Inferring climate from palaeodata frequently assumes a direct, linear relationship between the two, which is seldom met in practice. Here we simulate an idealized proxy characterized by a nonlinear, thresholded relationship with surface temperature, and we demonstrate the pitfalls of ignoring nonlinearities in the proxy–climate relationship. We explore three approaches to using this idealized proxy to infer past climate: (i) methods commonly used in the palaeoclimate literature, without consideration of nonlinearities; (ii) the same methods, after empirically transforming the data to normality to account for nonlinearities; and (iii) using a Bayesian model to invert the mechanistic relationship between the climate and the proxy. We find that neglecting nonlinearity often exaggerates changes in climate variability between different time intervals and leads to reconstructions with poorly quantified uncertainties. In contrast, explicit recognition of the nonlinear relationship, using either a mechanistic model or an empirical transform, yields significantly better estimates of past climate variations, with more accurate uncertainty quantification. We apply these insights to two palaeoclimate settings. Accounting for nonlinearities in the classical sedimentary record from Laguna Pallcacocha leads to quantitative departures from the results of the original study, and it markedly affects the detection of variance changes over time. A comparison with the Lake Challa record, also a nonlinear proxy for El Niño–Southern Oscillation, illustrates how inter-proxy comparisons may be altered when accounting for nonlinearity. The results hold implications for how univariate, nonlinear recorders of normally distributed climate variables are interpreted, compared to other proxy records, and incorporated into multiproxy reconstructions.

A principal goal of palaeoclimatology is to
infer climate information from geochemical, physical, or lithological signals
embedded in various proxy archives. Implicit in most analyses of palaeoclimate
records is the assumption that the observations are linearly related to the
target climate quantity, so that traditional calibration approaches

The climate system, however, is rife with nonlinearities, as are the
processes translating climate signals into proxy records

Even when the target climate quantity is well approximated by a normal distribution, nonlinearities often manifest themselves as non-normality in the proxy distribution. This is because a normally distributed proxy is only expected if the proxy is a linear recorder of a normally distributed climate variable, as the linear transform of a Gaussian random vector is also Gaussian. Temperature fluctuations, especially if averaged over a month or more, typically obey normal statistics, so linear temperature proxies tend to obey normal statistics as well. In contrast, tropical run-off proxies, for instance, are nonlinearly related to land precipitation, which is heavily skewed (thus non-normal) and only indirectly linked to sea-surface temperature (SST). Such records are not directly amenable to analysis using common techniques assuming linearity and normality (e.g. spectral analysis, principal component analysis, least-squares regression methods, correlation analysis, and parametric hypothesis testing).

One strategy to overcome this difficulty is to use generalized linear models

This article draws attention to the pitfalls of ignoring nonlinearity in the
proxy–climate relationship and explores a number of approaches to using
nonlinear proxies to infer climate variability when the underlying climate
obeys normal statistics. Our objective is not to consider all of the vagaries
of fitting a straight line through noisy data, including calibration vs.
regression, measurement errors in predictor variables, and various
regularization techniques. For in-depth discussions of these issues, relevant
and complementary to the problem at hand, see

Section 2 presents a simple toy model for a nonlinear proxy, used to illustrate the effects of nonlinearity on inferences about climate variability. Section 3 outlines various practical solutions to the problem, which we then apply to the Laguna Pallcacocha and Lake Challa records in Sect. 4. Section 5 provides discussion and concluding remarks.

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

To characterize ENSO state, we use December–January–February averages of the
NINO3.4 index (average sea-surface temperature anomaly in the region
(5

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

A proxy generated according to Eq. (

A nonlinear recorder of ENSO activity. Solid red:
December–January–February average of the NINO3.4 index from the ERSSTv3
data set

Our aim is to explore the quantitative information about climate variability
that can reliably be inferred from a proxy record generated according to
Eq. (

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.

Correcting for non-normality using inverse transform sampling. Solid
red: the normalized NINO3.4 index; dashed red: standardized NINO3.4 with
additive

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.

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. (

We compare the relative merits of three approaches to reconstructing past climate from a
proxy generated according to Eq. (

Distributions of estimated NINO3.4 variance ratios within the periods 1900–1949
and 1950–1999, using positive proxy excursions. Light blue: distribution of
ratios from 10 000 samples of

Method 1 (RAW): nonlinearity is ignored, and the proxy series

Method 2 (ITS): nonlinearity is recognized, empirically corrected via inverse transform sampling, and used as a predictor in a standard linear regression.

Method 3 (BPM): nonlinearity is recognized, and a probability model that allows for
Bayesian inversion of the structural dependence of the proxy on the climate (Eq.

The experimental sample is composed of 10 000 surrogate climate time series

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

For the two regression cases, uncertainties are quantified via 95 %
prediction intervals (PIs) following standard regression theory

For simplicity, we assume in the Bayesian treatment that all model parameters

Inferences on simulated

Results of performing inference on

We investigate the properties of the three inference methods, and the
sensitivity of the results to the ratio

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

Residuals (differences between estimated and true values of the
climate) as a function of the estimated climate. The three inference methods
(columns) are compared for three values of

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

To see how well such intervals encompass the true climate fluctuations, we
calculate their actual coverage rates

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

Inference diagnostics derived from 10 000 realizations of the
climate process

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

Finally we summarize how closely each method estimates the variance ratio
between the calibration (

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.

