guest post by Jan Galkowski
5. Trends Are Tricky
Trends as a concept are easy. But trends as objective measures are slippery. Consider the Keeling Curve, the record of atmospheric carbon dioxide concentration first begun by Charles Keeling in the 1950s and continued in the face of great obstacles. This curve is reproduced in Figure 8, and there presented in its original, and then decomposed into three parts, an annual sinusoidal variation, a linear trend, and a stochastic remainder.
The question is, which component represents the true trend, long term or otherwise? Are linear trends superior to all others? The importance of a trend is tied up with to what use it will be put. A pair of trends, like the sinusoidal and the random residual of the Keeling, might be more important for predicting its short term movements. On the other hand, explicating the long term behavior of the system being measured might feature the large scale linear trend, with the seasonal trend and random variations being but distractions.
Consider the global surface temperature anomalies of Figure 5 again. What are some ways of determining trends? First, note that by “trends” what’s really meant are slopes. In the case where there are many places to estimate slopes, there are many slopes. When, for example, a slope is estimated by fitting a line to all the points, there’s just a single slope such as in Figure 9. Local linear trends can be estimated from pairs of points in differing sizes of neighborhoods, as depicted in Figures 10 and 11. These can be averaged, if you like, to obtain an overall trend.
Lest the reader think constructing lots of linear trends on varying neighborhoods is somehow crude, note it has a noble history, being used by Boscovich to estimate Earth’s ellipticity about 1750, as reported by Koenker.
There is, in addition, a question of what to do if local intervals for fitting the little lines overlap, since these are then (on the face of it) not independent of one another. There are a number of statistical devices for making them independent. One way is to do clever kinds of random sampling from a population of linear trends. Another way is to shrink the intervals until they are infinitesimally small, and, so, necessarily independent. That definition is just the point slope of a curve going through the data, or its first derivative. Numerical methods for estimating these exist—and to the degree they succeed, they obtain estimates of the derivative, even if in doing do they might use finite intervals.
One good way of estimating derivatives involves using a smoothing spline, as sketched in Figure 6, and estimating the derivative(s) of that. Such an estimate of the derivative is shown in Figure 12 where the instantaneous slope is plotted in orange atop the data of Figure 6. The value of the derivative should be read using the scale to the right of the graph. The value to the left shows, as before, temperature anomaly in degrees. The cubic spline itself is plotted in green in that figure. Here it’s smoothing parameter is determined by generalized cross-validation, a principled means of taking the subjectivity out of the choice of smoothing parameter. That is explained a bit more in the caption for Figure 12. (See also Cr1979.)
What else might we do?
We could go after a really good approximation to the data of Figure 5. One possibility is to use the Bayesian Rauch-Tung-Striebel (“RTS”) smoother to get a good approximation for the underlying curve and estimate the derivatives of that. This is a modification of the famous Kalman filter, the workhorse of much controls engineering and signals work. What that means and how these work is described in an accompanying inset box.
Using the RTS smoother demands variances of the signal be estimated as priors. The larger the ratio of the estimate of the observations variance to the estimate of the process variance is, the smoother the RTS solution. And, yes, as the reader may have guessed, that makes the result dependent upon initial conditions, although hopefully educated initial conditions.
The RTS smoother result for two process variance values of 0.118 ± 002 and high 0.59 ± 0.02 is shown in Figure 13. These are 3 and 15 times the decorrelated variance value for the series of 0.039 ± 0.001, estimated using the long term variance for this series and others like it, corrected for serial correlation. One reason for using two estimates of the process variance is to see how much difference that makes. As can be seen from Figure 13, it does not make much.
Combining all six methods of estimating trends results in Figure 14, which shows the overprinted densities of slopes.
Note the spread of possibilities given by the 5 year local linear fits. The 10 year local linear fits, the spline, and the RTS smoother fits have their mode in the vicinity of the overall slope. The 10 year local linear fits slope has broader support, meaning it admits more negative slopes in the range of temperature anomalies observed. The RTS smoother results have peaks slightly below those for the spline, the 10 year local linear fits, and the overall slope. The kernel density estimator allows the possibility of probability mass below zero, even though the spline, and two RTS smoother fits never exhibit slopes below zero. This is a Bayesian-like estimator, since the prior is the real line.
