Liagkas_presentation

JEWRIA AKRAIWN TIMWN EFARMOGH SE BROQOPTWSH Sumperàsvmata
JEWRIA AKRAIWN TIMWN
EFARMOGH SE DEDOMENA BROQOPTWSHS
LIAGKAS GEWRGIOS
TMHMA STATISTIKHS & ANALOGISTIKWN -
QRHMATOOIKONOMIKWN MAJHMATIKWN
PANEPISTHMIOU AIGAIOU
PTUQIAKH ERGASIA, SAMOS 2013

1 JEWRIA AKRAIWN TIMWN
Eisvagwg† svth Jewr–a Akra–wn Tim∏n
H mËjodoc Block Maxima
H mËjodoc POT
H GenikeumËnh katanom† Akra–wn Tim∏n (GEV)
H GenikeumËnh Katanom† Pareto (GPD)
H mËjodoc MËgisvthc Pijanofàneiac (MLE)
H mËjodoc twn L-rop∏n (L-moments)
H mËjodoc twn TL-rop∏n (TL-moments)
2 EFARMOGH SE BROQOPTWSH
Perioq† melËthc kai dedomËnwn
Prosvarmog† twn dedomËnwn svthn GEV
Prosvarmog† twn dedomËnwn svthn GPD

H mËjodoc POT

H Jewr–a Akra–wn Tim∏n apotele– xeqwrisvtÏ klàdo thc
svtatisvtik†c
Anàptuxh jewrhtik∏n kai svtatisvtik∏n montËlwn pou
svqet–zontai me thn emfànisvh akra–wn parathr†svewn
UpologisvmÏc thc pijanÏthtac pragmatopo–hsvhc akra–wn †
svpàniwn gegonÏtwn
Montelopo–hsvh mËsvw thc GenikeumËnhc katanom†c Akra–wn
Tim∏n (GEV) kai thc GenikeumËnhc katanom†c Pareto (GPD)

H mËjodoc POT

'Esvtw anexàrthtec t.m X1, X2, . . . , Xm ⇠ F (àgnwsvth)
Mporo‘me na jewr†svoume Ïti oi X1, X2, . . . , Xm, m = nk
qwr–zontai sve k to pl†joc uposv‘nola (block) apÏ n
parathr†sveic to kajËna
H mËjodoc Block Maxima qwr–zei ta dedomËna sve megàla
kommàtia (blocks) kai svth svunËqeia epilËgei th mËgisvth
(elàqisvth) parat†rhsvh sve kàje block
Sun†jwc ta blocks kajor–zontai me bàsvh kàpoia qronik†
per–odo (hmËra, m†na, Ëtoc, ktl)

Sumbol–zoume me Y1, Y2, . . . , Yk tic mËgisvtec timËc sve kajËna
apÏ autà ta k uposv‘nola, svugkekrimËna
Yi = max X(i 1)n+1, X(i 1)n+2, . . . , X(i 1)n+m , i = 1, 2, . . . , k
EpomËnwc ta Yi ⇠ GEV , diÏti P
⇣
Yi dn
cn
 z
⌘
⇡ G (z) ,
efÏsvon upàrqoun oi akolouj–ec dn,cn
'Ara, h efarmog† thc mejÏdou proÙpojËtei thn l†yh twn
mËgisvtwn parathr†svewn apÏ isvomegËjh uposv‘nola twn
dedomËnwn kai thn prosvarmog† touc svthn GEV

Sq†ma : DedomËna broqÏptwsvhc tou svtajmo‘ thc Nàxou me thn
epilog† twn Block Maxima

Sq†ma : Et†svia mËgisvta broqÏptwsvhc tou svtajmo‘ thc Nàxou

Pleonekt†mata
Oi jewrhtikËc upojËsveic e–nai ligÏtero kr–svimec svthn pràxh
H anexarthsv–a twn meg–svtwn mpore– na epiteuqje– me thn
epilog† megàlou megËjouc block
Pio e‘kolo na efarmosvte–
Meionekt†mata
Oi abebaiÏthtec twn ektimht∏n mpore– na e–nai megàlec lÏgw
tou mikro‘ megËjouc tou de–gmatoc
(Mikr† apodotikÏthta)

