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Flagged observation analyses as a tool for scoping and communication in integrated ecosystem assessments
Flagged observation analyses as a tool for scoping and communication in integrated ecosystem assessments
https://orcid.org/0000-0002-0330-4670
Solvang Hiroko Kato Conceptualization Formal analysis Investigation Methodology Software Validation Writing – original draft 1 *
Arneberg Per Conceptualization Data curation Funding acquisition Investigation Writing – original draft 2
1 Marine Mammals Research Group, Institute of Marine Research, Bergen, Norway
2 Ecosystem Processes Research Group, Institute of Marine Research, Fram Centre, Langnes, Norway
Zambujal-Oliveira João Editor
University of Madeira / NOVA Lincs, PORTUGAL
Competing Interests: The authors have declared that no competing interest exist.

* E-mail: hiroko.solvang@hi.no
23 9 2024
2024
19 9 e030571631 10 2022
4 6 2024
© 2024 Solvang, Arneberg
2024
Solvang, Arneberg
https://creativecommons.org/licenses/by/4.0/ 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.

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.

The Research Council of Norway 299554 Arneberg Per Per Arneberg received “Sustainable multi-species harvest from the Norwegian Sea and adjacent ecosystems”, funded by The Research Council of Norway (pr. nr. 299554). Data AvailabilityAll relevant data are within the manuscript and its Supporting Information files.
Data Availability

All relevant data are within the manuscript and its Supporting Information files.
==== Body
pmcIntroduction

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: y(n)=t(n)+u(n),n=1,⋯,N, (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: ∇dt(n)=v(n), (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 v(n)∼N(0,σv2) where N is ‘n’ of a normal (Gaussian) distribution, the mean is 0 and the variance is σv2. 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, ti(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 systemmodel:z(n)=Fz(n−1)+Gv(n),observationmodel:y(n)=Hz(n)+w(n), (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 σv2, 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 σw2. 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 t(n)=1⋅t(n−1)+1⋅v(n).

When d = 2 in Eq (2),

z(n)=[t(n)t(n−1)], F=(2−110), G=[10], and the system model is given by

(t(n)t(n−1))=(2−110)(t(n−1)t(n−2))+(10)v(n). 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 z(n|j)≡E[z(n)|Y(j)]V(n|j)≡E[(z(n)−z(n|j))((z(n)−z(n|j))'] (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: z(n|n−1)=Fz(n−1|n−1)V(n|n−1)=FV(n−1|n−1)+GQG' (5)

filtering: K(n)=V(n|n−1)H'(HV(n|n−1)H'+w(n))−1z(n|n)=z(n|n−1)+K(n)(y(n)−Hz(n|n−1))V(n|n)=(I−K(n)H)V(n|n−1) (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 θ=(d,σv2,σw2). The log-likelihood function l(θ) of the model (1) is given by: l(θ)=∑n=1Nlogf(y(n)|Y(n−1),θ),=∑n=1Nlog{1(2π)2detΣ(n)exp(−12Δy(n)'Σ(n)−1Δy(n))}, (7)

where Y(n−1) = (y(1),y(2),⋯,y(n−1)), Δy(n) = y(n)−Hz(n|n−1), and Σ(n)=H(n)V(n|n−1)H'(n)+σw2. 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 σv2 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 σw2 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]: A(n)=V(n|n)F'V(n+1|n)−1z(n|N)=z(n|n)+A(n)(z(n+1|N)−z(n+1|n))V(n|N)=V(n|n)+A(n)(V(n+1|N)−V(n+1|n))A(n)' (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: z(n+i|n)=Fz(n+i−1|n),V(n+i|n)=FV(n+i−1|n)F'+GQG',i=1,⋯,j. (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: y(n+j|n)=E(Hz(n+j)+w(n+j)|Y(n))=Hz(n+j|n). (10)

The variance of y(n+j) is given by: u(n+j|n)=Cov(Hz(n+j)+w(n+j)|Y(n)),=HCov(z(n+j)|Y(n))H'+HCov(z(n+j),w(n+j)|Y(n))+Cov(w(n+j),z(n+j)|Y(n))H'+Cov(w(n+j)|Y(n)),=HV(n+j|n)H'+R(n+j). (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 u(n+j|n) [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 (put) 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 put, 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 r^(n+i)=y(n+i)−t^(n+i), i = 1,⋯,j, where t^(n+i) is given by Hz(n+i|n) in (10), that is, r^(n+i) is equivalent to the w^(n+i) obtained by y(n+i)−y^(n+i|n). The Gaussian distribution that the residual part follows, has a mean 0 and variance σ^w2. Therefore, we consider the upper tale probability P(X≥r^(n+i)) if the observations consecutively locate above the estimated trend and the lower tale probability P(X≤r^(n+i)) if the observations consecutively locate below the estimated trend, where X is the random variable given by the Gaussian distribution with 0 and variance σ^w2. The probability is also calculated for each time point, denoted as P1,⋯,Pj. Finally, put is obtained as P1×⋯×Pj. When interpreting put, it can be compared with a general statistical threshold, such as 0.05. However, because the value of put will depend on j, we recommend that the actual comparison should in this case be made with the threshold 0.05j.

The conceptual outlines of this study’s analysis procedure for FO analysis and estimation of put 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 put) 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 put) and S5 File (a version using numerical optimization for Q as an option). Example data are given in S6 File (yearly data of AMO).

10.1371/journal.pone.0305716.g001 Fig 1 Outline of proposed flagged observation (FO) analysis.

10.1371/journal.pone.0305716.g002 Fig 2 Outline of the proposed assessment for unexpected tendencies using probability multiplicated by the upper/lower tail probabilities at consecutive time points.

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.

10.1371/journal.pone.0305716.g003 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).

10.1371/journal.pone.0305716.t001 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.

Dataset	Variable	Q	Maximum log-likelihood	
AMO	AMO	0.01	-49.9	
Norwegian Sea ecosystem (WGINOR)	RHC	0.01	-86.9	
RFW	0.01	-84.7	
NAO	0.01	-170.3	
ZooB	0.01	-29.4	
MacB	0.01	-41.3	
MacR	0.01	-40.1	
MacW	0.01	-44.4	
MacL	0.01	-73.6	
HerB	0.01	-129.3	
HerR	0.01	-43.4	
HerW	0.01	-88.8	
HerL	0.01	-96.6	
BWB	0.01	-43.4	
BWR	0.02	-50.0	
BWW	0.01	-44.6	
BWL	0.01	-58.3	

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), put 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.

10.1371/journal.pone.0305716.t002 Table 2 The upper /lower probabilities at consecutive time points and the put for AMO.

Year	AMO	
2014	0.062	
2015	0.016	
2016	0.11	
2017	0.28	
2018	0.028	
2019	0.054	
2020	0.24	
p ut	1.1e-08	

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 σw2 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 put 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 put for the pattern was substantially higher than 0.053 (Table 3), suggesting that the NAO changes should not be prioritized in an IEA.

10.1371/journal.pone.0305716.g004 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).

10.1371/journal.pone.0305716.t003 Table 3 The upper /lower probabilities at consecutive time points and the put for RFW, NAO, ZooB, MacB, MacR, MacL, HerW, HerL, BWB, BWR, and BWW.

Year	RFW	NAO	ZooB	MacB	MacR	MacL	
2017	0.21	0.31	0.066	0.17	0.33	0.077	
2018	0.047	0.075	0.24	0.026	0.32	0.040	
2019	0.014	0.24	0.070	0.0016	0.18	0.0052	
p ut	1.4e-04	0.0055	0.0011	7.5e-06	0.019	1.6e-05	
Year	HerW	HerL	BWB	BWR	BWW	BWL	
2017	0.30	0.44	0.045	0.0041	0.033	0.20	
2018	014	0.15	0.051	0.0013	0.020	0.0014	
2019	0.20	0.12	0.13	0.0004	0.018	0.099	
p ut	0.0081	0.0082	2.9e-04	2.1e-09	1.2e-05	2.8e-04	

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 put 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 put 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 put 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 put 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 put 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 put 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 put 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, put 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, put 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 (put). 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 put for non-Gaussian cases, a local regression model [60] or a local linearization filter [61] may be applied.

