Page 43 - IJPS-8-2

International Journal of
Population Studies Used versus Offered densities of human population

too high are less sensitive to extreme values, they are taken 2.3.2. Relationship between the statistical moments of
as “robust statistics” (Wonnacott & Wonnacott, 1990). the two distributions
Thus, for non-negative real variables, the ratios between From (14) stems a generic relation between the statistical
corresponding pairs F O 1 () , F O 1 1 ( ) constitute moments of the two distributions (Kleinrock, 1975): At
robust indicators of heterogeneity in the distribution: any order r, it holds that
These include the inter-quartile ratio at level α = 1/4 as well 1
r
as the inter-decile ratio at level α = 1/10. E []x x E x r1 (15)
O
U
For instance, a log-normal distribution LN (m,s ) O
2
yields the following interquartile ratio at order α This relation corresponds to the zone-based formula
(cf. appendix C), with Φ (−1) the inverse CDF of the reduced in Stairs (1977). At order r = 1, the average density as
Gaussian variable: experienced by the users satisfies that

1
F 1 s 2 1 1 x 1 E O x 2 x 1 ( 2 O ) (16a)

O
O
U
F e x O
1

O
It is equivalent to a ratio of the average densities (RADs):
2.3. Human density as lived by the statistical
population of inhabitants RAD UO | x U 2 O (16b)
1
Any individual u inhabits a zone z and inside it a land unit x O
u
o(u). One defines the “used density” or “lived density” as Both relations correspond to formulas established by
the human density in the land unit inhabited by the user: Lewontin & Levins (1989) using zones as statistical units
x ≡ x o( u) (12) and Ottensmann (2018a) using land units. If the human
u
density is homogenous in the territory, then γ = 0 and the
O
Among the population of users, of size UP= Z , the average human densities according to either statistical
user-centric density is a random variable, with specific population are equal. However, the larger the heterogeneity
PDF and CDF denoted, respectively, as f and F . (as measured by the relative dispersion), the higher the
U
U
x of used to offered average human densities.
ratio x /
U
O
2.3.1. Relationship between f o and f U Figure 1 illustrates both the used and offered PDFs. Its
The PDF f of used human density is related to that f of assumptions are the following: That x is distributed log-
O
U
O
offered human density by a “consumption model” well- normal with parameters m = 3.24 (mean of ln(x )) and
o
O
known in probabilistic theory, especially in renewal theory s = 1.76 (standard deviation of ln(x )), stemming from
O
o
(Kleinrock, 1975): France-like conditions x =120 p/km² and 46. . The
O
O
related x is log-normal, too, with parameters m = 6.34
f x)
f x.( (13) and s =s , hence x = 2 661, p/km² and γ = 4.6, too. In
x
u
U
U
O
U
O
U
U
The reason is that the land units of which the human that particular instance, the ratio of average human

density belongs to xx, x , in proportion f (x).δx in densities, x / x , amounts to 22 – indeed a very large

O
U
O
their distribution, do contain x.a users each: Hence, their value.
1
total number of users amounts to a .. ().x f x δ O . x . These
O
1
users are those with user-centric density in xx, x 2.3.3. Indicators of used human density heterogeneity

and those users only: Thus, their number is also The indicators of heterogeneity recalled in §2.2.3 for the
f . x. U . On combining both formulas, as Oa. 1 = A random variable of offered density also apply to that of
x
U
Z
and UP= Z , it comes out that used density. Its variance stems from moments of the
offered density in a specific way: From (15) at order r=2,
A
x
x
f Z x. f 1
O
U
3
2
P Z E x x E x
O

U
Which implies (13) with proportionality coefficient of O
/
A / P =1 x . To sum up, Combining with (16a), it comes out that
O
Z
Z
2
1 1 E x
O
)
f x O x.( (14) V []x x O E x ( x O 2 (17)
x
3
f x)
O

U
U
O
Volume 8 Issue 2 (2022) 37 https://doi.org/10.36922/ijps.v8i2.297