The colour of environmental fluctuations associated with terrestrial animal population dynamics

Aim : The temporal structure (colour) of environmental variation influences population fluctuations, extinction risk and community stability. However, it is unclear whether environmental covariates linked to population fluctuations are distinguishable from a purely random process (white noise). We aimed to estimate colour coefficients and relative support for three models commonly representing coloured stochastic pro‐ cesses, in environmental series linked to terrestrial animal population fluctuations. Eurasia.


| INTRODUC TION
A major issue in population biology concerns how the colour of environmental variation interacts with population dynamics to drive observed patterns of population fluctuations (e.g., Dillon et al., 2016;Halley, 1996;Ruokolainen, Lindén, Kaitala, & Fowler, 2009;Steele, 1985). Here, colour refers to the temporal and/or spatial structure of variation that describes system dynamics. Analogous to the visible light spectrum, low frequencies (slow fluctuations) dominate in red environments, high frequencies (rapid fluctuations) dominate in blue environments, while no frequencies are dominant in white, purely random environments (Halley, 1996). Environmental colour is predicted to interact with population demographic processes, such as survival, reproductive and developmental rates, stage structure or age structure, the intensity of intra-and interspecific competition and the shape and strength of density dependence (Ruokolainen, Lindén, et al., 2009). These interactions will modify the size of and correlation between the environment and population fluctuations, affecting ecological factors such as the extinction risk of single species populations (Cuddington & Yodzis, 1999;Pimm & Redfearn, 1988; but see Fowler & Ruokolainen, 2013a) and in multispecies communities and spatially structured systems (Fowler & Ruokolainen, 2013b;Gonzalez & de Feo, 2007;Gudmundson, Eklöf, & Wennergren, 2015;Lögdberg & Wennergren, 2012;Ruokolainen, Ranta, Kaitala, & Fowler, 2009).
The interaction between environmental colour and population demographic processes will also modify species interactions (e.g., prey-predator, competition, host-parasites) and ecosystem stability and function (Fowler & Ruokolainen, 2013a, 2013bGreenman & Benton, 2005;Gudmundson et al., 2015;Lögdberg & Wennergren, 2012;Ripa & Heino, 1999;Ripa & Ives, 2003;Roughgarden, 1975;Ruokolainen & Fowler, 2008;Ruokolainen, Fowler, & Ranta, 2007;Ruokolainen, Ranta, et al., 2009). Furthermore, the response of species to coloured environmental variation depends on the time-scale considered, the extent to which it coincides with the life cycle of an organism and the particular life-history traits that are affected by environmental change (Heino & Sabadell, 2003). Additionally, environmental variables in reddened environments imply consecutive periods (days, weeks, years) of favourable or unfavourable conditions, which may decrease or increase extinction risk, respectively (Schwager, Johst, & Jeltsch, 2006). Populations with undercompensating growth tend to respond slowly to environmental changes; therefore, fluctuations in density are amplified under red environmental variation, which increases their extinction risk (Roughgarden, 1975). Large-scale changes in climate variables associated with the El Niño Southern Oscillation (SO) or North Atlantic Oscillation (NAO) indices, which may encapsulate overall fluctuations in local environmental fluctuations (Post & Forchhammer, 2002), also impact on the dynamics of populations (Coulson et al., 2001;Hallett et al., 2004).
