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information theory

  A mathematical approach in communication science which attempts to measure the amount of information or the degree of organization in a system. The theory and its associated methods have been used to describe settlement and population distributions in geographical space. Its mathematical formulation exhibits a close relationship to the mathematics of entropy in statistical thermodynamics.

The basic equation is due to Shannon (Shannon and Weaver, 1949):

{img src=show_image.php?name=bkhumgeofm16.gif }

Where {img src=show_image.php?name=bkhumgeofm40.gif } and H is the information statistic.

The individual xis might therefore be the probabilities of N possible outcomes of a stochastic experiment, proportions of a population in N census tracts, proportion of land in N counties, etc. It can be shown, by both example and mathematical proof, that H approaches zero as one of the xi approaches unity (statistically as one of the outcomes approaches a near certainty). On the other hand if all the xi are approximately equal (at 1ãN) then H approaches a maximum given by log N.

In information studies these results are used to define an index R:

{img src=show_image.php?name=bkhumgeofm15.gif }

which is variously termed a measure of redundancy (Shannon and Weaver, 1949) and order (von Foerster). Ambiguities in interpretation arise in geography because a highly concentrated spatial pattern (H = 0, R = 1) and a uniform spatial pattern (H = Hmax R = 0) both suggest order, albeit of quite different kinds.

Marchand (1972) pointed out that in cartography a choropleth map conveys maximum information if each of the class intervals has approximately the same number of mapping units, i.e. H is maximized. He also shows that the maximum amount of information carried by such a map is controlled by the number of mapping units and the number of classes utilized — however much raw data may have been used in the compilation. A similar approach uses the H statistic to partition within group from between group information as an aid to classification. (AMH)

References and Suggested Reading Marchand, B. 1972: Information theory and geography. Geographical Analysis 4: 234-57. Shannon, C.E. and Weaver, W. 1949: The mathematical theory of communication. Campaign, IL: University of Illinois Press. Thomas, R.W. 1981: Information statistics in geography. Concepts and techniques in modern geography 31. Norwich: Geo Books.



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Other Terms : possibilism | tragedy of the commons | Alonso model
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