Time series forecasting for global temperature: an outside view of climate forecasting

Note: In this blog post, I refer­ence a num­ber of blog posts and aca­demic pa­pers. Two caveats to these refer­ences: (a) I of­ten refer­ence them for a spe­cific graph or calcu­la­tion, and in many cases I’ve not even ex­am­ined the rest of the post or pa­per, while in other cases I’ve ex­am­ined the rest and might even con­sider it wrong, (b) even for the parts I do refer­ence, I’m not claiming they are cor­rect, just that they provide what seems like a rea­son­able ex­am­ple of an ar­gu­ment in that refer­ence class.

Note 2: Please see this post of mine for more on the pro­ject, my sources, and po­ten­tial sources for bias.

As part of a re­view of fore­cast­ing, I’ve been look­ing at weather and cli­mate fore­cast­ing. I wrote one post on weather fore­cast­ing and an­other on the differ­ent time hori­zons for weather and cli­mate fore­cast­ing. Now, I want to turn to long-range cli­mate fore­cast­ing, for mo­ti­va­tions de­scribed in this post of mine.

Cli­mate fore­cast­ing is turn­ing out to be a fairly tricky topic to look into, partly be­cause of the in­her­ent com­plex­ity of the task, and partly be­cause of the poli­ti­ciza­tion sur­round­ing An­thro­pogenic Global Warm­ing (AGW).

I de­cided to be­gin with a some­what “out­side view” ap­proach: if you were sim­ply given a time se­ries of global tem­per­a­tures, what sort of pat­terns would you see? What fore­casts would you make for the next 100 years? The fore­cast can be judged against a no-change fore­cast, or against the fore­casts put out by the widely used cli­mate mod­els.

Below is a chart of four tem­per­a­ture prox­ies since 1880, cour­tesy NASA:

Global Surface Temperature

The Hadley Cen­tre dataset goes back to 1850. Here it is (note that the cen­trings on the tem­per­a­ture axis are slightly differ­ent, be­cause we are tak­ing means of slightly differ­ent sets of num­bers, but we are any­way in­ter­ested only in the trend so that does not mat­ter) (source):


Eye­bal­ling, there does seem to be a sec­u­lar trend of in­crease in the tem­per­a­ture data. Per­haps the naivest way of calcu­lat­ing the rate of change is to calcu­late (fi­nal tem­per­a­ture—ini­tial tem­per­a­ture)/​(time in­ter­val) to calcu­late the an­nual rate of change. Us­ing that method, we get a tem­per­a­ture in­crease of about 0.54 de­grees Cel­sius per cen­tury.

But just us­ing fi­nal and ini­tial tem­per­a­tures over­weights those two val­ues and ig­nores the data in the other tem­per­a­ture read­ings. A some­what more so­phis­ti­cated ap­proach (albeit still a pretty un­so­phis­ti­cated ap­proach) is a lin­ear re­gres­sion model. I was won­der­ing whether I should down­load the data and run a lin­ear re­gres­sion, but I found a pic­ture of the re­gres­sion on­line (source):

Linear regression for temperatures

Note that the re­gres­sion line starts off a lit­tle lower than the ac­tual tem­per­a­ture in 1850, and also ends a lit­tle lower than the ac­tual tem­per­a­ture in the 2000s. The rate of growth seems even less here (about 0.4 de­grees Cel­sius per cen­tury). The rea­son the re­gres­sion gives a lower rate than sim­ply us­ing ini­tial and fi­nal tem­per­a­tures is that the tem­per­a­ture growth since the 1970s has been well above trend, and those well-above-trend tem­per­a­tures are given more weight if we just use fi­nal tem­per­a­ture than if we fit to a re­gres­sion line.

Lin­ear plus pe­ri­odic?

Another plau­si­ble story that seems to emerge from eye­bal­ling the model is that the tem­per­a­ture trend is the sum of an ap­prox­i­mately lin­ear trend and a pe­ri­odic trend, given by some­thing like a sine wave. I found one anal­y­sis of this sort by DocMar­tyn on Ju­dith Curry’s blog, and an­other in a pa­per by Syun Aka­sofu (note: there seem to be some prob­lems with both analy­ses; I am link­ing to them mainly as sim­ple ex­am­ples of the rough na­ture of this sort of anal­y­sis, not as some­thing to be taken very se­ri­ously). Note that both of these do more com­pli­cated things than look purely at tem­per­a­ture trends. While DocMar­tyn ex­plic­itly in­tro­duces car­bon diox­ide as the source of the lin­ear-ish trend, Aka­sofu iden­ti­fies “re­cov­ery from the lit­tle Ice Age” as the source of the lin­ear-ish trend and the Pa­cific Decadal Os­cilla­tion as the source of the sinu­soidal trend (but as far as I can make out, one could use the same graph and ar­gue that the lin­ear trend is driven by car­bon diox­ide).