H mËjodoc POT
H mËjodoc POT

H mËjodoc POT
'Esvtw X1, X2, . . . , Xn ⇠ F thc opo–ac jËloume na
melet†svoume thn dexià ourà
H mËjodoc POT basv–zetai svtic X1, X2, . . . , Xn pou
uperba–noun Ëna kat∏fli, Ësvtw u
Ja prËpei na basvisvto‘me svthn katanom† thc upËrbasvhc miac
t.m Xi pànw apÏ Ëna kat∏fli u, dedomËnou Ïti h Xi Ëqei
uperbe– to u:
Fu(y) = P (X u  y | X > u) = F(u+y) F(u)
1 F(u) , y > 0

H mËjodoc POT
Gia thn epilog† tou bËltisvtou katwfl–ou u antimetwp–zoume ta
probl†mata:
Eàn to u den Ëqei epilege– arketà megàlo, tÏte upàrqei
k–ndunoc h Fu na mhn prosvegg–zetai ikanopoihtikà apo thn
GPD
Gia mia pol‘ uyhl† tim† tou u, o arijmÏc twn tim∏n pou
uperba–noun e–nai pol‘ mikrÏc kai kata svunËpeia oi ektimhtËc
mac Ëqoun megàlh diak‘mansvh
Gia mia tim† tou katwfl–ou, oi ektimhtËc pou prok‘ptoun
parousviàzoun megàlh merolhy–a
Ja prËpei wc kat∏fli na epilËxoume to mikrÏtero u ∏svte
Fu ⇡ GPD

H mËjodoc POT
Gia thn bËltisvth epilog† tou katwfl–ou upàrqoun oi teqnikËc
pou basv–zontai sve katàllhla graf†mata:
Mean Residual Life plot
Threshold choice plot
L-moment plot
Dispersion index plot
Hill plot
'Eqontac prosvdior–svei to kat∏fli, mporo‘me na elËgxoume th
svumperiforà thc ouràc thc katanom†c mËsvw enÏc QQ-plot.
Sth svugkekrimËnh melËth ja parousviàsvoume d‘o mejÏdouc
epilog†c bËltisvtou u:

H mËjodoc POT
Mean Residual Life plot
Basv–zetai svto gegonÏc Ïti, an Ëna kat∏fli uo e–nai arketà
megàlo ∏svte Fuo ⇡ GPD, tÏte profan∏c Fu ⇡ GPD, 8u > uo.
'Ara kai oi mËsvec timËc twn Fu kai GPD ja prËpei na e–nai –svec
(mean excess function):
e(u) = E (X u | X > u) = sv⇤
1 k = sv+k(u m)
1 k , u > uo
H e (u) mpore– e‘kola na ektimhje– apÏ thn empeirik† mËsvh
uperbàllousva svunàrthsvh:
ˆe(u) = 1
nu
P
i:Xi >u
(Xi u)
Ïpou nu e–nai to pl†joc twn Xi pou uperba–noun to u

H mËjodoc POT
To gràfhma twn svhme–wn (u, ˆe (u)) ja prËpei na e–nai per–pou
grammikÏ gia u uo
Sq†ma : Mean residual life plot tou svtajmo‘ thc Nàxou

H mËjodoc POT
Threshold choice plot
Ektimo‘me tic paramËtrouc k, sv⇤ thc GPD gia diàforec timËc
tou katwfl–ou u
An isvq‘ei Ïti Fu ⇡ GPD ja prËpei h ekt–mhsvh tou k na mhn
ephreàzetai apÏ to kat∏fli, en∏ h ekt–mhsvh tou
sv⇤ = sv+k (u m) na metabàlletai grammikà wc proc to u
Mporo‘me kai pàli na epilËxoume to mikrÏtero u pou
ikanopoie– Ïla ta parapànw
Se arketà megàlo kat∏fli u:
H paràmetroc svq†matoc k kai h tropopoihmËnh paràmetroc
kl–makac (sv ku) e–nai anexàrthtec tou katwfl–ou

