level of abstraction which can be decomposed into smaller components and also from the point of view of individual attributes or variables in multivariate data representation where each attribute can be considered at different levels depending on the requirements of the application.

2.1 Hierarchies in complex systems

It has been argued by several researchers (Pattee 1973, Simon 1973) that complex systems can be considered to consist of simpler subcomponents at a level further down the scale and so on, forming a partial ordering which can be conceptually viewed as a hierachical tree at several levels of complexity. These properties are particularly evident in biological and physical systems where geospatial entities occur frequently. Extending the notion of hierarchies to computer systems, Simon notes that hierarchical natural systems are analogous to hierarchical computer systems where the hierarchies enable separation of the high frequency dynamics of components at lower levels from the low frequency interactions of components at the higher levels. These vertical couplings thus enable stable subassemblies to be treated as ‘givens’, only their equilibrium properties affecting system behaviour at the higher levels of abstraction. Horizontal couplings are also relevant when considering complex systems as they enable components to operate dynamically and independently of other subcomponents and reflect the larger aspects of systems as a whole. Hierarchical control levels are established by particular kinds of constraints that represent, not simply a structure, but a classification of the details of the lower level resulting in simplification of collective dynamics.

These ideas have been further elaborated by Ahl and Allen (1996) proceeding along similar lines of thought. They contend that the dynamics of complex systems such as environmental and ecological ones can be explained by considering the different hierarchies that pertain within and between the entities that comprise the system. Depending on the problem being considered, a single material system may require a different level of explanation for each scale and type of measurement. Hierarchy theory encompasses both the role of the observer and the process of observation in scientific discourse thus accommodating multiple perspectives.

As far as measurement and observation protocols are concerned, it is important to differentiate between related concepts of grain and extent, which determine the spatial and temporal limits of the observations. The grain or resolution determines what entities are accessible to the observation process. On the other hand, the extent determines the span of the sampling process in the spatial and temporal dimensions. The data collection process thus involves competition between detail and scope. These notions are intimately linked to the more familiar notion of scale in geographic information science; both resolution and extent determine scale and scale may be spatial and/or temporal. In the temporal dimension scale is equivalent to the frequency of behaviours of the entities being considered. Examples in the next subsection illustrate the relevance of these concepts to geographic phenomena in environmental systems.

According to Ahl and Allen, entities may be definitional, those that are postulated before measurements are made, or empirical, those that are observed and measured as part of the observation process. Definitional entities are scale and rate-independent but may be modified by the empirical entities discovered through the observation process. Furthermore, as Pattee states, the notion of categories of entities are aspects of the researcher building interpretive models of the problem domain. Gray (1997) argues that the problem of classification in geography is fraught with problems and suggests a disaggregate approach to dealing with complex categories. From the computational point of view, these notions map into common hierarchical data representations. Hirtle (1995) suggests that trees, ordered trees and semi-lattices may be used to represent the cognitive aspects of spatial knowledge.

The notion of interaction between levels is of particular interest in hierarchy theory. The multiple levels may be structurally and functionally ordered into nested or non-nested hierarchies. Both types of hierarchies exhibit characteristics that relate upper levels of the hierarchy to those at the lower level; the upper levels behave at lower frequencies, provide context and thereby constrain lower levels in the hierarchy. In addition, nested hierarchies exhibit the additional characteristic of containment or inclusion. Thus nested systems are determinable from knowledge of their components. Nested systems occur naturally in the spatial dimension (Car 1998, Timpf and Frank 1997). In addition, Timpf (1999) describes generation of different types of hierarchies depending on the abstraction mechanism used. Of these, the hierarchy generated by generalization/specialization abstractions is a nested one. It should be noted that hierarchies are powerful exploratory devices for reasoning about the structure and processes that characterize geographic phenomena (section 2.2).

