Autocorrelation within ecological time series and synchrony between them may provide insight into the main drivers of observed dynamics.
We here present methods that analyse autocorrelation and synchrony in ecological datasets using a spectral approach combined with Bayesian inference.
To exemplify, we implement the method on dendrochronological data of the pedunculate oak (Quercus robur). The data consist of 110 years of growth of 10 live trees and seven trees that died during a synchronized oak death in Sweden in c. 2002–2007. We find that the highest posterior density is found for a noise colour of tree growth of γ ≈ 0·95 (i.e. ‘pink noise’) with little difference between trees, suggesting climatic variation as a driving factor. This is further supported by the presence of synchrony, which we estimate based on phase-shift analysis. We conclude that the synchrony is time-scale dependent with higher synchrony at larger time-scales. We further show that there is no difference between the growth patterns of the alive and dead tree groups. This suggests that the trees were driven by the same factors prior to the synchronized death.
We argue that this method is a promising approach for linking theoretical models with empirical data.
Ecologists are increasingly recognizing the importance of studying the characteristics of environmental fluctuations (Inchausti & Halley 2003) and synchrony (Hastings 2010). Theoretical studies have shown that the autocorrelation of environmental fluctuations and synchrony have significant effects on population dynamics and extinction risks (Ripa & Lundberg 1996; Liebhold, Koenig & Bjørnstad 2004; Lögdberg & Wennergren in press). Theoretical studies have also demonstrated how autocorrelation and synchrony of ecological systems may be modelled by the spectral representation of time series (Halley 1996; Cuddington & Yodzis 1999). In this study, we demonstrate how this approach, combined with Bayesian inference, can be used for ecological time-series analysis. We show how model parameters may be estimated and how different hypotheses of fluctuation may be compared. Consequently we use analysis of autocorrelation and synchrony to make inference about processes that influence the dynamics observable in ecological datasets.
Ecological time-series data are usually positively autocorrelated (Steele 1985; Inchausti & Halley 2002), denoted ‘reddened’ because the spectral representation is dominated by low frequencies. Autocorrelation is commonly also present over multiple scales (Pimm & Redfearn 1988) and consequently first-order autoregressive (AR) models may neglect important aspects of autocorrelation. Consequently, environmental fluctuations are commonly analysed as Flicker noise, also known as ‘one over f’ (1/f) noise (Halley 1996). This assumes a fractal dynamic with self-similarity in the autocorrelation at different temporal scales and is described by a single-scale free colour parameter γ given by a powerlaw relationship of the power spectral density function (PSDF) of a time series with . Time series with γ = 0 have no autocorrelation (denoted ‘white noise’) whereas time series exhibiting random walk properties have γ = 2. Figure 1 (panels a and b) illustrates the difference between white and autocorrelated dynamics. Environmental fluctuations are generally estimated at γ ≈ 1, that is, ‘pink noise’ (Halley 1996). Pimm & Redfearn (1988) argued that ecological systems with similar dynamics are likely driven by the underlying environmental fluctuations.
However, internal processes may also cause autocorrelated dynamics of ecological systems. If studying individuals as part of a population, the growth rate of the individual may be autocorrelated by factors at the individual level (e.g. pathogens, pests, microclimatic factors). Similarly, local processes may cause autocorrelated dynamics if considering subpopulations as part of a metapopulation. To conclude that reddened dynamics of ecological processes are caused by some global factor (e.g. environmental fluctuation), it is therefore important to consider synchrony between time series. Synchrony is often defined as the co-fluctuation of two or more time series. Figure 1 (panels c and d) illustrates the difference between synchronized and non-synchronized dynamics. The synchrony of local populations is commonly attributed to either a classical ‘Moran effect’ where nearby patches are all influenced by a severe event, which makes their inherent local oscillations coordinated, or by a less strict interpretation of the ‘Moran effect’ where the populations are directly and continuously driven by fluctuations of environmental factors such as climatic variables (Heino et al. 1997). Furthermore, the synchrony of local populations may also arise as a result of between-patch dispersal (Abbott 2011) or by trophic interactions with other species already being synchronized (Liebhold, Koenig & Bjørnstad 2004). Individuals of a population may also be more or less synchronized in, for example, their growth. This synchrony is mainly attributed to the less strict interpretation of the ‘Moran effect’ where the growth rates are driven by environmental factors. If considering individuals as part of a population, synchrony may be caused by, for example, outbreaks of pathogens or pests within the population and common environmental factors. Different processes may be of different importance at different time-scales and synchrony may be more apparent at time-scales where global external factors are more influential. Further, when comparing two or more internally synchronous groups, the groups may be considered to actually co-fluctuate as one synchronous group (Fig. 1e) or as two different groups (Fig. 1f).
