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Monday, December 8, 2014

Change points in global surface temperature: and by what magic is earth warming?

Sou | 7:09 PM Go to the first of 28 comments. Add a comment

Quite often you'll read deniers claiming that the earth is warming by magic, or supernatural forces. No denier will ever use the words "magic" or "supernatural". They'll pick a euphemism, like "Little Ice Age" - though how an ice age caused warming I've yet to see anyone explain.

An ice age won't cause cooling or warming. An ice age is a state not a force. Sometimes a fake sceptic will add that it's because of a recovery from an ice age, as if warm was a natural state and cold an unnatural state. Yet that would require an explanation of how earth got into the unnatural state and what forced it out of that unnatural state. A lot of deniers think of climate as a bouncing ball minus the forcing that causes the ball to bounce. And they don't seem to mind that the bounce up in the global surface temperature is showing no sign of bouncing back down, although some of them swear that we're heading for an ice age - any day now.

Occasionally there'll be a vague explanation for the unnatural state, usually something to do with the sun - but with little enthusiasm.  (The sun may have played a minor role during the Little Ice Age, recent research points to the cause of it being heightened volcanic activity, which is thought to have driven changes in the ocean which caused more cooling. See papers below.) And when one asks, well what about the decline in solar forcing recently - why hasn't it cooled down? You'll be lucky to get a response.

The reason deniers want to reach for a non-explanation of warming, like a bounce from the Little Ice Age - is that they don't want to reduce waste CO2. Some of them know that the greenhouse effect is real and still deny it. Others just deny the greenhouse effect holus bolus.

There are deniers who go to some lengths to reject physics and chemistry. One such person, Jeff Patterson, wrote another article for WUWT today (archived here). He's been here before. Jeff likes playing with numbers but he's not very good at it. Last time he wrote that he thought there was a "parabolic temperature trend".  He didn't go so far as to predict when the parabola he sees would trend down again. Back then his conclusion was:

The climate record of the past 163 years is well explained as the integral second-order response to a triggering event that occurred in the mid-to-late 1870s, plus an oscillatory mode regulated by solar irradiance. There is no evidence in the temperature records analyzed here supporting the hypothesis that mankind has had a measurable effect on the global climate. 
Jeff declined to say what that "triggering event" was back in the mid to late 1870s. Going by his last sentence it's clear that he doesn't accept that increasing CO2 was a trigger let alone a cause.  As for his "oscillatory mode regulated by solar irradiance" - he doesn't explain himself other than to talk about some mysterious "60+ year mode", which he tries, unsuccessfully, to tie in with the eleven year solar cycle.

Today Jeff's writing about a recent article at by Stefan Rahmstorf, which was a very nice illustration of the surface temperature trend. Stefan was explaining what was meant by "statistical significance" of trends and confidence intervals. It's one of the clearest explanations I've come across for the layperson. Stefan also demonstrated and described change point analysis (see also here) as applied to the GISTemp temperature anomaly record. (The link has some further reading listed.) Here is the result:

Fig. 4. Global temperature (annual values, GISTEMP data 1880-2014) together with piecewise linear trend lines from an objective change point analysis. (Note that the value for 2014 will change slightly as it is based on Jan-Oct data only.) 
Graph by Niamh Cahill.

Stefan said the above analysis was by Niamh Cahill. She is a PhD candidate at the School of Mathematical Sciences, University College Dublin, who specialises in developing statistical models for sea level data. Stefan wrote:
It is the proper statistical technique for subdividing a time series into sections with different linear trends. Rather than hand-picking some intervals to look at, like I did above, this algorithm objectively looks for times in the data where the trend changes in a significant way. It will scan through the data and try out every combination of years to check whether you can improve the fit with the data by putting change points there. The optimal solution found for the global temperature data is 3 change points, approximately in the years 1912, 1940 and 1970. There is no way you can get the model to produce 4 change points, even if you ask it to – the solution does not converge then, says Cahill. There simply is no further significant change in global warming trend, not in 1998 nor anywhere else. 

Jeff Patterson is disputing the use of change point analysis, arguing that to apply it to global surface temperature is "naive" and that "as is commonly known, the CPA cannot be used on auto-regressive time series".