Local linear fits to HadCRUT4 time series were used by Fyfe, Gillet, and Zwiers in their 2013 paper and supplement. We do not know the computational details of those trends, since they were not published, possibly due to Nature Climate Change page count restrictions. Those details matter. From these calculations, which, admittedly, are not as comprehensive as those by Fyfe, Gillet, and Zwiers, we see that robust estimators of trends in temperature during the observational record show these are always positive, even if the magnitudes vary. The RTS smoother solutions suggest slopes in recent years are near zero, providing a basis for questioning whether or not there is a warming “hiatus”.
The Rauch-Tung-Striebel smoother is an enhancement of the Kalman filter. Let denote a set of univariate observations at equally space and successive time steps . Describe these as follows:
The multivariate is called a state vector for index . and are given, constant matrices. Equations (5.3) and (5.4) say that the noise component of observations and states are distributed as zero mean Gaussian random variables with variance and covariance , respectively. This simple formulation in practice has great descriptive power, and is widely used in engineering and data analysis. For instance, it is possible to cast autoregressive moving average models (“ARMA”) in this form. (See Kitigawa, Chapter 10.) The key idea is that equation (5.1) describes at observation at time as the result of a linear regression on coefficients , where is the corresponding design matrix. Then, the coefficients themselves change with time, using a Markov-like development, a linear regression of the upcoming set of coefficients, , in terms of the current coefficients, , where is the design matrix.
For the purposes here, a simple version of this is used, something called a local level model (Chapter 2) and occasionally a Gaussian random walk with noise model (Section 12.3.1). In that instance, and are not only scalars, they are unity, resulting in the simpler
with scalar variances and .
In either case, the Kalman filter is a way of calculating , given , values for and , and estimates for and . Choices for and are considered a model for the data. Choices for and are based upon experience with and the model. In practice, and within limits, the bigger the ratio
the smoother the solution for over successive .
Now, the Rauch-Tung-Striebel extension of the Kalman filter amounts to (a) interpreting it in a Bayesian context, and (b) using that interpretation and Bayes Rule to retrospectively update with the benefit of information through and the current state . Details won’t be provided here, but are described in depth in many texts, such as Cowpertwait and Metcalfe, Durbin and Koopman, and Särkkä.
Finally, commenting on the observation regarding subjectivity of choice in the ratio of variances, mentioned in Section 5 at the discussion of their choice “smoother” here has a specific meaning. If this ratio is smaller, the RTS solution tracks the signal more closely, meaning its short term variability is higher. A small ratio has implications for forecasting, increasing the prediction variance.
6. Internal Decadal Variability
The recent IPCC AR5 WG1 Report sets out the context in its Box TS.3:
Hiatus periods of 10 to 15 years can arise as a manifestation of internal decadal climate variability, which sometimes enhances and sometimes counteracts the long-term externally forced trend. Internal variability thus diminishes the relevance of trends over periods as short as 10 to 15 years for long-term climate change (Box 2.2, Section 2.4.3). Furthermore, the timing of internal decadal climate variability is not expected to be matched by the CMIP5 historical simulations, owing to the predictability horizon of at most 10 to 20 years (Section 11.2.2; CMIP5 historical simulations are typically started around nominally 1850 from a control run). However, climate models exhibit individual decades of GMST trend hiatus even during a prolonged phase of energy uptake of the climate system (e.g., Figure 9.8; Easterling and Wehner, 2009; Knight et al., 2009), in which case the energy budget would be balanced by increasing subsurface-ocean heat uptake (Meehl et al., 2011, 2013a; Guemas et al., 2013).
Owing to sampling limitations, it is uncertain whether an increase in the rate of subsurface-ocean heat uptake occurred during the past 15 years (Section 3.2.4). However, it is very likely that the climate system, including the ocean below 700 m depth, has continued to accumulate energy over the period 1998-2010 (Section 3.2.4, Box 3.1). Consistent with this energy accumulation, global mean sea level has continued to rise during 1998-2012, at a rate only slightly and insignificantly lower than during 1993-2012 (Section 3.7). The consistency between observed heat-content and sea level changes yields high confidence in the assessment of continued ocean energy accumulation, which is in turn consistent with the positive radiative imbalance of the climate system (Section 8.5.1; Section 13.3, Box 13.1). By contrast, there is limited evidence that the hiatus in GMST trend has been accompanied by a slower rate of increase in ocean heat content over the depth range 0 to 700 m, when comparing the period 2003-2010 against 1971-2010. There is low agreement on this slowdown, since three of five analyses show a slowdown in the rate of increase while the other two show the increase continuing unabated (Section 3.2.3, Figure 3.2). [Emphasis added by author.]
