Today in comments, JohnV suggested that, owing to the short time associated with my hypothesis test, I was neglecting temperature variability at long time scales, particularly those with periods longer than 7 years. Specifically, he wrote:
As we have tried to explain many times and in many ways, this is not a valid approach. Why? I’m glad you asked. Primarily because it excludes the low frequency component of weather noise (longer than ~7 years).
I realize JohnV has tried to explain this many times and in many ways. The difficulty is that, by my understanding of “excluding”, the 7 year limitation does not exclude the effect of low frequency components of weather noise on the estimate of the trend or its uncertainty. By using Cochrane-Orcutt I am doing something more subtle: I am assuming the proportion of energy associated with the internal variability (weather noise) at high and low scales is distributed in a manner consistent with AR(1) noise.
The both analysis methods I use, Cochrane-Orcutt (CO)or Ordinary Least Squares (OLS) corrected for red noise estimates the uncertainty intervals for the sample based on a specific assumption about the proportion of energy at longer time scales relative to the energy at shorter time scales. In fact, strictly speaking, it’s possible for the method either over or under estimate the amount of variability at low frequencies!
The type of assumption made can be best explained by examining graphs illustrating the energy spectral density (PSD) of the internal variability of the temperature. These will clearly show that methods do not neglect the existence of energy at frequencies lower than 7 years!
For the remaining portion of the article, I’m going to assume the reader has some general knowledge regarding energy spectral density, as discussing the all the idea underlying spectral decomposition of time series is more than can possibly be achieved in a blog post.
How much ‘energy’ is assumed to exist at large time scales when CO or OLS are used?
As a method, both OLS both Cochrane-Orcutt make assumptions about the proportion of energy at low and high frequencies. Regular OLS assumes the “noise” in the signal is “white”. CO assumes the noise is AR(1) or “red”.
The distributions for Cochrane-Orcutt with three different choice of time constant (which are associated with lag 1 autocorrelations in the residuals) and regular OLS are shown below.
The green line, hovering over “zero” represents the spectral energy density assumed when the variable component of the temperature in a trend analysis is treated as white noise. Notice that the same amount of energy is assumed to be contained in the interval from f=0.01 months-1 to 0.02 months-1 as for f=2.01 months-1 and 2.02 months-1. That is: the spectral energy density is constant at all frequencies.
When applying regular OLS to fit a trend to a time series, one assumes that the energy is distributed this way. Whether or not one measures data for 1, 100 or 1000 months, the method assumes there is energy at 0.000,…,001 months-1, which corresponds to a period of 1,000,…,000 months!
So, from the point of view of estimating the trend using OLS, low frequency components of weather variability are not excluded. One assume these components exist.
The difficult: The assumption that the energy spectral density is constant at all frequencies could be wrong. If so, the uncertainty estimates for the trend will be incorrect.
To avoid using a bad assumption, it is advisable to check whether the assumption of white noise applies. One way to check this is to examine the lag one correlation in the residuals. If lag-1 autocorrelation is statistically significant, but but the correlogram indicates the residuals do decay with lag, then you might try Cochrane-Orcutt, or try to correct OLS for red noise. That’s what I did. (Other tests can be done; other choices are possible.)
When you use Cochrane-Orcutt, you make a different assumption about the distribution of energy over different frequencies. The blue, red and yellow curves represent the spectral energy density assumed for three different time constants when the variable component of the temperature in a trend analysis is treated a “red” or AR(1) noise.
Comparing the blue, red or yellow curves to the green curve for white noise, you can easily see that CO, as a method, involves the tacit assumption that the energy spectral density for low frequency components is larger than the high frequency components.
So, not the formalism of CO used to estimate a trend does not exclude energy at these low frequencies. It assumes there is quite a bit of energy at those frequencies! In reality, the method actually assumes variability does exist at inverse frequencies of 100,…,000 months even if measurements are taken over shorter periods, like, say 89 months.
Of course, it is true the assumed shape of the energy spectral density could be incorrect. So, once again, after applying Cochrane-Orcutt, you test to see if the data are inconsistent with “red noise”.
Why do people keep suggesting the energy at low frequencies excluded?
Some may wonder why so many people assume the components of variability at low frequencies are excluded when one uses short time series. It is not.
However, there is a very well know issue where failure to measure over long time periods does matter a lot. Suppose you don’t know the shape of the spectrum, and you wish to discover (rather than assume) it. It is not possible to measure the amount of energy at the low frequenceis without taking long data records.
