The physical basis of climate change is understood through a combination of theory, numerical simulations and analyses of historical data. Climate change is driven by radiative forcing, a change in net radiation (downwelling minus upwelling, often specified at the top of atmosphere) resulting from an imposed perturbation of a climate in equilibrium, for example by increasing the atmospheric concentration of a greenhouse gas. The Earth's response to forcing is complex and not fully understood, in part due to physical uncertainties in important feedbacks such as cloud responses (see the assessment reports of the Intergovernmental Panel on Climate Change (IPCC), e.g., ).

Given the physical uncertainties inherent in all climate simulations, the observed temperature record since the late nineteenth century is often used as a source of empirical information about the Earth's systematic response to forcing. (Figure  shows one estimate of annually averaged global mean surface temperatures from the past 136 years, along with estimates of radiative forcings from various constituents during that period, with the data sources described in Sect. .) Analysis of the observed temperature record is complicated, however, by the short available record of direct measurements, by uncertainties in the historical radiative forcings themselves, and by the internal temperature variability that exists even in the absence of forcing. Statistical methods are therefore required to quantify the information in the historical record about the response to forcing: given the data, what do we know about how global temperatures have warmed in response to forcing, how much warming can we expect in plausible future forcing scenarios, and how do we expect uncertainties to change as we continue to observe the Earth's temperatures?

It can be helpful to divide approaches to using the observed temperature record to understand aspects of mean climate change into two categories. One common approach involves assuming a physical model of the system. (Here the term “model” encompasses anything from a simple energy balance model to a very complicated atmosphere–ocean general circulation model (GCM).) Analysis then may involve estimating statistical parameters in order to best fit the observed record. Estimated parameters have statistical uncertainty because of the finite observational record and the internal variability inherent in the climate system, even were the model a perfect representation of reality. Analyses of observed temperatures using simple or moderately complex physical models include , , , , , , , , , , , ,  and many others. Analyses using GCMs are summarized in chap. 10, and references therein.

Another category of approaches involves analyses that are more empirical and appear to de-emphasize assumptions about the underlying physics generating the observed temperatures. Many studies use regression models that treat time rather than radiative forcing as the covariate. This practice is often used, for example, to test for significant warming (e.g., ) or for changes in warming trends (e.g., ); in general, regressions in time are widespread in the literature (see also , especially chap. 2 Box 2.2, and references therein).

In both categories, analyses require characterizing internal variability for the purpose of quantifying uncertainty, a task that also involves assumptions. A typical approach is to assume a statistical model for the dependence structure of the noise, such as assuming an autoregressive moving average (ARMA) noise model with a small number of parameters, which is fit to the residuals of whatever trend model is being used. Some authors, however, argue for nonparametric (resampling or subsampling) methods for time series rather than parametric approaches (e.g., )

In this work we consider a parametric method to be any that makes explicit assumptions about the functional form of the probability distribution that the data come from, described with a finite number of statistical parameters. In particular, for the purposes of this work, we consider the class of low-order ARMA models to be a parametric class. We consider a nonparametric method to be one that attempts to make fewer distributional assumptions about the data and does not involve a parametrized statistical model.

However, methods that de-emphasize assumptions, be they physical or statistical, can be problematic in the climate setting. While regressions in time are simple to apply and do not appear to make explicit assumptions about how temperatures should respond to forcing, these models both limit what can be learned from the data and can result in misleading inferences. Regressions in time are sensitive to arbitrary choices (such as the start and end date of the data analyzed), cannot be expected to apply over even modestly long time frames and cannot in general reliably separate forced trends from internal variability. Furthermore, in accounting for internal variability, nonparametric methods for time series often require long data records to work well, and can be seriously uncalibrated in data-limited settings with strong temporal correlation, such as the setting we are discussing.

