Interannual variability in a plankton time series

ENVIRONMETRICS

Environmetrics 2003; 14: 73–86 (DOI: 10.1002/env.566)

Interannual variability in a plankton time series

Michael Dowd*,y, Jennifer L. Martin, Murielle M. LeGresley, Alex Hanke and Fred H. Page

Fisheries and Oceans Canada, Biological Station, 531 Brandy Cove Road, St. Andrews, N.B., Canada E5B 2L9

SUMMARY

Temporal changes in a plankton time series are examined, with an emphasis on interannual variability. Astochastic cycle model is used which describes an annual cycle with a fixed frequency, but a randomly varyingamplitude and phase. A state space representation is used with the Kalman filter, and associated fixed-intervalsmoother, to provide estimation of the time-varying state. Parameter estimation relies on maximum likelihoodmethods. A data set is considered comprised of an irregularly sampled time series of plankton (dinoflagellate)abundance over a 12 year period in the Bay of Fundy, off the east coast of Canada. Analysis of the log10-transformed data indicated timing changes in the seasonal cycle of up to 23 days. Significant variations inabundance relative to the mean cycle were found for some highly sampled summer periods. Case deletiondiagnostics identified two influential observations, one of which has a large impact on the estimated system noise.Examination of the sampling protocol, or monitoring design, indicates the need to reduce the observation errorvariance in order to improve detection of interannual variations in plankton abundance. Copyright # 2003 Crownin the right of Canada. Published by John Wiley & Sons, Ltd.

key words: interannual variability; cyclic time series; plankton; state space model; Kalman filter; fixed-intervalsmoother; irregular sampling; case deletion diagnostics; maximum likelihood

1. INTRODUCTION

Environmental monitoring of plankton populations has received increased attention in recent years, as

evidenced by its important role in the development of a global ocean observing system (International

Oceanographic Commission, 1999). Monitoring plankton provides basic information on the biological

state of the ocean, and its variability, in order to predict harmful algal blooms, identify ecosystem

changes associated with anthropogenic activities, such as coastal nutrient loading, as well as to detect

the impacts of global climate change. Emerging measurement technologies in optical and acoustical

oceanography hold promise for a revolution in plankton monitoring (Cullen et al., 1997; Gallager

et al., 1996; Batchelder et al., 1995). At present, long multi-year records of plankton species

abundance are relatively rare and it appears timely to consider the analysis of historical time series

of plankton abundance. Here, we examine one such 12 year data set obtained from a plankton

monitoring program in our region (Martin et al., 1995, 1999, 2001).

Received 16 August 2001

Copyright # 2003 Crown in the right of Canada. Published by John Wiley & Sons, Ltd. Accepted 22 April 2002

*Correspondence to: Michael Dowd, Fisheries and Oceans Canada, Biological Station, 531 Brandy Cove Road, St. Andrews,N.B., Canada E5B 2L9.yE-mail: [email protected]

Understanding the temporal variability in plankton populations is a scientific question central to

marine ecology and biological oceanography. Fluctuations of plankton populations at a variety of time

scales has been well established (Weibe, 1970; Platt et al., 1970). Of particular interest is the

interannual variability, or changes in the seasonal cycle from year to year. Extracting these signals

from plankton monitoring data is confounded by the ubiquitous presence of small scale variability,

generally referred to as plankton patchiness. It arises due to the redistribution of freely floating

plankton populations by a turbulent fluid environment (Bennett and Denman, 1985; Abraham, 1998).

Theory also suggests it has an origin in plankton population dynamics (Edwards, 2001) and their

interaction with environmental variation (Vance, 1990; Popova et al., 1997). Early studies highlighted

the effect that patchiness had on plankton abundance estimates (Wiebe and Holland, 1968). Clearly,

our ability to identify the low frequency signal in the background of environmental and sampling noise

is fundamental to the proper interpretation of these biological time series.

