Berkeley and the Long-Term Trend

On a recent thread which was not about the temperature trend, but about Judith Curry’s mischaracterization of it, “Dan H.” stated that what mattered was the long-term trend, which was a steady increase at a rate between about 0.006 and 0.0075 deg.C/yr, and that the Berkeley data reinforced this idea. He later said that it was a steady increase plus a cyclic variation with period about 60 years. Let’s examine those ideas closely, shall we?

Here’s the Berkeley data (minus the final two “don’t belong” data points), together with a lowess smooth on a 10-year time scale:

Here’s just the smooth, together with 10-year averages:

Visually, it certaintly doesn’t seem to be a steady rise plus cyclic variation. But let’s try those models anyway. First let’s fit a long-term linear trend by least squares. We’ll compare that (plotted in blue) to the smoothed curve (plotted in red) and the raw data (in black):

Note the sizeable departure of the data from the linear trend over the last several decades. Let’s expand the scale for a direct comparison of the linear fit (in blue) to the smoothed curve (in red):

Indeed the long-term linear model isn’t very good. In fact it isn’t right, which is easily confirmed statistically. The biggest difference between reality and the Dan H. model is the rapid upward trend over the last 30 years. Could it be … global warming?

The linear model doesn’t hold water. But what if we include a cyclic variation with period about 60 yr? It turns out that the best-fit linear-plus-cyclic model has a period of 70.9 yr. Here’s that model (plotted in blue), compared to the smoothed curve (in red) and the data (in black):

Note the sizeable departure of the data from the linear-plus-cyclic trend over the last several decades. Let’s expand the scale for a direct comparison of the model (in blue) to the smoothed curve (in red):

Indeed the long-term linear-plus-cyclic model isn’t very good. In fact it isn’t right, which is easily confirmed statistically. The most important difference between reality and the Dan H. model version 2 is the rapid upward trend over the last 30 years. Could it be … global warming?

We could also test the long-term linear model, and the linear-plus-cyclic model, by fitting them only to part of the data — say, data up to 1975 — then extrapolating that model to see how well is validates what followed. Care to guess how that comparision looks?

If the linear model is correct, then the warming rate will be constant over time. If the linear-plus-cyclic model is correct, then the warming rate will be constant-plus-cyclic with the same period. Let’s compute the warming rate using each 30-year segment of the Berkeley data, together with the estimated uncertainty in that rate, using an ARMA(1,1) model for the noise just to feed the “uncertainty monster.” Here’s the result, with a horizontal dashed line in red indicating the mean rate of the linear-plus-cyclic model over the entire time span:

The long-term linear model is nonsense. The long-term linear-plus-cyclic model is nonsense.

90 responses to “Berkeley and the Long-Term Trend

  1. If you like cycles, you’ll love these!! (elephant trunk wiggleing in the extreme):
    59.75years + trend
    or 60.1years + 317 and no trend
    are largest inputs

  2. The oscillations at the beginning are suspicious.

  3. But wait, you left out the Leprechauns, the sheep albedo and the kitchen sink.

    I thought my kitchen sink just gave me problems when the weather got hot, but Dr Spencer opened my eyes. I had the attribution backwards.

  4. Thanks, Tamino. This is a talking point I’ve been seeing quite a bit from denialists who like to pretend to be scientific. I think their idea is that if they can minimize the trend due to climate change, that they can get away with saying warming won’t be significant. The thing is that we have about a dozen separate lines of evidence that all favor a sensitivity of 3 degrees per doubling. If this estimate were wrong, it is far mor likely to be too low than too high. It is as if they think the only evidence is the temperature trend, and if they can come up with a Rube Goldberg model that explains that, the problem goes away.

  5. Could it be … Satan? (h/t to the Church Lady)

    These folks would be better served by looking at the physics, not messing around with cycles.

    • If they look to the physics rather than astrological fits, then they have to attribute the changes to something. Mechanisms like land use, albedo, and 7 billion souls worth of karmic crystal pyramid energy would still blame the Anthro part of the denied AGW. What mechanisms are left if you start with CO2 sensitivity being a hoax? Dark matter cloud nucleation? Phlogistonial inertia?

