To illustrate the challenges posed by nonlinear recorders of climate, we consider an idealized model for a run-off proxy that displays a univariate and stationary but nonlinear response to ENSO-induced rainfall. Sediment-based run-off proxies are expected to feature nonlinearities for at least four reasons: sediment mobilization is a nonlinear function of flow speed ; flow speed is a nonlinear function of rainfall e.g.; rainfall is a nonlinear function of sea-surface temperature e.g.; and finally, we assume that rainfall at the location of the proxy occurs only during the warm phase of ENSO, inducing an asymmetric relationship with climate.

To characterize ENSO state, we use December–January–February averages of the NINO3.4 index (average sea-surface temperature anomaly in the region (5∘ N–5∘ S; 170∘ W–120∘ E)) from the Extended Reconstructed Sea Surface Temperature (ERSSTv3) data set . Although NINO3.4 statistics are well known to be skewed at monthly scales, the seasonal averaging results in a nearly Gaussian distribution, so that for this example non-normality in the proxy arises on account of the proxy–climate relationship rather than the underlying climate. To add stochasticity, we assume that the idealized land-based proxy is sensitive to the local expression of NINO3.4 via c′(t)=c(t)+ϵ1(t), where c(t) is the NINO3.4 index, c′(t) is the local expression of ENSO variability, and for convenience ϵ1 is a zero-mean Gaussian white-noise process with variance δ2. An additive noise ϵ2, in the proxy units, represents measurement error. Our idealized model for the run-off proxy then takes the form p(t)=αc(t)+ϵ1(t)β+ϵ2(t)c′(t)≥0ϵ2(t)otherwise, where α>0 and β>1. The exact value of β is relatively unimportant in the context of our example; in the following, we choose α=1 and β=3, though our results are qualitatively insensitive to this choice.

As we expect the error introduced by the local vs. global ENSO signal to be much larger, we neglect measurement error in the remainder for the sake of simplicity of exposition. We also ignore chronological errors, though they would be important in nature. This example may be viewed as a best-case scenario: it is the most parsimonious representation of the fact that the ENSO signal is global but experienced locally by the proxy, with deviations controlled by δ.

A proxy generated according to Eq. () selectively records El Niño events, and it does so with great sensitivity – in this respect it is an ideal proxy. Fig.  shows the NINO3.4 index and a pseudo-proxy series derived from it using Eq. (). It is qualitatively similar to run-off proxy records commonly interpreted as reflecting ENSO variability e.g.. The pseudo-proxy correctly identifies maxima in the target NINO3.4 index but exaggerates their relative magnitudes, while the minima in the target series are clipped at zero. Analysis of this skewed pseudo-proxy will thus lead to correct conclusions regarding the timing of positive excursions in the NINO3.4 index, but the nonlinearity and thresholding can potentially result in overestimates of changes in ENSO variability. Quantitatively, the ratio of the NINO3.4 sample variances within the 1950–1999 period and the 1900–1949 period is 1.25, while the distribution of variance ratios for 10 000 realizations of the pseudo-proxy, differing in their noise alone, is peaked at higher values (Fig. ), with a median of 5.25 and interquartile range p. 26 of 6.20. Indeed, the variance ratio for 86 % of the realizations of p(t) is more than 50 % higher than that calculated from the original NINO3.4 series, indicating the large impact of the nonlinear transform (Eq. ) on the second-moment properties of the proxy as compared with the climate. Note that this assessment considered positive values of the proxies only, but results are nearly identical without this restriction.

Our aim is to explore the quantitative information about climate variability that can reliably be inferred from a proxy record generated according to Eq. (), and to explore the pitfalls of failing to account for the nonlinear relationship with climate. A full review of all existing inference strategies in the presence of nonlinearity and non-normality is beyond the scope of this paper; instead, we focus on three workable approaches that may be useful in palaeoclimatology.

Functional transformation: when the transfer function between proxy and climate is known a priori, it may be leveraged to infer climate. Indeed, in the absence of noise, Eq. () expresses a one-to-one mapping between proxy values and local climate when the latter displays positive excursions. This relationship over positive values is therefore invertible and may be used for inference, for instance by specifying this relationship as a link function within a GLM . Linear regression can then be used to perform inference on climate given the proxy in a manner that accounts for nonlinearities. However, as all negative excursions are mapped to zero, the GLM framework cannot be used over such points. A related approach is to embed the mechanistic model describing how climate signals are recorded in the proxy archives within a hierarchical model. Under some circumstances, such models may be inverted using Bayes' rule to yield quantitative information about the underlying climate e.g.. This is similar in spirit to the GLM framework, as both exploit the functional relationship between proxy and climate. A limitation common to both frameworks is that the parametric from of the proxy–climate relationship must be known, which may not be the case in practice.

