Guest post by Richard Saumarez
Correlations are often used to relate changes in climate variables in order to establish a potential relationship between them. These variables may have been processed in some way before they are correlated, possibly by filtering them. There has been some debate, which may have shed more heat than light, about the validity of combining these processes and whether they interact to make the conclusions drawn from them invalid. The object of this post is to explain the processes of correlating and filtering signals, the relationship between them and show that the results are predictable provided one takes care.
The importance of the Fourier Transform/Series.
Fourier analysis is of central importance in filtering and correlation because it allows efficient computation and gives theoretical insight into how these processes are related. The Fourier Transform is an analytical operation that allows a function that exists between the limits of – and + infinity to be expressed in terms of (complex) frequency and is a continuous function of frequency. However, a Fourier Transform of a real-World signal, which is sampled over a specific length of time – a record -, is not calculable. It can be approximated by a Fourier series, normally calculated through the discrete Fourier Transform algorithm, in which the signal is represented as the sum of a series of sine/cosine waves whose frequencies are an exact multiple of the fundamental frequency (=1./length of the record). Although this may seem to be splitting hairs, the differences between the Fourier Transform and the series are important. Fortunately, many of the relationships for continuous signals, for example a voltage wave form, are applicable to signals that are samples in time, which is the way that a signal is represented in a computer. The essential idea about the Fourier transform is that it takes a signal that is dependent on time, t, and represents it in a different domain, that of complex frequency, w. An operation performed on the time domain signal has an equivalent operation in the frequency domain. It is often simpler, far more efficient, and more informative to take a signal, convert it into the frequency domain, perform an operation in the frequency domain and then convert the result back into the time domain.
Some of these relationships are shown in figure 1.
Figure 1. The relationship between the input and output of a system in the time and frequency domains and their correlation functions. These are related through their (discrete) Fourier Transforms.
If a signal, x(t) passes through a linear system, typically a filter, the output, y(t) can be calculated from the input and the impulse response of the filter h(t), which is, mathematically, its response to an infinite amplitude spike that lasts for an infinitesimal time. The process by which this is calculated is called a convolution, which is often represented as “*”, so that:
y(t)=x(t)*h(t)
Looking at figure 1, this is shown in the blue upper panel. The symbol t, representing time, has a suffix, k, that indicates that this is a sampled signal at t0, t1, t2 …… Arrows represent the Discrete Fourier Transform (DFT) that convert the signal from the time to the frequency domain the inverse transform (DFT-1) that converts the signal from the frequency to the time domain. In the frequency domain, convolution is equivalent to multiplication. We can take a signal and transform it from x(tk) to X(wn). If we know the structure of the filter, we can calculate the DFT, H(wn), of its impulse response. We can write, using the symbol F as the forward transform and F-1 as the inverse transform, the following relationships to get the filter output:
X(w)=F[x(t)]
H(w)=F[h(t)]
Y(w)=X(w)H(w)
y(t)=F-1[Y(w)]
What we are doing is taking a specific frequency component of the input signal, modifying it by the frequency response of the filter to get the output at that frequency. The importance of the relationships shown above is that we can convert the frequency response of a filter, which is how filters are specified, into its effect on a period of a time domain signal, which is usually what we are interested in. These deceptively simple relationships allow the effects of a system on a signal to be calculated interchangeably in the time and frequency domains.
Looking at the lower panel in Figure 1, there is a relationship between the (discrete) Fourier Transform and the correlation functions of the inputs and outputs. The autocorrelation functions, which are the signals correlated with themselves are obtained by multiplying the transform by a modified form, the complex conjugate, written as X(w)*, (see below), which gives the signal power spectrum and taking the inverse transform. The cross correlation function is obtained by multiplying the DFT of the input by the complex conjugate of the output to obtain the cross-power spectrum, Gxy(w) and taking the inverse transform, i.e.:
Gxy(w)=X(w)Y*(w)= X(w)X*(w)H*(w)
Rxy(t)=F-1[Gxy(w)]
Therefore there is an intimate relationship between time domain signals representing the input and output of a system and the correlation functions of those signals. They are related through their (discrete) Fourier Transforms.
