Open Mind

Predicting the “Weather”

October 13, 2008 · 13 Comments

This isn’t really about predicting the weather, at least not about making a forecast for your locale. Instead I’ll make, just to have a little fun, a prediction for the GISS monthly average temperature for the next 15 months.


The GISS estimate of global average temperature has, since 1975, behaved (at least approximately) as the sum of a linear trend and a random process. The random part (the “noise”) is, by its nature, unpredictable over the long haul, so I don’t really expect this forecast to be very meaningful. But it’s likely to be interesting! So we’ll base a prediction on the data from 1975 onward. And here’s that data:

Careful analysis shows that there’s more to this data than just a linear trend and a random process. If we remove the linear trend, then do a Fourier analysis of the data, we get this:

The tall spectral peaks at very low frequencies aren’t actually significant of periodic behavior; they’re a sign of the autocorrelated structure of the noise process. But I’ve labelled two of the spectral peaks, at frequencies 1 and 2 cycles/yr, because they are significant; they signify that these data actually show a detectable annual pattern. How can that be? The GISS data are temperature anomaly, which is the difference between the actual temperature and the average value for the same month in the entire data set. Shouldn’t this remove the annual pattern?

In fact it removes the average annual pattern. If the annual pattern since 1975 isn’t quite the same as that during the entire time span, then the difference will still remain. And in fact, the average annual pattern since 1975 is not the same as during the entire time span covered by the data. Global warming has caused greater warming during northern hemisphere winter than summer, so not only has the average temperature changed, so has the annual pattern.

Hence the difference between the recent (since 1975) annual pattern, and the average since 1880, remains in this data, and it’s that “residual” annual cycle which shows in the Fourier spectrum. We can do a simultaneous fit of a linear trend and the residual annual cycle, which gives this:

This fit, all by itself, gives us a prediction for the future. But perhaps we can do better. If we subtract this fit from the data, we have residuals which indicate the progress of the noise over this time span:

We have, previously, modelled the noise as an ARMA(1,1) process in order to estimate the uncertainty in the trend rate. This gives a sufficiently precise estimate of the autocorrelations to estimate the trend rate uncertainty. But for prediction this is an imperfect approach, because it involves the “white noise” part of the process, which is inherently unknown. We can, however, attempt to predict the progress of the noise process by a higher-order AR process. So I fit AR processes of various orders to the residuals, and used the corrected Akaike Information Criterion (AICc) to select which order to use. I selected an AR(8) process, which may or may not be the best choice but this is just for fun so let’s run with it.

We therefore have a model consisting of the superimposition of a linear trend, an estimate of the “residual” annual pattern, and an AR(8) process for the noise component. More to the point, we can compute those processes for future times, generating a forecast of global temperature. And here is the fit to the data, together with the forecast through the end of 2009:

For a closer look at the recent time span, here’s the previous graph starting with the year 2000:

As you can see, the AR(8) process mimics the noise fluctuations at the start of the “prediction” interval, but dies down quickly, leaving only the linear trend and annual pattern. Hence our model for the noise helps predict the very near future, but doesn’t really help in the long term. That’s the nature of noise; its autocorrelation helps predict the short term, but eventually the noise becomes completely unpredictable. But we expect the trend and annual components to continue, so they help predict further into the future.

Frankly, I don’t put a lot of stock in this forecast. If my life were at stake, I’d attempt to model the el Nino southern oscillation and its impact on global temperature. I’d explore other methods for making the forecast, and test them all by reserving part of the the data not to be used for the prediction at all, but for verification. I’d also emphasize (to whoever held my life in the balance) that the randomness will still be there and will still be random. And of course I’d express as many caveats as possible, emphasizing that a single large volcanic eruption, or a cessation of industrial activity (and the cooling effect of sulfate emissions), will alter the very nature of the system itself, invalidating any prediction because the underlying process will have changed.

But it was fun!

Categories: Global Warming

13 responses so far ↓

  • Patrick Hadley // October 13, 2008 at 1:15 pm

    Is there any statistical or climatic reason for beginning the graph in 1975? What is special about a period of 33 years?

    On another point I know that if you toss a fair coin two hundred times and after the first 100 you find you have 45 heads and 55 tails that does not mean that in the next hundred there are likely to be more heads to even things out.

    However the physics of greenhouse gasses means that there must be extra heat on the planet somewhere and that a few years of below trend temperatures are therefore likely to be balanced by a rebound period of above trend warming. Do you not have to take into account the last few years being below trend when predicting the near future?

  • Gustave // October 13, 2008 at 2:31 pm

    I am doing a project on tornadoes of the last fifty years. Do you know a website for it?

