Flagged observation analyses as a tool for scoping and communication in integrated ecosystem assessments

Hiroko Kato Solvang; Per Arneberg

doi:10.1371/journal.pone.0305716

Abstract

Working groups for integrated ecosystem assessments are often challenged with understanding and assessing recent change in ecosystems. As a basis for this, the groups typically have at their disposal many time series and will often need to prioritize which ones to follow up for closer analyses and assessment. In this article we provide a procedure termed Flagged Observation analysis that can be applied to all the available time series to help identifying time series that should be prioritized. The statistical procedure first applies a structural time series model including a stochastic trend model to the data to estimate the long-term trend. The model adopts a state space representation, and the trend component is estimated by a Kalman filter algorithm. The algorithm obtains one- or more-years-ahead prediction values using all past information from the data. Thus, depending on the number of years the investigator wants to consider as “the most recent”, the expected trend for these years is estimated through the statistical procedure by using only information from the years prior to them. Forecast bands are estimated around the predicted trends for the recent years, and in the final step, an assessment is made on the extent to which observations from the most recent years fall outside these forecast bands. Those that do, may be identified as flagged observations. A procedure is also presented for assessing whether the combined information from all the most recent observations form a pattern that deviates from the predicted trend and thus represents an unexpected tendency that may be flagged. In addition to form the basis for identifying time series that should be prioritized in an integrated ecosystem assessment, flagged observations can provide the basis for communicating with managers and stakeholders about recent ecosystem change. Applications of the framework are illustrated with two worked examples.

Citation: Solvang HK, Arneberg P (2024) Flagged observation analyses as a tool for scoping and communication in integrated ecosystem assessments. PLoS ONE 19(9): e0305716. https://doi.org/10.1371/journal.pone.0305716

Editor: João Zambujal-Oliveira, University of Madeira / NOVA Lincs, PORTUGAL

Received: October 31, 2022; Accepted: June 4, 2024; Published: September 23, 2024

Copyright: © 2024 Solvang, Arneberg. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting Information files.

Funding: Per Arneberg received “Sustainable multi-species harvest from the Norwegian Sea and adjacent ecosystems”, funded by The Research Council of Norway (pr. nr. 299554).

Competing interests: The authors have declared that no competing interest exist.

Introduction

Against a background of increasing impact from climate change and other anthropogenic drivers, causing elevated rates of change in marine ecosystems [1–6] leading to patterns of variability beyond the range of the Holocene [7–12], ecosystem-based management (EBM) is increasingly identified as a needed framework for management of marine socio-ecological systems [13]. Integrated ecosystem assessments (IEA) have been developed to provide the scientific basis for EBM [14], and numerous groups of scientists working with IEA have been established, such as the regional IEA groups within the International Council for the Exploration of the Sea (ICES, [15]).

Among the core activities of IEA groups are analyses of time series to summarize changes that have occurred in recent decades in ecosystems, and attempts to highlight possible connections between physical, biological, and human ecosystem components [14, 16]. Emphasis is put on keeping an open communication with management and stakeholders [13, 17, 18]. As the groups typically have at their disposal a large number of time series [16, 19], it will often be necessary to prioritize a subset of them for more extensive analyses and communication purposes [20, 21]. Prioritization should preferably be done using a standardized framework applied to all time series. Here we present an approach which is based on analyses of patterns of recent change, where the aim is to identify time series in which the most recent values deviate significantly from an expected trend, possibly indicating unexpected change. This should be of high relevance for IEA groups, as they are often challenged with understanding and interpreting recent change [14].

Our approach is based on estimating trends of time series before assessing whether the most recent observations deviate significantly from these trends. Since temporal changes in ecosystems can take the form of long-term movements as well as short- or mid-term cyclic periods and noise components, different definitions of trends have been used in marine IEAs [16]. In the field of statistical time series analysis, the long-term movements are commonly classified as ‘trends’, while short- or mid-term cyclic periods are not, due to the different assumptions about the statistical properties. When investigating a trend in time series data, it can therefore be useful to separately identify non-stationary trends and stationary cyclic components. This decomposition is performed by a framework called ‘structural time series modelling’, which is using a state space representation where the state of each component is estimated by the Kalman filter algorithm [22]. The concept is different from applying an Autoregressive integrated moving average model to adjust with the aim of studying stationary processes from nonstationary time series data [23].

The Kalman filter algorithm can make one- or multistep-ahead predictions in the numerical procedure. The numerical procedure introduced in this paper uses prediction values and forecast uncertainty bands to assess the status of a recent observation, which determines whether the most recent observation follows the prediction or deviates from it [24], thus giving an indication whether change that is unexpected from the predicted trend, is occurring in the time series. We call the significant deviated observations a “Flagged Observation” and the approach “Flagged Observation analysis”, hereafter referred to as “FO” and “FO analysis”, respectively. The interpretation of a FO is not equivalent to the types of early warning signals that have been proposed with the aim of predicting critical transitions in marine populations or ecosystems [25, 26], nonlinear ecological change [27] or early warning signs based on theoretical framework in social-ecological networks [28]. An important difference is that while the latter frameworks address change caused by specific types of dynamics and often with specific types of outcomes, FO analysis focuses on identifying unexpected patterns of change that is not specific to any type of underlying dynamics or outcomes. Rather, as described above, FO analysis is a practical tool for IAE groups for prioritizing time series for in depth analyses, communication, and other purposes.

In addition to identify unexpected observations for single years, it can also be interesting to explore whether observations from the most recent years together form a pattern where all are consecutively either above or below the predicted trend in a way that is not expected. This is equivalent to asking whether there is an unexpected tendency for the most recent years, and a framework for this is also presented here.

In this article, we first introduce the numerical procedure and then demonstrate two examples using time series data for, respectively, the Atlantic Multi-decadal Oscillation (AMO) and the Norwegian Sea ecosystem.

Statistical method

The statistical method includes first a procedure for trend estimation and second a procedure for FO analysis based on multistep ahead prediction values. The methods we apply have been established in statistical time series analysis [22, 29–31]. The output from these analyses can be used to identify single observations (for example years) that deviate significantly from the expected trend, and, with the help of an additional procedure, explore whether there are unexpected tendencies seen across all of the most recent observations. The details are as given below.

Trend estimation procedure

The observation model of a time series is given by: (1) where t(n) is the trend component and u(n) is the residual component at time step n, assuming Gaussian white noise. In this article, we introduce a stochastic trend model given by a dth-order difference equation model and a method for estimating the trend [29, 30, 32, 33]. The observation we analyze is recorded at equally intervals (annually or monthly), thus, we consider y(n) as discrete time series.

The stochastic differential trend model is defined by the dth-order difference equation, which was posed as a smoothing problem by ref. [34]. This model allows for more flexible trends than does the polynomial regression model. The stochastic trend model for a variable is expressed in the following way: (2) where ∇ is defined as a difference operator given by ∇t(n) ≡ t(n)−t(n−1) and v(n) is assumed to be an identical and independent sequence with where N is ‘n’ of a normal (Gaussian) distribution, the mean is 0 and the variance is . If d = 1,

∇¹t(n) = v(n), that is, t(n) = t(n−1)+v(n), where the trend is known as a random walk model. If d = 2,∇²t(n) = v(n), that is, {t(n)−t(n−1)}−{t(n−1)−t(n−2)}=v(n), then, t(n)−2t(n−1)−t(n−2)=v(n). The notation for the difference operator is varied, e.g. ∇ is defined in [29, 35], but Δ is used in [30]. Provided that the variance of v(n) is sufficiently small, t_i(n) yields a smooth trend. If the variance is not small, t(n) yields a fluctuated trend. We choose the second order difference stochastic model to estimate the trend in this study.

The model can be represented in the following state space form [22, 29, 30, 32, 33], as (3) where z(n) is the state vector corresponding to the trend component t(n), v(n) is the system noise vector that is assumed to be a Gaussian white noise with mean 0 and unknown variance , F,G, and H indicate integers, vectors or matrices, and w(n) is observation error that is assumed to be a Gaussian white noise with mean 0 and unknown variance . We call Formula (3) a linear-Gaussian state space model. The trend component is modelled by the dth-order differential equation model. When d = 1 in Eq (2),

z(n) = [t(n)], F = G = H = 1, that is, the system model in (3) is given by When d = 2 in Eq (2),

, , , and the system model is given by

. The observation error w(n) corresponds to u(n) in Eq (1) in this case. A particularly important problem in state-space modelling is to estimate the state z(n) based on the observations of the time series y(n).