A commonly used heuristic for ordering hierarchical levels is according to the spatial and temporal characteristic frequencies of the empirical entities. Alternatively, levels can be conceptualized as information flows or filters. Filters attenuate, delay or integrate input signals between levels. In other words, filters inform categories (equivalent to categorization mapping functions as discussed in section 5). A natural counterpart to filters is that of surfaces. Filters identify successive levels, the hierarchical structure of which is reflected in a set of surfaces. Thus, the structure of (nested) hierarchies is represented by a set of surfaces surrounding more local surfaces within. For example, Varma (1999) proposes building multiple, hierarchically

related representations of spatial objects using HHCodes. These multidimensional codes enable fusion and interrelation of spatially referenced data using interlocking cells that are generated by aggregation techniques to form surfaces (or higher dimension structures).

2.2 Spatiotemporal hierarchies in environmental analysis

Forging links between scale and structure is central to hierarchy theory. Capturing the set of relationships between the whole, its parts below and the concepts above is the goal of hierarchical analysis. These general concepts embodied in hierarchical theory are extremely relevant to geographic phenomena that arise in environmental research. Marine environmental applications provide several examples of phenomena that occur in relatively discrete regions of space and time (Lucas 2000). It has been suggested (Johnson 1996) that ecological modeling typically requires three spatiotemporal scales, giving rise to a corresponding three level hierarchy: focal, at which the phenomenon of interest is observed, a higher level which defines the constraints and a lower level which encompasses the mechanism that generates the phenomenon being studied. In fact, the examples of plankton biomass abundance suggest that many more spatiotemporal scales are relevant (along the space-time continuum) to adequately represent all related phenomena. Diel vertical migration of plankton needs to be considered at 24-hour temporal intervals and over several kilometres of the spatial dimension. Consideration of the effects of eddies and upwellings range over relatively localised spatial extents of tens to hundreds of kilometres and temporal intervals spanning weeks to a year. The dynamics of events such as El Nino may require temporal scales of several years to decades and cover large areas of the marine environment and the study of very long term effects such as those of global warming and melting polar ice caps span global spatial scales and temporal scales spanning centuries. In other words, functional dynamics that define the phenomena determine appropriate space-time scales. Thomas Dickey (1992) suggests that biomass recruitment in large marine ecosystems is affected by a broad range (upto 9 orders of magnitude) of time and space scales of various oceanic parameters.

Another geoscientific domain that is very dependent on hierarchical representation and analysis is that of biodiversity research (Stoms and Estes 1993, Stoms et al 1996). Biological surveys and monitoring of species distributions reveal distribution patterns that vary as a function of space-time scale. Both inductive and deductive models have been derived to predict the spatial locations of specific species and the environmental attributes of those locations. Observed data sets are sensitive to, and therefore have to be analyzed with reference to the extent, grain and sampling intensity of the spatial and temporal dimensions. A major challenge is to be able to generalize sparse observation data collected at fine spatiotemporal resolutions.

3. Overview of system design

The discussion in the previous section postulates the requirement for a framework that enables the capture and exploration of how scale and categorization affect spatiotemporal analysis.

Figure 1. Hierarchical reasoning: scale and categorization affect analysis

Figure 1 expresses a high level conceptual view of the rationale underlying the framework. The system enables the exploratory analysis of geographic phenomena using spatiotemporally referenced observational data sets. The framework:

provides mechanisms that encapsulate the multilevel characteristics of hierarchy theory as identified in section 2

enables integration of the spatial, temporal and attribute dimensions of the observational data set

allows categorization hierarchies to be specified for required dimensions

seamlessly integrates aspatial properties with the dominant space and time dimensions

captures the notion of definitional and empirical entities

provides mechanisms to aggregate through scales by the provision of appropriate mapping functions.

4. Motivating example

A simplified version of a database for monitoring fishing activities is used as an example scenario. Figure 2 describes the conceptual model using UML notation.