The main purpose of this study is to examine an approach for detecting interannual variability in

observed plankton abundance time series. Wood and Horwood (1995) have pointed out the need to

develop robust statistical methods for analysing plankton data. We are motivated by the goal of

assessing the information content of the plankton monitoring data and retrospectively assessing the

sampling protocol. The basis of this work is a structural time series model cast in a state space

framework and using a fixed-interval smoother to derive estimates for plankton abundance. The paper

is organized as follows. Section 2 introduces the plankton time series data. Section 3 presents the

statistical framework. An application of the method is carried out in Section 4. A summary and

conclusions follow in Section 5.

2. OBSERVATIONS

As part of a regional toxic plankton monitoring program, plankton populations have been monitored

since 1988 at fixed stations in the western Bay of Fundy off Canada’s east coast. The plankton have

been collected at four depths (0, 10, 25 and 50 m) using 1 L water-sampling bottles. Organism counts

are enumerated for each species found. The sampling schedule is irregular: during the winter months

the station has been sampled monthly, in the spring and fall the station is sampled bi-weekly, and in the

summer it is sampled weekly. Details of the sampling schedules and procedures are found in Martin

et al. (1995). Recently, these data have been extensively quality controlled and organized into a

database. In this study, we make use of plankton adundance data from June 1988 to December 1999 as

reported in Martin et al. (1995, 1999, 2001).

The plankton data considered here are time series of organism counts per unit volume, or

concentrations. Of particular interest are data on species aggregates, or taxonomic groups, such as

the diatoms, dinoflagellates and zooplankton. For this study, we have elected to focus on a

dinoflagellate time series from one sampling station and a single depth (0 m). This series appears to

have statistical properties that are representative of a broad range of the regional plankton data.

Figure 1(a) shows the plankton abundance time series. The observation set contains 276 data points

and covers a period of 604 weeks (�11.6 years). Cell concentrations range over six orders of

magnitude. There is no suggestion of a trend, but seasonality in the form of an annual cycle is clearly

evident. High plankton concentrations are found in summer months and lower concentrations

(including zero) are seen in winter. The sampling effort reflects this seasonality in abundance. We

have chosen to work with log10-transformed data. Denoting the original data as y�k , the transformed

data takes the form yk ¼ log10ðy�k þ 1Þ, with the factor of one is introduced to deal with the zero

74 M. DOWD ET AL.

Copyright # 2003 Crown in the right of Canada. Published by John Wiley & Sons, Ltd. Environmetrics 2003; 14: 73–86

concentrations. These log10-transformed data are plotted in Figure 1(b) are used hereafter. The

log10-transform stabilizes the variance and does not permit negative concentrations.

As a preliminary step in model identification we fit an annual cycle to the time series using an

ordinary least squares harmonic regression (Figure 1b). Some scatter about the fitted curve is evident;

zero concentrations and some values near the peaks appear as outliers. (Note that results from this

regression are biased towards clusters of observations in the summer season.) However, it appears that

a cyclic model at the annual period proves a useful starting point for further analysis. In order to

identify interannual variations in plankton abundance we must carefully consider the problem of

extracting the time-varying seasonal signal from the noisy and irregularly sampled observations.

3. MODEL

In order to represent the time varying level �t of the random process, consider a model which

represents the annual cycle as a periodic function, �t ¼ ae�i!t, where a denotes the complex

amplitude, ! is the radial frequency of the annual cycle and i ¼ffiffiffiffiffiffiffi�1

p. The amplitude of the cycle

is jaj and its phase is � ¼ tan�1ð�Jfag=RfagÞ, where R and J denote the real and imaginary parts of

Figure 1. (a) Time series of plankton (dinoflagellate) counts per litre. These are reported on a log scale and a value of one has

been added for plotting purposes. (b) The log10-transformed plankton concentrations (dots), as well as the results from an

ordinary least squares harmonic regression at the frequency of the annual cycle (solid line). The time axis begins 1 June 1988 and

ends 14 December 14 1999

INTERANNUAL VARIABILITY IN A PLANKTON TIME SERIES 75


a complex number, respectively. To allow for evolving seasonality, the cyclic model for �t is expressed

as a first order Markov difference equation with additive white noise, i.e.