H mËjodoc POT
Sq†ma : Threshold choice plot tou svtajmo‘ thc Nàxou

H mËjodoc POT
Sq†ma : Diàgramma svhme–wn tou svtajmo‘ thc Nàxou me epilegmËno to
kat∏fli

H mËjodoc POT
Pleonekt†mata
Pio apotelesvmatik† an Ëna ‘mikrÏ’ an∏tato Ïrio
dikaiologe–tai
(PerisvsvÏterec anexàrthtec uperbàsveic apÏ ta block maxima)
Meionekt†mata
H paradoq† thc anexarthsv–ac e–nai kr–svimh svthn pràxh.
Qreiàzetai teqnikËc apÏ-omadopo–hsvhc (declustering)
Qreiàzontai ep–svhc diagnwsvtikà gia thn epilog† tou
katwfl–ou. H epilog† tou e–nai kàpwc asvaf†c svthn pràxh
LigÏtero e‘kolo na efarmosvte–

H mËjodoc POT

UpÏ orisvmËnec svunj†kec kanonikÏthtac, h sveirà twn mËgisvtwn
tim∏n (Block Maxima) akoloujo‘n asvumptwtikà thn GenikeumËnh
katanom† Akra–wn Tim∏n (GEV), me sv.p.p:
G(z : m, sv, k) =
8
<
:
exp

1 + k z m
sv
1
k
,
exp
⇥
exp z m
sv
⇤
k 6= 0
k = 0
Ïpou 1 + k z m
sv > 0, m✏R e–nai h paràmetroc jËsvhc, sv > 0 h
paràmetroc kl–makac kai k h paràmetroc svq†matoc.

H GEV qwr–zetai sve tre–c epimËrouc oikogËneiec katanom∏n :
an k > 0 h GEV pa–rnei th morf† thc Frechet me barià ourà
an k < 0 th morf† thc Weibull , fragmËnh svto m k/sv
an k = 0 th morf† thc Gumbel katanom†c me lept† ourà
Sq†ma : GEV oikogËneiec katanom∏n

Eàn X1, X2, . . . , XT e–nai t.m me koin† svunàrthsvh katanom†c
F, to u (T) = F 1 (1 1/T) , dhlad† prÏkeitai gia to
(1 1/T) posvosvtia–o svhme–o thc F katanom†c. Isvq‘ei h
svqËsvh P [X1 > u (T)] = 1 F (u (T)) = 1/T, to opo–o
onomàzetai ep–pedo epanaforàc T-qrÏnwn.
Gia thn katanom† GEV oi ektim†sveic twn akra–wn
p-posvosvtia–wn svhme–wn (Coles, 2001) upolog–zontai:
zp =
(
m sv
k
⇣
1 [ log (1 p)] k
⌘
,
m svlog [ log (1 p)] ,
k 6= 0
k = 0
Ïpou zp = F 1 (1 p) , 0 < p < 1. H metablht† zp onomàzetai
ep–pedo epanaforàc pou svqet–zetai me thn per–odo epanaforàc
1/p.

H mËjodoc POT

H GenikeumËnh Katanom† Pareto (GPD) gia Ëna arketà uyhlÏ
kat∏fli u (POT), me svunàrthsvh katanom†c uperbàsvewn (X u)
dedomËnou Ïti X > u, prosvegg–zetai apÏ th svqËsvh:
H(y) = P (X u | X > u) =
(
1 1 + k y
sv⇤
1
k
+
,
1 exp y
sv⇤ ,
k 6= 0
k = 0
pou or–zetai svto sv‘nolo y : y > 0, 1 + y y
sv⇤ > 0 me
sv⇤ = sv + k (u m).

