Kimura etal.: Kalman filtering the delay-difference equation 



691 



Wild, A. 



1986. Growth of yellowfin tuna, Thunnus albacares, in the 

 eastern Pacific Ocean based on otolith increments. Inter- 

 Am. Trop. Tuna Comm. Bull. 18:423-482. 



Appendix 1 : Time series properties of the 

 delay-difference equation 



If recruitments {R,} are assumed to be uncorrelated with 

 E(R,) = R t and V(R, ) = o 2 p , then the delay-difference equa- 

 tion (Eq. 2) is close to being a standard ARMA time series 

 model (Box and Jenkins, 1970). Let 0, = (l + p)s o ,0 2 = -ps%, 

 and 9 = p<us . Then Equation 2 can be written as 



B l =4> 1 B l _ 1 +4> x B,_ t +R,-aR,_ 1 . 



(!') 



Defining B l =(R l - GR l ) /( 1 - 0, - 2 ) , and substituting R t = 

 n, + R x and B, = B, + B\ into Equation 1', yields the process 



Appendix 2: Derivation of the log-likelihood 

 when only the variance ratio r= cr^/a^ is 

 known. 



Source: Pella, J. J. 1995. Auke Bay Laboratory, Na- 

 tional Marine Fisheries Service, NOAA, 11305 Glacier 

 Highway, Juneau, AK 99801. Personal communication. 



We start with the usual log-likelihood when variances 

 are assumed known 



ln(L(y,4'))=-|ln(2^)-|X ln( /',)-|Xf- (1 "> 

 Substituting f, =o 2 m f' into (1"), we have 



B, = 0,5,., + 2 B,_ 2 + n, - On,,, , 



(2') 



ln(L(y,«P)) = -|ln(2jr)-|ln«)- 



I n t rt 2 



* ,=1 *°m 1=1 /l 



(2") 



where n, the are now white noise. Thus under the assump- 

 tion of constant growth parameters (p, to), constant survival 

 s , and random recruitment (R,), the displaced process [B, I 

 is an ARMA time series model with the same statistical 

 properties of IB,}. 



The stationarity and invertibility of this time series can 

 easily be established by considering the roots of the char- 

 acteristic polynomials (see Box and Jenkins, 1970): 



1-0,X-0 2 X 2 =O, and 



i-ax = o. 



(3') 



(4') 



By differentiating with respect to ct 2 , and by setting 

 this result equal to zero, we obtain the estimator 



1 n 2 



" /=i it 



which we substitute into Equation 2" to arrive at the func- 

 tion to be maximized 



ln(L c (;y > 4>)) = --[ln(2;r)+l]- 

 l£ln(/:)-§ln(a*). 



(3") 



The stationarity and invertibility of IB, I follows from the 

 observation that the roots l/s and l/(ps ) of Equation 3' 

 and l/(pcos ) of Equation 4' are all almost surely greater 

 than unity. 



Let c = ( 1 - 00, + e 2 ) o 2 p , and c, = -6o 2 p . The autocovar- 

 iance terms (i.e. the y k in standard notation) of the pro- 

 cess |B,)can then be shown to be 



y o =[c o + (l + 2 )0,c 1 /(l-0 2 )]/[(l-0 2 2 )- 



(l+0 2 )0, 2 /(l-0 2 )], 



y, = (c, +0,7 o )/(l-0 2 ), 



7 2 = 0i Xi + <PiYo . and 



Yk = 0i7*-i + 02>V2 . for£>3. 



Two interesting correlation coefficients that can be es- 

 timated using the y k are 



p(B„R,) = Jo 2 /y and p(B,_„,B,) = y„ ly 



