William M. Briggs

Statistician to the Stars!

New Arcsine Climate Forecast: Hot and Cold!

If you weren’t worried before, then take a look at this shocking new climate forecast!
Arcsine climate forecast

No, only kidding. This is the real forecast:
Arcsine climate forecast

Sorry. Can’t help myself. Here are four more “forecasts”.

Arcsine climate forecast

Each of the “forecasts” were generated by what is called a “random walk.” Here is what that is. Grab a coin and go out and stand on a corner of some sidewalk that stretches for a long way in both directions. Call one direction “positive” and the other “negative”. The corner you start at will be called “zero”. Flip your coin: If it is heads, then take one step forward toward positive; if tails, then take one backward toward negative. Keep doing this for a long time and soon you will find…that your neighbors think you are crazy.

But that’s a random walk. If you do the coin flips and steps for a long enough time, you’ll find that you spend a heck of lot more time than you might have guessed on either the positive or the negative side. You will probably find that, when you quit, you are way up along the positive side, or way down along the negative. This is true even though the average of those coin flips, the +1s and -1s that make up your steps, is pretty near 0; and even though the average goes to 0 the longer you flip the coin.

The “climate model forecasts” generated above were done so by reference to a paper by A.H. Gordon, available here, called “Global Warming as a Manifestation of a Random Walk”. It is a very readable paper that bears attention.

Gordon proposed that a climate could be made by generating random “shocks” to a climate system. What’s that? Well, imagine the climate is going along peacefully, maintaining its temperature and minding it’s own business, when suddenly—bame!—some external force causes it to change its temperature up or down. An external force might be a change in the Earth’s orbit, or a shifting in cloud cover, or a flock of birds flying this way or that, or anything. This shock persists in the system for some time; little shocks build up and over the course of a year the climate—the mean temperature—changes. It is just as likely for this random-walk climate’s temperature to go up as it is to go down,.

Random walks have some surprising properties which, by virtue of being surprising, are not intuitive. The first is that we’d expect adding random ups and downs (1s and -1s) together would get us a bunch of no changes (or 0s). We don’t get 0s, but numbers which travel far from 0 as time goes on. In fact, it can be shown—via something called the arcsine law—that it’s more probable that this climate will be at an extreme value whenever the series stops, and will not be near 0. The pictures show this.

What about the real climate, the one we actually live in? It’s certainly true that the real climate experiences external shocks of every kind. Gordon found (over the period he looked and with one particular, often used data set) that temperatures went up about just as many times as it went down, just like what would be expected in a random walk climate. He found that the value of the temperature at the end of the series he had was an extreme one, just like we would be expect in a random walk climate. He made a lot of pictures, like we have, and noticed that a lot of them look just like our real climate.

The pictures that make up our and Gordon’s “arcsine forecast”, for technical reasons, aren’t 1s and -1s, but numbers simulated from a normal distribution with a central parameter of 0, which means the numbers are equally likely to be above 0 as below 0 just like in the -1/+1 random walk, but here they can be any number greater or less than 0 (the standard deviation parameter, for those who know of such things, is set at 0.12, which is the same as the estimated standard deviation parameter for actual global mean temperature; see Gordon’s paper for a fuller description).

What does all this say about the real climate? That it happens to look just like a bunch of random numbers. Gordon cautions, “That is not to say that the temperature record is a random walk, but that it does possess similar features.” The surface temperature records “also exhibits properties of the arc-sine law. It is concluded that the global series could have arisen from random fluctuations and could therefore be analogous to arc-sine law governed by random walks.”

This means the climate we have might be less controllable than we thought it was (controllable negatively or positively through man’s activities).

He ends with some sage advice:

It is important to examine all ways and means by which the observed data series develop trends before facing hard and fast conclusions that any particular activity is the one and only responsible agent.

