a v a i l a b l e a t w w w . s c i e n c e d i r e c t . c o m
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l
On the use of correlated beta random variables
with animal population modelling
Carlos Tadeu dos Santos Dias
a, A ri Sam aranay ak a
b, B ry an M anly
c,∗ aESALQ, University of Sao Paulo, Piracicaba, BrazilbUniversity of Otago, Dunedin, New Zealand
cWestern EcoSystems Technology Inc., Cheyenne, WY, USA
a r t i c l e
i n f o
Article history: R eceiv ed 2 M ay 2007 R eceiv ed in r ev is ed fo r m 6 M ar ch 2008 A ccept ed 13 M ar ch 2008 P u blis h ed o n lin e 15 M ay 2008 K eywords:Co r r elat ed n o n - n o r m al dis t r ibu t io n s D em o g r aph ic par am et er s
P o pu lat io n dy n am ics P o pu lat io n s im u lat io n
a b s t r a c t
W e g iv e r eas o n s w h y dem o g r aph ic par am et er s s u ch as s u r v iv al an d r epr o du ct io n r at es ar e o ft en m o delled w ell in s t o ch as t ic po pu lat io n s im u lat io n u s in g bet a dis t r ibu t io n s . In pr act ice, it is fr eq u en t ly ex pect ed t h at t h es e par am et er s w ill be co r r elat ed, fo r ex am ple w ith s u r v iv al r at es fo r all ag e clas s es t en din g t o be h ig h o r lo w in t h e s am e y ear. W e t h er efo r e dis cu s s a m et h o d fo r pr o du cin g co r r elat ed bet a r an do m v ar iables by t r an s fo r m in g co r r elat ed n o r m al r an do m v ar iables , an d s h o w h o w it can be applied in pr act ice by m ean s o f a s im ple ex am ple. W e als o n o t e h o w t h e s am e appr o ach can be u s ed t o pr o du ce co r r elat ed u n ifo r m , tr ian g u lar, an d ex po n en t ial r an do m v ar iables .
© 2008 E ls ev ier B .V . A ll r ig h t s r es er v ed.
1 .
Introduction
S t o ch as t ic po pu lat io n m o dellin g is fr eq u en t ly u s ed t o det er-m in e t h e lik ely fat e o f an an ier-m al po pu lat io n , in ar eas s u ch as pes t co n t r o l, h ar v es t m an ag em en t an d co n s er v at io n . F o r ex am ple, w it h pes t co n t r o l t h er e is in t er es t in w h et h er a pr o -po s ed co n t r o l m eas u r e w ill in fact lim it t h e s iz e o f t h e pes t po pu lat io n , w h ile w it h a t h r eat en ed s pecies t h er e is in t er-es t in t r y in g t o det er m in e t h e pr o babilit y t h at t h e po pu lat io n w ill s u r v iv e fo r a cer t ain n u m ber o f y ear s in t o t h e fu t u r e, an d t o as s es s t h e im pact o f m eas u r es in t en ded t o in cr eas e t h is pr o babilit y .
A n y po pu lat io n m o del is a co m pr o m is e bet w een a s im -plifi ed r epr es en t at io n o f t h e m o s t im po r t an t as pect s o f t h e po pu lat io n dy n am ics an d a co m plex m o del in co r po r at in g ev er y t h in g t h at co u ld affect t h e po pu lat io n n u m ber s . In pr ac-t ice, ac-t h e lev el o f r ealiac-t y r epr es en ac-t ed in a m o del depen ds o n
∗ Corresponding author. T el.: + 1 307 634 17 5 6; fax : + 1 307 637 69 8 1.
E - m ail addr es s :bm an ly @ w es t - in c.co m (B . M an ly ).
t h e m an ag em en t g o als an d t h e av ailable k n o w ledg e, an d w it h im pr o v em en t s in t h e u n der s t an din g s o f t h e u n der ly in g pr o -ces s es t h er e is a t en den cy t o u s e m o r e co m pr eh en s iv e m o dels t h at r epr es en t t h e po pu lat io n ch ar act er is t ics m o r e r ealis t i-cally (R ag en an d F o w ler, 19 9 2). T h is is do n e by in co r po r atin g ch ar act er is t ics s u ch as en v ir o n m en t al s t o ch as t icit y , den s it y depen den ce, s en es cen ce, po s s ible cat as t r o ph es , dis per s al an d s pecies in t er act io n s in t o t h e m o del (D ix o n et al., 19 9 7), an d u s in g s im u lat io n bas ed m et h o ds r at h er t h an an aly t ical m eth -o ds .
In t h is n o t e, w e co n cen t r at e o n t h e w ay t h at en v ir o n m en -t al s -t o ch as -t ici-t y can be allo w ed fo r in m o dels . In par -t icu lar, w e co n s ider t h e s it u at io n w h er e t h e dem o g r aph ic par am et er s o f a po pu lat io n ar e believ ed t o v ar y in a r an do m w ay w it h tim e, an d w h er e t h es e par am et er s m ay be co r r elat ed. T h u s , in a t y p-ical po pu lat io n m o del fo r an an im al w it h a life s pan in y ear s t h er e w ill be s ev er al ag e- s pecifi c y ear ly s u r v iv al r at es . T h es e
0304- 38 00/$ – s ee fr o n t m at t er © 2008 E ls ev ier B .V . A ll r ig h t s r es er v ed. do i:10.1016/j.eco lm o del.2008 .03.020
can be ex pect ed t o v ar y fr o m 1 y ear t o t h e n ex t , an d o ft en it w ill be des ir able t o m ak e t h e ag s pecifi c s u r v iv al r at es co r r e-lat ed t o allo w , fo r ex am ple, fo r s ev er e w in t er s t o affect all o f t h e s u r v iv al r at es in a s im ilar w ay . S im ilar ly , t h er e w ill o ft en be ag e- s pecifi c r epr o du ct io n r at es fo r t h e fem ales , w it h each o f t h es e bein g abo v e av er ag e in g o o d y ear s an d belo w av er ag e in bad y ear s . In s o m e cas es t h e s u r v iv al r at es an d r epr o du ct io n r at es m ay als o be co r r elat ed.
W e s u g g es t t h at a r elat iv ely s im ple w ay t o allo w fo r co r r e-lat ed dem o g r aph ic par am et er s is by as s u m in g t h at t h es e h av e bet a dis t r ibu t io n s . W e s h o w t h at t h e pr o per t ies o f t h e bet a dis -t r ibu -t io n m ak e -t h is v er y s u i-t able fo r m o dellin g dem o g r aph ic par am et er s , an d des cr ibe h o w co r r elat io n can be in t r o du ced.
2 .
