1. Introduction
Climate time series often contain abrupt changes and other nonlinearities in their behavior. Changepoints are times of abrupt shifts in a series’ characteristics, including means, trends, variances, and autocorrelations. For examples, a sudden change from a cooling period (i.e., decreasing trend) to a warming period can be characterized by a changepoint in the trend; a sudden increase due to the relocation of a station may be characterized as a changepoint in the mean. Abrupt changes may be caused by changes in climate forcings, related to climate variability in the ocean and atmosphere, or induced by artificial changes in measurement procedures such as station relocations or instrumentation changes.
It is crucial to know changepoint times in climate series, especially when assessing long-term trends, as their presence may grossly alter trend estimates, which impedes our understanding of external forcings and climate variability over the instrumental record (Lund et al. 2007; Beaulieu et al. 2012; Cahill et al. 2015; Beaulieu and Killick 2018). Series with artificial changes merit adjustment via homogenization methods, as trends and extreme quantiles are more accurately estimated from homogenized data (Hewaarachchi et al. 2017; Trewin et al. 2020; Vincent et al. 2020). On average, approximately six station relocations or instrumentation changes occur over a century in a randomly selected U.S. climate station (Mitchell 1953; Menne and Williams 2009). As such, a changepoint analysis of a climate series is often a worthy initial exploratory endeavor.
Statistical methods to detect changepoints have rapidly evolved over the last few decades. These include methods to detect a single shift in the series’ mean (Chernoff and Zacks 1964), in its variance (Hsu 1977), or in a general linear regression model (Quandt 1958; Robbins et al. 2016). In the climate literature, changepoint detection has most often been used to detect mean shifts. However, this may result in misinterpreting a long-term climate trend as a sequence of mean shifts that follows (approximates) the trend (Beaulieu and Killick 2018).
Much of the changepoint literature assumes independent and identically distributed model errors (termed “white noise” here). However, climate time series are often autocorrelated, inducing memory at time scales longer than the measurement frequency (Hasselmann 1976). This memory is often modeled as a first-order autoregressive [AR(1)] process in climate studies (Lund et al. 2007; Robbins et al. 2011; Hartmann et al. 2013). In an AR(1) model, autocorrelation geometrically decays to zero with increasing time, representing one type of short-term memory. In the climate setting, it is important to allow autocorrelation and mean shift model features in tandem as both can inject similar run patterns into a climate series. An alternative is to use prewhitening techniques that mitigate the effects of autocorrelation (Robbins et al. 2011; Serinaldi and Kilsby 2016). Beaulieu and Killick (2018), Shi et al. (2022), and Gallagher et al. (2022) show that changepoint inferences can be drastically wrong if autocorrelation in a series is ignored. The memory in climate series has also been modeled as a long-memory process, where autocorrelation decays as a power law (Yuan et al. 2015). Long-memory processes and changepoint models can be confused as they both have similar spectrums. Unfortunately, this ambiguity may lead to misleading inferences. Beaulieu et al. (2020) discuss how to distinguish changepoints and long memory in surface temperatures.
Multiple changepoints may be present in climate series. Methods designed to detect a single changepoint have been applied iteratively to estimate multiple changepoint configurations through a process known as binary segmentation (Scott and Knott 1974; Rodionov 2004). Binary segmentation is now known to perform poorly in multiple changepoint problems (Shi et al. 2022) [see Fryzlewicz (2014) for an interesting attempt to fix binary segmentation]. Penalized likelihood methods, the approach taken here, were developed in Davis et al. (2006), Lu et al. (2010), Killick et al. (2012), and Li and Lund (2012) and tend to perform better (Shi et al. 2022). Here, a likelihood, which measures the goodness of the statistical model fit, is balanced against a penalty that prevents fitting too many changepoints. Penalized likelihood methods can allow for autocorrelation. Bayesian approaches to the multiple changepoint problem also exist. Most of these place some sort of prior distribution on the changepoint times, for instance a spike and slab prior (see Barry and Hartigan 1993; Chib 1998; Fearnhead 2006; Cappello et al. 2021, and references within). Li et al. (2019) construct an informative prior on the changepoint times from the station’s metadata record. The references above are by no means exhaustive; indeed, the changepoint literature is vastly expanding.
As most methodological statistics papers are not written with user comprehension in mind, the technical changepoint literature can seem impenetrable to non-statisticians, making it challenging to select an appropriate approach for the climate scientist. Compounding difficulties, Lund and Reeves (2002) and Beaulieu and Killick (2018) show that spurious changepoint inferences easily occur when prominent data features (e.g., autocorrelation, long-term trend) are ignored—the choice of model and method is critical in changepoint analyses. Indeed, changepoint techniques can produce different results when the models and assumptions are only slightly changed.
The aim of this paper is to present, through an example, a comprehensive changepoint analysis of a climate series. To this end, we analyze the Central England temperature (CET) series by fitting different changepoint models capable of detecting shifts in trends. We also compare our changepoint fits with long-memory models. Our focus is on penalized likelihood multiple changepoint techniques, enabling us to compare several models while preventing overestimation of the number of changepoints. We also discuss mean shift models and how they fit data containing a long-term trend such as the CET series. Emphasis is placed on implementation and interpretation over the theoretical foundations of penalized likelihoods. Nonetheless, references to the formal statistical literature are provided.
The rest of this paper proceeds as follows. The CET series used here is introduced in the next section. Section 3 then provides some rudimentary background on changepoint models, describing the penalized likelihood methods used here. The next three sections present fits of various multiple changepoint models. Results for each type of model motivate the subsequent fits. Remarks about the optimal model are made in the final section along with concluding comments.
