Correlation, filtering, systems and degrees of freedom

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.

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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.

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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].

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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.

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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.

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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:

clip_image012,

which can be normalised to give the coefficient: clip_image014, 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.

clip_image016

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:

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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.

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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.

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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.

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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.

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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)

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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.

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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:

clip_image032

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.

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Great Greyhounds

A first for me! I understood the entire article! I had the chance to work in a field where using FFT, DFT, and ACF were an everyday occurrence. The biggest point in the article is in the conclusion, where he states: “Speaking from bitter experience, it is very easy to make mistakes in signal processing and it is difficult to recognise them.”
How true! And I’m not impressed with some of the mathematics errors being made by the ‘Climate Clergy’ as of late…

Ian H

Have Climate Audit and WUWT swapped places today? I’m more used to seeing this kind of heavy statistical analysis on Climate Audit. Meanwhile Climate Audit has a cartoon by Josh.

Richard Saumarez:
You have provided a fine article in that it provides a good overview of the subject and it contains much important detail. However, it reads as though it is the text for a tutorial lecture intended to introduce the subject.
The introductory paragraph of your article only states what you intend “the object” of your article to be
I write to respectfully suggest that you amend your first paragraph to convert it into an introductory abstract stating the main points which your article explains (perhaps as bullet points). I make this suggestion because I suspect few people who do not know its contents will read much of your article in its present form: they will be constantly thinking “So what?” before deciding to not bother reading further.
Richard

Jean Baptiste Joseph Fourier (1768 – 1830) was a brilliant Mathematician and Physicist who did not live to see the formulation of the Laws of Thermodynamics. His series expansions are used in calculators to derive the standard Trigonometry values. Fourier is misquoted as the creator of the greenhouse hypothesis. “The Most Misquoted and Most Misunderstood Scientific Papers in the Public Domain” by Timothy Casey, posted at http://geologist-1011.mobi has the explanation of the original Fourier statement as well as the Tyndall mistake of opacity and absorption. The Casey site also has the original translation from French for Fourier and the complete published work by Tyndall on the absorption experiments. The AGW hypothesis suffers from a lack of transparency, compounded by stupidity and coupled with dishonesty. These defects cannot be filtered out.

Useless grad student

When you calculate the magnitude and phase from the real and imaginary components with respect to the frequency does the phase, does the function look like the following?
sum(Mag*cos(frequency*time+phase)) from frequency 0 to n.

Mike McMillan

We have a straightforward CO2 signal since 1958, and spotty CO2 data points prior. Can we come up with correlation numbers, 0 to 1, with the various temperature records since then? Pretending that the records we have from GISS and Hadley are real, of course.

peter azlac

Do you think “Phil” will find this easier than Excel?

In order to use the Fourier transform for time series analysis, should one not first determine whether or not the time series data is stationary, at least in the weak sense
http://en.wikipedia.org/wiki/Stationary_process
so at to determine whether or not FT analysis is applicable?

rbabcock

Thank you Richard for a great explanation in understandable terms. It reminded me why I went into building things and let someone else do the math behind it.

Andor

I agree fully with the (k)= *1 but certainly not with the DFT amplitude! It has to correlate directly and if proportional to the mean frequency high-point. The relevance here is surely practical but to multiply the ACF spectrum. Now…processing the impulse spectrum assumption to zero which is in this case not possible but if, ACF=filtered points to a much wider mathematical spectrum.
Characterize the following: Composition of attenuation noise and the statistical noise, will not give cross-over to auto correlation! How can it be??The characters does not multiply. Even divide the sum of the 2 points directly showing a mean point of amplitude, it won’t sum-up? I would rather go for the Effix-drp method and will slowly trend it to fgg-trp analogy for this instance? Take Z=(*1) over the Fx1.3 -g2.4 where g=gravity and not velocity, it will surely put more pressure on the correlated values of the synoptical chart proportional to force(f-d*1)

A correlation coincidence, or maybe not.
http://www.vukcevic.talktalk.net/Peru-Ice.htm
North and the South hemisphere marching in step, warmth in the north with ice in the south.

Stephen Richards

Richard
Good try. I hated fourier analysis at uni and I hate it now.