Here’s DocMar­tyn’s fore­cast:

DocMartyn's forecast

Here’s Aka­sofu’s pic­ture:


Au­to­cor­re­la­tion and ran­dom walks

Sim­ple lin­ear re­gres­sion is un­suit­able for time se­ries fore­cast­ing for vari­ables that ex­hibit au­to­cor­re­la­tion: the value in any given year is cor­re­lated to the value the pre­vi­ous year, in­de­pen­dent of any long-term trend. As Ju­dith Curry ex­plains here, au­to­cor­re­la­tion can cre­ate an illu­sion of trends even when there aren’t any. (This may seem a bit coun­ter­in­tu­itive: if only tem­per­a­ture lev­els, and not tem­per­a­ture trends, ex­hibit the au­to­cor­re­la­tion, i.e., if tem­per­a­ture is ba­si­cally a ran­dom walk, then why should we see spu­ri­ous trends? So read the whole post). Not only can ap­par­ent spu­ri­ous lin­ear-look­ing trends be de­tected, so can ap­par­ent spu­ri­ous cycli­cal trends (see here).

Un­for­tu­nately, I don’t have a good un­der­stand­ing of the statis­ti­cal tools (such as ARIMA) that one would use to re­solve such ques­tions. I am aware of a few pa­pers that have tried to demon­strate that, de­spite the ap­pear­ance of a lin­ear trend above, the tem­per­a­ture se­ries is more con­sis­tent with a ran­dom walk model. See, for in­stance, this pa­per by Ter­ence Mills and the liter­a­ture it refer­ences, many of which seem to come to con­clu­sions against a clear lin­ear trend. Mills also pub­lished a pa­per in the Jour­nal of Cos­mol­ogy here that’s un­gated and seems to cover similar ground, but the Jour­nal of Cos­mol­ogy is not such a high-sta­tus jour­nal, so the pub­li­ca­tion of the pa­per there should not be treated as giv­ing it more au­thor­ity than a blog post.

Lin­ear in­crease is con­sis­tent with very sim­ple as­sump­tions about car­bon diox­ide con­cen­tra­tions and the an­thro­pogenic global warm­ing hypothesis

Here’s a sim­ple model that would lead to tem­per­a­ture in­creases be­ing lin­ear over time:

  • The only sec­u­lar trend in tem­per­a­ture oc­curs from ra­di­a­tive forc­ing due to a change in car­bon diox­ide con­cen­tra­tion.

  • The ad­di­tive in­crease in tem­per­a­ture is pro­por­tional to the log­a­r­ithm of the mul­ti­plica­tive in­crease in at­mo­spheric car­bon diox­ide con­cen­tra­tion (Wikipe­dia).

  • About 50% of car­bon diox­ide emis­sions from burn­ing fos­sil fuels is re­tained by the at­mo­sphere. The mag­ni­tude of car­bon diox­ide emis­sions is pro­por­tional to world GDP, which is grow­ing ex­po­nen­tially, so emis­sions are grow­ing ex­po­nen­tially, and there­fore, the to­tal car­bon diox­ide con­cen­tra­tion in the at­mo­sphere is also grow­ing ex­po­nen­tially.

Ap­ply a log­a­r­ithm to an ex­po­nen­tial, and you get a lin­ear trend line in tem­per­a­ture.

(As we’ll see, while this looks nice on pa­per, ac­tual car­bon diox­ide growth hasn’t been ex­po­nen­tial, and ac­tual tem­per­a­ture growth has been pretty far from lin­ear. But at least it offers some prima fa­cie plau­si­bil­ity to the idea of fit­ting a straight line).