In this study, we present a novel method to analyse fluctuations and synchrony from ecological time-series data. We implement the method on dendrochronological data of the pedunculate oak (Quercus robur) and demonstrate how it may be used to highlight the importance of factors at the local, individual (population) level or by global factors. The pedunculate oak is one of the most important tree species in Europe for invertebrates associated with veteran trees (Jansson et al. 2009). The annual growth of oaks is known to be a result of a complex interaction of abiotic and biotic factors, for example, precipitation, temperature and defoliating insects (Thomas, Blank & Hartmann 2002). Climatic factors are expected to be both temporally autocorrelated and have a synchronous effect on tree growth. Insect outbreak may also cause synchronization (Bjørnstad et al. 2002) but the temporal autocorrelation caused by this is likely to be acting on shorter time-scale and potentially be cyclical rather than fractal. Hence, if we find that the fluctuation in annual growth is synchronous as well as autocorrelated similarly to climatic variables (i.e. Flicker noise with γ ≈ 1), we may expect that climatic variations are the main drivers.
We may also expect that years of particularly unfavourable conditions are followed by a recovery period of low annual growth. Such periods of low growth cause a reddened spectrum as well as synchronization if the unfavourable conditions are affecting the whole population. If trees cannot recover, such bad years may lead to a prolonged decline in annual growth and ultimately synchronized mortality of individual oaks (Andersson, Milberg & Bergman 2011). A number of synchronized mortality events for European oaks (Quercus) have been documented throughout Europe, the earliest dating back to 1739–1748 in north-eastern Germany (Thomas 2008). The causes of the synchronized mortality events seem to vary, from large-scale factors as climate to defoliation by insect larvae. One such event took place in Sweden in c. 2002–2007. To investigate if the trees that died and those that did not were historically influenced by different factors, we compare annual growth of trees that died with trees that managed to recover. Using hierarchical Bayesian modelling, we compare the posterior distributions of the estimated autocorrelation parameters. This approach is beneficial in that we include the uncertainty at the individual level (i.e. in γ of individual trees) when we estimate the population-level parameters that are most relevant for comparison. If the oaks that died exhibited a higher degree of autocorrelation in annual growth, we may expect that these trees historically have had longer periods of recovering after years of bad conditions or been particularly sensitive to different environmental factors. We compare two competing models: M1 – all trees come from the same synchronous group, and M2 – live and dead trees are different and co-fluctuate differently. If M1 is more probable, we may conclude that live and dead trees have historically been influenced by the same environmental factors. If instead M2 is more probable, we may conclude that they have been sensitive to different factors. Recent studies suggest that phase-shift analysis is a promising approach to model synchrony (Vasseur 2007; Keitt 2008; Hastings 2010; Lögdberg & Wennergren in press). In systems with cyclic behaviour, the phases may be easily identified (Hastings 2010). Yet for many ecological systems, including tree growth, we do not expect cyclic dynamics. We will in our study show how phase-shift analysis can be combined with Bayesian inference and used for analysis of synchrony in systems without apparent oscillations.
The purpose of this paper is to present a methodology for analysis of fluctuations and synchrony based on spectral analysis and Bayesian inference. We apply these methods on data of oak tree growth and demonstrate how we may extract valuable insight into the dynamic behaviour by studying spectral colour of time series and synchrony between them. We further demonstrate how this approach may be used to distinguish the degree of synchrony at different time-scales. We employ hierarchical Bayesian modelling, an approach that is being increasingly used within ecological studies with the benefit that it allows for parameter uncertainty at different levels of the model (Clark 2005).