H GPD anàloga me thn tim† pou pa–rnei h paràmetroc k:
an k > 0 tÏte ekte–netai dexià wc to àpeiro
an k < 0 Ëqei ànw Ïrio svthr–gmatoc to u sv⇤
/k
an k = 0 ekful–zetai svthn Ekjetik† katanom† me paràmetro
1/sv⇤
Sq†ma : GPD oikogËneiec katanom∏n

Gia thn katanom† GPD, oi ektim†sveic twn pijanot†twn na
svumbo‘n gegonÏta pËra apÏ to e‘roc twn parathro‘menwn
dedomËnwn, dhlad† twn p-posvosvtia–wn svhme–wn (Coles, 2001)
upolog–zontai me qr†svh twn exisv∏svewn:
zp =
( sv⇤
k p k 1
sv⇤log
⇣
1
p
⌘ k 6= 0
k = 0
Ïpou zp = F 1 (1 p) , 0 < p < 1.

H mËjodoc POT

'Esvtw ta block maxima Y1, Y2 . . . , Yn ⇠ GEV . H svunàrthsvh
logarijmik†c pijanofàneiac twn Yi , ja e–nai:
an k 6= 0 tÏte
lGEV (m,sv,k) = n log (sv)
1 + 1
k
Pn
i=1 log
⇥
1 + k yi m
sv
⇤ Pn
i=1

1 + k yi m
sv
1
k
me 1 + k yi m
sv > 0 , 8i = 1, 2, . . . , n

an k = 0 tÏte
lGumbel (m,sv) = n log (sv)
Pn
i=1
yi m
sv
Pn
i=1 exp yi m
sv
Mia dusvkol–a gia thn ekt–mhsvh twn paramËtrwn tou montËlou
GEV me th mËjodo thc mËgisvthc pijanofàneiac, ofe–letai svto Ïti
den plhro‘ntai oi proupojËsveic gia thn efarmog† thc
kanonikÏthtac. AutÏ svhma–nei Ïti ta gnwsvtà asvumptwtikà
apotelËsvmata thc pijanofàneiac den efarmÏzontai autÏmata.

To prÏblhma autÏ Ëqei melethje– leptomer∏c (Smith, 1985) me
ta akÏlouja apotelËsvmata:
an k > 0.5 oi ektim†sveic thc mËgisvthc pijanofàneiac Ëqoun
tic svun†jeic asvumptwtikËc idiÏthtec
an 1 < k < 0.5 oi ektim†sveic sve genikËc grammËc mporo‘n
na epiteuqjo‘n, allà den Ëqoun tic asvumptwtikËc idiÏthtec
an k < 1 oi ektim†sveic den mporo‘n na upologisvto‘n

Me th megisvtopo–hsvh thc svunart†svhc logarijmik†c
pijanofàneiac, pa–rnoume tic mËgisvtec ektim†sveic pijanofàneiac
ˆj0 =
⇣
ˆm, ˆsv, ˆk
⌘
. H beltisvtopo–hsvh aut∏n g–netai me arijmhtik†
beltisvtopo–hsvh twn algor–jmwn. Dhlad† gia n ! 1, h katanom†
twn ektimht∏n mËgisvthc pijanofàneiac ˆj0 (d-diasvtàsvewn)
svugkl–nei svth kanonik† katanom†:
ˆj0 ⇠ Nd
⇣
j0, IE (j0) 1
⌘
me IE (j) na isvo‘tai me to ant–svtrofo tou parathro‘menou p–naka
plhrofor–ac pou upolog–svthke apÏ thn logarijmik† pijanofàneia
ei,j (j) = E
n
@2
@ji@jx
` (j)
o
:

IE (j) =
0
B
B
B
B
@
e1,1 (j) · · · e1,d (j)
...
... ei,j (j)
...
ej,i (j)
...
ed,1 (j) · · · ed,d (j)
1
C
C
C
C
A
kai svun†jwc o parapànw p–nakac upolog–zetai katà thn diàrkeia
thc diadikasv–ac beltisvtopo–hsvhc. Ta d.e upolog–zontai apÏ thn
ekt–mhsvh ˆwi,j tou parathro‘menou p–naka plhrofor–ac, sve (1-a)
diàsvthma empisvtosv‘nhc kai gia ˆj mpore– na upologisvte– wc:
ˆji ± z↵
2
p
ˆwi,i = ˆji ± z↵
2
r
Var
⇣
ˆji
⌘
Ïpou to z↵
2
e–nai to 1 ↵
2 posvosvthmÏrio thc tupopoihmËnhc
kanonik†c katanom†c kai to SEi =
r
Var
⇣
ˆji
⌘
e–nai to tupikÏ
svfàlma gia thn i-osvt† paràmetro.