    [Response: You can find data here.]

  • ron lobeck // October 13, 2008 at 2:54 pm

    Some time ago I asked about the effect of the ’tilt’of the Earth’s axis because a paper I had discovered by an astronomer from 1940 predicted that the NH was staring to ‘lean’ closer to the Sun in the NH winter, thus increasing the insolation at higher latitudes. Has this effect been considered in the analysis of the figures produced by GISS and which were used in the analysis ‘predicting the weather’…I raise this because of the comment ‘…global warming greater in NH in winter…’

    Also, has anyone claculated the amount of CO2 generated in the lifetime of a human(never mind all the animals)? In view of the population explosion it must has some contribution…? If it is significant perhaps we could arrange ” NATIONAL NON-BREATHING DAYS” which could go some way to alleviating the problem!

  • Dano // October 13, 2008 at 3:14 pm

    However the physics of greenhouse gasses means that there must be extra heat on the planet somewhere and that a few years of below trend temperatures are therefore likely to be balanced by a rebound period of above trend warming. Do you not have to take into account the last few years being below trend when predicting the near future? [emphasis added]

    No to the bolded. Please look at actual surface data. To wit:

    o The Met Office Hadley Centre and the University of East Anglia have today released preliminary global temperature figures for 2007, which show that the top 11 warmest years all occur in the last 13 years.

    The provisional global figure, using data from January to November, currently places 2007 as the seventh warmest on record since 1850.

    o Climatologists at the NASA Goddard Institute for Space Studies (GISS) in New York City have found that 2007 tied with 1998 for Earth’s second warmest year in a century.

    o Global warming stopped in 1998,” has become a recent mantra of those who wish to deny the reality of human-caused global warming. The continued rapid increase of the five-year running mean temperature exposes this assertion as nonsense.reality, global temperature jumped two standard deviations above the trend line in 1998 because the “El Niño of the century” coincided with the calendar year, but there has been no lessening of the underlying warming trend.

    [emphases added]

    That is: assuming La Niña cooling is a long-term trend is a mistake.

    Best,

    D

  • Patrick Hadley // October 13, 2008 at 6:04 pm

    Dano, I really did not intend to start an argument about the long term significance of recent temperature trends, and am prepared for the sake of argument to accept your points completely.

    The Hadley Centre report said that “Over the last ten years, global temperatures have warmed more slowly than the long-term trend. But this does not mean that global warming has slowed down or even stopped. It is entirely consistent with our understanding of natural fluctuations of the climate within a trend of continued long-term warming. . . . . . .Even then, due to natural variations in climate, we expect to see ten-year periods both globally and regionally with little or no warming and other ten-year periods with very rapid warming.”

    My point was simply to ask whether after a ten year period where temperatures have increased more slowly than expected, is it more likely that we will have ten years with very rapid warming?

    Is the weather for the next ten years like the coin tossing experiment where the second hundred is totally independent of the first hundred, or does the science tell us that, because of heat stored “in the pipeline,” a cooler than trend period is likely to be followed by a warmer than trend period?

  • transplant // October 14, 2008 at 12:26 am

    Ron Lobeck,
    First question:
    Look up “Milankovic Cycles”

    Second querstion:
    Humans do not breath fossil carbon, they burn it. Look up “carbon cycle”

  • David B. Benson // October 14, 2008 at 1:02 am

    Patrick Hadley // October 13, 2008 at 6:04 pm — Here are the five and ten years averages from the HadCRUTv3 global surface temperature product:

    http://tamino.files.wordpress.com/2007/12/5yrave.jpg

    http://tamino.files.wordpress.com/2008/04/10yave.jpg

  • Dano // October 14, 2008 at 3:24 pm

    My point was simply to ask whether after a ten year period where temperatures have increased more slowly than expected, is it more likely that we will have ten years with very rapid warming?

    Patrick,

    first, what does it mean that we are endinga ten year period where temperatures have increased more slowly than expected , especially when we are reading that events (temp [as David points out above, melt, droughts) are increasing more rapidly than expected?

    Second, the point of Tamino’s post is that making a statistical model based on past temps is hard.

    Lastly, the answer is difficult, as we don’t know how much extra heat is being held in the oceans, and when they will stop holding this heat (or when the tundra will release CH4, speeding it up).

    The point being that predicting/projecting the future is difficult, there are emergent properties, hence the need to be cautious and plan.

    Best,

    D

  • Patrick Hadley // October 14, 2008 at 4:45 pm

    Dano asks “What does it mean that ‘we are ending a ten year period where temperatures have increased more slowly than expected’ ,”. In my post I quoted the UK Met Office Hadley Centre which said “Over the last ten years, global temperatures have warmed more slowly than the long-term trend.”

