This paper is not to be cited without prior reference to the author

International Council for the Exploration of the Sea

CM 1999/T:02 Bayesian Approach to Fisheries Analysis

TIME SERIES MODELS IN FISH RECRUITMENT - A JOURNEY FROM CLASSICAL STATISTICS TO DYNAMIC MODELS AND BAYESIAN

FORECASTING

ABSTRACT

C.M. O'Brien

CEF AS Lowestoft Laboratory Pakefield Road, Lowestoft

Suffolk NR33 ORT United Kingdom

fax: +441502513865 e-mail: c.m.obrien@cefas.co.uk

The investigation of stock-recruitment (S-R) relationships can result in functional models that are appealing when depicted in 2-dimensions as the level of recruitment versus spawning stock biomass. Translation of a functional S-R model to the third dimension of time may produce an estimated sequence of recruitment that bears little resemblance to the time series of recruitment used to estimate the 2-dimensional functional S-R model. This difference may result from mis-specification of modelling assumptions that may not have taken due account of temporal effects. By considering recruitment data for plaice (Pleuronectes platessa L.) in the North Sea, an autoregressive moving average (ARMA) model is developed to represent the recruitment process. The classical time series ARMA models can be reformulated as dynamic linear models (DLMs) which allow the incorporation of prior beliefs and expert information. The connection between the classical and Bayesian models is explored and the fundamental principles used by a Bayesian forecaster in structuring forecasting problems through dynamic models are briefly discussed.

Key words: dynamic linear model, stationarity, stochastic process.

© 1999 Crown Copyright

.---... ~----- ..... --- .. ---. ---

INTRODUCTION

The stock-recruitment (S-R) relationship has been a central issue within the field of fish population dynamics (c.f. Clark, 1976; Rothschild, 1986; Shepherd and Cushing, 1990) but most S-R models provide a fairly poor fit to data (Myers et aI., 1995). The investigation of S-R relationships can result in functional models that are appealing when depicted in 2-dimensions as the level of recruitment versus spawning stock biomass. Translation of a fitted functional S-R model to the third dimension of time may produce an estimated sequence of recruitment that bears little resemblance to the time series of recruitment used to estimate the 2-dimensional functional S-R model (c.f. Anon., 1998). This difference might result from not taking due account of temporal effects.

Most available long-term time series of recruitment are based on fishery landings and generally rely upon sequential population analysis (SPA) to reconstruct population recruitment from catches (Pope, 1972; Fournier and Archibald, 1982; Megrey, 1989).

A special feature of time series data is that the datum values are generally not independent but correlated in time. In this paper, classical time series models are applied to the time series of plaice (Pleuronectes platessa L.) recruitment in the North Sea (Anon., 1996a; Anon., 1999). An autoregressive moving average (ARMA) model is developed to represent the recruitment process. The ARMA model can be reformulated as a dynamic linear model (DLM), allowing the incorporation of prior beliefs and expert information, if available. ARMA models assume that a time series is a linear combination of past values, and current and past values of an error term. They apply to time series with no systematic change in mean and variance (c.f. Makridakis et al., 1983).

MATERIAL AND METHODS

Classical time series models are reviewed for a stationary time series; i.e. a series for which there is no systematic change in mean (no trend) and no systematic change in variance. Often, however, time series are non-stationary (Box and Tiao, 1975; Cox, 1981) but repeated differencing of an observed time series can yield a stationary time series. Generally, first-order or second-order differencing handles problems of non­stationary mean and logarithmic (or power) transformation (c.f. Box and Cox, 1964) of the raw data handles non-stationary variance.

Classical time series

Schnute and Richards(1995) in their formulation of a stochastic catch-age model assigned process error to recruitment through the following formulation:

(1)

leading to recruitment equations derived from the log-normal autoregressive process

2

In R, = In R + r ( In R'_I -In R ) + 0"1 8, (2)

with parameters (R, y, (j"l), where noise is introduced by the independent standard normal variates i5t . The Equation (2) implies that In Rt has the following conditional means and variances:

E[ln Rt I Rt-1] = (1 - y) In R + Y In Rt-1

var[ln Rt I Rt- 1] = (j"12

and unconditional means and variances:

E[ln Rt ] = In R

When y=O, In Rt is obviously independent of Rt-1 and follows a normal distribution with mean In R and variance (j"? When y=1, Equation (2) corresponds to a random walk with finite conditional first-order and second-order moments but infinite unconditional variance. The Equation (2) provides a simple process for generating correlated recruitments; the non-stationary case y= 1 allows recruitments to drift toward high or low levels over long time periods.