One important difference between the two approaches concerns the choice of dependent vs. independent variable. Under the GLM framework, there is ambiguity in the set-up of the model, as the goal is to infer climate from proxies but the proxies are best understood as the dependent variable. Bayes' rule, in contrast, naturally inverts the etiologically correct specification of climate as the dependent variable to infer climate conditional on the proxies .

We now apply those approaches to better understand past ENSO variability as depicted by nonlinear proxies.

In the absence of information on the proxy–climate link, ITS is computationally simple and non-parametric, and allows the transformed time series to be analysed using classical tools. Fig.  shows the result of applying this transform to a pseudo-proxy record generated according to Eq. (). The amplitude of positive excursions in the transformed series is now comparable to the original NINO3.4 series, and in repeated experiments the distribution of variance ratios between the early and late twentieth century (Fig. ) is centred at 1.5, much closer to the true value of 1.25, and the interquartile range is now reduced to 0.75 (compare with 5.25 and 6.20, respectively, using the raw proxy values). Similarly, the probability of overstating the true change in variance by more than 50 % is only 22 % (vs. 86 % using the raw proxy values). Similar results are obtained by inverting Eq. (), showing that application of the correct transform can effectively correct for nonlinearity.

However, while the transform adequately corrects for the cubic nonlinearity, it cannot overcome the fundamental limitation that this idealized proxy only records El Niño events, while information about La Niña  events is irretrievably lost. Proxies generated according to Eq. () thus represent a lossy transformation of the underlying climate signal, one that no amount of statistical ingenuity can ever rectify. A good probabilistic analysis should nonetheless yield reasonable estimates of positive climate excursions, and accurate error bounds for all climate excursions.

Method 1 (RAW): nonlinearity is ignored, and the proxy series p(t) is used as a predictor in a standard linear regression (an assumption implicit to many palaeoclimate studies).

The experimental sample is composed of 10 000 surrogate climate time series c(t) of length n=300 years generated according to an AR(1) process, with mean zero and a lag-1 autocorrelation of γ=0.7 to mimic the serial correlation typical of observed sea-surface temperature time series; conclusions are not sensitive to this choice. The innovation standard deviation, σ, is chosen so that the standard deviation of c(t) is unity. We add ϵ1∼N(0,δ), with δ=0.5×σ, to produce c′(t), and the idealized proxy series is then generated according to Eq. (), with α=1 and β=3.

In the RAW and ITS cases, inference is made via linear regression, where the regression model is trained on a calibration interval formed from the most recent 150 time points. To minimize erroneous inference when p=0, the regression model is fit using only proxy observations that are greater than zero. We adopt a heuristic approach to the p(t)=0 observations in the prediction interval, assigning to them the value c¯-, the calibration sample mean of c(t) calculated over time points for which p(t)=0. While somewhat unorthodox in the regression context, the divergent treatments in the p=0 and p>0 cases is a reasonable ad hoc procedure warranted by the thresholding behaviour of the proxy.

For the two regression cases, uncertainties are quantified via 95 % prediction intervals (PIs) following standard regression theory e.g.Eq. 7.22, where the variance of residuals is again computed separately for p=0 and p>0. In the Bayesian model, the posterior mean serves as the optimal prediction of c(t), and 95 % credible intervals are readily obtained from the posterior samples.

For simplicity, we assume in the Bayesian treatment that all model parameters (α,β,μ,σ,δ) are known. Doing so permits for closed-form expressions for the required posterior distributions that aid the interpretation of results. In practical applications these parameters would be inferred from the data as well, via a hierarchical model and Markov chain Monte Carlo-based inference e.g.. By constructing the Bayesian model to match the proxy-generating process, in both structure and parameter values, we give it an unfair advantage over the other methods. We use this best-case scenario as a benchmark, expected to provide the best estimates of climate given the skewed proxy measurements, as well as reliable estimates of uncertainty. Appendix B explores the role played by the prior specification of the variance of the climate process, σ2.

Results of performing inference on c(t) using each of the three approaches are plotted in Fig.  for one realization of the climate and noise processes. The raw regression overestimates positive excursions (cf. time points 110 to 130), while in contrast the amplitudes of positive excursions as inferred from the regression on the transformed proxies and using the Bayesian inversion of the mechanistic model are in reasonable agreement with c(t). The raw regression could thus exaggerate any changes in climate variability prior to the calibration period, a direct consequence of linearizing the nonlinear climate-proxy relationship.