  6. Clear and to the point. Thanks.

  7. Great post.

    Question: Is it possible that the apparent cyclicity in the early part of the record is due to the station distribution? Those big swings have bothered me since BEST came out. The BEST animation ( shows that the “global” temperature anomaly is anything but globally defined between 1800 and 1850. Most of the records appear to be eastern U.S. and western Europe, all of which would presumably be influenced to some degree by the NAO (for example).

    With true global coverage later in the record the surface warming trend overwhelms any regional temperature cyclicity, as you’ve very convincingly demonstrated.

  8. Denial becomes increasingly difficult and complex. Requiring more and more energy to express.

  9. That sort of thing is an inevitable consequence of having higher uncertainties at those years (in addition to some real non-constant behavior, undoubtedly). The larger the uncertainties, the further you’ll occasionally deviate from the mean in a 10-year moving average and the longer it’ll take to get back to the mean, resulting in a time series with an apparent periodicity that’s related to the length of the moving average and the noise spectrum.

    • Thanks Zach. Apparently I wasn’t the only one wondering about those ‘oscillations’.

      Two questions for tamino (and anyone else):

      1. Can you show us the effect Zach refers to with some artificial data (decreasing noise amplitude with time)?

      2. Do those oscillations (or harmonics thereof) affect the AMO periodicity analysis?

      • At WUWT, Willis E. tried to claim that we have already seen a 2°C increase in global temperature based on the trough of one of the oscillations and the recent temperature peak.

        Shorter Wills: “Temperature has already risen 2°C and there has been no catastrophe.”

    • In an interview with AAAS (member central, subsc. required), Muller states:

      “Our analysis shows warming underway by 1800, large variations up and down throughout the 19th century, and that variability on the 3-15 year scale has been dramatically decreasing over the past two centuries.”

      Apparently he believes that the variability in the early data is real. What can be rigorously concluded about 19th century variability from BEST data? Can one demonstrate that that variability is an artifact of smoothing choices, as indicated above by Zach?

      [Response: Or perhaps, that the high level of variability is regional. The early data have far less geographical coverage, so they can exhibit high regional variability, while later data are more global and therefore reflect far less global variability.]

  10. Jarch, Dan H., and others:

    Tamino posits a change in the long-term linear trend around 1975. I did a Chow test on the Hadley CRU global annual dT data. Call b the linear trend, the coefficient of the year term, where I refer to linear regressions of dT on the calendar year.

    The combined regression had SSE 3.81111295 for N = 161, R2 = 65%, b = 0.004475889, t = 17, p < 1.0 x 10^-37.

    Group 1 had SSE 1.968239, N 125, R2 35%, b 0.002537, t 8.2, p < 2.6 x 10^-13.

    Group 2 had SSE 0.270003, N 36, R2 81%, b 0.016975, t 12, p < 1.2 x 10^-13

    The Chow statistic is therefore F = 55.16399459 with 2 and 157 degrees of freedom, respectively. The significance is off the scale. There was indeed a break at this time, and the linear trend is significantly different in the two time periods. For 1850-1969 we have 0.026 K per century. For 1975-2010 we have 0.17 K per century.

    Please let me know if you'd like to see the raw data.

    • Doing single Chow tests is extremely risky. If a linear fit to the entire dataset is poor, a linear model with breakpoint will often fit better, whether it’s actually a reasonable model or not. And even if the data [i]is[/i] described well by a linear model with breakpoint, testing only one year doesn’t tell you where the breakpoint really is. It can produce positive results even if the true breakpoint is a decent distance away. To get robust results, you need to test a range of potential breakpoints and look for a peak in significance. In doing so, you also need to account for the fact that you’re performing multiple tests, of course, and adjust your significance threshold accordingly.

  11. Sorry, b = 0.002573 for group 2, not 2537. And the first period is 1860-1974, not 1860-1969. And, of course, I use way more significant digits reporting the figures above than I have any right to.

    But the conclusions stands. The trend is not linear. It is accelerating.