Great post, Richard… Brought back Dif Eq nightmares!
The same principles that apply to electrical engineering and seismic signal processing are all too frequently ignored in climate signal processing.

banjo

Wanna hear something funny?
Just watching cnn weather here in the uk.
Rising co2 causes greater turbulence and is a danger to flights.
I quote “Better hang on tight!”
Now stop laughing! get up off the floor, and dry your eyes,you chaps have work to do:)

Paul Westhaver

Fantastic!!! Move over greenie biologists and let the EEs do the modelling. Great article.

DirkH

Joseph A Olson says:
April 9, 2013 at 4:20 am
“Jean Baptiste Joseph Fourier (1768 – 1830) was a brilliant Mathematician and Physicist who did not live to see the formulation of the Laws of Thermodynamics. His series expansions are used in calculators to derive the standard Trigonometry values. Fourier is misquoted as the creator of the greenhouse hypothesis. “The Most Misquoted and Most Misunderstood Scientific Papers in the Public Domain” by Timothy Casey”
That’s just so beautiful! Arrhenius misquotes Fourier! And to this day warmists claim Fourier claimed a greenhouse effect through CO2 – which would have amazed me as during his day the battle between atomists and non-atomists was still raging.
(Even in the 1890’s it still was:
“During the 1890s Boltzmann attempted to formulate a compromise position which would allow both atomists and anti-atomists to do physics without arguing over atoms.”
http://en.wikipedia.org/wiki/Boltzmann
)

William C Rostron

Articles like this are one reason I read WUWT daily. I was trained in electrical engineering, and currently do power plant simulation for a living. When one has fought through trying to tune a high order interdependent thermal-hydraulic model for both accuracy and stability, one can appreciate the practical implication of these mathematical laws.
One of the major problems with sampled data systems is the ability to know that samples are truly random with respect to the underlying data. Periodic sampling will always give rise to the problem of aliasing. Once alisasing occurs, the resulting information is forever corrupted, unless perhaps, and only in a very narrow sense, essential characteristics of the original data is somehow already known. For climate data this is problematic since periodicies exist both daily and seasonally. An obvious example of the aliasing problem is trying to tease out accurate numbers from periodic samples taken only twice a day, as most of the early temperature records were done. Only random sampling prevents aliasing and preserves some characteristics of the underlying data. Only random sampling can ensure that statistical analyses are valid. But how can we know that the sampling is truly random?
It is interesting to me that many essential characterists of periodic data can be teased out of random sampled data sets (tamino link). The question in my mind is: can anyone have much assurance that climate data has been randomized sufficiently, or sampled sufficiently, to have confidence that the underlying data is valid to the claimed accuracies? I cut my teeth in metrology before doing controls engineering, then doing math modeling. I have a hard time believing that temperatures over the world were known to tenths of a degree even a hundred years ago, much less before that. I’ve personnal experience with measurement bias and test equipment accuracy, and the effects on process measurement. It is not a trivial exercise to get temperature uncertainties to less than one degree C in a modern industrial environment, where everything about the problem that can be known is known. How much greater uncertainty exists for proxies that were developed over hundreds or thousands of years where no instruments or calibrations exist at all?
That’s not to say that there isn’t any useful information there. You use what you can use. We have a data archival system here that stores data in a compressed format, that only preserves accuracy at specific psuedo random sample points. The idea is to preserve useful underlying data without using up terrabytes of disc space. One of the things that happens with this is, the archived data is useful for some kinds of analysis but completely useless for other kinds of analysis. The trend of data over weeks, months, and years is useful, because the frequency of the archive data is fast relative to these periods. The data is useless for trying to analyse transient events that we know occured in the past, but have lost the original records. Because the compression criteria destroys the short term timing information, no valid analysis can be determined for periods shorter than, say, one hour. Things that we know have to be correlated due to design, can’t be proven in the archived data. No amount of sophisticated analytical techniques can fix this. The data is corrupted in some essential way, and the corruption can’t be characterized, so transient analysis always fails. This is generally true with any data corruption. I’ve seen this talked about on this blog and elsewhere, but I don’t know that it is sufficiently addressed in many climate studies.
-BillR

BTW….Fourier did not live to see the Periodic Table of Elements, the Electromagnetic Spectrum or Theory of Atoms. For an excellent history on EMR see, http://principia-scientific.org/publications/History-of-Radiation.pdf by Dr Matthias Kleespies. Fourier assumed that all energy within the Earth’s crust was residual heat of origin, which he calculated having the remaining capacity to melt a one meter square, three meter high block of ice per year. One glance at an active volcano would disprove this assumption. To extrapolate, from a single observation of a glass covered box, that this great scientist, with no knowledge of atoms, molecules, radiation or nuclear fission….was the FATHER OF GREENHOUSE GAS HYPOTHESIS….is shameless slander.