We investigate the properties of the three inference methods, and the sensitivity of the results to the ratio δ/σ, using residual plots (Fig. ). For small noise levels (δ=0.1σ), and for p>0, there is most structure in the residual plot for the RAW method and least for the BPM, indicating that the RAW and BPM approaches provide, respectively, the poorest and best fits to the data. The curvature of the residual plot for p>0 is positive for the RAW regression, as the nonlinearity magnifies large excursions. In contrast, the residual plot for the ITS regression displays negative curvature, indicating that the transform overly compresses large excursions and overly magnifies small ones. In the least noisy case (δ=0.1σ), the Bayesian model yields near-perfect estimates for p>0, while for p=0 the distribution of residuals is similar for all three methods.

As the noise level increases (middle and bottom rows), the residual plots become less structured and the vertical spread in each case becomes comparable to that of the p=0 case. For higher noise levels, there are large positive outliers in the residuals from the RAW proxy regression, as expected since the method makes no account of the nonlinearity. In contrast, the ITS regression and Bayesian inversion yield generally smaller residuals even for high noise levels.

We further investigate the diagnostic features of the inference procedures using the 10 000-member ensemble of realizations of the climate and noise processes for the intermediate noise level δ=0.5σ (Table ). To quantify the properties of the uncertainty intervals, we first evaluate their width, considering both the p=0 and p>0 cases for each inference method (top row). All three methods show larger uncertainties in the p=0 case (around 2.7 ∘C) with small differences between methods, consistent with the lack of information offered by the proxy in this regime. When p>0, uncertainties are reduced for all methods, as expected from the information provided by the proxy. The raw proxy regression results in the widest uncertainty intervals, while those from the Bayesian inversion are narrowest.

To see how well such intervals encompass the true climate fluctuations, we calculate their actual coverage rates , to be compared with a nominal coverage rate of 95 % (Table , rows 3–4). For the two regression approaches, the uncertainties are slightly too permissive in both cases, with actual coverage rates of about 94 %. The coverage rate for the mechanistic model is correct for the p=0 case but only about 90% when p>0. In contrast to RAW and ITS, the mechanistic model does not involve two separate calibrations and solutions for the p=0 and p>0 cases. As such, the coverage rates are more appropriately assessed without division into the two cases, yielding PIs with a coverage rate of 93 with a standard error of ±2 %, compatible with the nominal rate of 95 %.

While coverage rates can always be increased by widening PIs, a useful probabilistic model should yield predictions that are both sharp and on point. A complementary view comes from the interval score Eq. 43, which balances coverage with sharpness. That is, the score rewards the narrowest intervals that encompass the true values at the nominal rate (here, 95 %) and should be as small as possible. Computed scores (Table , rows 5–6) are lowest for the BPM, whose slightly permissive coverage is more than compensated for by smaller widths; the scores are also the least variable for this method. At the opposite end, the raw regression displays the largest and most variable scores, while the ITS regression displays scores much closer to the BPM, and with comparable variability, particularly when data speak (p>0).

The bias in the mean is quite small for all methods (rows 7–8) but is slightly more variable for the two regression methods (raw and transformed) than for the Bayesian model. The bias in the Bayesian inference for p>0 results from the balance between the erfc term in Eq. () giving greater weight to more positive values and the informative prior shrinking the posterior mean towards the prior mean (Eq. ); given the particular parameters, the erfc terms has a stronger effect and there is a slight positive bias. More appropriately, assessing the Bayesian inference jointly for the p=0 and p>0 cases results in a bias that is not inconsistent with zero (0.05 with standard error 0.08). Changing the prior sharpness for the climate process has competing effects on the bias (and other diagnostics) in the p=0 and p>0 cases; these issues are investigated further in Appendix B.

Finally we summarize how closely each method estimates the variance ratio between the calibration (t>151) and validation (t≤150) intervals (Table , rows 9–10). We calculate the ratio of sample variances between the two periods for each of 10 000 inferences of c(t) using each method. We restrict our attention to the case where p>0, as when p=0 each method estimates c(t) by a constant value and the variance of the corresponding climate cannot be estimated. To remove the effect of differences in sample variances calculated over the two intervals for realizations of the actual climate c(t), we divide the ratio inferred from each pseudo-proxy by the ratio calculated from the corresponding realization of c(t).

An ideal estimator of past climate should therefore result in a distribution of normalized variance ratios that is tightly distributed about unity, and indeed we find this to be the case for regression on the transformed proxies and for the Bayesian inversion (Fig. , blue and green curves; Table ). In contrast, regression on the raw proxies results in a much broader distribution, peaked at about 0.4, with a much heavier upper tail (Fig. , cyan curve; Table ). More tellingly, regression on the raw proxies results in a normalized variance ratio in excess of 1.5 in 29 % of experiments, more than 4 times as frequently as for the other two methods (Table , last row): in other words, regression on the raw proxies ups the risk of overestimating changes in variability by a factor of 4 as compared to the other methods.