  12. With respect to the oscillations at the beginning of the sequence, the wiki page List of large volcanic eruptions of the 19th Century contains the following commen on Mount Tambora:

    Greenland and Antarctic ice samples suggest an undocumented eruption roughly half the magnitude of Mount Tambora occurred, contributing to the 1810’s being the coldest decade in at least 500 years.

    with a citation.

  13. There is a typo after the linear-plus-cyclic picture, you don’t say linear-plus-cyclic, you just say linear.

    [Response: Thanks, fixed.]

  14. Tamino, do you have a default lowess bandwidth that you use for most of your examples, or do you vary it to suit the data at hand? I tend to do the latter, which also means I need to specify bw(__) in the legend or somewhere so others can replicate a graph if they want.

    [Response: I generally “default” to a 15-year time scale, but for this analysis I switched to 10.]

  15. Tamino, just as the uncertainties of the last two data points are two large to be included in any meaningful analysis, I would have thought the uncertainties before about 1880, and certainly 1850 are also to large. Antarctica is not the world, which is why the last two data points are no good. But neither is Europe, and hence the early data points are not meaningful either.

    [Response: I quite agree. But the task at hand was to test Dan H.’s claim that the Berkeley data reinforce the characterization of temperature change as “long-term linear trend” or “linear-plus-cyclic.” One can safely eliminate the final two data points without impeding that, but removing early data would hamper testing his hypothesis.]

  16. It’s cycles all the way down.

  17. The positive slope of anomalys show that the rate of warming is accelerating, not that warming is occuring, warming is proven to be occuring by all Annual anomalys simply being positive for the last twenty years.

    The anomalys slope could be flat or negative and warming would still be occuring, as long as the anomalys are positive.

    Why is it nobody realises that this discussion is focused on acceleration of warming, not warming? From Watts through to Phil Jones through to Ken Trenberth, everybody….everybody… has the wrong end of the vector.

    The focus on anomalys has distracted from the most relevant metric, Global Annual Average Temperature, which has been increasing every year for the last 10 and longer, meaning no ‘Plateau’.

    • Chris Ho-Stuart

      That’s not correct, and misunderstands anomalies. An anomaly is the difference between temperature and a base value. If anomaly slope is negative, then the temperature slope will also be negative.

      • An anomaly is the amount of change that occurs over say a year, analogous to velocity. The change in yearly anomalies indicated by slope is analogous to change in velocity, acceleration.

      • let me edit that…..

        …An anomaly is the amount of change that occurs relative to a baseline over say a year..

      • let me edit that again… anomaly is the sum of change relative to a baseline over say a year.

      • criminogenic: No…The anomaly is simply the temperature minus the baseline temperature. An anomaly is not a derivative…It is simply a temperature but a temperature that is referenced to a baseline (such as a 30-year average) for that particular station.

        There are important advantages of using anomalies over using temperatures: One is that the anomaly field has better behavior than the temperature field. For example, anomalies are correlated over fairly large distances: If it was warmer than average in NYC this year, it was also likely warmer than average in Boston and Montreal and Philadelphia even though the actual average value in this different cities might be fairly different. (In a particularly dramatic example: One would find a huge variation in temperature over just a few miles if one compared the temperature at the top of Mt. Washington to the temperature in a nearby valley; however, if you looked at temperature anomalies, the two stations would likely show similar behavior.)

        Another advantage of anomalies over absolute temperature is with temperature, you have to be worried about how representative your stations are. For example, some “skeptics” who don’t understand anomalies are worried that the dropout of lots of cold weather stations in Siberia over the last few decades has biased the record warm. This could indeed be a problem if one was trying to measure average global temperatures; however, if you measure global temperature anomalies that is not an issue.

      • Correlations aren’t altered by offsetting one series wrt another. Anomalies are correlated precisely because the temperatures are.

      • Joel Shore | November 8, 2011 at 1:43 am |


        The Anomaly is the sum of differences over a period, not just one measurement.

        So time is a factor.

        Anomaly = Sum of Differences/ Period

        So that the Anamoly can be compared to other Anomalies the period is 1.

        [Response: Please see this. Time is not a factor in the definition of anomaly (except in the choice of the reference period).]

    • Are you under the impression that temperature anomalies are changes in temperature? Because they’re not. They’re temperatures measured from a common baseline.