The state setting trend component shall now be considered to be based on the set of observations Y(j) = {y(1),y(2),⋯,y(j)}. In particular, for j<n, the state estimation problem results in the estimation of the future state based on the present and past observations and is called prediction. For j = n, the problem is to estimate the current state and is called a filter. On the other hand, for j>n, the problem is to estimate the past state z(j) based on the observations until the present time and is called smoothing [30]. The general approach to these state estimation problems is to obtain the conditional distribution of the state z(n) given the observations Y(j). The state-space model given by (3) is a linear model, and the system and observations noises and the initial state z(0) follow a Gaussian distribution. That is, all of these conditional distributions become Gaussian distributions. Therefore, to solve the state estimation problem for the state space model, it is sufficient to obtain the mean vectors and the variance (covariance matrices) of the conditional distributions [30]. In order to obtain the conditional joint distribution of states z(1),z(2),⋯,y(n) given the observations Y(n) = {y(1),y(2),⋯,y(n)} by applying the Kalman filter algorithm is known to be a computationally efficient procedure [22, 30, 31, 36]. The conditional mean and the variance (covariance matrix) of the state z(n) are denoted by (4) where ’ means transpose. The Kalman filter algorithm recursively operates the following one-step-ahead prediction (j = n−1) and filtering (j = n) to obtain the joint conditional distribution of the state based on the derivation shown in Appendix C of [30]. This is here given as:

prediction: (5) filtering: (6) Here, z(n|n−1) and V(n|n−1) correspond to the conditional mean and conditional variance of the state respectively, Q includes the variance of the system noise, and K is called the Kalman gain. Setting the initial state z(1|0) as zero and the initial variance V(1|0) of the state as an arbitrary real number (e.g. 0.1), the Kalman gain is calculated by using the initial variance and observation noise. The filtering value z(1|1) is obtained by using observation y(1) and the calculated gain from the filtering procedure (6). Then the next prediction values z(2|1) and V(2|1) are calculated by using z(1|1) and V(1|1) in the prediction procedure (5). The iterative calculation procedure for the state is continued until n = N. Recursive methods based on state space representations are known to be efficient for calculating the likelihood functions in discrete-time Gaussian processes, and the state space model and the Kalman filter therefore provide an efficient method for the computation of the likelihood of the time series models [22, 29, 30]. In our study, the model provided in (1) includes the parameter vector . The log-likelihood function l(θ) of the model (1) is given by: (7) where Y(n−1) = (y(1),y(2),⋯,y(n−1)), Δy(n) = y(n)−Hz(n|n−1), and . Recall that V(n|n−1) = FV(n−1|n−1)+GQG’ in the one-step-ahead prediction and the Q control the smoothness of the estimated trend. The variance of the system noise in Q can be optimized to obtain a better fitted trend to the data by using maximum likelihood within an arbitry variance range. The variance of the observation noise can be directly set to the variance of the observation in this study. These variances could also be estimated as one parameter of this model by the numerical optimization procedure, such as the Newton-Raphson method [29, 30] applied with a likelihood function. If it is necessary to compare differential stochastic models with different orders for the trend component, the optimum differential order d is identified by the AIC [30, 32, 37]. Based on the maximum log-likelihood, the number of parameters for d and the variances of system noise and observation noise, AIC = −2l(θ)+2×number of parameters.

After identifying the optimum trend model, the smoothed trend is estimated by a fixed-interval smoother algorithm [29, 30, 33]: (8) In this study, we set the differential order as d = 2 to obtain a smooth trend for all time series data as introduced in refs. [29, 30, 33]. In addition, we apply grid searching within a range for finding an optimum Q that can control smoothness for the trend.

FO analysis by multistep-ahead prediction values

The above Kalman filter algorithm provides one-ahead prediction. However, it can be expanded to give a long-term prediction of the state, which is a multistep-ahead (j-ahead) prediction for j = 1,2,3,⋯[22, 30]. Let us consider a relevant situation. With the Kalman filter, one-ahead prediction for z(n+1) is obtained by z(n+1|n) and variance V(n+1|n). If the future observation y(n+1) are not observed, the calculation is formally conducted by using the data observed until time point n [22, 30]. This gives z(n+1|n+1) = z(n+1|n) and V(n+1|n+1) = V(n+1|n). Then, two-ahead prediction z(n+2|n) and variance V(n+2|n) are obtained by z(n+1|n) and V(n+1|n), respectively. In general, j-ahead prediction based on the observations until time point n is obtained by repeating the prediction step j times. The algorithm used to predict states z(n+1),z(n+2),⋯,z(n+j), based on the data Y(n) observed until time point n, is expressed as follows: (9) [22, 30]. When Y(n) is observed, the relationship between the state and the time series is expressed by the observation model y(n) = Hz(n)+w(n) in (3). The mean and the variance for the distribution of the prediction y(n+j) are given as y(n+j|n)≡E(y(n+j)|Y(n)) and u(n+j|n)≡Cov(y(n+j)|Y(n)), respectively. Here,E(∙) and Cov(∙) are notations for the expectation vector and the variance-covariance matrix, respectively (but in this study only variance as the data are univariate) [30]. Using the observation Eq (3), the mean of y(n+j) is expressed by: (10) The variance of y(n+j) is given by: (11) Therefore, the prediction distribution for y(n+j) based on data Y(n) is a normal distribution with mean y(n+j|n) and variance u(n+j|n) or a standard deviation [30]. The forecast bands (FBs) are calculated by Eqs (6) and (7) [30, 38]. Note that we define the values given by the multistep-ahead prediction procedure as ‘forecast values’ because they are calculated by using the previous data in the algorithm.

Assessing unexpected tendencies

If all the observations from n+1 to n+j consecutively fall either above or below the predicted trend, this is recognized as a potentially unexpected tendency. Here we estimate the probability of an unexpected tendency (p_ut) by multiplying the upper or lower probabilities, where the upper probabilities are calculated if all the observations from n+1 to n+j time points are located above the predicted trend, and the lower probabilities are calculated if they are located below the predicted trend. Because these probability estimates have to be independent when calculating p_ut, they cannot be estimated from the observations directly, which have to be considered dependent in time series data. Therefore, upper/lower probabilities are calculated based on the residuals expressing the difference between the observations and the predicted trend, which can be assumed to follow an identical independent distribution. For the calculation of the upper/lower probability, the residual part is given as , i = 1,⋯,j, where is given by Hz(n+i|n) in (10), that is, is equivalent to the obtained by . The Gaussian distribution that the residual part follows, has a mean 0 and variance . Therefore, we consider the upper tale probability if the observations consecutively locate above the estimated trend and the lower tale probability if the observations consecutively locate below the estimated trend, where X is the random variable given by the Gaussian distribution with 0 and variance . The probability is also calculated for each time point, denoted as P₁,⋯,P_j. Finally, p_ut is obtained as P₁×⋯×P_j. When interpreting p_ut, it can be compared with a general statistical threshold, such as 0.05. However, because the value of p_ut will depend on j, we recommend that the actual comparison should in this case be made with the threshold 0.05^j.

The conceptual outlines of this study’s analysis procedure for FO analysis and estimation of p_ut are given in Figs 1 and 2 respectively. The numerical procedure has been implemented using MATLAB code [39] and R code [40]. MATLAB codes are given in S1 File (grid searching method for Q and the calculation of p_ut) and S2 and S3 Files (a version using numerical optimization for Q as an option). R code is given in S4 File (grid searching method for Q and the calculation of p_ut) and S5 File (a version using numerical optimization for Q as an option). Example data are given in S6 File (yearly data of AMO).

Download:

Fig 1. Outline of proposed flagged observation (FO) analysis.

https://doi.org/10.1371/journal.pone.0305716.g001

Download:

Fig 2. Outline of the proposed assessment for unexpected tendencies using probability multiplicated by the upper/lower tail probabilities at consecutive time points.

https://doi.org/10.1371/journal.pone.0305716.g002

Illustrative examples

The dataset for the Atlantic Multi-decadal Oscillation (AMO) is based on index monthly raw data [41], while for the Norwegian Sea ecosystem, we use the yearly data assembled by the ICES integrated ecosystem assessment working group for the Norwegian Sea (WGINOR, [42]). Abbreviations for the Norwegian Sea data used in this article are summarized in S1 Table together with a short summary of the objective for inclusion of each time series in the work of the group. In the examples, we do not attempt to fully interpret any FOs revealed but comment on the possible background for some of them to illustrate the context for further work in IEA groups. Most of the time series examples in this study are short (i.e., < 50 observations), as this typically what IEA groups have at their disposal [19, 43]. However, some longer time series have also been included.