When assessing the relevance of put estimates, we have recommended to use a threshold of, in the case of a 0.05 level, as 0.05j. This is done because put 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 put 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.

(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).

(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.

(PDF)

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

(M)

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

(M)

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

(M)

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

(R)

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

(R)

S6 File Yearly data of AMO.

(TXT)

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.

10.1371/journal.pone.0305716.r001
Decision Letter 0
Palumbo Pasquale Academic Editor
© 2024 Pasquale Palumbo
2024
Pasquale Palumbo
https://creativecommons.org/licenses/by/4.0/ 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.
Submission Version0
8 Feb 2023

PONE-D-22-30034Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem AssessmentsPLOS ONE

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Reviewer #1: I like this paper. To briefly summarize, Kato and Arneberg develop a tool based on the Kalman filter to identify anomalous observations in time series. This tool can be useful in integrated ecosystem assessments where large numbers of observations need to be considered. I do not have full overview of the literature on Integrated Ecosystem Assessments, but Kato And Arneberg provide a satisfying overview of relevant work and they presumably offer a valuable contribution.

The idea is sound and relatively straight forward: Estimate a trend with the Kalman filter and assess whether recent observations deviate sufficiently from this trend to flag them as low probability events. While the basic statistical method is developed and established in the literature and numerous textbooks, the presentation of the technicalities leaves something to desire (see list of comments below).

The approach to assess joint probabilities is non-standard (there is no established standard, as far as I know, unless all involved distributions are Gaussian). This part needs to be explained better. That 'the joint probability ... should be calculated by random variables' (lines 175-177) is cryptic. If I understand correctly, Kato and Arneberg represents the trend prediction uncertainty band by simulation. But if the distribution is assumed Gaussian, I think one can calculate the probability of a given observation from the estimated covariance. Regardless, how the simulated probability evaluation is done in a multivariate setting needs to be spelled out in detail. The relevant computer code is said to be available as supplementary material, but I only found a table of abbrevations in the supplement.

I do not have much to say about the examples, they seem to be executed in a satisfactory fashion. But I wonder why you restrict your sample to the period 1980-2020 (see line 202) when the data series goes back to 1869. And why do you use annual means rather than the monthly observations?

Specific comments:

- Eq. 2: Please provide the definition of the difference operator. The given expression for d=2 is not self-explanatory.

- Eq. 3: What is G? What is its interpretation?

- Eq. 4: What is Q and what is its interpretation? On line 146, you say that Q should be 'optimal'. What is the criteria for the optimal Q? Further, the procedure involving equations 4 and 5 could be explained better. The explanation on lines 123 and forward is unclear.

- Line 137: Why can the variance of w be set to the variance of the observation? Is the variance of the observation an expression of observation uncertainty? Is the observation uncertainty known?

- Line 154: Assuming Y(n+1)=Y(n), where Y(n)=(y(1), ..., y(n)) is cryptic. Presumably, Y(n+1) has the element y(n+1), which evidently is not in Y(n), so it is not clear what is assumed. It is further unclear how z(n+2|n) relates to z(n+1|n).

- Eq. 7: The d-notation is already used for something else (see eq. 2), should be changed.

- Line 203: Annual means for 1980-2020 presumably results in a time series with 41 observations.

Best of luck with further work on this paper!

Reviewer #2: This paper deals with the problem of assessing recent changes in ecosystems by suitably processing time series. The Authors propose a procedure aimed at detecting time series where the most recent observations are less expected on the basis of past data (large deviations from the prediction). The estimation of trends in a time series and the prediction on the basis of past data is obtained by exploiting a stochastic trend model and a Kalman filter/smoother/predictor algorithm. The assessment of unexpected changes is made by using estimated Forecast Bands and joint probabilities. Two examples that illustrate the proposed procedure are reported.

I would like to point out that my opinion on this paper is from the viewpoint of a person with expertise in data processing, and not on biological/ecological applications.

From my viewpoint, the methodology used by the Authors in this paper is well established (it is grounded on the works of Kalman (~1960), Akaike (~1980) and Kitagawa (~1980)). Thus, from a methodolgical point of view I don't see any contribution in this work.

In any case, the Authors should specify in what the proposed "Flagged Observation Analysis" is different from, and possibly an improvement over, the approaches in [25-28].

A possible contribution could be in the two case studies investigated, in the field of marine echosystems, which appear to be interesting, although I am not an expert in this field, so I cannot express a qualified opinion in this area.

Below some comments on data processing aspects of the paper.

After a rather detailed presentation, in the first 6 pages of the paper, of a well established method in the field of time series analysis, the Authors present the proposed methodology for computing "joint probabilities", aimed at "Flagging" unexpected observations and assessing unexpected changes in a time series.

The procedure is outlined in the section "Assessing unexpected tendencies - joint probability for detected FO" in the lines from 180 to 188 of the paper.

However, despite it importance in the paper, the procedure is not clearly introduced and properly justified.

The points that are obscure to me are listed below:

- it is not clear what kind of joint probability is computed in the procedure. It is understood that the computed probability concerns the observations from n+1 to n+J. However, it has not been explained what kind of probability is (is it the probability that ALL the observations from n+1 to n+J are outside a given Forecast Band? Or is the probability that AT LEAST ONE observation in the interval is outside the FB? Or what else?)

- in line 183 I can not understand the sentences "if y(n+j|n) is upper over FB / if y(n+j|n) is lower under FB".

According to my understanding, the predicted value y(n+j|n) is in the middle of any FB, thus how can it be upper or lower w.r.t. FB?

- there is no reported motivation on the chosen sizes of 10.000 (in the generation of random numbers, at step 1) and 1000 (for the averaging in step 3). Is there any quantitative reason for these specific choices?

- since all variables are assumed Gaussian, apparently there is no need to resort to a numerical Monte Carlo procedure, since all probabilities can be accurately computed. Am I missing something?

Throughout the paper the Authors refer to Table 1 and Table 2. However, these tables are not included in the paper.

A suggestion:

I see that the Atlantic Multi-Decadal Oscillation (AMO) are collected monthly, and that the Authors only employ their annual means in their procedure. In this way, the information about the yearly deviation from the mean is lost.

It would be interesting if the authors tested their procedure also on data averaged on a shorter time period (e.g., trimestral or semetral) and compared the obtained results.

Specific comments

-Line 124: specifying the initial variance equal to 0 is rather unusual, except for the particular case where the initial state is exactly known.

- Line 136: The sentence "The flexibility of the estimated trend..." the concept of flexibility should be explained

- In line 141, "c", the argument of AIC(c), should be explained.

- Line 145: if the value d=2 has been chosen exploiting the Akaike criterion (as the Authors say), then the values of AIC should be shown

Typos

- Line 49. Check the sentence starting with "Her we present...", something is wrong.

- Line 121. The Kalman gain in the equations (5) should always written as K(n)

- Line 130. Replace "proceeses" with "processes"

- Line 138. Replace "differernt" with "different"

- Line 146: square brackets should be used for the reference [33]

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10.1371/journal.pone.0305716.r002
Author response to Decision Letter 0
Submission Version1
1 Jun 2023

Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: I like this paper. To briefly summarize, Kato and Arneberg develop a tool based on the Kalman filter to identify anomalous observations in time series. This tool can be useful in integrated ecosystem assessments where large numbers of observations need to be considered. I do not have full overview of the literature on Integrated Ecosystem Assessments, but Kato And Arneberg provide a satisfying overview of relevant work and they presumably offer a valuable contribution.

The idea is sound and relatively straight forward: Estimate a trend with the Kalman filter and assess whether recent observations deviate sufficiently from this trend to flag them as low probability events. While the basic statistical method is developed and established in the literature and numerous textbooks, the presentation of the technicalities leaves something to desire (see list of comments below).