In the following, we illustrate two primary points. First, we show that targeted parametric mean models that incorporate even limited physical information can provide better-fitting, more interpretable, and more illuminating descriptions of the systematic response of interest compared to approaches that de-emphasize assumptions. Second, we show that parametric models for residual (i.e., internal) variation can provide for safer and more accurate uncertainty quantifications in this setting than do approaches that de-emphasize assumptions, even if the parametric model is misspecified, as long as the parametric modeling is done with particular attention towards the representation of low-frequency internal variability. We believe that the analysis that we present is informative, even if not maximally so, and we attempt to highlight both complications with our analysis as well as important sources of information about global warming that are ignored in our approach. Parts of our analysis share similarities with others listed above, especially with  and . In distinction to previous papers, here we are primarily interested in contrasting what can be learned using physically motivated models vs. those that de-emphasize assumptions, as well as in emphasizing the role that accounting for internal variability plays in inferring uncertainties in mean trends. Our goals are to indicate directions in which statisticians can incorporate explicit modeling to positive effect and to highlight what we view are some of the important sources of uncertainty and information in this problem.

This article is organized as follows. In Sect. , we introduce the data sources used in our analysis. In Sect. , we provide some background on modeling the historical global mean temperature record and contrast a simple, minimally physically informed model with a more empirical approach. In Sect. , we highlight insights that can be gained from the more informed approach, with an emphasis on probing different aspects of uncertainty in trends. In Sect. , through synthetic simulations, we compare the performance of various parametric and nonparametric methods of uncertainty quantification in the presence of temporal correlation in settings similar to that of the historical temperature record. In Sect. , we give some concluding remarks.

This analysis requires estimates of historical global mean temperatures and radiative forcings. To the extent that we are interested in how temperatures may evolve in the future (and how uncertainty in the response to radiative forcing evolves as more data are observed), we also need radiative forcings associated with a plausible future scenario.

Evaluating the systematic response of global mean surface temperatures to forcing is complicated by the long timescales for warming of the Earth system. Because the Earth's climate takes time to equilibrate, the near-term (transient or centennial-scale

Since the term transient climate response has a specific definition in the literature (see the Appendix for a discussion), we use the term centennial-scale response to describe the systematic response of temperatures to forcings on the mixing timescale of the mixed layer of the ocean but not of the deep ocean.

A common framework is to decompose observed temperatures into two components: a systematic component changing in response to past forcings and a residual component representing sources of internal variability. That is, for global mean temperatures T(t) at time t, T(t)|{F(t′);t′≤t}=f(F(t′);t′≤t)+ϵ(t), where f is an unknown functional of {F(t′);t′≤t}, the collection of past radiative forcings associated with each forcing agent, and ϵ(t) is a residual process that has mean zero and is correlated in time. Here we emphasize in our notation that the systematic response of interest is the response conditional on a given forcing trajectory. The problem, then, is how to estimate the systematic response f using the historical temperatures and forcings.

In this section, we illustrate what can be learned by applying the simple model () to observed temperatures. To do this, we must introduce an additional model to capture internal variability (ϵ(t) in the assumed true model Eq. ). We then use our full model to infer the parameters in model (), to evaluate their uncertainties given the data, and to explore the implications for understanding temperature trends.

To diagnose features of internal variability, spectral analysis is an intuitive framework, since the frequency properties of internal variability are tied to uncertainties in trends: uncertainty in smooth trends is more strongly affected by low-frequency than high-frequency internal variability. Figure , left, shows the raw periodogram associated with the residuals from model () and a smoothed estimate of the spectrum. We also show the spectra associated with fitting the residuals to two models for the internal variability. Both are standard autoregressive-moving average (ARMA) models (Supplement Sect. S3 includes a definition), allowing for dependence between the noise at each time step with the noise for past values. The two models we compare are an ARMA(4,1) and an AR(1) model, which were chosen according to common information criteria used for selecting time series models: respectively, the small sample-corrected Akaike information criterion (AICc)  and the Bayesian information criterion (BIC) . (See Table S1 in the Supplement for the coefficient estimates, innovation standard deviations, and AICc and BIC associated with these two models.) Both models appear to fit the data reasonably well; the ARMA(4,1) model arguably overfits at the higher frequencies, but the AR(1) model may be underestimating variability at the lowest frequencies. Since low-frequency variability is most important to uncertainties in smooth trends, we adopt the more conservative choice of using the ARMA(4,1) model. Figure , right, shows the normal quantile–quantile (Q–Q) plot for the sample innovations from this model; there is evidence that the innovations are somewhat more light-tailed than Gaussian, so standard errors based on a Gaussian assumption should not be overoptimistic.