Now if you're not a statistician you might be asking "what is an auto-regressive time series".  What I understand autoregression to mean is that past values affect current and future values. Obviously there is some autoregression in surface temperature anomalies. The global surface temperature next year will, in part, be dependent upon the global surface temperature this year (unless you are a David Archibald or a John McLean). But that's not all it depends upon. And if there were only a random fluctation about a mean, then temperatures would go up and down around a mean, they wouldn't be going up and up and up like they are. There are forces acting that are pushing the global surface temperature upwards (primarily increasing greenhouse gases) which are outweighing the forces pushing the temperature downwards (like volcanic eruptions and aerosols).

I have not come across any other article anywhere that states that you cannot apply change point analysis if there is any autoregression in the time series. I did find an article describing a technique to distinguish between shifts in the mean and autocorrelation. In any case, I would defer to an expert in statistics rather than someone who could be a process engineer (whose expertise appears to be an electronics production line), writing for a pseudo-science blog.

The "evidence" that Jeff provides seems to support the opposite of what he is claiming. He put up a random sample of an ARIMA[3,1,1] process. ARIMA stands for "autoregressive integrated moving average", although Jeff admits he is not implying that climate can be modeled as an ARIMA process, so it's not clear why he chose it. Here's his chart:

Figure 1 Simulated climate data from an ARIMA process
Source: WUWT

You'll notice that although Jeff says he is not implying that climate can be modeled as an ARIMA process, he still labelled the chart as "simulated climate data from an ARIMA process".  But compare that to the GISTemp chart above. They are nothing alike. To illustrate this, I've made an animation starting with Jeff's chart and merging through to the actual global surface temperature (GISTemp).

Sources: WUWT and NASA GISS

I don't know how many runs Jeff had to do to get so much of the chart above the zero line, but it strikes me that if he could have got a run anything like the actual surface temperature charts he would have used it. And if he did find one that looked anything like the surface temperature chart, would he have said how many runs he had to do to get it? Would he have computed the odds? Maybe, maybe not. The odds would have been very low. I haven't done it myself but if anyone has, do tell.

Jeff also claims that "CPA fails for any integrative process, a class which in all likelihood the climate falls within". I don't know that that is true, nor what he means by "in all likelihood". Someone did quiz him in the comments about this but so far Jeff hasn't elaborated.

As he did last time, Jeff makes illogical leaps. He wrote:
In short it is in my view incorrect to term the nearly 20 year slowing in the rate of warming as a pause. Rather it is the natural (and perhaps cyclical) variation around a warming trend that has remained constant at ~.008 °C/decade2 since the late 1800s. There is no empirical evidence from the temperature record that mankind has had any effect one way or the other. 

Do you spot the leap of faith?

First he argues that there has been a rise in global surface temperature, a "warming trend" that has "remained constant" at around 0.008°C/decade2.  He'd be wrong about that. Global surface temperatures have risen around 0.8°C since the 1920-30s. Since 1950, the global surface temperature has risen by an average of 0.122°C a decade. Since 1980 it has risen by 0.156°C a decade. It's not had a constant rate of increase of 0.008°C per decade2 (nor a constant increase of 0.08°C a decade).

The leap of faith is that according to Jeff, this large rise in surface temperature must be being caused by something other than human actions. Magic? If it were random, then temperatures would have gone up and down but on balance not moved far from a mean. Instead, what is happening is that the average surface temperature is heading up and up and up. There's no going back down again. The only time in the last 130 years that the global surface temperature fell was back at the turn of last century - between 1880 and 1910. That's an awfully long time between downshifts if all that was happening was a random fluctuation.

Jeff is thinking that it's somehow normal for the earth to get hotter and hotter. But he provides no explanation. No forcing agent. It's as if he drove his car to a the top of a hill and then put it in neutral gear, turned off the brakes and gave it a shove. Then tried to claim that it rolled downhill all by itself with no forcing - ignoring his initial push and the force of gravity.

Jeff talks of "no empirical evidence" yet the temperature data itself is evidence. The warming trend in the temperature data shows that something is acting on earth to cause it to heat up. Jeff seems to think earth gets hot by magic. It makes you wonder if he puts his potatoes in a pot of water and expects them to cook and soften all by themselves, without him having to put the pot on a hotplate. Or maybe he puts instant coffee in his mug, adds cold water and wonders why the coffee doesn't dissolve. Why, when someone else makes instant coffee it comes out hot, yet he can't get it to heat up no matter how long he leaves his mug of cold water sitting still. He's done all the ARIMA time analysis of coffee and cold water and he reckons it should get hot or turn to ice just by autoregression :)

Someone might kindly tell him the trick of cooking potatoes and making instant coffee. It is to boil water - by applying heat. That is, injecting energy - adding a forcing. It doesn't happen by magic.