The approach to assess joint probabilities is non-standard (there is no established standard, as far as I know, unless all involved distributions are Gaussian). This part needs to be explained better. That 'the joint probability ... should be calculated by random variables' (lines 175-177) is cryptic. If I understand correctly, Kato and Arneberg represents the trend prediction uncertainty band by simulation. But if the distribution is assumed Gaussian, I think one can calculate the probability of a given observation from the estimated covariance. Regardless, how the simulated probability evaluation is done in a multivariate setting needs to be spelled out in detail. The relevant computer code is said to be available as supplementary material, but I only found a table of abbrevations in the supplement.

Response: The concept is now described in the introduction. In the methods section, we have now added a description of what we want to estimate: “Thus, we want to estimate the joint probability that the location of the observations from the most recent years relative to the predicted trend could be the result of random variation. As the time series data is sampled as one sample at one time point, this should be calculated by random variables, which is generated from a Gaussian distribution. Thus, what we want to estimate, is the probability that random variation in a Gaussian distribution could have resulted in a lower estimated probability.”

With cross sectional data, the probability of a given observation can be calculated from estimated covariance, which corresponds to a p-value. With time series data, where each observation is from a single time point, this approach cannot be used. Therefore, an alternative approach must be applied, and here we have based this on simulations in the following way: we assume Gaussian distribution for the prediction with Hz(n+j|n) as mean and d(n+j|n) as variance. Furthermore, we conduct Monte Carlo simulations to calculate the p-value. The details are given in the methods section. See lines 185-206.

The computer code has now been added in supplementary files S1 and S2 with example data in supplementary file S3.

I do not have much to say about the examples, they seem to be executed in a satisfactory fashion. But I wonder why you restrict your sample to the period 1980-2020 (see line 202) when the data series goes back to 1869.

Response: We have mostly focused on short time series (i.e., < 50 observations), because this is typically what is available to IEA groups. In the paper, there are also examples of longer time series (e.g., for herring), and to give additional examples, we have also added analyses using a longer time series for the AMO data and also using monthly data for the AMO time series to provide an example of an even longer time series. These latter analyses have been added to the supplementary material. The main text has been revised accordingly. See lines 218-220, 225-228 and 248-252.

And why do you use annual means rather than the monthly observations?

Response: See above

Specific comments:

- Eq. 2: Please provide the definition of the difference operator. The given expression for d=2 is not self-explanatory.

Response: We have given the definition of the dth-order difference equation in lines 101-107.

- Eq. 3: What is G? What is its interpretation?

Response: G is integer when the trend is random walk model (d=1), and G is a vector leading that the first component corresponds to add v(n). Text describing this is added on lines 113-119.

- Eq. 4: What is Q and what is its interpretation? On line 146, you say that Q should be 'optimal'. What is the criteria for the optimal Q? Further, the procedure involving equations 4 and 5 could be explained better. The explanation on lines 123 and forward is unclear.

Response: We have added the explanation of Q. The criterium for the optimal Q is log-likelihood is as explained in lines143-144 and 153-157. Please note that Q-values had been set too high in the original submission. This has now been adjusted. A modification of the G vector has also been done. This has resulted in some adjustments to the results, most importantly that the decline in mackerel spawning stock biomass is now identified as a flagged observation (lines 317-322).

- Line 137: Why can the variance of w be set to the variance of the observation? Is the variance of the observation an expression of observation uncertainty? Is the observation uncertainty known?

Response: Because the state space model includes the observation model (the second equation) in formulae (3), the variance of the observation is an expression of observation uncertainty. The observation uncertainty is given as the variance of the observations in this study to estimate the variance of the system noise in the system equation (the first equation) in formulae (3), however; the observation uncertainty can be estimated as one parameter of the model by a numerical optimization. We have added an explanation for the flexibility to the observation noise (lines 144-146 and 156-157)

- Line 154: Assuming Y(n+1)=Y(n), where Y(n)=(y(1), ..., y(n)) is cryptic. Presumably, Y(n+1) has the element y(n+1), which evidently is not in Y(n), so it is not clear what is assumed. It is further unclear how z(n+2|n) relates to z(n+1|n).

Response: Since the future observation y(n+1) does not exist in fact, the prediction has to be estimated based on the time series Y(n) = (y(1), y(2), … , y(n)). In the sense, it is assumed that Y(n+1) = Y(n). This concept was referred in Section 9 [57]. We have added the reference in the text (line 164).

- Eq. 7: The d-notation is already used for something else (see eq. 2), should be changed.

Response: We have changed. (lines 177 and 179)

- Line 203: Annual means for 1980-2020 presumably results in a time series with 41 observations.

Response: text has been corrected (line 225)

Best of luck with further work on this paper!

Thank you very much for your kind words.

Reviewer #2: This paper deals with the problem of assessing recent changes in ecosystems by suitably processing time series. The Authors propose a procedure aimed at detecting time series where the most recent observations are less expected on the basis of past data (large deviations from the prediction). The estimation of trends in a time series and the prediction on the basis of past data is obtained by exploiting a stochastic trend model and a Kalman filter/smoother/predictor algorithm. The assessment of unexpected changes is made by using estimated Forecast Bands and joint probabilities. Two examples that illustrate the proposed procedure are reported.

I would like to point out that my opinion on this paper is from the viewpoint of a person with expertise in data processing, and not on biological/ecological applications.

From my viewpoint, the methodology used by the Authors in this paper is well established (it is grounded on the works of Kalman (~1960), Akaike (~1980) and Kitagawa (~1980)). Thus, from a methodolgical point of view I don't see any contribution in this work.

In any case, the Authors should specify in what the proposed "Flagged Observation Analysis" is different from, and possibly an improvement over, the approaches in [25-28].

Response: The approaches in [25-28] have different scopes, namely identifying change caused by specific types of dynamics and often with specific types of outcomes. Flagged Observation Analysis aims at identifying unexpected change in a way that is not specific to any type of dynamics and/or outcomes, and where the aim is to identify time series that may need to be followed up more closely in an IEA, possibly by addressing dynamics and consequences. We have added text explaining this (lines 75-78).

A possible contribution could be in the two case studies investigated, in the field of marine ecosystems, which appear to be interesting, although I am not an expert in this field, so I cannot express a qualified opinion in this area.

Response: Concerning the contribution of the paper, we would like to point out that the procedure for joint probability calculation is novel. In addition, the application of the methods in an IEA context in the way it is done here is also novel and should be of interest for a wide readership interested in IEAs and related frameworks.

Below some comments on data processing aspects of the paper.

After a rather detailed presentation, in the first 6 pages of the paper, of a well established method in the field of time series analysis, the Authors present the proposed methodology for computing "joint probabilities", aimed at "Flagging" unexpected observations and assessing unexpected changes in a time series.

The procedure is outlined in the section "Assessing unexpected tendencies - joint probability for detected FO" in the lines from 180 to 188 of the paper.

However, despite it importance in the paper, the procedure is not clearly introduced and properly justified.

Response: The need for this procedure has now been introduced and justified in the introduction. Additional description has been added to the methods section (see response to reviewer 1 above).

The points that are obscure to me are listed below:

- it is not clear what kind of joint probability is computed in the procedure. It is understood that the computed probability concerns the observations from n+1 to n+J. However, it has not been explained what kind of probability is (is it the probability that ALL the observations from n+1 to n+J are outside a given Forecast Band? Or is the probability that AT LEAST ONE observation in the interval is outside the FB? Or what else?)

Response: The joint probability is estimated when all observations from the recent years fall on one side of the predicted trend. The position relative to the Forecast Bands is not relevant but was erroneously described as this in the original submission. We have now corrected this and also give a more detailed explanation in the method section. Reviewer 1 raised the same issue, so please see response above.

- in line 183 I can not understand the sentences "if y(n+j|n) is upper over FB / if y(n+j|n) is lower under FB". According to my understanding, the predicted value y(n+j|n) is in the middle of any FB, thus how can it be upper or lower w.r.t. FB?

