1. Introduction

Quantifying the change in a variable over time is one of the most common calculations in climate science. Two standard techniques are 1) simple linear regression and 2) the epoch difference—that is, the difference between the average value over the end of the time series and the average value over the beginning of the time series. While simple linear regression directly provides an estimate of the linear trend over time, the epoch difference provides the net change between two periods. However, under the assumption that the underlying trend is in fact linear, one can divide the epoch difference by a characteristic time scale (discussed below) to also provide an estimate of the linear trend.

The summary for policymakers of the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report (AR5) quantified the change in global surface temperature over the historical record using both linear regression and an epoch difference (IPCC 2013). Ding and Steig (2013) used both linear regression and epoch differences to estimate trends in observed surface temperature at stations in Antarctica. Deser et al. (2012) explicitly took both approaches to quantify their model’s climate response over the 2005–60 period. They used 10-yr epoch differences and linear regression and commented that both methods produced similar results. However, it is not readily apparent whether both methods are necessary or whether the two methods will always yield similar results when the underlying trends are indeed linear.

In some cases, even if the trend is expected to be linear, simple linear regression is not an option. For example, phase 3 of the Coupled Model Intercomparison Project (CMIP3) requested that daily data be archived for specific 20-yr periods over the twenty-first century (e.g., 2046–65 and 2081–2100) (Meehl et al. 2007), and, thus, large gaps in the time series made linear regression impossible. In such situations, epoch differences are an obvious alternative. But what length of epoch should be used? One finds a wide range of epoch lengths in the literature for assessing changes in climate over the twentieth and twenty-first centuries—for example, 5 years (e.g., IPCC 2007), 10 years (e.g., Deser et al. 2012), 20 years (e.g., Collins et al. 2013), and 40 years (e.g., Kidston and Gerber 2010). Furthermore, in some studies, the epochs cover only the ends of the time series (e.g., Deser et al. 2012), while in others the epochs split the time series in half (e.g., Ding and Steig 2013).

Here, we pose two questions for the case of a linear trend:

1. Which method is better for trend estimation: simple linear regression or the epoch difference?
2. What is the optimal length of an epoch?

We demonstrate that under most circumstances, simple linear regression is preferred to an epoch difference in estimating a linear trend. We further demonstrate that if an epoch difference estimator is used, the optimal epoch length is approximately one-third the length of the time series when the memory is small.

Climate trends are often described by their linear component, and many variables (e.g., temperature and precipitation) exhibit linear trends in response to climate change over the twenty-first century in model simulations (e.g., Thompson et al. 2015). In addition, simple linear regression is relatively straightforward and easily replicated. For example, Hartmann et al. (2013) explicitly use linear regression to quantify trends over the historical record because of its simplicity and frequent use in climate research. However, there is no a priori reason to expect that climate variables will exhibit linear trends, and when they do not, simple linear regression may not be the most appropriate method (e.g., Seidel and Lanzante 2004). In this case, epoch differences are still an appropriate method for quantifying the net change over two periods of time. Thus, we stress that the comparisons made here between simple linear regression and epoch differences are applicable only in the specific case that the underlying trends are linear.

2. Problem setup

We begin with a first-order autoregressive [AR(1); red noise] time series x_n of length N (Box et al. 2008):

where n = 1, 2, …, N, x(0) = 0 by assumption, the coefficients α and β satisfy α² + β² = 1, and the ϵ_n are independent, identically distributed, Gaussian noise with mean 0 and variance of σ². The coefficient α is between 0 and 1 and captures the memory in the time series such that α = 0 denotes a time series with no memory from one time step to the next. The value of α can be estimated as the lag-1 autocorrelation of the time series (e.g., Box et al. 2008).

Next, we impose a trend upon x_n such that

where γ is the slope of the imposed linear trend [see (A1) in appendix A for y_n written as a function of y alone].

The question is, how should we estimate γ? Here, we explore two methods. The first method is simple linear regression; we denote the regression estimator of the trend (i.e., the slope) as . The second method is the epoch difference of length M converted into a trend—that is, the difference between the average value over the last M time steps and the average value over the first M time steps divided by N − M (see Fig. 1). We denote the epoch difference estimator of the trend as .

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(a) Example showing two methods for estimating the linear trend of 100 years of land surface–air temperature anomalies: linear regression (dashed black) and a 10-yr epoch difference estimator (dashed blue). The trend resulting from each method is given in the legend. (b) The distribution of estimated trends calculated from 100 000 synthetic white noise time series of the form of (2) with yearly variance σ² = 0.01°C² and linear trend γ = 0.08°C decade⁻¹.