Gia to montËlo thc GenikeumËnhc katanom†c Pareto (GPD)
lambànontac upÏyhn Ïti to de–gma mac (Y1, Y2 . . . , Yn) proËrqetai
apÏ to montËlo twn uperbàsvewn pànw apÏ Ëna uyhlÏ kat∏fli
(POT), h svunàrthsvh logarijmik†c pijanofàneiac d–netai apÏ tic
parakàtw svqËsveic:
an k 6= 0 tÏte
lGPD (sv⇤, k) = nu log (sv⇤) 1 + 1
k
Pnu
i=1
yi u
sv⇤
an k = 0 tÏte
lGPD (sv⇤) = nu log (sv⇤)
Pnu
i=1
yi u
sv⇤
Ïpou to u e–nai to kat∏fli, to nu o arijmÏc twn parathr†svewn
pou uperba–noun to Ïrio u kai to sv⇤ = sv + k (u m).

Ep–svhc, oi ektim†sveic thc mËgisvthc pijanofàneiac gia to montËlo
(GPD) Ëqei melethje– leptomer∏c (Smith, 1984):
an k < 0.5 oi ektim†sveic e–nai asvumptwtikà kanonikà
katanemhmËnec, me tic asvumptwtikËc diakumànsveic na ftànoun
to kàtw fràgma Cramer-Rao kàtw apÏ orisvmËnec
katàllhlec svunj†kec kanonikÏthtac, Ëqoume

ˆsvMLE
ˆkMLE
s N
✓
sv
k
, n 1

2sv2 (1 k) sv (1 k)
sv (1 k) (1 k)2
◆
an k 0.5 oi ektim†sveic qarakthr–zontai wc mh
kanonikopoihmËnec, dedomËnou Ïti oi svunj†kec kanonikÏthtac
den ikanopoio‘ntai kai svth svugkekrimËnh per–ptwsvh
upàrqoun probl†mata svthn sv‘gklisvh
an k > 1 oi ektim†sveic den upàrqoun, epeid† h svunàrthsvh
pijanofàneiac kontà svto akra–o svhme–o te–nei svto àpeiro wc
x prosveggisvtikà svto sv
k

H mËjodoc POT

Oi L-ropËc upolog–zontai apÏ grammiko‘c svunduasvmo‘c twn
taxinomhmËnwn tim∏n twn dedomËnwn
Parousviàzoun mikrÏterh deigmatik† metablhtÏthta apÏ tic
ektim†sveic twn kanonik∏n rop∏n
Oi L-ropËc Ëqoun orisvte– gia m–a katanom† pijanot†twn,
allà svth pràxh prËpei na ektimhjo‘n me bàsvh Ëna
peperasvmËno de–gma
Se Ëna diatetagmËno de–gma X1:n  X2:n  . . .  Xn:n , me n
to mËgejoc tou de–gmatoc, h ektim†tria thc pijanÏthtac twn
svtajmisvmËnwn rop∏n br e–nai amerÏlhpth ektim†tria twn br ,
me:
lr+1 =
Pr
k=0 P⇤
r,kbk
Ïpou to r = 0, 1, 2, . . . , n 1

EidikÏtera, h l1 metrà thn jËsvh tou mËsvou tic svugkekrimËnhc
katanom†c pou meletàme (L-jËsvhc), h l2 metrà thn kl–maka
(L-kl–makac), h l3 metrà thn asvummetr–a (L-asvummetr–ac) kai h l4
metrà thn k‘rtwsvh (L-k‘rtwsvhc). Genikà
br = n 1
Pn
j=r+1
(j 1)(j 2)...(j r)
(n 1)(n 2)...(n r) xj:n
Oi deigmatikËc L-ropËc, lr e–nai amerÏlhptoi ektimhtËc twn lr kai
ep–svhc oi L-ropËc twn analogi∏n d–nontai apÏ:
tr = lr
l2
me r = 3, 4, 5, . . . kai ep–svhc t = l1
l2
e–nai h L-svuntelesvt†c
diak‘mansvhc (L-Cv), t3 = l3
l2
h L-asvummetr–ac (L-Cs) kai t4 = l4
l2
h
L-k‘rtwsvhc (L-Ck).