Materials and methods
We reuse a part of the data used by Andersson, Milberg & Bergman (2011). It was collected from a single site (Tinnerö) within the county of Östergötland, south-eastern Sweden. Core samples were taken from oaks using a Swedish increment borer. Each tree was cored at chest height. Once sampled, the cores were air dried in the laboratory, before being mounted on wooden plates and polished. The cores were scanned (Nashuatec MP C3500, 400 dpi) and measured in a Cybis CooRecorder 7.1, Cybis Elektronik & Data AB, Saltsjöbaden, Sweden. To minimize the risk of errors caused by missing or false rings, the original ring widths of two cores from each individual tree, having a correlation value of ≥ 0·5, were averaged into one single tree ring series.
Our analysis requires that all time series are of the same length and for the analysis focusing on a single group of time series, the first part of the analysis we choose to use the 110 years between 1881 and 1990. Trees commonly grow faster in the early stages of life, and to reduce the impact of this, we discarded trees that had not reached the drilling height at least 20 years prior to 1881. Ten live and seven dead trees fulfilled these requirements and were used in the analysis. Tree increments are (naturally) positive and usually positively skewed and we therefore modelled log increments. Data requirements are further discussed in Appendix B in Appendix S2 of the Supporting information.
Spectral approach to time-series analysis
A time series can be described as a summation of cosine functions, here referred to as wave components. By fast Fourier transform (FFT), a series of length L is described with M wave components, M = m + 1 (where m is defined as the integer part of L/2) with frequencies 2πf/L, f = 0, 1, 2…m. Each wave component further has a phase and amplitude, where the latter is conveniently analysed by the periodogram.
Phases are here used to analyse synchrony between time series, while the periodogram is used for autocorrelation within time series. Denoting the periodogram and phases of frequency f as ψf and θm, respectively, the necessary calculations for FFT of a time series x = x1, x2…xL are
Note that the periodogram contains L elements while the θ contains m. However, the periodogram is symmetric with ψi = ψL-i for i = 1, 2, …m and the periodogram analyses are therefore performed on the m first elements.
Bayesian analysis of periodogram
We use ψ to indicate the vector of unique entries of the periodogram, with ψf,n denoting the periodogram of frequency f for time series n. We assume that the fluctuations follow the assumptions of 1/f noise (see Halley 1996 for a review). This has proven to be a good model for both temporal (Vasseur & Yodzis 2004) and spatial (Lindström, Håkansson & Wennergren 2011) analysis of ecological systems. Commonly, the noise colour parameter is obtained as the slope of least square regression of log (frequencies) vs. log (periodogram) (Solo 1992; Vasseur & Yodzis 2004). Rather than such point estimate, we are here interested in the full posterior distribution of parameters. We implement Markov chain Monte Carlo (MCMC) techniques as presented in Appendix A.1 in Appendix S1.
Assuming Flicker type noise, the expected value of the PSDF for frequency f of time series n is where an relates to the overall variance of the time series and γn is the colour parameter for time series n. In determining the likelihood function we use the Whittle approximation (Whittle 1957), which states that the periodogram ordinates are independent and exponentially distributed. Various smoothing methods implement Bayesian estimation of spectral densities at particular frequencies (e.g. Gangopadhyay, Mallick & Denison 1999). We are, however, mainly interested in the relationship between high- and low-frequency components. Hence we use the Whittle approximation to describe the residual distribution around the 1/f model.
One of the key elements of our method for analysis of the periodograms is the hierarchical Bayesian framework. Rather than assuming fixed priors for individual an and γn, these distributions are considered to be determined by a set of hyperparameters that are estimated in the model. This provides population-level parameters for population comparison that are estimated in a framework that recognizes the uncertainty at the individual level.
Defining and , the posterior for the hierarchical model of the periodogram is given by
where ξa and ξγ denote hyperparameters for the hierarchical priors of a and γ, respectively, and and denote the hyperpriors. indicates the exponential distribution with parameter 1/Yf,n evaluated at ψf,n. We model as normal distributions with mean and standard deviation σγ (i.e. ). Because an is logically positive, we model as a normal distribution on the logarithm of a with mean and standard deviation σalog (i.e. ).
We analyse periodograms separately for the two groups by eqn (eqn 2) and make inference about individual- and population-level parameters. To compare the temporal autocorrelation pattern of the two groups, we compare the marginal posterior distributions of .
The Whittle approximation is strictly appropriate only ‘under suitable regularity conditions’ (Solo 1992), and Contreras-Cristán, Gutiérrez-Peña & Walker (2006) suggest that the assumptions of the Whittle approximation are violated for highly autocorrelated processes. To justify the likelihood function, we therefore provide an analysis of residuals in 'Model checking'.