We now have to look in greater detail at the DFT, what we mean by a frequency component and what a cross-correlation function represents.
Figure 2 shows reconstruction of a waveform, shown in the bottom trace in bold by adding increasingly higher frequency components of its Fourier series. The black trace is the sum of the harmonics up to that point and the red trace is the cosine wave at each harmonic. It is clear that as the harmonics are summed, its value approaches the true waveform and when all the harmonics are used, the reconstruction is exact.
Up to this point, I have rather simplistically represented a Fourier component as being a cosine wave. If you compare harmonics 8 and 24 in figure 2, the peak of every third oscillation of harmonic 24 coincides with the peak in harmonic 8. In less contrived signals this does not generally occur.
Figure 2 A wave form, shown in bold, is constructed by summing its Fourier components shown in red. The black traces show the sum the number of Fourier components up to that harmonic.
The Importance of Phase
Each Fourier component has two values at each frequency. A sine and cosine waves are generated by a rotation of a point about an origin (Figure 3). If it starts on the y axis, the projection of the point is a sine wave and its projection on the x axis is a cosine wave. When the Fourier coefficients are calculated, the contribution of both a sine and a cosine wave to the signal at that frequency are determined. This gives two values, the amplitude of the cosine wave and its phase. The red point on the circle is at an arbitrary point and so its projection becomes a cosine wave that is shifted by a phase angle, usually written as j. Therefore the Fourier component at each frequency has two components, amplitude and phase and can be regarded as a vector.
Earlier, I glibly said that convolution is performed by multiplying the transform of the input by the transform of the impulse response (this is true since they are complex). This is equivalent to multiplying their amplitudes and adding their phases. In correlation, rather than multiplying X(w) and Y(w), we use Y(-w), the transform represented in negative frequency, the complex conjugate. This is equivalent to multiplying their amplitudes and subtracting their phases. Understanding the phase relationships between signals is essential in correlation[i].
Figure 3 The Fourier component is calculated as a sine and cosine coefficient, which may be converted to amplitude, A, and phase angle j. The DFT decomposes the time domain signal into amplitude and phases at each frequency component. The complex conjugate at is shown in blue.
Signal shape is critically determined by phase. Figure 4 shows two signals, an impulse and a square wave shown in black. I have taken their DFT, randomised the phases, while keeping their amplitudes the same, and then reconstructed the signals, shown in red.
Figure 4. The effect of phase manipulation. The black traces are the raw signal, and the red trace is the signal with a randomised phase spectrum but an identical amplitude spectrum
This demonstrates that phase has very important role in determining signal shape. There is a classical demonstration, which space doesn’t allow here, of taking broad band noise and imposing either the phase spectrum of a deterministic signal, while keeping the amplitude spectrum of the noise unaltered or doing the reverse: imposing the amplitude spectrum of the deterministic signal and keeping the phase of the noise unaltered. The modified spectrum is then inverse-transformed into the time domain. The phase manipulated signal has a high correlation with the deterministic signal, while the amplitude manipulated signal has a random correlation with deterministic signal, so underlining the importance of phase in determining signal shape.
Figure 5 The phase spectra of delayed impulses.
A very important concept is that phase represents delay in a signal. A pure time delay is a linear change in phase with frequency as shown in figure 5. The amplitude of the signal is unaltered, but in the case of a delay, there is increasing negative phase with frequency. However, any system that changes the shape of the input signal as it is passed through it, as is usually the case, will not have a linear phase spectrum. This is a particularly important concept when related to correlation.
We are now in a position to understand the cross-correlation function. Looking at the formula for correlation shown in figure 1, the CCF is:
,
which can be normalised to give the coefficient: 
, where N is the number of points in the record.
This rather formidable looking equation is actually quite straightforward. If k is zero, this is simply the standard formula for calculating the correlation coefficient and x is simply correlated with y. If k is one, the y signal is shifted by one sample and the process is repeated. We repeat this for a wide range of k. Therefore the function rxy is the correlation between signals two signals at different levels of shift, k and this tells one something about the relationship between the input and output.