Oi Hosking kai Wallis (1993, 1997) parËqoun odhg–ec gia thn
ektËlesvh thc qwrik†c anàlusvhc svuqnÏthtac qrhsvimopoi∏ntac tic
L-ropËc kai perigràfontai sve tËsvsvera b†mata:
1 exËtasvh twn dedomËnwn (tesvt asvumbatÏthtac, Di ),
(Discordancy test)
2 prosvdiorisvmÏ twn omoiogen∏n perioq∏n (testing of regional
homogeneity)
3 epilog† thc qwrik†c katanom†c
4 ekt–mhsvh thc qwrik†c katanom†c svuqnot†twn

Oi ektim†sveic twn paramËtrwn thc GEV me th mËjodo twn
L-rop∏n d–nontai wc:
ˆm = l1 +
ˆsv
h
G
⇣
1 + ˆk
⌘
-1
i
ˆk
ˆsv = l2
ˆk
(1 2 ˆk
)G(1+ˆk)
ˆk = 7.8590c + 2.9554c2
Ep–svhc, oi ektim†sveic twn paramËtrwn thc GPD e–nai:
ˆm = l1
⇣
2 + ˆk
⌘
l2
ˆsv =
⇣
1 + ˆk
⌘ ⇣
2 + ˆk
⌘
l2
ˆk = 1 3t3
t3+1

H mËjodoc POT

Oi TL-ropËc upolog–zontai apÏ grammiko‘c svunduasvmo‘c
twn taxinomhmËnwn tim∏n twn dedomËnwn
H mËjodoc twn TL-rop∏n e–nai ousviasvtikà h mËjodoc twn
L-rop∏n svthn opo–a h E (Xr k:r ) antikajisvtàtai apÏ thn
E (Xr+t k:r+2t), 8r 1 , Ïpou t = 1, 2, 3, . . . na e–nai h
perikommËnh (trimmed) rop† pou proËrqetai apÏ Ëna de–gma.
Oi TL-ropËc e–nai mia epËktasvh twn L-rop∏n, afo‘ gia t = 0
mac d–nei tic klasvikËc L-ropËc

'Eqei apodeiqje– (Elamir & Seheult, 2003) Ïti oi TL-ropËc
⇣
l
(t)
r
⌘
pou proËrqontai apÏ Ëna t.d X1, X2, . . . , Xn megËjouc n, me
ajroisvtik† svunàrthsvh Qx (u) = F 1
x (u) kai dedomËnou Ïti
X1:n  X2:n  . . .  Xn:n or–zontai wc ex†c:
l
(t)
r = 1
r
Pr 1
k=0 ( 1)k
✓
r 1
k
◆
E (Xr+t k:r+2t)
Ïpou,
E (Xi:r ) = r!
(i 1)!(r 1)!
R 1
0 x (F) Fi 1 (1 F)r i
dF
r = 1, 2, . . . , n 2t. AutÏ de–qnei Ïti oi l
(t)
r ropËc e–nai Ënac
amerÏlhptoc ektimht†c twn l
(t)
r rop∏n.