The Atlantic Multi-decadal Oscillation (AMO)

AMO is a pronounced signal of climate variability in the North Atlantic Sea surface temperature (SST) [44]. The monthly data recorded from December in 1869 to March in 2021 has been published under the NCAR CLIMATE data guide [45]. We extracted monthly raw data for the period 1980–2020, from which we calculated annual means, giving a time series with 41-time points. In addition, a time series with monthly data for the period 1980–2020 has been included to illustrate how seasonality can be taken into account, and a time series for the years 1900–2020 to provide an example of a long time series for these data (results for the latter two time series are shown in the S2 Fig).

To illustrate how inferences may differ for different time periods, both seven-years and three-years predictions are shown for this example. Thus, the three-years ahead prediction is made for the years 2018–2020 and the seven-years ahead prediction for 2014–2020. To optimize Q for the model, we set 0.01≤Q≤0.1 as a range of grid serach. The calculated maximum log-likelihood and optimum Q for each dataset are summarized in Table 1. Using the parameters of the model, the Kalman filter algorithm was run to calculate three-years- and seven-years-ahead predictions. Fig 3 presents the outputs of this.

Download:

Fig 3.

The estimated three-years (upper) and seven-years (lower) prediction values of sea surface temperature and the most recent observations for the dataset on the Atlantic Multi-decadal Oscillation. The solid grey lines and the black points indicate the observations used for making the forecast values, and the black points indicate the observations that were plotted for comparison with the prediction values (dotted blue line). The dotted grey line presents the prediction value Hz(n|n−1) and the solid blue lines present the smoothed trend estimates obtained by a fixed-interval smoother algorithm. The light-blue band presents the 95% FB (shown for the whole time series), the dark green band represents the 80% FB and light-green band the approximately 70% FB (the latter two shown for predicted years only).

https://doi.org/10.1371/journal.pone.0305716.g003

Download:

Table 1. Calculated log-likelihood (LL) and optimum Q by applying a stochastic trend model for d = 2 for the AMO and Norwegian Sea ecosystem datasets.

https://doi.org/10.1371/journal.pone.0305716.t001

For the case of seven-years-ahead prediction, none of the observations for the most recent years fall outside the 95% FB, but the observation for 2014 fall outside the 70% FB and for 2015 outside the 80% FB. Thus, only two single possible FOs are identified when looking at each of the seven most recent years individually, and this is due to a marked decrease in SST in 2014 and 2015 that contrasts the slightly increasing trend predicted for 2014–2020. Although all observations for the seven most recent years fall below the predicted trend (Fig 3), p_ut is not smaller than 0.05⁷ (Table 2), suggesting that there is not a tendency in the data that should be flagged for closer analyses or highlighted in communication with stakeholders. For the case of three-years-ahead predictions, none of the observations from the most recent years fall outside any of the FBs, and they are spread evenly around the predicted trend (Fig 3). Thus, while there are some indications that some of the seven most recent observations deviate from the trend predicted for the last seven years, no such pattern is seen for the trend predicted for the last three years, suggesting that an assessment group may need to look differently at change over these two time periods.

Download:

Table 2. The upper /lower probabilities at consecutive time points and the p_ut for AMO.

https://doi.org/10.1371/journal.pone.0305716.t002

Using a longer time series (1900–2020) to estimate the predicted trend for the last three or seven years, respectively, did not change these conclusions (S2 Fig). Analyses based on monthly data gave predicted trends that follows seasonal fluctuations, and no FOs were detected for the last three or seven months (based on a three-months-ahead and seven months-ahead predicted trend, respectively, S2 Fig).

Norwegian Sea ecosystem

The Norwegian Sea is located West and Northwest of Norway, bordered by the North Sea and the Atlantic Ocean to the south, the Greenland Sea to the west and the Arctic Ocean and Barents Sea to the north and east. It is a deep-sea area with three species of mainly planktivorous pelagic fish making up the economically most important fish stocks: mackerel (Scomber scombrus), Norwegian spring-spawning herring (Clupea harengus) and blue whiting (Micromesistius poutassou). Ocean currents are dominated by relatively warm and saline Atlantic water masses flowing in from the south and colder and fresher Arctic water masses flowing in from the northwest [46]. There is considerable negative density dependence acting on biomass within the three pelagic fish stocks, presumably through intraspecific competition over food [47]. While there are also indications of competition among the stocks, most strongly between mackerel and herring [47], other work has suggested that interspecific competition is less significant [48]. The combined biomass of the three species has increased over the last decades while zooplankton biomass has declined, and it has been hypothesized that the pelagic fish biomass may have exceeded the carrying capacity of the system [21]. The climate has historically varied between cold and warm phases, with plankton and fish productivity tending to increase in the warmer phases [49]. Here, we analyse time series on spawning stock biomass, recruitment, and growth (age and weight at age 6) for the three pelagic fish stocks, zooplankton biomass, and three key variables for the physical environment: heat content, freshwater content, and the North Atlantic Oscillation index. The observations have been recorded annually, although the starting/ending years of observation vary among the time series.

For the current work, we used time series with different start years and 2019 as the last year [42]. As one of the main aims of IEAs in the Norwegian Sea has been to provide background information for advisory work for operational fisheries management [50, 51], change over a short period of the most recent years is typically of interest. The conditions for making the prediction were therefore set to three-years-ahead predictions for 2017–2019 using the data observed up to 2016 to estimate the predicted trend. The calculated maximum likelihood, and optimum Q to search in the range for each data are summarized in Table 1. The Q for most time series were estimated as 0.01, which gives a smoothed predicted trend. A fixed estimate of was used, which was estimated using the observations until 2016.

Fig 4 presents the outputs of the analyses. Looking at variables for the physical environment, FOs for individual years were observed for relative freshwater content (RFW), where the observation for 2018 fall outside the 80% FB and for 2019 outside the 95% FB. In addition, observations for all the last three years fall well above the predicted trend and p_ut is close to 0.05³ (Table 3), indicating a possible unexpected tendency in the data. While freshening of the Norwegian Sea had been going on for nearly a decade before 2019 [52], these increases in freshwater content point to a recent intensification of this that might require the attention in IEAs of the Norwegian Sea. For the North Atlantic Oscillation (NAO), all observations for the last three years were lower than the predicted trend, but the p_ut for the pattern was substantially higher than 0.05³ (Table 3), suggesting that the NAO changes should not be prioritized in an IEA.

Download:

Fig 4. The estimated three-years prediction values and the three most recent observations for data on variables related to climate, plankton and pelagic fish in the Norwegian Sea ecosystem.

The solid grey lines indicate the observations used for making the forecast values and the black points indicate the observations that were plotted for comparison with the prediction values (dotted blue line). The dotted grey line presents the prediction value Hz(n|n−1) and the solid blue lines present the smoothed trend estimates obtained by a fixed-interval smoother algorithm. The light-blue band presents the 95% FB (shown for the whole time series), the dark green band represents the 80% FB and light-green band the approximately 70% FB (the latter two shown for predicted years only).

https://doi.org/10.1371/journal.pone.0305716.g004

Download:

Table 3. The upper /lower probabilities at consecutive time points and the p_ut for RFW, NAO, ZooB, MacB, MacR, MacL, HerW, HerL, BWB, BWR, and BWW.

https://doi.org/10.1371/journal.pone.0305716.t003

For zooplankton biomass (ZooB), the estimate for 2017 fall above the 70% FB. The estimates for 2018 and 2019 also fall above the predicted trend, but p_ut is considerably larger than 0.05³ (Table 3), indicating that there is no clear unexpected upward tendency in the data.