2. The CET series
The CET time series is perhaps the longest instrumental record of surface temperatures in the world, commencing in 1659 and spanning 362 years through 2020. The CET series is a benchmark for European climate studies, as it is sensitive to atmospheric variability in the North Atlantic (Parker et al. 1992). This record has been previously analyzed for long-term changes (Plaut et al. 1995; Harvey and Mills 2003; Hillebrand and Proietti 2017); however, to our knowledge, no detailed changepoint analysis of it has been previously conducted. Changepoints are plausible in the CET record for several reasons. First, artificial shifts near the record’s onset may exist at times when data quality was lower (Parker et al. 1992). Furthermore, an increase in the pace of climate warming arising globally during the 1960s–1970s (Beaulieu and Killick 2018; Cahill et al. 2015) may be present. The length of the CET record affords us the opportunity to explore a variety of temperature features.
The CET series, available at https://www.metoffice.gov.uk/hadobs/hadcet/, was provided by the U.K. Met Office. Measurements commenced in 1659 and were mostly compiled by Manley (1953, 1974) until 1973, and then continued and updated to 1991 in Parker et al. (1992). The series is now kept by the Hadley Centre Met Office. The CET time series is an annual composite of 15 stations in the United Kingdom, located over a roughly triangular area bounded by Lancashire, London, and Bristol. The series is thus representative of the climate of the English Midlands. The station locations used to form the composite series are depicted in the top graphic in Fig. 1. The CET temperatures, presented in the bottom graphic of Fig. 1, have been previously adjusted for inhomogeneities due to changes in measurement practices through time (Manley 1953, 1974; Parker et al. 1992), and for urban warming since 1960 (Parker and Horton 2005). However, until 1722, available instrumental records used in the CET time series did not overlap. As such, noninstrumental weather diaries and the Utrecht instrumental series were used to adjust the CET series and fill the gaps (Parker et al. 1992). Between 1722 and 1760, there are no gaps in the composite record of all stations, but observations were generally collected in unheated rooms as opposed to outdoors. A few outdoor temperature measurements were collected and used to establish relationships between temperatures in unheated rooms and outdoors. These relationships were then used to adjust the CET time series (Parker et al. 1992). The daily CET time series starts in 1772 and has been used to update the monthly series (Parker et al. 1992). As such, some authors use only the data post-1772 for their analyses (Hillebrand and Proietti 2017). In this paper, we conduct a changepoint analysis on both the full CET time series (1659–2020) and the truncated series (1772–2020) that excludes the poorer quality data at the beginning of the record.
Station locations and annual average temperatures of Central England.
Citation: Journal of Climate 35, 19; 10.1175/JCLI-D-21-0489.1
3. Structural change models
To explore structural changes in the CET series, a hierarchical changepoint analysis, gradually building on past findings, will be conducted. Let Xt denote the annual temperature observed at time t and suppose that data from the years 1, …, N are available. In general, a changepoint analysis partitions the series into m + 1 distinct regimes, each regime having homogeneous characteristics. The number of changepoints m is unknown and needs to be estimated from the series. Let τi denote the ith changepoint time; boundary conditions take τ0 = 0 and τm+1 = N.
Methods for handling multiple changepoint analyses without penalized likelihoods exist. One popular technique is termed binary segmentation (Scott and Knott 1974). Binary segmentation works with any single changepoint technique, termed an “at most one change” (AMOC) method. Many AMOC tests have been developed, including cumulative sums (CUSUM) (Page 1954), likelihood ratios (Jandhyala et al. 2013), Chow tests (Chow 1960), and sum of squared CUSUM tests (Shi et al. 2022). Binary segmentation first analyzes the entire series for a changepoint. If a changepoint is found, the series is split into subsegments about the identified changepoint time and the two subsegments are further scrutinized for additional changepoints. The procedure is repeated iteratively until no subsegments are deemed to have changepoints. While simple and computationally convenient, binary segmentation is one of the poorer performing multiple changepoint techniques (Shi et al. 2022), often being fooled by changepoints that occur close to one another or multiple shifts that move the series in opposite directions. There have been attempts to fix binary segmentation; see the wild binary segmentation and related methods in Fryzlewicz (2014) and Eichinger and Kirch (2018). Unfortunately, these techniques typically assume independent model errors or are restricted to single parameter changes per regime (e.g., mean shifts only). Perhaps worse, wild binary segmentation tends to overestimate changepoint numbers when they are in truth infrequent (Lund and Shi 2020).
The notation here is as follows: L*(m; τ1, …, τm) is the optimal Gaussian likelihood that can be achieved from a model with m changepoints that occur at the times τ1, …, τm. Here, the data sample X1, X2, …, XN is regarded as fixed. To determine L*(m; τ1, …, τm), one must estimate all parameters in the mean function f and the AR(1) model errors assuming that m changepoints occur at the times τ1, …, τm. This procedure will be discussed further below. The quantity P(m; τ1, …, τm) is the penalty for having a model with m changepoints at the times τ1, …, τm. As more and more changepoints are added to the model, the overall fit gets better [
Many penalty structures have been proposed in the statistics and climate literature. These include the Akaike information criterion (AIC), the Bayesian information criterion (BIC), the modified Bayesian information criterion (mBIC), and minimum description lengths (MDL). We will use BIC and MDL here. These two penalties were judged as “winners” in a recent changepoint detection comparison in Shi et al. (2022). AIC penalties are not considered here because they often erroneously estimate an excessive number of changepoints (Shi et al. 2022). The BIC penalty for having m changepoints at the times τ1, …, τm is
Penalized likelihoods. The boxed terms are the penalties, with the unboxed terms constituting
Then pi is the inferred probability that model gi is the quasi-true model in the model set under a prior where all R models are equally likely (prior probabilities are 1/R for each model). These BIC posterior model probabilities highlight uncertainties in our model comparisons.