Oi deigmatikËc TL-ropËc gia t = 1, 2, 3, . . . d–nontai anologikà:
l
(t)
r =
1
r
Pn t
i=t+1
2
6
6
6
4
Pr 1
k=0( 1)k
0
@
r 1
k
1
A
0
@
i 1
r + t 1 k
1
A
0
@
n i
t + k
1
A
0
@
n
r + 2t
1
A
3
7
7
7
5
xi:n
Oi TL-ropËc twn analogi∏n (t = 1, 2, 3, . . .) ekfràzoun akrib∏c
ta –dia me aut† twn analogi∏n twn L-rop∏n,
t
(t)
2 =
l
(t)
2
l
(t)
1
e–nai h TL-svuntelesvt†c kl–makac (TL-Cv)
t
(t)
3 =
l
(t)
3
l
(t)
2
h TL-svuntelesvt†c asvummetr–ac (TL-Cs)
t
(t)
4 =
l
(t)
4
l
(t)
2
e–nai h TL-k‘rtwsvhc (TL-Ck).

H mËjodoc POT

Hmer†svia dedomËna broqÏptwsvhc (mm) pËnte Ellhnik∏n
pÏlewn gia qronik† per–odo apÏ to 1955 e∏c 2010 (20.451
timËc)
Perilambànontai h pÏlh twn Kuj†rwn, Làrisvac, M†lou,
Mutil†nhc kai Nàxou
Sq†ma : Grafik† apeikÏnisvh twn jËsvewn gia touc pËnte Ellhniko‘c
svtajmo‘c

H mËjodoc POT

56 Block 1955-2010 ˆm ˆsv ˆk
K‘jhra MLE 43.8 16.6 0.24
mm L-moms 44.6 18.5 -0.12
TL-moms 43.4 /43.0 19.0 /19.1 -0.21 /-0.27
Làrisva MLE 33.4 13.5 0.31
mm L-moms 33.5 14.0 -0.26
TL-moms 32.7 /32.6 12.6 /11.9 -0.45 /-0.55
M†loc MLE 39.4 12.8 0.16
mm L-moms 39.2 12.8 -0.17
TL-moms 39.7 /40.1 12.7 /12.5 -0.11 /-0.03
Mutil†nh MLE 56.7 18.5 0.02
mm L-moms 56.0 17.0 -0.11
TL-moms 56.7 /57.2 14.6 /14.0 -0.11 /-0.12
Nàxoc MLE 34.5 11.8 0.22
mm L-moms 34.8 12.8 -0.13
TL-moms 34.5 /34.5 13.3 /13.9 -0.14/-0.07

Sq†ma : Diagnwsvtikà diagràmmata gia Ëlegqo kal†c prosvarmog†c me
MLE

AsvumbatÏthta Di D
(1)
i /D
(2)
i
L-moms TL-moms
K‘jhra 1.07 0.82/0.53
Làrisva 1.27 1.31/1.18
M†loc 0.64 1.20/1.33
Mutil†nh 1.18 1.33/1.25
Nàxoc 0.83 0.34/0.70
Z-tesvt ZGEV
L-moms 0.12
TL-moms 1.69 / 3.48
OmoiogËneia L-moms TL-moms
H1 0.94 3.63 /5.88
H2 -0.90 0.15 /0.79
H3 -0.37 0.36 /0.42
P–nakac : Tesvt asvumbatÏthtac (an Di > 1.33 asvumbatà), tesvt
omoiogËneiac (an Hj < 1 gia j = 1, 2, 3, omoiogen†c) kai Z-tesvt (an
ZGEV
< 1.64 tÏte prosvarmÏzetai svthn katanom†)

Sq†ma : Diàgramma analogi∏n twn L-rop∏n

H mËjodoc POT

POT 1955-2010 Uperbàsveic ˆsv ˆk
K‘jhra MLE 17.4 0.11
32 mm L-moms 117 17.3 -0.13
TL-moms 15.2 /14.5 -0.26 /-0.31
Làrisva MLE 12.1 0.26
31 mm L-moms 83 11.6 -0.27
TL-moms 10.4 /11.0 -0.37 /-0.29
M†loc MLE 14.0 0.08
36 mm L-moms 74 11.6 -0.17
TL-moms 12.4 /14.3 -0.06 /0.13
Mutil†nh MLE 16.9 0.10
49 mm L-moms 80 14.9 -0.17
TL-moms 14.6 /16.3 -0.18 /-0.04
Nàxoc MLE 12.3 0.10
26 mm L-moms 117 12.4 -0.10
TL-moms 12.5 /13.0 -0.09/-0.04