Below is the code where you can generate your own arcsine climate model forecast in R:

# Climate random walk software
# Cut and paste everything at once into R
# http://www.r-project.org

## Make changes in this block

# You must specify the standard deviation.
# The one here is from Gordon (1990).
s = 0.1238

# Now specify the number of "years".
# It's fun to pick large n: try "n=1e5", (hundred thousand)
# The waves you see in the data with larger n are predicted by the
# arcsine law. The increase in height and length by well known
# mathematical equations. See Feller: An Introduction to Probability Theory and its Applications: Vol. 1
n = 100


## Do not touch anything below unless
## you know how to code in R.

# This is the random walk climate. Simple, no?
y = cumsum(rnorm(n,0,s))

# This makes the pretty picture
plot(1:n,y,type='l', xlab="Year", ylab="Temperature Anomaly (C)", axes=F)


  1. Richard S Courtney

    November 13, 2008 at 4:31 am

    A H Gordon is reported as saying; “It is concluded that the global series could have arisen from random fluctuations and could therefore be analogous to arc-sine law governed by random walks.”

    Gordon’s finding is confirmatory evidence that the climate models are based on assumptions that warrant confirmation.

    The basic assumption used in the General circulation Models (GCMs) is that change to climate is driven by change to radiative forcing. And it is very important to recognise that this assumption has not been demonstrated to be correct. Indeed, it is quite possible that there is no force or process causing climate to vary. I explain this as follows.

    The climate system is seeking an equilibrium that it never achieves. The Earth obtains radiant energy from the Sun and radiates that energy back to space. The energy input to the system (from the Sun) may be constant (although some doubt that), but the rotation of the Earth and its orbit around the Sun ensure that the energy input/output is never in perfect equilibrium.

    The climate system is an intermediary in the process of returning (most of) the energy to space (some energy is radiated from the Earth’s surface back to space). And the Northern and Southern hemispheres have different coverage by oceans. Therefore, as the year progresses the modulation of the energy input/output of the system varies. Hence, the system is always seeking equilibrium but never achieves it.

    Such a varying system could be expected to exhibit oscillatory behaviour. And, importantly, the length of the oscillations could be harmonic effects which, therefore, have periodicity of several years. Of course, such harmonic oscillation would be a process that – at least in principle – is capable of evaluation.

    However, there may be no process because the climate is a chaotic system. Therefore, the observed oscillations (ENSO, NAO, etc.) could be observation of the system seeking its chaotic attractor(s) in response to its seeking equilibrium in a changing situation.

    Very, importantly, there is an apparent ~900 year oscillation that caused the Roman Warm Period (RWP), then the Dark Age Cool Period (DACP), then the Medieval Warm Period (MWP), then the Little Ice Age (LIA), and the present warm period (PWP). All the observed rise of global temperature in the twentieth century could be recovery from the LIA that is similar to the recovery from the DARP to the MWP. And the ~900 year oscillation could be the chaotic climate system seeking its attractor(s). If so, then all global climate models and ‘attribution studies’ utilized by IPCC and CCSP are based on the false premise that there is a force or process causing climate to change when no such force or process exists.

    As a caveat, I add that the assumption that climate change is driven by radiative forcing may be correct, but that would not be evidence that the ‘projections’ of GCMs are correct. If climate change is driven by radiative forcing, then it is still extremely improbable that – within the foreseeable future – the climate models could be developed to a state whereby they could provide reliable predictions. This is because the climate system is extremely complex. Indeed, the climate system is more complex than the human brain (the climate system has more interacting components – e.g. biological organisms – than the human brain has interacting components – e.g. neurones), and nobody claims to be able to construct a reliable predictive model of the human brain. It is pure hubris to assume that the climate models are sufficient emulations for them to be used as reliable predictors of future climate when they have no demonstrated forecasting skill.

    Gordon’s conclusion is yet more evidence that the underlying assumptions utilised in models that ‘project’ future climate may not be correct and, therefore, those assumptions warrant confirmation.

    Richard S Courtney

  2. Nice proposal. I wonder what it gets to it being tested and approved as a better model than the current GCMs? I do think that it is not enough to say that everything is chaotic, therefore you can’t forecast, people are always searching for patterns and we don’t like being said that we don’t know as much as we want to or think we do. It’s a terrible carachteristic of ours, but it’s ours nevertheless (and the observer effect or wishcasting comes from this), but mainly, even if this proves to be more correct than the current mainstream solutions, it won’t be adopted until it produces better forecasts than the current solutions.

    It will take decades for that to be recognized. Perhaps even centuries.