    • The baseline is the average annual Global temperature, between 1951-1980 for GISS, etc.

      My understanding is that an Anomaly is the average temperature variation over a period relative to a baseline, if the temperature is greater it results in a positive anomaly, if it is equal to the baseline then there is zero anomaly and if it is less than the base line it is a negative anomaly.

      • Chris Ho-Stuart

        It is better to say the anomaly is the average temperature DIFFERENCE, not the temperature variation.

        If the temperature is less than the baseline, then the anomaly is negative. If the anomaly is reducing, then the temperature is reducing. Your original claim that a positive anomaly means increasing temperature was wrong. It only means that the temperature is above whatever baseline is chosen.

        Falling anomaly means falling temperature, and conversely.

      • Temperature anomaly is simply the temperature measured in, say, Kelvins, minus the baseline temperature. The difference between anomalies and true temperatures is constant. They’re direct measures of T, not dT.

      • I’ll clarify what I think:

        An Anomaly is the sum of change relative to a baseline over a period of time.

        Positive Anomaly over one period doesn’t mean increasing temperature, periods of positive anomaly do, as the ongoing average increases.

      • My kingdom for an edit button: change should be diference.

      • An Anomaly is a measure of dT, if a year starts off above baseline but ends below baseline the exact same amount then the sum of diference over the time period is zero meaning the anomaly is zero.

        Anamoly = sum of diference/time

        time = 1 as 1 year is the time period most suited to minimise smoothing needs.

        Therefore Anomaly is a function of time and change in Anomaly is a function of time.

      • No, it’s as Joel Shore said: “The anomaly is simply the temperature minus the baseline temperature.” There’s no term in there for time.

        So, for instance, it is meaningful to say that “The anomaly for Metro Atlanta at 3 PM today was ___.” (Or it would be if I knew what the actual value is.)

        You might object that time is cached in there somehow, since in practice it’s the mean of 30 years absolute temperatures, normalized to zero. But in principle it could be anything convenient, say absolute zero (as in the Kelvin example) or the freezing point of water (which, however varies inconveniently with salt content) or for that matter your birthday, suitably formatted–39.85, say, if you were born March 9 and are now 26. (Arguably convenient for you–but for me, not so much.)

        So, no time in there at all. Just a difference from the temperature, period.

      • :
        An Anamoly describes the sum of difference over a year, when this sum is added to the baseline Temperature, average annual global Temperature for the year is described, when this figure is added to the population the average is increased, if the Anomaly is positive.

      • criminogenic … is there any particular reason you think this site should be an interactive dictionary?

        You definition is wrong, deal with it, and you can learn why by hitting google a time or two, rather than waste our time (and diminish your reputation here, as a result).

      • Kevin,

        I understand a single Anomaly has no term for time.

        A group of sequential Anomalies do involve a term for time.

      • “A group of sequential Anomalies do involve a term for time.”

        By the same logic, a group of sequential temperatures would, too, or a group of any suitable measurements which occur over time [assume light-hearted examples here–kids’ heights as they grow, or something.]

        In which case, the nature of “anomaly” would seem to be irrelevant, and we could make dhogaza happy by letting this go–since (I think) the main question, that of ‘acceleration,’ has been adequately dealt with.

      • Crim,

        Why are you persisting in this misunderstanding? Why complicate something fairly simple? An anomaly is a simple difference from a baseline average and is not a rate of change. Using your analogy, an anomaly is a displacement, not a velocity.

      • True,

        I see how you are saying Anomaly is analoguous to displacement, it is a measure of difference at a moment of time. Thus a series of Anomaly over time is analoguous to a series of Displacement over time, velocity.

        The reason I am persisting with this is that I think there has been no plateau in Global warming over the last 10 years, as every year has been above average thus the average of the population is increasing, meaning Annual Global Average Temperature has been increasing. Meaning there’s something wrong with analysis of Anomaly as ‘flattening’ is used to say Global warming has stopped.

        If Anomaly is seen as Displacement, Anomalies can be seen as Velocity and change in Anomalies can be seen as Acceleration, thus the ‘Platueau’ describes no acceleration.