�t ¼ �t�1e�i! þ �t: ð1Þ

The system noise is given by �t � CNIDð0; 2�2nÞ, i.e. a complex valued and normally distributed,z

serially independent random variable with zero mean and variance 2�2n. This model allows the

complex amplitude of the cyclic process to vary stochastically, but with a fixed frequency. At any time

t, the amplitude and phase of the process is diagnosed by expressing it in polar form. The stochastic

cycle model is generally expressed in its real-valued representation and in this form it is reviewed by

Haywood and Wilson (2000).

Consider the real valued state space representation of the dynamics (1) and its observation vector

xt ¼ Dxt�1 þ nt; yt ¼ Htxt þ �þ et ð2Þ

for t ¼ 1; . . . ; T. The state evolution equation has elements

xt ¼�rt

�it

� �; D ¼ cos! sin!

�sin! cos!

� �

where �rt ¼ Rf�tg and �i

t ¼ Jf�tg. The system noise nt � NIDð0; �2nIÞ. The observation equation is

based on univariate measurements of Rf�tg at times k, where k is a subset of the model times t. Hence,

we map irregularly sampled observations onto model times to define the observation vector yt. Missing

observations are treated by defining Ht ¼ ð1 0Þ�t;k, with � representing the Kronecker delta, i.e.

�t;k ¼ 1 for t ¼ k, and 0 for t 6¼ k (see also Kohn and Ansley, 1983). A time independent level is

incorporated into the observation equation as �. The measurement error et � NIDð0; �2eÞ. Initial

conditions are specified as x0 � Nð½�r0; �

i0�0;P0Þ. The stochastic cycle model in the context of (2) may

be viewed as a time domain generalization of the frequency domain technique of complex

demodulation (e.g. Priestley, 1981, Section 11.2.2).

The goal of fixed-interval smoothing is to produce optimal estimates of the time evolution of the

state xt. Let Yt ¼ ðy1; . . . ; ytÞ0 be a vector created by stacking its sub-components. The complete set of

observations is given as Y ¼ YT . Define

x̂tj� ð�Þ ¼ EfxtjY�g; Ptj� ð�Þ ¼ En

xt � x̂tj� ð�Þ� �

xt � x̂tj�ð�Þ� �0jY�o

where these quantities depend on an underlying parameter set � and are computed using (2). Fixed-

interval smoothing defines estimators for the state x̂tjT and its error covariance PtjT for t ¼ 1; . . . ; T.

Standard algorithms for evaluating these quantities are derived in Jazwinski (1970) and Anderson and

Moore (1979); extensions are found in Kohn and Ansley (1989) and de Jong (1989). These algorithms

generally rely on a forward sweep in time of the Kalman filter to compute x̂tjt and Ptjt, after which a

backwards sweep updates these to yield x̂tjT and PtjT . Filter innovations (one-step ahead prediction

zDefinition: a complex valued variate is called complex normal with zero and �2 variance when its real and imaginary parts areindependent normal variates with mean zero and variance �2=2 (Brillinger, 1981, Addendum).

76 M. DOWD ET AL.


errors) are central diagnostics for assessing the validity of the underlying assumptions. Unknown

parameters � can be determined using maximum likelihood methods (Harvey, 1990, Section 3.4), or

alternatively using the EM algorithm (Watson and Engle, 1983). With reference to the former, the log-

likelihood function Lð�jYÞ associated with the Kalman filter is numerically maximized with respect to

�. Its maximum achieved is Lð�̂jYÞ, where �̂ is the maximum likelihood estimate for the parameter �.The sensitivity of the estimation problem to individual observations can be assessed using case

deletion diagnostics. Let Y ðiÞ denote Y with the ith observation removed. The log-likelihood function

based on Y ðiÞ is Lð�jY ðiÞÞ; the parameter �̂ðiÞ is associated with its maximum. The discrepancy in the

log-likelihood at its maxima, with and without the ith observation, is

�LðiÞ ¼ L �̂jY� �

� L �̂ðiÞjY ðiÞ� �

ð3Þ

Case-deletion parameter anomalies are also readily computed as ��ðiÞ ¼ �̂� �̂ðiÞ. Another diagnostic

is focused on the smoother state. Let the vector of stacked smoother states be X̂ ¼ðx̂1jTð�̂Þ; . . . ; x̂T jTð�̂ÞÞ0. Denote X̂ðiÞ as the corresponding vector with components x̂