Using the fully parametric model, combining Eq. () and a Gaussian ARMA(4,1) noise model, we proceed with uncertainty quantification through a parametric bootstrap

A parametric bootstrap involves generating repeated, synthetic simulations under the fitted statistical model and then refitting the model to each simulated time series to obtain new estimates of model parameters. The distribution of those estimates then gives a measure of the uncertainty in the original estimates.

In Sects.  and , we showed that analyses of the Earth's systematic temperature response are better informed by making physical assumptions than by an ostensibly more empirical approach. In this section, we ask a similar question about characterizing internal variability, in settings similar to that discussed, where data are temporally correlated and limited in length. We used a Gaussian ARMA(4,1) noise model in Sect. ; while we do not claim a physical motivation for this model, the model does make relatively strong statistical assumptions. Is it more robust to assume a low-order parametric model for the noise, as we have done, or to adopt a nonparametric approach? The answer to this question depends on the length of the data record and the nature of the internal variability.

For the purposes of this illustration, we will use not the actual temperature record but rather some simple synthetic examples. We consider several artificial, trendless time series (the true mean of the process is constant) with temporally correlated noise, and evaluate the results of testing for a linear time trend (i.e., fitting model Eq.  and testing against the null hypothesis that β=0) using different parametric (both correctly specified and not) and nonparametric methods for estimating uncertainties. The ordinary least-squares standard errors, which assume uncorrelated noise, tend to be anti-conservative (too small) and time series methods are supposed to ameliorate this overconfidence, but may or may not be successful in this regard depending on the context. The illustrations are simple but our conclusions should be relevant to actual data analysis and to more informed models.

We consider a few parametric approaches common in time series analysis. The typical practice is to assume that the noise follows a low-order model, such as an ARMA model. Ideally, the noise model would be chosen (as in Sect. ) in consultation with diagnostic plots and an information criterion like the AICc that balances model fit with the number of statistical parameters. However, it is common practice to automatically select a model either solely by minimizing the AICc or another criterion, or just to assume a simple time series model, often the AR(1) model. Uncertainty in the trend parameter(s) can then be estimated in several ways. Usually, this is done assuming asymptotic normality for maximum likelihood estimators, and a test might be carried out using a t statistic. If asymptotic normality is not viable, as for model (), an alternative is to resort to using a parametric bootstrap (again as we do in Sect. ) and to perform the test using bootstrap p values.

We also evaluate the perhaps most typical nonparametric method for accounting for dependence in a time series, the block bootstrap . Here, residuals are resampled in blocks to generate bootstrap samples that retain much of the dependence structure in the original data. A popular variant is the circular block bootstrap , in which blocks can be overlapping and blocks starting at the end of the time series wrap back to the beginning. The block bootstrap has been a common method in the climate and atmospheric sciences for at least 2 decades and has been applied previously to test for time trends (or changes thereof) in the global mean surface temperature record. For example, argued that the circular block bootstrap gives better uncertainty estimates than does a parametric analysis in the setting of testing for a trend in a 16-year segment of the global mean temperature record.

While the block bootstrap works very well in some settings, the procedure is not free of assumptions. Like other variants of the nonparametric bootstrap, its justification is based on an asymptotic argument; for the block bootstrap to work well, the size of the block has to be small compared to the overall length of the data but large compared to the scale of the temporal correlation in the data. When the overall data record is short and internal variability is substantially positively correlated in time, as for the historical global mean temperature record, these dual assumptions may not both be met and we should not expect the block bootstrap to perform well.