Taken together, the results of our numerical experiments lead to a number of general observations about the potential pitfalls of ignoring nonlinearity in proxies, as well as the strengths and weaknesses of potential solutions. A failure to account for strong nonlinearity in a proxy often presents itself in structured residuals (Fig. ) and incurs a high risk of overstating changes in climate variability (Table , last row). Since palaeoclimate proxies are almost never considered in isolation, but (rightfully so) often stacked against comparable records, it is necessary that all proxies be approximately linearly related before interpreting discrepancies between records. This caveat is especially applicable to multiproxy analyses, as a climate reconstruction based on linear regression that includes such nonlinear proxy predictors would inherit the same potential pitfalls, though they may be attenuated by the presence of other sources of information.

Applying ITS to the nonlinear proxy record prior to regression analysis results in a much better fit according to the residual plots, as well as more faithful estimates of changes in variance. Such a transformed proxy series will thus be less likely to lead to spurious conclusions about past climate variability if included in a multiproxy reconstruction or a proxy intercomparison.

An important shortcoming of the regression methods used here is that inference is generally limited to a best-estimate of past climates and an uncertainty interval, whereas the Bayesian framework naturally provides the full distribution of climate conditional on the proxy observations for further discussion see. However, we note that the excellent performance of the Bayesian inversion in all numerical experiments performed here is to be expected given that the inference model is constructed using the correct mechanistic model and parameter values. In practical applications, the parametric form of the mechanistic model would likely not be an exact representation of the data-generating mechanism, while the parameters themselves would need to be estimated from the data. This can be computationally demanding when no closed-form solution exists for the posterior distribution of model parameters (as in this case) and would naturally broaden the posterior distributions, i.e. lead to more uncertainties about the reconstructed climate.

We now investigate how inferences made from the iconic Laguna Pallcacocha record of river run-off from the southern Ecuadorian Andes are affected by different considerations of nonlinearity. The primary variable of interest here is the red colour intensity of the sediment, which is controlled by sediment delivery to the lake. Episodes of alluvial deposition are hypothesized to occur predominantly during El Niño events , similar to our idealized proxy (Sect. 2). Our intent here is not to dispute qualitative conclusions made from the Laguna Pallcacocha record – they turn out to be robust across the analyses we consider – but to show how nonlinearity may affect the quantitative inference made from such a proxy, and we view the analysis primarily as an illustrative example.

The time series of Pallcacocha red colour intensity displays pronounced positive skew (Fig. a), closely following a generalized extreme value distribution (GEV, Fig. b). In contrast, the distribution of December–January–February averages of NINO3 values is fairly close to a Gaussian, so using the Pallcacocha red colour record to infer changes in ENSO variability requires inferring a normally distributed climate variable from a strongly non-normal proxy time series – as per the experiments in Sects. 2 and 3. Fitting a mechanistic model would necessitate determining both its structural form and parameters, so for simplicity we explore the consequences of an empirical transform (ITS). By design, applying the transform to the empirical distribution produces a nearly Gaussian series (Fig. d).

We investigate how the transform affects the time series of sample variances computed on disjoint, 100-year segments (Fig. ). While the 100-year variance time series calculated from the raw series suggests an abrupt increase in ENSO-related variance at about 5000 years BP (grey dashed line), the series calculated from transformed values displays a more gradual increase. To establish if either or both series record significant increases in variance around 5000 years BP, we compute the variance ratio between two segments delineated by the grey line (centred at 3150 BCE), with lengths that vary between 200 and 3600 years. A window length of 200 years, for instance, means that we compare variances over the period 3350–3151 and 3150–2951 BCE. As the statistic of interest is a variance ratio, results are unchanged under linear transformation of the series: while accounting for the nonlinearity is necessary for meaningful inference, calibrating to climate units is not.