      • Chris Ho-Stuart

        Anomalies are not in any sense a velocity. That’s been your problem from the start. It’s just wrong.


        Actually, not, logically speaking, since velocity requires a change in displacement. If your displacement numbers stay roughly the same, you ain’t movin’–or not much, anyway.

        I agree with your intuition that “warming hasn’t stopped,” and have argued that it’s ridiculous to claim it has, just because the same sorts of (relatively high) temperatures have been coming up repeatedly for a few short years.

        However, should we see the same anomaly numbers–off the top of my head, roughly .3-.6 C per NCDC algorithm–recycle quasi-randomly for another decade or more, it likely would approach statistical significance. Then we really would be justified in calling it a “pause” or something similar. “Displacement” would be remaining more or less constant, “velocity” would would have reached zero within the bounds of measurement/variability, and (to paraphrase an old comedy bit) acceleration would be “right out.” We’d know something significant (and, given the political near-stasis on mitigation, very helpful) had happened.

        But given other physical indicators–the state of the cryosphere being a leading one to my mind–that “ain’t gonna happen.” I’m not a betting man, but I would bet that the present decade will be significantly warmer than the last. (Not least because I expect to see ice-free Arctic summers within that span–“ice-free” defined as sub-1 million km2 extent–and I expect that there will be a warming feedback from that event. Indeed, I believe there’s some thought that we’re seeing feedback from the declining extent already.)

      • Crim,

        Several have attempted to correct your misunderstanding without success, so I don’t know what else to try but an example, in fact real figures from GISTEMP Global.

        The baseline for annual temperatures is defined as the mean for 1951-1980, 14.0 °C in fact.

        For 2001-10 we get annual (J-D) figures of:-
        This is what that looks like.

        You seem to think that even a constant anomaly would indicate a steady increase. Not so, as shown by a fictional alternative to the above:-
        This is what that would look like (obviously I’ve tricked WFT because they aren’t real figures). Can you see that there is no increase for a constant anomaly?

      • I’ll take it outside to the open thread.

  18. But … this means the Loehle and Scafetta magic cycles so well-received in Currytown are … nonsense?

    The Landscheidt solar power wave so beloved in climastrology is … nonsense?

    The ramp and sine so popular in Watt$land is … nonsense?

    There are a lot of broken hearts in deniersville this night.

    • But at least it won’t make you go blind. . . oh, wait a minute. . .

    • They can still invent different cycles with periods such as 120yr, 200 yr, 437yr and so on. It seems to be tamino’s work to check those hypothesis for them. But they still have a large bag “full of cycles”

      • Wonders of reckless extensions of Fourier transforms to where Fourier didn’t went (with some reasons, but let us not be hindered by mere mathematical realities …)

      • I’m getting this vision of miscellaneous ‘cyclists’–h/t to Connie Willis for the wordplay involved–coming to Tamino, hat in hand, and plaintively asking, “Please, sir, can I have some more–analysis?

        But it was only a fantasy. . .

      • yea, we’re roughly on the 13.75 billion year mark of a highly complex cycle of initial singularities if one wants to be holistic in scientific mattress (intentional misspelling).

      • Horatio Algeranon

        they still have a large bag “full of cycles”

        Ahh, so that’s why they call it “re-cycling” when you use the cow manure mulch for your garden.

  19. Thus far unremarked to my knowledge (and without going into the cyclic stuff very early) is the major divergence between CRUTEM3 and Berkeley in the common overlap period before the other two data records join the party in 1880. Recognizing that in this era many of the instruments were not yet standardized stevenson screen measures (largely being N/S facing wall exposures (N/S depends upon the hemisphere, so mainly North) and unshielded) *if* Berkeley are doing better at identifying and accounting for the non-climatic influences the implication is that there wasn’t some mid-to-late nineteenth Century cooling as implied by CRUTEM3 and that therefore the long-term trend since pre-industrial times by the imperfect linear trend metric since 1850 is potentially (and I stress potentially) much greater than previously reported and understood. Somewhere in the range of maybe 30% larger because of the end-point influence of this potentially spurious cooling tendency very early in an 1850-2010 trend period calculation in CRUTEM3.