Sq†ma : Diagnwsvtikà diagràmmata gia Ëlegqo kal†c prosvarmog†c me
MLE

Sq†ma : Diàgramma analogi∏n twn L-rop∏n

Sumperàsvmata - Katanom∏n
H ekt–mhsvh thc paramËtrou k
e–nai jetik† me th mËjodo MLE
arnhtik† me th mËjodo L-rop∏n, TL-rop∏n
Gia th jetik† tim†
h GEV pa–rnei th morf† thc Frechet katanom†c
h GPD ekte–netai dexià wc to àpeiro
Gia th arnhtik† tim†
h GEV pa–rnei th morf† thc Weibull katanom†c
h GPD Ëqei ànw Ïrio svthr–gmatoc to u sv⇤
/k

Sumperàsvmata - Qwrik†c Anàlusvhc
H qwrik† anàlusvh svuqnot†twn twn mejodologi∏n L-rop∏n kai
TL-rop∏n mac Ëdeixe Ïti
oi pËnte svtajmo– e–nai svtatisvtikà apodekto– wc omoiogenËc
(tesvt asvumbatÏthtac kai eterogËneiac)
me to Z-tesvt oi L-ropËc paràgoun kal‘tera apotelËsvmata
gia th montelopo–hsvh et†sviwn broqopt∏svewn svth GEV (oi
TL-ropËc Ïqi kal† prosvarmog†)
gia timËc pànw apÏ Ëna kat∏fli oi L-ropËc kai oi
TL-ropËc prosvarmÏzontai kal‘tera svth GPD

Sumperàsvmata - EpipËdou epanaforàc
Nàxoc (GEV) 5et∏n 10et∏n 20et∏n 50et∏n 100et∏n
MLE 55.6 69.1 84.4 108.4 129.9
L-moms 55.6 68.7 81.9 101.0 116.9
TL-moms (t=1) 56.9 70.0 83.9 104.2 121.2
TL-moms (t=2) 56.7 68.8 81.0 97.7 111.0
Nàxoc (GPD) 5et∏n 10et∏n 20et∏n 50et∏n 100et∏n
MLE 58.8 70.1 82.3 99.8 114.2
L-moms 47.3 57.9 69.3 85.8 99.4
TL-moms (t=1) 47.3 57.8 69.0 85.1 98.2
TL-moms (t=2) 47.1 56.9 67.0 80.8 91.6
P–nakac : Ep–pedo epanaforàc (mm) tou svtajmo‘ thc Nàxou

Bibliograf–a
Coles, S.G. (2001). An introduction to statistical modelling
of extreme values. Springer series in Statistics, Springer,
Berlin.
Elamir, E.A.H. and Seheult, A.H. (2003). Trimmed
L-moments. Computational Statistics & Data Analysis, 43,
299-314.
Hosking, J.R.M. (1990). L-moments: analysis and
estimation of distributions using linear combinations of
order statistics. Journal of the Royal Statistical Society,
Series B, 52, 105-124.
Hosking, J.R.M. and Wallis, J.R. (1993). Some statistics
useful in regional frequency analysis. Water Resources
Research, 29, 271-281.

Bibliograf–a
Hosking, J.R.M. and Wallis, J.R. (1997). Regional frequency
analysis: an approach based on L-moments. Cambridge
University Press, Cambridge, U.K.
Kotz, S. and Nadarajah, S. (2000). Extreme Value
Distributions: Theory and Applications. Imperial College
Press, London.
Reiss, R. and Thomas, M. (2001;2007). Statistical Analysis
of Extreme Values, with Application to Insurance, Finance,
Hydrology and Other Fields. 2nd edition; 3nd edition,
Birkhuser Verlag.
Shabri, A.B., Daud, Z.M. and Ariﬀ, N.M. (2011). Regional
analysis of annual maximum rainfall using TL-moments
method. Theor Appl Climatol, 104, 561-570.

Liagkas_presentation

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Andere mochten auch

Andere mochten auch (19)

Liagkas_presentation