ðiÞtjTð�̂Þ ¼

EfxtjYðiÞg; for t ¼ 1; . . . ; T. These are the smoother state estimates using �̂ but with the ith observation

deleted. The diagnostic is defined

�XðiÞ ¼ X̂ � X̂ðiÞ� �0

��1 X̂ � X̂ðiÞ� �

ð4Þ

where � is the smoother error covariance which can be determined using the modified fixed-interval

smoothing algorithm of de Jong (1989) (also, see below). The form (4) is motivated by the widely used

regression diagnostic of Cook (1977). Here, it is expressed in terms of the smoother state rather than

the predicted observations. The diagnostics (3) and (4) can be regarded as special cases of predictive

influence functions (Johnson and Geisser, 1983; Cavanaugh and Johnson, 1999).

We are also interested in a retrospective assessment of the sampling protocol, or the observing array

design, used in the plankton monitoring program. At the observation times, Ht ¼ ð1 0Þ and the system

(2) clearly satisfies the observability criteria (e.g. Harvey, 1990, Section 3.3). Hence, our focus is on

assessment of the sampling protocol (e.g. the irregular sampling) associated with (2), making use of

the estimated parameters �̂ as well as the design matrices D and Ht (but ignoring the observations yt).

Two approaches for assessing the observing array are considered: (i) The relation between the

sampling interval and the smoother error variance. This is done by defining Ht to correspond to

observations every jth time step (for j ¼ 1; 2; . . .) and computing PtjT as a function of j; (ii) the

influence of a single observation on the estimated state of the system. That is, we seek a representation

of the form X̂ ¼ �krkyk, where rk is a vector of weights which determines the influence of the kth

observation yk on the estimation of X̂. The summation is over the complete observation set. We refer to

the rk as representer functions following Bennett (1992, Chapter 5), which has roots in the representer

theorem of Kimeldorf and Wahba (1971). To compute the representers note that, conditional on �̂, the

fixed-interval smoother can be expressed as generalized least-squares linear regression problem

(Duncan and Horne, 1972). Ignoring issues of dimensionality, smoother state estimates are readily

determined as X̂ ¼ RY (where R is derived from Ht and D, using �̂). (Smoother covariances � can also

be determined.) The influence of the kth observation on bX is just the kth column of R (and equals rk).



4. APPLICATION

The state space model (2) was applied to the plankton time series. Sampling intervals are multiples of

one week and define a unit time step in (2). The analysis period covered 604 weeks, with 276 data

points available. A so-called ‘diffuse prior’ was used for the initial conditions such that x0 was based

on the harmonic regression of Section 2, but a relatively large variance was assumed. (Alternatively,

Ansley and Kohn (1985) offer a technique that eliminates the dependence on the initial conditions.)

The vector of unknown parameters was defined as � ¼ ð�2e ; �

2n; �Þ

0. Maximization of the

log-likelihood function Lð�jYÞ using a quasi-Newton method yielded �̂ ¼ ð5:16 � 10�1;4:29 � 10�3; 2:19Þ0.

Kalman filter results are shown in Figure 2. Figure 2(a) shows a time series of the filter state

estimate �̂rtjtð�̂Þ (the first element of x̂tjtð�̂Þ), plus the mean level �̂. A few outliers are evident, some of

which are associated with zero concentrations. Discontinuities resulting from forecast errors over

observation voids are seen for this physically realizable filter. The standardized innovations (Figure 2b)

are not inconsistent with a sample drawn from a population of normally and identically distributed,

Figure 2. Results from the Kalman filter using the maximum likelihood parameter estimates for �. (a) Filter estimates for the

state �rtjt plus the mean level � (solid line), the 90% confidence interval (gray shaded area) and the observations yt (dots). (b)

Standardized innovations vs

78 M. DOWD ET AL.


serially independent random variables. The apparent clustering is due to the seasonally varying

sampling interval and should not be confused with non-stationarity or serial correlation.