    As an aside as a science community and society we are always far too asymmetric in our view of end-point effects endlessly obsessing over the recent data impacts rather than the impact of the choice of start point.

    Certainly there are papers from a couple of recent studies replicating the old instrumentation that suggest these early instruments were substantially warm biased which would lend a degree of credence towards this possibility. This would then have obvious implications for our understanding of things such as the climate sensitivity and natural climate variability.

    • Peter, or others – just a civilian here and things are not obvious to me. So I am trying to take a little step here in grasping what you are saying.

      Is this about right, or totally off.

      • More like this which is the OLS trend periods for the two datsets from 1850-2010 assuming that interface does it properly. Certainly the difference between the two slopes looks reasonable given the multi-decadal timeseries differences.

        Failing that the OLS with AR1 DOF correction according to Santer et al. approach is 0.61+/-0.06K/century for CRUTEM(not 3v – this will be a secondary effect) and 0.86+/-0.06K/century for Berkeley over Jan 1850 to Dec 2010. That is a substantial difference driven almost entirely by the mid-to-late 19th Century differences between them.

      • Peter,

        I don’t think it’s valid to derive a least squares fit from a rolling mean as you have done. I think you should use the raw figures like this. Your point stands, though.

    • Timothy (likes zebras)


      It would certainly help in terms of natural climate variability, but I don’t think it is going to help with climate sensitivity. According to the Keeling curve I can find on wikipedia, CO2 levels were less than 320ppm in 1960. I can’t seem to find out when CO2 levels went above 300ppm [about 1900?].

      I would have thought that the pre-1900 surface temperatures are not likely to be important for determining climate sensitivity, because CO2 levels would still not then have risen that far from pre-industrial values.

      • Timothy (likes zebras):

        Law Dome

        Looks to like it’s between between 1910 and 1917 on the above graph. I am curious where this is going as Fred Moolten has said on Climate Etc. that Issac Held thinks anthropogenic atmospheric CO2 may have had a more significant role earlier than current science indicates (hope I haven’t botched this.)

  20. David B. Benson

    My goodness but that became confused and confusing. The so-called anomalies are simply adjustments of the record to a new position for zero. Exactly as one converts between kelvin and degrees Celcius. Mathematically a


    BPL: That doesn’t sound right. Reading too warm is not the same as warming up faster.

    • If your early records were warm biased and the previous records did not adequately adjust for it then they would have been warm biased early in the record yielding an under-estimate of the *real* warming trend. In the same way as modern cold biased records would yield an under-estimate. Think of a see-saw …

  22. An entire thread devoted to my postings. I am honored. I see it even attracted my old friend to post. BPL, it has ages since we last sparred. Congradulations on your publication.

    If you are going to use the Berkeley data in your presentation, then you should include all of it. Specifically, this paper:

    Here, they show that global temperatures show a positive correlation with ENSO, and an even stronger correlation with the AMO, which they characterize as an ~70 cycle. Of course, they are not the first to show this correlation. Others have shown this previously:

    [Response: I guess you missed the part about them only showing correlation with short-term fluctuations — nothing at all to do with any “70-yr cycle” — in temperature over land only. You also seem to have missed my having posted about that paper. Twice.

    All of which illustrates another problem with the global warming “debate.” When your nonsensical ramblings are shown to be nonsensical, rather than simply nurse your wounded arse you just throw up some more nonsense, this time on a paper which you appear not to understand.]

    They also identified a 9.1-year period, which can easily be seen in this graph:

    [Response: It shows no such thing. You’ve been fooled by chronic mathturbators — this is even more nonsensical than your “trend-plus-cyclic” model.]

    The trend was accelerating, which will occur repeatedly in any oscillating system. The maximum temperature slope (60-month linear regression) occurring in Jan, 1996. Since then, the temperature rise has been decelerating, with a current 60-month slope of 0 (using your similar GISS data).

    Contrary to Tamino’s early statement, this in no way implies that global warming is not happened. Only that other forces have been overlooked, and in Muller’s own words, “the human component of global warming may be overestimated.”

    [Response: It’s abundantly clear that your net contribution to reasoned discussion is less than zero. You do not discuss, you babble.]

    • Discobabble; a symptom of excessive mathturbation?