Looking at variables for pelagic fish stocks, the most substantial evidence for unexpected change is seen for blue whiting. For spawning stock biomass (BWB), one of the three most recent years fall above the 80% FB, another above the 70% FB, and the third also well above the predicted trend (Fig 4). The p_ut of this pattern is slightly larger but close to 0.05³ (Table 3), suggesting that there may be a tendency for an increase in biomass beyond the expected. At the same time, there are indications of unexpected declines in blue whiting individual growth, shown for weight at age 6 (BWW), where all recent observations fall below the 70% FB (Fig 4) and p_ut is substantially smaller than 0.05³ (Table 3). For the other measure of growth, length at age 6 (BWL), all the three most recent observations are lower than the predicted trend and one also outside the 80% FB (Fig 4), and with p_ut close to 0.05³ (Table 3), pointing to a similar unexpected tendency as for weight growth. The changes in biomass and growth may be linked, as increases in biomass tend to be associated with decreases in growth, possibly through intraspecific competition over food [47]. In addition, for blue whiting recruitment (BWR), two observations fall below the 70% FB and one below the 80% FB (Fig 4) with p_ut falling well below 0.05³ (Table 3), indicating that the decline in recruitment for the most recent years indeed represents an unexpected tendency that should be prioritized in an IEA. Although pelagic fish recruitment remains hard to forecast (e.g. [53]), it is interesting to note that considerable progress has been made in predicting variation in the geographical location of blue whiting spawning habitat, which may be linked to recruitment success [54–56], thus offering a possible avenue for more detailed assessments and studies following up the changes in blue whiting recruitment.

For mackerel, an unexpected decline is seen in spawning stock biomass (MacB), for which two of the three most recent years fall below the 80% FB and the third also below the predicted trend, with p_ut well below 0.05³ (Fig 4, Table 3). As the stock has been fished well above recommended levels since 2013 due to the termination of an international quota sharing agreement [51], a decline in mackerel stock spawning biomass was observed a few years into this period, which is not expected from the predicted trend—indeed an issue that should be flagged for prioritization by an IEA group. Consistent with the decline in mackerel spawning stock biomass, there are indications of an increase above the expected in individual growth measured as length at age 6 (MacL), with observations for one of the three most recent years falling above the 80% FB and the two others above the predicted trend (Fig 4) and with p_ut smaller than 0.05³ (Table 3). While all of the observations for the three most recent years for mackerel recruitment (MacR) fall above the predicted trend, p_ut is substantially larger than 0.05³ (Table 3), suggesting this variable should not be flagged for prioritization in an IEA.

Considering the four variables related to herring, the only FOs for a single year is one estimate of recruitment (HerR), which falls above the 70% FB (Fig 4). As pelagic fish recruitment is highly variable, a single observation falling outside the expected trend may not be a reason for flagging this variable for prioritization in an IEA. Although all observations for herring weight and length at age 6 (HerW and HerL) from the three most recent years fall below the predicted trend, p_ut estimates are clearly larger than 0.05³, suggesting these variables should not be flagged for prioritization.

Discussion

We have introduced a time series analysis procedure for making predictions for a specific time period using a structural time series model including a trend model. Based on this, we have outlined a framework for investigating whether the most recent observations deviate from the predicted trend for this time period and thus represent possible flagged observations (FOs). This includes assessing both whether single years represent FOs or whether all the observations from the recent years together represent an unexpected tendency that is classified as a FO. The trend was estimated using a stochastic trend model and observed time series data, and the specific-years-ahead predictions were systematically calculated according to the iterative procedure of the Kalman filter algorithm. The statistical analysis is followed by a qualitative evaluation of each FO, where it may be decided to follow some of them up by more detailed analyses within an integrated ecosystem assessment (IEA). We have also introduced a procedure for assessing the probability that the most recent observations together represent an unexpected tendency (p_ut). Thus, FO analysis corresponds to a scoping exercise in an IEA that aims at guiding the assessment work and that can also be used for communication purposes. We have also illustrated how FO analysis can be applied using two examples, which illustrate how FOs with different probabilities can be used to guide the work in an IEA. We note that applications of FO analyses may also extend beyond IEAs to other areas of science and advisory processes where an overview of recent change is required across multiple time series.

The time series available from marine ecosystems that are relevant for analyses described here are typically short (i.e., < 50 time points) [19, 43]. This puts constraints on the types of time series analyses that can be performed. For example, null hypothesis testing with time dependency based on autoregressive model or auto-correlation for such short time series can produce misleading results, including false positive and negative results [57]. The procedure described here does not include null hypothesis statistical testing when the trend is estimated, and the type of structural time series model used by us is not based on a frequentist framework but corresponds to a Bayesian approach in the state space representation and Kalman filter algorithm [58], which may properly assess trends in short time series [57]. An alternative method to the one used here could have been the Box Jenkins model, which transforms a non-stationary mean time series to a stationary process [59]. However, effective fitting using this method, again, requires longer time series than what is normally available for marine ecosystems [19, 43]. Thus, for short time series, the Bayesian framework used here appears to be more theoretically appropriate than alternative approaches.

The linear state space model shown in (3) assume a Gaussian distribution for the system noise and observation noise. If a Gaussian distribution cannot be assumed, FOs for single years can still be estimated by applying algorithms such as the extended Kalman filter [31], the non-Gaussian filter and smoothing algorithms by numerical approximating the non-Gaussian distributions by step function, a piecewise linear function, a spline function, and a particle filter [29]. To estimate p_ut for non-Gaussian cases, a local regression model [60] or a local linearization filter [61] may be applied.

When assessing the relevance of p_ut estimates, we have recommended to use a threshold of, in the case of a 0.05 level, as 0.05^j. This is done because p_ut will depend on the number of recent years considered (j). For example, with predictions made for the three last years, this threshold becomes 0.000125 at the 0.05 level. It may be argued that this gives a conservative approach for identifying unexpected tendencies. However, it should be noted that we still identified several unexpected tendencies here (Tables 2 and 3). It is important to note that p_ut should not be used in testing for statical significance, but, as for the FOs, as a tool for prioritisation and communication for IEA groups.

To study and assess recent change, IEA groups often rely on examinations of anomaly plots. In such plots, recent change appears as deviations from a long-term mean, often estimated for the whole length of the time series (see e.g. Bulgin, Merchant [62] for an application to global sea surface temperatures, SST). By focusing on deviations from the expected trend for the most recent years (where the expected trend is estimated by using information from the whole time series prior to the recent years), FO analysis can provide a different perspective of recent change. For example, using a seven-years prediction, observations from two of these years fall outside the expected (Fig 3). We argue that the same interpretation is less evident from an anomaly plot of the same time series (see S1 Fig). Thus, while positive and negative values indicate how observations deviate from a constant mean value in an anomaly plot, the trend changes through time, making it harder to assess how the most recent observations deviate from the trend that should be expected for these recent years. Recognizing that anomaly plots are important for a large range of purposes within IEAs, we emphasize that FO analysis can provide useful additional information for the practical work in IEA groups, in particular in the light of the challenge they are often faced with of understanding and assessing the most recent development of an ecosystem [14].

Since the cumulative output of FO analysis also aim at giving a sweeping overview of the recent dynamics of all the measured elements in an ecosystem by highlighting the variables that exhibit unexpected change while at the same time showing trends and data for those that do not, the approach should be useful for facilitating the necessary dialogue between scientists and stakeholders about recent ecosystem change within the process of an IEA. Such overviews can also contribute to the scientific output used to educate and inform the public and the political system during parts of policymaking processes related to for example ecosystem-based management [63].

Supporting information

S1 Fig. Bar plots for anomalies of AMO’s yearly data from 1980 to 2020.

The y-axis indicates the value subtracting mean value of yearly data from 1980 to 2020. The x-axis indicates year. The negative/positive values correspond lower/higher temperature to the mean value.

https://doi.org/10.1371/journal.pone.0305716.s001

(PDF)

S2 Fig. Outputs for monthly time series data and longer time series data.