  3. The Earth’s climate system has never been, and will never be, in equilibrium. The enormous ranges of important spatial, physical phenomena and processes, and temporal scales almost ensures that conclusion. By the Earth’s climate system I mean the atmosphere, oceans, biological, human-kind activities, and all the important physical and chemical phenomena and processes that occur both within each subsystem and between subsystems. All corrections appreciated.

    The concept of radiative equilibrium overlooks the fact that a thermodynamically heterogeneous system can attain different internal states for a given energy level. The internal states attained determine the “average” temperature of the system. And the heterogeneity ensures that inter-system processes continue to be obtained. While an equilibrium state is the preferred natural direction of change, a truly constant state is very likely never attained.

    The hypothesis that the Earth’s systems are chaotic is just that and nothing more; an hypothesis. It is an untested and untestable hypothesis. Extrapolation from the numerical solutions of extremely simple systems of non-linear ODEs that exhibit chaotic behavior to the physical systems of Earth’s climate is not wise, in my opinion. The chaotic ODE systems are classic examples of ill-posed initial value problems. There is some speculation that chaotic response might in fact be periodic, but having extremely long periods. So long that the equations haven’t been solved sufficiently long to observe the periodicity. Pure speculation, and I probably should not have brought this up.

  4. Briggs:
    Pretty coloured graphs! The squiggles spoil them though, strange.

    Seriously, this term random walk, I have heard banded about but didn’t understand it. Random noise in electronics has these properties, so I am lead to believe, but couldn’t quite see how this, or anything could be truly random but with the coin example and this explanation it makes more sense.
    I have been wondering whether computer models could run their outputs many times a second so that the output would look like a graphic equaliser or a flickering flame. Would this help in recognising ‘beats’ or ‘pulses’ of change or would this just produce a similar effect as a static line? I don’t know. It might quieten the fear mongers if peaks and troughs could be shown in such a way that would make them look less significant. If the sun behaves like a nuclear fire, there should be some patterns in its output. It must flicker (flacker) like a giant candle.

    I don’t understand exactly what you mean, the part after “but mainly” To the end.

  5. I hate to rain on this parade but the Gordon work is pretty old. We have learned something about the nature of solar and terrestrial fluctuations in the intervening years. Scafetta and West [Physical Review Letters 90, 248701 (2003)] showed that both solar flare and terrestrial climate fluctuations obey a 1/t^a scaleless power spectrum, where t is the time between events and a ~ 2.1. This is strong evidence of coupling between solar fluctuations and terrestrial climate. It also suggests that a simple random walk step model is too simple but can be improved upon. It would be useful to revisit the Gordon approach, incorporating these findings and bringing it up to date. It would be very interesting, if not sensational, if the climate system can be shown to be represented by a Markov process.

  6. Richard S Courtney

    November 13, 2008 at 1:00 pm

    Noblesse Oblige:

    It seems that I have failed to understand your message because I am confused by it.

    You write:
    “I hate to rain on this parade but the Gordon work is pretty old.”
    Are you thus attempting to claim that Gordon’s work should be ignored because others have found the same as he has? If so, then that seems to be a very strange claim because replication is supporting evidence.

    And is your reference to the work of Scafetta and West a claim that solar variation is the driver of climate change? Perhaps it is, because several others have made such claims. But I fail to see how this pertains to Gordon’s work until those claims have been demonstrated to be correct. So, I fail to see how your reference to the work of Scafetta and West equates to your assertions that:
    “It also suggests that a simple random walk step model is too simple but can be improved upon. It would be useful to revisit the Gordon approach, incorporating these findings and bringing it up to date.”

    So, I would much appreciate clarification of your points. And I thank you for the clarification in anticipation of its provision.


  7. Richard/// I apologize for not being clearer and for not creating the wrong impresssion. I do think that Gordon’s work is potentially significant, but that its INTELLECTUAL curiosity interest can be enhanced, possibly into something that could INFLUENCE current climate research, if it is updated to take account of what has been learned since 1990. I mentioned the Scafetta and West finding, which is an empirical one, that suggests that the earth-sun system is a weakly coupled complex system. Therefore, random events on the sun translate into random shocks on the earth, providing a physical (non heuristic) basis for Gordon’s observations. Furthermore, the probablility distribution of these events has actually been measured (the 1/t^a), so that it is no longer necessary to assume a simple Brownian motion step model. One can simulate this by taking Monte Carlo samples from this distribution. If the result is consistent with the climate record, and if Markovian independence of climate steps is verified, that would be strong evidence of the sun as a significant player in climate change, and by implication a reduced role for greenhouse gases.