    • Dan H., I have given this example before. Look at the periodicity of the following series of ordered pairs:


      Now predict the next entry. If you said 4, you are correct, as that is the 11th digit in the base of Napierian logarithms, e. One needs to be very careful attributing periodicity to a time series unless one has many, many periods or a good reason for there being a periodicity at the specified period.

      In the case of temperature time series, we do not have enough data to establish a period of ~70 years, and the physics says you are wrong. Me, I’ll go with the physics.

    • Dan, I’ve done Granger causality analysis of dT and the AMO, since I noticed the close correlation too. Cause runs unequivocally from temperature to the AMO, with a lag time of about 8 months.

      • David B. Benson

        Thanks. I need that particular datum.

        Puts the death knell to assuming the AMO explains global temperature.

  23. Another reason you can tell that “60 year cycles” are mathematical babble is by understanding some of the properties of the Fourier transform. In particular, in the presence of a linear trend, low frequency (long period) cyclical features are obscured.

    Here I plotted fourier transforms of GISTEMP (1880 to present) vs. a linear trend (0.57 K/century). The fourier space points corresponding to 130, 65, and 44 year cycles are equally well-explained by a linear trend.

    • You should really plot that on a log-log power spectral density plot.

      Then do an FFT of ANY linear trend.

      You will see 1/f^2 at the low frequency limit (e. g. red noise) and at the upper limit 1/f^0 (e. g. white noise), in between it goes from red -> pink -> white.

      Also, this is very important, all corresponding phases are the same in incremental value.

      The only way to construct a linear trend using FFT is to carry both the amplitude and the phase.

      FFT are also periodic, and the assumption of a stationary ergodic time series holds, although that has never stopped anyone from misapplying that basic assumption.

      In real world harbor resonance data you first need to remove the tidal cycle, from the raw data, a linear detrend is minimum standard practice for any wave measurements exceeding ~17 minutes (4096 @ 4 Hz) in the nearshore.

      Also, using an FFT as a digital filter (by zeroing out coefficients) is very bad practice vs a proper time domain IIR filter (e. g. sine Butterworth).

  24. For anyone who hasn’t seen it, this post by Nick Stokes is simply fantastic

    Wonderful illustration

  25. Tamino, and others — if you’ll allow a brief aside from the serious statistics — I’ve found showing amateurs the woodfortrees site, including the site’s caution about fooling yourself with trends, does encourage people to poke at this themselves. Often people will cherrypick at first, but if willing they can convince themselves fairly quickly that cherrypicking _is_ deceptive.
    The experience may open some eyes and ears.

    Got a moment to criticize choosing this chart/set of intervals?
    On the screen the result has some shock value.

    I don’t know any statistical test can be done to check the appearance:

    • This is similar to a chart which the IPCC produced for the AR4 SPM.

    • I’ve done similar things with other data sets.

      Isn’t WfT a great resource for us all? I don’t think that Paul Clark gets anything like enough credit for his work.

      • It’s very similar to the graph I posted here several weeks ago. At the time TrueSceptic taught me ho to find raw data on Wood for Trees!

        If you start a 30-year trend starting in 1970, and add another 30-year trend at 1980 (picking up “the pause” decade), and then add a 20-year trend at 1990 (almost half of which is “the pause”), then something surprising, to me anyway, happens.

      • I don’t remember that but thanks.

        I might also have been responsible (in an epic Deltoid thread) for introducing a certain Girma Orssengo to WFT, but the thing is that we can clearly see what anyone’s done: it’s all in the linked page.

        (Not that WFT compares with what Tamino does, but quite often we can get similar results with limited knowledge of statistical techniques. It’s a shame that some who could benefit, Stephen Wilde being one recent example, seem unable to do so.)

      • Bart R. over at Climate Etc. can make some excellent graphs on WfTs.

      • Yes!

    • I still have trouble using R. I’m writing my own statistics software.

    • Nice. If you click the ‘raw data’ link at the bottom of the graph, you will get a text file showing the datapoints. Since you only have linear trends, you get the slope of each. Put enough of these – perhaps a 50 year interval starting every 25 years? – into a spreadsheet and you ought to be able to get some statistics.