Monthly time series data: The estimated three-months (upper) and seven-months (lower) prediction values of sea surface temperature (blue lines) and the most recent observations for the dataset on the Atlantic Multi-decadal Oscillation from January 1980 to December 2020. The solid grey lines indicate the observations used for making the forecast values and the black points indicate the observations that were plotted for comparison with the prediction values (dotted blue line). The dotted grey line presents the prediction value Hz(n|n−1) and the solid blue lines present the smoothed trend estimates obtained by a fixed-interval smoother algorithm. The light-blue band presents the 95% FB (shown for the whole time series), the dark green band represents the 80% FB and light-green band the approximately 70% FB (the latter two shown for predicted years only).The estimated trends are not smooth as seen for data based on annual means but fluctuate periodically. This is because the monthly data includes seasonal fluctuations; Longer time series data: The estimated three-years (upper) and seven-years (lower) prediction values of sea surface temperature (blue lines) and the most recent observations for the monthly dataset on the Atlantic Multi-decadal Oscillation from 1990 to December 2020. The estimated trend pattern is smoothed, like the trend shown for the shorter time series (1980–2020) in Fig 3, and the FOs are the same as those identified using the shorter time series (Fig 3).

https://doi.org/10.1371/journal.pone.0305716.s002

(PDF)

S1 Table. Abbreviations used in figures and tables for data from the ICES integrated ecosystem assessment working group for the Norwegian Sea (WGINOR) and a short summary of the objective for inclusion of each of the time series in the work of the group.

https://doi.org/10.1371/journal.pone.0305716.s003

(PDF)

S1 File. Matlab code for grid searching method for Q and the calculation of p_ut.

https://doi.org/10.1371/journal.pone.0305716.s004

(M)

S2 File. Matlab code of a version using numerical optimization for Q.

https://doi.org/10.1371/journal.pone.0305716.s005

(M)

S3 File. Matlab code of a function used in S2 File.

https://doi.org/10.1371/journal.pone.0305716.s006

(M)

S4 File. R code for grid searching method for Q and the calculation of p_ut.

https://doi.org/10.1371/journal.pone.0305716.s007

(R)

S5 File. R code for a version using numerical optimization for Q.

https://doi.org/10.1371/journal.pone.0305716.s008

(R)

S6 File. Yearly data of AMO.

https://doi.org/10.1371/journal.pone.0305716.s009

(TXT)

Acknowledgments

We would like to thank Benjamin Planque, who motivated us to consider this approach for analysing the time series data compiled by the ICES integrated ecosystem assessment working groups and Mette Mauritzen and Daniel Howell for comments on an earlier draft of the paper.