Choice of hyperpriors
The conjugate priors for the mean and standard deviation of the normal distribution are the normal and scaled inverse χ2 distribution (denoted Inv−χ2), respectively. While in principle we may wish to be as uninformative as possible, at least for the hyperparameters of , we can argue for a slightly more informative prior. Fluctuations in ecological systems generally show colour parameter values between 0 and 2 (Vasseur & Yodzis 2004). Hence, it is reasonable to expect our estimate of to lie within that range and our choice of prior distribution should reflect that belief. However, we do not exclude the possibility of outside this range and use a normal distribution with mean 1 and standard deviation 1 as hyperprior for . This distribution has 68% of its central density within the range [0, 2].
The Inv−χ2 hyperprior for is parameterized by degrees of freedom, ν0, and expected variance, . While elicitation of precise prior beliefs about Vγ is not straightforward, we argue that it is unlikely to have some trees showing completely random growth pattern while others show high autocorrelation. When choosing a prior, we followed the suggestion of Gelman et al. (2004) and elicit the distribution by the value we believe to be most likely (i.e. the prior mode) and some upper value, C, below which we believe 95% of the density is located. We chose a modal value of 0·0001, hence indicating a priori beliefs of little variation between the noise colours of different trees. However, we do not exclude high variability and use C = 1, indicating that we leave 5% probability of variance larger than one. The corresponding values of ν0 and are 1·05 × 10−4 and 1·9, respectively.
The periodogram for f = 0 is defined as the mean of a time series. This is not an essential part of our analysis here but its definition is required to sample from the posterior predictive distribution (PPD). We assume that and use priors and . Both priors are defined on the real line.
Analysis of synchrony
We are here interested in synchrony between different time series and indicate the phase of frequency f of time series n by θf,n, for n = 1, 2…N and f = 1, 2…m. If for every f, θf,n are identical for all n we say that the time series are completely synchronous. Alternatively, if all phases are distributed randomly on a uniform circular distribution [denoted ], we may conclude that there is no synchrony.
The first step in our analysis is to test for synchrony of a single group. This is performed by comparing two models: the null model (M0), which assumes randomly distributed phases, and model M1, which assumes that θf,n for n = 1, 2…N are centred at some mean phase μf. Because phases are angular data, we use the wrapped Cauchy distribution (WCD) to model θ under M1, where the probability distribution of θf,n is given by
where ρf (0 ≤ ρf ≤ 1) models the angular dispersion. Other circular distributions (e.g. von Mises or wrapped normal) may also be appropriate but the WCD is advantageous because it has an easily interpretable dispersion parameter defined on finite range, hence facilitating prior elicitation. For ρf = 1, we obtain complete synchrony for the phases of frequency f and eqn (eqn 3) approaches a random angular distribution as ρf → 0. If the length of the considered time series is even, the phase of the highest frequency, θm,n, is either 0 or π and was modelled as
where μm is either 0 or π. This provides a comparable measure of synchrony that allows for a more straightforward extension to hierarchical modelling (see below).
We compare models by computing ratios of posterior model probabilities R1 = P(M1|data)/P(M0|data) = B1P(M1)/P(M0) where B1 is the Bayes factor given by
where and are the priors of μf and ρf, respectively, and and are the priors for each model. The ratio of posterior model probabilities describes the evidence in favour of each model, for example, R1 = 2 indicates that M1 has twice as much probability under the posterior distribution as M0. We specify the priors on each model as , hence R1 = B1. We further specify priors that are uniform on the supported range, that is, and and calculate R1 separately for the group of live and dead trees. Appendix A.2.1 in Appendix S1 addresses computation of B1.
Synchrony between two groups
In determining whether two sets of time series are better modelled as one synchronous group or as two separate groups, we compare two competing models by the ratio of posterior model probabilities, R2. Model M1 is as above, hence modelling all phases as coming from the same WCD with parameters μf,0 and ρf,0. Model M2 instead model phases of the two groups by separate WCD, each distribution i with parameters μf,i and ρf,i. Assuming and defining θf,n,i as the phase of frequency f for individual n of group i, we calculate (by reversible jump MCMC, see Appendix A.2.1 in Appendix S1) the Bayes factor
Analogous to the single-group analysis, R2 = B2 and we use priors and .