We have a signal x(t) which has been passed through a physical system, with specific characteristics, which results in an output y(t) and we are trying to deduce the characteristics of the system, h(t). Since, from Figure1, the DFT, Y(w) of the output is the product of the DFTs of the input X(w) and the impulse response, H(w), could we not simply divide the DFT of the output by the DFT of the input to get the response? In principle, we can, providing the data is exact.
However most real world measurements contain noise, which is added to the inputs and outputs, or even worse other deterministic signals, and this renders the process somewhat error prone and the results of such a calculation are shown below (figure 6), illustrating the problem:
Figure 6. Left: Input (black) and output (red) signals for a system. Right: the calculated impulse response with 2.5% full scale amplitude noise added to the input and output (black) compared with the true impulse response (red). The low pass filtered response is shown in green.
This calculation illustrates another very important concept: linear physical systems store and dissipate energy. For example, a first order system, which could be the voltage output of a resistor/capacitor network or the displacement of a mechanical spring/damper system, absorbs energy transients and then releases the energy slowly, resulting in the negative exponential impulse response shown in figure 6. The variables which fully define the first order system are its gain and the time constant. The phase spectrum of the impulse response in distinctly non-linear. Attempts to measure another variable, for example delay, which implies a linear phase response, is doomed to failure because it doesn’t really mean anything. For example, if one is looking at the relationship between CO2 and temperature, this is likely to be a complex process that is not defined by delay alone and therefore the response of the system should be identified rather than a physically meaningless variable.
Noise and Correlation
Correlation techniques are used to reduce the effects of noise in the signal. They depend on the noise being independent of, and uncorrelated with, the underlying signal. As explained above, correlation is performed by shifting the signal in time, multiplying the signal by itself (auto-correlation) or with another signal (cross-correlation), summing the result, and performing this at every possible value of shift.
Figure 7. Broad band noise ( black) with the autocorrelation function( red) superimposed.
In broad band noise, each point is, by definition uncorrelated with its neighbours. Therefore, in the auto-correlation function, when there is no shift, there will be perfect correlation between it and its non-shifted self. For all other values of shift, the correlation is, ideally, zero, as shown in figure 7.
The auto correlation function of a cosine wave is obtained in the same manner. When it is unshifted, there will be a perfect match and the correlation will be 1. When shifted by ¼ of its period, the correlation will be zero, be -1 when shifted by ½ a period and zero when shifted by ¾ of period.
The ACF of a cosine wave is a cosine wave of the same frequency with an amplitude of the square of the original wave. However if there is noise in the signal, the value of the correlation will be reduced.
Figure 8 shows the ACF of broadband noise with two sine wave embedded in it. This indicates recovery of two deterministic signals that have serial correlation and are not correlated with the noise. This is a basis for spectral identification in the presence of noise.
Figure 8 The ACF of a signal containing two sine waves of different frequencies embedded in noise. The ACF (red) is the sum of two cosine waves with the same frequencies.
A very important feature of the ACF is that if destroys phase information. Referring to Figure 1, the DFT of the ACF is X(w) (or Y(w)) multiplied by its complex conjugate, which has the same amplitude and negative phase. Thus when they are multiplied together, the amplitudes are squared and the phases are added together making the resultant phase zero. This is the “power spectrum” and is the ACF is its inverse DFT. Therefore the ACF is composed entirely of cosine waves and is symmetrical about a shift of zero.
However, the cross-correlation function, which is the inverse DFT of the cross-power spectrum contains phase. By multiplying the complex conjugate of the output by the input in the frequency domain, one is extracting the phase difference and the delays at each frequency between the input and the output and the cross-correlation function reflects this relationship. If the power spectrum, e.g.: the DFT, of rxx(t) is Gxx(w) and the cross-power spectrum of rxy(t) is Gxy(w), then:
H(w)=Gxy(w)/Gxx(w)
Figure 9 shows the same data as used in figure 6 to calculate the impulse response and the error is very much reduced because signal correlation is a procedure that separates the signal from noise.