In particular, there is always a frequency below which you absolutely cannot estimate the energy spectral density based on data. It’s shown in purple above. To the right, you technically can estimate the energy spectral density. Moreover, even if you are to the right of the purple line, if you are too close, your estimate of the energy spectral density likely be inaccurate.
So, if your goal is to measure the spectrum, to discover the shape, you must use a very long data record to determine the energy density at low frequencies.
But the fact that you can’t measure the shape of the spectral energy density, doesn’t mean the energy in that region is excluded from trend analysis. Energy at low frequency components is included in the trend analysis.
Issue
Now that I’ve explained the energy from low frequency components is included in trend analysis, I should also note there are issues of concern.
It’s possible for noise to be neither red nor white. The shape of the energy spectrum could be different in a way that causes the uncertainty interval using Cochrane-Orcutt to be either too large or too small. The sign of the error depends on how the correct distribution compares to the assumed distribution. So questions about whether or not AR(1) applies, the magnitude of the lag-1 autocorrelation or other features involving the autocorrelation of the spectrum do matter, but the claim that energy at low frequencies are excluded is incorrect.
Our disagreement here is probably because of my loose definition of “weather noise”. I assume your ESD plot is showing the energy at each frequency due to the assumed auto-correlation alone. That seems reasonable. However, it neglects known processes that we typically call weather when discussing climate: ENSO, AMO, Schwabe Cycle, etc.
Boris’ analogy to falsifying the arrival of summer based on a month of data in the spring is useful. Please, really consider this:
Given the central tendency for the rate of warming in USA48 in March, what daily temperature measurements from March 2008 would falsify the arrival of summer?
Your approach would be to use C-O to fit a line to the March 2008 data. You would define the confidence limits strictly based on the uncertainty in the curve fit. If the central tendency did not fall in the confidence limits for March 2008, you would conclude that the arrival of summer is falsified.
Is this a fair characterization of your approach? Does it seem valid to you? What would happen if every day in March 2008 had nearly the same temperature? Your confidence limits would be very small.
My approach to defining the confidence limits would start with computing the trends from every previous March. I would be concerned about the arrival of summer only if the trend from this March was outside the limits of the historical trends.
The historical trends are able to include slower processes (low-frequency energy) because they are sampled at different points in the slow processes.
JohnV–
Yes. The technique assumes a specific distribution. If the real ESD has a different shape, then the uncertainty intervals could be wrong.
What I have been saying is this JohnV: Qualitatively,this sort of argument can have merit. But you need to show that there is some large cyclic variability. It is, of course, the exact argument denialists have been advancing for years. But no one took it seriously unless they could suggest the amount of energy in the oscillation in question and estimate the time scale.
This is entirely different from discussing well characterized cyclic features like the annual cycle.
I’ve looked at the specific identifyable cycles you and other suggest. ENSO doesn’t reverse. The Sun is very arguable. If you want to use the PDO, go find evidence that it’s not just a marker that affects the relative spatial distribution of temperature on the earth, but the GMST itself.
Unless you have concrete evidence, and estimated magnitude of the energy for a speculative cycle, I’m going to treat that speculation the same way I treat the speculated 200 years “Little Ice Age/ Medieval Warming Period” cycles. I’ll treat it as speculative hand waving enthusiasts wish to believe in to save their theory.
In the meantime, I’m comfortable assuming the various cycles (PDO, AMO, AO blah, blah, blah) result in energy spread over a distribution of frequencies, and that the collective behavior approximated using the AR(1). But if you have evidence to the contrary, bring it forward. I am also in the process of looking for it. But, in the meantime, I’m going with “balance of the evidence”.
Should you, or I or anyone come up with remotely plausible estimates of a bulge of energy at some specific long periodicity, we can apply it equally to testing the “no warming” hypothesis since 1990 and the 2C/century hypothesis now. In fairness, that’s the only way to do these things currently.
So, if you have evidence of these bulges, find them. Don’t just suggest there always might be something uncharacterized, and undreamed of somewhere that saves the IPCC projections from falsification!
lucia,
Many people have been telling you for months that your method of estimating the uncertainty in the underlying trend based on a single 7-year trend is wrong. Your confidence limits define the range of 7-year trends, and it’s true that the 7-year trend is less than 2C/century. That’s the only question your C-O confidence limits can answer.
The important question in my mind is what this says about the underlying trend. To determine this, we need to know the variability in 7-year trends about the underlying trend. This can be estimated empirically (as we’ve both attempted) or from models (as Gavin has done).
