Galton looks at movement as requiring integration of theories of time, space, objects and position. The combined theory, which he refers to as a theory of dominance is presented as a tool for investigating the structure of qualitative space-time in an underlying continuous space. Elmasri and Lee cover relational and object-oriented temporal data storage. Their temporal approach extends that of Galton with concepts of temporal anchorage and determinacy.

This is a necessary extension for any research into interrelationships, but the technique of these authors presumes that the interrelationships are already understood and modelled. It presumes a versioning approach to TGIS, using time-stamping. The authors claim that their work is extensible to many temporal dimensions. Renolen, devises a complete set-based theory of temporal objects, which he asserts can handle multiple temporal dimensions, although current publications are restricted to two spatial ones.

Erwig et al propose a less-general approach using the concept of abstract data types ADT for temporal objects. Spatio-temporal objects are a special case of temporal objects. For either, objects could be embedded directly as attributes referencing the relevant ADT type.

The spatial alternative is to record co-ordinates as conventional data attributes. For temporal data, the alternative is to store a time stamp and a flag indicating the future behaviour. The review notes that causal logics are nonmonotonic subsequent information can change the truth of existing information. The frame and qualification problems arise in relation to large sets of propositions. However, much of the work referenced in this section is system development rather than data analysis work.

Examples relate to conventional and spatial databases, constraint databases and to time-series data derived from spatial as well as temporal sources. They describe how these can be used to derive geometric representations.

This approach introduces the unit-of-time sequence UTS to represent an observation of a phenomenon over a unit of time. The UTS is a quadruple of an identifier, some spatial or aspatial variable value, a time unit, and a behavioural function B a known law or one derived from the shape of an interpolated curve. The behavioural function models the entity's behaviour throughout the UTS time unit. This powerful approach allows the functions to take many forms punctual for events, stepped, linear or spline - the latter is also suggested by Erwig et al and by Halls and Miller , , whose version was partially implemented by Burbidge At the join between UTS, functions may be continuous representing continuity of observed behaviour , or discontinuous.

Discontinuities, referred to as accidents, equate to knots in Halls and Miller , , and in Peuquet et al , These can be visualised as plotted curves, in which a knot or accident is a point at which the smoothness of the curve is interrupted, representing an altered rate of change noting that the scale of measurement or representation can affect the visual identification of knots. The approach is viable if it is known that the observations relate to a herd.

However, if the data were observations of the location of a single cow or even a part of the herd , it would not be possible to determine whether the observed phenomena represented one cow in different locations over time, a herd in one place over time, or a moving herd. It is fundamental to the model presented that the observed cow represents a herd of determined size. When working with historical data it is often not possible to determine how the observed sample represents the whole, and, indeed, potential relationships may be the subject of the research.

Many curves might be interpolated from the same data points, representing different levels of certainty in the position of these data points.

Halls and Miller , , and the current authors, are interested in precisely this problem, of how to work with imprecise data and underspecified relationships. The representation is polyhedra, which permits the use of a unifed manipulation algebra. Spatial dimensionality is expressed as cartographic 2-D representation and primitives. The Chernikova algorithm Chernikova is used to define the intersection, union and difference of the polyhedra, but is limited to convex representations.

However, this extended model allows spatial phenomena to merge and divide, extending the expressive power of the earlier models. A prototype implementation, GeoLis, is described This is a mathematically elegant but complex approach. The mapping between the polyhedra operators and the spatio-temporal questions of what happened here then, when and where did that change, or Adid anything else change around that time and place is not clear.

The approach is similar to that of Galton or Elmasri and Lee , above. Halls and Miller, however, assume incomplete and inexact data. From the data, possible states of the model at any past time can be interpolated, using splines and recording the functions or equations for these, after the manner of Yeh and de Cambray Rather than considering UTS sequences, Halls and Miller derive a single spline curve for a prolonged period referred to as a worm based on the temporal nodes todes.

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This is analogous to the timeline concept of Peuquet et al A single curve might model a number of different equations, and display the same discontinuity representations as occur between UTS. However, the continuous spline can be manipulated, using the associated data on the relative precision of each tode. Todes are considered to exert a 'pull' upon the curve, with the strength of the pull related to the subjective precision assigned to the recorded attribute value. Statistical descriptors can be used to describe the precision of the spline fit. Although some knots represent lack of precision in the todes before or after the discontinuity, others suggest the presence of temporally-related events that could be of interest to the researcher.

## Geographic information systems and spatial analysis

He added representation of the temporal value a date, for example and precision of values. This simple approach allowed McKay to correctly identify known points in time relating to the management of areas of bracken. This point was found to correlate with management activities on an adjacent plot of heather burning. Subsequent work has demonstrated that this is a previously unrecognised influence in bracken ecology Scott, , personal communication.