Figure 9. Estimate of the impulse response using the data in figure 6 via cross-correlation (black) and pre-filtering the data (green).
These calculations are based on a single record of the input and output. When available, one uses multiple records and calculates the estimated response E[H(w)] from the averaged power spectra:
E[H(w)]=< Gxy(w)>/<Gxx(w)>
Where < x> means the average of x. This leads to a better estimate of the impulse response. It is possible to average because correlation changes the variable from the time domain to relative shift between the signals so aligning them.
One simple check that can be performed to check that one is getting reasonable results, assuming that one has enough individual records, is to calculate the coherence spectrum. This is effectively the correlation between the input and output at each frequency component in the spectrum. If this is significantly less than 1.0, it is likely that there is another input, which hasn’t been represented, or the system is non-linear.
One of the major problems in applying signal processing methods to climate data is that there is only one, relatively short, record and therefore averaging cannot be applied to improve estimates.
Improving resolution by record segmentation and filtering.
One can improve estimates of the response if one has a model of what the signal represents. If one is dealing with a short term process, in other words the output varies quickly and transiently in response to a short term change in input, one can estimate the length of record that is required to capture that response and hence the frequency range of interest. This enables one to segment the record into shorter sections. Each segment has the same sampling frequency, therefore the highest frequency is preserved. By shortening the length of each record we have thrown away low frequencies because the lowest frequency is 1/(record length). However, we have created more records containing high frequencies, which can be averaged to obtain a better estimate of the response.
The other strategy is filtering. This, again, involves assumptions about the nature of signal. Figure 10 shows the same data as in figures 7 & 8 after low-pass filtering. The ACF is no longer an impulse but is expanded about t=0. However the ACF of the deterministic signal is recovered with higher accuracy.
This can be done here because the signal in question has a very small, low frequency, bandwidth and is not affected by the filtering (figure 11). The effects of the filter are easily calculable. If it has a frequency response of A(w), the input and output signals become X(w)A(w) and Y(w)A(w). The cross correlation spectrum is therefore simply:
Gxy(w)= A(w)X(w)A*(w)Y*(w)=A2(w)X(w)Y*(w)
Figure 10 The ACFs using the same data as in Figure 6. Note that the ACF of noise is no longer an impulse at t=0 and that there has been a considerable improvement in the ACF of the two sine waves as they now represent a higher fraction of the total power in the signal.
A2(w)is the autocorrelationfunction of the filter, which has no phase shift and will not affect the phase of the cross-power spectrum, provided the same filter is used on the input and output. This is because the phase reversal of the complex conjugate of the filter in the output cancels out that applied to the input, so the timing relationships between the input and output will not be affected. Provided the ACF spectrum is in the pass band of the filter, it will be preserved. In figure 9, the estimated impulse responses are shown using filtered (green) and non-filtered data. If one wishes to characterise the response, by assuming it is a first order system (which this is), one can fit an exponential to the data so getting its gain and time constant. The filtered result gives marginally better estimates but one has to design the filter rather carefully, appreciate that filtering modifies the impulse response and correct the results for this.
Thus, it is possible to filter signals and perform correlations, provided that the frequency band being filtered does not overlap to much the system response, as illustrated in figure 11, and one is careful.
Figure 11. The signal spectrum is composed of the true signal and noise spectra. A good filter response (green) will attenuate (grey) some of the noise but preserve the true signal, while a bad filter will modify the true signal spectrum and hence the ACF.
In practice, however there is likely to be an overlap between the noise and signal spectra. If the system response is filtered, the correlation functions will be filtered and become widened and oscillatory. In this case, the results won’t mean much and almost certainly will not be what you think they are! There are more advanced statistical methods of determining which part of the spectra contain deterministic signal but, in the case of climate data, the short length of the modern record and the poor quality of the historical record makes this very difficult.
Degrees of Freedom.