Turn­ing on the heat: the time se­ries of car­bon diox­ide concentrations

So how have car­bon diox­ide con­cen­tra­tions been grow­ing? Since 1958, the Mauna Loa ob­ser­va­tory in Hawaii has been track­ing at­mo­spheric car­bon diox­ide con­cen­tra­tions. The plot of the con­cen­tra­tions is termed the Keel­ing curve. Here’s what it looks like (source: Wikipe­dia):

Keeling curve

The growth is suffi­ciently slow that the dis­tinc­tion be­tween lin­ear, quadratic, and ex­po­nen­tial isn’t visi­ble to the naked eye, but if you look care­fully, you’ll see that growth from 1960 to 1990 was about 1 ppm/​year, whereas growth from 1990 to 2010 was about 2 ppm/​year. Un­for­tu­nately the Mauna Loa data go back only to 1958. But there are other data sources. In a blog post at­tempt­ing to com­pute equil­ibrium cli­mate sen­si­tivity, Jeff L. finds that the 1832-1978 Law Dome dataset does a good job of match­ing at­mo­spheric car­bon diox­ide con­cen­tra­tion val­ues with the Mauna Loa dataset for the pe­riod of overal (1958-1978), so he splices the two datasets for his (note: com­menters to the post pointed out many prob­lems with it, and while I don’t know enough to eval­u­ate it my­self, my limited knowl­edge sug­gests that the crit­i­cisms are spot on; how­ever, I’m us­ing the post just for the car­bon diox­ide graph):

law dome

Note that it’s fairly well-es­tab­lished that car­bon diox­ide con­cen­tra­tions in the 18th cen­tury, and prob­a­bly for a few cen­turies be­fore that, were about 280 ppm. So even if the speci­fics of the Law Dome dataset aren’t re­li­able, the broad shape of the curve should be similar. No­tice that the growth from 1832 to around 1950 was fairly slow. In fact, even from 1900 to 1940, the rel­a­tively fastest-grow­ing part of the pe­riod, car­bon diox­ide con­cen­tra­tions grew by only 15 ppm in 40 years. From what I can judge, there seems to have been an abrupt shift around 1950, to a rate of about 1 ppm/​year. A lin­ear or ex­po­nen­tial curve doesn’t ex­plain the shift. And as noted ear­lier, the rate of growth seems to have gone up a lot around 1990 again, to about 2 ppm/​year. The rea­son for the shift around 1950 is prob­a­bly post-World War II global eco­nomic growth, in­clud­ing in­dus­tri­al­iza­tion in the now-be­com­ing-in­de­pen­dent colonies, and the rea­son for the shift around 1990 is prob­a­bly the rapid take-off of eco­nomic growth in In­dia, com­bined with the ac­cel­er­a­tion of eco­nomic growth in China.

To the ex­tent that the AGW hy­poth­e­sis is true, i.e., the main source of long-term tem­per­a­ture trends is ra­di­a­tive forc­ing based on changes to car­bon diox­ide con­cen­tra­tions, per­haps look­ing for a lin­ear trend isn’t ad­vis­able, be­cause of the sig­nifi­cant changes to the rate of car­bon diox­ide growth over time (speci­fi­cally, the fact that car­bon diox­ide con­cen­tra­tions don’t grow ex­po­nen­tially, but have his­tor­i­cally ex­hibited a piece­wise growth pat­tern). So per­haps it makes sense to di­rectly regress tem­per­a­ture against the log­a­r­ithm of car­bon diox­ide con­cen­tra­tion? Two such ex­er­cises were linked above: DocMar­tyn on Ju­dith Curry’s blog, and a blog post at­tempt­ing to com­pute equil­ibrium cli­mate sen­si­tivity by Jeff L. Both seem like de­cent first passes but are also prob­le­matic in many ways.

One of the main prob­lems is that the tem­per­a­ture re­sponse to car­bon diox­ide con­cen­tra­tion changes doesn’t all oc­cur im­me­di­ately. So the mem­o­ryless re­gres­sion ap­proach used by Jeff L., that ba­si­cally just asks how cor­re­lated tem­per­a­ture in a given year is with car­bon diox­ide con­cen­tra­tions in that year, fails to ac­count for the fact that tem­per­a­ture in a given year may be in­fluenced by car­bon diox­ide con­cen­tra­tions over the last few years. Ba­si­cally, there could be a lag be­tween the in­crease in car­bon diox­ide con­cen­tra­tions and the full in­crease in tem­per­a­tures.

Still, the prima fa­cie story doesn’t seem to be bod­ing well for the AGW hy­poth­e­sis:

  • Car­bon diox­ide con­cen­tra­tions have not only been ris­ing, they’ve been ris­ing at an in­creas­ing rate, with no­table changes in the rate of in­crease around 1950 and then again around 1990.

  • Tem­per­a­ture ex­hibits fairly differ­ent trends. It was about flat from 1945-1978, then grew very quickly around 1978-1998, and then has been about flat (with a very minor warm­ing trend) 1998-pre­sent.

So, even a story of car­bon diox­ide with lag doesn’t provide a good fit for the ob­served tem­per­a­ture trend.