Response: As described above, positions relative to FBs are not relevant, but position relative to the predicted trend, and text has been corrected (line 199) .

- there is no reported motivation on the chosen sizes of 10.000 (in the generation of random numbers, at step 1) and 1000 (for the averaging in step 3). Is there any quantitative reason for these specific choices?

Response: Since the state space model assumes a Gaussian distribution, we use a large number as 10000 to generate the distribution to avoid bias in in the estimates. Also, to avoid biased estimation for the joint probability using one Gaussian distribution, we take the averaged joint probability. However, as you pointed, it is arbitrary. Therefore, we have not suggested a specific number of randomizations and iterations but left this open for each investigator to decide (lines 205-206).

- since all variables are assumed Gaussian, apparently there is no need to resort to a numerical Monte Carlo procedure, since all probabilities can be accurately computed. Am I missing something?

Response: This issue was also raised by reviewer 1, so please see the response above.

Throughout the paper the Authors refer to Table 1 and Table 2. However, these tables are not included in the paper.

Response: This must have been a technical problem with the original submission. We are sorry about this. The tables have been uploaded with the revised submission.

A suggestion:

I see that the Atlantic Multi-Decadal Oscillation (AMO) are collected monthly, and that the Authors only employ their annual means in their procedure. In this way, the information about the yearly deviation from the mean is lost.

It would be interesting if the authors tested their procedure also on data averaged on a shorter time period (e.g., trimestral or semetral) and compared the obtained results.

Response: We thank for this suggestion and have added analyses using monthly data (lines 225-227 and 249-252). See also response to the same issue from reviewer 1 above.

Specific comments

-Line 124: specifying the initial variance equal to 0 is rather unusual, except for the particular case where the initial state is exactly known.

Response: No, the initial state z(1|0) is zero and the initial variance V(1|0) is an arbitrary real number (e.g. 0.1). We have rephrased the sentence (lines 130-131).

- Line 136: The sentence "The flexibility of the estimated trend..." the concept of flexibility should be explained

Response: We have corrected ‘smoothness’ from ‘flexibility’ (line 143 - 144).

- In line 141, "c", the argument of AIC(c), should be explained.

Response: We have corrected (line 149).

- Line 145: if the value d=2 has been chosen exploiting the Akaike criterion (as the Authors say), then the values of AIC should be shown

Response: Since we want to estimate the trend as smooth as possible, we fixed d = 2 as indicated in reference [33]. Therefore, in this study, it is not necessary to show AIC to compare the trend model for d = 1 and the trend model for d = 2. We have added the explanation (lines 146-149).

Typos

- Line 49. Check the sentence starting with "Her we present...", something is wrong.

Response: Revised by adding “on” (line 50).

- Line 121. The Kalman gain in the equations (5) should always written as K(n)

Response: We have corrected (line 127).

- Line 130. Replace "proceeses" with "processes"

Response: revised (line 137).

- Line 138. Replace "differernt" with "different"

Response: revised (line 146).

- Line 146: square brackets should be used for the reference [33]

Response: revised (line 154).

Attachment Submitted filename: Reviewers comments and how they are dealt with.docx

10.1371/journal.pone.0305716.r003
Decision Letter 1
Palumbo Pasquale Academic Editor
© 2024 Pasquale Palumbo
2024
Pasquale Palumbo
https://creativecommons.org/licenses/by/4.0/ 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.
Submission Version1
19 Jun 2023

PONE-D-22-30034R1

Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem Assessments

PLOS ONE

Dear Dr. Solvang,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we have decided that your manuscript does not meet our criteria for publication and must therefore be rejected.

Specifically:

I am sorry that we cannot be more positive on this occasion, but hope that you appreciate the reasons for this decision.

Kind regards,

Pasquale Palumbo

Academic Editor

PLOS ONE

Additional Editor Comments (if provided):

Both Reviewers have clearly stated that the revised manuscript did not improve the original submission. 

[Note: HTML markup is below. Please do not edit.]

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: (No Response)

Reviewer #2: (No Response)

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2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Partly

Reviewer #2: No

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3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: I Don't Know

Reviewer #2: No

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4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

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5. Is the manuscript presented in an intelligible fashion and written in standard English?

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Reviewer #1: No

Reviewer #2: Yes

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: Review, PONE-D-22-30034_R1

The authors have revised their manuscript. As I wrote in my previous report, I think the study is worthwhile documenting in a publication. Reviewer #2 points out that the methodology is known, but I think the application is novel and valuable. Consideration of the behavior of time series needs support by statistical tools. It now occurs to me, however, that the reliance on the normal distribution may be problematic. That the authors do so in their examples may be ok, but I recommend them to add a discussion of issues such as fat tail risk that should be accounted for in real applications. The involvement of non-Gaussian distributions would further highlight the value of the simulation framework they promote, which, as I try to argue below, is unnecessary when working with Gaussian distributions.

My main concern in my previous report was that the presentation of the method needed several clarifications. The authors have tried, presumably, but there are still points that remain unclear (see below). Fortunately, the authors have now supplied their computer code, but I have not had the opportunity to study it. Maybe some of my questions could have been resolved there, but I nevertheless think the manuscript needs more work. Another concern is that the authors claims to have revised the manuscript on specific points, but they have not. In particular, I requested, related to eq. 2, the definition of the difference operator, which I admittedly likely can look up in a statistics textbook. The authors reply that the definition is given in the revised manuscript, but it is not! Further, related to eq. 3, I asked what ‘G’ is and about its interpretation. The authors provide some presumed interpretation in their letter, claiming it is added to the manuscript. One word has been added to the manuscript (‘vectors’), which supplies nothing of the sort. Thus, one may question their motive for engaging in peer-review publication.

My first comment regarded to method for calculating joint probabilities, and I suggested that if distributions are Gaussian, probabilities can be estimated. If I remember correctly, the joint distribution of Gaussian distributions is also Gaussian. The authors claim that with time series, 'where each observation is from a single time point', whatever that is supposed to mean (can an observation be from non-single time points?), the joint distribution stuff does not work. I am not sure, I think considering the joint distribution for consecutive instances of time series are regularly considered in time series statistics (the other reviewer makes the same point), but I am no expert, and I am not sure I contribute to clarifying the situation. Notwithstanding, the authors say they assume a Gaussian distribution for predictions, with some presumably well-defined mean and variance and use Monte Carlo simulations to calculate the p(robability)-value. In their manuscript, the authors write that ‘what we want to estimate with the joint probability here, is the probability that random variation in a Gaussian distribution at each time point, could have resulted in a lower estimated joint probability’ (lines 191-193). While I find this unclear, I maintain that as long as a well-defined Gaussian distribution is assumed, simulations are not necessary to calculate probabilities. Monte Carlo simulations obviously provide a number that is close enough as long as it is appropriately carried out.

With regard to the explanation of the Monte Carlo simulations, I have two comments: (i) On point 2, ‘upper over FB’ and ‘lower under FB’ remains unclear. I failed to comment specifically on this in my previous report; I just requested clarifications in general; I apologize, but these statements are cryptic. The point is further made by reviewer #2, but the statements remain in the manuscript. (ii) On point 4, the summation should run from ‘b’ equals one to M, I presume.

I asked about ‘Q’ in relation to eq. 4, which should be optimal with regard to, it turns out, the log-likelihood (of what?). The authors write that Q-values had been ‘set too high’ in their original analysis. But it is supposed to be optimal, not ‘set’. This is unsettling.

I pointed out that ‘Y(n+1) = Y(n)’ was inconsistent. I realize it is only a notational issue, but it is confusing and looks wrong. The authors insist on sticking to the notation, presumably on grounds that it is used in some reference. Adding this reference does not add clarity to the confusing notation. I find this unfortunate. My related comment that it was unclear how z(n+2|n) relates to z(n+1|n), the authors have simply chosen to disregard.