Let Yt denote the term In Rt, /..l denote In R and Et denote (j"l i5t in the earlier Equation (2). Then the Equation (2) may be re-written in the familiar form of a first-order autoregressive model, abbreviated to AR(1), given by

(3)

This is a simple example of a stochastic process in which the uncertainty in the time series derives from the variable Et - a purely random disturbance term with a mean of zero and a variance of (j"12. Under an assumption that the variable Et follows an independent normal distribution with the specified mean and variance then the parameters (p, y, (j"l) may be estimated by maximum likelihood (Ansley, 1979; Ljung and Box, 1979). Residuals from a model fit may be used to validate the model and to suggest alternative models that may perform better. The Equation (3) may be generalized to a p th-order autoregressive model, denoted by AR(P) given by

(4)

Under an assumption that the variable Et follows an independent normal distribution with a mean of zero and a variance of (j"12 then the parameters {j1, {Yj}, (j"l) may, once again, be estimated by maximum likelihood.

3

A very simple modification to the AR(1) model given by Equation (3) is to introduce a single lagged value of Et; namely,

(5)

where 8 is a moving average parameter. This model is an autoregressive moving­average (ARMA) model of order (1,1) or more generally, an autoregressive integrated moving-average (ARlMA) model of order (1,0,1). An explanation follows presently! Firstly, however, one can think of this time series model as an autoregressive model in which the residual errors are themselves correlated, the correlations being generated by a moving average of independent terms. Failure to recognize such observation error would lead either to concluding that a time series follows an incorrect model or, if an AR(1) model is fit to observed data, to biased estimation of the true autoregressive parameter. Additional higher-order moving average terms may be introduced, if necessary; as may further higher-order autoregressive terms.

The general form of the ARlMA model is referred to as ARlMA(p,d,q) where p is the order of the autoregressive term, d is the degree of differencing involved to achieve stationarity and q is the order of the moving average term. Identification of the appropriate number of terms to include in a model may be guided by the examination of the autocorrelation (ACF) and partial autocorrelation (PACF) functions (c.f. Kendall, 1976). Box and Jenkins(1976) present a paradigm for fitting ARIMA models, which is to iterate through the following steps: (a) model identification; (b) estimation of model parameters; and (c) diagnostic checking. These steps are repeated until a satisfactory model is found. Initial model identification is achieved through the interaction of theory and practice leading to the fit of a tentative model. Diagnostic checks are applied with the object of uncovering possible lack of fit and diagnosing the cause. If any inadequacy is found, the iterative cycle of identification, estimation and diagnostic checking is repeated until a suitable representation is found. It is important, however, that in practice one employs the smallest possible number of parameters for adequate representation of a time series - the principle of parsimony.

The parsimonious structure of a parametric time series model is classically investigated using the minimum Akaike information criterion (AIC) (Akaike, 1973). AIC penalizes -2 In (maximized likelihood) of a model by twice the number of parameters included in the model and a model with the smallest AIC is selected. Hurvich and Tsai(1989) demonstrated that in small samples the bias of the AIC may not be negligible, especially as the dimension of the model approaches the sample size. The modified small-sample statistic, AICc, penalizes AIC by the addition of 2(P+1)(p+2)/(n-p-2) where p represents the number of model parameters and n is the sample size (length of the time series). However, as Jenkins (1982) showed, abandoning judgement altogether in (automated) model selection can lead to the potential for selecting inappropriate models and care must be taken not to over-fit a model to data.

Once a particular time series model has been identified, the residuals must be examined to see if they indicate a departure from randomness. Two main tools are

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used to check on the randomness of a sequeflceof residuals - the cumulative (residual) periodogram plot (Durbin, 1969), and portmanteau lack-of-fit test (Ljung and Box, 1978; Davies and Newbold, 1979). The former provides a visual diagnostic check, whilst the latter allows a test under the null hypothesis that a model is correctly specified. The portmanteau test involves the calculation of a weighted sum of squared autocorrelations, followed by comparison with a reference X2 distribution. If the application of these tools indicates that the model is inadequate, the complete cycle of identification, estimation and diagnostic checking is repeated.

Simulation is relatively straightforward for an ARIMA model and merely requires that an ARIMA model structure be specified, that parameter values be either given a priori or estimated from a model fit and that the innovations Et be either specified or generated under an appropriate assumption (c.f. Jones, 1980).