You completely missed the point about monthly trends (again) by switching to discussion of annual cycles. The question was not about annual cycles. It was about weather in the month of March that affects the March temperature trend. From year-to-year, March weather can be very different. The uncertainty in the trend for March 2008 says nothing about the year-to-year variability.
Can you see the parallels to uncertainty in a single 7-year trend and uncertainty across 7-year trends? They are completely different things. You can’t use one to validate the other — that’s like using the average weight of neighbourhood dogs to validate the weight of neighbourhood cats. (To borrow your analogy from another thread).
JohnV–
Trend analysis is a method to estimate the variabilty of 7 year trends based on the estimate itself.
I’m not missing the point about March. March is a subset of a full year, and each month may be different. March is different from July. We know this.
My answer is: We specifically know this feature about the annual season. It is well characterized. But you are now speculating there is some feature about 2001-2007 that somehow makes the characteristics of this weather “different” the population of all possible 2001-2007 years, or possibly from the weather we will have in 2008-2015, if there are no major volcanic eruptions or other surprises (like say Mars Attacks!)
Unlike March or April, there is no identifyable reason why 2008-2015 should have different statsitical properties form 2001-2007!
Your analogy is wrong. What I am doing is more like finding the average temperature for March by averaging over 7 years of March data. I compute my error bars based on the march data.
As for the idea that we can’t find the variance of all possible 7 year trends drawn from the same population from the uncertainty calculated from the CO fit.. that’s bunk. The 7 year trends are all from the same population (so no volcanos etc.) we can do this.
Gavin suggested I test this with AR(1). I did as he suggested, and posted. That was the subject of the previous post. The method works. Do the montecarlo and check!
lucia,
I said nothing about comparing March to July. I am talking about comparing March of this year to March of previous years. Each year is a unique realization of the annual cycle. The variability across years defines the confidence limits on March trends. It is a completely different property than the uncertainty in the trend for March of one year.
We seem to be arguing past each other again. I am saying that we can’t find the variance of all 7-year trends (relative to the underlying trend) using the confidence limits from a single 7-year trend. You’re talking about multiple trends from the same population (no volcanoes, etc).
I will try to re-pose my question:
What does the uncertainty in the trend from March of this year say about the uncertainty in trends for March of all years?
Is this rhetorical?
Why don’t you go do the problem, JohnV? Get the data for the past 7 year which seem to be from similar populations, do the fits, for each year and compare the variabilities you get both ways.
Having lived in Chicago a while, it seems to me that March has a generally warming trend, and that happens pretty much every year!
lucia,
No, it’s not rhetorical. Your whole assumption is that the C-O uncertainty in a single trend is indicative of the uncertainty between trends. Do you believe that assumption is valid for March temperature trends?
So, the question is:
Do you believe the uncertainty in a C-O trend for March of a single year is the same thing as the uncertainty in the trends for March of all years?
All I need from you is a simple yes or no. Then I’ll do the work of testing it.
Unfortunately, I think we’re still talking past each other. Why are you suggesting I get data from 7 years? I’ve already done that. I’m proposing a parallel analysis using monthly trends for March only. Why? Because March is not controversial. Because more data is available (I can use March of every year going back 50 or more years).
JohnV–
My impression is you are asking if the trend from March 1- March 31 for one year is the same for all years in a similar population (e.g. no volcanos). Is that what you are asking? Or are you asking something else?
lucia,
Thanks for stepping back to clarify. I will try to explain better. First, ignore volcanoes and 7-year trends — those are specifics that we have been discussing on another thread. I am trying to approach the more general problem of determining trend uncertainty using the specific example of March temperature trends in the USA48.
I will refine the example a little to make the comparison to 7-year trends and underlying trends a little more clear. It’s primarily a thought experiment, but it can be analyzed with real numbers if necessary.
Pretend it’s the end of March and we want to make a prediction for the temperature trend from March 1 to May 31 (this is the underlying trend). For various reasons we will restrict ourselves to using daily March temperatures to make the prediction. For the sake of simplicity, assume the long-term average trend for March is the same as the long-term average trend for March-April-May.
You choose to use C-O to fit a line to the daily March temperatures. The procedure gives you confidence intervals. Based on this analysis you provide a range of likely underlying temperature trends for March 1 to May 31.