Different processes may cause different levels of synchronization at different time-scales, here expressed as a relationship between ρf and f. We focus on a single group and because ρf is defined as 0 ≤ ρf ≤ 1, we model , where . The posterior is then given as
where , and the hyperpriors , and are chosen to be proportional to one over the support of the parameters. Computations are addressed in Appendix A.2.2 in Appendix S1.
The likelihood function assumes that the ordinates of the periodogram are independent and exponentially distributed. It is credible that the assumptions of the Whittle approximation may be violated under true Flicker noise (which is fractal over all scales and hence non-stationary). These violations may be more problematic if the focal interest is estimation of the ‘true’ density of some specific frequencies rather than the overall relationship between amplitudes of different frequencies. Here we use the 1/f model to describe the relationship between amplitudes of different frequencies. We therefore perform a residual analysis to validate the choice of likelihood function (i.e. independent and exponentially distributed ordinates of the periodogram). Unlike frequentistic methods, the Bayesian estimation provides a probability distribution of parameters and this uncertainty needs to be accommodated in the residual analysis. We therefore analyse the posterior distribution of model residuals based on random draws from the posterior distribution (as given by the MCMC sampler). For each replicate, we calculate the Kolmogorov–Smirnov test statistic to investigate potential deviations from the assumption of exponentially distributed residuals and the Pearson correlation for autocorrelation of lag one, five and 10 to investigate if residuals are independent. This yields a PPD of P-values for each test and we enumerate the posterior probability of P < 0·05 as indicative of violations of the assumptions in the likelihood function. The PPD of P-values was based on 10 000 joint draws of an, γn from the posterior distribution.
We also provide a picture for visual comparison of the PPD of the periodogram with the observed periodograms of the analysed data. If the observed pattern deviates from the predicted for some frequencies, this would reveal deviance from the 1/f assumption. We plot the PPD (based on 10 000 posterior draws) and observed periodogram on a log–log scale, under which the 1/f PSDF is linear.
We also test the accuracy of our model in the time domain and compare predictive summary statistics of the data to the corresponding PPD of the same statistics. That is, we simulate a large number of replicates with the described model, each parameterized with a random draw from the posterior, and obtain predictive sets of time series with the inverse Fourier transform
We simulated 10 000 predictive time series of length L = 110 from the PPD of the single-group analysis, each with 10 and seven individuals, respectively (hence comparable to the data). For each simulated time series, we calculated the mean (between simulated trees) standard deviation, skewness and kurtosis. We also calculated ‘region-wide synchrony’, a measure based on cross-correlation (Bjørnstad, Ims & Lambin 1999).
We present estimates of the posterior distribution by the mean of the marginal posterior distribution, followed by the 95% central credibility interval (CCI).
The marginal posterior distribution of and γ is shown in Fig. 2. The posterior distributions are clearly separated from γ = 0, where white noise is defined (using an even flatter prior with standard deviation 100 gave almost identical results). Hence, we may conclude that the time series are evidently autocorrelated. Also, the marginal posterior distributions of γ are alike, indicating that the pattern of autocorrelation is similar between trees. This is also expressed in σγ, which models the between-tree variation and was estimated at 0·039 [0·008, 0·11] and 0·049 [0·008, 0·16] for the analysis of live and dead trees, respectively.
The ratio of posterior model probabilities R1 was estimated at 3·4 × 1016 and 1·2 × 1024 for analysis of live and dead trees, respectively. This is interpreted as very strong evidence to support the hypothesis that there is synchrony in the growth of the analysed oak trees. The analysis of between-group synchrony estimated R2 at 5·1 × 1025, hence giving very high support in favour of model M1 (both live and dead trees come from the same synchronous group) compared with M2 (live and dead trees come from different synchronous groups). This indicates that prior to the synchronized death event, the growths of trees that survived and those that died were driven by the same external factors.
Further, the marginal posteriors of β1 (modelling the trend in synchrony over different frequencies) were estimated at −0·015 [−0·029, −0·0005] and −0·023 [−0·044, −0·004] for live and dead trees, respectively. Because the distributions have most of the densities for values lower than zero, we may further conclude that there is a trend such that higher synchrony is obtained for lower frequencies. The estimates of ρ for the live tree group are illustrated in Fig. 3, which shows a clear trend with higher posterior densities of large ρi for low i.