References

1. Bindoff NL, Cheung WW, Kairo JG, Arístegui J, Guinder VA, Hallberg R, et al. Changing ocean, marine ecosystems, and dependent communities. In: Pörtner H-O, Roberts DC, Masson-Delmotte V, Zhai P, Tignor M, Poloczanska E, et al., editors. IPCC special report on the ocean and cryosphere in a changing climate2019. p. 477–587.
- View Article
- Google Scholar
2. Hoegh-Guldberg O, Cai R, Poloczanska ES, Brewer PG, Sundby S, Hilmi K, et al. The ocean. In: Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, et al., editors. Climate Change 2014: Impacts, Adaptation, and Vulnerability Part B: Regional Aspects Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press; 2014. p. 1655–731.
3. Meredith M, Sommerkorn M, Cassota S, Derksen C, Ekaykin A, Hollowed A, et al. Polar regions. In: Pörtner H-O, Roberts DC, Masson-Delmotte V, Zhai P, Tignor M, Poloczanska E, et al., editors. IPCC special report on the ocean and cryosphere in a changing climate2019.
4. O’Hara CC, Frazier M, Halpern BS. At-risk marine biodiversity faces extensive, expanding, and intensifying human impacts. Science. 2021;372(6537):84–7. pmid:33795456
- View Article
- PubMed/NCBI
- Google Scholar
5. Pörtner H-O, Karl DM, Boyd P, Cheung W, Lluch-Cota S, Nojiri Y, et al. Ocean systems. In: Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, et al., editors. Climate Change 2014: Impacts, Adaptation, and Vulnerability Part B: Regional Aspects Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press; 2014. p. 411–84.
6. Koul V, Sguotti C, Årthun M, Brune S, Düsterhus A, Bogstad B, et al. Skilful prediction of cod stocks in the North and Barents Sea a decade in advance. Communications Earth & Environment. 2021;2(1):140.
- View Article
- Google Scholar
7. McLeod KL, Lubchenco J, Palumbi SR, Rosenberg AA. Scientific Consensus Statement on Marine Ecosystem-Based Management. Signed by 217 academic scientists and policy experts with relevant expertise and published by the Communication Partnership for Science and the Sea at http://compassonline.org/?q=EBM. 2005.
- View Article
- Google Scholar
8. O’Boyle R, Jamieson G. Observations on the implementation of ecosystem-based management: Experiences on Canada’s east and west coasts. Fisheries Research. 2006;79(1):1–12. https://doi.org/10.1016/j.fishres.2005.11.027.
- View Article
- Google Scholar
9. Tallis H, Levin PS, Ruckelshaus M, Lester SE, McLeod KL, Fluharty DL, et al. The many faces of ecosystem-based management: Making the process work today in real places. Marine Policy. 2010;34(2):340–8. https://doi.org/10.1016/j.marpol.2009.08.003.
- View Article
- Google Scholar
10. Winther J-G, Dai M, Rist T, Hoel AH, Li Y, Trice A, et al. Integrated ocean management for a sustainable ocean economy. Nature Ecology & Evolution. 2020;4(11):1451–8. pmid:32807947
- View Article
- PubMed/NCBI
- Google Scholar
11. Fulton EA, Punt AE, Dichmont CM, Harvey CJ, Gorton R. Ecosystems say good management pays off. Fish and Fisheries. 2019;20(1):66–96. https://doi.org/10.1111/faf.12324.
- View Article
- Google Scholar
12. Holsman KK, Haynie AC, Hollowed AB, Reum JCP, Aydin K, Hermann AJ, et al. Ecosystem-based fisheries management forestalls climate-driven collapse. Nat Commun. 2020;11(1):4579. pmid:32917860
- View Article
- PubMed/NCBI
- Google Scholar
13. Dickey-Collas M, Link JS, Snelgrove P, Roberts JM, Anderson MR, Kenchington E, et al. Exploring ecosystem-based management in the North Atlantic. Journal of Fish Biology. 2022;101(2):342–50. pmid:35841280
- View Article
- PubMed/NCBI
- Google Scholar
14. Levin PS, Fogarty MJ, Murawski SA, Fluharty D. Integrated Ecosystem Assessments: Developing the Scientific Basis for Ecosystem-Based Management of the Ocean. Plos Biology. 2009;7(1):23–8. WOS:000262811000004. pmid:19166267
- View Article
- PubMed/NCBI
- Google Scholar
15. Walther YM, Möllmann C. Bringing integrated ecosystem assessments to real life: a scientific framework for ICES. Ices Journal of Marine Science. 2014;71(5):1183–6. WOS:000338630800019.
- View Article
- Google Scholar
16. ICES. Report of the Workshop on integrated trend analyses in support to integrated ecosystem assessment (WKINTRA). ICES CM 2018/IEASG: 15: 23pp. 2018. https://doi.org/10.17895/ices.pub.8278.
- View Article
- Google Scholar
17. Dickey-Collas M. Why the complex nature of integrated ecosystem assessments requires a flexible and adaptive approach. ICES Journal of Marine Science. 2014;71(5):1174–82.
- View Article
- Google Scholar
18. ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; outputs from 2020 meeting). ICES Scientific Reports. 3:35. 114 pp. 2021.
- View Article
- Google Scholar
19. ICES. The second workshop on integrated trend analyses in support to integrated ecosystem as-sessment (WKINTRA2). ICES Scientific Reports. 1:86. 2019.
- View Article
- Google Scholar
20. Holsman K, Samhouri J, Cook G, Hazen E, Olsen E, Dillard M, et al. An ecosystem-based approach to marine risk assessment. Ecosystem Health and Sustainability. 2017;3(1):e01256-n/a.
- View Article
- Google Scholar
21. ICES. Report of the Working Group on Integrated Ecosystem Assessments for the Norwegian Sea (WGINOR). ICES WGINOR REPORT 2018 26–30 November 2018. Reykjavik, Iceland. ICES CM 2018/IEASG:10. 123 pp. 2019.
22. Harvey AC. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press; 1990.
23. Hamilton JD. Time Series Analysis. New Jersey, United States: Princeton University Press; 1994.
24. Mitchell PL, Sheehy JE. Comparison of predictions and observations to assess model performance: a method of empirical validation. In: Kropff MJ, Teng P, Aggarwal PK, Bouma J, Bouman BAM, Jones JW, et al., editors. Applications of Systems Approaches at the Filed Level: Kluwer Academic Publishers; 1997. p. 437–51.
25. Clements CF, McCarthy MA, Blanchard JL. Early warning signals of recovery in complex systems. Nat Commun. 2019;10(1):1681. pmid:30975997
- View Article
- PubMed/NCBI
- Google Scholar
26. Clements CF, Ozgul A. Indicators of transitions in biological systems. Ecol Lett. 2018;21(6):905–19. pmid:29601665
- View Article
- PubMed/NCBI
- Google Scholar
27. Litzow MA, Hunsicker ME. Early warning signals, nonlinearity, and signs of hysteresis in real ecosystems. Ecosphere. 2016;7(12). WOS:000390136700014.
- View Article
- Google Scholar
28. Suweis S, D’Odorico P. Early Warning Signs in Social-Ecological Networks. PLoS One. 2014;9(7). WOS:000339378400043. pmid:25013901
- View Article
- PubMed/NCBI
- Google Scholar
29. Kitagawa G, Gersch W. Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics 116. New York: Springer-Verlag 1996.
30. Kitagawa G. Introduction to time series modeling with Applications in R. Second Edition, London, New York: Chapman & Hall/CRC; 2021.
31. Anderson B.D.O. and Moore J.B., Optimal Filtering. 1976. Prentice Hall, New Jersey.
32. Kato H, Naniwa S, Ishiguro M. A bayesian multivariate nonstationary time series model for estimating mutual relationships among variables. Journal of Econometrics. 1996;75(1):147–61. https://doi.org/10.1016/0304-4076(95)01774-7.
- View Article
- Google Scholar
33. Kitagawa G, Gersch W. A smoothness priors state-space modelling of time-series with trend and seasonality. Journal of the American Statistical Association. 1984;79(386):378–89. WOS:A1984SW65000024.
- View Article
- Google Scholar
34. Whittaker E. On a new method of graduation. Proceedings of the Edinburgh Mathematical Society. 1923;41:63–75.
- View Article
- Google Scholar
35. Brockwell P.J. and Davis R.A. Time Series: Theory and Methods, Second edition. 1991. Springer-Verlag New York Inc.
36. Kalman R.E. A new approach to linear filtering and prediction problems, Transactions of the American Society of Mechanical Engineers–Journal of Basic Engineering, 1960; 82(1):35–45.
- View Article
- Google Scholar
37. Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control. 1974;19(6):716–23.
- View Article
- Google Scholar
38. Hyndman RJ, Athanasopoulos G. Forecasting. Principles and practice. 2nd Edition (May 6, 2018). Melbourne: OTexts; 2018.
39. MATLAB. Natick, Massachusetts: The MathWorks Inc. ver. R2018b.
40. R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria 2023.
41. Trenberth KE, Shea DJ. Atlantic hurricanes and natural variability in 2005. Geophys Res Lett. 2006;33(12). https://doi.org/10.1029/2006GL026894.
- View Article
- Google Scholar
42. ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; outputs from 2019 meeting). ICES Scientific Reports. 2:29. 46 pp. 2020.
- View Article
- Google Scholar
43. Samhouri JF, Andrews KS, Fay G, Harvey CJ, Hazen EL, Hennessey SM, et al. Defining ecosystem thresholds for human activities and environmental pressures in the California Current. Ecosphere. 2017;8(6):e01860.
- View Article
- Google Scholar
44. Dijkstra HA, te Raa L, Schmeits M, Gerrits J. On the physics of the Atlantic Multidecadal Oscillation. Ocean Dynamics. 2006;56(1):36–50.
- View Article
- Google Scholar
45. Schneider DP, Deser C, Fasullo J, Trenberth KE. Climate Data Guide Spurs Discovery and Understanding. Eos, Transactions American Geophysical Union. 2013;94(13):121–2. https://doi.org/10.1002/2013EO130001.
- View Article
- Google Scholar
46. Skjoldal HRe. The Norwegian Sea Ecosystem. Trondheim: Tapir Academic Press; 2004.
47. Huse G, Holst JC, Utne K, Nottestad L, Melle W, Slotte A, et al. Effects of interactions between fish populations on ecosystem dynamics in the Norwegian Sea—results of the INFERNO project Preface. Marine Biology Research. 2012;8(5–6):415–9. WOS:000303560300001.
- View Article
- Google Scholar
48. Planque B, A . F, B H, Mousing E, C H, C B, et al. Quantification of trophic interactions in the Norwegian Sea pelagic food-web over multiple decades. ICES J Mar Sci. 2022. https://doi.org/10.1093/icesjms/fsac111.
- View Article
- Google Scholar
49. Drinkwater KF, Miles M, Medhaug I, Otterå OH, Kristiansen T, Sundby S, et al. The Atlantic Multidecadal Oscillation: Its manifestations and impacts with special emphasis on the Atlantic region north of 60°N. Journal of Marine Systems. 2014;133:117–30. https://doi.org/10.1016/j.jmarsys.2013.11.001.
- View Article
- Google Scholar
50. ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; outputs from 2021 meeting). ICES Scientific Reports. 4:35. 48pp. https://doi.org/10.17895/ices.pub.19643271. 2022.
- View Article
- Google Scholar
51. ICES. Working Group on Widely Distributed Stocks (WGWIDE). ICES Scientific Reports. 3:95. 874 pp. http://doi.org/10.17895/ices.pub.8298. 2021.
- View Article
- Google Scholar
52. Mork KA, Skagseth Ø, Søiland H. Recent Warming and Freshening of the Norwegian Sea Observed by Argo Data. Journal of Climate. 2019;32(12):3695–705.
- View Article
- Google Scholar
53. Garcia T, Planque B, Arneberg P, Bogstad B, Skagseth Ø, Tiedemann M. An appraisal of the drivers of Norwegian spring-spawning herring (Clupea harengus) recruitment. Fisheries Oceanography. 2020;30(2):159–73.
- View Article
- Google Scholar
54. Miesner AK, Payne MR. Oceanographic variability shapes the spawning distribution of blue whiting (Micromesistius poutassou). Fisheries Oceanography. 2018;27(6):623–38.
- View Article
- Google Scholar
55. Hatun H, Payne MR, Jacobsen JA. The North Atlantic subpolar gyre regulates the spawning distribution of blue whiting (Micromesistius poutassou). Canadian Journal of Fisheries and Aquatic Sciences. 2009;66(5):759–70. WOS:000267811500005.
- View Article
- Google Scholar
56. ICES. Interim Report of the Working Group on Seasonal to Decadal Prediction of Marine Ecosystems (WGS2D), 27–31 August 2018, ICES Headquarters, Copenhagen, Denmark. ICES CM 2018/EPDSG:22. 42 pp. 2018.
- View Article
- Google Scholar
57. Hardison S, Perretti CT, DePiper GS, Beet A. A simulation study of trend detection methods for integrated ecosystem assessment. ICES Journal of Marine Science. 2019.
- View Article
- Google Scholar
58. West M, Harrison J. Bayesian Forecasting and Dynamic Models. New York: Springer; 1997.
59. Box G, Jenkins G. Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day; 1970.
60. Fox J. and Weisberg S. An R Comparison to Applied Regression. Third edition. 2018. SAGE Publications, Inc.
61. Ozaki T. Time Series Modeling of Neuroscience Data. 2012. Chapman & Hall/CRC Press.
62. Bulgin CE, Merchant CJ, Ferreira D. Tendencies, variability and persistence of sea surface temperature anomalies. Scientific Reports. 2020;10(1):7986. pmid:32409718
- View Article
- PubMed/NCBI
- Google Scholar
63. Cormier R, Kelble CR, Anderson MR, Allen JI, Grehan A, Gregersen O. Moving from ecosystem-based policy objectives to operational implementation of ecosystem-based management measures. Ices Journal of Marine Science. 2017;74(1):406–13. WOS:000397136400039.
- View Article
- Google Scholar

[ref1] 1. Bindoff NL, Cheung WW, Kairo JG, Arístegui J, Guinder VA, Hallberg R, et al. Changing ocean, marine ecosystems, and dependent communities. In: Pörtner H-O, Roberts DC, Masson-Delmotte V, Zhai P, Tignor M, Poloczanska E, et al., editors. IPCC special report on the ocean and cryosphere in a changing climate2019. p. 477–587.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Hoegh-Guldberg O, Cai R, Poloczanska ES, Brewer PG, Sundby S, Hilmi K, et al. The ocean. In: Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, et al., editors. Climate Change 2014: Impacts, Adaptation, and Vulnerability Part B: Regional Aspects Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press; 2014. p. 1655–731.

[ref3] 3. Meredith M, Sommerkorn M, Cassota S, Derksen C, Ekaykin A, Hollowed A, et al. Polar regions. In: Pörtner H-O, Roberts DC, Masson-Delmotte V, Zhai P, Tignor M, Poloczanska E, et al., editors. IPCC special report on the ocean and cryosphere in a changing climate2019.

