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Reply to Carsten Haaland's Comment

Haaland makes three major points about my paper. I will respond to them in the order in which he treats them. His first point concerns my choice of functions. He argues that the exponential function is an appropriate one for population density. However, I was not studying population density. Had I been doing so, the use of the negative exponential function would have been clearly appropriate. Then the constant in the exponent would have been treated as a Fourier series. Equivalently, I could have treated the logarithm of density as a linear function of distance, with the slope being a Fourier series in direction. Since I was not studying density but percentages of various kinds defined on areas, there was no reason to expect an exponential function or any other. So I decided to use the simple linear model when it was appropriate and to use a second-degree quation when the relationship of a characteristic to distance had an inflection point within the range of the data. Haaland takes me to task for not choosing a function for distance with the same flexibility as that for direction. In response I would point out that I did not use the full generality of the Fourier series but limited it to three terms generating a function with seven parameters. This number is, of course, greater than the three parameters in the second-degree equation expressing the effect of direction, but it is considerably less than the full flexibility of the Fourier series allows. I pointed out that this was the approximate quivalent of using seven sectors and that the use of this number of sectors conforms roughly to what investigators have usually done in using the discrete sector model. I further pointed out that the choice of the limits on each of these functions was made partly because higher limits did not add much to the explanation. I was not trying to catch every nuance of the data by providing a maximally flexible model. Instead, I was trying to capture its regularity with a parsimonious model. Haaland's second major point is that my restriction to a single center of dominance, the city center, was not reasonable. When he argues that my diagram shows two centers of dominance, he is misinterpreting what I said and what is generally understood by the term "center of dominance." It is not used to mean a place where a characteristic attains a relative maximum or minimum. Instead, it means a point with a use such that it exerts a causal influence on the organization and distribution of population characteristics. The model has the actual city center as the only point of dominance in that all characteristics are predicted by reference to their location with respect to the city center. Were I to expand the model to consider a second point of dominance, I would specify that point prior