RESULTS

North Sea plaice

A time series model is investigated for North Sea plaice recruitment based upon values previously reported (Anon., 1996a). Three main features of the S-R pairs are considered to be important: (i) there is no information about the relationship between stock and recruitment at low spawning stock biomass; (ii) there is no information about the level of spawning stock biomass at which recruitment would be expected to start to show a decline; and (iii) inter-annual variability in recruitment is rather low, except for the occurrence of three exceptionally strong year-classes in 1963, 1981 and 1985. A time series plot of recruits at age 1 (recruitment) and spawning stock biomass (ssb) is shown in Figure 1. Clearly, any time series model for recruitment that eliminates the necessity to treat the three strong year-classes differently from the other year-classes would be desirable.

I ::= ~~~ruitmen(

(960 1970 1980 1990

time in years

Figure 1. A time series plot of recruits at age 1 (recruitment) and spawning stock biomass (ssb).

To begin the process of time series model identification, fitting and checking for North Sea plaice recruitment it is necessary to propose an initial tentative model from which to start the iterative model selection strategy. In the absence of a suitable

5

-~ ~---~.---.

biological model, the statistical model represented by Equation (2) would seem to be an appropriate model from which to begin. If the natural logarithm of recruitment can be represented by an AR(1) model then the residuals from the autoregressive fit should be scattered about a straight line joining the points (0, 0) and (0.5, I) in the cumulative periodogram. A diagnostic plot is shown in Figure 2 for both an AR(1) and an AR(2) model. The cumulative periodogram of the residuals shows that the AR(1) model has not removed all the serial correlation since a downward curvature (relative to a 45° line) is apparent in the plot. This curvature may be removed, however, by considering an AR(2) model for the natural logarithm of recruitment but this is not the only approach possible.

AR(1) fit AR(2) fit

~ ~

~ ~

~ ~

~ ~

~ :;1

~ ~ O~O 0.1 02 O~' OA 0.5 0.0 0.1 0.2 0.3 0.4 05

frequency frequency

Figure 2. Cumulative periodogram plots for residuals of autoregressive models fitted to the natural logarithm of recruitment. 95% confidence bands are given by the region bounded by the two parallel straight lines in each plot.

A plot of Ale against the order p of a sequence of autoregressive models (arbitrarily up to the order 6) is shown in Figure 3. The values of Ale shown for the autoregressive models with orders ° through to 6 have the minimum value subtracted from all of them so that the minimum is always zero for the chosen model. It is worth noting that while an AR(1) model has the smallest value of Ale, the value for an AR(2) model is only slightly greater and thus should not be excluded from further consideration. In fact, as previously remarked, the AR(2) model is to be preferred to the simpler AR(1) model but neither is wholly satisfactory.

If the natural logarithm of recruitment can be represented by an AR(1) model in which the residual errors are themselves correlated by a moving average of independent terms then the residuals from the autoregressive moving-average fit should be scattered about a straight line joining the points (0, 0) and (0.5, I) in the cumulative periodogram. A diagnostic plot is shown in Figure 4 for the ARMA(1,l)/ARIMA(1,O,I) model; and a similar plot given for the ARMA(2,1)/ARIMA(2,0,1) model. The cumulative periodogram of the residuals shows that the ARMA(1, I) model has lessened the effect of the serial correlation present in the earlier AR(1) model but not rerrtoved it completely. It is worth noting

6

AR(p) fit

Figure 3, Akaike's information criterion for autoregressive models of order p fitted to the natural logarithm of recruitment.

also that the cumulative periodogram of the residuals from the ARMA(2, 1) model is little changed from that produced for the previous AR(2) model but the Ljung-Box test did not indicate significant (probability> 0.05) error autocorrelation in the model. On this basis, an autoregressive integrated moving-average model of order (2,0,1) appears to give a parsimonious description for North Sea plaice recruitment.