In contrast to the BIC penalty, the MDL penalty is more complex in form, also accounting for the changepoint location times τ1, …, τm. The MDL penalty depends on the form of f and is rooted in information theory, quantifying the computer memory needed to store the model (good fitting models use minimal space). MDL penalties have previously proven useful in changepoint detection (Davis et al. 2006; Li and Lund 2012). Posterior model probabilities are not available for the MDL information criterion. Other penalties used in the climate literature for changepoint problems include those in Caussinus and Mestre (2004).
In contrast to GAs, binary segmentation is a greedy algorithm that often becomes trapped at a local penalized likelihood minimum. Killick et al. (2012) and Maidstone et al. (2017b) produced two rapid dynamic programming–based multiple changepoint configuration optimizers that currently cannot handle our needs: Maidstone et al. (2017b) assumes independent model errors and Killick et al. (2012) assumes all parameters change at each changepoint time [including the AR(1) correlation parameter ϕ and error variance σ2]. GAs are the only optimization method that reasonably handle all models considered in this paper.
4. Models fitted
a. Trend shift models
More compactly, one can write E[Xt] = f(t) = μr(t) + βr(t)t, where r(t) ∈ {1, 2, …, m + 1} denotes the regime being used at time t; for example, r(t) = 1 for 1 ≤ t ≤ τ1.
The changepoint literature has focused primarily on detecting mean shifts; fewer studies have been dedicated to detecting trend shifts. However, Maidstone et al. (2017a) present a dynamic programming approach that estimates trend shift configurations using a penalty based on absolute distances that is neither the MDL nor BIC. Their {ϵt} must be white noise (uncorrelated) with a zero mean and constant variance. See Bai and Perron (1998), Bai and Perron (2003), and their related R package “strucchange” by Zeileis et al. (2015) for more details.
The underbraced constant term above does not change over distinct changepoint configurations and can be neglected in the changepoint configuration comparisons. The above equations show how to estimate model parameters and evaluate model likelihoods given the changepoint configuration; the optimal changepoint configuration is found by a GA search. The penalized likelihoods obtained with two different penalties, MDL and BIC, are presented in Table 1 for the various models used here. Since regression lines are described by two parameters, all regimes are required to be at least three years long (so that fits in any single regime are not perfect).
On the full CET series, GA optimizations of the BIC and MDL penalized likelihoods estimate identical trend shift configurations, both flagging three breaks at the times 1700, 1739, and 1988 (Table 2). This methodological agreement is convenient, but is not typical in changepoint analyses. Figure 2 graphically depicts our model fit. Cooling occurs during the first 39 years, followed by an increasing-trend second regime, with subsequent shifts to two warming trend regimes. The last regime, which starts in 1989, is warming with a trend of 1.1°C per century. When fitting trend shift models to CET series on post-1772 data only, we find a single changepoint in 1987 (Table 3), which is consistent with our analysis on the full series.
Estimated CET trend shift structure. BIC and MDL flag the same changepoints in both the CET series (1700, 1739, 1988; red solid line) and truncated CET (1987; blue dashed line) series when assuming either AR(1) or white noise errors.
Citation: Journal of Climate 35, 19; 10.1175/JCLI-D-21-0489.1
Model fitting results. Here,
Model fitting results based on truncated CET series. Here,
In both cases, the AR(1) correlation estimate is very small (
Other assumptions made on the model errors include normality and a constant variance in Xt. To assess normality, we apply a Shapiro–Wilk test to the model residuals. This test does not reject normality (Tables 2 and 3) at any common levels of statistical significance. To investigate the constant variance assumption, we apply Levene’s test to the residuals. This test does not find evidence of a changing variance in the residuals of the trend shifts models fitted to the CET series at any appreciable levels of statistical significance. Normality and constant variance assumptions in all future fitted models (Tables 2 and 3 list these) is investigated—these features are not rejected in any of the models compared here.
b. A fixed slope mean shift model
A GA was used to estimate this configuration, which is plotted against the data in Fig. 3. For the full CET series, both BIC and MDL flag a single mean shift in 1988, while the single detected shift moves to 1990 in the truncated series (post 1772). Fewer changepoints are detected in this model than with the trend shift models of the previous section, but the time of the single change detected here is consistent with the last changepoint found in the trend shifts models. Since the BIC and MDL penalized likelihoods in Tables 2 and 3 are larger for the constant slope model than for the regime-varying trend slope model, the inference is that regime-varying slopes are preferable.
The estimated CET trend shift structure for the full (red solid line) and truncated CET (blue dashed line) series when a constant regime trend slope is imposed. Both BIC and MDL flag a single changepoint in 1988 for the full series and 1990 for the truncated series.
Citation: Journal of Climate 35, 19; 10.1175/JCLI-D-21-0489.1
c. Joinpin models
There is debate over whether trend models should impose continuity in E[Xt] at the changepoint times in temperature series (Rahmstorf et al. 2017). These so-called joinpin models require E[Xt] = f(t) to be continuous in time t. Here, we compare a joinpin model to the trend shifts and fixed slope mean shift models fitted in the previous sections. Unfortunately, it is not clear what an appropriate MDL penalty is for this case, nor does this seem to be an easy matter to rectify; hence, we proceed with BIC penalties only.
In the formulation of Maidstone et al. (2017a), the white noise variance is fixed and needs to be estimated. While median absolute deviations could be used for this purpose, we instead use the estimated error variance of 0.29 (Table 2), taken from the discontinuous model fits and BIC penalties of the last section. This fit assumes IID errors, which seems plausible given the results of the previous sections. The fitted model flags a single changepoint in 1973 in the full CET series and none in the truncated series; see Tables 2 and 3 and Fig. 4. These fits are stable against changes from 0.29 in the white noise variance. Compared to our previous model fits, the joinpin model has a much higher BIC than the trend shift and fixed slope mean shifts models (Tables 2 and 3). As such, joinpin models do not appear to be competitive.