Out of the total 17 trees considered, the Pearson test for autocorrelation at lag one, five and 10 found that only one tree of each lag (yet different trees) had more than 50% of the PPD of P-values below 0·05. The Kolmogorov–Smirnov test for deviations from exponentially distributed residuals showed that none of the trees had more than 50% of the posterior predictive probability density of P-values below 0·05. This indicates no substantial deviations from the assumptions of the likelihood function. Figure 4 further reveals a good fit between PPD and data, hence supporting the use of the 1/f model. A similar fit was found for the dead tree group and the corresponding figure is provided in the Supporting information.
All summary statistics of the observed data in the time domain were within the 95% CCI of the PPD. We here present the observed values followed by the posterior predictive 95% CCI in brackets for live and dead trees, respectively. Region-wide synchrony: 0·25 (0·09, 0·31) and 0·34 (0·15, 0·45), STD: 0·47 (0·29, 0·70) and 0·51 (0·36, 0·87), Skewness: 0·92 (0·55, 1·1) and 0·70 (0·60, 1·44), Kurtosis: 3·7 (3·1, 5·3) and 3·5 (3·2, 6·9).
The first part of our analysis focuses on the two groups separately and demonstrates how analysis of synchrony and autocorrelation in ecological datasets may reveal valuable information regarding which processes are the main drivers of the fluctuations. The periodogram analysis shows that the analysed tree growths are autocorrelated in time, with noise colour parameters for both individuals (γ) and population level () clearly separated from 0 (see Fig. 2). The highest posterior density is found for and the individual estimates (γn) exhibit very little differences between trees, indicating very similar noise colour of the growth of oak trees. The posterior estimates of γ are considered ‘pink noise’ (i.e. 0 < γ < 2). Pink noise is commonly a good model for ecological systems (Halley 1996; Vasseur & Yodzis 2004), with the colour parameter usually estimated around one for climatic data. The estimates found here are close to one, suggesting that climatic fluctuations are plausible drivers for the fluctuation in tree growth. However, as the trees are from the same area, we may not exclude the possibility of other, more local factors.
Climatic factors are expected to also cause synchronous growth, and through R1, we find strong support for synchrony in both groups. Dendrochronological studies are largely based on the assumption that individual trees react similarly to external factors and this result was therefore expected. The analysis, however, formally demonstrates how phases can reveal details of the synchronous behaviour. We further conclude that synchrony is more apparent for lower frequencies (Fig. 3 and posterior estimates of β1) indicating a higher degree of synchrony at larger time-scales. Hence, at shorter time-scales, the growth is more influenced by factors at the individual level like herbivorous insects, pathogens and individual differences in soil conditions while factors that influence the whole population (such as climatic variation) are more influential on longer time-scales. In this system, the latter are also more influential on the overall dynamics of the individual tree because of the dominance of lower frequency in the fluctuations (as given by γ).
The second part of our study shows how the analysis may be extended to compare groups, and in our example with the oak growth, we want to investigate whether the trees that died and those that did not have historically fluctuated differently. A marked difference in parameters estimating the autocorrelation pattern would suggest that one group historically have gone through longer periods of low growth rates. By comparing the posterior estimates of parameters in analyses of autocorrelation, we, however, show that these are similar and hence the groups are not likely to have varied differently in terms of their ‘colour’. Further, the comparison of model M1 (both live and dead trees come from the same synchronous group) and M2 (live and dead trees come from different synchronous groups) indicated a strong support for M1. This suggests that the two groups have historically co-fluctuated, influenced by the same global factors. Consequently, before the synchronized death occurred, there was nothing in the growth pattern of the trees that died which suggests that they would have been more sensitive to previous climatic conditions. These results indicate one of two possibilities (or both). First, the cause of synchronized death may have been something that had not occurred previously for the time of the analysed period and the trees that died were more sensitive to this. Secondly, the death of some oaks may be a combination of global external factors and more unpredictable, seemingly random factors at the individual level, such as pests or microscale variation in climatic or hydrologic factors. This, perhaps more likely explanation, is unfortunate in terms of conservation planning as nothing in the previous growth pattern suggests which trees are more sensitive and hence likely to die. However, our method provides additional tools in identifying synchrony in population dynamics. This is particularly important for conservation in fragmented landscapes with scarce important dynamic substrates such as old hollow oaks that harbour a large number of specialized species (Jansson et al. 2009).