On account of the non-normal and serially correlated nature of both series, a standard F test for the significance of the observed variance ratio is not appropriate. Instead, we assess the significance of the variance ratio for each series via a block bootstrap resampling plan , using a block length of 25 years (Table ). That is, we randomly sample with replacement the original time series in 25-year blocks, compare variances before and after the year 3150, and do so N=2000 times to obtain a null distribution, smoothed by kernel averaging e.g.Chap 4. The P value for each observed ratio (the probability of observing a ratio at least as large as that observed from chance alone) is then estimated from the bootstrap-derived null distributions. The results (Table ) show two salient features: firstly, the variance ratio is generally larger with the raw than with the transformed series, as would be expected from Sects. 2 and 3. Secondly, the variance ratio estimated from the raw series is deemed significant at the 5 % level for all segment lengths, while the transformed series requires segment lengths of at least 1000 years to make this conclusion. Hence, with the raw series one would detect a significant shift at the boundary in a matter of 200 years, while when using the power-transformed series, this transition would be seen to occur much more gradually, taking at least 1000 years to become manifest. We note that the change in variance is largely driven by the record's accumulation rate , so we refrain from interpreting this aspect of the record.

Following , we next apply Morlet wavelet analysis , modified to account for energy conservation , to both the raw and transformed series. Results are qualitatively similar in both cases (Fig. ): interannual variability is muted prior to the mid-Holocene, while millennial and centennial variability is relatively persistent. However, the two wavelet transforms differ in detail, in particular regarding which “islands” of multidecadal power are significant with respect to a red noise benchmark . In the latter 5000 years of the record, high-frequency variability appears more consistently significant with the empirical transform than without, indicating that the mid-Holocene shift for high-frequency variance is more consistently significant under the transform, even though the shift in total variance appears less substantial.

Our analysis does not contradict the qualitative conclusions of the original study by . Indeed, the transformed series also suggests that ENSO events “become more frequent over the Holocene until about 1200 years ago, and then decline towards the present” . Wavelet analysis accounting for energy conservation also suggests a statistically significant modulation in the 2000-year range, though it is nearly equal in amplitude to a ∼ 500 yr broadband peak and lies mostly within the cone of influence. Nonetheless, results of the statistical analyses are markedly changed from a quantitative standpoint when nonlinearity is recognized and the proxy series transformed to approximate normality. Applying classical estimators to the raw series may lead to erroneous conclusions about the magnitude of changes in the variability of the underlying climate, as well as their localization in frequency space. The empirical transform counterbalances the nonlinearities inherent to run-off proxies such as the Pallcacocha red colour intensity record, and the resulting transformed series is nearly normal and more likely to have an approximately linear relationship with climate (cf. the numerical examples of Sects. 2 and 3). One is therefore able to interpret changes in variability estimated from the transformed proxy record as indicating changes in climate variability (cf. Fig. ).

Recently, used varved thickness at Lake Challa (equatorial east Africa) to retrace changes in ENSO activity since the Last Glacial Maximum. The proxy is interpreted as follows: El Niño (La Niña) events are associated with wetter (drier) conditions in east Africa and decreased (increased) surface wind speeds, which drive turbulent mixing in the lake, thereby bringing nutrients to the surface and boosting primary productivity. Thicker varves are therefore indicative of La Niña conditions, and thinner varves of El Niño conditions, resulting in a large anti-correlation (r=-0.48) between varve thickness and NINO3.4 SST . The raw varve thickness data are plotted in Fig. a for the past 3000 years. The histogram is highly skewed and closely fits a log-normal distribution with parameters (μ,σ)=(0.33,0.31) (Fig. b). The transformed series (Fig. c) closely follows a standard normal distribution (Fig. d).

As the raw Lake Challa series has been used as a proxy for ENSO activity , it is worth asking how much it resembles the Lake Pallacacocha red colour intensity. Fig.  plots the two time series, as well as their standard deviations on sliding 40 yr windows, before and after transforming each record to normality. While Fig. c shows a fair amount of structure in each time series (peaks near 500 BCE and 2000 CE in Lake Challa varve thickness variability, quasi-periodic modulations of red colour intensity variability with an approximate recurrence time of 300 years), the transformed series (Fig. d) show different features: variability in red colour intensity now displays more pronounced late Holocene excursions than its untransformed counterpart, while variations in varve thickness now appear to recur at centennial, rather than millennial, scales.

This illustrative comparison is necessarily limited, as the proxies are located at either end of the Indo-Pacific system and are sensitive to different aspects of tropical Pacific SST (Challa responds more to central Pacific anomalies; Pallcacocha responds more to variability along the Peruvian coast, though see for a non-climatic interpretation), which are not simply related to each other. In addition, the Pallcacocha sensitivity to SST is highly nonlinear, as noted before, as is the Challa relationship: note that “the varve thickness record is particularly sensitive to La Niña  conditions and shows some evidence for saturation during El Niño years”. The main message is that the visual impression changes with the application of the transforms. Given that each proxy is non-normal due to a nonlinear relationships with normally distributed SSTs, there is a strong rationale for comparing the transformed rather than the untransformed series.