There are a few differ­ent ways of re­solv­ing this. One is to re­turn to the point made ear­lier about how the ac­tual tem­per­a­ture is a sum of the lin­ear trend (driven by green­house gas forc­ing) plus a bunch of pe­ri­odic trends, such as those driven by the PDO, AMO, and so­lar cy­cles. This sort of story was de­scribed by DocMar­tyn on Ju­dith Curry’s blog and in the pa­per by Syun Aka­sofu refer­enced above.

Another com­mon ex­pla­na­tion is that the 1945-1978 non-warm­ing (and, ac­cord­ing to some datasets, mod­er­ate cool­ing) is ex­plained by the in­creased con­cen­tra­tion of aerosols that blocked sun­light, and that there­fore can­celed the warm­ing effect of car­bon diox­ide. In­deed, in the early 1970s, there were con­cerns about global cool­ing due to aerosols, but there were also a few voices that noted that over the some­what longer run, as aerosol con­cen­tra­tions were con­trol­led bet­ter, the green­house effect would dom­i­nate and we’d see rapid tem­per­a­ture in­creases. And given the way tem­per­a­tures un­folded in the 1980s and 1990s, the peo­ple who were call­ing for global warm­ing in the 1970s seemed un­usu­ally pre­scient. But the pause (or at any rate, sig­nifi­cant slow­down) in warm­ing af­ter 1998, de­spite the fact that the rate of car­bon diox­ide emis­sions has been ac­cel­er­at­ing, sug­gests that there’s more to the story than just aerosols and car­bon diox­ide.

UPDATE: Some peo­ple have ques­tioned whether there was a pause or slow­down at all, and whether us­ing 1998 as a start year may be mis­guided be­cause it was an un­usu­ally hot year due to a strong El Nino. 1998 was un­usu­ally hot, and in­deed the lack of warm­ing rel­a­tive to 1998 for the next few years was ex­plain­able in terms of 1998 be­ing an anomaly. But the time pe­riod since then is suffi­ciently long that the slow­ness of warm­ing can’t just be ex­plained by 1998 be­ing very warm. For a list of the range of ex­pla­na­tions offered for the pause in warm­ing, see here.

Should we start us­ing ac­tual cli­mate sci­ence now?

The dis­cus­sions above were very light on both cli­mate sci­ence the­ory and heavy­brow statis­ti­cal the­ory. We just looked at global tem­per­a­ture and car­bon diox­ide trends, eye­balled the graphs, and tried to rea­son what sort of growth pat­terns were there. We didn’t talk about what the the­ory says, what in­de­pen­dent lines of ev­i­dence there are for it, what sort of other in­di­ca­tors (such as re­gional tem­per­a­tures) might be used to test the the­ory, and what his­tor­i­cal (pre-1800) data can tell us.

A more se­ri­ous anal­y­sis would con­sider all these. But here is what I be­lieve: if a more com­pli­cated model can­not con­sis­tently beat out sim­ple mod­els such as those based on per­sis­tence, ran­dom walk, sim­ple lin­ear re­gres­sion, ran­dom walk with drift, etc., then the model has not re­ally ar­rived at prime time for fore­cast­ing. There may still be in­sights to be gleaned from the model, but its abil­ity to fore­cast the fu­ture is not one of its sel­l­ing points.

The his­tory of cli­mate mod­el­ing so far sug­gests that such suc­cess has been elu­sive (see this draft pa­per by Kesten C. Green, for in­stance). In hind­sight from a 1990s van­tage point, those in the 1970s who bucked the “global cool­ing” trend and ar­gued that the green­house effect would dom­i­nate seemed very pre­scient. But the con­sid­er­able slow­down of warm­ing start­ing around 1998, even as car­bon diox­ide con­cen­tra­tions grew rapidly, took them (and many oth­ers) by sur­prise. We should keep in mind that there are many sto­ries in fi­nan­cial mar­kets of trad­ing strate­gies that ap­pear to have been suc­cess­ful for long pe­ri­ods of time, far ex­ceed­ing what chance alone might sug­gest, but then sud­denly stop work­ing. The fi­nan­cial mar­kets are differ­ent from the cli­mate (in that there are hu­mans com­pet­ing, and eat­ing away at each other’s strate­gies) but the prob­lem still re­mains that some­thing (like “the earth is warm­ing”) may have been true over some decades for rea­sons quite differ­ent from those posited by peo­ple who suc­cess­fully pre­dicted them.