Citation: Journal of Climate 28, 24; 10.1175/JCLI-D-15-0032.1

An example of these two methods is given in Fig. 1a, where we plot the annual mean land surface–air temperature anomalies (base period 1951–80) from GISTEMP (Hansen et al. 2010; downloaded from http://data.giss.nasa.gov/gistemp/) between 1913 and 2012. A similar linear fit and epoch difference was calculated in the AR5 summary for policymakers, although the epochs were of different lengths and the epoch difference was not converted into a trend (IPCC 2013). The two methods of estimating the trend over the 100 years yield slightly different answers; the 10-yr epoch difference estimator yields = 0.1°C decade⁻¹ and the linear regression estimator = 0.08°C decade⁻¹.

To determine which trend estimator is optimal, we define the “optimal estimator” as the estimator with the smallest variance. As an example, Fig. 1b shows the distribution of estimated trends calculated from 100 000 synthetic white noise time series (α = 0) of the form of (2) with yearly variance σ² = 0.01°C² (estimated from the detrended temperatures over 1970–2012 in Fig. 1a) and linear trend γ = 0.08°C decade⁻¹. The different colors denote the different estimation methods. The width of the distribution of trends estimated using simple linear regression (dashed black) is smaller than the widths of all of the epoch difference estimators, with M = 5 yr exhibiting the largest spread and M = 30 yr the smallest spread. In appendixes A, B, and C, we analytically derive the variance of these distributions (i.e., the variance of and ); however, we present only the final results here for brevity.

3. Results

Figure 2 shows the variance of divided by the variance of . This ratio is a function of time series memory α and the epoch difference length M for a time series of length N = 100. The most striking feature of this figure is that the variance of is greater than the variance of for nearly all combinations of α and M (warm colors). In some cases (small M), the variance of the epoch difference estimator is more than 4 times that of the variance of the linear regression estimator. Only for time series with large memory (α 0.85) does the linear regression estimator variance exceed that of the epoch difference. The exact cutoff at which the linear regression estimator variance exceeds that of the epoch difference is a function of N, and we find that as long as N ≥ 10 and α ≤ 0.5, simple linear regression always exhibits the smaller variance (not shown).

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The ratio of the variances of the epoch difference trend estimator to the linear regression trend estimator as a function of memory α and epoch difference length M for a time series of length N = 100.

Citation: Journal of Climate 28, 24; 10.1175/JCLI-D-15-0032.1

Further inspection of Fig. 2 reveals that for α 0.85 there is a minimum in the ratio of estimator variances when the epoch length is around 30 time steps. This minimum is more easily seen in Fig. 3 (orange line), where we plot a cross section of Fig. 2 along α = 0 (the special case of no memory). Figure 3 also shows results for time series of length N = 25, 50, and 75. In all cases, the ratio of variances exhibits a minimum—that is, there is an optimal M, denoted M*, where the variance of the epoch difference estimator is minimized.

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The ratio of the variances of the epoch difference trend estimator to the linear regression trend estimator as a function of epoch difference length M for a white noise time series. Colors denote time series of different lengths N. The minimum of each curve is denoted by colored circles, and the dashed vertical lines denote the location of M = N/3.

Citation: Journal of Climate 28, 24; 10.1175/JCLI-D-15-0032.1

In the special case of no memory (α = 0), the variance of the epoch difference estimator is given by

[see (C19)]. Differentiating (3) with respect to M and setting the resulting expression equal to 0 yields a remarkably simple form for M*:

Thus, in the case of no memory, the optimal epoch length is one-third the length of the time series. This general result is shown in Fig. 3.

When α ≠ 0, one can use the more general equation for the variance of [see (C16) in appendix C] to determine the optimal epoch length. Results of this calculation are shown in Fig. 4, where the optimal M is plotted as a function of time series length N. The different colors denote different values of α. For small memories (α ≤ 0.5; purple and blue dots) the optimal epoch length remains near N/3; however, when α increases to 0.7 (green dots) the optimal epoch length decreases for short time series. In the rather extreme case of α = 0.9 (orange dots), the optimal epoch length is M* = 1 for all N. In this limit, the autocorrelation is so large that adding additional data points to the epoch difference does not help the trend estimation.