Estimated CET joinpin shift structure for full (red solid line) and truncated (blue dashed line) series. BIC flags one shift in 1973 in the full series and none for the truncated series.
Citation: Journal of Climate 35, 19; 10.1175/JCLI-D-21-0489.1
While a changepoint seems plausible toward the end of the record due to an increased warming rate, the joinpin fit to the earliest data is poor, similar to the fixed slope mean shifts model. This is graphically evident in the Fig. 4 fits, but is also reflected by the higher BIC scores in Tables 2 and 3. A joinpin model should be used when a discontinuous mean function is unlikely or physically implausible. With the CET series, it is not evident whether the estimated mean function should be continuous or discontinuous. Elaborating, for series containing “only a single station,” mean discontinuities are physically expected. However, when more and more station records are averaged into a composite record, mean function discontinuities are reduced, becoming less pronounced with an increasing number of stations. Should a discontinuous mean function be deemed possible, a trend shift model provides greater flexibility since it can simultaneously approximate a joinpin continuous structure as well as discontinuous shifts (Beaulieu and Killick 2018).
d. Long-memory models
A body of climate literature argues that climate time series exhibit long memory, where the series’ autocorrelation decays slowly in lag, often via a power law (Yuan et al. 2015; Blender and Fraedrich 2003; Franzke 2012). Long-memory correlation and changepoint features can inject similar run properties into a climate series, which is appreciated in the statistical and econometric literatures (Diebold and Inoue 2001; Granger and Hyung 2004; Mills 2007; Yau and Davis 2012). The daily CET series may exhibit long memory (Syroka and Toumi 2001; Franzke 2012).
To fit ARFIMA models, the R package fracdiff (Maechler 2020) was used. A BIC penalty was calculated and is listed in Table 1. An MDL penalty is not informative since this model does not have any changepoints. Long-memory model fits to the full and truncated CET series are described in Tables 2 and 3. The long-memory models have the largest BIC score among all models compared on the full CET time series. On the truncated series, they are also among the least plausible, although joinpin models have higher BIC scores. These results suggest that changepoints, rather than long memory, are more plausible in the CET series. For additional evidence that changepoints are preferred over long-memory features, we applied the time varying wavelet spectrum methods in Norwood and Killick (2018) to the CET series. These methods were used on surface temperatures in Beaulieu et al. (2020) and shown to discriminate changepoint and long-memory models well in long series. The results confirm that a changepoint model is more appropriate than a long-memory model. The fitted model of autoregressive order zero was also preferred to the fitted model of order one, reinforcing that correlation aspects in the CET series are minimal.
e. Model selection uncertainty
Among the six models compared, the trend shift model with white noise is judged the most plausible, as suggested by both BIC and MDL scores. The BIC posterior probabilities for all models fitted above are presented in Table 4. For the full series, the model probability for the trend shift model with white noise is 0.64, followed by the joinpin model with probability 0.12 and the trend shift model with AR(1) errors with probability 0.11. The three other models all have a posterior probability of 0.05 or less. This highlights the uncertainty in the model selected, although the trend shifts models with AR(1) and white noise errors are very similar [the autocorrelation estimated in the AR(1) model is small and both configurations identify the same shifts]. As for the joinpin model, the fit at the start of the record seems poor.
BIC posterior probabilities for models fitted to the full and truncated CET series.
Moving to the truncated series, the trend shift model with white noise has a posterior probability of 0.68. The next most plausible models are the fixed slope mean shift model with AR(1) errors and the trend shift model with AR(1) errors, having posterior probabilities of 0.1 and 0.09, respectively (Table 4). These models are similar in that estimated changepoint times are very close, giving further evidence for a shift in the late 1980s. However, this suggests that a fixed slope model should not be entirely discarded. Unlike results for the full CET series, the joinpin model ranks very low (0.02) on the truncated CET series. This is not surprising given that no changepoint is detected under the joinpin model in the truncated series (Fig. 4).
5. Trends versus mean shifts
The model’s mean structure is compactly written as f(t) = E[Xt] = μr(t), where r(t) ∈ {1,2, …, m + 1} denotes the regime being used at time t.
We discuss only results on the full series here, but conclusions are consistent (i.e., the same changepoints are detected after 1772) if we repeat the analysis on the truncated series only. Fitting this model, seven changepoints are flagged with both MDL and BIC (Fig. 5).
The estimated CET mean shift structure for full (red solid line) and truncated (blue dashed line) series. BIC and MDL detect the same changepoints for both the CET and truncated CET series assuming white noise errors.
Citation: Journal of Climate 35, 19; 10.1175/JCLI-D-21-0489.1
Both penalties pinpoint 1989 as a changepoint time, which is consistent with results of the previous section. Here, MDL and BIC both deem the “cold year” in 1740 an outlier, bracketing this time by two changepoints. Because MDL methods are based on information theory (Rissanen 1978) and not large sample statistical asymptotics, they often flag outliers. Shifts are more frequent at the beginning of the record, perhaps suggesting that the data during these times are less reliable. Evident in the fits is that the last three regimes act to move the series higher in a “staircase,” which is expected for a series experiencing a long-term warming trend (Fig. 5).