Figure 2. Conceptual Model of fisheries observational data

The main attributes of interest are catch and discards, which are organised according to their respective tows/trawls, time, marine species, vessel, gear and port of origin. Tow trajectories are recorded by periodically sampling a GPS. A general-purpose data repository such as this one is used by various groups with different objectives. The phenomena of interest include analyses of ‘catch surfaces’ by species, temporal range and so on. Measures of interest in the multidimensional problem space include aggregates, counts, averages, or CPUE (catch per unit effort) which may be calculated using a variety of attributes and formulae. Variations in the temporal dimension may be used to elicit trends in total catch or individual species or any combination of categories of species, changes in the spatial distribution of areas fished and level of activity by port of departure. It may also be useful for fisheries scientists to investigate the average length/time per tow/trawl by species, time of year, fishing area and so on. It should be noted that the spatial attributes may be in vector format (coastlines, trawl trajectories etc.) or in gridded format to represent surfaces of required multidimensional measures.

As noted in section 2.2 the spatiotemporal scales may vary and be composed of hierarchies of dimensions. For example, the gridded surfaces may be aggregated at varying cell resolutions perhaps depending on the distance of the fishing area from the shore (on the assumption that fishing activity is more intense in inshore rather than offshore areas). Alternatively, areas of interest may be designated fishing areas or identified spawning grounds, represented in vector format.

Likewise, the temporal dimension may be represented by a points in time, temporal intervals, sequencesof temporal intervals or functionally determined temporal intervals (for example the temporal designation ‘spring’ could be interpreted using a function with spatial arguments). The point to note is that as far as the multidimensional view of data is concerned, the space and time attributes are represented by identifiers or nominal index values. The aggregation and disaggregation operations on these attributes require special spatiotemporal mapping functions and cannot be aggregated using standard statistical/numerical operators. This is illustrated in section 5 below.

4. Multidimensional analysis

4.1 Basic concepts

Recently, there has been much interest in the database research community on modeling and querying multidimensional databases (Agrawal 1996, Chaudhuri 1997, Codd 1993, Harinarayan 1996, Ho 1997, Zhao 1997). On-line analytical processing (OLAP) and the requirements of large statistical and scientific databases have been converging and one of the most influential developments in dimensional analysis is the CUBE operator (Gray et al 1997). A variant of this operator has been included in the recent SQL3/SQL1999 standard. Although the operator was originally designated for the relational database, the concept is easily and perhaps more naturally realized as a multidimensional array (MDA). The CUBE operator provides a concise syntax for specifying a set of aggregate operations between two or more attributes. A detailed exposition of multidimensional concepts is beyond the scope of this paper but we focus on a few basic concepts.

A cube is an n-dimensional construct arranged by the dimensions of the data. It represents a logical view of multidimensional data.

A multidimensional data array is characterized by two types of attributes - parameters and measures. The set of numeric values (stored in the cells of the multidimensional array) are the measures. They represent the extensional objects of the analysis.

A dimension consists of one or more attributes. Each of the measures depends on a set of dimensions which provide the context for the measure. Thus a measure is a value in the multidimensional space of dimensions.

The attributes of a dimension may be related via a hierarchy of relationships and thus represent different levels of aggregation of the same dimension. The relationships between the different levels are defined by a mapping function. The mapping between a nested hierarchy of species may be realized by a simple aggregate function; the catches of individual pelagic species can be simply accumulated or averaged to give an aggregate total of all_pelagic) whereas the aggregation of individual fishing_areas into regions would require a complex spatial mapping such as the resel-based model suggested by Tobler (1995). In the context of spatial hierarchies, space in the form of gridded nested hierarchies (section 2.1) are much easier to instantiate than non-nested grids or vector representations. However, the point to note is that this is only of relevance when considering the performance of hierarchies; at the conceptual level, the underlying representation is transparent and does not affect the design of the framework. Figure 3 illustrates an example of a 3-dimensional MDA and the hierarchical categorization pertaining to the space dimension.