A n ex ample population model
A s an ex am ple o f a s im ple po pu lat io n m o del, s u ppo s e t h at r epr o du ct io n o ccu r s o n a y ear ly bas is fo r fem ales t h at ar e at leas t 3 y ear s o ld, w it h t h e r epr o du ct io n r at e R o t h er w is e n o t depen din g o n ag e. S u ppo s e als o t h at t h e y ear ly s u r v iv al r at e is
S1fo r fem ales in t h eir fi r s t y ear o f life, S2fo r fem ales in t h eir s eco n d y ear o f life, an d S3fo r fem ales o lder t h an t w o y ear s . T h en if t h e dem o g r aph ic par am et er s v ar y fr o m y ear t o y ear t h e n u m ber s o f fem ales ag e 0, 1 an d 2 o r m o r e in y ear t + 1 w ill be g iv en by N1,t+1= RtS3,tN3,t, N2,t+1= S1,tN1,t, an d N3,t+1= S2,tN2,t+ S3,tN3,t, w h er e Rtis t h e r epr o du ct io n r at e in y ear t, an d S1,t, S2,tan d S3,t ar e t h e s u r v iv al r at es in t h at y ear.
T h es e eq u at io n s s ay t h at t h e fem ales ag e 0 in y ear t + 1 co m e fr o m t h e fem ales ag e 2 o r m o r e t h at s u r v iv e an d r epr o -du ce, t h e fem ales ag ed 1 in y ear t + 1 ar e t h e s u r v iv o r s fr o m t h o s e ag e 0 in t h e pr ev io u s y ear, an d t h e fem ales ag e 2 o r m o r e in y ear t + 1 ar e t h e s u r v iv o r s fr o m t h is g r o u p in y ear t, plu s t h e r ecr u it s t h at w er e ag e 1 in y ear t t h at ar e s t ill aliv e.
In a cas e lik e t h is it is r eas o n able t o m o del t h e s u r v iv al r at es
S1,t, S2,tan d S3,tby bet a r an do m v ar iables w it h a r an g e fr o m z er o t o o n e, an d t o m o del t h e r epr o du ct io n r at e Rtby a bet a r an do m v ar iable w it h a r an g e fr o m z er o u p t o t h e m ax im u m r epr o du ct io n r at e t h at is ev er lik ely t o o ccu r. T h e s u r v iv al r at es ar e t h en lik ely t o h av e a po s it iv e co r r elat io n , w h ich can be in t r o du ced q u it e eas ily in t o t h e m o del. T h e r epr o du ct io n r at e can als o be co r r elat ed w it h t h e s u r v iv al r at es if n eces s ar y .
M o dels s im ilar t o t h e o n e abo v e ar e v er y co m m o n ly u s ed in eco lo g y . F o r ex am ple, co n s ider t h e N ew Z ealan d s ea lio n po p-u lat io n w h ich is t h e o n ly en dem ic N ew Z ealan d pin n ipeds . T h e In t er n at io n al U n io n o f t h e Co n s er v at io n o f N at u r e an d N at u r al R es o u r ces h as clas s ifi ed t h is h ig h ly lo caliz ed s pecies as v u ln er able, an d t h e N ew Z ealan d g o v er n m en t h as a leg al o blig at io n t o pr o t ect it . A ll s ea lio n s pu ps ar e h eav ily depen -den t o n t h eir m o t h er s an d liv e m ain ly o n lan d fo r t h e fi r s t y ear o f life. T h ey liv e in w at er du r in g t h e s eco n d y ear o f life an d beco m e s ex u ally m at u r e at t h e ag e o f abo u t 3–5 y ear s . M at u r ed fem ales pr o du ce o n e pu p du r in g t h e an n u al br eed-in g s eas o n w it h a pu ppeed-in g eed-in t er v al o f abo u t 1–2 y ear s . S u r v iv al
r at es ar e lo w es t fo r pu ps an d h ig h es t fo r adu lt s . T h er e is a co m -m er cially i-m po r t an t -m id- w at er t r aw l fi s h er y fo r s q u id in t h e s am e ar ea as t h e s ea lio n s , w h ich ar e s u bject ed t o acciden -t al by ca-t ch . Co n s ider in g -t h e lev el o f by ca-t ch -t h er e is co n cer n abo u t t h e s u s t ain abilit y o f t h e s pecies , bu t an y r es t r ict io n o n fi s h in g act iv it ies is eco n o m ically s en s it iv e, h en ce s afeg u ar d-in g t h e s pecies h as beco m e a s er io u s m an ag em en t is s u e.
In o r der t o iden t ify s u it able m an ag em en t m eas u r esH ilbo r n an d W ade (19 9 9 ),M an ly an d W als h e (19 9 9 ), an dB r een et al. (2003)all u s ed s t o ch as t ic po pu lat io n m o dels s im ilar t o th e o n e des cr ibed abo v e t o as s es s t h e s u s t ain abilit y o f t h e s ea lio n po pu lat io n u n der v ar io u s s cen ar io s . T h ey u s ed u n co r r e-lat ed dem o g r aph ic r at es in t h eir s im u e-lat io n s bu t it is h ig h ly lik ely t h at t h e dem o g r aph ic r at es (s t ag e s pecifi c s u r v iv al r at es an d t h e r ecr u it m en t r at e) all beco m e lo w er in bad y ear s an d h ig h er in g o o d y ear s . F o r ex am ple, failu r e o f t h e g o v er n m en t t o clo s e t h e fi s h er y w h en t h e by cat ch t h r es h o ld is r each ed (D O C, 19 9 9), dis eas e o u t br eak s (B ak er, 19 9 9), o r u n fav o u r able w eat h er co n dit io n s m ay r es u lt in lo w s u r v iv al r at es fo r all s t ag es . O n t h e o t h er h an d, pr o t ect iv e m eas u r es s u ch as u s in g s ea lio n ex clu der dev ices o r en fo r cin g r es t r ict io n o n fi s h in g co u ld im pr o v e t h e s u r v iv al r at es o f all s t ag es . In ex am ples lik e t h is it is t h er efo r e appr o pr iat e t o u s e co r r elat ed dem o g r aph ic r at es fo r s im u lat io n s .
3 .
M ethods for introducing environmental
stochasticity into population models
E n v ir o n m en t al s t o ch as t icit y co m es fr o m t h e u n pr edict able effect s o f dem o g r aph ic an d en v ir o n m en t al fact o r s . T h e effect o f dem o g r aph ic fact o r s o n t h e dy n am ics o f a po pu lat io n is m o r e im po r t an t in s m all po pu lat io n s , w h er eas en v ir o n m en -t al fac-t o r s h av e a lar g er in fl u en ce o n lar g e po pu la-t io n s . T h e u s e o f in div idu al bas ed m o dels is an ex cellen t w ay o f r epr e-s en t in g dem o g r aph ic v ar iabilit y bet w een in div idu ale-s (G r o s s et al., 19 9 2). H o w ev er, t h is appr o ach is s eldo m u s ed ex cept w ith v er y s m all po pu lat io n s du e t o t h e diffi cu lt y o f t r ack in g each in div idu al in t h e m o del t h r o u g h o u t t h eir life. S t r u ct u r ed po p-u lat io n m o dels , s p-u ch as t h e ex am ple in t h e las t s ect io n , th en pr o v ide a r eas o n able alt er n at iv e.