My reading of the current manuscript is that it remains unclear on several methodological points. While I acknowledge that state space models and the Kalman filter are complex issues that are hard to explain in clear detail, it just makes it more important to be clear and concise. Worse, the authors seem unwilling to revise their work for clarity and provide an honest account of their revisions.

Reviewer #2: As I wrote in my previous review, my opinion on this paper is from the viewpoint of a person with expertise in data processing, and not on biological/ecological applications.

My opinion is that the revised version of the paper has not improved the clarity in the exposition and justification of the proposed method.

My major criticisms concern the section "Assessing unexpected tendencies - joint probability for detected FO", starting at line 184 of the manuscript.

In this section (lines 187-188) the Authors say that the location of the true observations with respect to the predicted trend "could be" the result of a random variation. My point is that of course such locations are the result of a random variation, otherwise the process would be deterministic! The Authors should better characterize the type of randomness that they have in mind. It is a matter of what type of "random model" the future observations are more likely to be drawn from. It is a classical "Hypothesis Testing" problem in statistics.

It is not clear what are the joint probabilities that are estimated by the algorithm from line 195 to line 204.

The given explanation (lines 191 to 193) is not at all clear.

The Authors shoud give a more formal and rigorous definition of the probabilities p_j that are being estimated by the computations at line 199 (step 2 of the algorithm): what are the events captured by such probabilities? Moreover, what are the FO_{n+j}?

Next, the Authors should also give an exact defintion of the joint probability P(p_1,...,p_J) and justify the reason why the events described by the probabilities p_j are independent.

Without a formal definition it is difficult to understand if the estimation procedure is correct, or at least acceptable.

Another point which is not at all discussed, is about the threshold to be considered on the estimated joint probability: when this estimated probability is to be considered so small to flag the observations?

Moreover, as I pointed out in my previous review, since all variables are assumed Gaussian, apparently there is no need to resort to a numerical Monte Carlo procedure, since all probabilities can be accurately computed.

The answer that the Authors gave to this issue is unclear and not satisfactory. A sequence of observations of a Gaussian process is a Gaussian vector. Since the means and covariances are known, all probabilities can be exactly computed.

As a final remark on this section, in my opinion the right methodological framework for the problem investigated by the authors is the Statistical Hypothesis Testing. Indeed, the problem stated by the Authors consists in quantitatively assessing whether the recent observations are in accordance with the estimated past model (Null Hypothesis) or not (Alternative Hypotesis).

Other issues

Lines 155-157: The authors write: "Furthermore, it is possible to identify the optimum Q and R using a numerical optimization based on the log-likelihood function (6)". Thus, it is not clear if they use or not this identification procedure or not (they only say: "is it possible")

Line 159: The section "FO analysis by multistep-ahead prediction values" is not well written.

I understand that observations are available until time n, and the state prediction for the following times is needed.

However, it is not clear the reason why it is formally assumed that Y(n)=Y(n+1)=...=Y(n+j) when j-ahead prediction has to be computed.

When observations are not available, the Kalman filter considers only the computation of the prediction step, without the correction step, and therefore no "formal" output is needed. It would be wrong to compute the Kalman correction step using past values of the observations.

Indeed, the Authors correctly compute the state predictions using the equations at line 170, without the use of any "fake" observation.

Lines 171-174 can be improved. The Authors do not clearly distinguish between the true observation y(n+j) (unavailable) and its prediction y(n+j|n) (computable). The expectation at line 171 is indeed the prediction of the observation at n+j. The covariance at line 172 should be replaced with the error covariance Cov(y(n+j)-y(n+j|n)|Y(n)).

Typos:

Line 119, Either H is a row vector or a transpose symbol is missing in eq. (3)

Line 204. The upper limit of the summation should be M, and not 1000.

Line 230. Maybe 2013 should be 2014.

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Reviewer #1: Yes: Sturla F. Kvamsdal

Reviewer #2: No

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For journal use only: PONEDEC3

10.1371/journal.pone.0305716.r004
Author response to Decision Letter 1
Submission Version2
14 Aug 2023

Dear PLOS ONE Straive Editorial Assistant, Bernadith Millamina,

This document contains our response to the comments and concerns provided to us by two reviewers on our submitted manuscript:

Re: PONE-D-22-30034

Solvang and Arneberg: Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem Assessments

The comments from the two reviewers are given in black – how we responded are given in red. We submit the version of the manuscript showing all tracked changes and the line number we have described are corresponding to the version with tracked changes.

Reviewer #1: Review, PONE-D-22-30034_R1

The authors have revised their manuscript. As I wrote in my previous report, I think the study is worthwhile documenting in a publication. Reviewer #2 points out that the methodology is known, but I think the application is novel and valuable. Consideration of the behavior of time series needs support by statistical tools. It now occurs to me, however, that the reliance on the normal distribution may be problematic. That the authors do so in their examples may be ok, but I recommend them to add a discussion of issues such as fat tail risk that should be accounted for in real applications. The involvement of non-Gaussian distributions would further highlight the value of the simulation framework they promote, which, as I try to argue below, is unnecessary when working with Gaussian distributions.

While we have constructed the linear state space representation by (3) to describe the proposed model (1) and (2) and the state component is estimated by linear Kalman filter, the state space model is not necessary Gaussian distribution. We have explained about the expansion to non-Gaussian model and filtering algorithms in lines 457-467.

My main concern in my previous report was that the presentation of the method needed several clarifications. The authors have tried, presumably, but there are still points that remain unclear (see below). Fortunately, the authors have now supplied their computer code, but I have not had the opportunity to study it. Maybe some of my questions could have been resolved there, but I nevertheless think the manuscript needs more work. Another concern is that the authors claims to have revised the manuscript on specific points, but they have not. In particular, I requested, related to eq. 2, the definition of the difference operator, which I admittedly likely can look up in a statistics textbook. The authors reply that the definition is given in the revised manuscript, but it is not!

We have described the definition of the difference operator in lines 111-119. is used as a difference operator . The notation is a general form to express the differenciation of the variabes in statistical time series analysis. We have also added the literatures to confirm the notation, e.g. pages 28, 33-34 in [29] and page 19 in [59].

Further, related to eq. 3, I asked what ‘G’ is and about its interpretation. The authors provide some presumed interpretation in their letter, claiming it is added to the manuscript. One word has been added to the manuscript (‘vectors’), which supplies nothing of the sort. Thus, one may question their motive for engaging in peer-review publication.

G is a role to operate how the system noise v(n) contributes to which component in the state vector z(n). If we assume random walk trend model, G becomes just scalar 1. If we assume the trend model with d=2, the state vector becomes (t means transpose) and v(n) contribute to only the first component in this state. In the case, G indicates , that is, in (3) is represented by and the first component of contributes the first component of z(n). The system model is given by . We have added concrete explanation about G in lines 128-135.

My first comment regarded to method for calculating joint probabilities, and I suggested that if distributions are Gaussian, probabilities can be estimated. If I remember correctly, the joint distribution of Gaussian distributions is also Gaussian. The authors claim that with time series, 'where each observation is from a single time point', whatever that is supposed to mean (can an observation be from non-single time points?), the joint distribution stuff does not work. I am not sure, I think considering the joint distribution for consecutive instances of time series are regularly considered in time series statistics (the other reviewer makes the same point), but I am no expert, and I am not sure I contribute to clarifying the situation.

Time series data is recorded as one sample at one time point. For each time point, it is assumed that the data is modelled by the stochastic process. Therefore, ‘mean’ of time series data corresponds to the ‘trend’ component, which is not same for overall time points. In this study, we apply the trend model in (2), which is time dependent, and it is not appropriate to calculate ‘joint probability’ using each probability based on the trend component. Therefore, in this revision, we use the residual part obtained by subtracting trend from the observation, which can be assumed identical and independent Gaussian distribution. Using the residual value, the upper-tale probability (if observations consecutively appear above the trend) probability is calculated using a computational function of the Gaussian distribution function. If consequently time points are three years, we obtain three upper-tale probabilities, e.g. and the joint probability becomes . We have revised the calculation procedure in lines 228-259.