Figure 3. Example of an MDA and a dimension hierarchy

4.2 Hierarchies in the framework

Hierarchical reasoning is regarded as spatiotemporal analysis with an awareness of how scale and categorisation can reveal/affect the perceived patterns and characteristics underlying a given data set. That in fact, all analytical deductions are relative to some category and scale, explicit or implicit. In the analytical framework, hierarchies are used for knowledge representation. Dimensions and associated hierarchies are equivalent to the definitional entities in hierarchy theory and the classification or categories of these entities at various levels of analysis. The framework considers a dimension hierarchy as a collection of dnodes and edges. Examples of dnodes include ‘month’ = {January, February, etc.}, and ‘quarter’ = {Q1, Q2, Q3, Q4}. The edge between ‘month’ and ‘quarter’ define a partition mapping function where ‘Q1’ ® ‘January, February, March, April’; ‘Q2’ ® ‘May, June, July, August; and so on. All dimensions participating in a CUBE operation may be perceived (by default) as a hierarchy consisting of the base data and the ‘ALL’ aggregate.

A dimension hierarchy H is a partial ordering of dnodes, in increasing coarseness of scale, organised according to one or more paths:

The term V or u is used to denote the node or vertex of a hierarchy. Each dnode v is also associated with a domain of elements. A dimension path is a linear totally ordered list of dnodes, All the occurrences of a dnode in a single path must be unique, although a dnode may partake roles in several different paths. Finally, every adjacent vertex pair in a path denoted by the precedent symbol is also associated with a partition mapping function:

The partition mapping function is a special case that defines the mapping of the dnode with the finest granularity in each path to the query data set domain(s). An example can help clarify the definition above: time is defined as a partial ordering of dnodes V = {day, week, month, quarter, year}. This contains two dimension paths: = (day month quarter year) and = (dayweek). The domain of elements for ‘month’ is ‘January’, ‘February’, etc. The mapping categorizes data record timestamps according to daily intervals. Once a dimension hierarchy is defined, it is meant to be reusable in different queries and contexts. This helps provide a common frame of reference when different client applications refer to a particular aggregation scale. Note that a dimension hierarchy is not complete.

The partition mapping, or categorization function is defined as:

where, is the categorization mapping function, are the parent and child dnodes respectively, i.e.. For example, ‘month’ is the parent of ‘quarter’. Child nodes derive aggregates from their parents, so child nodes are coarser in scale. Note that m is a two-way mapping, thus enabling operations to traverse up or down a hierarchy.

A categorization function m maps each and every element from the domain of to one or more elements in the domain of . Although m is defined as a bidirectional mapping between the two dnodes, it is more accurate to distinguish between them. Therefore, is taken to mean the mapping from child to parent, while m implies the parent to child mapping . That m has two definitions should not cause any ambiguity since their connotations are context dependent, and the correct application should be obvious — in any case, the explicit definition can be specified. Categorization functions reflect a one-to-many relationship between the parent and child domains with respect to a given path. This means that each element in the child is associated with one or more parent cells (aggregation), and each parent cell is associated with one child cell. Parent dnodes may partake in several paths, each with its own relationships. The mappings are also complete, in the sense that every element is associated with at least one element in the other domain.

5. System architecture

Data consolidation or summarization refers to the process of summarizing information by aggregating the data. From a computational perspective, this process is independent of whether a hierarchical categorization is meaningful or not. Similarly, it is irrelevant whether data is stored in a relational database or a simple text file. The theme of this research involves developing a comprehensive computational framework to facilitate hierarchical reasoning comprising a set of techniques and tools that enable properties of data categorization and scale-oriented processes to be quantified. The conceptual model and analytical operations generally make up an algebra or query language. However, many queries involving complex hierarchical aggregations are often non-trivial. Therefore, it is useful to view query processing as a flow of data. That can be branched for parallel processing, or transformed to suitable formats for visualization or other applications.

Figure 4 depicts the high level architecture of the system which is composed of four layers.

Figure 4. System architecture

Physical organization: This layer is concerned with implementation issues such as data structures and organization.