S t o ch as t icit y can be added t o a po pu lat io n m o del by m ak -in g t h e dem o g r aph ic par am et er s v ar y w it h t im e. T h e r es u lt -in g m o del t h en r eq u ir es m o r e dat a t o det er m in e v ar ian ces as w ell as m ean v alu es fo r t h e par am et er s . In addit io n , few an aly t ical r es u lt s ar e av ailable co n cer n in g t h e pr o per t ies o f m o dels , s o t h at t h es e h av e t o be det er m in ed by s im u lat io n .
In t h e pas t v ar io u s appr o ach es h av e been pr o po s ed t o r ep-r es en t en v iep-r o n m en t al s t o ch as t icit y in po pu lat io n m o dels . O n e w ay in v o lv es es t im at in g a s epar at e s et o f v it al r at es fo r each t im e in t er v al.D ix o n et al. (19 9 7 )u s ed t h is appr o ach by es t i-m at in g a t ii-m e s er ies o f i-m at r ices o f dei-m o g r aph ic par ai-m et er s t o des cr ibe t h e in fl u en ce o f r an do m en v ir o n m en t al v ar ia-t io n . T h e adv an ia-t ag es o f ia-t h is appr o ach ar e ia-t h aia-t iia-t allo w s fo r bo t h dem o g r aph ic an d en v ir o n m en t al s t o ch as t icit y , av o ids t h e n eed fo r s im u lat io n m et h o ds , allo w s co m plex s t o ch as -t ic m o dels -t o be ev alu a-t ed u s in g an aly -t ical -t ech n iq u es , an d it in co r po r at es t h e au t o - co r r elat io n s an d cr o s s - co r r elat io n s t h at ex is t in fi eld dat a. T h e dr aw back s w it h t h e appr o ach ar e t h e co m plex it y in t h e an aly s is an d t h e im po s s ibilit y o f
es t im at in g t im e- s pecifi c dem o g r aph ic par am et er s w it h m o s t po pu lat io n s .
A s eco n d appr o ach is po s s ible if lo n g t im e s er ies o f es t i-m at es ar e av ailable fo r t h e dei-m o g r aph ic par ai-m et er s . In t h at cas e t h e dem o g r aph ic par am et er s fo r a s im u lat ed y ear can be det er m in ed by r an do m ly ch o o s in g t h e v alu es fo r o n e o f t h e y ear s w it h es t im at ed r at es . T h is is a t y pe o f n o n - par am et r ic bo o t s t r appin g (D av is o n an d H in k ley , 19 9 7), u s ed, fo r ex am -ple by K ay e an d P y k e (2003). A n adv an t ag e o f t h is m et h o d is t h at an y cr o s s - co r r elat io n r efl ect ed in t h e es t im at es o f t h e dem o g r aph ic par am et er s ar e au t o m at ically in co r po r at ed in t o t h e po pu lat io n pr o ject io n s . T h e m et h o d do es , h o w ev er, ig n o r e an y au t o co r r elat io n pr es en t in t h e t im e s er ies o f v it al r at es . A ls o , fo r m an y po pu lat io n s lo n g t im e s er ies o f es t im at es o f dem o g r aph ic par am et er s do n o t ex is t .
A t h ir d appr o ach t h at is co m m o n ly u s ed in v o lv es a t y pe o f par am et r ic bo o t s t r ap pr o cedu r e. In t h is cas e, t h e s t o ch as -t ic v ar iabili-t y o f each v i-t al r a-t e is ex pr es s ed by a s u i-t able s t at is t ical dis t r ibu t io n , w it h t h e par am et er s o f t h e dis t r ibu -t io n es -t im a-t ed fr o m -t h e v ar ia-t io n in es -t im a-t ed dem o g r aph ic par am et er s . D em o g r aph ic par am et er s fo r each t im e s t ep o f a po pu lat io n s im u lat io n ar e s am pled fr o m t h e es t im at ed dis t r i-bu t io n s .
T h e t h ir d m et h o d r eq u ir e t h e s pecifi cat io n o f s t at is t ical dis t r ibu t io n s fo r dem o g r aph ic par am et er s . In t h e abs en ce o f s u ffi cien t dat a t o det er m in e t h e r eal dis t r ibu t io n s , m o deller s o ft en t en d t o ju s t s elect a dis t r ibu t io n t h at s eem s r eas o n -able (K ay e an d P y k e, 2003), w h er e t h e s elect io n is s u bject iv e, an d depen den t o n pr ev io u s pr act ice an d co n v en ien ce fo r t h e m o deller. F o r ex am ple, t r u n cat ed n o r m al dis t r ibu t io n s w it h m ean s an d s t an dar d dev iat io n s s et eq u al t o o bs er v ed v alu es ar e m o s t co m m o n ly u s ed, w it h o u t t ak in g in t o acco u n t t h e po s s ible dis t o r t io n cau s ed by t r u n cat io n . In fact , t h e s elec-t io n o f dis elec-t r ibu elec-t io n s is v er y r ar ely ju s elec-t ifi ed in elec-t h e lielec-t er aelec-t u r e, w h ich r ais es t h e pr o blem o f w h et h er t h e m o del o u t co m es ar e s en s it iv e t o t h e s elect io n o f t h e dis t r ibu t io n .
S am ar an ay ak a (2005 )dev elo ped a s et o f cr it er ia t o co n s ider w h en s elect in g a dis t r ibu t io n t o r epr es en t en v ir o n m en t al v ar i-at io n , as fo llo w s :
(a) A n y dis t r ibu t io n is defi n ed by it s par am et er s , an d h en ce t h o s e par am et er s s h o u ld be able t o be s pecifi ed fr o m t h e av ailable in fo r m at io n abo u t t h e dem o g r aph ic par am et er bein g co n s ider ed. T h e m o s t co m m o n ly av ailable in fo r m a-t io n abo u a-t a-t h e s a-t o ch as a-t ic v ar iaa-t io n is a s er ies o f es a-t im aa-t es o v er a t im e per io d, o r s o m et im es s im ply t h e es t im at es fo r t h e m ean an d v ar ian ce o f t h e dem o g r aph ic par am e-t er. T h er efo r e, e-t h e par am ee-t er s fo r e-t h e s elece-t ed dis e-t r ibu e-t io n n eed t o be s pecifi ed fr o m t h is in fo r m at io n .