Distribution of variances ratios between the calibration and
validation intervals, as inferred from each method and normalized by the same
ratio calculated from the actual realization of

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.

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

We now investigate how inferences made from the iconic Laguna Pallcacocha
record of river run-off from the southern Ecuadorian Andes

The time series of Pallcacocha red colour intensity displays pronounced
positive skew (Fig.

Raw and transformed time series of the Laguna Pallcacocha record

We investigate how the transform affects the time series of sample variances
computed on disjoint, 100-year segments (Fig.

On account of the non-normal and serially correlated nature of both series, a
standard

Variances calculated over non-overlapping 100-year segments of the Pallcacocha red colour intensity record (thick cyan line) and after applying the empirical transform (thin blue line). Both series were standardized prior to analysis.

Following

Wavelet analysis of the Laguna Pallcacocha record

Our analysis does not contradict the qualitative conclusions of the original
study by

Significance of variance changes across the 3150 BCE boundary (grey
line in Fig.

Recently,

Raw and transformed time series of Lake Challa varve thickness

Comparison of Lake Challa and Laguna Pallcacocha records. Top row: raw (untransformed) records. Bottom row: transformed records. Left: time series interpolated at biannual resolution. Right: standard deviation on sliding 40 yr windows.

As the raw Lake Challa series has been used as a proxy for ENSO activity

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

Much of palaeoclimatology is concerned with the characterization of climate variability in the pre-instrumental era. Many proxy archives exhibit nonlinear relationships to the underlying climate, whereas many commonly used tools assume, either explicitly or implicitly, that the proxies are linearly related to a normally distributed climate variable. As illustrated for an idealized proxy (Sects. 2 and 3), and for ENSO-sensitive proxies (Sect. 4), direct variance estimates on different segments of nonlinear proxy records generally give a misleading picture of climate variability.

There are a number of techniques that allow for meaningful conclusions about changes in the underlying climate from such nonlinear proxy archives. Applying an empirical transform to such a series can render it approximately normal, so that standard data-analytic tools may be applied. In the context of the pseudo-proxy study (Sect. 2), the inverse transform sampling approach corrected for nonlinearity and reduced the risk of overstating variance changes. In the context of inferring climate values from proxy values (Sect. 3), the transform provided a regression fit comparable to that of an optimal benchmark and led to normally distributed residuals, in accordance with the assumptions of the regression framework. These examples also highlighted the dangers of ignoring nonlinearity in the proxy–climate relationship.

Applying the transform to the Lake Pallcacocha record
(Sect.

We thus stress that generic recipes are no substitute for a mechanistic
understanding, and ultimately the scientific interpretation of a proxy in
terms of its climatic, ecological, or geological controls should guide the
choice of statistical methodology. The example of Sect. 3 illustrates that
explicitly modelling the mechanism giving rise to non-normality yields much
better estimates of the underlying climate. In the example considered here,
both the functional form and parameters of the mechanistic model were assumed
known, while in real applications the form would likely be an approximation
and the parameters would be estimated as part of a fully Bayesian analysis

In instances where many proxies are used as predictors of past climates, such
as climate reconstructions of the Common Era

Finally, while this article has focused on nonlinear proxy records with
power-law type relationships, it is worth pointing out that a number of other
valuable climate proxies may deviate from linearity in other ways. In
particular, records based on proportions (e.g. pollen counts, lithological
fractions, fractions of certain faunal assemblages), being in the range

In this article we showed that nonlinearity fundamentally alters the
information content of climate proxies and that it must be dealt with, lest
some erroneous inferences be made. Though we advocate for mechanistic
modelling whenever possible, we showed that a simple empirical transform
(ITS) can often be sufficient to remedy many of the ills of nonlinearity, and
we recommend that palaeoclimatologists working with nonlinear proxies adopt such
simple transforms in their work. Matlab code for implementing the transform
is available at

Posterior probability density functions of climate, according to
Eqs. (6, 7), with

Consider the proxy model of Eq. (

We are interested in inferring

In what follows, we set the prior on the climate states,

On account of the thresholding behaviour of the proxy, it is necessary to
distinguish two cases in calculating the posterior distribution of climate.

Case 1:

With some rearrangements, the integral may be rewritten with the help of

Case 2:

The posterior is once more the product of two terms. The first is a normal
density that combines the information from the prior and the likelihood of
the observed value of

Examples of the posterior distribution are shown in Fig.

The choice of prior affects both point estimates and uncertainty intervals
and is the dominant control on the inference in data-poor situations – such
as the

To build intuition into the role of the prior, consider the case where

Table

Although the uncertainty intervals are wider under the DBL as compared with
the CTL prior, the coverage rates are largely unchanged between the two.
There is a trade-off in this case, as the wider prior results in an increased
probability of small magnitude values of

Inference diagnostics derived from 1000 realizations of the climate
process

J. Emile-Geay and M. Tingley are grateful to Peter Craigmile and Carl Wunsch for comments on
an earlier version of this manuscript. J. Emile-Geay acknowledges funding from NSF
grant DMS-1025464. The code and data necessary to reproduce the results of
this study, including figures and tables, will be made available at