Conceptual model: The conceptual model provides an abstract view of the underlying database.

Analytical operations: This layer is concerned with the flexible data management operations that are available to the researcher during the analytical process.

Process management: Queries/operations are treated as a sequence of data flows, which facilitate retrieval optimization and information system interoperability.

5.1 Taxonomy of operators

As noted earlier, in the framework, data is modelled multidimensionally, and often conceptualised as a hypercube where each dimension relates to one or more attributes. During the analytical process, each dimension must either be a parameter or measure: parameter dimensions define the array indexing to the hypercube. e.g. with three dimensions, any data record could be accessed by mdb[i, j, k]; and measure dimensions are the data that is actually stored in the multidimensional array, which is the return value of querying the database; e.g. mdb[i, j, k] ® val, where val is the attribute measure.

It is useful to think of an operator as a function op(arg1, arg2,...,). Since operators tend to be either unary or binary, they are utilised in more familiar forms: op arg1 (e.g. -12) or arg1 op arg2 (e.g. 1 + 1). In the context of the framework, arguments are typed, that is all the arguments must belong to some known domain. Thus, an operator can be classified according to the type of its operands: base array, MDA, and lattice. The base array is equivalent to the array constructs typically found in imperative programming languages such as C/C++, Java and Pascal. There are of course a huge number of possible operations that are useful in different applications, as well as particular classes of base arrays (e.g. 1-dimensional, 2-dimensional arrays). This leads to the introduction of a key aspect of the multidimensional data model: matrix operations. MDA operators are more closely related to database oriented operations. Some operations are closely affiliated with those in relational databases, whilst others are specific to the multidimensional database.

Common operations include:

OLAP operators roll up/drill down, slice/dice and pivot.
Roll up (increasing the level of aggregation) and drill down along one or more of the parameter hierarchies.
Slicing (selection, i.e. extracting a subset of the database by range and constraints) and dicing (projection, i.e. extracting only the relevant parameters).
Pivoting. Switching parameters to measures, and vice-versa.
Restructuring operators allows parameters to become measures and measures to become parameters: fold and unfold.
Querying operators MDA to be sliced, diced, classified and aggregated: selection, projection, categorization, aggregation. There is also the CUBE and HCUBE, which are an extension of the categorization and aggregation operators.
Joining operators allows data to be combined in a variety of ways from two arrays: cartesian product, join.
Set-theoretic operators: union, difference and intersection.

In the more theoretical database references, many of these operators are described as a calculus or algebra, which are then implemented in an operational form as the more familiar SQL query:

SELECT (projection) (dimension hierarchy) (aggregation)
FROM
WHERE (selection)
GROUP BY (categorisation) CUBE ... HAVING (selection)

Each of these blocks return an MDA, and implies that if all these operators are well-defined then an MDA query language is at least as expressive as standard SQL.

6. Conclusions

The framework described in this paper is predicated on the fact that the concepts embodied in hierarchy theory are extremely relevant for analysis in the spatiotemporal domain. The design of the spatiotemporal framework owes much to recent developments in OLAP and database research. These ideas have been used to realize a system that encompasses the structural and behavioural concepts embodied in hierarchy theory to provide a flexible exploratory environment for research. As the system is intended to reason about geographic phenomena, the space and time dimensions play a central role. Unlike aspatial attributes that are easier to categorise, these two properties are continua and are consequently less easy to classify into hierarchies. The implementation of operations on hierarchies relating to these two dimensions require special spatial functions and more work needs to be done to consider the performance implications for GIS that enable hierarchical spatiotemporal reasoning. Further developments will also provide the user with the ability to specify more flexible hierarchies and the ability to modify the hierarchy structure in various ways. Another major requirement is to integrate visualization functionality into the interface of the framework to enable easier user interaction.

Acknowledgements

Howard Lee was partly funded by an E.B. Spratt bursary awarded by the Computing Laboratory, University of Kent.

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