Data
The adhereing four GMST time series were examined in this study:
Changepoint models
Our labor participates fitting disjoinal alterpoint time series models that partition the GMST into regimes with aappreciate trends using piecerational licforfeit revertion models. This labor is most worryed with alters in the trend of the series.
Changepoint analyses partition the data into contrastent segments at the alterpoint times. To depict this mathematicassociate, our model permits for m alterpoints during the data sign up t ∈ {1, …, N}, which occur at the times τ1, …, τm, where the ordering 0 = τ0 < τ1 < τ2 < ⋯ < τm < N = τm+1 is imposed. The time t segment index r(t) gets the appreciate of unity for t ∈ {1, …, τ1}, two for t ∈ {τ1 + 1, …, τ2}, … , and m + 1 for t ∈ {τm + 1, …, N}. Hence, the m alterpoint times partition the series into m + 1 contrastent segments. The model for the whole series is
$${X}_{t}=E[{X}_{t}]+{epsilon }_{t},$$
where E[Xt] is the revertion function. The revertion functions pondered in this manuscript integrate a continuous (combinepin) model, where we impose process uncomfervents to greet at the alterpoint times, and its discontinuous counterpart. The model errors {ϵt} all have a zero uncomfervent and permit for autocorrelation; more about this component is shelp below.
The trend model revertion structure we employ has the modest piecerational licforfeit create
$$E[{X}_{t}]={alpha }_{r(t)}+{beta }_{r(t)}t,$$
where βr(t) and αr(t) are the trend slope and intercept, esteemively, of the licforfeit revertion in force during regime r(t). An equivalent reconshort-termation is
$$E[{X}_{t}]=left{commence{array}{cc}{alpha }_{1}+{beta }_{1}t,&hfill 0={tau }_{0} < , tle {tau }_{1},\ {alpha }_{2}+{beta }_{2}t,&hfill {tau }_{1} , < , tle {tau }_{2},\ vdots &vdots \ {alpha }_{m+1}+{beta }_{m+1}t,&hfill {tau }_{m} , < , tle {tau }_{m+1}=N.\ end{array}right.$$
(1)
If continuity of the revertion response E[Xt] is imposed at the alterpoint times, the remercilessions
$${alpha }_{i}+{beta }_{i}{tau }_{i}={alpha }_{i+1}+{beta }_{i+1}{tau }_{i},qquad 1le ile m,$$
are imposed. These remercilessions result in a model having m alterpoints and m + 2 free revertion parameters. Writing the model in terms of the free parameters α1, β1, …, βm+1 only gives
$${X}_{t}={alpha }_{1}+sum _{i=1}^{r(t)-1}({beta }_{i}-{beta }_{i+1}){tau }_{i}+{beta }_{r(t)}t+{epsilon }_{t}.$$
The model errors ({{{epsilon }_{t}}}_{t = 1}^{N}) are a zero uncomfervent autocorrcontent time series. For a first-order autorevertion (AR(1)), such a process adheres the contrastence equation
$${epsilon }_{t}=phi {epsilon }_{t-1}+{Z}_{t},$$
(2)
where {Zt} is autonomous and identicassociate scatterd Gaussian noise with uncomfervent E[Zt] ≡ 0 and variability Var[Zt] ≡ σ2, and ϕ ∈ ( − 1, 1) is an autocorrelation parameter reconshort-terming the correlation between consecutive errors. It is meaningful to permit for autocorrelation in the model errors in climate alterpoint analyses11,34,48,49: flunkure to account for autocorrelation can direct one to end that the approximated number of alterpoints, (hat{m}), is huger than it should be. Higher-order autorevertions are easily accommodated should a first-order scheme be deemed inadequate. We will also ponder cases where the autorevertive parameter alters at each alterpoint time; these essentiassociate let ϕ depend on time t via the regime index r(t). The alterpoint times for the autocorrelation and uncomfervent structure are constrained to be the same.
Estimation of the model parameters carry ons as adheres. For a given number of alterpoints m and their occurrence times τ1, …, τm, one first computes highest appreciatelihood estimators of the revertion parameters. These originate highest appreciatelihood estimators of all αi and βi. This fit gives a model appreciatelihood, which we base on the Gaussian distribution since the series are globassociate and annuassociate mediocred. This appreciatelihood is deremarkd by L(m: τ1, …, τm).