(b) T h e s elect ed dis t r ibu t io n s h o u ld h av e t h e abilit y t o m at ch t h e s h ape o f t h e dis t r ibu t io n o f fi eld dat a, o r t h e s h ape t h at is co n s ider ed appr o pr iat e fo r t h e applicat io n . T h is do es n o t ju s t r efer t o t h e po s s ibilit y o f h av in g v ar io u s s h apes . It r efer s r at h er t o t h e abilit y o f t h e dis t r ibu t io n t o ch an g e it s s h ape t o s u it t h e v ar iat io n r epr es en t ed by dat a. T h e in t en ded s h ape w ill be u n im o dal in m o s t cas es , bu t m u lt i- m o dal s h apes , J s h apes , an d U s h apes can als o s o m et im es be des ir able, pr o v ided t h at t h ey ar e n o t cr eat ed u n in t en t io n ally as a co n s eq u en ce o f s elect in g a dis t r ibu t io n .
(c) T h e v alu es o f dem o g r aph ic v ar iables g en er at ed fr o m t h e dis t r ibu t io n s h o u ld h av e t h e in t en ded m ean an d v ar iabil-it y . It h as been fo u n d t h at t h e v alu es g en er at ed fr o m s o m e dis t r ibu t io n s do n o t h av e t h e co r r ect m ean an d s t an dar d dev iat io n as a co n s eq u en ce o f eit h er t h e diffi cu lt y o f s pec-ify in g t h e par am et er s fo r t h e s elect ed dis t r ibu t io n w it h t h e r eq u ir ed accu r acy o r t h e diffi cu lt y o f g en er at in g v alu es fr o m t h e dis t r ibu t io n (S am ar an ay ak a, 2005).
(d) It s h o u ld be po s s ible t o s im u lat e dat a fr o m t h e s elect ed dis t r ibu t io n fo r all r ealis t ic co m bin at io n s o f t h e m ean an d s t an dar d dev iat io n , u s in g o n e o f t h e s t an dar d alg o r it h m s (e.g .,F is h m an , 19 7 3; K en n edy an d G en t le, 19 8 0; R ipley , 19 8 7 ; S w ar t z m an an d K alu z n y , 19 8 7; an dB u r g m an et al., 19 9 3).
(e) T h e s elect ed dis t r ibu t io n s h o u ld be able t o s im u lat e v al-u es w it h in appr o pr iat e lim it s , w h ich ar e z er o t o o n e in t h e cas e o f a s u r v iv al r at e, an d n o n - n eg at iv e in t h e cas e o f a r epr o du ct io n r at e. In fact , it is im po s s ible t o en s u r e t h at v alu es ar e w it h in t h e appr o pr iat e r an g e fo r s o m e co m m o n ly u s ed dis t r ibu t io n s , s u ch as t h e n o r m al o r lo g -n o r m al w it h o u t im po s i-n g s o m e t r u -n cat io -n . I-n s o m e cas es t h e pr o per t ies o f dat a g en er at ed fr o m t r u n cat ed dis t r ibu -t io n s w ill n o -t be -t h e s am e as in -t en ded. D epen din g o n -t h e deg r ee o f t r u n cat io n an d it s as y m m et r y , t r u n cat io n can co n s ider ably dis t o r t t h e dis t r ibu t io n o f g en er at ed v ar iat es fr o m t h e in t en ded dis t r ibu t io n , as s h o w n byCas w ell (2001, p. 412)fo r ex am ple.
(f) O n ce a po s s ible r an g e fo r a dem o g r aph ic v ar iable is det er m in ed, s u ch as z er o t o o n e fo r a s u r v iv al r at e, t h e co m bin at io n s o f t h e m ean an d t h e s t an dar d dev iatio n o f t h e v ar iable ar e au t o m at ically r es t r ict ed. T o be s pecifi c, it is eas y t o s h o w t h at , if a v ar iable m u s t be w it h in t h e r an g e (a, b) t h en t h e co n s t r ain t
2≤ (b − )( − a) (1)
m u s t apply , w h er e is t h e m ean an d 2is t h e v ar ian ce o f t h e dis t r ibu t io n . It is t h en des ir able t h at t h e dis t r ibu t io n can h av e all o r m o s t o f t h e co m bin at io n s o f t h e m ean an d v ar ian ce t h at ar e po s s ible, s u bject t o t h is co n s t r ain t. In t h is r es pect , t h e s h ape o f t h e dis t r ibu t io n is als o im po r-t an r-t . F o r ex am ple, if a u n im o dal dis r-t r ibu r-t io n is des ir ed t h en t h e v ar ian ce m u s t be les s t h an t h at o f a u n ifo r m dis -t r ibu -t io n w i-t h in -t h e r an g e (a, b), i.e. i-t is n eces s ar y -t h a-t
2≤(b − a) 2
12 .
(g ) B ecau s e t h er e ar e lim it at io n s t o co m bin at io n s o f t h e m ean an d v ar ian ce t h at can o ccu r w it h a s pecifi c s h ape fo r a dis t r ibu t io n it is u s efu l t o h av e t h e abilit y t o det er m in e w h at t h e s h ape w ill be fr o m t h e v alu es o f t h e m ean an d v ar ian ce. T h is is n o t po s s ible w it h all dis t r ibu t io n s . (h ) A s n o t ed in S ect io n 1, dem o g r aph ic v ar iables ar e o ft en
co r r elat ed fo r v ar io u s r eas o n s . F o r ex am ple, fav o u r able en v ir o n m en t al co n dit io n s can im pr o v e bo t h t h e s u r v iv al an d r epr o du ct io n r at es fo r a po pu lat io n (po s it iv e co r r e-lat io n ), a lo w s u r v iv al r at e fo r an im als m ay lead t o an in cr eas ed r epr o du ct io n r at e fo r t h e s u r v iv in g an im als (n eg at iv e co r r elat io n ), an d o n e h ealt h y co h o r t can co n
-t r ibu -t e -t o im pr o v ed r epr o du c-t io n in s ev er al y ear s in a r o w (co r r elat io n in t im e). It is t h er efo r e des ir able t o be able t o g en er at e co r r elat ed r an do m v ar iables fr o m t h e ch o s en dis t r ibu t io n s fo r dem o g r aph ic v ar iables .
O n t h e bas is o f t h e abo v e pr o per t ies t h e bet a dis t r ibu t io n is bet t er s u it ed t h an m o s t o t h er co m m o n ly u s ed dis t r ibu t io n s fo r r epr es en t in g en v ir o n m en t al s t o ch as t icit y in po pu lat io n dy n am ics m o dels . T h er efo r e, w e co n s ider m o r e abo u t t h e pr o per t ies o f t h e bet a dis t r ibu t io n in t h e fo llo w in g s ect io n .
4 .
