GDP per Capita versus FIFA Rank III

In earlier post titled GDP per Capita versus FIFA Rank I made the observation that the data must follow a binomial distribution. In a second post on the same subject I observed that the split between high and low GDP per capita countries is quite balanced.

To verify the above observations I will follow my posted axiom and transform the data!
I used MATLAB to generate a data plot, a lag plot, a histogram, and a normal probability plot:

y = GDPcapita *1e-3;


subplot(221)
plot(y, '.-', 'markersize', 10)


subplot(222)
plot(y(2:n),y(1:n-1), '*b')


subplot(223)
hist(y, 8)


subplot(224)
normplot(y) % This requires the Statistics Toolbox



Observations:

1. The run sequence plot shows a clear sinusoidal pattern (fixed location, fixed variation.)
2. The lag plot also shows a non-random sinusoidal pattern.
3. The histogram shows a bimodal distribution with noise.
4. The normal probability plot is useless since we already know that the data are not random.

Since the data follows a sinusoidal pattern then the histogram of the data ought to be symmetric and bimodal. The two extremes are the max and min amplitude and the rest of the data points fall in between the two extreme bins. For example:


However, the histogram of the GDP per capita data for the WC2010 countries does not have such a clean shape. It actually looks like a binomial distribution mixed with an error component. It turns out that the error component is described fairly well by a uniform distribution.

To test this it is sufficient to create a sinusoidal pattern mixed with uniformly distributed pseudo-random noise.
 
% MODEL
t = 0:pi/2:15*pi;
v = cos(t);


%ADD PSEUDO-RANDOMNESS
r = rand( 1,length(t) );
vv = v + r; vv = vv ./max(vv);


% DATA
y = GDPCapita;
yy = y - mean(y); % DEMEAN
yy = yy ./max(yy); % NORMALIZE AMPLITUDE



I run the model a few times using the pseudo-random number generator and the results were reasonable. Though not sophisticated, the model seems to be sufficient to show that at least for WC2010 countries the GDP per capita  and FIFA rank are related through a sinusoidal function that results in a bimodal distribution with a uniformly distributed error component!

I cannot imagine that such behaviour is common for past World Cups. However, it is one more observation which shows that money makes little-to-no difference in football at the national level.