To reduce contention, I also decide to use C-O to fit a line to the daily March temperatures. However, I choose not to use the C-O confidence intervals. Instead, I use historical temperature data for March, April, and May. I find the difference between the March trend and the March-April-May trend for every year. I then compute the standard deviation of these differences and use that to define my confidence intervals.
My question then is this:
Do you expect your confidence intervals to be the same as mine? Do they measure the same thing?
John V–
I must not understand you, because as far as I can see, what you are suggesting isn’t comparable to what I am testing.
Ok, so you want to predict the rate of change of temperature from March 1 to May 31? (For what, someplace?)
You assume linear for simplicity? Does anyone claim the average rate of change of temperature in March-April-May should be linear? (That is, if you average daily temperatures for 30 or 50 or 100 years, the trend from March 1- May 31 is linear in whatever record you have is linear?)
I’d say the underlying trend during this period is non linear. We could estimate it by betting data for 30 years, calculating the daily avearge for each of the 30 days and plotting. The evolution of that average better represents an “underlying trend” in an ensemble sense. I doubt very much if it’s linear.
So far, you want to impose a hypothesis no one has advanced, and which I suspect no on believes!
This would only make sense if someone somewhere had a hypothesis that the underlying trend was linear.
March April May are almost certainly in different portions of any “sinusoid” and each have different underlying climate trends from each other. Just eyeballing a sinusoid and drawing a line suggest the trend are likely to be noticeably different.
I’m puzzled by why you think this has something to do with what I am testing.
In contrast, the trend shown on the IPCC graph appears to show a linear trend from 2000-2030. If the central tendency on the IPCC graph did not look so very straight, I wouldn’t fit a straight line. I’d fit something else.
If the purpose of this analysis is to convince me that CO doesn’t properly get the variation in trends when the underlying trend is thought to be straight, why don’t you do this (which I think is more equivalent to what I do)
* Pick years with no volcanos using my criteria.
* For each of those “N” years, get Jan 1-Jan 31 data.
* Fit each of those years, using CO get the mean trend. “m”
* Calculate the “N” CO standard errors: sm
* Set that aside for the moment.
* Also, check for lag-1 residuals to the CO fit. Make sure they are less than… oh… 0.2. (There’s a formula. I don’t know it, so I’m just guessing.)
* Possibly, look at the whole correlogram, because you know people might look at that later. CO doesn’t work if the correlogram shows signs that the noise isn’t AR(1). I have no idea if this case is or isn’t. It might be. But if the correlogram clearly shows it’s not, CO won’t work.)
Now
* Calculate the standard deviation of the “N” trends to get σm.
* Calculate the average of the “N” trends to get smave.
Compare the the two. Maybe do a Chi-Square to find the uncertainty in the standard deviation over the “N” trends. Maybe look at the standar devation of the “N” sm’s for each year.
The, figure out if you can falsify the hypothesis that the two standard deviations are the same.
But, in short, the test you seem to suggest is not applicable because there is no hypothesis that the trend underlying March-May temperature is linear. In contrast, the IPCC document shows a straight line for the underlying trend. I’m testing that trend.
Then repeat for all 12 months.
January is January. Feb is Feb. March is March. etc.
I do not have a hypothesis that the “underlying” trend for March is the same as in April or May. As far as I’m aware, no one does. In contrast, the IPCC projections show a constant underlying trend from 2000-2003. It is only because this
lucia, you’re being way too literal.
It’s a thought experiment. It is intentionally not related to your falsification. Trends for March, April, and May do not matter. I’m trying to get you to think about the uncertainty in the trend estimate. Step back from your “falsification” and see if what you’re doing makes sense in other contexts.
Let me simplify even further. Forget about March and May. Think about just April. Restrict it to the USA lower 48. You estimate the trend and the uncertainty in the trend for April 2008 using C-O. Call this sdCO.
I estimate the trend for April 2008 and every previous April going back 50 years. That is, I have 50 independent realizations of April weather. From these 50 trends I calculate a standard deviation. Call this sd50.
Do you expect sdCO and sd50 to be the same? Are they measuring the same thing? I say they are not.
JohnV–
What do you mean I’m being to literal? You were suggesting a thought experiment involving March-April-May, which would likely tell us the temperature trend in March is different from April, and different from May. You seemed to think this would tell us something about Cochrane Orcut. I explained why the test you proposed wasn’t going to work.