One of the main challenges when studying the impact of coloured environmental variation on population dynamics is the accuracy of colour coefficients estimated from environmental time series. Autocorrelated processes have been observed and studied in many different fields, and the terminologies and methodologies used for their description and detection vary across disciplines. For example, within the geophysical and climatological sciences, variability correlated at all time-scales is often referred to as long-term persistence (e.g., Halley, 2009;Rybski et al., 2006) or the 'Hurst effect' (Hurst, 1951;reviewed by O'Connell et al., 2016), where strong long-term persistence characterizes a domination of low-frequency (slow or reddened) fluctuations. In ecology, two methods are commonly used to generate and/or statistically model coloured stochastic processes; autoregressive [AR(k), where k gives the order of the process, most frequently performed in the time domain] or sinusoidal [1/f β , where β is the spectral exponent, describing the linear relationship between log(signal power) and log(frequency) in the frequency domain, referred to hereafter as 1/f processes]. Analysis in the time domain investigates correlations between values of the series at different times, whereas analysis in the frequency domain is a standardized method for decomposing the overall variance of a time series into contributions from different frequencies (e.g., Chatfield, 1996;Dillon et al., 2016). These processes differ in some respects; for example, the variance grows continuously with observation time in a 1/f process, whereas variance growth will eventually cease in an autoregressive process (Halley, 2005). The methodology used and the time-scales at which observations of the focal variable are made will impact our perception of the true variability of, for example, climate covariates (Király & Jánosi, 2002;Ruokolainen, Lindén, et al., 2009). For example, in a white noise process, the total variance is independent of observation sampling frequency but scales linearly with the amount of sample aggregation, whereas in a coloured process the total variance depends on time-series length, sample timing and aggregation (Halley, 2007). In 1/f models, low sampling frequency can result in a systematic 'whitening' (or 'aliasing') of the power spectrum, which yields estimates closer to zero compared with the true colour (Kirchner, 2005). However, aliasing can be offset by sampling aggregation (e.g., seasonal averages calculated from daily samples; Halley, 2007Halley, , 2009Kirchner, 2005). The detection of processes other than white noise (avoiding type II statistical errors) is especially difficult when time series are short and the true colour is close to zero (Gerrodette et al., 1987;Miramontes & Rohani, 2002;Pimm & Redfearn, 1988), which has led to the development of several alternative approaches for overcoming such difficulties  (Cannon, Percival, Caccia, Raymond, & Bassingthwaighte, 1997;Halley, 2009;Miramontes & Rohani, 2002;Pimm & Redfearn, 1988;Rohani, Miramontes, & Keeling, 2004).
Inaccuracies in estimation and differences across approaches can potentially filter through to generate statistical artefacts and important differences in population dynamics, affecting, for example, extinction risk forecasts (Halley & Kunin, 1999). Given the short time-scales typically available for ecological time-series data, two important questions in population biology are therefore: (a) what model form (e.g., white noise, autoregressive or 1/f process) best characterizes, and (b) what statistical power do we have for correctly detecting autocorrelation in the environmental variables that drive natural population fluctuations? Identifying the role of coloured environmental noise in driving population dynamics, moderating species interactions and shaping community structure is especially important in the context of climate change and the future challenges faced by biodiversity. Climate warming is expected to change the spatio-temporal structure of natural environmental fluctuations (Wigley, Smith, & Santer, 1998). For example, recent evidence suggests that El Niño years are increasing in frequency (Power, Delage, Chung, Kociuba, & Keay, 2013; i.e., becoming less red). In contrast, Northern Hemisphere weather systems are expected to start changing more slowly (i.e., becoming redder) because of amplified arctic warming causing weakened jet streams (Hurrel & Loon, 1997;Mann et al., 2017).
Previous efforts have been made to estimate the colour parameters of natural environmental variables (e.g., García-Carreras & Reuman, 2011;Vasseur & Yodzis, 2004). To date, however, no study has attempted to do this for multiple environmental variables that have been linked explicitly and robustly to natural animal population time series. Here, using abiotic environmental variables that have been appropriately coupled to population fluctuations across a wide range of terrestrial animal taxa and geographical locations (Knape & de Valpine, 2011), we have evaluated the statistical support for different colours and the underlying processes in the natural environmental variables associated with animal populations. This represents a useful step forward given (a) the lack of statistical support for many purported power-law relationships, such as those investigated here (Stumpf & Porter, 2012), and (b) the predicted sensitivity of population responses to the colour of environmental noise and the corresponding wide-ranging effects on species and their interactions with other species. Our initial aim was to investigate whether animal populations respond to a broad range of abiotic environmental drivers or to some subset of possible variables. We then assessed the level of statistical support and power in natural environmental time series for three models commonly used to characterize or simulate (coloured) stochastic processes: white noise (a purely random process without any temporal structure), AR(1) and 1/f models. Support for these methods was assessed by analysis in both the frequency and time domains (Chatfield, 1996;Dillon et al., 2016) to allow meaningful comparison across the methods. We also evaluated the multi-segmenting method (MSM) of Miramontes and Rohani (2002), especially designed for estimating 1/f colour coefficients from short time series. We recorded the frequency of different environmental coefficient estimates (including their confidence intervals) to (a) establish the evidence for different coloured environmental covariates associated with population fluctuations in different animal taxa, and (b) assess whether results are characterized by lack of statistical power using the relatively short (≤ 50-year) length of available animal population time series through comparison with longer (101-year) versions of the same associated environmental series.