  8. The random walk postulate is interesting but global mean temperature is most certainly not a random walk. We can deduce this from the physical constraints – random walks extend arbitrarily from the origin, yet temperature is bounded at 0K. From physical intuition, the random walk cannot be physically correct.

    Indeed, some of the tests applied by Gordon are very weak tests. For example:

    Gordon found that temperatures went up about just as many times as it went down, just like what would be expected in a random walk climate

    This is true, but if this was our test, we’d be seeing random walks everywhere!

    I still regard the work of people like Timothy Cohn and Demetris Koutsoyiannis to be at the forefront of the debate on the stochastic nature of geophysical processes; for an interesting discussion relevant to this topic I’d recommend the following presentation, which describes the issues better than I ever could:


  9. Richard S Courtney

    November 13, 2008 at 7:09 pm

    Noblesse Oblige:

    Thankyou for the clarification. I now think I understand your argument.


  10. Stochastic (random) is just another word for “unknown.” Matt, you have made that point many times.

    Einstein said, “God does not play dice with the Universe.”

    Yes, we can use a random number generator inside a mathematical model to create model outputs that seem to mimic real world phenomena. However, the simplicity of such models belies their veracity.

    Perhaps the most we can conclude from Gordon’s work is that climatology is still in its infancy, so much so that rolling the dice produces as good a model as anything else.

  11. I fully agree with R.Courtney when he says :

    “However, there may be no process because the climate is a chaotic system. Therefore, the observed oscillations (ENSO, NAO, etc.) could be observation of the system seeking its chaotic attractor(s) in response to its seeking equilibrium in a changing situation.”

    Spence_UK :

    I also consider D.Koutsoyiannis as having the deepest insights in this matter and I had some very interesting and intellectually rewarding e-mail correspondence with him .
    This : http://www.itia.ntua.gr/en/docinfo/673/ is for me his best paper . Brilliant .

    Dan :

    The deterministic chaos property , at least for strictly temporal chaos (like the Lorenz system , a diode circuit , damped pendulum etc) is testable .
    Prove that at least one Lyapounov coefficient is positive and you have a chaotic system .
    It can be readily done for low dimensionnal systems like those above .

    The problem with the Earth system is that its behaviour is spatio-temporal chaos which is notoriously more difficult because both spatial and temporal structures behave chaotically . We have no more only ODEs that describe the temporal chaos but PDE that describe spatio-temporal chaos . The space matters .
    That’s why the Navier-Stokes problem is “so difficult” as Terry Tao says 🙂

    I find this post and the Gordon’s paper significant insofar it is another element of an ensemble of insights that all point to the same direction – namely that there is no qualitative difference between the behaviour of the Earth system at short time scales (“weather”) and at large scales (“climate”) .

    Whether it is the chaos theory approach that shows positive Lyapounov coefficents .
    Whether it is the Markovian approach that observes that the system doesn’t seem to have a memory what only says in other words that the Lyapounov coefficients are positive (aka the trajectories diverge exponentially thus “forgetting” the initial conditions) .
    Whether it is D.Koutsoyannis stochastical scaling approach .

    They all say the same thing .

    However I do not believe that the climate (or more specifically the GMT which is supposed to be somehow a relevant metrics) is a random walk or even a Markovian process which are for me only suitable approximations of a chaotic system for shorter time scales where it is hard to see the difference between a random system and the deterministic chaos . The difference should appear at higher scales (centuries and more ?) but we have unfortunately no dependable data for those scales .