Then, I suggested another test you might do. Now, you come forward “simply” and the simplification involves a)stripping the test of all the checks required to make sure CO is a valid method,
b) switching January to April and then eliminating “repeat for all 12 months”.
c) including years with evolving dust veils from volcanos.
d) Not averaging the standard errors in the trend from the “N” applications of CO (when the theory of CO only says it’s the average of that standard error that should match the standard devation of the trend.) and
e) Not doing statistical tests to see if the difference in the two final values is statistically significant.
These changes aren’t “simplifying”. They are stripping the test of validity so that your results will tell us nothing about whether or not Cochrane Orcutt does what it is supposed to do either because you are not checking if you are applying it where it’s valid, you are including samples from different populations (volcano vs. no volcano years) and you are, for some reason, dropping out the final averaging step.
So, if you think this calculation will prove the point you are trying to make, go ahead, do the computation. When you are done, you will presumably have the answer to your own question.
If you want to tell others what you’ve found, you’ll need to explain what you did and your results. So, bear in mind the caveats already given (volcanos, check correlogram, average the sd’s etc.) Remember that when you are done there still may be open questions. For example, I noticed I forgot to tell you that if you use land/ocean data, you should exclude the bucket years too. For all I know, I forgot other caveats, (most of which we have already discussed at this blog.)
I’ll be interested to learn what you have found when you are done.
lucia,
Take two giant steps away from volcanoes. Step away from your “falsification”. Think abstractly. Consider whether uncertainty in a short-term trend is the same as uncertainty in the difference between a short-term trend and the long-term trend. Use something other than temperature if you must.
I’ll try to meet you half way by generating and working with AR(1) + White Noise data. My early results show that the uncertainty intervals calculated from C-O on a single sample are much too small. I will report back after checking my results.
JohnV–
I’m only responding to what you are suggesting as your thought experiment. If you want to do the AR(1), why not discuss that in the first place? That was the discussion in yesterdays post!
With generated AR(1), preliminarily I have found.
1) If I assume the lag 1 residuals are known, I get precisely 5% rejections using CO. That’s what I reported here
Notice that I also said I had written the scripts for another purpose and am looking at other questions. That’s what I’m doing.
2) If I compute the lag 1 autocorrelation based on the data, the CO over rejects. However, the amount varies.
For small autocorrelations, the rejection rate is close just above 5%– so yes, the error bars are a little too small. For those cases, Tamino’s Lee & Lund correction under rejects. So, it make the error bars too small.
For large autocorrelation, the rejection rate can be as high as 10% when it should be 5%. But in that case the Tamino Lee & Lun correction gets a 5% rejection.
For autocorrelations near those exhibited by the data, the CO rejection rate, which should be 5% is roughtly 7%. So, the error bars are a little larger. However, the Tamino Lee and &Lund correction under rejects slightly.
So, what we get is the Tamino-Lee&Lund error bars are sometimes too large, but never too small (for the few cases I’ve done.) The CO ones, with ρ known are absolutely perfect. The CO ones with iteration (which is the real why to do it) are a little too small in some cases.
So, yes, I am exploring this, and I’m planning to report fully when I’ve looked at number more carefully.
But— and this is important: the IPCC projections have been rejecting under BOTH CO and OLS with Tamino error bars. I also tested “reject by both” (because the correlation is imperfect.)
If we compare the standard deviations estimated based on the trend using CO and those based on the 10,000 generated series, the difference has stayed below 20%! So, you see, I have some basis for believe they aren’t off by a factor of 2 as suggested by Gavin.
I’m still checking this.
lucia,
I only offered to fall back on concrete AR(1) data because I’ve given up trying to get you to step back and do the thought experiment. Maybe it’s me, maybe it’s you, but we were clearly not getting anywhere. I hope to come up with a non-temperature analogy that demonstrates my point. Or maybe somebody who understands what I’m trying to say can give it another shot.
Either way, I’m likely out for the weekend. The mountains are calling and my bike and I must go.
JohnV–
Have fun.
I suggest if you think of an statistical analysis that can be done it would be wiser for you to simply do it rather than suggest we all debate what the answer would be if someone bothered to do it. You doing the analysis would result in two things a) permit us all to know for sure what the heck the analysis is you are describing and b) give you the answer to the question. Both are better than having me try to figure out just what analysis you are suggesting could be done, and also guess what the result of the analysis would be!
I’m going to look at this in an entirely different way. It would appear that the 8 year time peroid
between 2001 and 2008 will have a negative slope. That is, the 8 year slope will be negative, assumming that the first 6 months of 2008 is a good predictor of the last six months ( june, july is good predictor)
Anyways, assume that at the end of 2008 we have a negative slope for the 8 year span of 2001 -2008.
