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Fishery Bulletin 105(1) 



(1976) and Box et al. (1978). Framing the problem in 

 terms of power spectra, we also offer some additional 

 new insights into this noise reduction approach. Next, 

 we apply the ARIMA-based noise reduction approach 

 to the time series data and present the results. We 

 have implemented Box et al.'s (1978) algorithm, not 

 Pennington's (1985). Finally, we discuss our perceptions 

 of the utility of this approach in light of our results and 

 overall experience with it. 



Materials and methods 



General characteristics of ARIMA models 



In this section, we first briefly review ARIMA models 

 for stochastic processes. Then we review the approach 

 of Box et al. (1978) for obtaining maximum likelihood 

 estimates for an underlying ARIMA time series from a 

 time series of observations with independent and identi- 

 cally distributed (IID) observation noise. 



ARIMA models are parsimonious models that can 

 adequately represent many stochastic time series (Box 

 and Jenkins, 1976). Stochastic time series that can be 

 represented by ARIMA models are essentially the out- 

 put of a linear filter applied to an input time series of 

 white noise (Box and Jenkins, 1976). We will refer to 

 such time series as ARIMA processes. 



For a zero-mean stochastic time series |2,| that can 

 be expressed as an ARIMA model, we denote the model 

 (using the notation of Box and Jenkins, 1976) as 



(p(B)z, =a(B)ai, 



(1) 



where z 



(p(B) 



the value of the time series at time t; 

 the generalized autoregressive (AR) opera- 

 tor; 

 a(B} = the moving average (MA) operator; 

 B = the backward shift operator; and 

 a, = IID normally distributed random variables 

 with mean zero and variance o~. 



The backward shift operator B has the property that 

 B 2f = 2, p hence B'" 2, = 2,_,„. The operators q>{B) and 

 a(B) are polynomials in B of order p+d and q (respec- 

 tively) such that 



(p(B) = l-Y,<PjB' 



.1=1 



1 

 a(B) = l-^a^B'. 



(2) 



Furthermore, (p(B) can be factored such that ()n(B) = 

 0(B)(1-B)'*, where the (ordinary) AR operator 0(B) has 

 a form similar to (p(B) but is of order p. The operator 

 (l-B)"^ represents d sequential applications of the back- 

 ward difference operator 1-fi such that the original time 

 series I2,) is self-differenced d times before application 

 of the AR operator ^{B). 



As a shorthand, an ARIMA model that consists of an 

 AR operator of order p, d backward difference opera- 

 tions, and a MA operator of order q is abbreviated as 

 (p,d.q). A (p,0,0) model is referred to as an AR model of 

 order p, or AR(p) in shorthand notation, and a (0,0,(7) 

 model is referred to as a MA model of order q iMMq) in 

 shorthand). A {p,0,q) model is referred to as an autore- 

 gressive-moving average model {ARMA{p,q) in short- 

 hand) and a (0,d,q) model is referred to as an integrated 

 moving average model (IMA(d,q) in shorthand). Finally, 

 a random walk model is (0,1,0), while a random-walk- 

 plus-uncorrelated-observation noise (RWPUN) model 

 is (0,1,1). 



In general, ARIMA models represent nonstationary 

 time series. However, the time series that results from 

 applying the backward difference operator d times to 

 a {p,d,q) ARIMA process is a stationary ARMA(p,(/) 

 process. Typical constraints imposed on ARIMA models 

 are that, when B is regarded as a complex variable, 

 the polynomial in B representing the generalized au- 

 toregressive operator has zeros on or outside the unit 

 circle (IS 1=1), whereas the polynomials representing 

 the ordinary autoregressive operator and the moving 