Suppose we have two signals and we want to determine if they have different means. They both have a normal distribution and the same variance. Can we test the difference in means by performing a “Student’s t” test? This will almost certainly be wrong, because in most simple statistical tests, there is an assumption that each observation is independent. In figure 7, the ACF is an impulse and nominally zero elsewhere, showing that each point is independent of each other. If the signal is filtered, the points are no longer independent because we have convolved the signal with the impulse response of the filter, as shown in figure 10. Looking at figure 1, the time domain convolution is given by:
This is similar to the correlation formula, except that the impulse response is reversed in time. It shows that the output at any point is a weighted sum of the inputs that have preceded it and are therefore no longer independent. Therefore in applying statistical tests to signal data, one has to measure the dependence of each sample on others by using the autocorrelation of the signal to calculate the number of independent samples or “degrees of freedom”.
Conclusion.
Correlation and filtering are highly interdependent through a set of mathematical relationships. The application of these principles is often limited because of signal quality and the “art” of signal processing is to try to get the best understanding of a physical system in the light of these constraints. The examples shown here are very simple, giving well defined results but real world signal processing may be messier, require much more statistical characterisation and give results that may be limited statistically by inadequate data.
One always has to ask what is the goal of processing a signal and does this make any sense physically? For example, as discussed earlier, cross correlation is often used interchangeably with “delay and it is only meaningful if one believes that phase response of the system in question has a linear phase response with frequency. If one is estimating something that is not meaningful, additional signal processing will not be helpful.
Rather, if one has a model of the system, one can then calculate the parameters of the model and, having done this, one should look carefully at the model to see if it accounts for the measured signals. Ideally, this should be tested with fresh data if it is available, or one can segment the raw data and use one half to create the model and test it with the remaining data.
Modern scripting programs such as “R” allow one to perform many signal processing calculations very easily. The use of these programs lies in not applying them blindly to data but in deciding how to use them appropriately. Speaking from bitter experience, it is very easy to make mistakes in signal processing and it is difficult to recognise them. These mistakes fall into three categories, programming errors, procedural errors in handling the signals and not understanding the theory as well as one should. While modern scripting languages are robust, and may largely eliminate straight programming errors, they most certainly do not protect one from making the others!
[i] I have used the electrical engineering convention in this post, i.e.: the basis function of the Fourier Transform is a negative complex exponential.
Related articles
- Fourier series before Fourier (johndcook.com)
- Retrotechtacular: The Fourier Series (hackaday.com)
- Sampling Rate (tamino.wordpress.com)
Well I had a longer post, but somehow M$ UNI-Virus blew it away.
While I understand Richard Courtney and other’s suggestions; my preferencs Richard (author), is that you are best writing in the manner that is comfotable to you. Your various audiences, just have to adapt to you; not verse vicea.
A nice presentation, and a good refresher, after too many years. I’ll save it for reference.
Now for an encore (in your spare time); the single biggest absence from climate science is the theory of sampled data systems.
All your nice transform stuff pre-supposes, that you do have actual real data, and not just noise.; which is about ALL there is in climate “data”. So a primer on the Nyquist sampling theorem, would be a good mate to this current dissertation.
Thanks for taking the time; it was well worth it.
George E. Smith
RCSaumarez: “I agree that if you are familiar with DSP, this is a very simple exposition.”
Of course it is to DSP types (if my assumptions based on the first diagram and first few paragraphs are correct). But that certainly does not mean that your post has no value to much of this site’s audience. My experience suggests that, even to many of this site’s brighter habitues, it would indeed. So you are to be commended for the effort that you have expended.
My criticism–which I intended to be constructive–was not that your post is simple. My purpose simply was to help you help your readers. My guess is that of this site’s regulars the subset who are familiar with digital signal processing may be larger than you imagine. (For example, my own exposure to it arose from the practice of law, not from engineering) So, if there is indeed something in your post that you think even DSP types would find of interest, it would be helpful to say so up front. That way such readers can know whether they would be right to do what I did: allocate their time to other things.
Be that as it may, I join you in your belief that certain of the climate discussions we’ve seen would have benefited significantly from DSP-theory results. And, again, you are to be commended for your effort.
My thanks to RIchard for a very understandable explanation of the Fourier transform. As a self-taught mathematician, such expository work is very valuable to me.