What to make of this?
1. Climate coolists: Its the end of AGW, AGW is wrong, models are wrong, radiative physics is wrong,
we are entering an ice age. That hand is Doyle Brunson’s 10-2.
2. Climate Warmists: it’s the weather. we have no explaination. It happens all the time. there is no
information in an 8 year trend. none.
is there information in an 8 year trend? Well, one approach to that problem is to bicker about
error bars. Another approach is to see in the actual record, lets say the past 100 years, how often
we see a negative trend over an 8 year peroid. Is it common? is it rare. If its RARE, then information
theory tells me it has a HIGH information content. But thats just my take on things
So. I went to look at all 8 year trends from 1900 to 2007 ( 2008 isnt done) Here is what you find
1. Every Batch of them ( save 1) is associated with volcanic activity. In the early 1900s, in the 6os,
in the 70s, in the 80s in the 90s. If you find a 8 year negative slope in GSMT, You had a volcano.
This is a good thing. It tells is the science of GW understand things.
2. The SOLE exception is the batch of 8 year negative trends in the mid 40s. Now, until recently
GCM had not been able to match these negative trends (hmm) BUT now we find that the observatiion
record, the SST bucket/inlet problem, may be the cause of this apparent cooling trend.
So, from 1990 to 2000, a time when C02 was increasing we find that on rare occasions we will see
8 year trends that are negative. The cause: volcanos, and bad SST data.
Now, look at beyond 2000 and the last 8 years. negative trend. Any volcano? nope. any bucket problems?
err nope. So for the first time in 100 years you have a negative slope that is not correlated with either volacanoes or bad observation data. That looks interesting. Wave your arms and cry weather?
That’s not science. Thats like waving your arms and crying weather when it gets warmer. The appeal to
ignorance. We have a cooling regime. a cooling regime that is not associated with volcanoes and not associated with data errors. I think Thats interesting and meaningful. Dont know what it means, but its the kind of thing you want to investigate rather than shrug off
The low frequency stuff is absolutely critical to the significance of the trend, and of course it is really the low frequency stuff that is important to understanding the weather/climate conundrum. All too often climate scientists gloss over this but it is essential to interpretation of significance. And I would thank Lucia for quite a nice discussion on the topic.
The reason this stuff interests me is because the issues of LTP, self-similarity etc. are really all about how low frequency parts of the spectrum develop. If you plot your frequency graph log/log instead of linear scale (even semi-log X axis would do) you notice that on these scales, all Markovian processes become essentially flat as the log frequency scale becomes large negative. LTP processes continue linearly in log/log space for arbitrarily large negative log frequency. This is a profound and important difference, and nobody really knows what the right answer is as far as I can tell. Lucia makes the assumption that Markovian descriptions are valid (which is fair when testing IPCC claims, because they implicitly make that assumption also), but this is an assumption, and certainly not set in stone. (This relates to the assumption of ergodicity Lucia discussed on another post in an answer to Boris, and again kudos to Lucia for explicitly stating that, an assumption rarely explicitly stated by climate scientists in my experience).
The problem climate scientists are going to run into is that there is only a narrow band of low frequency weather spectral power that their argument holds up under, and that band is rapidly shrinking. Under LTP assumptions (large low frequency variations), IPCC predictions don’t falsify, but the 20th century warming is insignificant also. Under Markovian assumptions (relatively small low frequency variations), the 20th century warming is significant, but IPCC predictions falsify.
Those who assume valid the strong-positive-feedback-AGW hypothesis are currently between a rock and a hard place. Will the weather over the next few years save their position, or squeeze the tiny space within which their assumptions hold water disappear to nothing? I think I’m going to get through a lot of popcorn in the next few years watching this play out in the blogs 🙂
SteveM–
I think you have “the man on the street’s” (T.C. Mitts) argument perfectly stated. Absent volcano eruptions, these these zero trends have been rare.
So, the trend fitting I am doing, (which is not exactly “sophisticated” statistics, and can be done on EXCEL), are simply supporting the general impression T.C. Mitts would have gotten from the data: 8 year flat trends are not really what one expects if there is a 2C/century trend. The exception is should a volcano erupt. (In which case, all the climate scientist better include caveats in their bets for warming. 🙂 )
We can add to that the fact that, likely as not, “T. C. Mitts” was not under the impression that the temperature flat lining for 8 years was consistent with warming– absent volcanos. These sorts of caveats were not communicated quite this way until after the weather flatlined. Moreover, those advocating this large degree of variability now weren’t rushing over to Tamino’s blog, requesting he be more careful when, last August, he performed best fit trends to prove that “No statistically significant warming” was falsified since Jan 2000. That analysis involved 91 months of data.