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Optimal epoch difference length M* as a function of time series length N. Different colors denote different time series memories α. The M = N/3 line is drawn in black.

Citation: Journal of Climate 28, 24; 10.1175/JCLI-D-15-0032.1

Returning to Fig. 3, it is also apparent that the ratio of variances has a minimum limit of 1.125 in the case of α = 0. Even when the optimal epoch length is chosen, the variance of the epoch difference estimator is 12% larger than that of the linear regression estimator. This limit is straightforward to derive when α = 0 using (3) and the equation for the variance of [see (B17) in appendix B]:

Taking the ratio of (3) to (5) and setting M = N/3 leads to a minimum ratio of 1.125 in the limit of large N.

When performing an epoch difference, one is not limited to defining the two epochs to be the same length. For example, the AR5 summary for policymakers analyzes changes in observed surface air temperature between 1850 and 2012 using one epoch of length 51 years (1850–1900) and the other of length 10 years (2003–12) (IPCC 2013). The ratio of variances of the estimated trend computed using epochs of different lengths M₁ and M₂ and simple linear regression is shown in Fig. 5a (for N = 100 and α = 0). As one might suspect from Fig. 4, the minimum variance is found when M₁ = M₂ = N/3 ≈ 33 [(C20)]. Also evident is that a larger M₁ does not compensate for a smaller M₂. Choosing M₁ = 45 and M₂ = 21, for example, does not produce an estimator variance as small as that when M₁ = M₂ = 33.

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(a) The ratio of the variances of the epoch difference trend estimator to the linear regression trend estimator for epochs of different lengths M₁ and M₂ for a time series of length N = 100 and no memory. (b) The optimal epoch length of M₂ for a given M₁ with the one-to-one denoted by the gray dashed line. Colors denote the ratio of variances.

Citation: Journal of Climate 28, 24; 10.1175/JCLI-D-15-0032.1

Finally, Fig. 5a suggests that if one has certain restrictions on the length of M₁, there is an optimal length of the second epoch M₂. The optimal M₂ given M₁ (denoted as ) is given by (C21):

Solutions to this equation are plotted in Fig. 5b, and it is clear that using equal-length epochs (dashed line) is not the optimal solution unless M₁ = M₂ = 33. For example, if M₁ = 10, then setting M₂ = 24 will minimize the variance of the epoch difference estimator of the trend. The general relationship is that the optimal increases nonlinearly with M₁.

4. Discussion

In this short note we show that using simple linear regression to estimate a trend in a red noise time series with an imposed linear trend is almost always preferred over the use of an epoch difference trend estimator, although the degree to which the regression estimator is optimal is strongly dependent on the epoch length. We further demonstrate that if an epoch difference is used, the optimal epoch length is approximately one-third the length of the time series (N/3) under most circumstances. These conclusions break down in the limit that the time series exhibits extremely large autocorrelation. Finally, we demonstrate that if the epoch length at one end is constrained to a suboptimal value then the optimal epoch length at the other end is, in general, neither N/3 nor symmetric.

It is perhaps not surprising that simple linear regression is superior to the epoch difference trend estimation under most circumstances. While simple linear regression uses all of the data in the time series, maximizing the degrees of freedom, the epoch difference estimation uses only the data at the beginning and end of the time series and thus ignores information. In most cases, more information provides a tighter constraint on the trend. This is, however, not the case when the autocorrelation of the time series is large. In this instance, the data in the middle of the time series are no longer independent of the data at the end points; the epoch difference estimation then outperforms the linear regression estimation of the trend.

As discussed in the introduction, the results presented here are based on the assumption that the trend being estimated is linear and that the variability superposed on this linear trend is first-order autoregressive [AR(1)]. While the climate system is known to exhibit variability that is bimodal or oscillatory (e.g., the Madden–Julian oscillation), many aspects of the climate system are well modeled as AR(1) (e.g., Hartmann and Lo 1998; Feldstein 2000; Newman et al. 2003). Spectral methods may be used to carry out an analysis similar to that presented in this paper when considering autoregressive models of higher order (e.g., Bloomfield 1992; Bloomfield and Nychka 1992).

In addition, the assumption of a linear climate trend is often made in climate research, and, in fact, it is embraced by all of the IPCC reports. If, however, the trend is not linear or the underlying variability is not AR(1), simple linear regression may not be the most appropriate method (e.g., Seidel and Lanzante 2004).