M ÓDULOS COMBINADOS TARIFA PLANA ZONAL + NF CUOTAS

raphy with NMI

Communities dete ted with various methods were seen to orrelate with the ge-

ographi al divisions of nodes to west, enter and east when visualized using the

MST of Figure 3.9. The division of nodes to lo ations was also seen to orre-

late with the geneti LM distan e [29℄. Although these orrelations were learly

visible in MST visualizations, it was also lear that the orresponden e was not

perfe t. The NMI framework is now used to quantify these orrelations in both

ases.

Cal ulating the mutual information between results of ahierar hi al ommunity

dete tion methodand thelarge-s alegeographi aldivision isnotstraightforward

for two reasons. First of all, the mutual information approa h requires division

ofthe nodesintogroups,whi h an bedoneforhierar hi al ommunity stru ture

tree by looking at one level of the hierar hy at the time. This is done for the

blo k diagonalization of the NSA distan e matrix by dening hierar hy levels

with respe t to the number of splitting events. For example, at the third level

of hierar hy,all ommunities are three linksaway from the root node. This is of

ourse not a unique nor ne essarily the best way to dene the hierar hy levels.

One ould, for example, dene a distan e between the nodes in the tree. The

distan es ouldberelatedtothe blo kdiagonalization pro ess, orusesomeextra

biologi al information. Tryingout allthe waysofdividingthe treeintohierar hy

levelswouldleadtoaverylargenumberofdierent ombinationsof ommunities,

and would not be a feasible solution. The se ond problem is a variation of the

data than itisfor ahierar hi al tree. Multiplehierar hylevelsare not used here

for the geographi al lo ations. Instead, only two divisions are used: the rst is

the most a urate geographi al division feasible, that is, the division of nodes

a ording to individual sampling lo ations. The se ond is the rude division of

nodes tothe three areas dis ussed earlier: west, enter and east.

Thenormalizedmutualinformationforea hhierar hylevelof ommunitystru ture

dete ted with the blo k diagonalization method is shown in Figure 3.10. The

normalized mutual information of node lo ations and hierar hy levels is seen to

in rease asfun tion ofhierar hy level inpanel a). This meansthat the last divi-

sions madeby theblo k diagonalizationmethodare not ompletely randomwith

respe t to the lo ations. However, the blo k diagonalization method wasseen to

nd ommunities inrandomizeddata,whi hmightsuggestthat thelast levelsof

the tree might be noisyalso for the real data.

(a) alllo ations (b) 3division

Figure 3.10: Normalized mutual information of ommunities produ ed with the

blo kdiagonalizationmethodusingtheNSAdistan ematrixanda)thelo ations

of the spe imen and b) division of the nodes to west, enter and east. The

hierar hylevelinthe ommunity stru ture isonthe horizontal axisand theNMI

onthe verti alaxis. The NMIbetween the twogeographi division(west- enter-

east, sampling lo ations)is approximately

0.37

.

The NMIofthe large-s alegeographi aldivisionandthe hierar hi al ommunity

stru ture, on the other hand, is at its maximum after the rst split to blo ks,

and is seen to slowly de rease thereafter, almost saturating at the last levels of

hierar hy. Thisbehavior an beexplainedwiththe helpof Figure3.7 illustrating

the bran hing pro ess. The rst split separates west from enter and east, and

the enter and east separate onlyat the nextlevel. As the west is more densely

NMI, as italso furtherdivides the western omponent. It isworth noti ing that,

as dis ussed earlier, the number of bran hing events might not be the optimal

way of dening dierent hierar hy levels. This is highlighted by the fa t that

if a ommunity division is hosen from the hierar hy tree in su h a way that

ommunities with mainly western nodes are hosen from the rst level and the

rest fromthe se ondlevel,asillustratedinFigure3.11, theoverallNMIimproves

from the originalrst level value of

0.468

toa value of

0.527

.

Figure 3.11: The rst two levels of the ommunity hierar hy tree produ ed by

dete tion using the blo k diagonalization method using the NSA distan e mea-

sure. The green shading represents an alternative way of hoosing a division of

nodes to ommunities from the tree. This parti ular hoi e ree ts better the

division of nodes to the three geographi al groups: west, enter and east. The

NMIof this divisionand the large-s alegeographi aldivision is

0.527

, whereasif the original levels of the tree would be used, the orresponding values would be

0.468

for the rst and

0.446

for the se ond level.

In document Contratos y tarjetas no disponibles para nuevas altas (página 147-160)