Smoother state estimates �̂rtjTð�̂Þ, plus the mean level �̂, are given in Figure 3(a). Differences

between the filter and smoother are small; the largest deviations occur in the first cycle of the time

series (where the filter has its largest error variance due to the diffuse prior) and smoother state

estimates are more continuous. Amplitude and phase changes associated with the smoother estimates

are shown in Figures 3(b) and 3(c). The amplitude anomaly (relative to the mean) has a range that is of

order one. The temporal pattern shows an initial rapid increase in amplitude to mid-1989, after which

we see some oscillations and a general decline. The phase anomaly shown is relative to the entering

phase, with the mean removed. It ranges over � 0.4 rad, corresponding to a timing change of 23 days.

The lack of seasonal oscillations in the amplitude and phase suggest that the cyclic model provides a

good fit to the data. Low frequency amplitude and phase anomalies correspond to interannual

variability. There is little evidence for persistent, long term trends in either the magnitude or the

timing.

Annual cycles of the smoother estimated plankton abundance, along with the mean of these cycles,

are plotted in Figure 4. Variability about the seasonal mean appears largest in the summer and winter.

Seasonal anomalies about the mean annual cycle, along with their 90 per cent confidence intervals, are

given in Figure 5. For most of the analysis period, the 90 per cent confidence interval of smoother

estimated annual cycles includes the mean cycle. Exceptions to this occur in the summers of 1988 and

1998, which exhibit anomalously low values of abundance. If 80 per cent confidence intervals are

used, anomalies about the mean are also found for winter 1989 and the summers of 1991, 1994 and

1997. Note that increased sampling in the summer period reduces the standard error of the smoother

state estimates.

Influential observations were examined using the case deletion diagnostics of (3) and (4), and are

shown in Figure 6. Two observations are identified as being influential in terms of both �L and �X:

one in fall 1989 (obs45 at t¼ 70), and another in spring 1995 (obs156 at t¼ 364). Both observations

appear as outliers in the smoother residuals and filter innovations. Although �L is similar for these two

observations, obs45 causes very little change in the ��, whereas obs156 causes relatively large

changes in �� (not shown). Notably, the estimated system noise variance �2n increases by 29 per cent

when obs156 is omitted. The observation obs156 also has a higher value for �X, indicating a greater

overall influence in determining the smoother state. This results largely from having fewer surround-

ing observations to mediate its overall effect on the smoother state estimation.

The monitoring design was examined using representer functions. Suppose that observations are

available every time step so that stationarity in PtjT is achieved in regions away from end-effects.

Figure 7 shows the representer function attributed to a single observation in the middle of the analysis

period. The function is symmetric for �̂rtjT with a peak at the observation time (Figure 7a), and is anti-

symmetric for �̂itjT (Figure 7b). An alternative representation is given in Figures 7(c) and 7(d) in terms

of the amplitude and phase. The amplitude of the representer has a measurable effect for more than a

year surrounding the observation time. The phasing of the representer is periodic; zero-crossings of

annual cycle centered at the observation time remain in phase, while at other times a phase shift is

imposed. The main conclusion is that, given the estimated �̂, the system has a long memory, or

decorrelation time, and observations have a very non-local effect on state estimation.

The relation between the sampling interval on the smoother variance is shown in Figure 8. In the

case where observations are available every jth time step, a periodic steady state for PtjT results (again,

away from the end-effects). In Figure 8, the standard error is shown in terms of its mean steady state

value plus the variation resulting from error growth between observation times (where the smoother is



Figure 3. Results from the fixed-interval smoother. (a) Smoother estimates for �rtjT plus the mean level � (solid line), the 90%

confidence interval (gray shaded area), and the observations yt (dots). (b) Amplitude drift, where jaj denotes the amplitude

anomaly about its mean level. (c) Phase drift, where � denotes the phase anomaly relative to the entering phase, with the mean

removed (positive phase anomaly implies the series lags the reference cycle)

80 M. DOWD ET AL.


not directly constrained by measurements). The standard error of the smoother state estimate increases

with larger sampling intervals, but at a decreasing rate. Note that the estimated standard error of

smoother estimated plankton abundance ranges from 0.19 to 0.27 and varys seasonally.