Before getting to the WUWT comments, I'll add that I'm not a statistician. Any stats knowledge I once had is rusty and extremely under-used. For stats I defer experts like Tamino and Niamh Cahill. I know some readers are also experts, so feel free to add commentary.

From the WUWT comments

Patrick B was the second person to comment, and asked the question about CPA supposedly failing for any integrative process etc:
December 7, 2014 at 6:22 pm
CPA fails for any integrative process, a class which in all likelihood the climate falls within.
These seems to be a basic assumption in your analysis. Please elaborate. Thanks.

Peterkar picked Jeff up on his misuse of the word "infer"
December 7, 2014 at 6:57 pm
Infer/imply. Get ‘em right. Please.

Martin C picked up Jeff on his mistake in the trend:
December 7, 2014 at 7:41 pm
. . .warming trend that has remained constant at ~.008 °C/decade . . . ”
Shouldn’t that be per YEAR ? OR 0.08 °C/decade . . ? 

Martin C corrected his previous comment and wrote:
December 7, 2014 at 7:50 pm. .OK. sorry, I see it is ‘an INCREASE in the rate of warming . . not just ‘the rate of warming. I was thinking of the warming of 0.8°C over about a century, when I made the previous comment.

Bill 2
December 7, 2014 at 7:56 pmWhether it is due to humans or not, at least we all can agree that it’s been warming and there is no pause. 

And to support my long-held contention that deniers are link-averse, Typhoon replied to Bill 2
December 7, 2014 at 8:10 pmTwo words: statistical significance.

majormike1 is an historical and current and a climate ignoramus, writing:
December 7, 2014 at 8:56 pm
It has been cooling since the Holocene Climatic Maximum 6,000 years ago. Each of warming periods that followed – Minoan, Roman, Medieval, and the one we are in now – was not as warm as its predecessor. Current warming is just a natural rebound from the coldest period of the past 10,000 years, the Little Ice Age (1450-1850 AD). Historical climate “ignorati” don’t recognize that climate change did not begin with Al Gore’s birth. 

There aren't too many comments so far. Only 15 at last count. You can see the latest archive here.

Reeves, Jaxk, Jien Chen, Xiaolan L. Wang, Robert Lund, and Qi Qi Lu. "A review and comparison of changepoint detection techniques for climate data." Journal of Applied Meteorology and Climatology 46, no. 6 (2007): 900-915. doi: (open access)

Schurer, Andrew P., Simon FB Tett, and Gabriele C. Hegerl. "Small influence of solar variability on climate over the past millennium." Nature Geoscience 7, no. 2 (2014): 104-108.  doi:10.1038/ngeo2040

Lehner, Flavio, Andreas Born, Christoph C. Raible, and Thomas F. Stocker. "Amplified inception of European Little Ice Age by sea ice–ocean–atmosphere feedbacks." Journal of Climate 26, no. 19 (2013): 7586-7602. doi:  (paper here)

Miller, Gifford H., Áslaug Geirsdóttir, Yafang Zhong, Darren J. Larsen, Bette L. Otto‐Bliesner, Marika M. Holland, David A. Bailey et al. "Abrupt onset of the Little Ice Age triggered by volcanism and sustained by sea‐ice/ocean feedbacks." Geophysical Research Letters 39, no. 2 (2012). DOI: 10.1029/2011GL050168 (open access)


  1. This comment has been removed by a blog administrator.

    1. Anonymous, welcome to HotWhopper. Just for next time - see the comment policy. Also this HW article and more particularly
      this article by Tamino. (PS the article you linked to doesn't fit the "denier" definition but the blog does.)

    2. SouHere is Anonymous comment, for the curious and because I think he or she may be a first time poster, with the live link replaced to an archived version:

      This is tangentially relevant ....

    3. That is a very relevant and interesting post. It shows that even when you see a clear trend by eye, you simply do not get a statistically significant trend over a short period. In other words, Brandon Shollenberger shows how irrelevant a short-term small fluctuation like the pause is. A clear visual for the statistically inclined. Interesting that that comes from him.