During the 15-year period beginning in 1998, the ensemble of HadCRUT4 GMST trends lies below almost all model-simulated trends (Box 9.2 Figure 1a), whereas during the 15-year period ending in 1998, it lies above 93 out of 114 modelled trends (Box 9.2 Figure 1b; HadCRUT4 ensemble-mean trend per decade, CMIP5 ensemble-mean trend per decade). Over the 62-year period 1951-2012, observed and CMIP5 ensemble-mean trends agree to within per decade (Box 9.2 Figure 1c; CMIP5 ensemble-mean trend per decade). There is hence very high confidence that the CMIP5 models show long-term GMST trends consistent with observations, despite the disagreement over the most recent 15-year period. Due to internal climate variability, in any given 15-year period the observed GMST trend sometimes lies near one end of a model ensemble (Box 9.2, Figure 1a, b; Easterling and Wehner, 2009), an effect that is pronounced in Box 9.2, Figure 1a, because GMST was influenced by a very strong El Niño event in 1998. [Emphasis added by author.]
The contributions of Fyfe, Gillet, and Zwiers (“FGZ”) are to (a) pin down this behavior for a 20 year period using the HadCRUT4 data, and, to my mind, more importantly, (b) to develop techniques for evaluating runs of ensembles of climate models like the CMIP5 suite without commissioning specific runs for the purpose. This, if it were to prove out, would be an important experimental advance, since climate models demand expensive and extensive hardware, and the number of people who know how to program and run them is very limited, possibly a more limiting practical constraint than the hardware.
This is the beginning of a great story, I think, one which both advances an understanding of how our experience of climate is playing out, and how climate science is advancing. FGZ took a perfectly reasonable approach and followed it to its logical conclusion, deriving an inconsistency. There’s insight to be won resolving it.
FGZ try to explicitly model trends due to internal variability. They begin with two equations:
is the model membership index. is the index of the model’s ensemble. runs over bootstrap samples taken from HadCRUT4 observations. Here, and are trends calculated using models or observations, respectively. and denote the “true, unknown, deterministic trends due to external forcing” common to models and observations, respectively. and are the perturbations to trends due to internal variability of models and observations. denotes error in climate model trends for model . denotes the sampling error in the sample. FGZ assume are exchangeable with each other as well, at least for the same time . (See [Di1977, Di1988, Ro2013c, Co2005] for more on exchangeability.) Note that while the internal variability of climate models varies from model to model, run to run, and time to time, the ‘internal variability of observations’, namely , is assumed to only vary with time.
The technical innovation FGZ use is to employ bootstrap resampling on the observations ensemble of HadCRUT4 and an ensemble of runs of 38 CMIP5 climate models to perform a two-sample comparison [Ch2008, Da2009, ]. In doing so, they explicitly assume, in the framework above, exchangeability of models. (Later, in the same work, they also make the same calculation assuming exchangeability of models and observations, an innovation too detailed for this present exposition.)
So, what is a bootstrap? In its simplest form, a bootstrap is a nonparametric, often robust, frequentist technique for sampling the distribution of a function of a set of population parameters, generally irrespective of the nature or complexity of that function, or the number of parameters. Since estimates of the variance of that function are themselves functions of population parameters, assuming the variance exists, the bootstrap can also be used to estimate the precision of the first set of samples, where “precision” is the reciprocal of variance. For more about the bootstrap, see the inset below..
In the case in question here, with FGZ, the bootstrap is being used to determine if the distribution of surface temperature trends as calculated from observations and the distribution of surface temperature trends as calculated from climate models for the same period have in fact similar means. This is done by examining differences of paired trends, one coming from an observation sample, one coming from a model sample, and assessing the degree of discrepancy based upon the variances of the observations trends distribution and of the models trends distribution.