The BIC and MDL scores obtained on the full CET series are 648.17 and 656.09, respectively. Should this model be included in our main comparison, one would still prefer the trend shift model should the MDL penalty be used to make conclusions. However, the BIC mean shift score is smaller than the BIC trend shift score in the previous section, indicating a preference for the mean shift model. A model containing only mean shifts will flag a sequence of shifts in an attempt to follow a long-term trend should the data have a trend and it not be included in the model. If the trend is not steep, as is the case here, it is especially challenging to distinguish between trends and mean shifts. To illustrate this, we conducted a simulation study where 500 synthetic series with the same trend magnitude and variability (as estimated in the truncated CET time series over 1772–2020) were generated. The mean shifts plus white noise and trend shifts plus white noise models were fitted to each series. In only 18% of the synthetic series, the correct model with a long-term trend was selected by BIC. Figure 6 presents a histogram of the difference between the two fitted models’ BIC scores, further demonstrating the bias BIC has for the erroneous mean shifts model. Should there be any suspicion about a trend or “staircase feature” in the record, we recommend using techniques that incorporate trends, as done here.
Histogram of differences in BIC scores between the trend and mean-shift models. The correct model is the trend model; however, BIC selects the mean-shift model the majority of the time.
Citation: Journal of Climate 35, 19; 10.1175/JCLI-D-21-0489.1
6. Comments, conclusions, and discussion
This study compared and contrasted several common changepoint model fits for data containing trends, as well as a long-memory autocovariance model, to the CET time series. To our knowledge, this is the first time a detailed changepoint analysis has been conducted on this long record. Starting with a trend shift model, several different changepoint structures were fitted, illustrating the techniques and salient points of changepoint analyses.
Tables 2 and 3 present the log-likelihood, BIC, and MDL scores of all model fits. Depending on the model configuration, we detect either three changepoints (trend shifts models) or one changepoint (fixed slope mean shifts and joinpin models) in the full series. This changepoint count discrepancy traces to the large variations in the series during roughly the first century of the record.
Most models agree on a change to a rapidly warming regime circa 1988, except for the joinpin model (this is also true for the truncated series). Among all fitted models, the optimal one has trend shifts in 1700, 1739, and 1988 (full series), and one in 1988 (truncated series). Table 5 provides estimates of the best fitting model’s intercept and slope parameters by regime. While the best fitting model is the trend shifts model, other models are also plausible (Table 4). Models with higher posterior probabilities tend to be consistent in their flagged changepoint times, but highlight that a fixed slope model (as opposed to the varying slopes in the trend shifts models) may be plausible. Long-memory models yield the highest BIC scores, and are less plausible than all other models compared. The results of the full and truncated CET series are consistent, showing that our post-1772 changepoint inferences are not overly sensitive to inclusion of the first century of the series.
Parameter estimates of the best fitting model: Trend shifts with white noise errors.
Having both BIC and MDL penalties agree on the model type and changepoint configuration adds robustness to our conclusions, suggesting that the fitted segmentations are stable. According to Lavielle (2005), changepoint segmentations that are stable over a range of penalty values should be preferred. Overall, models with shifts were deemed preferable to models having autocorrelated errors.
While our aim is not necessarily directed to the causes of the detected shifts, we provide some interpretations here. Shifts flagged during the first century of the record are likely due to inferior data quality over this early period (Hillebrand and Proietti 2017). Due to lack of overlapping instrumentation coverage before 1722, noninstrumental weather diaries were used to adjust the series (Parker et al. 1992). Observations were generally collected in unheated rooms until 1760 and adjusted by calibrating indoor and outdoor observations later (Parker et al. 1992). Even with the most careful adjustments, one cannot guarantee that all biases were removed from the data. Some authors omit the first century of data altogether due to this issue (Hillebrand and Proietti 2017).
The trend shifts model on the earlier part of the data detects two changepoints in 1700 and 1739, characterizing a steep cooling trend followed by a warming trend. The mean shifts model fitted on the earlier part of the data flags multiple changes (1691, 1699, 1727, 1740, 1741), calling for a closer examination of the earlier part of the record. In data with inhomogeneities, BIC penalties favor mean shift models over trend shift models, even if the trend shifts model is truth. A mean shift model characterizes a warming trend as a staircase of increasing steps. This issue can be troublesome if the trend in the data is weak, as demonstrated in our simulation study (see Fig. 6).
The changepoint flagged in 1988 (from multiple models and in both the full and truncated CET series) is not surprising given the warming seen on the global level in the 1960/70s in a range of surface temperature records, as discussed in studies using both trend shift and joinpin models (Cahill et al. 2015; Beaulieu and Killick 2018; Rahmstorf et al. 2017; Ruggieri 2013). While the more recent part of the CET series is considered more reliable and has been adjusted for inhomogeneities, we cannot entirely discard issues in this era either. Overall, it is possible that a combination of natural and artificial causes contributes to shifts in the CET series.
To further rule out artificial changes, one could subtract all
Residual analyses were conducted to ensure that the underlying assumptions of the model were met. With the CET series, residuals of the trend shift model fit were judged to be uncorrelated (white noise). However, climate time series often exhibit autocorrelation that should be taken into account. We stress the importance of verifying the underlying assumptions in any changepoint model. Indeed, neglecting positive autocorrelation raises the risk of detecting spurious shifts. Also, the series’ autocorrelation may be more complex than an AR(1) process and may itself contain shifts (Beaulieu et al. 2012; Beaulieu and Killick 2018). Some climate series may also contain long-memory autocorrelations (Vyushin et al. 2012). An additional challenge lies with the ambiguity between long-memory and changepoint models: both features can produce series with similar run structures. Because of this, a long-memory model was included as part of our comparison. We found that the CET time series is best represented by a multiple trend shift changepoint structure and not a long-memory model. Such a comparison is not possible for all climate series since lengthy records are required to analyze long-memory series (Beaulieu et al. 2020). The CET time series, which is the longest publicly available surface temperature series, enables this comparison. Other assumptions that were made include constant variance temperatures and normally distributed observations. Both assumptions cannot be rejected in any models fitted (Tables 2 and 3).