  2. That series looks heavily auto regressive to me... Does this "expert" give the parameters, or say how he made it? Also I think for a while bishop hill or someone similar was claiming ARIMA(3,1,1) is a better fit to the BOM's data series than whatever they were using... That would be the source of this idiot's decision.... Data dredging!

    1. A Doug Keenan wannabe?

    2. for a while bishop hill or someone similar was claiming ARIMA(3,1,1) is a better fit to the BOM's data series than whatever they were using
      Yes, it was Doug Keenan who was promoting this kind of idea. I don't even think it was quite that simple, since it wasn't that ARIMA(3,1,1) would be a better fit, I think it was more that one could produce a time-series using an ARIMA(3,1,1) process that would be a better fit. There would, of course, be many that would fit very poorly. I also don't understand how one would report that result "if I select a particular random number seed, I can produce a timeseries that matches some dataset". So what, what does that tell you about that data? Absolutely nothing as far as I can tell.

      I did try pointing out to Doug Keenan that his idea were nonsense, and he called me a Troll. Maybe I shouldn't have been quite so blunt, but I don't it would have made any difference.

  3. One of the more odd-ball aspects of Pattersons post is the line "CPA fails for any integrative process, a class which in all likelihood the climate falls within."

    An integrative process is one that is non-stationary, such as a random walk - Patterson is claiming here that climate varies randomly, with echos to the Beenstock "Polynomial Cointegration" paper and other such claims. Which are absoluteun-physical nonsense, as global temperatures are demonstrably trend-stationary, with internal variations +/- around the response to forcing, that regress to that forcing response over time. See Tamino here and here for a discussion on this.

    'Random walk' claims like this are just another variation on "It's not our fault" denial, wrapped in this instance with complicated math. Complicated, but still denial of physics, in this case conservation of energy; temperatures _cannot_ randomly vary despite what energy imbalances are doing.

    Incidentally, Patterson's choice of ARIMA(3,1,1) is unjustified for just that reason - Foster and Rahmstorf 2011 (Appendix) demonstrated that global temperatures have an autocorrelation pattern of ARMA(1,1). Using ARIMA(3,1,1) is an implicit assumption that there are at least 1 unit root (the second parameter), making the series non-stationary, and is an incorrect choice on his part. In fact, he's just begging the question in that regard.

    1. Looking through the WUWT comment thread confirms my previous post - Patterson is operating from an assumption of temperatures being non-stationary, and he is wrong. There are even commentors bringing up the Beenstock et al papers, which have utterly non-physical conclusions such as temperatures being inversely correlated with methane concentrations. Which, seriously, should be a red flag of a terrible error to anyone with any physical knowledge whatsoever.

      Forcings themselves are driven by solar changes, orbital inclination, volcanic activity, and anthropgenic factors. While hard to exactly predict, these are not random series. Temperature is demonstrably covariant with those forcings (Foster and Rahmstorf 2011, also Lean and Rind 2008 among others), meaning trend-stationary, returning to the trend after variations occur. The non-stationary model Patterson uses is wrong from the start.

      [ Note that this leaves aside other issues with his post, such as rather arbitrary time constants, no description of the parameters he fed into the R functions he used, and in fact the lack of any justification for his ARIMA(3,1,1) model - his post is unsupportable math-babble that begs the question ]

    2. OK, I just spent some time laughing out loud.

      Patterson says he chose his model and approach based on Statistical models and the global temperature record from the Met Office. In that paper Julia Slingo first discusses external forcings and the climate, showing results from several papers that demonstrate temperatures follow those forcings.

      Next, she discusses several statistical models that can produce tracks and statistics similar to the observed climate without any external forcings, including the model Patterson used. That is to say, with no cause-effect relationship to the external world. Slingo says:

      "These results have no bearing on our understanding of the climate system or of its response to human influences such as greenhouse gas emissions and so the Met Office does not base its assessment of climate change over the instrumental record on the use of these statistical models.

      Nor do the results provide any reason to disregard observational evidence of global warming. "

      Patterson is begging the question even more than I originally thought - his math assumes that there are no external forcings, that temperatures are a random walk with integrative drift. In short, he's in denial of just about all of physics.

      I don't know if he didn't fully read that paper, and/or if he didn't understand the implications (D-K syndrome), or whether he deliberately chose the model in denial of the physics. But the WUWT post is beyond wrong.