The equations (6.1) and (6.2) can be rewritten:
moving the trends in internal variability to the left, calculated side. Both and are not directly observable. Without some additional assumptions, which are not explicitly given in the FGZ paper, such as
we can’t really be sure we’re seeing or , or at least less the mean of . The same applies to and . Here equations (6.5) and (6.6) describe internal variabilities as being multivariate but zero mean Gaussian random variables. and are covariances among models and among observations. FGZ essentially say these are diagonal with their statement “An implicit assumption is that sampling uncertainty in [observation trends] is independent of uncertainty due to internal variability and also independent of uncertainty in [model trends]”. They might not be so, but it is reasonable to suppose their diagonals are strong, and that there is a row-column exchange operator on these covariances which can produce banded matrices.
7. On Reconciliation
The centerpiece of the FGZ result is their Figure 1, reproduced here as Figure 15. Their conclusion, that climate models do not properly capture surface temperature observations for the given periods, is based upon the significant separation of the red density from the grey density, even when measuring that separation using pooled variances. But, surely, a remarkable feature of these graphs is not only the separation of the means of the two densities, but the marked difference in size of the variances of the two densities.
Why are climate models so less precise than HadCRUT4 observations? Moreover, why do climate models disagree with one another so dramatically? We cannot tell without getting into CMIP5 details, but the same result could be obtained if the climate models came in three Gaussian populations, each with a variance 1.5x that of the observations, but mixed together. We could also obtain the same result if, for some reason, the variance of HadCRUT4 was markedly understated.
That brings us back to the comments about HadCRUT4 made at the end of Section 3. HadCRUT4 is noted for “drop outs” in observations, where either the quality of an observation on a patch of Earth was poor or the observation was missing altogether for a certain month in history. (To be fair, both GISS and BEST have months where there is no data available, especially in early years of the record.) It also has incomplete coverage [Co2013]. Whether or not values for patches are imputed in some way, perhaps using spatial kriging, or whether or not supports to calculate trends are adjusted to avoid these omissions are decisions in use of these data which are critical to resolving the question [Co2013, Gl2011].
As seen in Section 5, what trends you get depends a lot on how they are done. FGZ did linear trends. These are nice because means of trends have simple relationships with the trends themselves. On the other hand, confining trend estimation to local linear trends binds these estimates to being only supported by pairs of actual samples, however sparse these may be. This has the unfortunate effect of producing a broadly spaced set of trends which, when averaged, appear to be a single, tight distribution, close to the vertical black line of Figure 14, but erasing all the detail available by estimating the density of trends with a robust function of the first time derivative of the series. FGZ might be improved by using such, repairing this drawback and also making it more robust against HadCRUT4’s inescapable data drops. As mentioned before, however, we really cannot know, because details of their calculations are not available. (Again, this author suspects this fault lies not with FGZ but a matter of page limits.)
In fact, that was indicated by a recent paper from Cowtan and Way, arguing that the limited coverage of HadCRUT4 might explain the discrepancy Fyfe, Gillet, and Zwiers found. In return Fyfe and Gillet argued that even admitting the corrections for polar regions which Cowtan and Way indicate, the CMIP5 models fall short in accounting for global mean surface temperatures. What could be wrong? In the context of ensemble forecasts depicting future states of the atmosphere, Wilks notes (Section 7.7.1):
Accordingly, the dispersion of a forecast ensemble can at best only approximate the [probability density function] of forecast uncertainty … In particular, a forecast ensemble may reflect errors both in statistical location (most or all ensemble members being well away from the actual state of the atmosphere, but relatively nearer to each other) and dispersion (either under- or overrepresenting the forecast uncertainty). Often, operational ensemble forecasts are found to exhibit too little dispersion …, which leads to overconfidence in probability assessment if ensemble relative frequencies are interpreted as estimating probabilities.
In fact, the IPCC reference, Toth, Palmer and others raise the same caution. It could be that the answer to why the variance of the observational data in the Fyfe, Gillet, and Zwiers graph depicted in Figure 15 is so small is that ensemble spread does not properly reflect the true probability density function of the joint distribution of temperatures across Earth. These might be “relatively nearer to each other” than the true dispersion which climate models are accommodating.