| Data sources
We investigated abiotic environmental variables that were ex-  (Mitchell & Jones, 2005). These environmental variables provided the main data source for our present analysis, and details of how corresponding population and environmental time series were initially filtered are fully described by Knape and de Valpine (2011). We further filtered the available environmental series for analysis by selecting only those covariates that were selected in the most parsimonious model describing population fluctuations by Knape and de Valpine (2011) according to Akaike information criterion corrected for small sample size (AIC C ; 371 series out of 492) and had ≥ 10 consecutive (annual) data points.
We excluded 10 further environmental time series that were not suitable for spectral analysis because they showed identical values for all except one data point. We also excluded 135 repeated series; if the same environmental variable was associated with the dynamics of several species, overlapping in time, only the part of the environmental variable with the longest corresponding population time series was used.
We performed the analyses both on the time interval for which population data were available in the matched GPDD natural population time series [following Knape & de Valpine (2011), hereafter referred to as the 'short' series] and on the entire series from the CRU TS v.2.1 database (years 1901-2002, the 'long' series). The latter approach increased statistical power under the assumption that the same covariates also affected the corresponding populations before (or after) the populations had been surveyed. Given that winter series were composed of environmental data from December, January and February, the final year of the long series missed the last 2 months and was excluded. For consistency, we also removed the last year from the other environmental covariate types. Finally, 21 series were removed where no match between the original, short series and the long CRU series could be found. This left us with 163 unique short (≤ 50-year) and 160 unique long (101-year) environmental time series for our main analysis (Figure 1; Supporting Information Appendix S1 Table S1). All environmental time series were standardized to have zero mean and unit variance, and detrended (see 'Time series analysis and model comparison').
The terrestrial animal population time series were composed of annual abundance estimate data from birds, mammals and insects. We

| Time series analysis and model comparison
We compared three statistical models commonly used to generate and/or characterize stochastic coloured time series: white noise, AR(1) and 1/f models. The white noise model (i.e., a random process with no temporal structure) is a special case of both latter models, but is described using one parameter less. To evaluate the strength of support for these models, we analysed data in the frequency domain, where we fitted the theoretical power spectra to the empirical periodograms of the detrended time series (detrending removed the least-squares straight-line fit with respect to year from the environmental time series, then used the residuals for further analysis). Given that detrending has the potential to 'whiten' power spectra, we also performed the main analysis using non-detrended data (see Supporting Information Appendix S2). This differs from the normal approach for AR(1) models, where parameters are generally estimated in the temporal domain. However, we first estimated the autocorrelation coefficients in the frequency domain to ensure that F I G U R E 1 Characteristics of the environmental time series used in the study. (a) The geographical positions in U.S.A. and Canada (left), Eurasia (middle) and U.K. and Ireland (zoomed in, right) of the Global Population Dynamics Database (GPDD) natural population time-series data, whose fluctuations are associated with the geo-spatially linked environmental covariates analysed here (a small jitter is added to allow overlapping locations to be shown). (b) Three example environmental time series, coloured based on their associated colour parameters, estimated from the frequency domain (spectral exponent, β; autocorrelation coefficient, ρ f (± 95% CIs); middle long series is defined as 'blue' based on β alone; black series denote white noise). Upper, middle and lower environmental series show summer frost day frequency, summer precipitation and winter minimum temperature, associated with population dynamics of Euphydryas editha (bay checkerspot butterfly), Contopus virens (eastern wood-pewee) and Libellula quadrimaculata (four-spotted chaser), respectively (thick part of lines shows dates corresponding to available animal population data).

Length of time series
Frequency the competing models compared the same data. The empirical periodograms (power spectra) were obtained with a standard fast Fourier transform algorithm using the 'fft' function in MATLAB (v.2014a), where the number of frequencies evaluated equals ceil (n + 1)∕2 − 1 (ceil rounds a number to the next larger integer).
The theoretical power spectrum of the AR(1) model is (Chatfield, 1996, p. 100): where ρ is the autocorrelation parameter and σ is the standard deviation of the (environmental) time series. For 1/f (short for 1/f β ) processes, the corresponding function is: where β is the spectral exponent (slope) and α is a constant regulating the intercept of the spectrum and thus the strength of fluctuations.