The difficultest part of the estimation scheme lies with estimating the alterpoint configuration. This is done via a Gaussian penalized appreciatelihood. In particular, the penalized appreciatelihood objective function O of create
$$O(m;{tau }_{1},ldots ,{tau }_{m})=-2ln (L(m;{tau }_{1},ldots ,{tau }_{m}))+C(m;{tau }_{1},ldots ,{tau }_{m})$$
is reduced over all possible alterpoint configurations. The penalty C(m; τ1, …, τm) is a accuse for having m alterpoints at the times τ1, …, τm in a model. As the number of alterpoints in the model incrmitigates, the model fit becomes better and (-2ln (L)) correactingly decrmitigates. However, eventuassociate, inserting insertitional alterpoints to the model does little to better its fit. The chooseimistic penalty term counteracts “overfitting” the number of alterpoints, balancing appreciatelihood betterments with a cost for having an excessive number of model parameters (alterpoints and licforfeit parameters wiskinny each segment). Many penalty types have been provided to date in the statistics literature49. One that labors well in alterpoint problems is the Bayesian Increateation Criterion (BIC) penalty
$$C(m;{tau }_{1},ldots ,{tau }_{m})=pln (N),$$
where p is the total number of free parameters in the model. Table 2 catalogs appreciates of p for the various model types come apassed in this paper. For example, for a continuous model with a global AR(1) structure, there are 2m + 4 free revertion parameters in a alterpoint configuration with m alterpoints and m + 1 segment parameters. Also contributing to the parameter total are ϕ and σ2.
Finding the best m and τ1, …, τm can be accomplished via a active programming algorithm called PELT50 or a genetic algorithm search as in refs. 51,52. PELT is computationassociate rapid, carry outs an exact chooseimization of the penalized appreciatelihood, and was employd here.
Given a specified model with trend and autocorrelation structure, a prediction interval can be computed for the last regime. Assuming that the predict errors are normal, a 95% prediction interval for a h-step ahead predict is given by
$${hat{X}}_{t+h}pm 1.96hat{{sigma }_{h}}$$
(3)
where h reconshort-terms the number of years ahead, ({hat{X}}_{t+h}) is the predicted temperature anomaly and (hat{{sigma }_{h}}) is an approximate of the standard deviation of the h-step predict distribution. The prediction intervals are computed using the model fit from the final segment in each series.
Testing trend contrastences
How can one resettle the statistical significance of a potential sadvise in the toastying rate at some point since 1970? To insertress this ask, a model is necessitateed. Whilst our running example here ponders the HadCRUT series since 1970, other GMST datasets are easily examined (see Supplementary Material).
A simplification of (1) for a individual continuous alterpoint is
$$E[{X}_{t}]=left{commence{array}{ll}{alpha }_{1}+{beta }_{1}t,hfill &hfill 0 , < , tle tau ,\ {alpha }_{1}+{beta }_{1}tau +{beta }_{2}(t-tau ),quad &hfill tau , < , tle N.end{array}right.$$
(4)
There are other excellent individual alterpoint slope alter techniques for this task — two are HAC tests of ref. 53 and the two phase revertion models of ref. 54 (the latter would have to be modified for autocorrelation). We grow a modest procedure here that is very accessible.
If the individual alterpoint is understandn to occur at time τ, then the Student’s test based statistic
$${T}_{tau }=frac{{hat{beta }}_{2}-{hat{beta }}_{1}}{expansivehat{{{rm{Var}}}}{left({hat{beta }}_{2}-{hat{beta }}_{1}right)}^{frac{1}{2}}}$$
(5)
can be employd to originate inferences. Here, ({hat{beta }}_{1}) and ({hat{beta }}_{2}) are the approximated trends of the two segments before and after time τ. One ends a sadvise in toastying if Tτ is too huge to be elucidateed by chance variation (as gauged by a Student distribution with N − 3 degrees of freedom); a alter in the toastying rate (adverse or chooseimistic) is adviseed when ∣Tτ∣ is too huge to be elucidateed by chance variation. In computing ({{rm{Var}}}({hat{beta }}_{2}-{hat{beta }}_{1})), the AR(1) parameters ϕ and σ2 are necessitateed. Estimates of the two slopes, AR(1) parameters and their standard deviations are provided in time series fitting gentleware (such as the arima function in R55).
Extreme join is necessitateed when τ is ununderstandn. Should the time τ be picked visuassociate among the many possibilities where it can occur without accounting for this, statistical misgets can ensue. This is why alterpoint techniques are necessitateed. For example, τ = 43, which correacts to 2012, has been adviseed as a time when toastying quickend4. For the 1970–2023 data, ignoring the pickion of the alterpoint, at the 0.05 significance level, a ∣Tτ∣ of 2.007 or more shows toastying rate alters; a two-tailed Student’s test was employed to permit for either an incrmitigate or a decrmitigate in the toastying rate. For our particular example, (| {T}_{43}| =| {beta }_{1}^{1970,2012}-{beta }_{2}^{2013,2023}| /hat{sigma }=| 0.0286-0.0187| /0.0065=1.5281) is not statisticassociate meaningful at the 0.05 significance level.