FrankK

A highly mathematical and interesting article and I commend the author.
I often do peer review of earth science reports and often pull up authors for using “length of time”.
I know it has now become part of the colloquial language, but why use it in a scientific or engineering report?
Time does not have length. They are two different physical units – Time [T} and Length [L]. Why not use “time period” of “duration of time”
Cheers.

John Blake

As noted by statistical cognoscenti, Global Climate Models (GCMs) universally assume that simplistic linear rather than non-linear/complex relationships govern atmospheric/oceanic circulation patterns, especially when discussing CO2. Fourier’s math/statistical ingenuities aside, this root error invalidates all conventional conclusions whatsoever– regardless of computer power, input data, weighted variables, results are neither right nor wrong but meaningless, sheer bumpf.
Modelers’ sovereign neglect of basic physics –hot air rises, water flows downhill– merely aggravates their systemic failure to assess real-world phenomena by valid technical means. Given most sad-sack politicians’ abusive neglect of critical thought despite consequences affecting tens of million human souls, it’s hard to say who is more culpable– the pseudo-scientists who push this nonsense, or the obtuse jacks-in-office who use AGW Catastrophism as a club to beat down opposition, blithely sabotaging global energy economies in the process.

OldWeirdHarold

I’m waiting with bated breath for the next article which will tie this in to the hockey schtick business.

Jeff L

I think the most important & interesting part of this post is figure 6 & this text :
” 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.”
This statement implies that a convolutional model for climate forcing might be applicable.
That is an absolutely profound statement as it is a completely different way of looking at climate.
The convolutional model for climate is a model I have contemplated investigating (coming at it from a geophysical data perspective & the signal processing we do on seismic data) & I have wondered if anyone in the climate community (or weather forecasting community) has ever attempted. Using the convolutional model defined in the paper :
y(t)=x(t)*h(t)
Temperature could be looked at as the system output y(t). CO2 forcing (or any other forcing of choice) could be viewed as the filter x(t), and underlying it there is some sort of underlying climatic impulse response h(t).
Now this is where it gets interesting in that we have a measurement for y(t) – our temperature time series & x(t) – and our CO2 time series and what we are interested in is h(t) – the climatic impulse response function. Now we need to assume stationarity & linearity (and those assumptions ultimately would need to be tested) but in theory, using the convolutional model, we can calculate h(t) . This is very similar to what we do in seismic inversion, although we are solving for the x(t), the earth filter (we know our time series output, which is a convolution of a known impulse response h(t) (known because we generate it) and the earth filter x(t).
As I haven’t attempted this, I have no idea what h(t) might look like, but it opens up a huge set of questions for investigation. Would h(t) vary in different place around the earth (local time series vs a global average time series)? What would be the physical significance of h(t) to the underlying physics of the atmosphere? Would this fundamentally change how we view the climate? Could climate be represented as the sum of a series of forcings & impulse responses? Could you isolate the effects of various forcings in this manner (treating other forcings as noise while isolating the forcing of interest) ? Would h(t) be the same for different forcings or would it be different? This could be applied not just to temperature as the output function y(t) but could be also applied to any other measurable time series, such as humidity, pressure, heights , thicknesses, etc. What would the h(t) functions of these other output time series look like ? What would be the physical meaning & relationships between these various h(t) functions ? There are tons of different y(t) functions to investigate (think about all the climate datasets out there that are expressed as a function of time – atmospheric temp data, oceanic temp data, humidity data, pressure data, global time series, sub-regional time series, local time series, etc etc) and lots of different potential forcings ( CO2, solar variability (in various forms), ENSO, PDO, AMO, etc etc). In theory, one could look for h(t) functions between every combination of response & forcing. How would these h(t) vary? Would there be systematic spatial variation? What would be the physical significance of these variations with the underlying atmospheric physics?
The most important question is : Could we use these various h(t) functions to develop models to predict future climate & maybe even weather ? IF the answer yes, then climate science might be turned on end by this approach.
This is a short list of questions that could be investigated using a convolutional model for climate (and potentially weather) forecasting. It is daunting , which is why I have never dove into it – it is a full time research job for someone (and I already have a full time job ).
I hope this inspires this post inspires someone with the time, skills & interest to dive into this research.
As far as I know, no one has ever approached climate research from this perspective ( if anyone knows of research taking this approach, please post for all to see & a summary of the results). This is such a fundamentally different way of looking at climate that the results have the chance to be revolutionary & transformational, especially if h(t) can be used in a predictive manner. Of course, as it goes with science, this line of research may completely bomb out just as easily (a failed experiment, if you will) , but no one will know until someone does the work. I wish it could be me, but I just don’t have time.
Again, I hope this inspires someone with the skills & time to dive into this line of research.

