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| % GINI computes the Gini coefficient and the Lorentz curve.
%
% Usage:
% g = gini(pop,val)
% [g,l] = gini(pop,val)
% [g,l,a] = gini(pop,val)
% ... = gini(pop,val,makeplot)
%
% Input and Output:
% pop A vector of population sizes of the different classes.
% val A vector of the measurement variable (e.g. income per capita)
% in the diffrerent classes.
% g Gini coefficient.
% l Lorentz curve: This is a two-column array, with the left
% column representing cumulative population shares of the
% different classes, sorted according to val, and the right
% column representing the cumulative value share that belongs to
% the population up to the given class. The Lorentz curve is a
% scatter plot of the left vs the right column.
% a Same as l, except that the components are not normalized to
% range in the unit interval. Thus, the left column of a is the
% absolute cumulative population sizes of the classes, and the
% right colun is the absolute cumulative value of all classes up
% to the given one.
% makeplot is a boolean, indicating whether a figure of the Lorentz
% curve should be produced or not. Default is false.
%
% Example:
% x = rand(100,1);
% y = rand(100,1);
% gini(x,y,true); % random populations with random incomes
% figure;
% gini(x,ones(100,1),true); % perfect equality
%
% Explanation:
%
% The vectors pop and val must be equally long and must contain only
% positive values (zeros are also acceptable). A typical application
% would be that pop represents population sizes of some subgroups (e.g.
% different countries or states), and val represents the income per
% capita in this different subgroups. The Gini coefficient is a measure
% of how unequally income is distributed between these classes. A
% coefficient of zero means that all subgroups have exactly the same
% income per capital, so there is no dispesion of income; A very large
% coefficient would result if all the income accrues only to one subgroup
% and all the remaining groups have zero income. In the limit, when the
% total population size approaches infinity, but all the income accrues
% only to one individual, the Gini coefficient approaches unity.
%
% The Lorenz curve is a graphical representation of the distribution. If
% (x,y) is a point on the Lorenz curve, then the poorest x-share of the
% population has the y-share of total income. By definition, (0,0) and
% (1,1) are points on the Lorentz curve (the poorest 0% have 0% of total
% income, and the poorest 100% [ie, everyone] have 100% of total income).
% Equal distribution implies that the Lorentz curve is the 45 degree
% line. Any inequality manifests itself as deviation of the Lorentz curve
% from the 45 degree line. By construction, the Lorenz curve is weakly
% convex and increasing.
%
% The two concepts are related as follows: The Gini coefficient is twice
% the area between the 45 degree line and the Lorentz curve.
%
% Author : Yvan Lengwiler
% Release: $1.0$
% Date : $2010-06-27$
function [g,l,a] = gini(pop,val,makeplot)
% check arguments
assert(nargin >= 2, 'gini expects at least two arguments.')
if nargin < 3
makeplot = false;
end
assert(numel(pop) == numel(val), ...
'gini expects two equally long vectors (%d ~= %d).', ...
size(pop,1),size(val,1))
pop = [0;pop(:)]; val = [0;val(:)]; % pre-append a zero
isok = all(~isnan([pop,val]'))'; % filter out NaNs
if sum(isok) < 2
warning('gini:lacking_data','not enough data');
g = NaN; l = NaN(1,4);
return;
end
pop = pop(isok); val = val(isok);
assert(all(pop>=0) && all(val>=0), ...
'gini expects nonnegative vectors (neg elements in pop = %d, in val = %d).', ...
sum(pop<0),sum(val<0))
% process input
z = val .* pop;
[~,ord] = sort(val);
pop = pop(ord); z = z(ord);
pop = cumsum(pop); z = cumsum(z);
relpop = pop/pop(end); relz = z/z(end);
% Gini coefficient
% We compute the area below the Lorentz curve. We do this by
% computing the average of the left and right Riemann-like sums.
% (I say Riemann-'like' because we evaluate not on a uniform grid, but
% on the points given by the pop data).
%
% These are the two Rieman-like sums:
% leftsum = sum(relz(1:end-1) .* diff(relpop));
% rightsum = sum(relz(2:end) .* diff(relpop));
% The Gini coefficient is one minus twice the average of leftsum and
% rightsum. We can put all of this into one line.
g = 1 - sum((relz(1:end-1)+relz(2:end)) .* diff(relpop));
% Lorentz curve
l = [relpop,relz];
a = [pop,z];
if makeplot % ... plot it?
area(relpop,relz,'FaceColor',[0.5,0.5,1.0]); % the Lorentz curve
hold on
plot([0,1],[0,1],'--k'); % 45 degree line
axis tight % ranges of abscissa and ordinate are by definition exactly [0,1]
axis square % both axes should be equally long
set(gca,'XTick',get(gca,'YTick')) % ensure equal ticking
set(gca,'Layer','top'); % grid above the shaded area
grid on;
title(['\bfGini coefficient = ',num2str(g)]);
xlabel('share of population');
ylabel('share of value');
end
end |