A dvantages of using beta distributions
T h e pr o babilit y den s it y fu n ct io n o f bet a dis t r ibu t io n o v er t h e r an g e a–b is defi n ed as
f(x) ={(x − a)˛−1(b − x)ˇ−1} {B(˛, ˇ) (b − a)˛+ˇ−1}
,
w h er e ˛ > 0, ˇ > 0, an d B(˛,ˇ) is t h e bet a fu n ct io n (Jo h n s o n et al., 19 9 4). H er e, a an d b ar e t h e t w o bo u n dar ies o f t h e dis t r ibu t io n , s o t h at g en er at ed v alu es o f r at es w ill be w it h in t h at r an g e. T h is elim in at es t h e pr o blem s as s o ciat ed w it h t r u n cat ed dis t r ibu -t io n s fo r s u r v iv al an d r epr o du c-t io n r a-t es . In pr ac-t ice, m o s -t be-t a r an do m n u m ber g en er at o r s pr o du ce r an do m n u m ber s o v er t h e r an g e (0, 1). It is t h er efo r e u s u al t o g en er at e v alu es o v er t h is r an g e an d t r an s fo r m t h em t o an o t h er r an g e if n eces s ar y . T h e bet a dis t r ibu t io n w it h a = 0 an d b = 1 is called t h e s t an -dar d bet a dis t r ibu t io n . T h e t w o s h ape par am et er s fo r t h is dis t r ibu t io n can be s pecifi ed in t er m s o f t h e des ir ed v alu es fo r t h e m ean an d v ar ian ce fr o m t h e eq u at io n s ˛ = (1 − ) 2 − 1
!
, (2) an d ˇ = (1 − ) (1 − ) 2 − 1!
. (3)T h e par am et er s pecifi cat io n is t h er efo r e ex act , r at h er t h an appr o x im at e as it is w it h s o m e dis t r ibu t io n s .
T h er e ar e s t an dar d alg o r it h m s av ailable in t h e lit er at u r e t h at can be u s ed in g en er at in g r an do m n u m ber s fr o m a bet a dis t r ibu t io n o n ce t h e t w o s h ape par am et er s ar e k n o w n (e.g . s eeK en n edy an d G en t le, 19 8 0; M o r r is an d D o ak , 2002; S t at L ib, 2004). H o w ev er, car e is n eces s ar y w h en s elect in g an alg o r it h m becau s e it h as been fo u n d t h at t h e pr o per t ies o f t h e v ar iables g en er at ed u s in g s o m e o f t h e pu blis h ed alg o r it h m s differ fr o m w h at w as in t en ded (S am ar an ay ak a, 2005).
F r o m E q .(1)it is k n o w n t h at t h e s t an dar d dev iat io n o f an y dis t r ibu t io n w it h in t h e r an g e (0, 1) m u s t s at is fy t h e co n s t r ain t t h at
2≤ (1 − ).
T h e s t an dar d bet a dis t r ibu t io n is defi n ed fo r all co m bin a-t io n s o f a-t h e m ean an d v ar ian ce a-t h aa-t s aa-t is fi es a-t h is co n s a-t r ain a-t . It can t h er efo r e be u s ed t o r epr es en t v ar iat io n w it h all t h eo -r et ically po s s ible co m bin at io n s o f t h e m ean an d v a-r ian ce.
F ig. 1 – T he shape of the beta distribution for different combinations of the mean and standard distribution (U , U - shaped; R , reverse J- shaped; U ni, unimodal; and J, J- shaped). B eta distributions are not defi ned above the semi- circular region.
T h e bet a dis t r ibu t io n can t ak e w ide v ar iet y o f s h apes in clu din g u n ifo r m , t r ian g u lar, u n im o dal, J s h aped, an d U -s h aped, depen din g o n t h e v alu e-s o f t h e par am et er -s ˛ an d ˇ. T h er efo r e, t h is dis t r ibu t io n can acco m m o dat e alm o s t all r eq u ir ed s h apes fo r t h e dis t r ibu t io n o f dem o g r aph ic par am -et er s . F u r t h er m o r e, becau s e t h e par am -et er s ˛ an d ˇ ar e fu n ct io n s o f t h e m ean an d t h e s t an dar d dev iat io n , t h e s h ape o f t h e dis t r ibu t io n is det er m in ed ex act ly by t h e co m bin at io n o f m ean an d s t an dar d dev iat io n u s ed, as s h o w n in F ig . 1. T h is m ak es it po s s ible t o be s u r e in adv an ce w h at t h e s h ape o f th e g en er at ed dis t r ibu t io n w ill be fo r a g iv en ch o ice o f t h e m ean an d s t an dar d dev iat io n .
T h e diffi cu lt y o f g en er at in g r an do m v ar iat es w it h a s peci-fi ed co r r elat io n s t r u ct u r e w as dis cu s s ed by T o dd an d N g (2001). A lt h o u g h t h ey co n s ider ed t h e bet a dis t r ibu t io n t o be m o s t s u it able fo r r epr es en t in g en v ir o n m en t al s t o ch as t icit y , t h ey s u g g es t ed t h e u s e o f a m et h o d fo r g en er at in g co r r elated s u r-v ir-v al r at es bas ed o n t h e P r o bit dis t r ibu t io n becau s e t h ey w er e u n aw ar e h o w t o g en er at e co r r elat ed v ar iat es fr o m t h e bet a dis t r ibu t io n .
A t leas t t w o m et h o ds h av e been u s ed in t h e lit er at u r e to g en er at e co r r elat ed v ar iables fo r po pu lat io n m o dellin g .D o ak et al. (19 9 4)as s u m ed t h e dem o g r aph ic r at es ar e dr iv en by a co m m o n en v ir o n m en t al fact o r, an d s im plifi ed t h e co r r elat io n s t r u ct u r e bet w een dem o g r ah ic r at es by u s in g t h e in div idu al co r r elat io n s w it h t h at en v ir o n m en t al fact o r, w h ileG r o s s et al. (19 9 8 )fi r s t g en er at ed s t an dar d n o r m al v ar iat es w ith th e r eq u ir ed co r r elat io n s , an d t h en t r an s fo r m ed t h em t o bet a v ar i-at es w it h t h e r eq u ir ed m ean an d s t an dar d dev ii-at io n . T h e G r o s s et al. t y pe o f appr o ach is q u it e g en er al an d w e dis cu s s t h is appr o ach in t h e fo llo w in g s ect io n fo r t h e g en er at io n o f co r r elat ed v alu es fr o m a v ar iet y o f dis t r ibu t io n s .
5 .
T he generation of correlated random
variables
W e co n s ider t h e appr o ach fo r g en er at in g co r r elat ed r an do m v ar iables u s ed by G r o s s et al. (19 9 8 ), bu t g en er aliz ed t o a n u m
-ber o f differ en t t y pes o f dis t r ibu t io n . T h e idea in all cas es is t o fi r s t pr o du ce u n ifo r m ly dis t r ibu t ed r an do m v ar iables o v er t h e r an g es (0,1) w it h t h e r eq u ir ed co r r elat io n s . T h es e ar e t h en t r an s fo r m ed t o v alu es fr o m t h e dis t r ibu t io n o r dis t r ibu t io n s o f in t er es t if t h es e ar e n o t u n ifo r m . In s o m e cas es adju s t m en t s t o t h e co r r elat io n s ar e n eeded s o as t o o bt ain t h e r eq u ir ed co r-r elat io n s fo r-r t h e fi n al v ar-r iables . A ls o , in s o m e cas es t h e r-r an g e o f co r r elat io n s t h at can be h an dled is r es t r ict ed. W e beg in by co n s ider in g t h e g en er at io n o f t h e p u n ifo r m r an do m v ar iables o v er t h e r an g e (0,1) w it h s pecifi ed co r r elat io n s .