Next, FrankK says:
April 9, 2013 at 7:29 am
Man, you grammar nazis must like being frustrated. “Length of time” is a good, understandable, and perfectly valid English expression. Yes, it does not make literal sense … so what? It makes perfect sense and is totally without ambiguity, which is why it is used in English. When someone says it you know exactly and precisely what they mean.
So I don’t care if you beat your head against that wall for a hundred years … we’ll still be saying “length of time”, just like we have been for centuries.
But good luck with your project … how about you leave off correcting meaningless mistakes until you’ve succeeded with that one? To give you a sense of the size of the task you’ve set yourself, consider that there are TWENTY THREE MILLION separate pages on the web that use that phrase, so you’d best get going …
Of course, once you finish that, you’ll have to go fight the grammar criminals that talk about a long span of time. Span, of course, originally didn’t have anything to do with time, so it’s exactly as illogical as a length of time … and despite that, it’s been used to mean a length of time since the 1500’s … see, that’s how English changes, Frank.
I know you and many others would like to keep English the same forever, and to force it into a logical straight-jacket. But here’s the ugly truth about not only English but many other languages:
LANGUAGES ARE NOT LOGICAL, NEVER WERE, AND NEVER WILL BE
So it doesn’t matter how long your futile attempts last, Frank … you’re not going to succeed.
Finally, you should understand that such pedantry as you peddle will rarely be appreciated by authors. See, we choose our words very carefully to frame exactly the thought we’re trying to explain. And many authors, like myself, don’t give a damn whether our language is logical.
We may want it to be effective, arresting, quotable, strong, bathetic, sad, or any one of a number of things, and while we are doing that to the best of our abilities, having some jumped-up joker come along to tell us a phrase isn’t logical is … well … unappreciated at best, and much worse at worst.
You like that? “Worse at worst”? It’s a kind of truism, but then it expresses my meaning exactly.
And that is all I care about, that my words express and convey my meaning as clearly as possible. And in that quest, I have no interest at in whether one of the more common phrases in the English language is logical or not.
w.
PS—Did you notice my use of another evil phrase above, “how long your futile attempts last”? I suppose once you’ve exterminated “length of time”, you’ll have to start in on things like “How long did the concert drag on?” and “Will you be gone long?” … because as you correctly point out, time doesn’t have length, it has duration.
I suspect a phrase like “Don’t stay too long” is used, not because it is logical, but because it is economical. We don’t say “Don’t stay for an over-extended duration of time” as you might recommend, which means the same thing and is indeed logical, because it is clumsy.
And language needs to be efficient at times, for practical reasons. Generally, if two phrases mean the same thing, the shorter one will win out, whether it is logical or not.
Short version? Don’t bother fighting to make language logical … it isn’t, and good authors and critics just live with that as long as the meaning is clear.
FrankK says:
April 9, 2013 at 7:29 am
Time does not have length.
It will be a long time before you convince me of that.
But it only took a short time to type this comment.
This is a great article! Thanks.
Of time and duration ….
O gentlemen, the time of life is short;
To spend that shortness basely were too long
William Shakespeare
Good for Shakespeare, good for most of us.
Greg Goodman says:
April 9, 2013 at 12:34 pm
“It will or it could do ?”
It will. The equation
dCO2/dt = k*(T – To)
is equivalent to the statement. I actually did such an analysis to see if any other features might pop up. But, except for normal, to-be-expected variation from data with random errors, this is the essential feature. You can see the correlation in the plots with your naked eyes. It is really trivially true. And yet, incredibly, the debate rages on, and the scientific illiterates of the AGW movement press forward.
Very good overview of the basics in filtering and correlation, and you could add the problems of undersampling e.g. spurious responses.
I wonder if you or others could make a simple article about the matrix operations that are used from time to time. Like SVD or Principal Components.
I have played a little with SVD via some net application, and it can certainly extract a signal, but it can for sure also show some signal even where no signal exist, or it can show a completely different signal. I am not able to look through the finer details, but i have a natural scepticism when these methods are used instead of plain averages.
Willis Eschenbach says:
April 9, 2013 at 2:51 pm
My thanks to RIchard for a very understandable explanation of the Fourier transform. As a self-taught mathematician, such expository work is very valuable to me.