Note that even with­out the abil­ity to make ac­cu­rate or use­ful cli­mate fore­casts, many tenets of the AGW hy­poth­e­sis may hold, and may use­fully in­form our un­der­stand­ing of the fu­ture. For in­stance, it could be that the cyclic trends and sources of ran­dom vari­a­tion are big­ger than we thought, but the part of the in­crease in tem­per­a­tures due to in­creas­ing car­bon diox­ide con­cen­tra­tions (mea­sured us­ing the tran­sient cli­mate re­sponse or the equil­ibrium cli­mate sen­si­tivity) is still quite large. Which ba­si­cally means we will see (large in­crease) + (large vari­a­tion). In which case the large in­crease still mat­ters a lot, but would be hard to de­tect us­ing cli­mate fore­cast­ing, and would be hard to use to make bet­ter cli­mate fore­casts. But if that’s the case, then it’s im­por­tant to be all the more sure of the other lines of ev­i­dence that are be­ing used to at­tain the equil­ibirum cli­mate sen­si­tivity es­ti­mate. More on this later.

Cri­tique of insularity

I want to briefly men­tion a cri­tique offered by fore­cast­ing ex­perts J. Scott Arm­strong and Kesten Green (I men­tioned both of them in my post on gen­eral-pur­pose fore­cast­ing and the as­so­ci­ated com­mu­nity). Their Global Warm­ing Au­dit (PDF sum­mary, web­site with many re­sources) looks at many cli­mate fore­cast­ing ex­er­cises from the out­side view, and finds that the cli­mate fore­cast­ers pay lit­tle at­ten­tion to gen­eral fore­cast­ing prin­ci­ples. One might de­tect a bit of a self-serv­ing el­e­ment here: Arm­strong isn’t happy that the cli­mate fore­cast­ers are en­gag­ing in such a big and mon­u­men­tal ex­er­cise with­out con­sult­ing him or refer­ring to his work, and an un­char­i­ta­ble read­ing is that he is feel­ing slighted at be­ing ig­nored. On the other hand, if you be­lieve that the fore­cast­ing com­mu­nity has come up with valuable in­sights, their cri­tique that cli­mate fore­cast­ers didn’t even con­sider the in­sight ob­tained by the fore­cast­ing com­mu­nity in their work is a fairly pow­er­ful crit­i­cism. (Things may have changed some­what since Arm­strong and Green origi­nally pub­lished their cri­tique). Broadly, I agree with some of Am­strong and Green’s main points, but I think their cri­tique goes over­board in some ways (to quite an ex­tent, I agree with Nate Silver’s treat­ment of their cri­tique in Chap­ter 12 of The Sig­nal and the Noise). But more on that later. Also, I don’t know how rep­re­sen­ta­tive Arm­strong and Green are of the fore­cast­ing com­mu­nity in their view on the state of cli­mate fore­cast­ing.

I have also heard anec­do­tal ev­i­dence of similar cri­tiques of in­su­lar­ity from statis­ti­ci­ans, ge­ol­o­gists, and weather fore­cast­ers. In each case, the claim has been that the work in cli­mate sci­ence re­lied on meth­ods and in­sights bet­ter de­vel­oped in the other dis­ci­plines, but the cli­mate sci­en­tists did not ad­e­quately con­sult ex­perts in those do­mains, and as a re­sult, made el­e­men­tary er­rors (even though these er­rors may not have af­fected their fi­nal con­clu­sions). I cur­rently don’t have a clear pic­ture of just how wide­spread this crit­i­cism is, and how well-jus­tified it is. I’ll be dis­cussing it more in fu­ture posts, not so much be­cause it is di­rectly im­por­tant but be­cause it gives us some idea of how au­thor­i­ta­tive to con­sider the state­ments of cli­mate sci­en­tists in do­mains where di­rect ver­ifi­ca­tion or ob­ject-level en­gage­ment is difficult.

Look­ing for feedback

Since I’m quite new to cli­mate sci­ence and (largely, though not com­pletely) new to statis­ti­cal anal­y­sis, it’s quite pos­si­ble that I made some el­e­men­tary er­rors above. Cor­rec­tions would be ap­pre­ci­ated.

It should be noted that when I say a par­tic­u­lar work has prob­lems, it is not a defini­tive state­ment that that work is false. Rather, it’s sim­ply a state­ment of my im­pres­sion, based on a cur­sory anal­y­sis, that de­scribes the amount of cred­i­bil­ity I as­so­ci­ate with that work. In many cases, I’m not qual­ified enough to offer a cri­tique with high con­fi­dence.