5 .1 . G eneration of correlated uniform random
v ariables
L et (X1, X2,. . ., Xp) be a r an do m s am ple fr o m a m u lt iv ar iat e n o r m al dis t r ibu t io n w it h t h e co v ar ian ce m at r ix
V =
1.0 12 13 . . . 1p 21 1.0 23 . . . 2p · · · · · · · · · · p1 p2 p3 . . . 1.0
,an d z er o m ean s fo r all v ar iables . T h is r an do m s am ple can be g en er at ed, fo r ex am ple, by t h e alg o r it h m o f B edall an d Z im m er m an (19 7 6). N o t e t h at becau s e o f t h e u n it v ar ian ces t h e co v ar ian ce m at r ix is als o t h e co r r elat io n m at r ix .
G iv en t h is s it u at io n , it is w ell k n o w n t h at if F(x) is t h e cu m u -lat iv e dis t r ibu t io n fu n ct io n o f a s t an dar d n o r m al v ar iable t h en
Ui= F(Xi) h as a u n ifo r m dis t r ibu t io n o n t h e in t er v al (0,1), i.e., t h e pr o babilit y o f a v alu e les s t h an o r eq u al t o Xiis u n ifo r m ly dis t r ibu t ed fo r all i. F u r t h er m o r e, t h e co r r elat io n bet w een Ui an d Ujw ill eq u al Uij=
(
6)
ar cs in(
ij 2)
(W ilk s , 19 62). A lt h o u g h it is n o t appar en t fr o m t h is las t eq u a-t io n , ia-t do es s h o w a-t h aa-t ijan d Uijar e v er y s im ilar, becau s e o v er t h e r an g e o f ijfr o m z er o t o o n e t h e lar g es t abs o lu t e differ en ce fr o m Uij is o n ly abo u t 0.02. F u r t h er m o r e, if t h e co r r elat io n bet w een Xian d Xjis s et eq u al t o
ij= 2 s in
(
Uij6
)
t h en t h is en s u r es t h at t h e co r r elat io n bet w een Uian d Ujeq u als Uij.
5 .2 . G eneration of correlated beta v ariables
O n ce co r r elat ed u n ifo r m ly dis t r ibu t ed v ar iables o v er t h e r an g e (0,1) h av e been g en er at ed t h es e can be co n v er t ed t o s t an dar d bet a r an do m v ar iables u s in g t h e in v er s e o f t h e bet a dis t r i-bu t io n fu n ct io n . T h u s , if Ui is t h e it h u n ifo r m ly dis t r ibu t ed r an do m v ar iable t h en t h e eq u at io n Ui=
*
Yi 0 {(x − a)˛−1(b − x)ˇ−1} {B(˛, ˇ)(b − a)˛+ˇ−1} dxis s o lv ed t o det er m in e t h e v alu e o f Yi. T h e v alu es o f Y1, Y2, . . .
Ypw ill t h en be co r r elat ed v ar iables w it h bet a dis t r ibu t io n s . A lt h o u g h t h is is t h e cas e, t h e n u m er ical m et h o d fo r der iv in g
F ig. 2 – T he relationship between the correlation of two standard beta distributions and the correlation between the two standard normal variables used to generate them. T he points in the fi gure have been jiggered slightly horiz ontally to show the distribution of the beta correlations better for the normal correlations of −0 .9 5 , −0 .9 0 , −0 .7 5 , −0 .5 0 , −0 .2 5 , −0 .1 0 , 0 .0 0 , 0 .1 0 , 0 .2 5 , 0 .5 0 , 0 .7 5 , 0 .9 0 and 0 .9 5 used to generate the beta variables. T he fi tted regression line shown gives B eta C orrelation = 0 .9 3 1 (N ormal C orrelation).
t h e v alu es o f t h e bet a v ar iables m ean s t h at t h eir jo in t dis tr i-bu t io n do es n o t n eces s ar ily ag r ee w it h a s t an dar d t h eo r etical bet a dis t r ibu t io n .
It m ig h t w ell be ex pect ed t h at t h e co n v er s io n o f t h e u n i-fo r m r an do m v ar iables t o bet a r an do m v ar iables w ill s er io u s ly m o dify t h e co r r elat io n s bet w een t h e v ar iables . H o w ev er, it t u r n s o u t t h at t h is is n o t n eces s ar ily t h e cas e. A s s h o w n in F ig . 2, t h e co r r elat io n bet w een t w o g en er at ed bet a v ar iables is appr o x im at ely eq u al t o t h e co r r elat io n bet w een t h e tw o n o r m al v ar iables u s ed t o g en er at e t h em , m u lt iplied by 0.9 31. T h e appr o x im at io n is fair ly g o o d ex cept fo r v er y h ig h o r v er y lo w n o r m al co r r elat io n s . F o r ex am ple, if t h e n o r m al co r-r elat io n is 0.9 5 t h en t h e bet a co r-r r-r elat io n m ay be an y w h er-r e bet w een abo u t 0.6 t o 0.9 5 .
T o pr o du ceF ig . 2, a t o t al o f 7 29 3 s it u at io n s w er e co n s id-er ed. T h e fi r s t bet a v ar iable Y1 h ad o n e o f 33 dis tr ibu t io n s w it h a m ean 1 o f 0.05 , 0.15 , 0.25 , 0.35 , 0.45 , 0.5 0, 0.5 5 , 0.65 , 0.7 5 , 0.8 5 o r 0.9 5 , an d a s t an dar d dev iat io n 1 eit h er lo w (0.05 M in [1,1 − 1]), m ediu m (0.5 M in [1, 1 − 1]), o r h ig h (M in (1,1 − 1]). T h e s eco n d bet a v ar iable Y2t h en h ad a m ean 2an d a s t an dar d dev iat io n 2ch o s en fr o m t h e s am e lis t o f 33 dis t r ibu t io n s as fo r v ar iable 1, bu t w it h o u t t h e s am e pair o f dis t r ibu t io n s r epeat ed in a r ev er s ed o r der. T h is t h en g av e 33C
2+ 33 = 5 61 differ en t pair s o f dis t r ibu t io n s , in clu din g 33 s it -u at io n s w h er e t h e t w o bet a dis t r ib-u t io n s h ad eq -u al m ean s an d s t an dar d dev iat io n s . F o r each o f t h es e pair s o f dis t r ibu t io n s t h e 13 co r r elat io n s −0.9 5 , −0.9 0, −0.7 5 , −0.5 0, −0.25 , −0.10, 0.00, 0.10, 0.25 , 0.5 0, 0.7 5 , 0.9 0 an d 0.9 5 w er e co n s ider ed fo r t h e n o r m al v ar iables u s ed t o g en er at e t h e bet a v ar iables . F o r each o f t h e 5 61 × 13 = 7 29 3 s itu atio n s co n s ider ed a to tal o f 10,000 pair s o f bet a v alu es w er e g en er at ed in o r der t o es t im at e t h e co r r elat io n g en er at ed fo r t h e bet a v ar iables .