  12. Mike,
    The word random has often been followed up by a caveat that the word is being used in a certain sense, “i.e. unknown”. I took this to mean that there was another type, the one that most people think it means, which is something else, that no one can explain!
    ‘Stochastic’, on the other hand is banded about liberally, being far fancier than the word unknown and therefore evidently preferable when one is trying to sound intelligent. Now I know they mean the same.
    This is as I had suspected. When I tried to look up the definition after witnessing an ‘RC’ discussion, there was no clear explanation to be found. I had surmised that the word meant random, but why use the word, when random will do? Why use random when unknown will do?
    It’s an intellectual dilemma if they are all the same word. Stochastic is a dreadful word and ought to be outlawed.
    When I studied computing I recall the teacher saying, “it’s technically impossible for a computer to generate random numbers.”
    It is a relief to find that there is no such thing as random in any other sense. This means that I was right. I wish he’d said “random means unknown”. He was a great teacher so I was probably daydreaming and missed it.
    Einstein, bless him, was not defining random well with those words.
    If, at this stage, it is still apparent that I am confused, then I suggest you leave me in blissful ignorance: )

  13. Briggs

    November 14, 2008 at 6:52 am


    Right: random means unknown and nothing more. Gordon is implying that the climate is more unpredictable than believed.

    Obviously, this “random walk” model isn’t a real model of the climate, because there are many aspects of climate that we can predict with skill. But we can take his point that we still can’t predict very well and that it’s likely we are fooling ourselves with our explanatory models to some extent.

  14. Joy,

    The first two words I learned as a neophyte freshman at the Big U were “stochastic” and “heuristic.” They are Faculty Club words, to be dropped in intellectual conversations. They give the (false) impression that the speaker is educated.

    They mean “unknown” and “incapable of being known”. I dunno and I don’t care.

    When I heard the latter phrases emanate from the mouth of my smart aleck 8-year-old, I knew immediately that he was genius material.

  15. Mike D–
    Didn’t you also learn “ad hoc” right away?

    I thought heuristic meant something like “facilitating discovery” or “pointing to the truth”. Heuristic methods are valuable. Their short coming is they aren’t “TRUTH” in all it’s glory. (That said, the word is used by those who claim they have found the “TRUTH”. So, it’s often used as a slam.)

    Ad hoc is similarly a slam. Ideally, to figure out a combustion problem, you solve the navier stokes equations, will a million dollars for the prize use that to fund your efforts to solve all chemistry problems, and find a way to solve the heat equation, navier stokes, thermo and chemistry problems all at once. In practice, you might create a method that gives you usable answers but only when dealing with combustion in a typical furnace design. But, that practical solution is, (sneer) ad hoc because you relied on assumption limiting it to a special case.

  16. Yes, heuristic means all those things, but pared down is equivalent to “raises more questions than it answers.” The common usage is in developing inferences when none exist, as in, “We can infer absolutely nothing from my multivariate, eigen-vectored, principal component, stat analysis, but it is heuristic.” It is used in the “I give up” phase of a study, and so can be roughly translated as “I don’t care anymore.”

    I have a pottery kiln, that I built myself, and fire with propane to cone 10 (very hot), so am familiar with furnace design and combustion. I use the ad hoc de facto methodology, so am well-versed there too. It seems to work, the proof being that I haven’t blown myself up or burned the neighborhood down, yet.

  17. Off-topic question.
    Really love the blog, beginner at statistics although I did a lot of maths once, long time ago.
    I have a question which I raised at another climate blog about the difference between noise and the signal. The answer I got said that either my idea is way off or the statistics guru whose blog it is didn’t get the point. Have seen many of your posts including “You cannot measure a man” from March which make me think you might like to comment on it.
    Can I email you the question (and the response I got)?

  18. random, stochastic, ad hoc, de facto, heuristic, nonstationarity, Markovian, aperiodic, n. b., et al., ipso post facto, a prior, a posterior

    BINGO !!!*

    I think the above are accurate descriptions of almost all the untestable and untested hypotheses invoked in investigations of The Great Global Climate Crisis.

    * Maybe some here aren’t familiar with this game played in Productive Meetings **

    ** An Oxymoron

  19. Dan
    Yes, I have heard of the ‘bingo’ played by naughty staff in meetings.
    This is why I could never work in management. The vocab would hurt my ears.

  20. Briggs

    November 15, 2008 at 1:06 pm


    About eight years ago some of us were playing meeting Bingo when we decided to come up with our own annoying useless buzzwords. We’d each push our creations and see whose caught on.

