[EM] mathematical norms

[Stephane Rouillon]

Dear Olli and Diego,

to measure discrepancies and correlations between two sets of measures,
mathematicians use norms. I will not give you the general form of norms
(I even though there are more complex than the ones I know), however
among these, three norms are usually used. The norm-1 (L1 also called H0),
the norm-2 (L2) and the infinite norm (Linf). Examples will follow the
definitions
I provide below without and with mathematical symbols.
We assume X and S two sets of measures.

The norm-1 is the sum of the absolute values of the differences between the
values in each set for every elements. L1 = (for i=1 to n) Sigma |Xi-Si|.
The norm-2 is the square root of the sum of the squares of the differences
between the values in each set for every elements. L2 = sqrt( (for i=1 to n)
Sigma (Xi-Si)^2).
The least square method minimizes norm L2.
The norm-N is L(N)= ( (for i=1 to n) Sigma (Xi-Si)^N)^(1/N).
The infinite norm is the limit of norm-N when N tends to infinite: Linf =
(over i) max {|Xi-Si|}.

A basic example: the Hockey pool.
Let's try to predict the number of points some fictional players will make
next year:
Player            Prediction            Reality
Gretzky                     99                     0
Saku Koivu               11                   32
Mario Lemieux          66                   78

Let's evaluate the errors L1, L2 and Linf:
L1 = 99 + 21 + 12 = 132
L2 = sqrt(99^2 + 21^2 + 12 ^2) = 101.9
L3 = max(99, 21, 12) = 99

Comparing different norm is generally useless. What is useful is comparing
errors generated
by different methods with the same norm. An election-method example:
Political parties        Popular support(X)     Seat distrib.1        Seat
distrib.2        Seat distrib.3
Party A                                    43%
47%                        46%                    50%
Party B                                    33%
29%                        32%                    30%
Party C                                    24%
24%                        22%                    20%

For Sd1: L1 = 8%, L2 = 5.6%, L3 = 4%;
For Sd2: L1 = 6%, L2 = 3.7%, L3 = 3%;
For Sd3: L1 = 14%, L2 = 8.5%, L3 = 7%.

A method is better when the norm of the error is nearer to zero.
All norms say that the Seat distribution 2 is the most proportional, then n.1
and finally
Seat distribution 3 is the worst. But rarely norms do not agree:
Back to the hockey example. New predictions
Player            Prediction 2            Reality
Gretzky                     50                    0
Saku Koivu               82                   32
Mario Lemieux          28                   78
L1 = 150, L2 = 86.6, L3 = 50.
Norm L1 says the first prediction set was the best, norms L2 and Linf say set
number 2 is the best!
Which one is right?

All this to say that from what I understood, Sainte-Lague minimizes norm L1
and not norm L2 as Olli said...

Steph
PS: Finois, Franceuzish, English, Espanol, Cantonnais or Mandarin, Olli,
spretchs du Deutch ?

Olli Salmi escribio :

> At 14:18 -0500 19.8.2003, Diego Tello wrote:
> >¿Cómo disminuir la desproporcionalidad electoral en un sistema de
> >reparto proporcional?.
> >
>
> Una solución posible es el método de Sainte-Laguë. No soy matemático,
> pero creo que minimiza el índice de mínimos cuadrados.
>
> Los países con el método Sainte-Laguë tienen una barrera legal de 4%.
>
> Olli Salmi
>
> >Se convierte en un problema de programación lineal entera,
> >minimizando un índice, en particular el índice de mínimos cuadrdo de
> >Gallagher que tiene virtudes ya que ocupa una función trascendente
> >que es la raíz cuadrada.
> >
> >¿Que otras variables se podrían sugerir para la elaboración de un
> >nuevo método de reparo proporcional, además del número de votos?.
> >¿Podría ser determinar una barrera electoral que dependa del número
> >de electores?..................
>
> ----
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