Model selection based on criteria does not guarantee that the selected model is “truth.” All models are an approximation of reality, and multiple models can plausibly represent the data. To quantify this, one can calculate posterior model probabilities with BIC that each fitted model is the “quasi-truth.” This assumes that all models included in the comparison have the same prior weight, which may not be reasonable. One must also note that this measure is relative to the models included in the comparison, and does not reflect the uncertainty that the “true” model may not be part of the model set. Similarly, uncertainty in the total number of changepoints and their individual occurrence times is a difficult statistics problem. Bayesian methods, which were not considered here, can in principle place uncertainty margins on the number of changepoints and their locations. When several distinct models have similar penalized likelihood scores, inferences about the number of changepoints are likely to be less reliable. Recent statistics work is now studying this issue (Li et al. 2019; Cappello et al. 2021).
Ultimately the choice of “best model” should be arrived at from a judgment made by the researcher(s) based on objective statistical metrics, such as presented in this work, combined with understanding of the data recording practices and physics of the natural system.
Acknowledgments.
Rebecca Killick gratefully acknowledges funding from EP/R01860X/1 and NE/T006102/1. Robert Lund and Xueheng Shi acknowledge funding from NSF DMS-2113592. The comments of three referees and the editor substantially improved this manuscript.
Data availability statement.
The Central England data used in this study are available at https://www.metoffice.gov.uk/hadobs/hadcet/. We used the annual means from 1659 to 2020.
REFERENCES
Bai, J., and P. Perron, 1998: Estimating and testing linear models with multiple structural changes. Econometrica, 66, 47–78, https://doi.org/10.2307/2998540.
Bai, J., and P. Perron, 2003: Computation and analysis of multiple structural change models. J. Appl. Econ., 18, 1–22, https://doi.org/10.1002/jae.659.
Barry, D., and J. A. Hartigan, 1993: A Bayesian analysis for change point problems. J. Amer. Stat. Assoc., 88, 309–319, https://doi.org/10.2307/2290726.
Beaulieu, C., and R. Killick, 2018: Distinguishing trends and shifts from memory in climate data. J. Climate, 31, 9519–9543, https://doi.org/10.1175/JCLI-D-17-0863.1.
Beaulieu, C., J. Chen, and J. L. Sarmiento, 2012: Change-point analysis as a tool to detect abrupt climate variations. Philos. Trans. Roy. Soc., 370A, 1228–1249, https://doi.org/10.1098/rsta.2011.0383.
Beaulieu, C., R. Killick, D. Ireland, and B. Norwood, 2020: Considering long-memory when testing for changepoints in surface temperature: A classification approach based on the time-varying spectrum. Environmetrics, 31, e2568, https://doi.org/10.1002/env.2568.
Blender, R., and K. Fraedrich, 2003: Long time memory in global warming simulations. Geophys. Res. Lett., 30, 1769, https://doi.org/10.1029/2003GL017666.
Brockwell, P. J., and R. A. Davis, 1991: Time Series: Theory and Methods. Springer-Verlag, 580 pp.
Burnham, K. P., and D. R. Anderson, 2004: Multimodel inference: Understanding AIC and BIC in model selection. Sociol. Methods Res., 33, 261–304, https://doi.org/10.1177/0049124104268644.
Cahill, N., S. Rahmstorf, and A. C. Parnell, 2015: Change points of global temperature. Environ. Res. Lett., 10, 084002, https://doi:10.1088/1748-9326/10/8/084002.
Cappello, L., O. H. M. Padilla, and J. A. Palacios, 2021: Scalable Bayesian change point detection with spike and slab priors. 33 pp., https://doi.org/10.48550/arXiv.2106.10383.
Caussinus, H., and O. Mestre, 2004: Detection and correction of artificial shifts in climate series. J. Roy. Stat. Soc., 53, 405–425, https://doi.org/10.1111/j.1467-9876.2004.05155.x.
Chernoff, H., and S. Zacks, 1964: Estimating the current mean of a normal distribution which is subjected to changes in time. Ann. Math. Stat., 35, 999–1018, https://doi.org/10.1214/aoms/1177700517.
Chib, S., 1998: Estimation and comparison of multiple change-point models. J. Econometrics, 86, 221–241, https://doi.org/10.1016/S0304-4076(97)00115-2.
Chow, G. C., 1960: Tests of equality between sets of coefficients in two linear regressions. Econometrica, 28, 591–605, https://doi.org/10.2307/1910133.
Davis, R. A., T. C. M. Lee, and G. A. Rodriguez-Yam, 2006: Structural break estimation for nonstationary time series models. J. Amer. Stat. Assoc., 101, 223–239, https://doi.org/10.1198/016214505000000745.
Diebold, F. X., and A. Inoue, 2001: Long memory and regime switching. J. Econometrics, 105, 131–159, https://doi.org/10.1016/S0304-4076(01)00073-2.
Eichinger, B., and C. Kirch, 2018: A MOSUM procedure for the estimation of multiple random change points. Bernoulli, 24, 526–564, https://doi.org/10.3150/16-BEJ887.
Fearnhead, P., 2006: Exact and efficient Bayesian inference for multiple changepoint problems. Stat. Comput., 16, 203–213, https://doi.org/10.1007/s11222-006-8450-8.
Franzke, C., 2012: Nonlinear trends, long-range dependence, and climate noise properties of surface temperature. J. Climate, 25, 4172–4183, https://doi.org/10.1175/JCLI-D-11-00293.1.
Fryzlewicz, P., 2014: Wild binary segmentation for multiple change-point detection. Ann. Stat., 42, 2243–2281, https://doi.org/10.1214/14-AOS1245.
Gallagher, C. M., R. Killick, R. Lund, and X. Shi, 2022: Autocovariance estimation in the presence of changepoints. J. Korean Stat. Soc., https://doi.org/10.1007/s42952-022-00173-5.
