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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
S=CR^(-α)
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.
R=BS^(-α)
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

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

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