Senadores

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Voting Blocks[edit | edit source]

<math> \def\ave#1{\langle{#1}\rangle} </math>


we take all votes from the Brazilian senate [1]. The votes are converted into 'Sim' = 1, 'Não' = -1 and the rest (no voting, missing, etc) = 0. The correlation between two senators can be calculated as

<math>

C(i,j) = \frac{ \ave{ (v_i - \ave{v_i})(v_j - \ave{v_j}) } }{ \sqrt{ \ave{(v_i - \ave{v_i})^2} \ave{(v_j - \ave{v_j})^2} } } </math>

This measures the level of agreement of a pair of senators. In practice, if one senator votes 1 (or -1) and the other also, the correlation increases. The quantity is normalized by the standard deviation of both senators.

However, the fact that the above formula uses the mean creates problems with our interpretation of the votes. In fact, if a senator votes more 1 than -1, it will have a positive mean. When computed against another that tend to vote more on -1, even if both miss the voting session (v=0) there is a negative contribution to the correlation. In other words, the non-voting behaviour is being computed as votes.

In order to completely disregard the missing votes, we propose to use

<math>

C'(i,j) = \frac{ \ave{ v_i v_j } }{ \sqrt{ N_i N_j } } </math>

where N is the number of voting sessions each senator expressed a valid vote, and it comes from <math>N=\ave{v^2}</math>. We still have a problem, for instance, if a senator k had only given one valid vote and he is compared to another to which he agreed.

<math>

C'(k,j) = \frac{ \ave{ v_j } }{ \sqrt{ N_j } } </math>

If we want to compute the agreement between senators, these two senators agree on every time both voted we need to change the formula

<math>

A(i,j) = \frac{ \ave{ v_i v_j } }{ \ave{ |v_i v_j| } } </math>

where <math>\ave{ |v_i v_j| }</math> is the number of voting session both gave valid votes simultaneously.

Furthermore the ordering of the matrix is interesting to see the structure in it. We compute the eigenvalues and eigenvectors of the matrix. The sum of all eigenvalues corresponds to the number of senators.

Correlation matrix for the 54th Senate (2011-2014) arranged for the largest block.
File:C54maxaf.png
Affinity for the 54th Senate (2011-2014) arranged for the largest block.
Agreement matrix for the 54th Senate (2011-2014) arranged for the largest block.
File:A54maxaf.png
Affinity for the 54th Senate (2011-2014) arranged for the largest block.
File:C54max2.png
Correlation matrix for the 54th Senate (2011-2014) arranged for the second largest block.
File:C54max2af.png
Affinity for the 54th Senate (2011-2014) arranged for the second largest block.
Agreement matrix for the 54th Senate (2011-2014) arranged for the second largest block.
Affinity for the 54th Senate (2011-2014) arranged for the second largest block.
File:C54all.png
Correlation matrix for the 54th Senate (2011-2014) arranged for all blocks.
File:C54allaf.png
Affinity for the 54th Senate (2011-2014) arranged for all blocks.
File:A54all.png
Agreement matrix for the 54th Senate (2011-2014) arranged for all blocks.
File:A54allaf.png
Affinity for the 54th Senate (2011-2014) arranged for all blocks.