In the following, we compare five methods (four parametric and one nonparametric) for generating nominal p values testing for a significant linear time trend in trendless synthetic data: (a) a t test assuming independent noise, (b) a t test assuming Gaussian AR(1) noise, (c) bootstrap p values from a parametric bootstrap assuming Gaussian AR(1) noise, (d) a t test assuming Gaussian ARMA(p,q) noise with the order chosen by minimizing AICc, and (e) bootstrap p values from a circular block bootstrap. We generate synthetic data from three different time series models, but in cases (b) and (c), the assumed parametric model is always AR(1). (Table S2 summarizes the models from which we simulate, which were chosen in part to share a similar strength of correlation with the observed global mean temperature record.) In cases (a), (b), and (d), the t test degrees of freedom are approximated by n-2 minus the number of parameters in the noise model. The parametric models in cases (a) through (d) are all estimated via maximum likelihood. In case (e), we estimate p values using a few different block lengths and show the results most favorable to the block bootstrap.

After generating nominal p values, we evaluate the performance of the different methods. Since the null hypothesis is true in this artificial setting (the synthetic series are trendless), a correct p value would be uniformly distributed. Deviations from the uniform distribution would then be a sign that the procedure generating the nominal p value is uncalibrated. We therefore use Q–Q plots to compare the distribution of the simulated nominal p values with the theoretical uniform distribution. Uncertainties are underestimated when the nominal p value quantiles are smaller than the theoretical quantiles (i.e., when the Q–Q plot is below the one-to-one line). In this situation, inferences are anti-conservative and the tests using the selected method will have type I error rates that are larger than the nominal rate.

We have sought to show here that targeted parametric modeling of global mean temperature trends and internal variability can provide more informative and accurate analyses of the global mean temperature record than can more empirical methods. Since all analyses involve assumptions, it is important to consider the role that assumptions play in resulting conclusions. In the setting of analyzing historical global mean temperatures, where the data record is relatively short and temporal correlation is relatively strong, ostensibly more empirical methods can fail to distinguish between systematic trends and internal variability, and can give seriously uncalibrated estimates of uncertainty. While linear-in-time models can be used for some purposes when applied to moderately narrow time frames (and with careful uncertainty quantification), the demonstrations shown here suggest that they do not have an intrinsic advantage over more targeted analyses. Targeted analyses can be used over longer time frames – allowing for better estimates of both trends and noise characteristics – and can address a broader range of questions within a single framework.

The model we use in our analysis provides insights about the information contained in the historical temperature record relevant to both shorter-term and longer-term trend projections. The limited historical record of global mean temperature can provide information about shorter-term trends but unsurprisingly cannot constrain long-term projections very well. The past 136 years of temperatures simply do not alone contain the relevant information about equilibration timescales that would be required to constrain long-term projections, especially when aggregated to a single global value. (Use of spatially disaggregated data may provide additional information.) The distinction between uncertainties in shorter-term and longer-term projections, itself not easily made using a time trend model, serves to further illustrate that while the historical data record is an important source of information, it alone cannot be expected to answer the most important questions about climate change without bringing more scientific information to bear on the problem.

We believe that our discussion is illustrative of broader issues that arise in applied statistical practice, and will have particular relevance to problems involving trend estimation in the presence of temporally correlated data and in relatively data-limited settings, common in climate applications. We suspect that many applied statisticians have personally felt the tension between targeted modeling on the one hand and more empirical analyses on the other. One lucid discussion of the broader issues surrounding this tension can be found in the discussion of model formulation in . Empirical approaches have very successfully generated new insights and predictions in important areas, but there is a wide range of scientific problems where these approaches do not perform well and where targeted, domain-specific modeling is required. Statisticians can bring important insights to scientific problems; a crucial role for the statistician is to consider the modeling choices that will be both the most illuminating and the most reliable given the scientific questions and data at hand.