Man, you grammar nazis must like being frustrated. “Length of time” is a good, understandable, and perfectly valid English expression. Yes, it does not make literal sense … so what? It makes perfect sense and is totally without ambiguity, which is why it is used in English. When someone says it you know exactly and precisely what they mean. etc etc etc
—————————————————————————————————————
Goodness me Willy you do get very excited and aggro sometimes. I just disagree with you. OK.
Mr Saumarez,
Thank you for a very nice summary of frequency analysis and correlation. Well written and as short as can reasonably be expected for a complicated topic.
You wrote (regarding the applicability of these analysis techniques to this problem (climate modeling));
“3) the process does not vary with time (stationarity)..”
This leaves me a little concerned. It should be remembered that all of the material properties of the materials acting in the climatic system (thermal conductivity, thermal capacity, thermal diffusivity, density, refractive index, etc.) all vary with temperature (and pressure for some of them). Since temperature varies with time (the Sun still rises and sets) all of these parameters vary with time.
So I suspect that stationarity is not to be assumed for this analytical problem.
While these analytical techniques have some applicability, the results must be viewed very carefully, especially when extrapolating into future decades.
I maintain that modeling the climate of this complex system with a view towards making forward looking “projections” is actually an intractable problem. There are far too many unknowns and of course the errors bars must of necessity widen after each subsequent time interval (ie; if the projection for tomorrow’s energy content is +/- 1%, the projection for the day after must necessarily be 101% + 1% or 99% – 1%, or +/- 2.01%) thus weather forecasts are generally only good for a few days.
In aerospace engineering we have identified those problems that are “intractable” from an analytic perspective. For example, when a satellite is launched that pesky rocket shakes holy H—L out of it. It will likely never again vibrate that hard on orbit, but customers prefer that the satellite survive the necessary launch sequence. While it is conceivable that an analytic approach could possibly predict that all the bolts will stay in place, from a pragmatic view nobody would believe such a complex model (incorporating friction, stiction, bending, surface properties, proper assembly techniques, etc.). So instead, a qualification model (qual model, or QM) is built. This represents the final design down to the last detail (same structure, same bolt torques, same assembly sequence, etc.). And then we shake that model even harder than the final Flight Model (FM). If it survives, (90%+) we build an exact copy for flight. If not, we figure out why and then rinse and repeat.
I maintain that climate modeling is an “intractable” analytic problem and we should never expect the “predictions” to be worth much, if anything at all. Funny that almost two decades of empirical evidence (what the climate is really doing, ie; NOT MUCH) seem to align with my belief.
Cheers, Kevin (MSEE 1980, with DSP experience)
KevinK
You need to fix the incorrect use of bring/take before you tackle “length of time.” Worse yet. supposedly college educated public school (and even college) English teachers are using and/or allowing the use of the incorrect use of bring/take, thus another generation is learning the incorrect use.
KevinK says:
April 9, 2013 at 5:48 pm
“So I suspect that stationarity is not to be assumed for this analytical problem.”
Statistical stationarity does not mean the process does not vary with time, it means the joint probability distribution does not vary with time. Moreover, there is a relaxed qualification for applying these methods, that of “wide sense stationarity”, which means that the correlations between the state variables do not vary with time. And, even when a process is not strictly stationary, it is often stationary in its increments (e.g., Brownian motion). These concepts are widely applicable to natural phenomena.
usurbrain;
I searched my posting for the following terms; “bring”, “take” and “length of time”. Sorry, but those terms do not appear in my posting. Perhaps you meant to respond to a different poster ?
Thanks, Kevin
Bart;
I maintain that climate modeling is an “intractable” analytic problem and we should never expect the “predictions” to be worth much, if anything at all.
Cheers, Kevin
To relate the Laplace transform, to the Fourier, which may bridge the atmospheric transfer function, with the stocastic data observed…on a mechanistic level, I recommend knowing the RADON transform: http://frontcom.ing.uniroma1.it/~gaetano/texware/Radon.MI.pdf
My brain hurts, Brian.