The theoretical power spectral densities were fitted to the periodograms on the decibel scale (any logarithmic scale would be valid and will not affect our conclusions), using a least-squares approach.
The decibel transformation is a special case of logarithmic transformation, namely: That is, the power was decibel transformed (the dB operator is scaled logarithmically) before model fitting, making the multiplicative model additive.
To evaluate the white noise model, we fitted a linear regression with an intercept only: where f i is the empirical periodogram and ε f are normal, independent, identically distributed residuals.
For the AR(1) process, we fitted the model: solving for the maximum likelihood (i.e., least-squares) solution for parameter ρ, using the function 'fminbnd' in MATLAB (v.2014a) for one-dimensional nonlinear minimization. We used ordinary least squares to fit the remaining parameters in each iteration. The intercept α 2 then corresponds to dB(σ 2 ) (see Equation 1).
The power spectrum for the 1/f process is linear on the log-log scale and can be fitted as: where the intercept α 3 corresponds to 10α (see Equation 2).
We recorded parameter estimates from the frequency domain for the spectral exponent (β) and autocorrelation coefficient (ρ f ) for each time series, and symmetric 95% confidence intervals (CIs) for each estimate, based on the t-distribution. The statistical significance of parameter estimates was determined by assessing whether the range of the 95% CIs of the estimate spanned zero. Furthermore, we estimated the autocorrelation coefficient (± 95% CIs) from the temporal (time) domain (ρ T ), in common with the majority of previous work. This allowed us to determine whether the loss of power associated with estimating autocorrelation in the frequency domain (where ca. n/2 data points are used in the coefficient estimation) had an important influence on our results and conclusions. We also compared colour coefficients with a null hypothesis of intermediately reddened colour, generally referred to as pink noise in the frequency domain (e.g., Halley, 1996), here considered to represent ecologically non-trivial temporal structure on the annual scale. Given that the true underlying process for environmental fluctuations is unknown, this was done by checking whether CIs of the estimates spanned β = −1 for the 1/f models and ρ f = ρ T = 0.7 for the AR(1) models.
To compare relative model performance in the frequency domain, we recorded AIC C values from the 1/f, AR(1) and white noise models. Using AIC C values, we calculated model weights (w i ) and evidence ratios (E i ) as follows: and where ΔAIC Ci = AIC Ci -AIC Cmin , and R is the total number of models being compared (Anderson, 2008).
Finally, given that the detection of colour in short time series is difficult, we also evaluated the MSM of Miramontes and Rohani (2002), suggested to improve accuracy of 1/f colour estimates from time series as short as 47 time points (see Supporting Information Appendix S3).

| Power analysis for colour estimates
The power for detecting trends in, for example, time-series data, using least-squares linear regression, can be derived from the relationship between the slope, b (corresponding here to either ρ or β), the number of data points, n, the variance of the residuals, σ 2 res , and of the independent variable, σ 2 x , of the regression (Gerrodette, 1987):  1 -B) is then the probability of avoiding type II errors (correctly rejecting H 0 ).
Using Equation 9 and the normal cumulative density function, Φ, power (1 -B) can be calculated as a function of b, n, σ 2 x , σ 2 res and z A/2 (A): To evaluate our power to conclude correctly that colour coefficients (b in Equations 9 and 10) from regressions failing to reject our null hypothesis are not significantly different from zero, we performed a retrospective power sensitivity analysis (Gerrodette, 1987;Peterman, 1990;Thomas, 1997), varying b from zero to two in increments of 0.01. By rearranging Equation 9 we also evaluated the smallest detectable colour and the time-series length needed to recognize a specific colour, for a given level of power. The analyses were performed using the observed residual and independent variable variances from the linear least-squares regressions in the frequency (1/f; Equation 6) and time domains, using data from the short and long environmental time series, with A = 0.05.
Our power analysis is based on standard least-squares linear regression (Gerrodette, 1987). Estimates for AR models have a different sampling distribution, and the power formulas (Equations 9 and 10) are not strictly correct when |ρ T | approaches one and the AR (1) process is no longer stationary. We therefore performed a simulation-based power analysis using long artificial time series of known colour to confirm our results. Given that the AR(1) process in the frequency domain (Equation 5) is nonlinear, this was not included at all in the power analysis. were bird populations, 69 (31%) insects and 29 (13%) were mammals, respectively (Figure 2b). Given the potential for taxonomic selection bias (over-representation of some species/taxa) present in the original GPDD database, we did not compare our results with the underlying percentages from the GPDD or CRU data sets.