Granger, C. W. J., and N. Hyung, 2004: Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. J. Empir. Finance, 11, 399–421, https://doi.org/10.1016/j.jempfin.2003.03.001.
Hartmann, D., and Coauthors, 2013: Observations: Atmosphere and surface. Climate Change 2013: The Physical Science Basis, Cambridge University Press, 159–254, doi:10.1017/CBO9781107415324.008.
Harvey, D. I., and T. C. Mills, 2003: Modelling trends in central England temperatures. J. Forecasting, 22, 35–47, https://doi.org/10.1002/for.857.
Hasselmann, K., 1976: Stochastic climate models part I. Theory. Tellus, 28, 473–485, https://doi.org/10.3402/tellusa.v28i6.11316.
Hewaarachchi, A. P., Y. Li, R. Lund, and J. Rennie, 2017: Homogenization of daily temperature data. J. Climate, 30, 985–999, https://doi.org/10.1175/JCLI-D-16-0139.1.
Hillebrand, E., and T. Proietti, 2017: Phase changes and seasonal warming in early instrumental temperature records. J. Climate, 30, 6795–6821, https://doi.org/10.1175/JCLI-D-16-0747.1.
Hsu, D.-A., 1977: Tests for variance shift at an unknown time point. J. Roy. Stat. Soc., 26, 279–284, https://doi.org/10.2307/2346968.
Jandhyala, V., S. Fotopoulos, I. MacNeill, and P. Liu, 2013: Inference for single and multiple change-points in time series. J. Time Ser. Anal., 34, 423–446, https://doi.org/10.1111/jtsa.12035.
Karoly, D. J., and P. A. Stott, 2006: Anthropogenic warming of Central England temperature. Atmos. Sci. Lett., 7, 81–85, https://doi.org/10.1002/asl.136.
Kendon, M., M. McCarthy, S. Jevrejeva, A. Matthews, T. Sparks, and J. Garforth, 2021: State of the UK climate 2020. Int. J. Climatol., 41, 1–76, https://doi.org/10.1002/joc.7285.
Killick, R., P. Fearnhead, and I. A. Eckley, 2012: Optimal detection of changepoints with a linear computational cost. J. Amer. Stat. Assoc., 107, 1590–1598, https://doi.org/10.1080/01621459.2012.737745.
Lavielle, M., 2005: Using penalized contrasts for the change-point problem. Signal Process., 85, 1501–1510, https://doi.org/10.1016/j.sigpro.2005.01.012.
Lee, J., and R. Lund, 2012: A refined efficiency rate for ordinary least squares and generalized least squares estimators for a linear trend with autoregressive errors. J. Time Ser. Anal., 33, 312–324, https://doi.org/10.1111/j.1467-9892.2011.00768.x.
Li, S., and R. Lund, 2012: Multiple changepoint detection via genetic algorithms. J. Climate, 25, 674–686, https://doi.org/10.1175/2011JCLI4055.1.
Li, Y., R. Lund, and A. Hewaarachchi, 2019: Multiple changepoint detection with partial information on changepoint times. Electron. J. Stat., 13, 2462–2520, https://doi.org/10.1214/19-EJS1568.
Lu, Q., and R. Lund, 2007: Simple linear regression with multiple level shifts. Can. J. Stat., 35, 447–458, https://doi.org/10.1002/cjs.5550350308.
Lu, Q., R. Lund, and T. C. M. Lee, 2010: An MDL approach to the climate segmentation problem. Ann. Appl. Stat., 4, 299–319, https://doi.org/10.1214/09-AOAS289.
Lund, R., and J. Reeves, 2002: Detection of undocumented changepoints: A revision of the two-phase regression model. J. Climate, 15, 2547–2554, https://doi.org/10.1175/1520-0442(2002)015<2547:DOUCAR>2.0.CO;2.
Lund, R., and X. Shi, 2020: Short communication: Detecting possibly frequent change-points: Wild binary segmentation 2 and steepest-drop model selection. J. Korean Stat. Soc., 49, 1090–1095, https://doi.org/10.1007/s42952-020-00081-6.
Lund, R., X. L. Wang, Q. Q. Lu, J. Reeves, C. M. Gallagher, and Y. Feng, 2007: Changepoint detection in periodic and autocorrelated time series. J. Climate, 20, 5178–5190, https://doi.org/10.1175/JCLI4291.1.
Maechler, M., C. Fraley, F. Leisch, V. Reisen, A. Lemonte, and R. Hyndman, 2020: Fracdiff: Fractionally differenced ARIMA aka ARFIMA(P,d,q) models. R package version 1.5-1, 14 pp., https://cran.r-project.org/web/packages/fracdiff/fracdiff.pdf.
Maidstone, R., P. Fearnhead, and A. Letchford, 2017a: Detecting changes in slope with an L0 penalty. J. Comput. Graph. Stat., 28, 265–275, https://doi.org/10.1080/10618600.2018.1512868.
Maidstone, R., T. Hocking, G. Rigaill, and P. Fearnhead, 2017b: On optimal multiple changepoint algorithms for large data. Stat. Comput., 27, 519–533, https://doi.org/10.1007/s11222-016-9636-3.
Manley, G., 1953: The mean temperature of central England, 1698–1952. Quart. J. Roy. Meteor. Soc., 79, 242–261, https://doi.org/10.1002/qj.49707934006.
Manley, G., 1974: Central England temperatures: Monthly means 1659 to 1973. Quart. J. Roy. Meteor. Soc., 100, 389–405, https://doi.org/10.1002/qj.49710042511.
Menne, M. J., and C. N. Williams Jr., 2009: Homogenization of temperature series via pairwise comparisons. J. Climate, 22, 1700–1717, https://doi.org/10.1175/2008JCLI2263.1.