[ref4] 4. O’Hara CC, Frazier M, Halpern BS. At-risk marine biodiversity faces extensive, expanding, and intensifying human impacts. Science. 2021;372(6537):84–7. pmid:33795456
View Article
PubMed/NCBI
Google Scholar

[7] View Article

[8] PubMed/NCBI

[9] Google Scholar

[ref5] 5. Pörtner H-O, Karl DM, Boyd P, Cheung W, Lluch-Cota S, Nojiri Y, et al. Ocean systems. In: Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, et al., editors. Climate Change 2014: Impacts, Adaptation, and Vulnerability Part B: Regional Aspects Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press; 2014. p. 411–84.

[ref6] 6. Koul V, Sguotti C, Årthun M, Brune S, Düsterhus A, Bogstad B, et al. Skilful prediction of cod stocks in the North and Barents Sea a decade in advance. Communications Earth & Environment. 2021;2(1):140.
View Article
Google Scholar

[12] View Article

[13] Google Scholar

[ref7] 7. McLeod KL, Lubchenco J, Palumbi SR, Rosenberg AA. Scientific Consensus Statement on Marine Ecosystem-Based Management. Signed by 217 academic scientists and policy experts with relevant expertise and published by the Communication Partnership for Science and the Sea at http://compassonline.org/?q=EBM. 2005.
View Article
Google Scholar

[15] View Article

[16] Google Scholar

[ref8] 8. O’Boyle R, Jamieson G. Observations on the implementation of ecosystem-based management: Experiences on Canada’s east and west coasts. Fisheries Research. 2006;79(1):1–12. https://doi.org/10.1016/j.fishres.2005.11.027.
View Article
Google Scholar

[18] View Article

[19] Google Scholar

[ref9] 9. Tallis H, Levin PS, Ruckelshaus M, Lester SE, McLeod KL, Fluharty DL, et al. The many faces of ecosystem-based management: Making the process work today in real places. Marine Policy. 2010;34(2):340–8. https://doi.org/10.1016/j.marpol.2009.08.003.
View Article
Google Scholar

[21] View Article

[22] Google Scholar

[ref10] 10. Winther J-G, Dai M, Rist T, Hoel AH, Li Y, Trice A, et al. Integrated ocean management for a sustainable ocean economy. Nature Ecology & Evolution. 2020;4(11):1451–8. pmid:32807947
View Article
PubMed/NCBI
Google Scholar

[24] View Article

[25] PubMed/NCBI

[26] Google Scholar

[ref11] 11. Fulton EA, Punt AE, Dichmont CM, Harvey CJ, Gorton R. Ecosystems say good management pays off. Fish and Fisheries. 2019;20(1):66–96. https://doi.org/10.1111/faf.12324.
View Article
Google Scholar

[28] View Article

[29] Google Scholar

[ref12] 12. Holsman KK, Haynie AC, Hollowed AB, Reum JCP, Aydin K, Hermann AJ, et al. Ecosystem-based fisheries management forestalls climate-driven collapse. Nat Commun. 2020;11(1):4579. pmid:32917860
View Article
PubMed/NCBI
Google Scholar

[31] View Article

[32] PubMed/NCBI

[33] Google Scholar

[ref13] 13. Dickey-Collas M, Link JS, Snelgrove P, Roberts JM, Anderson MR, Kenchington E, et al. Exploring ecosystem-based management in the North Atlantic. Journal of Fish Biology. 2022;101(2):342–50. pmid:35841280
View Article
PubMed/NCBI
Google Scholar

[35] View Article

[36] PubMed/NCBI

[37] Google Scholar

[ref14] 14. Levin PS, Fogarty MJ, Murawski SA, Fluharty D. Integrated Ecosystem Assessments: Developing the Scientific Basis for Ecosystem-Based Management of the Ocean. Plos Biology. 2009;7(1):23–8. WOS:000262811000004. pmid:19166267
View Article
PubMed/NCBI
Google Scholar

[39] View Article

[40] PubMed/NCBI

[41] Google Scholar

[ref15] 15. Walther YM, Möllmann C. Bringing integrated ecosystem assessments to real life: a scientific framework for ICES. Ices Journal of Marine Science. 2014;71(5):1183–6. WOS:000338630800019.
View Article
Google Scholar

[43] View Article

[44] Google Scholar

[ref16] 16. ICES. Report of the Workshop on integrated trend analyses in support to integrated ecosystem assessment (WKINTRA). ICES CM 2018/IEASG: 15: 23pp. 2018. https://doi.org/10.17895/ices.pub.8278.
View Article
Google Scholar

[46] View Article

[47] Google Scholar

[ref17] 17. Dickey-Collas M. Why the complex nature of integrated ecosystem assessments requires a flexible and adaptive approach. ICES Journal of Marine Science. 2014;71(5):1174–82.
View Article
Google Scholar

[49] View Article

[50] Google Scholar

[ref18] 18. ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; outputs from 2020 meeting). ICES Scientific Reports. 3:35. 114 pp. 2021.
View Article
Google Scholar

[52] View Article

[53] Google Scholar

[ref19] 19. ICES. The second workshop on integrated trend analyses in support to integrated ecosystem as-sessment (WKINTRA2). ICES Scientific Reports. 1:86. 2019.
View Article
Google Scholar

[55] View Article

[56] Google Scholar

[ref20] 20. Holsman K, Samhouri J, Cook G, Hazen E, Olsen E, Dillard M, et al. An ecosystem-based approach to marine risk assessment. Ecosystem Health and Sustainability. 2017;3(1):e01256-n/a.
View Article
Google Scholar

[58] View Article

[59] Google Scholar

[ref21] 21. ICES. Report of the Working Group on Integrated Ecosystem Assessments for the Norwegian Sea (WGINOR). ICES WGINOR REPORT 2018 26–30 November 2018. Reykjavik, Iceland. ICES CM 2018/IEASG:10. 123 pp. 2019.

[ref22] 22. Harvey AC. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press; 1990.

[ref23] 23. Hamilton JD. Time Series Analysis. New Jersey, United States: Princeton University Press; 1994.

[ref24] 24. Mitchell PL, Sheehy JE. Comparison of predictions and observations to assess model performance: a method of empirical validation. In: Kropff MJ, Teng P, Aggarwal PK, Bouma J, Bouman BAM, Jones JW, et al., editors. Applications of Systems Approaches at the Filed Level: Kluwer Academic Publishers; 1997. p. 437–51.

[ref25] 25. Clements CF, McCarthy MA, Blanchard JL. Early warning signals of recovery in complex systems. Nat Commun. 2019;10(1):1681. pmid:30975997
View Article
PubMed/NCBI
Google Scholar

[65] View Article

[66] PubMed/NCBI

[67] Google Scholar

[ref26] 26. Clements CF, Ozgul A. Indicators of transitions in biological systems. Ecol Lett. 2018;21(6):905–19. pmid:29601665
View Article
PubMed/NCBI
Google Scholar

[69] View Article

[70] PubMed/NCBI

[71] Google Scholar

[ref27] 27. Litzow MA, Hunsicker ME. Early warning signals, nonlinearity, and signs of hysteresis in real ecosystems. Ecosphere. 2016;7(12). WOS:000390136700014.
View Article
Google Scholar

[73] View Article

[74] Google Scholar

[ref28] 28. Suweis S, D’Odorico P. Early Warning Signs in Social-Ecological Networks. PLoS One. 2014;9(7). WOS:000339378400043. pmid:25013901
View Article
PubMed/NCBI
Google Scholar

[76] View Article

[77] PubMed/NCBI

[78] Google Scholar

[ref29] 29. Kitagawa G, Gersch W. Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics 116. New York: Springer-Verlag 1996.

[ref30] 30. Kitagawa G. Introduction to time series modeling with Applications in R. Second Edition, London, New York: Chapman & Hall/CRC; 2021.

[ref31] 31. Anderson B.D.O. and Moore J.B., Optimal Filtering. 1976. Prentice Hall, New Jersey.

[ref32] 32. Kato H, Naniwa S, Ishiguro M. A bayesian multivariate nonstationary time series model for estimating mutual relationships among variables. Journal of Econometrics. 1996;75(1):147–61. https://doi.org/10.1016/0304-4076(95)01774-7.
View Article
Google Scholar

[83] View Article

[84] Google Scholar

[ref33] 33. Kitagawa G, Gersch W. A smoothness priors state-space modelling of time-series with trend and seasonality. Journal of the American Statistical Association. 1984;79(386):378–89. WOS:A1984SW65000024.
View Article
Google Scholar

[86] View Article

[87] Google Scholar

[ref34] 34. Whittaker E. On a new method of graduation. Proceedings of the Edinburgh Mathematical Society. 1923;41:63–75.
View Article
Google Scholar

[89] View Article

[90] Google Scholar

[ref35] 35. Brockwell P.J. and Davis R.A. Time Series: Theory and Methods, Second edition. 1991. Springer-Verlag New York Inc.