Notwithstanding, the authors say they assume a Gaussian distribution for predictions, with some presumably well-defined mean and variance and use Monte Carlo simulations to calculate the p(robability)-value. In their manuscript, the authors write that ‘what we want to estimate with the joint probability here, is the probability that random variation in a Gaussian distribution at each time point, could have resulted in a lower estimated joint probability’ (lines 191-193). While I find this unclear, I maintain that as long as a well-defined Gaussian distribution is assumed, simulations are not necessary to calculate probabilities. Monte Carlo simulations obviously provide a number that is close enough as long as it is appropriately carried out.

The previous description was to manually calculate upper- or lower-tale probability using random number generated by a Gaussian distribution function. As you pointed out, it is not necessary to use the procedure and we should use the computational function to directly obtain upper- or lower-tale probability of the Gaussian distribution. Furthermore, it is not related to ‘probability value’ that means ‘significant probability’ to support the null hypothesis for statistical testing. We have revised the description in lines 228-259.

With regard to the explanation of the Monte Carlo simulations, I have two comments: (i) On point 2, ‘upper over FB’ and ‘lower under FB’ remains unclear. I failed to comment specifically on this in my previous report; I just requested clarifications in general; I apologize, but these statements are cryptic. The point is further made by reviewer #2, but the statements remain in the manuscript.

We have excluded the parts related to point 2 and revised the previous procedure, which is better both statistically and computationally. Please see the revised procedures in lines 228-259 and the outline of the procedure in Fig.1b.

(ii) On point 4, the summation should run from ‘b’ equals one to M, I presume.

We have not used the notation in the revised version.

I asked about ‘Q’ in relation to eq. 4, which should be optimal with regard to, it turns out, the log-likelihood (of what?). The authors write that Q-values had been ‘set too high’ in their original analysis. But it is supposed to be optimal, not ‘set’. This is unsettling.

Q is related to used in of the log-likelihood function (6). in is calculated using Q in the ‘prediction’ procedure (4). The log-likelihood is for the model (1). We have added the explanation in lines 173-180. ‘set too high’ meant that the optimum Q value was found within the range we set higher (0.05 to 0.5), however, we re-set lower range (0.01 to 0.1) to obtain more smoothed trend.

I pointed out that ‘Y(n+1) = Y(n)’ was inconsistent. I realize it is only a notational issue, but it is confusing and looks wrong. The authors insist on sticking to the notation, presumably on grounds that it is used in some reference. Adding this reference does not add clarity to the confusing notation. I find this unfortunate. My related comment that it was unclear how z(n+2|n) relates to z(n+1|n), the authors have simply chosen to disregard.

Since the future observation y(n+1) is not observed, the calculation is formally conducted by assuming approximately Y(n+1)=Y(n), which was taken from [34] in the provided procedure. The literature [22] also described that the state and variance for multi-ahead prediction are estimated by the observation until n (Y(n)). This is not our own idea, but is used as a general assumption for multistep-ahead prediction using the linear Kalman filter. We have added the explanation and references in lines 199 - 227.

My reading of the current manuscript is that it remains unclear on several methodological points. While I acknowledge that state space models and the Kalman filter are complex issues that are hard to explain in clear detail, it just makes it more important to be clear and concise. Worse, the authors seem unwilling to revise their work for clarity and provide an honest account of their revisions.

We apologize for previous unclear explanation. We have added more details to clarify the state space representation and the Kalman filter algorithm for FO detection lines in 136 - 227.

Reviewer #2: As I wrote in my previous review, my opinion on this paper is from the viewpoint of a person with expertise in data processing, and not on biological/ecological applications.

The statistical methodology we used in this study is not novel. As the application using existing statistical methods, we newly provide the FO analysis, which is a practical approach in the Integrated ecosystem assessment for marine resource. Therefore, this paper is about biological/ecological time series application.

My opinion is that the revised version of the paper has not improved the clarity in the exposition and justification of the proposed method.

We have revised the paper carefully, especially the previous procedure for assessing unexpected tendencies.

My major criticisms concern the section "Assessing unexpected tendencies - joint probability for detected FO", starting at line 184 of the manuscript.

In this section (lines 187-188) the Authors say that the location of the true observations with respect to the predicted trend "could be" the result of a random variation. My point is that of course such locations are the result of a random variation, otherwise the process would be deterministic! The Authors should better characterize the type of randomness that they have in mind. It is a matter of what type of "random model" the future observations are more likely to be drawn from. It is a classical "Hypothesis Testing" problem in statistics.

It is not clear what are the joint probabilities that are estimated by the algorithm from line 195 to line 204.

We apologize for previous unclear explanation. We have revised the previous procedure for the section in lines 228-259 and have made the outline of the procedure in Fig.1b.

The given explanation (lines 191 to 193) is not at all clear.

We have excluded the explanation.

The Authors shoud give a more formal and rigorous definition of the probabilities p_j that are being estimated by the computations at line 199 (step 2 of the algorithm): what are the events captured by such probabilities? Moreover, what are the FO_{n+j}?

We have revised the description to obtain p_j in lines 228-259.

Next, the Authors should also give an exact defintion of the joint probability P(p_1,...,p_J) and justify the reason why the events described by the probabilities p_j are independent.

Without a formal definition it is difficult to understand if the estimation procedure is correct, or at least acceptable.

As you pointed out, we didn’t explain the independent condition to calculate the joint probability. When the observations locate above the predicted trends at e.g. three time points, for calculating the probability for each time point, the residual obtained by subtracting trend (time-dependent model) from the observation is assumed the variable that obeys identical and independent (Gaussian) distribution. Using the residual value for each time point, the upper-tale probability of the value is calculated by a computational function using Gaussian distribution function. The upper-tale probabilities are independent.

Another point which is not at all discussed, is about the threshold to be considered on the estimated joint probability: when this estimated probability is to be considered so small to flag the observations?

The revised procedure applies an arbitrary threshold 0.05 which means that the event occurs by 5%-tail probability (found in the tail of the distribution). If j = 3, is assessed if .

Moreover, as I pointed out in my previous review, since all variables are assumed Gaussian, apparently there is no need to resort to a numerical Monte Carlo procedure, since all probabilities can be accurately computed.

The answer that the Authors gave to this issue is unclear and not satisfactory. A sequence of observations of a Gaussian process is a Gaussian vector. Since the means and covariances are known, all probabilities can be exactly computed.

The previous description was to manually calculate upper or lower tale probability using random number generated by a Gaussian distribution function. As you pointed out, it is not necessary to use the procedure and we should use the computational function to directly obtain upper or lower tale probability of the Gaussian distribution. We have revised the description in lines 228-259.

As a final remark on this section, in my opinion the right methodological framework for the problem investigated by the authors is the Statistical Hypothesis Testing. Indeed, the problem stated by the Authors consists in quantitatively assessing whether the recent observations are in accordance with the estimated past model (Null Hypothesis) or not (Alternative Hypotesis).

As you suggested, the procedure is not related to statistical testing considering null hypothesis. It is just calculation of probability based on the Gaussian distribution function. We have excluded ‘probability value’ that means significant probability to assess the test statistics.

Other issues

Lines 155-157: The authors write: "Furthermore, it is possible to identify the optimum Q and R using a numerical optimization based on the log-likelihood function (6)". Thus, it is not clear if they use or not this identification procedure or not (they only say: "is it possible")

In this paper, the variance of the system noise in Q is found by grid search within a range and the variance of the observation noise in R is given by the observation until the year before the year for multi-ahead prediction. On the other hand, there is another way to identify these variances by the numerical optimization procedure (e.g. Newton-Raphson method introduced in [34]). To avoid misunderstanding, we have modified the sentence in lines 181-183. We have also added the program using the numerical optimization procedure in Supplementary file S2.

Line 159: The section "FO analysis by multistep-ahead prediction values" is not well written.

I understand that observations are available until time n, and the state prediction for the following times is needed.