So, the impression that has been conveyed is that this method is considered perfectly fine and could be applied to 91 month records provided the out come proved a positive trend is statistically significant. Applying the exact same analysis now indicates “No 2 C/century warming over 89 months”, suddenly, the method doesn’t work.
So, collectively, the impression that is being conveyed is that the validity of a method depends on the answer obtained. In fact, so far as I can determine, the method I am using is imperfect, but still pretty good. As I’ve said before, my uncertainty bars may be a little too small, and I’m looking at things. (As far as I can tell, if I’m off, it’s not by very much, and the weather data itself, and that from the volcano-free period, can hardly be said to “support” Gavin’s error bars.
I think people would consider the data to mostly contradict him.
Of course, the argument over the uncertainty intervals, the variability of “(weather!)”, whether we give primacy to data over model predictions will continue. Because statistics can be used to obscure the reality, particularly when the person using statitics refuses to state the hypothesis their analysis tests in words or to admit other questions might exist.
But T.C. Mitts knows that statistics and graphs can obscure. That’s why tactics like proving “(weather!)” is variable with blogs posts that amount to, “Look at the graphs with all these downtrends.” (which 5 minutes of research reveals were all caused by volcanic eruptions,) looks really bad to T.C. Mitts!
RE:4112
JohnV
Lucia, too literal? As Mosher points out, we are dealing with nearly a decade in which the global climatic system has been shedding joules unattributable to any of the traditional possible “explanations” or dogmatic AGW hand waving. You deserve an A for your efforts to repeatedly redefine what Lucia is showing, but no cigar I’m afraid. Fact is, ALL the relevant global temperature metrics are down: land surface, sea surface, deep ocean, lower troposphere, and more troubling the tropical troposphere as well as the antarctic and arctic. Thought experiment? You wish. Cognitive dissonance your end? More likely.
Looks like Tamino is looking to rebut the falsification with Bayesian statistics. He has started with a post explaining the theory but I suspect more will follow.
I am sure his choice of priors will be interesting but I doubt that they will be able to address the fact that an 8 year flat trend without volcanoes is an unprecedented event (at least during the last 100 years).
Raven–
If he posts on Bayesian, that will be interesting. Bayesian requires a person to make assumptions about what you think is true in the first place (prior probabilities). Presumably, he would get that from the previous data, and then find the best estimate for something now. (Or something like that.)
So… will the “previous” data start in 1975? Or 1880? 1900?
I spent a little time this morning looking at the standard deviation of the slope calculated from a Cochrane-Orcutt (C-O) regression. The results surprised me, but they do match theory (which was new to me) and lucia’s results. In particular, for a pure AR1 + White Noise time series:
– the standard deviation of the slope for C-O regression on a single interval is equal to
– the standard deviation of the mean slopes from C-O regression on many intervals
Or, using lucia’s words:
“the distribution of a random variable (the trend)†== “the uncertainty in defining the trend in a single realisationâ€
(http://rankexploits.com/musings/2008/developing-estimates-for-uncertainty-intervals/)
I still believe the error bars are too small because the assumption of AR1 + White Noise implicitly excludes known cycles such as solar, ENSO, AMO, etc. My poorly explained “spring warming” thought experiment from last week was an attempt to look at non-random cycles from a different context.
However, the point of this comment is to emphasize something on which we agree. That is, under the assumption of AR1 + White Noise the C-O error bars from a single interval are appropriate. I’ve learned something. I’m happy about that.
JohnV–
The issue with your analogies is you are not being careful to pick examples that match what I am doing. Otherwise, if the features are such that I would never use CO, the fact that CO wouldn’t give good results is meaningless!
Now, on the lower frequency weather noise:
Yes, as I say in the post, if the energy spectral density function doesn’t look like the one assumed by CO, the uncertainty intervals will be wrong.
However, if we don’t know the true shape, it is difficult to speculate whether the CO error bars are too small or too large.
If, for example, the only “major bulge” sticking out of distribution above is ENSO, then the CO error bars may be too small. The reason would be there would then be more variability at scales below our sampling cut off than anticipated by CO. CO (as a method) sort of “knows” where the time cutt off is. It “sees” too much energy in proportion to the real amount, and then estimates too much uncertainty!