If Earth’s climate is thought of as a dynamical system, and taking note of the suggestion of Kharin that “There is basically one observational record in climate research”, we can do the following thought experiment. Suppose the total state of the Earth’s climate system can be captured at one moment in time, no matter how, and the climate can be reinitialized to that state at our whim, again no matter how. What happens if this is done several times, and then the climate is permitted to develop for, say, exactly 100 years on each “run”? What are the resulting states? Also suppose the dynamical “inputs” from the Sun, as a function of time, are held identical during that 100 years, as are dynamical inputs from volcanic forcings, as are human emissions of greenhouse gases. Are the resulting states copies of one another?
No. Stochastic variability in the operation of climate means these end states will be each somewhat different than one another. Then of what use is the “one observation record”? Well, it is arguably better than no observational record. And, in fact, this kind of variability is a major part of the “internal variability” which is often cited in these literature, including by FGZ.
Setting aside the problems of using local linear trends, FGZ’s bootstrap approach to the HadCRUT4 ensemble is an attempt to imitate these various runs of Earth’s climate. The trouble is, the frequentist bootstrap can only replicate values of observations actually seen. (See inset.) In this case, these replications are those of the HadCRUT4 ensembles. It will never produce values in-between and, as the parameters of temperature anomalies are in general continuous measures, allowing for in-between values seems a reasonable thing to do.
No algorithm can account for a dispersion which is not reflected in the variability of the ensemble. If the dispersion of HadCRUT4 is too small, it could be corrected using ensemble MOS methods (Section 7.7.1.) In any case, underdispersion could explain the remarkable difference in variances of populations seen in Figure 15. I think there’s yet another way.
Consider equations (6.1) and (6.2) again. Recall, here, denotes the model and denotes the run of model . Instead of , however, a bootstrap resampling of the HadCRUT4 ensembles, let run over all the 100 ensemble members provided, let run over the 2592 patches on Earth’s surface, and let run over the 1967 monthly time steps. Reformulate equations (6.1) and (6.2), instead, as
Now, is a common trend at time tick and and are deflections from from that trend due to modeling error and internal variability in the model, respectively, at time tick . Similarly, denotes deflections from the common trend baseline due to internal variability as seen by the HadCRUT4 observational data at time tick , and denotes the deflection from the common baseline due to sampling error in the patch at time tick . are indicator variables. This is the setup for an analysis of variance or ANOVA, preferably a Bayesian one (Sections 14.1.6, 18.1). In equation (7.1), successive model runs for model are used to estimate and for every . In equation (7.2), different ensemble members are used to estimate and for every . Coupling the two gives a common estimate of . There’s considerable flexibility in how model runs or ensemble members are used for this purpose, opportunities for additional differentiation and ability to incorporate information about relationships among models or among observations. For instance, models might be described relative to a Bayesian model average [Ra2005]. Observations might be described relative to a common or slowly varying spatial trend, reflecting dependencies among patches. Here, differences between observations and models get explicitly allocated to modeling error and internal variability for models, and sampling error and internal variability for observations.
More work needs to be done to assess the proper virtues of the FGZ technique, even without modification. A device like that Rohde used to compare BEST temperature observations with HadCRUT4 and GISS, one of supplying the FGZ procedure with synthetic data, would be perhaps the most informative regarding its character. Alternatively, if an ensemble MOS method were devised and applied to HadCRUT4, it might better reflect a true spread of possibilities. Because a dataset like HadCRUT4 records just one of many possible observational records the Earth might have exhibited, it would be useful to have a means of elaborating what those other possibilities were, given the single observational trace.
Regarding climate models, while they will inevitably disagree from a properly elaborated set of observations in the particulars of their statistics, in my opinion, the goal should be to strive to match the distributions of solutions these two instruments of study on their first few moments by improving both. While, statistical equivalence is all that’s sought, we’re not there yet. Assessing parametric uncertainty of observations hand-in-hand with the model builders seems to be a sensible route. Indeed, this is important. In review of the Cowtan and Way result, one based upon kriging, Kintisch summarizes the situation as reproduced in Table 1, a reproduction of his table on page 348 of the reference [Co2013, Gl2011, Ki2014]:
|NASA data set||0.080|
|HadCRUT data set||0.046|
|Table 1. Getting warmer.
New method brings measured temperatures closer to projections. Added in quotation: “Climate models” refers to the CMIP5 series. “NASA data set” is GISS. “HadCRUT data set” is HadCRUT4. “Cowtan/Way” is from their paper. Note values are per decade, not per year.