| Temporal analysis of environmental time series
The distribution of (a) environmental variable types, associated with (b) fluctuations in different terrestrial animal population taxa. All environmental variables related to temperature [Temp (mean, maximum and minimum seasonal averages)] were pooled because they were based on the same temperature data (Mitchell & Jones, 2005). Precip and Frost denote precipitation and frost day frequency, respectively. There were n = 224 unique population time series associated with both (a)  Class Environmental variable Count 0.053 ± 0.052; μ(ρ f L ) = 0.047 ± 0.023; μ(ρ T S ) = 0.028 ± 0.036 and μ(ρ T L ) = 0.102 ± 0.020.
Specifically, when colour exponents were estimated in the frequency domain, a white noise model was considered the best and most parsimonious fit to the short data series in 151 (93%) cases, while 1/f and AR(1) models were considered best in seven (4%) and five (3%) of cases, respectively (Figures 3 and 4). Only two (1%) of the parameter estimates were found to differ significantly from zero under both 1/f and AR(1) models. These general patterns are reflected in the longer time series; 106 (66%) were best described by white noise, 33 (21%) by 1/f and 21 (13%) by AR(1) models ( Figures   3 and 4), with 11 (7%) of the longer time series differing significantly from zero under both 1/f and AR(1) models. Of these, the majority (10) were reddened, and only one was blue. We also performed this analysis without detrending the weather data, to ensure our analyses of colour coefficients were not biased by our method of detrending the time series. We found a weak whitening effect attributable to our detrending approach in our main analysis (Supporting Information Appendix S2, Figure S2.1), more so in the short series compared with the long series. The means of the estimated colour coefficients (± 95% CIs) using the un-detrended (raw) data were  (Figure 4c; Supporting Information Appendix S1, Figure S1.1).
The AIC C weights consistently supported white noise models in the shorter time series, with an increase in relative likelihood for both 1/f and AR(1) models describing the longer time series (Figure 4c; Supporting Information Appendix S1, Figure S1.1).
We found that larger fractions of the coloured environmental covariates were associated with mammals, whereas smaller fractions were associated with birds, compared with the white covariates (Supporting Information Appendix S1, Figure S1.3). Averaged over all models, 20 (14%) and seven (32%) cases of the long environmental time series characterized as white or coloured, respectively, were associated with population time series of mammals, 88 (64%) and 10 (46%) cases, respectively, with birds, and 30 (22%) and five (22%) cases, respectively, with insects (Supporting Information Appendix S1, Figure S1.3). Given that sample sizes for coloured fractions were very low (Supporting Information Appendix S1, Figure S1.3), we did not test for significant differences between fractions.

| Power analysis for colour estimates
The statistical power to identify a coloured signal from white noise correctly was, as expected, higher for the long time series compared with the short, and in the time domain compared with the frequency domain ( Figure 5a). Mean power values (± 95% CIs) for pink environments (|β| = 1, ρ T = 0.7) were 0.575 ± 0.041 and 0.865 ± 0.016 (short series), and 0.995 ± 0.002 and 1.000 ± 0.000 (long series), respectively.
Consequently, the smallest detectable colour is lowest in the time domain using the long time series and highest in the frequency domain using the short series ( Figure 5b). Moreover, the time-series length required to detect colour correctly increases rapidly for colour close to white noise (Supporting Information Appendix S1, Figure S1.4). Correct detection of, for example, pink noise with power 0.8 would require at least 47 ± 6 data points in the frequency domain (β = 1) or 16 ± 1 points in the time domain (ρ T = 0.7) based on the short series, whereas correct identification of a signal with a colour exponent |β| = 0.1 with power 0.8 would require at least 4,637 ± 583 points in the frequency domain.

| DISCUSSION
We have shown that the majority (93%) of environmental variables that have previously been linked to terrestrial animal population fluctuations do not appear to show any recognizable temporal structure (colour), when estimated over a maximum of 50 years.