However, it is not clear the reason why it is formally assumed that Y(n)=Y(n+1)=...=Y(n+j) when j-ahead prediction has to be computed.

When observations are not available, the Kalman filter considers only the computation of the prediction step, without the correction step, and therefore no "formal" output is needed. It would be wrong to compute the Kalman correction step usin

Attachment Submitted filename: Reply to reviewers comments second.docx

10.1371/journal.pone.0305716.r005
Decision Letter 2
Zambujal-Oliveira João Academic Editor
© 2024 João Zambujal-Oliveira
2024
João Zambujal-Oliveira
https://creativecommons.org/licenses/by/4.0/ 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.
Submission Version2
22 Feb 2024

PONE-D-22-30034R2Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem AssessmentsPLOS ONE

Dear Dr. Solvang,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please submit your revised manuscript by Apr 07 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

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Academic Editor

PLOS ONE

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Reviewer #1: (No Response)

Reviewer #3: (No Response)

**********

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Reviewer #3: Yes

**********

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Reviewer #1: Yes

Reviewer #3: Yes

**********

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Reviewer #1: Yes

Reviewer #3: Yes

**********

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PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: No

Reviewer #3: Yes

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The manuscript has been revised heavily, mainly in the methodological part, and is now more aligned with standard expositions of the Kalman filter and related algorithms. I still find think the presentation can be written better and be more accessible given that it is an established method. The paper should be read by a copy editor to improve the overall writing and the odd typo. I will not comment in great detail on this section now, the authors have revised in accordance with my earlier comments and comments from the evidently more qualified other reviewer.

I have one comment on an issue that has been discussed earlier, and one question about the relevant probability threshold.

The earlier discussed issue is the case with 'approximately Y(n+1) = Y(n)'. I still find this a strange way of formulating what the authors are doing. I think what they do is to calculate forecasted values over several time steps conditional on Y(n). That is, the forecast for n+2 is conditional on Y(n), and not on Y(n+1) with approximately Y(n+1)=Y(n). As said in earlier correspondence, this may mainly be a notational issue, but the way the authors states it in the present version is cumbersome and could perhaps be wrongly interpreted. I see that the other reviewer also comments on this issue, and importantly points out that no Kalman correction or update should be made without new observations available. This is the kind of misunderstanding that could occur with the authors' notation.

I also wonder about the probability threshold that is used. For predictions over three time steps, the authors compare with the threshold 0.05^3, which is a very small number. Thus, at least given an approximately correct prediction model, very few observations will be flagged. And maybe this is the whole point, if the prediction model is approximately correct, there should be a small chance for flagging observations (false positives). Nevertheless, some discussion along these lines would be valuable, I think.

Reviewer #3: A thorough research has been carried out, which has applied significance. However, some author's assumptions should be given broader explanations, which are given in the review.

Thank you!

**********

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Reviewer #1: Yes: Sturla Kvamsdal

Reviewer #3: Yes: Lupak Ruslan

D.Sc. (Economics), Professor

Department of Economy

Lviv University of Trade and Economics

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Attachment Submitted filename: Review_PONE-D-22-30034_reviewer.pdf

10.1371/journal.pone.0305716.r006
Author response to Decision Letter 2
Submission Version3
17 Apr 2024

>Reviewer #1

>The manuscript has been revised heavily, mainly in the methodological part,

>and is now more aligned with standard expositions of the Kalman filter and related

>algorithms. I still find think the presentation can be written better and be more accessible

>given that it is an established method. The paper should be read by a copy editor to

>improve the overall writing and the odd typo.

We have gone through the manuscript carefully to improve the overall writing.

>I will not comment in great detail on this

>section now, the authors have revised in accordance with my earlier comments and

>comments from the evidently more qualified other reviewer.

>I have one comment on an issue that has been discussed earlier, and one question about the

>relevant probability threshold.

>The earlier discussed issue is the case with 'approximately Y(n+1) = Y(n)'. I still find this a

>strange way of formulating what the authors are doing. I think what they do is to calculate

>forecasted values over several time steps conditional on Y(n). That is, the forecast for n+2

>is conditional on Y(n), and not on Y(n+1) with approximately Y(n+1)=Y(n). As said in

>earlier correspondence, this may mainly be a notational issue, but the way the authors

>states it in the present version is cumbersome and could perhaps be wrongly interpreted. I

>see that the other reviewer also comments on this issue, and importantly points out that no

>Kalman correction or update should be made without new observations available. This is

>the kind of misunderstanding that could occur with the authors' notation.

The expression ‘assuming approximately Y(n+1)≈Y(n)’ may lead to misunderstandings. What we meant is that:

Consider the case that I(j) is as a set of actual observed time points in the time points 1,2,⋯,j, Y(j)≡{y(i)|i∈I(j)}.

For one-ahead prediction, the state z(n) and the variance v(n) are then obtained by the Kalman filter using actual observed data series Y(n)= {y(1),y(2),⋯,y(n)}, that is, z(n+1│n)=Fz(n|n) and v(n+1|n)= Fv(n│n) F^'+GQG'. Here, since the future observation y(n+1) is unavailable, Y(n+1)= {y(1),y(2),⋯,y(n)}=Y(n).

For long-term prediction, we consider j-ahead prediction of the state, z(n+j), based on Y(n+j), that is, z(n+j│n+j)=Fz(n+j-1|n+j-1) and v(n+j|n+j-1)= Fv(n+j-1│n+j-1) F^'+GQG'.

Since y(n+1),y(n+2),⋯,y(n+j) are not available after observed y(n), Y(n+j)= {y(1),y(2),⋯,y(n)}=Y(n).

Therefore, the state and variance for j-ahead prediction are z(n+j│n)=Fz(n+j-1|n) and v(n+j|n)= Fv(n+j-1│n) F^'+GQG'.

The point is that j-ahead prediction based on the observations until y(n) is performed by repeating the prediction step j times, using the relation that Y(n)= Y(n+1)=⋯=Y(n+j). This was from references [22, 34], and is not our own idea.

To avoid a wrong interpretation, we have rephrased the paragraph by using appropriate sentences and references and not using the notation that could lead to misunderstandings (lines 193 - 199).

>I also wonder about the probability threshold that is used. For predictions over three time

>steps, the authors compare with the threshold 0.05^3, which is a very small number. Thus,

>at least given an approximately correct prediction model, very few observations will be flagged. And maybe this is >the whole point, if the prediction model is approximately correct, there should be a small chance for flagging >observations (false positives). Nevertheless, some discussion along these lines would be valuable, I think.

This threshold value is used to assess the relevance of the put estimates, that is, the probability that the combined evidence of all recent observations indicates that there is an unexpected tendency across all of the most recent (i.e., predicted) years. Therefore, this threshold is not used to assess observations for single years. We agree that using the general formula for a threshold at the 0.05 level for j predicted years as 0.05j may give conservative estimates of what should be considered unexpected tendencies across all recent years. We have added a paragraph in the discussion and thank the reviewer for pointing out this issue (lines 407 - 413).

>Reviewer #3

>A thorough research has been carried out, which has applied significance.

>However, some author's assumptions should be given broader explanations, which are

>given in the review.

> Ecosystems influence the existing conditions of human life, either improve or worsen it, and accordingly require a >person's constant adaptation to such changes. Thanks to observations, it is possible to achieve the necessary >predictability of ecosystems, reliably forecast and satisfy needs while under its influence. All this confirms the >relevance of the article, and despite its significant advantages, it is appropriate to take into account a number of >recommendations:

>- the article should indicate the objectives of the observations with the overall goal of determining the scope and >communications in ecosystem assessments and the expected results from such observations;

A summary of the objectives for inclusion of each of the time series in the work of the ICES integrated ecosystem assessment working group for the Norwegian Sea (WGINOR) has been added to Supplementary Table 1 (see lines 249 - 251in the main text). The objective behind the AMO data should be evident from the main text (i.e., it is described as a pronounced signal of climate variability, line 257 - 258).