In our model, we use the Whittle approximation for the likelihood of periodogram for a long-term memory process such as the 1/f model. Violation of this assumption may be problematic if we were attempting to estimate the spectral density of some specific frequencies or to determine if the process was ‘true’ Flicker noise. As such, the fractal properties of the 1/f model would hold for all frequencies. As pointed out by Halley et al. (2004), most ecological studies consider only one or two orders of magnitude and we should be careful when stating that the system is fractal. Vasseur & Yodzis (2004) further concluded that there often is some time-scale where the 1/f model is a good approximation whereas flattening may occur at larger time-scales. In many ecological systems, there is also some minimum and/or maximum time-scale of relevance. In our analysis, we know that the within-year growth pattern is entirely different to the between-year variation and the pattern would be cyclic rather than fractal at the yearly scale. There is also some upper relevant limit in terms of the life span of the tree. We therefore use the 1/f model to obtain an easily interpretable measure of autocorrelation based on the spectral representation. It is, however, still important to validate the model and we therefore analyse the residuals of the periodogram to see if our assumption of independent and exponentially distributed ordinates of the periodogram holds. The residual analysis revealed no apparent autocorrelation, hence supporting that the residuals appear adequately modelled as independent, and further showed that the exponential distribution is appropriate. Figure 4 further demonstrates a good fit between PPD and data, suggesting that 1/f noise is a reasonable model for fluctuations in tree growth at the considered time-scale. Insect outbreaks typically show oscillatory behaviour (Bjørnstad et al. 2002), and if oak tree growth was determined mainly by such outbreaks, we would expect deviations with a peak of the periodogram at the scale of the oscillations. No such peak is observed in Fig. 4, and this further supports the notion that abiotic factors are more important for the growth of Q. robur.
The novelty of the presented analysis lies in the combination of spectral analysis with hierarchical Bayesian modelling. Previous studies of noise colour (Halley 1996; García-Carreras & Reuman 2011) consider point estimates of the colour parameter. We have instead formulated a hierarchical Bayesian model and thereby included the uncertainty at the individual level when estimating the population-level parameter and this uncertainty is included when comparing the distributions for the two groups.
We have also introduced new phase-based measures of synchrony and demonstrated how the presence of synchrony may be evaluated by the ratio of posterior model probabilities. Previous methods have mainly averaged over pairwise measures of correlation in series (Buonaccorsi et al. 2001). To test for synchrony, or to estimate confidence intervals, these previous methods are sensitive to the degree of autocorrelation in the time series. Buonaccorsi et al. (2001) recommended tests based on residuals where the degree of autocorrelation is specifically defined by AR models. Still, as they point out, these are AR-model dependent while our method for analysing synchrony is model free with regard to autocorrelation. In the Bayesian approach, we also obtain a posterior distribution of ρ (which quantifies synchrony) and hence may present parameter estimates in the presence of uncertainty. Further, our spectral approach to time-series analysis does not merely consider synchrony as the similarity between time series at every point in time. Rather, we focus on synchrony by comparison of the whole time series. Through hierarchical Bayesian modelling, we may also address differences in synchrony at different time-scales, which in the present analysis revealed higher synchrony at larger time-scales (Fig. 3).
Several theoretical papers (Ripa & Lundberg 1996; Cuddington & Yodzis 1999; Liebhold, Koenig & Bjørnstad 2004; Lögdberg & Wennergren in press) have studied the importance of noise colour and synchrony in population dynamics. Future studies within this field need a more applied focus where empirical data are included and we argue that Bayesian inference is particularly promising when doing so. Firstly, knowledge about the specific system may be included in the parameter estimation as prior information. Secondly, rather than just considering point estimates, posterior probability distributions of parameters are obtained. Thirdly, the uncertainty of parameters may be directly included in simulation models by sampling from the PPD. Ecological theory is facing a huge challenge in dealing with the next level of complexity, namely the time series of population as a response to climate change. Theory of population dynamics has already pointed out both synchrony and noise colour as two major characteristics of time series affecting population processes. We believe that the methodology presented here (i.e. combining spectral analysis with Bayesian inference) is a promising approach to take on this challenge.
The authors would like to thank Marie Andersson who collected the data used in this study. Tom Lindström was partly funded by the Swedish Research Council (VR).