However, there is considerable uncertainty associated with the colour estimates for these short series, indicated by wide confidence intervals (Supporting Information Appendix S1, Table S1).
Our power analysis ( environmental time series could also be why we found such small differences between the 1/f and the AR models (Halley, 2005) any cycles with a period shorter than 1 year are therefore not present in the data. Organisms with shorter life spans, for example, having multiple generations within a single year, might therefore show different results. In the case of red environmental noise, extinction risk decreases in annually reproducing species, but increases for semelparous and perennial species (Heino & Sabadell, 2003). The long environmental time series analysed here ran from 1901 to 2002, and as longer time series (with a higher temporal resolution) become available, we recommend re-analysis based on the approach used here.
Power law relationships are often proposed to describe natural patterns, although few of them are backed up statistically (Stumpf & Porter, 2012 (2011) suggested that there was a correlation between animal population and environmental spectral exponents, also using data from the GPDD. However, a correlation analysis between the colour parameter of an environmental signal and that of raw population sizes ignores density-dependent feedback mechanisms and uncertainty in parameter estimates, and can easily generate spurious correlations (Royama, 1981). We avoided those problems here using only the environmental covariates identified as meaningful components in population models by Knape and de Valpine (2011). Although part of this coupling identified may be weak owing to possible model overfitting (Knape & de Valpine, 2011), it is the only study we are aware of in which a robust statistical model selection framework has been used to link empirical population time-series fluctuations with envi- found small effects of autocorrelated environments on population dynamics and concluded that using white noise approximations is often appropriate. This is also argued by van de Pol et al. (2011), who found that the extinction risk of a shorebird population was largely insensitive to noise colour because of the poor tracking ability of demographic rates on the colour of the environment. None of the three studies above analysed the temporal structure of environmental variables per se. However, taken together, it is clear that a new perspective on the role of coloured environments is starting to emerge.
Given that many ecological factors, such as species extinction risk and ecosystem function and stability, depend on the colour of the environment that populations are exposed to (e.g., Gudmundson et al., 2015;Ripa & Ives, 2003;Ruokolainen, Lindén, et al., 2009), incorporating empirically supported noise processes into predictive models of population and community dynamics is crucial for their reliability. Furthermore, when non-white colours have been detected reliably in environmental series, we can ask whether this coloured noise is ecologically relevant. This will depend on, for example, the demographic details of the population being affected, such as the intrinsic growth rate and shape of density dependence (Ruokolainen, Ranta, et al., 2009;Fowler & Ruokolainen, 2013b), but it is largely still an open question of considerable ecological interest.
Our results have implications for understanding the temporal structure of environmental variation driving terrestrial animal population dynamics. Temperature was shown to be the most important weather variable associated with population dynamics, but our data sets were confined to the Northern Hemisphere with strong seasonal effects, as also evidenced by the important role of frost days. Environmental time series from a wider range of geographical locations might reveal that other variables also have an important impact. Our main analysis did not include long-term environmental change (owing to our detrending approach) or its impact on species in marine environments. In contrast to the terrestrial environment, marine environmental change is largely red because of the inertia in large bodies of water to rapid and/or dramatic fluctuations (Steele, 1985;Vasseur & Yodzis, 2004). For both environments, however, there is increasing concern about how climate change will alter the colour of environmental noise and the corresponding impact this will have on the dynamics of species.
Recent evidence suggests that the colour of some weather variables is becoming both more blue (e.g., El Niño years increasing in frequency; Power et al., 2013) and red (e.g., Northern Hemisphere weather systems start changing more slowly; Hurrel & Loon, 1997; Mann et al., 2017). Such changes could have profound implications for biodiversity across the globe.
Based on the results presented here, we propose that for time spans typical for ecological time series, white noise or weakly reddened processes often best describe the annually measured abiotic environmental variables that are associated with fluctuations in many terrestrial animal populations, and analyses of coloured processes are best performed in the time domain to maximize confidence in parameter estimates. Given that population trajectories themselves are often autocorrelated, this implies that intrinsic and extrinsic biotic factors such as density dependence, age structure and species interactions are the most likely cause for such fluctuations (see e.g., Akçakaya, Halley, & Inchausti, 2003;Ferguson et al., 2016).
However, more effort is needed to collect coupled population and environmental data over longer time spans to ensure that the importance of coloured environmental processes is not underestimated.