>- it is worth citing the advantages and disadvantages of conducting the analysis of observations with labels, in >particular, comparing them with some frequently used statistical methods;

The meaning of the term «label», as used here, is not clear to us. As this could not be clarified after contact with the editor, we are unfortunately not able to address this comment.

>- it is necessary to develop a set of proposals for the active implementation of the observation analysis tool with >labels, addressing them to international institutions, associations of countries or others.

Again, as the meaning of the term «label», as used here, is not clear to us, we are unable to also address this comment.

Attachment Submitted filename: Reply to reviewers comments third.docx

10.1371/journal.pone.0305716.r007
Decision Letter 3
Zambujal-Oliveira João Academic Editor
© 2024 João Zambujal-Oliveira
2024
João Zambujal-Oliveira
https://creativecommons.org/licenses/by/4.0/ 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.
Submission Version3
7 May 2024

PONE-D-22-30034R3Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem AssessmentsPLOS ONE

Dear Dr. Solvang,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

Please take into account the remaining typos indicated by one of the reviewers.

Please submit your revised manuscript by Jun 17 2024 11:59PM. If you will need more time than this to complete your revisions, please reply to this message or contact the journal office at plosone@plos.org. When you're ready to submit your revision, log on to https://www.editorialmanager.com/pone/ and select the 'Submissions Needing Revision' folder to locate your manuscript file.

Please include the following items when submitting your revised manuscript:A rebuttal letter that responds to each point raised by the academic editor and reviewer(s). You should upload this letter as a separate file labeled 'Response to Reviewers'.

A marked-up copy of your manuscript that highlights changes made to the original version. You should upload this as a separate file labeled 'Revised Manuscript with Track Changes'.

An unmarked version of your revised paper without tracked changes. You should upload this as a separate file labeled 'Manuscript'.

If you would like to make changes to your financial disclosure, please include your updated statement in your cover letter. Guidelines for resubmitting your figure files are available below the reviewer comments at the end of this letter.

If applicable, we recommend that you deposit your laboratory protocols in protocols.io to enhance the reproducibility of your results. Protocols.io assigns your protocol its own identifier (DOI) so that it can be cited independently in the future. For instructions see: https://journals.plos.org/plosone/s/submission-guidelines#loc-laboratory-protocols. Additionally, PLOS ONE offers an option for publishing peer-reviewed Lab Protocol articles, which describe protocols hosted on protocols.io. Read more information on sharing protocols at https://plos.org/protocols?utm_medium=editorial-email&utm_source=authorletters&utm_campaign=protocols.

We look forward to receiving your revised manuscript.

Kind regards,

João Zambujal-Oliveira

Academic Editor

PLOS ONE

Journal Requirements:

Please review your reference list to ensure that it is complete and correct. If you have cited papers that have been retracted, please include the rationale for doing so in the manuscript text, or remove these references and replace them with relevant current references. Any changes to the reference list should be mentioned in the rebuttal letter that accompanies your revised manuscript. If you need to cite a retracted article, indicate the article’s retracted status in the References list and also include a citation and full reference for the retraction notice.

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Reviewer #1: All comments have been addressed

Reviewer #3: All comments have been addressed

********** 

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #3: Yes

********** 

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #3: Yes

********** 

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #3: Yes

********** 

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #3: Yes

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6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: The manuscript looks pretty good. I suspect a remaining typo: An 'L' appears in some of the math where one may expect dots. See for example lines 136, 146, 147, and further on. May be some autocorrect playing games. It seems to have carried over from earlier versions if the track-changes version is to be believed. If so, I apologize for not spotting this earlier.

Reviewer #3: I thank the authors for considering the recommendation and wish them further scientific growth.

Regarding the concept of "labels", we mean "Flagged"

********** 

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Reviewer #1: No

Reviewer #3: Yes: Ruslan Lupak,

Dr Sc (Doctor of Economic Sciences), Prof.,

Prof. of the Department of Economy

Lviv University of Trade and Economics, Ukraine

ORCID ID: https://orcid.org/0000-0002-1830-1800

Scopus Preview ID: https://www.scopus.com/authid/detail.uri?authorId=57189037710

Web of Science Researcher ID: https://publons.com/researcher/2106107/lupak-ruslan

Google Scholar ID: https://scholar.google.com.ua/citations?hl=uk&user=UliL_9wAAAAJ&scilu

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10.1371/journal.pone.0305716.r008
Author response to Decision Letter 3
Submission Version4
13 May 2024

Dear PLOS ONE Academic Editor, João Zambujal-Oliveira,

This document contains our response to the comments and concerns provided to us by two reviewers on our submitted manuscript:

Re: PONE-D-22-30034R2

Solvang and Arneberg: Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem Assessments

The comments from the two reviewers are shown in black font, while our responses are given in red font. In the manuscript, all changes are shown using the “Track Changes” function of MS Word.

Reviewer #1

>The manuscript looks pretty good. I suspect a remaining typo: An 'L' appears in some of >the math where one may expect dots. See for example lines 136, 146, 147, and further >on. May be some autocorrect playing games. It seems to have carried over from earlier >versions if the track-changes version is to be believed. If so, I apologize for not spotting >this earlier.

This error was introduced when our word file with the manuscript was converted to a pdf file during the submission process. We have identified this as a problem with an old version of the editor used for equations and will see to that the problem is fixed when resubmitting.

Reviewer #3

>I thank the authors for considering the recommendation and wish them further scientific >growth.

>

>Regarding the concept of "labels", we mean "Flagged"

>- it is worth citing the advantages and disadvantages of conducting the analysis of >observations with (labels) flagged, in particular, comparing them with some frequently >used statistical methods;

We thank the reviewer for clarifying his comments. One of the most common approaches to address change in time series is by using plots and analyses of anomalies. In line 409-423 and Supplementary Fig. 1, we discuss how FO analyses may give useful additional information to that held by analyses of anomalies. In addition, it has recently been shown that commonly used methods for trend detection based on frequentist approaches may produce erroneous results (see ref 54). We point out that 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, which may properly assess trends in short time series (lines 383-391). We therefore think we have covered the two main advantages over other relevant frequently used statistical methods. We see no disadvantages of FO analyses when compared with the above mentioned frequently used methods, but emphasise that for analyses of anomalies, FO analyses represents a supplement rather than a replacement (lines 419-423).

>- it is necessary to develop a set of proposals for the active implementation of the >observation analysis tool with (labels) flagged, addressing them to international >institutions, associations of countries or others.

FO analysis is primarily developed as a tool for scoping in IEA groups (see lines 50-54, 79-80 and 377-378). Thus, the primary uses of the results are the IEA groups themselves. We also point out that the results from FO analyses can be used in communication with stakeholders and that this can also be relevant for policymaking processes (lines 424-430). However, how this should be done in practice, will, in our view, be context dependent and has to be developed by IEA groups and/or users of their results. While we agree that it is highly interesting to discuss this, we feel it is beyond the scope of this manuscript, as it would mean expanding the text considerably by for example including considerable literature on the science-policy interface.

Attachment Submitted filename: Reply to reviewers comments forth.docx

10.1371/journal.pone.0305716.r009
Decision Letter 4
Zambujal-Oliveira João Academic Editor
© 2024 João Zambujal-Oliveira
2024
João Zambujal-Oliveira
https://creativecommons.org/licenses/by/4.0/ 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.
Submission Version4
5 Jun 2024

Flagged observation analyses as a tool for scoping and communication in Integrated Ecosystem Assessments

PONE-D-22-30034R4

Dear Dr. Solvang,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

Within one week, you’ll receive an e-mail detailing the required amendments. When these have been addressed, you’ll receive a formal acceptance letter and your manuscript will be scheduled for publication.

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Kind regards,

João Zambujal-Oliveira

Academic Editor

PLOS ONE

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Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

**********

2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

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10.1371/journal.pone.0305716.r010
Acceptance letter
Zambujal-Oliveira João Academic Editor
© 2024 João Zambujal-Oliveira
2024
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https://creativecommons.org/licenses/by/4.0/ 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.
24 Jun 2024

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