    Nobody would argue that average surface temperature tells the whole story about the amount of extra heat retained by the planet as a result of increased greenhouse gasses. It makes sense that some of this heat could be held in the oceans. In the words of James Hansen this warming is “in the pipeline” delayed by the inertia of the great oceans.

    Would I be right in assuming that a constantly increasing amount of heat being retained by the planet as a result of increasing greenhouse gasseses would mean that any period when surface temperatures rise below the underlying trend will probably be balanced by a period of rebound when the temperatures will warm above the trend.

    I do not expect anyone to be able to make accurate predictions for the next ten years, but wonder if when trying to make a reasonable guess one should take into the below trend warming of the last ten years. Should we believe it to be more likely than not that surface temperatures will soon be rising above the long term underlying trend?

  • george // October 14, 2008 at 8:24 pm

    Patrick Hadley asks

    “I do not expect anyone to be able to make accurate predictions for the next ten years, but wonder if when trying to make a reasonable guess one should take into the below trend warming of the last ten years. Should we believe it to be more likely than not that surface temperatures will soon be rising above the long term underlying trend?

    I think the answer to your question is in this post.

    As Tamino showed, it is quite possible for noisy artificial data for which there is a “built-in” constant upward trend to show an apparent negative trend over a ten year period.

    Essentially, the temperature keeps rising due to the underlying trend, but this rise gets obscured by the noise over the short term.

    So I guess the real question is this:

    How do we know the measured trend (pick one: HadCRU, GISS, etc) for the past ten years is not simply apparent?

    Or asked somewhat differently, how do we know the noise has not simply obscured an underlying trend?

  • Cthulhu // October 14, 2008 at 9:22 pm

    I think that prediction will be close to the mark regarding the climb upwards back around 2002-2007 level.

    The SST anomoly trend is interesting wrt to immediate temps considering we are at ENSO neutral:
    http://www.woodfortrees.org/plot/hadsst2gl/from:1975/plot/hadsst2gl/from:1975/trend

    Satellite lower troposphere typically lag SST by a few months and UAH and RSS are already picking up the slack. Of course there’s every chance we’ll go into ENSO negative significantly again mid next year, but it’s just a matter of time.

    Here’s another 18 month prediction challenge.

    How will skeptic blogs deal with temperatures returning to around 2002-2007 levels? Will they keep exagerating the cooling effect of PDO and the solar minimum in spite of high temps, or do they start downplaying both to retain claims that the Earth still isn’t warming?

    Any guesses? I predict we’ll start to hear a lot more about La Nina in coming months and how the recent warming is just “coming out of the la nina”, about how they “all knew this was going to happen”, etc etc.

    I doubt we’ll hear anything like “WOW 0.7C warming in 12 months! That’s equal to the entire warming over the 20th century!”

    Of course if the solar cycle cranks up soon you can bet any higher temperatures will be put down to the Sun…

    [Response: I agree with your "predictions."]

  • Joel Shore // October 14, 2008 at 9:42 pm

    ron lobeck: What you have to understand about changes in the tilt of the earth’s axis is such effects happen on much longer timescales…so they can’t have any great effect over periods of decades. In fact, the rate of global temperature change for the warming out of glacial periods is something like 0.1 C / century…and the decent into glacial periods tends to be even slower.

    As for CO2 generated from breathing, that is CO2 that is being released from plants that took up CO2 when they were growing…so the entire lifecycle is approximately carbon-neutral. Note the distinction with CO2 being liberated from the burning of fossil fuels where we are taking vast stores of carbon that have been locked away from the atmosphere for millions of years and rapidly (on geologic timescales) liberating these stores into the atmosphere.

  • S2 // October 15, 2008 at 9:13 pm

    Just to put some figures on Joel’s post:

    I asked about the effect of the ’tilt’of the Earth’s axis because a paper I had discovered by an astronomer from 1940 predicted that the NH was staring to ‘lean’ closer to the Sun in the NH winter, thus increasing the insolation at higher latitudes.

    It hasn’t just started.

    The axial tilt varies between 22.1° and 24.5° on a cycle of about 41,000 years. We’re currently at 23.44° and decreasing - and yes, this would lead to an increase in winter insolation on high latitudes (and a decrease in summer insolation). But this decrease in tilt started around 10,000 years ago (and will continue for another 10,000 years before the trend reverses), so on a scale of several decades it is insignificant.

    And it doesn’t affect the total insolation over he entire planet one iota (although other variations in our orbit do).

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