[ref36] 36. Kalman R.E. A new approach to linear filtering and prediction problems, Transactions of the American Society of Mechanical Engineers–Journal of Basic Engineering, 1960; 82(1):35–45.
View Article
Google Scholar

[93] View Article

[94] Google Scholar

[ref37] 37. Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control. 1974;19(6):716–23.
View Article
Google Scholar

[96] View Article

[97] Google Scholar

[ref38] 38. Hyndman RJ, Athanasopoulos G. Forecasting. Principles and practice. 2nd Edition (May 6, 2018). Melbourne: OTexts; 2018.

[ref39] 39. MATLAB. Natick, Massachusetts: The MathWorks Inc. ver. R2018b.

[ref40] 40. R Core Team. R: A language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria 2023.

[ref41] 41. Trenberth KE, Shea DJ. Atlantic hurricanes and natural variability in 2005. Geophys Res Lett. 2006;33(12). https://doi.org/10.1029/2006GL026894.
View Article
Google Scholar

[102] View Article

[103] Google Scholar

[ref42] 42. ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; outputs from 2019 meeting). ICES Scientific Reports. 2:29. 46 pp. 2020.
View Article
Google Scholar

[105] View Article

[106] Google Scholar

[ref43] 43. Samhouri JF, Andrews KS, Fay G, Harvey CJ, Hazen EL, Hennessey SM, et al. Defining ecosystem thresholds for human activities and environmental pressures in the California Current. Ecosphere. 2017;8(6):e01860.
View Article
Google Scholar

[108] View Article

[109] Google Scholar

[ref44] 44. Dijkstra HA, te Raa L, Schmeits M, Gerrits J. On the physics of the Atlantic Multidecadal Oscillation. Ocean Dynamics. 2006;56(1):36–50.
View Article
Google Scholar

[111] View Article

[112] Google Scholar

[ref45] 45. Schneider DP, Deser C, Fasullo J, Trenberth KE. Climate Data Guide Spurs Discovery and Understanding. Eos, Transactions American Geophysical Union. 2013;94(13):121–2. https://doi.org/10.1002/2013EO130001.
View Article
Google Scholar

[114] View Article

[115] Google Scholar

[ref46] 46. Skjoldal HRe. The Norwegian Sea Ecosystem. Trondheim: Tapir Academic Press; 2004.

[ref47] 47. Huse G, Holst JC, Utne K, Nottestad L, Melle W, Slotte A, et al. Effects of interactions between fish populations on ecosystem dynamics in the Norwegian Sea—results of the INFERNO project Preface. Marine Biology Research. 2012;8(5–6):415–9. WOS:000303560300001.
View Article
Google Scholar

[118] View Article

[119] Google Scholar

[ref48] 48. Planque B, A . F, B H, Mousing E, C H, C B, et al. Quantification of trophic interactions in the Norwegian Sea pelagic food-web over multiple decades. ICES J Mar Sci. 2022. https://doi.org/10.1093/icesjms/fsac111.
View Article
Google Scholar

[121] View Article

[122] Google Scholar

[ref49] 49. Drinkwater KF, Miles M, Medhaug I, Otterå OH, Kristiansen T, Sundby S, et al. The Atlantic Multidecadal Oscillation: Its manifestations and impacts with special emphasis on the Atlantic region north of 60°N. Journal of Marine Systems. 2014;133:117–30. https://doi.org/10.1016/j.jmarsys.2013.11.001.
View Article
Google Scholar

[124] View Article

[125] Google Scholar

[ref50] 50. ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; outputs from 2021 meeting). ICES Scientific Reports. 4:35. 48pp. https://doi.org/10.17895/ices.pub.19643271. 2022.
View Article
Google Scholar

[127] View Article

[128] Google Scholar

[ref51] 51. ICES. Working Group on Widely Distributed Stocks (WGWIDE). ICES Scientific Reports. 3:95. 874 pp. http://doi.org/10.17895/ices.pub.8298. 2021.
View Article
Google Scholar

[130] View Article

[131] Google Scholar

[ref52] 52. Mork KA, Skagseth Ø, Søiland H. Recent Warming and Freshening of the Norwegian Sea Observed by Argo Data. Journal of Climate. 2019;32(12):3695–705.
View Article
Google Scholar

[133] View Article

[134] Google Scholar

[ref53] 53. Garcia T, Planque B, Arneberg P, Bogstad B, Skagseth Ø, Tiedemann M. An appraisal of the drivers of Norwegian spring-spawning herring (Clupea harengus) recruitment. Fisheries Oceanography. 2020;30(2):159–73.
View Article
Google Scholar

[136] View Article

[137] Google Scholar

[ref54] 54. Miesner AK, Payne MR. Oceanographic variability shapes the spawning distribution of blue whiting (Micromesistius poutassou). Fisheries Oceanography. 2018;27(6):623–38.
View Article
Google Scholar

[139] View Article

[140] Google Scholar

[ref55] 55. Hatun H, Payne MR, Jacobsen JA. The North Atlantic subpolar gyre regulates the spawning distribution of blue whiting (Micromesistius poutassou). Canadian Journal of Fisheries and Aquatic Sciences. 2009;66(5):759–70. WOS:000267811500005.
View Article
Google Scholar

[142] View Article

[143] Google Scholar

[ref56] 56. ICES. Interim Report of the Working Group on Seasonal to Decadal Prediction of Marine Ecosystems (WGS2D), 27–31 August 2018, ICES Headquarters, Copenhagen, Denmark. ICES CM 2018/EPDSG:22. 42 pp. 2018.
View Article
Google Scholar

[145] View Article

[146] Google Scholar

[ref57] 57. Hardison S, Perretti CT, DePiper GS, Beet A. A simulation study of trend detection methods for integrated ecosystem assessment. ICES Journal of Marine Science. 2019.
View Article
Google Scholar

[148] View Article

[149] Google Scholar

[ref58] 58. West M, Harrison J. Bayesian Forecasting and Dynamic Models. New York: Springer; 1997.

[ref59] 59. Box G, Jenkins G. Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day; 1970.

[ref60] 60. Fox J. and Weisberg S. An R Comparison to Applied Regression. Third edition. 2018. SAGE Publications, Inc.

[ref61] 61. Ozaki T. Time Series Modeling of Neuroscience Data. 2012. Chapman & Hall/CRC Press.

[ref62] 62. Bulgin CE, Merchant CJ, Ferreira D. Tendencies, variability and persistence of sea surface temperature anomalies. Scientific Reports. 2020;10(1):7986. pmid:32409718
View Article
PubMed/NCBI
Google Scholar

[155] View Article

[156] PubMed/NCBI

[157] Google Scholar

[ref63] 63. Cormier R, Kelble CR, Anderson MR, Allen JI, Grehan A, Gregersen O. Moving from ecosystem-based policy objectives to operational implementation of ecosystem-based management measures. Ices Journal of Marine Science. 2017;74(1):406–13. WOS:000397136400039.
View Article
Google Scholar

[159] View Article

[160] Google Scholar

Figures

Abstract

Introduction

Statistical method

Trend estimation procedure

FO analysis by multistep-ahead prediction values

Assessing unexpected tendencies

Illustrative examples

The Atlantic Multi-decadal Oscillation (AMO)

Norwegian Sea ecosystem

Discussion

Supporting information

S1 Fig. Bar plots for anomalies of AMO’s yearly data from 1980 to 2020.

S2 Fig. Outputs for monthly time series data and longer time series data.

S1 Table. Abbreviations used in figures and tables for data from the ICES integrated ecosystem assessment working group for the Norwegian Sea (WGINOR) and a short summary of the objective for inclusion of each of the time series in the work of the group.

S1 File. Matlab code for grid searching method for Q and the calculation of put.

S2 File. Matlab code of a version using numerical optimization for Q.

S3 File. Matlab code of a function used in S2 File.

S4 File. R code for grid searching method for Q and the calculation of put.

S5 File. R code for a version using numerical optimization for Q.

S6 File. Yearly data of AMO.

Acknowledgments

References

S1 File. Matlab code for grid searching method for Q and the calculation of p_ut.

S4 File. R code for grid searching method for Q and the calculation of p_ut.