ARIMA(1,D,1) fit .c .c

~ ri

w 0 ~

~ ;l

;:: ~

;J ;J 0.0 0.1 02 0.3 0.4 05 0.0 0.1 02 03 O. 0.5

Figure 4. Cumulative periodogram plots for residuals of autoregressive moving-average models fitted to the natural logarithm of recruitment. 95% confidence bands are given by the regions bounded by the two (solid) parallel straight lines in each plot. The dotted straight line connects the point (0,0) to the point (0.5,1),

Furthermore, simulations can be generated from the fitted model to judge whether or not this model produces sensible patterns in recruitment over time. In an attempt to aid the interpretation of the output generated for the simulated ARMA(2, 1) model for North Sea plaice recruitment, three lines have been superimposed on all the graphs of simulated sequences of recruitment. The locations of the three lines are shown in Figure 5 and each is identified by a letter a, b or c. Th~ir significance is explained in the legend to Figure 5. The relevance of the line (b) is that it corresponds to the level

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of recruitment that is the lowest of the three large year-classes of 1963, 1981 and 1985.

o ---------------------------------- ----------i

(e)

'" N

1960 1970 1980 1990

time in years

Figure 5. A time series plot of In [recruitment at age 1]. Th{! horizontal (dashed) line identified as (a) depicts the value of In [minimum of recruitment], the horizontal (dotted) line identified as (b) depicts the value of In[ recruitment for the 1963 year-class] and the horizontal (dashed) line identified as (c) depicts the value of In [maximum of recruitment].

Figure 6 shows four simulations of an ARMA(2,1)/ARIMA(2,O,I) model for the natural logarithm of recruitment, with parameters estimated for the North Sea plaice data. Each simulation represents a sequence of 37 (In-transformed) recruitment values generated over a consecutive time period. Simulated sequences of recruitment generated under the fitted model appear to exhibit the right number of low and high levels of recruitment. The graphs in Figure 6 have been chosen to represent the range of possible sequences that can be produced. On a number of rare occasions, a level of recruitment lower than any of those used to fit the model can be generated (see Figure 6( c)) and a level of recruitment higher than any of those used to fit the model will be generated (see Figure 6(d)). However, the vast majority of simulated sequences are constrained within the region bounded above and below by the lines (c) and (a), respectively, identified in Figure 5.

For the ARMA(2,1)/ARIMA(2,0,1) model fitted to North Sea plaice recruitment, the autoregressive parameters have estimated values of 0.646 and 0.134, the moving average parameter has an estimated value of 0.463 and the innovations have an estimated variance of 0.125. The (upper triangular) variance-covariance matrix for the autoregressive and moving average coefficients has the following estimated values:

AR(l) AR(2) MA(I)

AR(l) 0.221 -0.105 0.202

AR(2) 0.064 -0.087

MA(l) 0.211

The model parameters and variance-covariance matrix have been estimated on the natural logarithmic scale of recruitment, after correction with a mean of 12.97.

8

(a) ARIMA(2,O,1) simulation

10 20

simulated sequence

30

(e) ARIMA(2,O,1) simulation

10 20 30

simulated sequence

~

(b) ARIMA(2,O,1) simulation

I;:::-=-=-=-:::;-=--=-=-=-::;:-=--=-=-=-;:=-=-=--=.l 10 20 30

(d) ARIMA(2,O,1) simulation

10 20 30

simUlaledsequence

Figure 6. Four simulations of an ARIMA(2,O,l) process for In [recruitment] with parameters estimated for the North Sea plaice data. The dashed and dotted horizontal lines superimposed on the plots correspond to those previously identified in Figure 5.

A time series plot of the assessed level of recruitment and the estimated levels of recruitment under the ARMA(2,1)/ARIMA(2,O,1) model are shown in Figure 7 for completeness. The fitting of the time series models presented in this paper and the diagnostic techniques employed have been implemented within the programming environment of S-PLUS (MathSofi, 1998). A number of diagnostic functions available in the MASS library distributed with Venables and Ripley(1994) have been used.

! = ~~rurlmenl

1960 1970 1960 1990

time In years

Figure 7. A time series plot of recruits at age 1 and the ARIMA(2,O, 1) fit.

DISCUSSION

Traditionally, time series analysis has neither been widely used nor been of general utility (c.f. Hilborn and Walters, 1992) in the analysis of fisheries stock dynamics. The situation is, however, changing (Gudmundsson, 1994) and the importance of addressing time series issues has now been recognized. In this paper, a time series

9

model has been investigated for North Sea plaice recruitment based upon statistical analyses of the stock assessment data reported by the ICES Working Group on the Assessment of Demersal Stocks in the North Sea and Skagerrak. Box-Jenkins methods have been employed in the identification of a time series model and its subsequent diagnostic checking. An autoregressive integrated moving-average model of order (2,0,1) has been chosen to give a parsimonious description for North Sea plaice recruitment. Model parameters are estimated by maximum likelihood and residuals from the model fit are checked using the cumulative periodogram. Simulated sequences of recruitment generated under the fitted model have been presented and appear to exhibit the right number of low and high levels of recruitment. The three exceptionally strong year-classes in 1963, 1981 and 1985 for North Sea plaice recruitment are consistently modelled using the simple time series approach and the necessity to treat their occurrence as extreme (Anon., 1996b) has been removed from the modelling presented.