If there are major bulges at lower frequencies then the CO error bars could be too small. But the argument needs to be more sophisticated than “you’ve missed the low frequencies”. You actually need to come up with estimates of how much energy there might be in the pdo! Or, come up with an estimate of the time scale etc!
lucia,
Just for a laugh, try adding a small sinusoidal solar cycle to your Monte Carlo simulations. I extended my Monte Carlo work from this morning and the results were interesting. I’ll summarize here but my results do still require checking.
My procedure was:
(1) Generate AR1 + White Noise time series (rho=0.75, Gaussian noise, stdev = 0.12, slope = 0);
(2) Use Cochrane-Orcutt to obtain the slope for every 90 sample interval (mi);
(3) Use Cochrane-Orcutt to obtain the stdev of the slope for every 90 sample interval (si);
(4) Compute the standard deviations of the slopes, mi, from step 2 (Si);
(5) Found that Si ~ si (as you have said and as predicted by theory for pure AR1 + White Noise);
(6) Computed the z-score for every mi and si (Zi = mi/si);
(7) Found that |Zi| > 2 approximately 5% of the time (5% false rejections, as expected);
I wanted to see the effect of a small external cycle on the AR1 + White Noise. To do so, I added a hypothetical 11-year sinusoidal solar cycle to the time series from (1), and went through the same steps. Of particular interest are the results from (5) and (7). I summarize them below for a couple of solar cycle peak-to-trough amplitudes:
—
Peak-to-Trough = 0.05C (half of accepted value):
(5) Si = ~1.5 si
(7) ~20% false rejections (|Zi| > 2 in approx 20% of intervals)
—
Peak-to-Trough = 0.10C (accepted value):
(5) Si = ~2.5 si
(7) ~55% false rejections (|Zi| > 2 in approx 55% of intervals)
—
That is, if the solar cycle temperature peak-to-trough effect is only half the “accepted value” of 0.1C then your error bars should be ~1.5x larger. If the solar cycle is as large as the “accepted value”, then your error bars should be ~2.5x larger.
This is why I say your error bars computed from a single interval are questionable. All it takes is a very small cycle with a period longer than your interval to significantly affect your results.
If you’re like me, you’d probably rather update your own script than slog through my spreadsheet. I can email it if you’d like though.
JohnV says:
“All it takes is a very small cycle with a period longer than your interval to significantly affect your results”
And variations in the lengths of the solar cycle could introduce trend into the data if the earth’s time constant is larger than length of solar cycle.
I find it curious that Gavin has not brought up the solar cycle as a reason for the falsification.
This is because, according to models, the effect is miniscule. If the effect is 0.1C peak to trough, most models are are not detecting it. Basedon what I’ve seen of averaged model E results, this includes Model E results.
Though… I have to check that.
JohnV–
Very good. I’m planning to add the cyclic things– but I’d planned outan order, and that’s not next on my list. (The red& white are. I’m trying to ramp up in increasing levels of complication.)
Now you need to check a few things:
1) Look at the correlogram. See if the correlogram would have looked like AR(1). There are some tests to check for serial autocorrelation of the residuals to CO. I discussed that on one post. If the fit wouldn’t survive this, I wouldn’t use CO because we would know, based on the data, it doesn’t apply. (Also, if you find “finger prints” of the 11 year cycle, we can look at the data and see if they are there. That would be useful.)
2) See whether if the properties of the case you ran were true, the “weather noise” we actually experienced would have happened with any reasonable degree of probability. (I’m doing this with the case gavin called “closer” to the models than the data. Basically, gavin’s case fails.)
3) Run a case adding “ENSO” noise, and see what that does to the uncertainties intervals (and the correlogram).
If you don’t do these, I’ll end up doing them– but not for a month.
lucia,
I did the above using an Excel spreadsheet but Excel is pretty clunky for this sort of analysis. Before implementing the checks I will be moving to Matlab/Octave, R, or maybe C#. Give me a few days and I’ll report back. I might even surprise you and write it up on my own site.
JohnV–
Excel is very clunky for this! That’s why I broke down and wrote programs! That’s also why testing in a certain order. I want to make sure I’ve got each module correct before I move on to the next bit.
So, for me, adding red& white noise comes first. I know there is white noise in the system. I’m not sure the solar has a 0.1 C peak to trough amplitude!