5. SUMMARY AND CONCLUSIONS

In this study, we have considered the analysis of a time series of plankton abundance with an emphasis

on interannual variability. A stochastic cycle model was used to allow for drift in the timing and

magnitude of the seasonal cycle over time. This was considered in a state space framework, with a

fixed-interval smoother used to estimate the time evolution of the state. Maximum likelihood

estimation was used to determine unknown parameters, which included both the system noise and

observation error, as well as the mean level of the process. This stochastic cycle model appeared to

adequately fit the plankton abundance time series.

Analysis of the log10-transformed plankton time series showed low frequency variations in

adundance associated with interannual variability. Peak abundance varied by up to an order of

magnitude in concentration units, and the timing of the seasonal cycle over the analysis period

changed by 3–4 weeks. Higher frequency variations which were superimposed on the annual cycles

were not distinguishable from white noise. We speculate that these have their origin in sampling

variability associated with plankton patchiness. The estimated observation error variance is consistent

with reported sampling errors due to cell counting procedures (Smayda, 1978). The estimated system

noise variance is small and implies a system with a long decorrelation time. The representer functions

Figure 4. Estimated seasonal cycles for plankton abundance derived from the smoother (i.e. �þ �rtjT ). These are plotted for the

individual years 1988–1999 (solid lines), and for the overall seasonal mean cycle (dashed line). The time axis corresponds to the

calendar year



reflect this and emphasize the non-local impact of the act of measurement on the estimation of the state

of the system. Two influential observations were identified, one of which strongly affected estimates of

system noise variance.

Detection of interannual variability in plankton abundance relies on estimates of the state as well as

its error variance. The 90 per cent confidence intervals indicate that for some summer periods plankton

abundance was significantly different from that of the mean cycle. However, for most of the analysis

period we must conclude that the observed plankton abundance time series was simply a fixed and

repeating annual cycle. Examination of the monitoring, or sampling, design relied on system matrices

Figure 5. Seasonal anomalies of plankton abundance for the years 1988–1999, as estimated from the smoother. Gray shaded

areas correspond to 90% confidence intervals. Plots for each year are offset by 1.5 units. The time axis corresponds to the

calendar year

82 M. DOWD ET AL.


and the estimated observation and system noise variance. Analysis of the relation between the standard

error and the sampling interval, as well as the representer functions, suggests that identifying the mean

annual cycle is easily achieved. However, the detection of interannual variability about this mean is

much more difficult. It requires a reduction in the estimated error variance of the state. In practice, this

could be brought about by decreasing the sampling intervals (one week or better). A more viable

alternative would likely involve the design of sampling procedures which reduce the apparent

observation error variance by taking proper account of small-scale plankton patchiness, guided, say,

through stochastic field simulation.

An important goal of this study was to assess analysis techniques for plankton time series data. The

state space representation appears useful. It is a flexible framework that readily incorporates irregular

sampling (Kohn and Ansley, 1983) and a variety of models. In this instance, we found the cyclic model

to be satisfactory but have also experimented with more general (and periodic) shape functions within

the state space framework. State estimation via the Kalman filter, and its associated fixed interval

smoother, is well established and many excellent reference texts exist (e.g. Jazwinski, 1970; Anderson

and Moore, 1979; Harvey, 1990). A key feature for practical application is the explicit error estimates

that are produced as part of the analysis. Extension to non-linear and non-Gaussian state space models

is possible, but requires adopting a probabilistic perspective (Kitagawa, 1987; Carlin et al., 1992;

Figure 6. Case deletion diagnostics for each observation. (a) Likelihood anomaly �L. (b) Smoother anomaly �X. A yearly

reference is given by the dotted line



Figure 7. Influence of an observation at middle of analysis period (week 300) on the smoother state estimate. Panel (a), (b):