    3. If temperature would fluctuate like a random walk, would randomly go up or down a certain amount every year, one would expect that by now the Earth would be as hot as the sun or colder than 0 K. It would help a little if the Earth were only 6000 years old.

    4. I wonder what the biosphere would look like if it had evolved in a world with a temperature that genuinely did random walks? Though I guess that question would be verboten at WUWT, since it uses the e word...

    5. Captain FlashheartDecember 9, 2014 at 2:47 PM

      A brief web search uncovers multiple change point methods for ARIMA series, and suggests that the method this muppet used for the ARIMA(3,1,1) is designed only to detect changes in mean level, so no wonder it found change points in a non-stationary series. There are many other methods for change point analysis in regression and time series.

      There's nothing wrong with being ignorant of statistics, in fact many would consider themselves better off for it, but criticizing someone else's stats when you yourself know nothing is arrogant, deceitful and stupid. So, just another day at WUWT ...

  4. Sou,

    This method does have the nice feature of being agnostic to desired trend outcomes. I think it's a cool approach. From the comments at WUWT, RC and here, the downside of it as a communication tool is already showing -- discussions of advanced or novel statistical approaches quickly become arcane. One more thing for WUWTers to toss out the window with a completely false sense of understanding.

    I say the following as a non-stats expert; I think trend analysis is good for mainly two things:

    1) As a benchmark to use for being aware of state changes.

    2) As a last-ditch projecting tool when the underlying phenomena are essentially unknown and unknowable.

    As such, I'm natively averse to trend anlaysis in most things, no more so than in climate science. When I feel moved to throw a chart in front of someone, I want it to convey something about the underlying physics. Being a stats simpleton, whenever I use them they are dirt-basic simple.

    One approach I've been using to discuss Le Grande Pause is to show that it's not all that grand. The method is not unique -- I cribbed some ideas from Tamino and others. I pick a temperature time series and regress it against the log of CO2 concentration. Then I calculate the standard deviation of the residual and use that to make an envelope around the "Tco2" prediction from my ultra-easy model. The result looks like this:

    There's practically no way I can cheat. [1] The model is fairly INsensitive to start and end points, no fancy math or statistical black arts in sight. And I can tell a number of stories with it that don't involve repeating "the warming hasn't stopped". I can point out that temperature deviations outside the 1-sigma prediction envelope are not an exception but the rule. Temps go above and below the envelope all the time, then come right back in. Hey look, we're inside the envelope right now. When temperatures do deviate from the center of the prediction curve, we understand a lot about why that is happening due to things like volcanic eruptions, ENSO, AMO, etc.... [points to the bottom plot].

    As a novel technique or ground-breaking science this is nothing of course. I like it as a communication tool because, well I made it and am proud of it, and because I understand everything that went into it so I can explain and defend it without feeling like I'm stretching (read: exceeding) the limits of my knowledge. Best part of all, not a trendline or cherry-pick in sight, just all the available instrumental data related to the most fundamentally basic mechanism of all. The WUWTers I've dropped it on hate it but they've got very few angles on it me other than to call me a silly lunatic for "making things up". That's nearly a verbatim quote.


    [1] In this model I did "cheat" by pushing CO2 forward 20 years for a better fit since I'm using it for something else that needs the best one I can get.

  5. "Someone might kindly tell him the trick of cooking potatoes and making instant coffee."

    Now that is magic! I've cooked potatoes many times but never managed to produce instant coffee - I always just get cooked potatoes.

    1. You don't know what you're missing :D

  6. To detect breakpoints in a series such as the global mean temperature you have to take into account that the temperature of one year is corrected with the temperature of the next, in other words that temperature is an "auto-regressive time series".

    However, I see no reason to assume that this was not done by Niamh Cahill and I am not sure whether that would have made a significant difference, whether the number of breakpoints would be different, the solution looks reasonable.

    Like you wrote, Sou, the year to year correlations in the example of Jeff Patterson shown above is clearly different and thus not suited to study this question. One could maybe also simply ask Cahill what method was used before writing an attack post.