Mills, T. C., 2007: Time series modelling of two millennia of Northern Hemisphere temperatures: Long memory or shifting trends? J. Roy. Stat. Soc., 170A, 83–94, https://doi.org/10.1111/j.1467-985X.2006.00443.x.
Mitchell, J. M., Jr., 1953: On the causes of instrumentally observed secular temperature trends. J. Atmos. Sci., 10, 244–261, https://doi.org/10.1175/1520-0469(1953)010<0244:OTCOIO>2.0.CO;2.
Norwood, B., and R. Killick, 2018: Long memory and changepoint models: A spectral classification procedure. Stat. Comput., 28, 291–302, https://doi.org/10.1007/s11222-017-9731-0.
Page, E. S., 1954: Continuous inspection schemes. Biometrika, 41, 100–115, https://doi.org/10.2307/2333009.
Parker, D., and B. Horton, 2005: Uncertainties in central England temperature 1878–2003 and some improvements to the maximum and minimum series. Int. J. Climatol., 25, 1173–1188, https://doi.org/10.1002/joc.1190.
Parker, D., T. P. Legg, and C. K. Folland, 1992: A new daily central England temperature series, 1772–1991. Int. J. Climatol., 12, 317–342, https://doi.org/10.1002/joc.3370120402.
Plaut, G., M. Ghil, and R. Vautard, 1995: Interannual and interdecadal variability in 335 years of central England temperatures. Science, 268, 710–713, https://doi.org/10.1126/science.268.5211.710.
Quandt, R. E., 1958: The estimation of the parameters of a linear regression system obeying two separate regimes. J. Amer. Stat. Assoc., 53, 873–880, https://doi.org/10.1080/01621459.1958.10501484.
Rahmstorf, S., G. Foster, and N. Cahill, 2017: Global temperature evolution: Recent trends and some pitfalls. Environ. Res. Lett., 12, 054001, https://doi.org/10.1088/1748-9326/aa6825.
Rissanen, J., 1978: Modeling by shortest data description. Automatica, 14, 465–471, https://doi.org/10.1016/0005-1098(78)90005-5.
Robbins, M., C. Gallagher, R. Lund, and A. Aue, 2011: Mean shift testing in correlated data. J. Time Ser. Anal., 32, 498–511, https://doi.org/10.1111/j.1467-9892.2010.00707.x.
Robbins, M., C. Gallagher, and R. Lund, 2016: A general regression changepoint test for time series data. J. Amer. Stat. Assoc., 111, 670–683, https://doi.org/10.1080/01621459.2015.1029130.
Rodionov, S. N., 2004: A sequential algorithm for testing climate regime shifts. Geophys. Res. Lett., 31, L09204, https://doi.org/10.1029/2004GL019448.
Ruggieri, E., 2013: A Bayesian approach to detecting change points in climatic records. Int. J. Climatol., 33, 520–528, https://doi.org/10.1002/joc.3447.
Scott, A. J., and M. Knott, 1974: A cluster analysis method for grouping means in the analysis of variance. Biometrics, 30, 507–512, https://doi.org/10.2307/2529204.
Scrucca, L., 2013: GA: A package for genetic algorithms in R. J. Stat. Softw., 53, 1–37, https://doi.org/10.18637/jss.v053.i04.
Serinaldi, F., and C. G. Kilsby, 2016: The importance of prewhitening in change point analysis under persistence. Stochastic Environ. Res. Risk Assess., 30, 763–777, https://doi.org/10.1007/s00477-015-1041-5.
Shi, X., C. Gallagher, R. Lund, and R. Killick, 2022: A comparison of single and multiple changepoint techniques for time series data. Comput. Stat. Data Analysis, 170, 107433, https://doi.org/10.1016/j.csda.2022.107433.
Syroka, J., and R. Toumi, 2001: Scaling of central England temperature fluctuations? Atmos. Sci. Lett., 2, 143–154, https://doi.org/10.1006/asle.2002.0047.
Trewin, B., and Coauthors, 2020: An updated long-term homogenized daily temperature data set for Australia. Geosci. Data J., 7, 149–169, https://doi.org/10.1002/gdj3.95.
Vincent, L. A., M. M. Hartwell, and X. L. Wang, 2020: A third generation of homogenized temperature for trend analysis and monitoring changes in Canada’s climate. Atmos.–Ocean, 58, 173–191, https://doi.org/10.1080/07055900.2020.1765728.
Vyushin, D. I., P. J. Kushner, and F. Zwiers, 2012: Modeling and understanding persistence of climate variability. J. Geophys. Res., 117, D21106, https://doi.org/10.1029/2012JD018240.
Wang, X. L., 2003: Comments on “Detection of undocumented changepoints: A revision of the two-phase regression model.” J. Climate, 16, 3383–3385, https://doi.org/10.1175/1520-0442(2003)016<3383:CODOUC>2.0.CO;2.
Yau, C. Y., and R. A. Davis, 2012: Likelihood inference for discriminating between long-memory and change-point models. J. Time Ser. Anal., 33, 649–664, https://doi.org/10.1111/j.1467-9892.2012.00797.x.
Yuan, N., M. Ding, Y. Huang, Z. Fu, E. Xoplaki, and J. Luterbacher, 2015: On the long-term climate memory in the surface air temperature records over Antarctica: A nonnegligible factor for trend evaluation. J. Climate, 28, 5922–5934, https://doi.org/10.1175/JCLI-D-14-00733.1.
Zeileis, A., F. Leisch, K. Hornik, C. Kleiber, B. Hansen, E. C. Merkle, and M. A. Zeileis, 2015: Package ‘strucchange’. R package version 1.5-1, https://cran.r-project.org/package=strucchange.