Representer functions for the real and imaginary part of the smoother state, respectively. Panel (c), (d): Associated amplitude and

phase (note the compressed time axis)

Figure 8. Relation between the sampling interval and standard error of smoother state estimate. Both the standard error (dots)

and its range (line) are shown (see text)

84 M. DOWD ET AL.


Kitagawa, 1998). The development of diagnostics for state space models, which assess the observation

influence (Cavanaugh and Johnson, 1999) as well as the observing array design, provide important

extensions.

In conclusion, it is hoped that this approach to the detection of interannual variability can be

usefully applied to other plankton time series and provide guidance for the improvement of the

sampling design of plankton monitoring programs. Ultimately, we hope to see the fusion of process-

oriented plankton models with observational data sets (Lawson et al., 1995; Vallino, 2000). This

exercise will require state estimation methods not unlike the one presented in this study. Population

biology has begun to consider the effects of stochastic environment variations on populations

(Marion et al., 2000). We would encourage further development of stochastic models for plankton

populations, as well as their rigourous comparison to data.

ACKNOWLEDGEMENTS

We would like to thank Dr. Bruce Smith and Takayoshi Ikeda of the Department of Mathematics and

Statistics, Dalhousie University, as well as Drs. Trevor Platt and Brian Petrie of the Bedford Institute of

Oceanography, for useful discussions and for reviewing the manuscript.

REFERENCES

Abraham ER. 1998. The generation of plankton patchiness by turbulent stirring. Nature 391(6667): 577–580.Anderson BDO, Moore JB. 1979. Optimal Filtering. Prentice-Hall: Englewood Cliffs, NJ; 357.Ansley CF, Kohn R. 1985. Estimation, filtering and smoothing in state space models with incompletely specified initial

conditions. Annals of Statistics 13: 1286–1316.Batchelder HP, Van Keuren JR, Vaillancourt R, Swift E. 1995. Spatial and temporal distributions of acoustically estimated

zooplankton biomass near the Marine Light-Mixed Layers station (59 degree 300N, 21 degree 000W) in the North Atlantic inMay 1991. Journal of Geophysical Research 100(C4): 6549–6563.

Bennett AF. 1992. Inverse Methods in Physical Oceanography. Cambridge University Press: New York; 346.Bennett AF, Denman KL. 1985. Phytoplankton patchiness: inferences from particle statistics. Journal of Marine Research 43:

307–355.Brillinger DR. 1981. Time Series. Data Analysis and Theory. Holden-Day: San Francisco; 540.Carlin BP, Polsen NG, Stoffer DS. 1992. A Monte Carlo approach to nonnormal and nonlinear state space modelling. Journal

of the American Statistical Association 87: 493–500.Cavanaugh JE, Johnson WO. 1999. Assessing the predictive influence of cases in a state space model. Biometrika 86(1):

183–190.Cook RD. 1977. Detection of influential observations in linear regression. Technometrics 19: 15–18.Cullen JJ, Ciotti AM, Davis RF, Lewis MR. 1997. Optical detection and assessment of algal blooms. Limnololgy and

Oceanography 42: 1223–1239.DeJong P. 1989. Smoothing and interpolation with the state-space model. Journal of the American Statistical Association

84(408): 1085–1088.Duncan DB, Horn SD. 1972. Linear dynamic recursive estimation from the viewpoint of regression analysis. Journal of the

American Statistical Society 67: 815–821.Edwards AM. 2001. Adding detritus to a nutrient-phytoplankton-zooplankton model: a dynamical systems approach. Journal

of Plankton Research 23: 389–413.Gallager SM, Davis CS, Epstein AW, Solow A, Beardsley RC. 1996. High-resolution observations of plankton spatial

distributions correlated with hydrography in the Great South Channel, Georges Bank. Deep-Sea Research: Part II 43(7–8):1627–1663.

Harvey AC. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press: Cambridge;554.