F o r all o f t h e s it u at io n s co v er ed in F ig . 2t h e co effi cien t s o f v ar iat io n fo r Y1, 1 − Y1, Y2an d 1 − Y2w er e alw ay s les s t h an o r eq u al t o o n e. S im u lat io n r es u lt s n o t s h o w n h er e dem o n s t r at e
t h at if o n e o f t h es e co effi cien t s o f v ar iat io n is lar g er t h an o n e t h en t h e r elat io n s h ip bet w een t h e bet a co r r elat io n an d t h e n o r m al co r r elat io n beco m es a co m plicat ed fu n ct io n o f t h e n o r-m al co r r elat io n , 1, 1, 2an d 2. If a s it u at io n lik e t h is n eeds t o be s im u lat ed t h en t h e n o r m al co r r elat io n r eq u ir ed t o g en er-at e t h e bet a der-at a can be det er m in ed u s in g a W in do w s pr o g r am B et aR V 2.ex e t h at is av ailable fr o m t h e W es t er n E co S y s t em s T ech n o lo g y w eb s it e as des cr ibed belo w . T h is pr o g r am can als o be u s ed if t h e r elat io n s h ip
B et a Co r r elat io n = 0.9 31(N o r m al Co r r elatio n ) (4) s h o w n in F ig . 2is n o t co n s ider ed t o be accu r at e en o u g h fo r an applicat io n .
A s an ex am ple o f t h e g en er at io n o f bet a v ar iables co n s ider t h e po pu lat io n m o del des cr ibed in S ect io n 2, an d s u ppo s e t h at it is decided t h at t h e s u r v iv al r at es S1, S2 an d S3, an d t h e r epr o du ct io n r at e R s h o u ld h av e co r r elat ed bet a dis t r i-bu t io n s . S pecifi cally , s u ppo s e t h at it is des ir ed t h at S1s h o u ld h av e a bet a dis t r ibu t io n w it h m ean 1= 0.2 an d s t an dar d dev i-at io n 1= 0.2, S2 s h o u ld h av e a bet a dis t r ibu t io n w it h m ean 2= 0.5 an d s t an dar d dev iat io n 2= 0.1, an d S3s h o u ld h av e a bet a dis t r ibu t io n w it h m ean 3= 0.9 an d s t an dar d dev iat io n 3= 0.15 , w it h all t h r ee o f t h es e dis t r ibu t io n s o v er t h e r an g e (0,1). S u ppo s e als o t h at it is des ir ed t h at t h e r epr o du ct io n r at e s h o u ld h av e a bet a dis t r ibu t io n o v er t h e r an g e (0,0.5 ), w it h a m ean o f 0.2 an d a s t an dar d dev iat io n o f 0.1. In t h at cas e v al-u es o f 2R can be g en er at ed fr o m t h e bet a dis t r ibal-u t io n o v er t h e r an g e (0,1) w it h a m ean o f 4= 0.4 an d a s t an dar d dev iat io n o f 4= 0.2. T h e par am et er s o f t h e bet a dis t r ibu t io n co r r es po n d-in g t o t h es e m ean s an d s t an dar d dev iat io n s ar e t h en ˛1= 0.6, ˇ1= 2.4, ˛2= 12.0, ˇ2= 12.0, ˛3= 2.7 , ˇ3= 0.3, ˛4= 2.0 an d ˇ4= 3.0, r es pect iv ely .
T h e co r r elat io n s bet w een S1, S2an d S3m ig h t be ex pect ed t o be fair ly h ig h , w it h lo w er co r r elat io n s bet w een t h e s u r v iv al r at es an d t h e r epr o du ct io n r at e. S u ppo s e, t h er efo r e, t h at it is decided t h at t h e co r r elat io n m at r ix s h o u ld be eq u al t o t h e o n e s h o w n in par t (a) o f T able 1. A pply in g E q .(4)t h en s u g g es t s t h at t h e co r r elat io n m at r ix r eq u ir ed fo r t h e g en er at in g n o r-m al dis t r ibu t io n s s h o u ld be as g iv en in par t (b) o f t h e t able. H o w ev er, par t (c) o f t h e t able s h o w s t h e ex act n o r m al co r r e-lat io n s t h at w ill pr o du ce t h e r eq u ir ed bet a co r r ee-lat io n s . It can be s een t h at E q .(4)g iv es appr o x im at ely t h e co r r ect n o r m al co r r elat io n s ex cept fo r t h e co r r elat io n bet w een S1an d S3(0.5 4 in s t ead o f 0.7 9 ) an d t h e co r r elat io n bet w een S2an d S3 (0.8 6 in s t ead o f 0.9 5 ).
F ig. 3 – T he relationship between the correlation of two triangular distributions and the correlation between the two standard normal variables used to generate the triangular variables. T he fi tted regression line shown gives T riangular C orrelation = −0 .0 0 1 + 0 .9 9 5 (N ormal C orrelation).
5 .3 . G eneration of correlated triang ular v ariables
T h e appr o ach u s ed t o g en er at e co r r elat ed bet a v ar iables can als o be u s ed t o g en er at e co r r elat ed t r ian g u lar v ar iables . F o r s im plicit y w e as s u m e t h at t h e t r ian g u lar v ar iables ar e all w it h in t h e r an g e fr o m z er o t o o n e. In pr act ice t h ey can th en be co ded t o h av e an y o t h er r eq u ir ed r an g es . A s s u m e t h at Ti, t h e it h o f t h e t r ian g u lar v ar iables is r eq u ir ed t o h av e a m o de at
mi. T h en t h is can be g en er at ed fr o m t h e it h u n ifo r m r an do m v ar iable Uiby s o lv in g t h e eq u at io n
Ui= F(Ti)
w h er e F(t) is t h e cu m u lat iv e dis t r ibu t io n fu n ct io n o f t h e t r i-an g u lar dis t r ibu t io n . T h e eq u at io n r edu ces t o Ti=√(miUi), fo r 0 ≤ Ui≤ mi, an d Ti= 1 −√{(1 − Ui)(1 − mi)}, fo r mi< Ui≤ 1.
