2.2 Underpinnings of Zipf’s law
To understand Zipf’s law as a test of spatial segmentation, its theoretical and empirical underpinnings are first to be discussed.
Zipf’s law is defined as
or in its logarithmic transformation: log(S)=log(C)-α∙log(R), where S means size, R is rank and C is a constant. It is expected that α ≈ 1. The graph of the logarithmic form is then a straight line with a gradient of -1. If α (-- removed HTML --) 1 the larger cities are larger than predicted and/or the smaller are smaller than predicted; the distribution is then stretched. It is also usual to work with the reversed form, i.e. the Pareto distribution. Then α behaves contrary to the above described pattern.
Here rank is regressed on size. B is a constant. The probability Pr(X≥x)=〖BS〗^(-α) is then equivalent to the number of cities X larger than city x. The probability density of city sizes is derived by R=αBS^(-α-1) and indicates the number of cities with a probability X=x. In case of a perfect rank-size distribution in accordance to Zipf, the exponent is in each case the same (both -1) just as the constants are identical B=C, while deviations from the expected exponent tend into the opposite direction, just depending on the distribution form selected.
The derivation of Zipf’s law for cities…show more content… It is interesting that the mean estimate for natural cities is (-- removed HTML --) -1 suggesting that for large cities the natural size is larger than Zipf’s law would predict. A possible explanation could be that by estimation of the Zipf (or Pareto) function on administrative population, cities may appear too small, while estimating the function on luminosity with a possible major blooming effect (darker areas outside a city illuminated by over-glow from light sources in the city) may define cities too