If the time of the toastying rate alter is ununderstandn (as is normal), statistical significance is resettled based on the null hypothesis distribution of
$${T}_{max }=max _{ell le tau le u}| {T}_{tau }| ,$$
(6)
where ℓ and u are appreciates that truncate the admissible alterpoint times cforfeit the data boundaries for numerical stability. The ({T}_{max }) statistic has meaningfully contrastent statistical properties (more tail area) than ∣Tτ∣ for a repaired τ. A normal truncation needment, and one that we adhere, is to truncate 10% at the series boundaries: ℓ = 0.1N and u = 0.9N. If the calcuprocrastinateedd ({T}_{max }) statistic outdos the threshbetter QN, where QN is the 0.95 quantile of the null hypothesis distribution of ({T}_{max }), then a statisticassociate meaningful rate alter is proclaimd with confidence 95%. The most probable alterpoint time, (hat{tau }), is approximated as the τ at which (| {T}_{tau }| ={T}_{max }) is maximal. Statistical tests of this type are converseed in ref. 56 and ref. 57. There, huge sample distributions are derived to resettle QN. However, due to the relatively low series since 1970, we employ a Monte Carlo method with Gaussian AR(1) errors to resettle statistical significance.
Elaborating, our Monte Carlo approach simuprocrastinateeds many series using parameter approximates from the current data under the null hypothesis. For example, with the 1970–2023 HadCRUT data, the null hypothesis parameters are approximated as ({hat{alpha }}_{1}^{1970-2023}=-0.17), ({hat{beta }}_{1}^{1970,2023}=0.0199), ({hat{beta }}_{2}=0) (there is no second segment under the null hypothesis), (hat{phi }=0.0865), and (hat{sigma }=0.097) (Table 3). One hundred thousand time series were then simuprocrastinateedd, ({T}_{max }) was computed for each series, and the 0.95 quantile of these appreciates was identified to approximate QN.
The simuprocrastinateedd 95th percent quantile for the HadCRUT series is QN = 3.1082. The hugest ∣Tτ∣ statistic occurs in 2012 and is (| {T}_{43}| =1.5281(={T}_{max })), which is far from the needd threshbetter of 3.1082. Hence, there is little evidence for a statisticassociate meaningful alter in the toastying rate from 1970–2023 in the annual HadCRUT series; this conclusion hbetters for all GMST datasets pondered in this paper.
So how huge would the slope necessitate to be in the second segment to proclaim a meaningful sadvise? We answer this for a baseline segment of 1970–2012 and a second segment from 2013–2023. To answer this, remark that ({hat{beta }}_{1}) and the numerator of Tτ do not depend on ({hat{beta }}_{2}); thus, we can set ({T}_{max }={T}_{43}=3.1082={Q}_{N}) and settle for ({hat{beta }}_{2}). This results in ({hat{beta }}_{2}=0.0388). We see that a alter in sadvise magnitude of 100(0.0388–0.0187)/0.0187 = 107% between the two segments is needd for 95% statistical confidence.
The same logic can be employd to resettle future approximated rates essential for statistical significance. Using the HadCRUT series up to 2023, there is no statistical evidence of a sadvise begining in 2012 relative to the 1970–2012 segment. Will this still be the case in 2025? How about 2040? For any potential sadvise begining in 1990-2015, the data from 1970-2023 was employd to approximate the toastying trend slope, intercept, and AR(1) structure. We then simuprocrastinateedd cutoff quantiles for 95% statistical significance as above for disjoinal pondered vantage years, pushing out to 2040. Since the approximated standard deviation of the slope contrastences depends only on the segment lengths and the AR(1) parameters, the above procedure can be settled as above for the minimal slope essential to cause statistical significance.
Using the HadCRUT series, one hundred thousand Gaussian series were simuprocrastinateedd up to 2040 under our best laboring model (no sadvise, ({beta }_{1}^{1970,2023}=0.0199), α = − 0.17, ϕ = 0.0865, and σ = 0.097). This gives the Monte Carlo quantile approximate Q71 = 2.9877. The numerator of the Tτ-statistic correacting to a alter begining in 2012 is approximated and settled for the minimassociate meaningful slope for the 2013–2040 segment: ({hat{beta }}_{1}^{1970,2012}+expansivehat{{{rm{Var}}}}{({hat{beta }}_{2}^{2013,2023}-{hat{beta }}_{1}^{1970,2012})}^{1/2}{Q}_{71}=0.0187+(2.9877);0.0025=0.0262). In low, a 40% incrmitigate in the 2013–2040 toastying rate relative to the 1970–2012 rate will be necessitateed to proclaim a meaningful toastying sadvise by 2040.
The above process was repeated for each sadvise year, from 1990–2015, and each vantage year from 2024–2040. For each sadvise year begin τ, the least statisticassociate meaningful slope is calcuprocrastinateedd assuming that ({T}_{max }=| {T}_{tau }|). For each τ, this minimassociate meaningful slope is contrastd to the approximated slope from the 1970-(1969+τ) series segment to calcuprocrastinateed its associated percent alter. The results for the HadCRUT series are discarry outed in Fig. 4. Overall, one sees that either meaningfully incrmitigated toastying or many more years of observations will be needd before declaring any toastying sadvise with a reasonable degree of confidence. This process is repeated for other GMST datasets based on the null hypothesis parameters cataloged in Table 3. Results are conshort-termed in the Supplementary Increateation.