    Mine was look into the fire. I pushed it pretty hard, but it never stuck.

  21. Matt,

    I don’t know how far off-topic you’ll let this go, but.

    Another fun thing is to adapt Management’s use of the -ize ending to change nouns to verbs. Example, “We had a great gross income this week. Let’s annualize this and project the profits.”

    One application we came up with is, “Come in and chairize yourself.”

  22. I find great similarity between these two statements:

    Any technology sufficiently advanced will appear to be MAGIC to someone insufficiently educated.

    Any system of high complexity which we are unable to reasonably monitor and quantify appears chaotic.

    I’ve had a couple glasses of wine so will throw in an partially OT statement also.

    Scientists attempt to quantify the universe so that they can manipulate and possibly duplicate it, yet, many vehemently deny that any BEING could have DESIGNED or CREATED it!!


  23. Kuhnkat:

  24. Sorry to destroy the apparent proximity of the first 2 statements 🙂

    Chaotic systems can be both monitored and quantified .
    They only can’t be deterministically predicted .
    So the second statements comes probably from somebody who doesn’t know what chaotic systems are .

    Magic (or very advanced technology) on the other hand is supposed to be perfectly deterministic – you wave a hand and what you wish happens .

  25. Tom Vonk:
    I’m not convinced.
    Monitored simply means observed, consistently noted.
    Quantified means counted doesn’t it?
    These factors together do not prove that a chaotic system is understood.
    Einstein said, “Not everything that counts can be counted and not everything that can b counted counts.” How do we know were not counting things because we can as opposed to because we should to understand?
    For example, Brain activity can be monitered and quantified. It is not understood.
    I am with Kuhnkat.

  26. I didn’t see the word “understand” in the post I commented on and my purpose was to correct a factual error in a statement to avoid misleading impressions .
    Not to enter a philosophical discussion about the meaning of “understand” .

    So I commented on the “monitored” and “quantified” .
    As far as chaotic system go , and to be complete , they are not only “monitored” , “quantified” and “controlled” but also “understood” by the tool called chaos theory like relativistic systems are “understood” by the the tool called relativity theory .
    Of course the prerequisite to “understanding” is to learn the relevant theory .
    And of course there are still things within that need to be “understood” better .

    I have often met people who look at chaotic systems like at a sort of magics but that’s only because they don’t know what it is about .
    Once I explain it , they are surprised that it is hamiltonian mechanics and fluid dynamics with special and rather sophisticated mathematical tools .
    Sure , 99,9 % of people are not familiar with working in the phase space but that is not an argument that it should be particularly mysterious .

    One more time a very important caveat : understand and predict are 2 very different things like Heisenberg would say .

  27. Tom Vonk:
    Feel free to avoid entering into any philosophical discussions that are not to your pleasing.
    I am not here to waste anyone’s time

    “As far as chaotic system go , and to be complete , they are not only “monitored” , “quantified” and “controlled” but also “understood” by the tool called chaos theory”
    Yes,theEarth’s climate system, as any chaotic system is not understood. Whether you want to ignore this is another issue for if they were understood they could be predicted, exactly. If they are not predictable, they are not understood. This is not the same as saying that the words are synonymous.

    “Of course the prerequisite to “understanding” is to learn the relevant theory .”
    Of course

    “And of course there are still things within that need to be “understood” better .”
    Why better? If they are understood by the tool called “chaos theory”.

    “Once I explain it , they are surprised that it is hamiltonian mechanics and fluid dynamics with special and rather sophisticated mathematical tools .”
    Why does it not surprise me that the tools are rather special and sophistocated? They could not be rather mundane. It’s all relative.

    “Sure , 99,9 % of people are not familiar with working in the phase space but that is not an argument that it should be particularly mysterious .” I don’t know what the phase space is, but it sounds like “the bit that no one understands(or is familiar with) bar one or two. If this is the case, I suspect that it might indeed be mysterious.

    “One more time a very important caveat : understand and predict are 2 very different things like Heisenberg would say .” I would not confuse these two words and I appologise for using the word understand in it’s place. However in order for prediction to occur, understanding in it’s fullest sense is necessary.
    Like Monday then Tuesday.

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