      • This is a graph often presented by skeptics. I think Judith Curry gives this argument some credence. If you look for Girma’s article on WUWT, you can read the argument.

        So I made this graph to try help me ascertain the validity of what they are saying. Raw data:

        HadCrut –

        1910 – 1940 = 0.0152788 per year

        1970-2000 = 0.0160176 per year
        1980-2010 = 0.0159837 per year
        1990-2010= 0.0164756 per year

        With respect to “the pause”, note that 1980 – 2010 dips, but 1990-2010, around half of which is “the pause”, is the strongest trend of the three.

        GisTemp –

        1910 – 1940 = 0.0114178 per year

        1970-2000 = 0.0149702 per year
        1980-2010 = 0.0164691 per year
        1990-2010 = 0.0189519 per year

        Here, even greater strengthening into the teeth of “the pause”. I think you guys call this acceleration.

        BEST –

        1910 – 1940 = 0.0138951 per year

        1970-2000 = 0.0241013 per year
        1980-2010 = 0.0272477 per year
        1990-2010 = 0.0317388 per year


        Of course, if you run similar trends starting in 1920 and 1930, the trend 1910-1940 collapses.

    • I agree whole-heartedly on the sentiment.
      Pretty graph.
      If I may add my own, this one tends to cut through some clatter, or rather, I find that conversations tend to stop or go off on another track after I post it.

      • That’s good, but any reason for the 33 year smooth on GISTEMP? Why that period rather than some other?

  26. If you are going to leave off the last two data points as their uncertainty is too large, you may also want to leave out some of the earlier plot points as well (especially in the early 1800’s). Some of the uncertainties are over 3! (best to be consistent in these things)

    [Response: See the response to this.]

  27. Tamino,

    This is OT, but I’m hoping for an open thread topic on temperature time series (either here or perhaps over at Nick Stokes website).

    Both surface and satellite (land only) time series, both as anomalies and as the actual absolute temperature time series (from which the anomalies are calculated). If not the actual time series than a monthly anomaly array (with the base period specified (e. g. 1901-2000 (NCDC), 1981-2010 (UAH), 1979-2004 (RSS, or whatever RSS is currently using for their base period), BEST (1950-1980 (don’t know if the end years are inclusive as this would make the base period 31 years)..

    What with all the various discussions of the BEST temperature time series and analyses thereof, I’ve become interested in a comparison of NCDC, UAH, RSS, and BEST (all land only, given that’s all BEST provides to date)).

    Using a common time base (e. g. 1980-2009 (or 30 years), or 1979-2008 (again 30 years)).

    So, for example BEST states in Full_Database_Average_complete.txt that;

    “Estimated 1950-1980 absolute temperature: 7.11 +/- 0.50”

    Similar numbers exist for NCDC at;

    And in looking at the UAH data, the TLT absolute cycle temperatures are way below the surface temperaure time series because they are actually measuring a few KM (But exactly how many KM’s?) above the surface so those measurements are quite a bit colder in absolute terms.

    This all eventually boils down to the differences in trend slope between UAH/RSS and BEST/NCDC, the unexplained ~0.1C difference, and the denier (totally unfounded) conjecture that that difference is all due to a UHI bias in the surface temperature measurements. Something I have argued strongly against, if a proper area weighting scheme is used. As I’ve now done some (now getting rather serious from a rather simple start) urban vs rural population/population density estimates, as urban population area, as a proportion of total land surface area, is well below 1% (closer to 0.5% than it is to 1%).

    But I think that the various anomaly time series with a common time base and the absolute temperature added back into the respective anomaly time series, would clearly expose the denier BIG LIE since it has become quite obvious that the satellite and land surface datasets, while interesting to compare (given we only see anomaly time series comparisons) are in fact measuring two entirely different sets of temperatures (surface vs a few KM above the surface).

    Sorry for going OT, and I will persue this on my own regardless (it just will take a few emails to obtain the anomaly curves and/or absolute temperature time series).

  28. Bart R. made this graph in this recent exchange with Girma.

    In that a thread a statistician comments on this artcle, and claims Girma’s theory should be tested using quantile regression.