I’m with you FrankK at 7:29 on that
http://wattsupwiththat.com/2013/04/09/correlation-filtering-systems-and-degrees-of-freedom/#comment-1270240
I’d use “elapsed time”, or just “duration”, ‘time’ being implicit in “duration”. However, “length of time” is in common usage so you will never eradicate it.
If the first 3 dimensions can have a length, then why can’t the fourth, time ? It’s been a long time since I’ve spent such a short time considering the length of time. It’s inescapable.
Bart says: “A cross correlation done between the rate of change of CO2 and temperature will yield a flat spectrum and essentially an impulse for the impulse response.”
Greg “It will or it could do ?”
“It will. The equation
dCO2/dt = k*(T – To)
is equivalent to the statement.”
Saying the same thing twice does not make it any truer. So “will yield a flat spectrum” was a totally speculative remark that you can not back up. Disappointing. I thought you had something interesting to show.
” I actually did such an analysis to see if any other features might pop up. But, except for normal, to-be-expected variation from data with random errors, this is the essential feature. You can see the correlation in the plots with your naked eyes. It is really trivially true. And yet, incredibly, the debate rages on,”
Well if the best you can to is come up with some plot on WTF.org using a crappy 12 month running mean that looks to show some rough similarity I’m not surprised ” the debate rages on”.
Is that the sum total of your “analysis” ? I thought you meant you’d actually done something.
That is about as weak as the weakest “climate science” . With efforts like that on both sides, it is hardly surprising thet ” the debate rages on”.
Bart says: “Statistical stationarity does not mean the process does not vary with time, it means the joint probability distribution does not vary with time.”
It also requires that the mean remains constant over time. There is no point in doing FFT on ramp like 20th c temp or anything dependant on it. The rate of change does not ramp up.
This is just one basic error that Grant Foster aka Tamino makes in his attempt to “school” me. He challenges by graph of rate of change by doing an FFT on the actual ice area which is plunging downwards for a good part of the record. LOL
When I gently asked wether he had a done a test of stationarity like Dicky-Fuller test he avoided ansering and replied that I was “just showing off”, then refused to answer any more on the subject he had chosen to make two full articles out of. Seems like school’s out early this year then Grant?
One thing I learnt at school is that teachers don’t know half of what they claim to teach. They just get away with it because most children have not worked that out yet.
A expression that sums this up nicely: Those who can’t do, teach.
The follow up is ; Those who can’t teach , teach in university.
Dear thisisgettingtiresome , this is getting tiresome. Not to mention OT !
KevinK: ” Since temperature varies with time (the Sun still rises and sets) all of these parameters vary with time. ”
If something does not “vary with time” we would not be trying to work out if FFT !
If you have sufficiently long sample in relation to the period it can be regarded as “stationary”. However, if you have 30 or 35 years of statellite data on a system with circa 60y pseudo periodic change it is not.
Oops, “Dr Foster” needs to go back to school.
I believe control engineers in things like rocketry tended to make use of the Kalman filter for transfer function parameter and state variable estimation.
This climate change controversy suggests the application of Bayes theory, where different camps such as the skeptics and the AGWists could propose differing a-priori assumptions, then given the data, see how the assumptions develop. At least people might understand each others positions better.
Greg
Can I just add, and perhaps what Dr Foster is alluding to this, is that there may be structural drift and while the DFT can deal with this it should be removed as it will drown out the “harmonics”, obviously this will be expressed in the autocorrelation as well.
The other point to note is that the signal itself may not be stationary. That is the frequency of the signal may increase or decrease along the chronology. Hence, DFT approach will not be much use. You’ll need to use something more sophisticated such as a wavelet transform. I have much less experience of these so I don’t know how you might use them for doing the sort of stuff discussed above – perhaps as some type of mask prior to running other routine analysis.
@Bart.
As far as I can see from your differential equation, it is an integrator: i.e.; the CO2 will be proportional to the integral of the temperature excursion. The Laplace Transform of an integrator is 1/s and therefore one would not expect to see a flat cross-correlation function if you are correct.