The technique correctly identified the time points at which rainfall and river flow rates approached those critical for the onset of flooding.

## Geographic information systems and spatial analysis | SpringerLink

The project explored data smoothing to remove the less significant knots and concluded that automated discrimination was needed. In particular, neural networks are gaining favour for pattern recognition and classification applications Fitzgerald and Lees, , O. Keefe and Austin, Neural networks have the ability to handle fuzziness, especially in the implementation form adopted for the AURA architecture Zhou et al, The AURA architecture handles partial match database queries very well and has also been used to implement statistical classifiers in a fast and robust way.

This section presents an analysis of the spatio-temporal geometry implicit in Halls and Miller's approach to todes and worms, and outlines a means of addressing this requirement by application of the AURA neural network techniques. The data should be represented by a model that expresses the temporal geometry of the process in an appropriate way.

The model should describe the evolution of the process over time for any arbitrary point in time. The points of change in the current variable are identified by the todes. Similarly, todes in other variables represent events that may have some influence on the behaviour of the current variable.

The body of data must therefore be queried to find the related variable or variables that might influence the current variable. The data is queried by looking for correlations between the sets of todes for the current and related variables. Variables that have the same geometry may have a relationship to the current variable. That is, if two variables have the same temporal pattern of todes, it indicates the possibility that either one variable is influencing the other, directly or indirectly, or that both variables are influenced by a third unobserved variable.

The influence of one variable on another may not be instantaneous, so allowance must be made for temporal displacements between the geometric objects of different variables. The measurement of the point in time corresponding to any particular todes may be subject to noise, adding an element of uncertainty to the temporal location of todes. This allows for the possibility of missing data B the particular sub-sequences including the missing point will not be matched, but the remainder will.

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As an example, consider a sequence of todes e and intervals between them t figure 1. In temporal-geometric terms, the sub-sequence is a point in a 2-dimensional space spanned by the intervals t 1 ,t 2 figure 2.

This set of interval pairs is represented by a set of points in the 2-dimensional space figure 3. The data for the whole collection of variables is represented and stored in this manner. Note that by storing sub-sequences of events, the ordering of the whole sequence is not preserved. That is, the information that sub-sequence a precedes sub-sequence b is not stored. The set of points for the current variable thus becomes the query with which to extract matching variables from the data. The space in which the sub-sequences of data are embedded is referred to as the query space.

If the data for each variable were stored as a set of intervals between one point and the next, the probability is high that another variable will have some similar intervals between events. Querying the database with such a general query would not produce much useful information. As the length of the sub-sequence is increased, more structure is included in each point comprising the query set and the precision of the query increases.

In the example cited above, the data are represented as pairs of successive intervals, giving a 2-dimensional query space. Triples of successive intervals would give a 3-dimensional query space, and so on. However, longer sequences are less likely to match exactly because of the possibility of missing todes, and the uncertainty in the measurement of intervals between todes. The CMM operates by storing relationship between inputs and outputs, and retrieving outputs when presented with an input.

The input may be inexact or noisy, in which case the CMM will retrieve the closest matching output from those it has been trained with. By suitable coding of the inputs, a set of queries may be presented together to the CMM, and the output will be the superimposition of the corresponding outputs, with some element of noise added. AURA systems and their predecessors have been used in a number of processing tasks, including image processing and molecular matching Austin, , Lees, et al, , Austin, , O'Keefe, and Austin The AURA system can store and access very large numbers of associations of the order of millions of associations.

Retrieval time is dependent on the size of the database stored, but can be accelerated using hardware developed at York Kennedy and Austin, Inputs and outputs are in the form of vectors. Each element of the input vector is an input node, and each element of the output vector is an output node. The network is fully connected, in that the elements of the matrix represent the weights from each input node to each output node. The weights are binary, that is they represent the presence or absence of each particular connection. The input and output vector are binary, that is each element of the vector is represented by a single bit.

For most applications, some form of mapping is required to transform the inputs and outputs into vectors having this form. The vector v k itself has integer elements, the value of each element representing the strength of activation of that element.

### 1st Edition

The threshold process sets to zero elements that are not strongly activated, removing noise from the output to leave a pattern of elements representing the required output. The thresholding process may apply either a fixed or an adaptive threshold, depending on the properties required of the memory. Fixed thresholds are used where the properties of the input string are known and an exact match is required. If an approximate or best match is required, an adaptive threshold is used. In practice, a number of superimposed binary codes will be generated by the recall and threshold process.