Note that these estimates of trends, once divided by 10 years/decade to convert to a per year change in temperature, all fall well within the slope estimates depicted in the summary Figure 14. Note, too, how low the HadCRUT trend is.
If the FGZ technique, or any other, can contribute to this elucidation, it is most welcome.
As an example Lee reports how the GLOMAP model of aerosols was systematically improved using such careful statistical consideration. It seems likely to be a more rewarding way than “black box” treatments. Incidently, Dr Lindsay Lee’s article was runner-up in the Significance/Young Statisticians Section writers’ competition. It’s great to see bright young minds charging in to solve these problems!
|The bootstrap is a general name for a resampling technique, most commonly associated with what is more properly called the frequentist bootstrap. Given a sample of observations, , the bootstrap principle says that in a wide class of statistics and for certain minimum sizes of , the sampling density of a statistic from a population of all , where is a single observation, can be approximated by the following procedure. Sample times with replacement to obtain samples each of size called , . For each , calculate so as to obtain . The set so obtained is an approximation of the sampling density of from a population of all . Note that because is sampled, only elements of that original set of observations will ever show up in any . This is true even if is drawn from an interval of the real numbers. This is where a Bayesian bootstrap might be more suitable.
In a Bayesian bootstrap, the set of possibilities to be sampled are specified using a prior distribution on [Da2009, Section 10.5]. A specific observation of , like , is use to update the probability density on , and then values from are drawn in proportion to this updated probability. Thus, values in never in might be drawn. Both bootstraps will, under similar conditions, preserve the sampling distribution of .
Various geophysical datasets recording global surface temperature anomalies suggest a slowdown in anomalous global warming from historical baselines. Warming is increasing, but not as fast, and much of the media attention to this is reacting to the second time derivative of temperature, which is negative, not the first time derivative, its rate of increase. Explanations vary. In one important respect, 20 or 30 years is an insufficiently long time to assess the state of the climate system. In another, while the global surface temperature increase is slowing, oceanic temperatures continue to soar, at many depths. Warming might even decrease. None of these seem to pose a challenge to the geophysics of climate, which has substantial support both from experimental science and ab initio calculations. An interesting discrepancy is noted by Fyfe, Gillet, and Zwiers, although their calculation could be improved both by using a more robust estimator for trends, and by trying to integrate out anomalous temperatures due to internal variability in their models, because much of it is not separately observable. Nevertheless, Fyfe, Gillet, and Zwiers may have done the field a great service, making explicit a discrepancy which enables students of datasets like the important HadCRUT4 to discover an important limitation, that their dispersion across ensembles does not properly reflect the set of Earth futures which one might wish they did and, in their failure for users who think of the ensemble as representing such futures, give them a dispersion which is significantly smaller than what we might know.
The Azimuth Project can contribute, and I am planning subprojects to pursue my suggestions in Section 7, those of examining HadCRUT4 improvements using MOS ensembles, a Bayesian bootstrap, or the Bayesian ANOVA described there. Beyond trends in mean surface temperatures, there’s another more challenging statistical problem involving trends in sea levels which awaits investigation [Le2012b, Hu2010].
Working out these kinds of details is the process of science at its best, and many disciplines, not least mathematics, statistics, and signal processing, have much to contribute to the methods and interpretations of these series data. It is possible too much is being asked of a limited data set, and perhaps we have not yet observed enough of climate system response to say anything definitive. But the urgency to act responsibly given scientific predictions remains.
- Credentials. I have taken courses in geology from Binghamton University, but the rest of my knowledge of climate science is from reading the technical literature, principally publications from the American Geophysical Union and the American Meteorological Society, and self-teaching, from textbooks like Pierrehumbert. I seek to find ways where my different perspective on things canhelp advance and explain the climate science enterprise. I also apply my skills to working local environmental problems, ranging from inferring people’s use of energy in local municipalities, as well as studying things like trends in solid waste production at the same scales using Bayesian inversions. I am fortunate that techniques used in my professional work and those in these problems overlap so much. I am a member of the American Statistical Association, the American Geophysical Union, the American Meteorological Association, the International Society for Bayesian Analysis, as well as the IEEE and its signal processing society.
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