The time series recruitment model has been implemented within a North Sea plaice scenario model (Kell et aI., 1999) for the evaluation of management procedures. Application of the model to the time-series of recruitment reported in the latest North Sea plaice stock assessment (Anon., 1999) yields parameter estimates that are little changed from those given here. To motivate the link with Bayesian methods, the classical ARMA time series models can be expressed in state space form (c.f. Kalman and Bucy, 1961; Gardner et at., 1980). This formulation has been extensively developed (c.f. Harrison and Stevens, 1971, 1976) by exploiting the Kalman filter (e.g. Meinhold and Singpurwalla, 1983) and adopting a Bayesian approach in which knowledge of certain key parameters is assumed. In this way, the classical time series ARMA models are reformulated as dynamic linear models (DLMs) (c.f. West and Harrison, 1997) which allow the incorporation of prior beliefs and expert information, if available, and establishes a formal connection between the classical time series and Bayesian models.

Bayesian time series

A DLM is a system of equations specifying: (a) how observations of a process are stochastically dependent on the current process parameters; and (b) how the process parameters evolve in time, both as a result of the inherent process dynamics and from random disturbances.

Observation equation:

System equation: \1ft = G \lft-l + Wt, {Wt ~ N( 0, Wt)}

where Yt is an (n x 1) vector of observations made at time t, Ft is a known (n x r) matrix at time t, \1ft is an (r x 1) vector of process parameters at time t, Vt is an (n x 1) random normal vector with zero mean and variance known at time t, G is a known (r x r) matrix, Wt is an (r x 1) random normal vector with zero mean and variance known at time t, and Vt, W t are known variance-covatiance matrices. Examples of F t and G matrices for AR(P) , MA(q) and ARMA(p,q) models are given in Harrison and Stevens(1976), and West and Harrison(1997}.

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Conventionally, observations begin at t=l, the series developing as Yt (t=1,2, ... ). The model is stated in terms of discrete, equally spaced intervals of time but may be extended to incorporate unequal intervals of observations and missing values. All the available relevant starting information that is used to form initial views is identified by Do and statements made at time t are based on Dt = { Yt, Dt-r}.

Initial information: (\lfol Do) - N(mo, Co)

for some prior moments mo and Co. The observational and evolution error sequences are assumed to be mutually independent, and are independent of (\lfo 1 Do). One-step forecast and posterior distributions for any time t> 0 can be obtained sequentially:

posterior at (t - 1) given by (\1ft-II Dt-l); prior at t given by (\lftl Dt-l);

one-step forecast given by CYt 1 Dt-l); posterior at t given by (\lftl Dt);

and retrospective recurrence relationships yield the joint distribution of the historical parameters (\lfl, \lf2, ... , \lftl Dt). The k-step ahead forecast and the k-step lead time forecast exist. A range of models, rather than a single model, may be specified with associated probabilities that are up-dated after each new observation (c.f. Harrison and Stevens, 1976) or Bayes' factors may be calculated (Kass and Raftery, 1995). Alternatively, model selection might be based on a goodness-of-fit criterion such as the Bayes' information criterion (BIC) (c.f. Schwarz, 1978; Buckland et al., 1997), defined in a similar way to AIC, but with the '2' of the penalty term replaced with 'In (number of observations in the time series)'. Techniques of model assessment and diagnostic checking involving a retrospective analysis of the influence of past observations are available (Harrison and West, 1991). Models with non-linear components and non-normal structure (Smith, 1979) have been proposed but may increase computational complexity (c.f. Gamerman, 1997).

The incorporation of prior beliefs and expert information, if available, is feasible within the time series modelling of fish recruitment, if one adopts dynamic linear models and recruitment follows an ARMA-like process. The Bayesian forecasting technique allows one to specify a range of models, rather than a single model, with associated probabilities that are up-dated after each new observation. The method is able to cope with outliers, step changes in mean and trend, and changes in the process parameters. The approach warrants further investigation within fisheries analysis and biologically the models may have a sensible interpretation (c.f. Stenseth et al., 1999). The problem remains, however, of prior elicitation!

ACKNOWLEDGEMENT

This paper was prepared with funding support provided by the Ministry of Agriculture, Fisheries and F ood (contract MF0316).

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.-~ '1

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