# inequality / inequality / inequality-1.3.3

Released by inequality on 2004-08-01 at 17:16.

filesize
inequality-1.3.3.zip17.5 KB
ineqIndexSpreadsheets.zip680.6 KB
```Notes:
--- General remarks ---
*  Some changes for experimenters implemented.
*  Packet contains sources and a packet for experimenting
(ineqIndesSpreadsheets.zip).
*  The programs are written in Lua and in Python. This could
be interesting for comparison e.g. of class implementations.

--- Platon inequality ---
Statements:
*  "Platon" inequality: ZP
*  "Kullback-Leibler" redundancy (or distance): KLR
*  E[i] is e.g. total income in a quantile[i]
*  A[i] is e.g. the population in a quantile[i]
*  sumE = sum(E[i]), e.g. income of a society
*  sumA = sum(A[i]), e.g. population of a society
*  i is the index of the quantiles (containing A[i] and E[i])
in a set of quantiles which e.g. describes the resource
distribution in a society.
*  KLR = sum(ln(sumE/sumA)*(E[i]/sumE-A[i]/sumA)/2
*  "Kullback-Leibler" inequality: KLI = 1 - exp(-KLR)
*  "Pareto" equality: q
Ranges:
*  ZP and KLR are relative inequalities between 0 and 1
*  q is a relative equality between 0 and 0.5
*  As q is between 0 and 0.5 and indicates equality, the transformation
ZP = 1 - 2*q is between 0 and 1 and indicates inequality.
*  KLR is a redundancy >= 0
(redundancy = max.entropy - entropy)
*  pi = 3.1415....
Definition of Platon inequality:
*  KLI = 1 - (2/(ZP+1)-1)^(ZP).
Approximation (error < 0.002 below ZP=0.78) yields inverse function:
*  ZP = 1 - arcsin((1-KLI)^(KLI*0.06+0.61))*2/pi
(The program reduces the error to <0.00001)
Meaning (if q = (ZP+1)/2 with q between 0 and 0.5):
*  A society in which
a share of q people hold a share of 1-q of the resources and
a share of 1-q people hold a share of q of the resources
is entropywise equivalent to a society with
KLI = 1 - (1/q-1)^(2*q-1).

Changes:
Release Name: inequality-1.3.3 and 1.3.2
Changes (from 1.3.1 and 1.3.2 implemented in 1.3.3):
In extreme cases the Platon inequality was not exact enough.
In 1.3.3 I improved the computation of this inequality index.
I discovered that during playing with random data inputs to
the program (histo.zip).
For real life 1.3.1 and 1.3.2 are fine. But I just like to
prepare the program in a way, that it also works in experiments
with completely weird data.
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Release Name: inequality-1.3.1
Notes:
Formulas for most (not all) of the coefficients are here:
(1) http://poorcity.richcity.org/maugis.gif
That was in the web a few years ago. In Maugis' excellent list
you find two inequality measures from Atkinson.
Out of these two, I am interested only in the entropy measure.
(2) http://poorcity.richcity.org/entkiss.htm
Some names I still have to change.
I developped the entropy coefficients myself in 1995
and later found, that others did that already earlier:-).
(2a) The "Demand" coefficient is Atkinson's entropy measure,
which again is 1-exp(Theil_redundancy(E)).
(2b) The "D&R" coefficient is 1-exp(-Kullback_Leibler_redundancy).
Now I modestly call it Kullback_Leibler_inequality.
(2c) The Kullback_Leibler_redundancy is
(Theil_redundancy(A)+Theil_redundancy(E))/2.
In the literature It usually is called "Kullback Leibler Distance".
(As for A and E, here an example: A is the population of a quantile
and E is the sum of the incomes in that quantile.)
(2d) I use the term "redundancy" in the sense of ISO/IEC DIS 2382-16.
Changes:
Release Name: inequality-1.3.1
Stripping Coulter off its divider /2 wasn't a good idea.
Now it is back.
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Release Name: inequality-1.3.0
Notes:
(1) The Williamson variation has been fixed.
(2) Gini and Coulter are scaled between 0 and 1.
In the literature you sometimes find Gini
scaled between 0 and 1/2 and the Coulter
scaled between 0 and sqrt(1/2).
Changes:
Herfindahl disabled (needs rework)
Coulter: Division by 2 (in the square root) disabled.
Some features for experimenting added.
-----------------------------------------------------------------
Release Name: inequality-1.2.4
Notes:
"Any city
however small,
is in fact divided into two,
one the city of the poor,
the other of the rich;
these are at war
with one another."  (Plato, The Republic, 370 BC)
The Platon inequality (1999-04-25) is for use
by the author. It is based on the concept a
entropywise equivalent reference distribution
with only two quantiles as a representation
of a given measured distributen.
-----------------------------------------------------------------
Release Name: inequality-1.2.3
Notes:
The Platon inequality is for use by the author.
(You may test it empirically wether it reflects
perceptions of inequality. But as for now I
won't promote this inequality measure too much.)
Changes:
Platon inequality added to inequality-1.2.2
-----------------------------------------------------------------
Release Name: inequality 1.2.2
Notes:
(1) Experimenting with epsilon (societal inequality
aversion) has been implemented.
(2) Output example:
# Kullback-Leibler inequality: 32.3%
# Atkinson inequality: 32.8%
# Gini inequality: 42.2%
# Hoover inequality: 29.7%
# Coulter inequality: 13.4%
# MacRae equality: 67.2%
# Kullback-Leibler redundancy: 0.391
# Theil redundancy: 0.398
(3) The program is maintained in Lua and in Phyton.
Changes:
(1) No functionale bug fixes were necessary.
(2) You find the Kullback-Liebler distance and
the Kullback-Leibler distance in the Internet.
It's Kullback-Leibler,
and I call it Kullback-Leibler redundancy.
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