Haywood J, Wilson GT. 2000. An improved state space representation for cyclical time series. Biometrika 87(3): 724–726.International Oceanographic Commission. 1999. IOC-WMO-ICSU-FAO Living Marine Resources Panel of the Global Ocean

Observing System (GOOS). GOOS Report No. 74. UNESCO: Paris; 71.



Jazwinski AH. 1970. Stochastic Processes and Filtering Theory. Academic Press: New York; 376.Johnson W, Geisser S. 1983. A predictive view of the detection and characterization on influential observations in regression

analysis. Journal of the American Statistical Association 78(381): 137–144.Kimeldorf GS, Wahba G. 1971. Some results on Tchebycheffian spline functions. Journal of Mathematical Analysis andApplications 33: 82–95.

Kitagawa G. 1987. Non-Gaussian state-space modeling of nonstationary time series. Journal of the American StatisticalAssociation 82: 1032–1041.

Kitagawa G. 1998. A self-organizing state space model. Journal of the American Statistical Association 93(443): 1203–1215.Kohn R, Ansley CF. 1983. Fixed-interval estimation in state space models when some of the data are missing or aggregated.Biometrika 72: 683–688.

Kohn R, Ansley CF. 1989. A fast algorithm for signal extraction, influence and cross-validation in state space models.Biometrika 76(1): 65–79.

Lawson LM, Sptiz YH, Hofmann EE, Long RL. 1995. A data assimilation technique applied to a predator-prey model. Bulletinof Mathematical Biology 57: 593–617.

Marion G, Renshaw E, Gibson G. 2000. Stochastic modelling of environmental variations for biological populations. TheoreticalPopulation Biology 57: 197–217.

Martin JL, LeGresley MM, Strain PM. 2001. Phytoplankton monitoring in the Western Isles Region of the Bay of Fundy during1997–98. Can. Tech. Rep. Fish. Aquat. Sci. 2349: 91.

Martin JL, LeGresley MM, Strain PM, Clement P. 1999. Phytoplankton monitoring in the southwest Bay of Fundy during1993–96. Can. Tech. Rep. Fish. Aquat. Sci. 2265: 132.

Martin JL, Wildish DJ, LeGresley MM, Ringuette MM. 1995. Phytoplankton monitoring in the southwest Bay of Fundy during1990–92. Can. Manuscr. Rep. Fish. Aquat. Sci. 2277: 157.

Platt T, Dickie LM, Trites RW. 1970. Spatial heterogeneity of phytoplankton in a nearshore environment. Journal of theFisheries Research Board of Canada 27: 1453–1473.

Pepova EE, Fasham MJR, Osipov AV, Ryabchenko VA. 1997. Chaotic behaviour in an ocean ecosystem model under seasonalexternal forcing. Journal of Plankton Research 19: 1495–1515.

Priestley MB. 1981. Spectral Analysis and Time Series. Academic Press: London; 890.Smayda TJ. 1978. Estimating Cell Numbers. In Phytoplankton Manual, Sournia A (ed.). UNESCO Monographs on

Oceanographic Methodology, 165–180.Vallino JJ. 2000. Improving marine ecosystem models: use of data assimilation and mesocosm experiments. Journal of MarineResearch 58: 117–164.

Vance RR. 1990. Population growth in a time-varying environment. Theoretical Population Biology 37: 438–454.Watson MW, Engle RF. 1983. Alternative algorithms for the estimation of the dynamic factor, MIMIC and varying regression

coefficient. Journal of Econometrics 23: 385–400.Weibe PH. 1970. Small scale spatial distribution in oceanic zooplankton. Limnology and Oceanography 15: 315–321.Wiebe PH, Holland WR. 1968. Plankton patchiness: effects on repeated net tows. Limnology and Oceanography 13(2): 315–321.Wood SN, Horwood JW. 1995. Spatial distribution functions and abundances inferred from sparse noisy plankton data: an

application of constrained thin-plate splines. Journal of Plankton Research 17(6): 1189–1208.

86 M. DOWD ET AL.


Documents

Interannual variability in a plankton time series