W h en t h e t r an s fo r m at io n fr o m u n ifo r m r an do m v ar iables t o t r ian g u lar r an do m v ar iables w as applied m an y t im es w ith t h e m o des o f t h e t r ian g u lar dis t r ibu t io n s r an g in g fr o m 0.05 t o 0.9 5 an d co r r elat io n s r an g in g fr o m −0.9 9 to + 0.9 9 it w as fo u n d t h at t h e co r r elat io n o f t h e t r ian g u lar dis t r ibu t io n s w as v er y s im ilar t o t h e co r r elat io n fo r t h e n o r m al dis t r ibu t io n s u s ed fo r g en er at in g t h e co r r elat ed u n ifo r m dis t r ibu t io n s , as s h o w n in F ig . 3. T h e fi t t ed r eg r es s io n r elat io n s h ip is
T r ian g u lar Co r r elat io n = −0.001 + 0.9 9 5 (N o r m al Co r r elatio n ).
T able 1 – T he desired correlation matrix for beta variables for the ex ample described in S ection 2 , the normal distribution matrix based on the eq uation B eta C orrelation = 0 .9 3 1 (N ormal C orrelation), and the ex act normal distribution matrix as determined using the W indows program B etaR V 2 .
(a) D es ir ed bet a co r r elat io n s (b) N o r m al co r r elat io n s bas ed o n eq u at io n (c) E x act n o r m al co r r elat io n s S1 S2 S3 R S1 S2 S3 R S1 S2 S3 R S1 1.00 0.8 0 0.5 0 0.20 1.00 0.8 6 0.5 4 0.21 1.00 0.8 6 0.7 9 0.21 S2 0.8 0 1.00 0.8 0 0.30 0.8 6 1.00 0.8 6 0.32 0.8 6 1.00 0.9 5 0.30 S3 0.5 0 0.8 0 1.00 0.35 0.5 4 0.8 6 1.00 0.38 0.7 9 0.9 5 1.00 0.43 R 0.20 0.30 0.35 1.00 0.21 0.32 0.38 1.00 0.21 0.30 0.43 1.00
F ig. 4 – T he relationship between the correlation of two ex ponential distributions and the correlation between the two standard normal variables used to generate the ex ponential variables. T he fi tted regression line for positive correlations is E x ponential
C orrelation = −0 .0 2 1 + 0 .9 9 3 (N ormal C orrelation), while for negative correlations the line is E x ponential
C orrelation = −0 .0 1 7 + 0 .6 4 9 (N ormal C orrelation).
5 .4 . G eneration of correlated ex p onential v ariables
A pply in g t h e s am e appr o ach as u s ed fo r g en er at in g co r r elat ed bet a an d t r ian g u lar v ar iables t o ex po n en t ial v ar iables in v o lv es fi n din g t h e it h ex po n en t ial r an do m v ar iable Eiby s o lv in g t h e eq u at io n
Ui= F(Ei),
w h er e Uiis t h e it h co r r elat ed u n ifo r m v ar iable an d F(Ei) is t h e cu m u lat iv e dis t r ibu t io n fu n ct io n o f t h e ex po n en t ial dis t r ibu -t io n w i-t h m ean i. T h is eq u at io n s im plifi es t o
Ei= −ilo ge(1 − Ui)
w h er e i is t h e m ean o f t h e ex po n en t ial dis t r ibu t io n . T h e s t an dar d dev iat io n o f t h e dis t r ibu t io n is als o eq u al t o i. If n eces s ar y t h e ex po n en t ial v ar iables can be t r an s fo r m ed t o h av e an y o t h er des ir ed m ean s an d s t an dar d dev iat io n s w it h -o u t ch an g in g t h e c-o r r elat i-o n s bet w een t h em .
F r o m g en er at in g m an y s et s o f dat a w it h co r r elat io n s bet w een t h e n o r m al v ar iables u s ed t o g en er at e t h e v ar y in g fr o m −0.9 9 to + 0.9 9 it w as fo u n d th at th e r elatio n s h ip betw een t h e n o r m al co r r elat io n an d t h e ex po n en t ial co r r elat io n is g iv en appr o x im at ely by
E x po n en t ial Co r r elat io n = −0.017 + 0.649 (N o r m al Co r r elatio n ) fo r n eg at iv e n o r m al co r r elat io n s , an d
E x po n en t ial Co r r elat io n = −0.021 + 0.9 9 3(N o r m al Co r r elatio n ) fo r po s it iv e n o r m al co r r elat io n s , as s h o w n in F ig . 4. H en ce, it is po s s ible t o g en er at e ex po n en t ial r an do m v ar iables w it h co r r elat io n s fr o m abo u t −0.63 to + 0.9 7 in th e m an n er des cr ibed.
6 .
D iscussion
T h e appr o ach fo r g en er at in g co r r elat ed r an do m v ar iables dis -cu s s ed h er e is r elat iv ely s im ple t o u s e, an d s eem s par ti-cu lar ly s u it able fo r m o dellin g dem o g r aph ic par am et er s u s in g co r r e-lat ed bet a v ar iables .
A co n s ider able adv an t ag e is t h at t h is appr o ach allo w s m an y s im u lat io n s t o be s et u p in a s pr eads h eet . S u ppo s e, fo r ex am ple t h at p co r r elat ed bet a v ar iables n eed t o be g en -er at ed in a s pr eads h eet . T h en t o beg in w it h p s t an dar d n o r m al v ar iables can be g en er at ed fr o m p u n ifo r m r an -do m n u m ber s in t h e r an g e (0,1) u s in g t h e in v er s e n o r m al fu n ct io n . T h e n o r m al v ar iables can t h en be t r an s fo r m ed t o s t an dar diz ed co r r elat ed r an do m v ar iables by lin ear t r an s -fo r m at io n s , an d t h e cu m u lat iv e n o r m al fu n ct io n u s ed t o pr o du ce co r r elat ed u n ifo r m r an do m v ar iables . F in ally , t h e in v er s e bet a dis t r ibu t io n fu n ct io n can be u s ed t o co n v er t t h e u n ifo r m r an do m v ar iables t o p co r r elat ed bet a r an do m v ar iables .
In m an y cas es t h e t r an s fo r m at io n o f v ar iables ch an g es t h e co r r elat io n bet w een t h e v ar iables by o n ly a s m all am o u n t , as s h o w n in F ig s . 2–4. In o t h er cas es t h e ch an g e in co r r elat io n s m ay be s u bs t an t ial bu t can be det er m in ed if n eces s ar y by s im -u lat in g t r an s fo r m at io n s -u n der a w ide r an g e o f co n dit io n s . In par t icu lar, a W in do w s pr o g r am B et aR V 2.ex e can be u s ed t o det er m in e t h e n o r m al dis t r ibu t io n co r r elat io n s t h at pr o du ce s pecifi ed bet a dis t r ibu t io n co r r elat io n s . T h is an d a pr o g r am B et aR V .ex e t h at can be u s ed fo r g en er at in g m an y s am ples o f co r r elat ed bet a r an do m v ar iables ar e av ailable fr o m t h e w eb s it eh t t p://w w w .w es t - in c.co m . T h is w eb s it e als o des cr ibes t h e calcu lat io n s car r ied o u t by t h es e pr o g r am s , w it h ex am ples o f t h eir u s e.
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