Intro to multivariate AR(1) models estimating interaction strengths, aka the B matrix

Transcription

1 Iro o ulivariae AR(1) odels esiaig ieracio sreghs, aka he B ari Eli Holes FISH 507 Applied Tie Series Aalysis 23 Feruary 2017

2 Mea reverig processes I lecure, I ill alk aou esiaig ea-reversio i he coe of desiydepedece ad species ieracios, u ea-reverig sochasic processes are uiquious. The Orsei-Uhleeck process is he classic coiuous ie ea-reverig sochasic process. I he populaio dyaics lieraure, he Goperz odel is he classic discree ie ea-reverig process (alhough he Goperz odel also refers o a coiuous ie versio).

3 Uivariae ad ulivariae Goperz odels Uivariae odels Esiaig desiy depedece or er 1 u Siple 2 spp odel 22 B ari Large ulivariae odels Big B arices

4 Desiy depedece uivariae discree epoeial groh o i log space 1 r 1 ep( u) uivariae discree desiy depede groh o i log space geeral f 1 1 Specific ype ep( u f ( 1)) 1 The shape of f(_ 1) deeries he dyaics of he syse: sale or usale equiliriu Speed a hich equiliriu is approached Equiliriu level Sesiiviy o peruraios

5 The Goperz odel uivariae discree ie deeriisic Goperz odel o i log space 1 ep u 1 l 1 < 1 egaive desiy depedece = 1, o desiy depedece > 1, posiive desiy depedece (los up) The closer is o 0, he sroger he desiydepedece (sroger he pull ack o he ea). If =1, here is o pull ack o he ea (he ea is o i fac defied for his case).

6 Goperz odel rie i log space AR(1) 1 ep u 1 l 1 Takig he aural log of oh sides l l u l l 1 u l 1 l 1 u l 1 Susiuig for l u 1 AR(1) ius he oise er

7 Eaples of he Goperz for differe Our ieres u/(1-)

8 Equiliriu for he deeriisic Goperz odel 1 u he odel reaches equiliriu a =, so e ca rie u Ad via soe algera, e arrive a: u 1 (provided 1) The equiliriu is a fucio of BOTH u ad. This is raher uforuae.

9 Equiliriu for he deeriisic Goperz odel u=.5; =.9; =seq(0.1,500,.1) plo(,ep(u+( 1)*log()),ype="l") alie(h=1,col="red"); alie(v=ep(u/(1 )),col="lue") e(ep(u/(1 )), ep(u+( 1)*log([1])), pase("u/(1 ) =",ep(u/(1 ))),pos=4)

10 Add sochasiciy (process error) Addig sochasiciy yields a uivariae, lag 1 auoregressive or AR(1) process: 1 u 2 ~ N 0, If < 1, he process is saioary If = 1, he process is a rado alk & osaioary Ko as he discree ie Orsei Uhleeck process i physics u as Goperz or sochasic Goperz odel i populaio dyaics.

11 Eaple realizaios rado alk =1 2 p oscillaio u/(1-)

12 Equiliriu for he sochasic Goperz process I has a saioary disriuio proailiy disriuio of X as give < 1 Norally disriued ih ea ad variace Fig. 1 Ives e al. (2003)

13 Properies of he saioary disriuio Assuig < 1 (i.e. a saioary process) ea u 1 (provided 1) variace v q /(1 2 )

14 Basic feaures of he Goperz process Mai properies Mea reverig, aka desiy depede Saioary, so i flucuaes aroud a ea Poi equiliriu as opposed o a cycle equiliriu like Loka Volerra (Ly & hare) odels you sudied (aye) i Ecology 101 Ca e see as a locally liear approiaio of oher ypes of desiy depede ieracio odels locally liear is jargo for oly holds for sure if does chage oo uch. I our case, = log() = log audace. S. E. Hapo, NCEAS, UCSB, hapo@ceas.ucs.edu, 7 July 2007