  7. I have responded to Sou's remarks here

    Jeff Patterson

    1. Yeah! That’s the ticket! (Jeff Patterson channeling Tommy Flanagan)

    2. You've made a few basic, and unsupported, assumptions.

      First, that temperatures are integrative (nonstationary) series. They are not - temperatures (albeit with a considerable amount of internal variation) are feedback driven by energy balance, primarily the emissivity relationship embodied in the Stefan–Boltzmann law, and are hence long term trend-stationary. And this can be mathematically demonstrated by applying the covariant augmented Dickey-fuller test (CADF) to see that temperatures are convariant and trend-stationary with respect to climate forcings, that the temperature series do not have unit roots.

      Second, in your look at a statistical model rather than the actual climate (which could be considered a strawman argument on your part), you claim that changepoint analysis cannot be applied to autoregressive series. A quick look at the literature indicates that your claim is simply incorrect - I found >5,000 hits for identifying changepoints in autoregressive series. But that's really irrelevant due to your first error, in claiming a trend-stationary series is instead integrative.

      You've spent quite a bit of time on the subject, but the vast majority of your arguments simply don't apply to climate.

    3. Jeff, rather than expect people to read you misunderstanding yet another article (in this case mine), first explain here (in 25 words or less) on what grounds you reject the physics of greenhouse forcing.

      That way readers can decide (if they haven't already) if you're an utter nutter denier or a budding nobel physicist.

  8. How could I resist such a charming invitation? The physics of GHG is well established. Unfortunately that does little to inform us about their effect on climate dynamics which are governed by multiple, non-linear feedback mechanisms, many of which are not well understood. If proper account is taken of the ADO and assuming present temperatures lag the CO2 concentration (even that cannot be definitively proven), the transient sensitivity to a doubling of CO2 is approximately 1.5 degC. But even this (benign) effect says nothing about attribution. CO2 has been rising since long before we had any substantial contribution. Even now, our emissions pale in comparison to nature and not enough is known about feedback in the carbon cycle to untangle the two effects. In any case, the available empirical evidence simply does not support the CAGW hypothesis and until a clear and convincing signature can be found, rewiring the global economy is unwarranted.

    1. Thanks Jeff. Based on your comment (and your previous writing), it's a dead cert that you'll not be in the running for a Nobel Prize :(

    2. Two quick questions, Jeff.

      1) No-one other than people denying AGW call it "CAGW". It automatically disqualifies your opinion on everything. Why use it?

      2) Do you deny that the measured increase in CO2 is consistent with about 50% of human emissions?

  9. @KRDecember 18, 2014 at 6:30 AM

    If you claim a system comprised of multiple, non-linear, cross-coupled subsystems many of which are known to contain integrative elements and transport delays, when subjected to random variations both external and internal (not to mention impulsive events- volcanoes, comet hits, earthquakes etc.) produces a stationary time series then I say show me the large regulative mechanism (i.e. negative feedback) that dynamics demands exist. And if such a mechanism exists, why do we suppose it can discern our niggling effect from that of nature?

    1. The negative feedback is Stefan-Boltzmann.

    2. On the whole argument from incredulity (hiding behind the argument from from ignorance) that JP is promoting, we've been there, done that.

      Instead of 101 Denier talking points refuted, it should be 101 formal/informal logical fallacies Deniers use.

      Just because JP can't possibly imagine something dose not mean that that something is impossible.

      I would rather argue from a position of knowledge rather than a position of ignorance.

    3. As I noted above, the primary negative feedback causing temperatures to be trend-stationary (returning to track net forcings, even as they change,) is the Stephan-Boltzmann. Combined with the First Law of Thermodynamics, energy conservation, the stronger any imbalance between incoming and outgoing energy the stronger the feedback towards reducing that imbalance. Scaled by T^4, a _very_ strong feedback.

      And given our rather considerable knowledge of how anthropogenic and natural forcings have changed over time, attribution is quite clear. I suggest looking at IPCC AR5, WG1, Chapter 10, on detection and attribution, to see our current state of knowledge.

      You have presented multiple arguments from ignorance (claiming we don't know what is in reality quite well understood), claimed CO2 rise isn't anthropogenic (mass balance alone, we're emitting less than the atmospheric increase, without those emissions CO2 levels wouldn't be rising, not to mention all the other evidence), made flatly erroneous claims about change point analysis (copious literature on change points and integrative series), asserted against all known physics and thermodynamics that temperatures are a 'random walk', and in your initial post on the subject used an R function for changing _mean_ while making claims about changing trends - showing that you don't know what you're doing.

      In short, you have less than zero support for anything you've argued, and are clearly in denial of the science.


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