AAEI?ONNEEE AINOAeA?NOAAIIUE OIEAA?NEOAO EIOI?IAOEEE E ?AAeEIYEAEO?IIEEE
Eaoaae?a ?YN
?aoa?ao ia oaio:
«Yio?iiey neiaeiuo niiauaiee, ecauoi/iinoue enoi/ieea. Oeaeue naeaoey
aeaiiuo e oeiu nenoai naeaoey»
IEINE, 2009
Yio?iiey neiaeiuo niiauaiee, ecauoi/iinoue enoi/ieea
?anniio?aiiua auoa oa?aeoa?enoeee enoi/ieea – eiee/anoai eioi?iaoeee e
yio?iiey – ioiineeenue e iaeiiio enoi/ieeo, au?aaaouaathuaio iioie
iacaaeneiuo eee i?inouo niiauaiee, eee e enoi/ieeo aac iaiyoe.
Iaeiaei a ?aaeueiuo oneiaeyo iacaaeneiinoue yeaiaioa?iuo niiauaiee,
au?aaaouaaaiuo enoi/ieeii, – yaeaiea aeiaieueii ?aaeeia. *aua auaaao eae
?ac ia?aoiia – neeueiay aeaoa?ieie?iaaiiay eee noaoenoe/aneay naycue
iaaeaeo yeaiaioaie niiauaiey iaeiiai eee ianeieueeeo enoi/ieeia.
Iai?eia?, i?e ia?aaea/a oaenoa aa?iyoiinoe iiyaeaiey ioaeaeueiuo aoea
caaenyo io oiai, eaeea aoeau ei i?aaeoanoaiaaee. Aeey ?onneiai oaenoa,
iai?eia?, anee ia?aaeaia aoeaa “I”, aa?iyoiinoue oiai, /oi neaaeothuae
aoaeao “A”, ai?acaei auoa, /ai “I”, iinea aoeau “U” ieeiaaea ia
ano?a/aaony “H” e o.ae. Iiaeiaiay aea ea?oeia iaaethaeaaony i?e
ia?aaea/a ecia?aaeaiee – ninaaeiea yeaiaiou ecia?aaeaiey eiatho iau/ii
ii/oe iaeeiaeiaua y?einoue e oeaao.
I?e ia?aaea/a e o?aiaiee aeaiiuo /anoi oaeaea eiatho aeaei n
ianeieueeeie enoi/ieeaie, oi?ie?othueie noaoenoe/anee naycaiiua ae?oa n
ae?oaii niiauaiey. Niiauaiey, au?aaaouaaaiua oaeeie enoi/ieeaie,
iacuaathony neiaeiuie niiauaieyie, a naie enoi/ieee – enoi/ieeaie n
iaiyoueth.
I/aaeaeii, /oi i?e ii?aaeaeaiee yio?iiee e eiee/anoaa eioi?iaoeee a
niiauaieyo, yeaiaiou eioi?uo noaoenoe/anee naycaiu, iaeuecy
ia?aie/eaaoueny oieueei aaconeiaiuie aa?iyoiinoyie – iaiaoiaeeii
iaycaoaeueii o/eouaaoue oaeaea oneiaiua aa?iyoiinoe iiyaeaiey
ioaeaeueiuo niiauaiee.
Ii?aaeaeei yio?iieth neiaeiiai niiauaiey, au?aaaouaaaiiai aeaoiy
caaeneiuie enoi/ieeaie (iiaeiaiui aea ia?acii ii?aaeaeyaony yio?iiey
neiaeiiai niiauaiey, au?aaaouaaaiiai iaeiei enoi/ieeii n iaiyoueth).
Ionoue niiauaiey ia?aiai enoi/ieea i?eieiatho cia/aiey x1, x2, x3,….xk
n aa?iyoiinoyie, niioaaonoaaiii, P(x1 ), P(x2 ),….. P(xk ), niiauaiey
aoi?iai – y1, y2,…..ym n aa?iyoiinoyie P(y1 ), P(y2 ),….. P(ym ).
Niaianoioth yio?iieth aeaoo enoi/ieeia X e Y iiaeii ii?aaeaeeoue
neaaeothuei ia?acii: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, (1)
aaea P(xi,yj ) – aa?iyoiinoue niaianoiiai iiyaeaiey niiauaiee xi e yj .
Iineieueeo niaianoiay aa?iyoiinoue P(xi,yj ) ii oi?ioea Aaeana
ii?aaeaeyaony eae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, (2)
oi au?aaeaiea aeey niaianoiie yio?iiee iiaeii caienaoue a neaaeothuai
aeaea: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 (3)
Oae eae ia?aaea/a niiauaiey xi iaycaoaeueii niioaaonoaoao ia?aaea/a
iaeiiai ec niiauaiee (ethaiai) ec ainaiaey Y , oi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 (4)
e niaianoiay yio?iiey H(X,Y) ii?aaeaeeony eae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, (5)
aaea H ( Y/xi ) – oae iacuaaaiay /anoiay oneiaiay yio?iiey, io?aaeathuay
yio?iieth niiauaiey Y i?e oneiaee, /oi eiaei ianoi niiauaiea xi. Aoi?ia
neaaaaiia a iineaaeiai au?aaeaiee i?aaenoaaeyao niaie on?aaeiaiea H (
Y/xi ) ii anai niiauaieyi xi e iacuaaaony n?aaeiae oneiaiie yio?iieae
enoi/ieea Y i?e oneiaee ia?aaea/e niiauaiey X. E ieii/aoaeueii:
H (X,Y ) = H (X) + H (Y/X) . (6)
Oaeei ia?acii, niaianoiay yio?iiey aeaoo niiauaiee ?aaia noiia
aaconeiaiie yio?iiee iaeiiai ec ieo e oneiaiie yio?iiee aoi?iai.
Iiaeii ioiaoeoue neaaeothuea iniiaiua naienoaa yio?iiee neiaeiuo
niiauaiee:
1. I?e noaoenoe/anee iacaaeneiuo niiauaieyo X e Y niaianoiay yio?iiey
?aaia noiia yio?iiee eaaeaeiai ec enoi/ieeia:
H (X,Y) = H (X) + H (Y) , (7)
oae eae H (Y/X) = H (Y).
2. I?e iieiie noaoenoe/aneie caaeneiinoe niiauaiee X e Y niaianoiay
yio?iiey ?aaia aaconeiaiie yio?iiee iaeiiai ec niiauaiee. Aoi?ia
niiauaiea i?e yoii eioi?iaoeee ia aeiaaaeyao. Aeaenoaeoaeueii, i?e
iieiie noaoenoe/aneie caaeneiinoe niiauaiee oneiaiua aa?iyoiinoe
P(yj/xi) e P(xi/y j) ?aaiu eee ioeth, eee 1, oiaaea
P(xi /yj )*log P(xi /yj ) = P(yj /xi )*log P(yj /xi ) = 0 (8)
e, neaaeiaaoaeueii, H (X,Y) = H (X) = H (Y).
3. Oneiaiay yio?iiey eciaiyaony a i?aaeaeao
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