15 Paraeer esiaio i R Ope up R ad follo afer e Goperz_eaple_0.R

16 We eed o e careful if he daa are NONsaioary MANY (os) fiig algorihs assue he daa are dra fro he saioary disriuio ad hus are oly applicale if your daa are saioary disriuio. MARSS does o assue he saioary disriuio, fyi. o saioary saioary

17 oservaio error is ko a prole os error = spurious desiy depedece

18 Paraeer esiaio accouig for os error Goperz_eaple_1.R esiaio echically easy.. Goperz_eaple_2.R replicaio Goperz_eaple_3.R ML o he edge

19 Esiaig R ari is o so easy, u replicaio helps A LOT MARSS odels (hoever you fi he) allo you o easily icorporae replicaio.

20 Esiaio uch easier if you ca assue ha your daa are a saple fro he sochasic equiliriu Process has reached equiliriu

21 Ho o esiae he you are illig o assue he daa coe fro soc. equil.? Idea #1 Ipose he cosrai ha W()=(y()-y(-1)) ad W()-W(-1) Have he variace-covariace srucure of a sochasic Goperz oserved ih error. Copue Q fro he oal saple variace ad he esiae of

22 Ho o esiae he you are illig o assue he daa coe fro soc. equil.? Idea #2 If you surac E(()) he U=0 Use ea(daa) as E(()) Has he added value of reovig a oo! Used i Ives e al 2003 Goperz_eaple_4.R

23 Ipora essages u ad B are cofouded. Likelihood is aaa shaped, so e eed o cosrai u de ea he daa; se u=0; se ii=1 do deea daa; se u=0; esiae a; se iis=1 do deea daa; use covariaes i os o odel level e careful i ha covariaes you iclude i he process odel (you re iroducig u via Cc) Wha happes he e add oservaio error? Esiaio is ore difficul. Replicaio ill help us esiae R vs Q

24 2 species: Predaor Prey 1, 1,, 1, 1,, u u Moose Wolf

25 B = ieracio ari MAR(1): =B 1 + u + u u 1, 1,,, Process variaio MVN(0,Q) spp audace

26 Meaig of he B ari B p Ier specific effecs (ca e se o zero) 1p pp Ier specific effecs (ca e se o zero) Ira specific effecs effec of spp i o iself, aka desiy depedece

27 Oservaio error causes Spurious desiy depedece, i.e. appare sroger effec of self o self Spuriously lo species ieracio sreghs, i.e. appare loer effec of oher o self B p p pp Ier specific effecs (go o 0) Ira specific effecs go o 0

28 B = ieracio ari Addig covariaes covariae effec covariaes Process variaio o fro covariaes ( ueplaied ) c c c c c c c c c u u 3, 2, 1, , 1,,,

29 Loka Volerra predaor prey ieracios S. E. Hapo, NCEAS, UCSB, 7 July 2007

30 Siple 2 species syse Predaor & Prey Daa are siulaed usig a discree ie versio of a Loka Volerra odel ih desiy depedece i he herivore easy o chage ieracio sregh dh/d= H(1 H/K) a H P + h dp/d = e(aph) sp + p H = herivore sp P = predaor sp. = herivore irh rae K = herivore carryig capaciy a = per capia aack rae e = coversio efficiecy (cosued prey urig io e predaors) s = deah rae for predaors S. E. Hapo, NCEAS, UCSB, hapo@ceas.ucs.edu, 7 July 2007

31 This odel ca display a variey of dyaics To Herivore Predaor Herivore Predaor 0 To S. E. Hapo, NCEAS, UCSB, hapo@ceas.ucs.edu, 7 July 2007

32 Esiae sregh of desiy depedece ad ieracio sregh usig MARSS LV_eaple_1.R chage coversio efficiecy of predaor LV_eaple_2.R add oservaio error LV_eaple_3.R covariae affecs K of herivore LV_eaple_4.R covariae affecs coversio efficiecy of predaor

33 Copuer la: he oose ad olf dyaics o Isle Royale daa ad iages fro.isleroyaleolf.org