Spatial Mathematics: Theory and Practice through Mapping


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Math Lesson

The major difficulty which arises with the classical theory of inferential statistics is that it often appears to be primarily inductive. Its major interpretation is a 'frequency' view in which probabilities are associated with events arising from experiments which can be repeated : hence, probability is interpreted as relative frequency over a large number of repetitions.

Given certain properties of the experiment lack of bias, random sampling, etc. It is necessary, therefore, in this approach, to specify the population distribution from which given samples experiments are drawn. This in turn requires substantial assumptions about the underlying generating process which usually cannot be upheld. The major problem with classical. Inferential Statistics. The emphasis on the classical theory deriving from Neyman, Pearson and Fisher has been tenacious in Anglo-American geography and only recently have alternative approaches been explored.

The major alternatives are 1 a more ad hoc approach to data analysis following procedures outlined by Tukey as so-called exploratory data analysis, and 2 Bayesian approaches. It is this second approach which is argued here to hold the greatest future benefits for the subject see Wilson and Bennett, Bayesian theory argues that observed relative frequencies cannot be interpreted directly as specifying the population distribution, and hence the extent to which a particular outcome is plausible, instead, a statement is required as to the degree of belief we have in a given confidence statement.

This, in practice, requires that Bayesian methods specify a degree of belief as a prior probability distribution of possible states or outcomes. This prior degree of belief is then modified when an actual observation or experiment is made; the resulting change in degree of belief then defines a posterior probability distribution which then may be used as a prior for a subsequent experiment. It is argued by Bayesians that the Neyman-Pearson theory does not allow real inference about hypotheses, rather it is a solution to a decision problem relative to certain criteria.

In contrast, it is argued, Bayesian methods are genuine inferences which admit genuine confidence intervals and more truly reflect the manner in which knowledge, especially statistical knowledge, is obtained about phenomena and the underlying processes which are assumed to govern them.

The methodological issues which are raised by the distinction between Bayesian and Neyman- Pearson theory underpins a major problem of geographical theory : that the spatial sample or pattern observed is usually the whole universe or population of information which is available.

This creates some considerable methodological difficulties for the 'classical' approach. The major problem with the primarily emphasis on statistical inference and Neyman-Pearson theory, however, has been that this has opened up Anglo-American quantitative geography to attack as positivist science.

Thus Gregory has argued that geography had attained by the s much of the structure outlined by Comte as positivism, i. It is his largely valid contention of the extremes of this quantitative tradition in Britain and America that quantitative geography was alienating and distracting to researchers : first, it creates a false sense of objectivity by attempting to remove the observer from the observed, which in turn lends an ability to control and manipulate society; second, its use of computers, machines and techniques. Hence, the primary use of Neyman-Pearson theory has left the whole of quantitative geography vulnerable to methodological attack, whilst alternative statistical approaches which do not suffer from these problems have been ignored.

This paper has developed the arguments stated by Bennett that there needs to be a much greater methodological development of the core of geography. The view stated is that this core must draw from both French and German traditions of geography and link these with Anglo-American spatial science : to establish a social view of spatial science. In achieving this the role of spatial science itself must be reappraised and with it the role of statistical method. This suggests the following conclusions : first, much better understanding of spatial process is required, rather than map pattern, second techniques must be improved to differentiate map pattern; spatial process and spatial structure; third, a reestablishment of the subject matter of geography is needed which includes a social and humanistic approach to spatial situations including locational, environmental and governmental influences; and fourth, the statistical methods used must be extended by exercising extreme care in statistical inference supplemented by exploratory and Bayesian methods of approach.

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Bennktt R. London, Methuen. London, Pion. Cambridge, Cambridge University Press.

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London, Routledge and Kegan Paul. A reappraisal of the role of spatial science and statistical inference in geography in Britain [article] R. Plan 1. Introduction [link] 2. Spatial Science [link] 3. Inferential Statistics [link] 4. Conclusions [link] Bibliographie [link]. Introduction Since the s the ascendancy of the Franco- German regional method and traditions in Geography have become increasingly eclipsed in Britain by the influence of North American research.

Bennett Buttimer, Spatial Science Spatial science was central to much of the quantitative and theoretical geography developed in the Anglo-American tradition up to the s. Spatial pattern is now usually interpreted not in 2. Spatial science and statistical inference 25 terms of geometric or topological structures, but instead as one realization of an underlying generating process.

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