Statistical Inference_Quadrat_Analysis

Statistical Inference_Quadrat_Analysis - Andrei Rogers and...

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Andrei Rogers and Norbert G. Gomar Statistical inference in Quadrat Analysis The growing recognition of the need for establishing a systematic and quantitative means for describing and analyzing, the spatial dispersion of activities in urban areas has generated an interest in the potential applicability of cell-counting or quadrat methods in urban analysis [6,8,16]. These exploratory efforts to apply quadrat methods in analyses of urban spatial structure have justifiably emphasized probability theory and its use in model formulation. This paper strives to complement such efforts by focusing on the other, equally important, half of quadrat analysis: statistical inference and its use in model verification. In quadrat analysis, we may distinguish between efforts which focus on the development of stochastic models of point distributions and efforts which deal with the problem of inference regarding the model that may be said to “account” for an observed point distribu- tion. Efforts of the first kind address questions such as: given that 150 points are randomly placed inside a grid of 100 squares, what is the expected number of unoccupied squares, squares containing one point, and so on? Efforts of the second kind consider questions such given that in a grid of 100 squares, superimposed over a point pattern, 35 squares are empty, 31) squares contain a single point, and so on, what is the likelihood that the stochastic model that generated this point distribution was a random spatial process? This paper deals with the latter class of problems and, in particular, focuses on the use of the chi-square goodness of fit test to measure the adequacy of fit, to an observed frequency distribution, that is provided by alternative quadrat models of point dispersion. Let XI ,x2 , - - . , xN denote independent observations on a random variable which has an unknown distribution function F(x). The problem of testing whether F(x) = Fo(z), where Fo(x) is some par- Andrei Rogers is associate professm of city planning with the Depart- ment City and Regional Planning, at the University of California at Berkeley. Nmbert G. Gomar is associated with the Institute of Trans- portation and TraB Engineering at the University California at Berkeley.
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Rogers and Gomar 37 7 ticular distribution function, is called a goodness of fit problem, and any test of the null hypothesis: a goodness of fit test. Hypotheses of fit, like parametric hypotheses, may be divided into two classes: simple and composite. In (l), HO is a simple hypothesis if F&) is completely specified. Otherwise a composite hypothesis. For example, the hypothe& that the N observations of a random sample have been generated by a Poisson spatial process with mean equal to m a simple hypothesis. The hypothesis that the N ob- servations were produced by a Poisson spatial process whose param- eter is unknown, however, is a composite hypothesis. Tests of com- posite hypotheses, therefore, require estimates of the parameters of Fo(z).
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.

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Statistical Inference_Quadrat_Analysis - Andrei Rogers and...

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