Jeff L

A quick follow up comment on my comment :
” Now we need to assume stationarity & linearity (and those assumptions ultimately would need to be tested)”.
We always assume that climate & weather is non-linear but perhaps it just appears chaotic / non-linear to us because there is a series of forcings x(t) & associated impulse responses h(t) that are all interwoven into the total climatic response but in fact each individual x(t)*h(t) system is actually linear & it’s just that h(t) is a complicated function so that the system just appears chaotic to us.
… just more food for thought & additional researched needed associated with the convolutional climatic model.

James at 48

RE: “The same principles that apply to electrical engineering and seismic signal processing are all too frequently ignored in climate signal processing.”
Yes! (As a side note, I knew all those upper division and graduate EE and Calculus courses would come in handy some day!)

David in Texas

Richard,
“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.”
Thank you for this article. I find it most in lighting, but I’m confused. Above is what you “wrote” and below is what I “heard”.
“If it starts on the y axis” > Then I image a point in the diagram on the little circle at x = 0, y = -1 (or minimum), because the angle measurement seem to be starting from down position.
“the projection of the point [on to the y-axis] is a sine wave” > here I image the point will be projected again to x =0 (zero because it is the starting point) and to y = -1 (or minimum because that appears what the projection would look like to me and what appear in the black trace in the figure), but sin(0) is not -1.
“its projection on the x axis is a cosine wave.” > the projection I believe would be x = 0, y = 0 (rotating the x and y axes), but cosine(0) is not 0.
What am I doing wrong?
Thanks again,
David

DesertYote

Very nice review, but it makes me miss my last job. What I am doing now is rather boring. The most complicated math I use now is some basic graph theory 🙁
BTW, I have been getting interested in wavelet analysis.

Greg Goodman

Excellently written article, clear and concise.

Greg Goodman

Jeff L ” Now we need to assume stationarity & linearity (and those assumptions ultimately would need to be tested) but in theory, using the convolutional model, we can calculate h(t) .”
Temperature will not be stationary, however, dT/dt probably will be. That is the first step I usually take when looking at temperature time series.

Thanks, Richard.
Your excellent article transported me back to electronics engineering school.
This is one way to look at climate data and clearly see that we know almost nothing yet.
I hope more knowledge will bring back sanity to the climate debate.

jhborn

I second the comments of richardscourtney and OldWierdHarold.
My reaction upon seeing the post was the same as that of, I suspect, a not insignificant minority: the post seems unlikely to contain anything I don’t already know (well, knew at one time), and, since its introduction fails to identify a specific question before the house to which the post would apply its basic signal processing, reading it does not appear worth my time. So I haven’t read it.
This is not to say that it may not be a valuable post for many here; it’s just to let you know how it plays to one segment of your audience.

Greg Goodman

vukcevic says:
A correlation coincidence, or maybe not.
http://www.vukcevic.talktalk.net/Peru-Ice.htm
North and the South hemisphere marching in step, warmth in the north with ice in the south.
===
How about ANTARCTIC melting period (inverse) correlating with ARCTIC Oscillation index?
http://climategrog.wordpress.com/?attachment_id=208
Ironically it appears correlate somewhat better than the Arctic melting period does. That was quite a surprise.
This is still work in progress, more details once I’ve checked it more thoroughly.
Any suggestions on how to get a numerical measurement of correlation between daily AO and once per year melting periods?

usurbrain

Excellent article. I have always wondered why FFT analysis has not been applied to the GW data. My only guess is that they want the scam and not the truth.
It is amazing what a FFT can do. In 1977 I was a nuclear engineer (with a second degree in applied mathematics) responsible for starting up a nuclear power plant. I had spent weeks trying to find the cause of a problem with the flow of one of the pumps. One of the engineers was using the HP 5420A Digital Signal Analyzer (I believe this was the very first FFT analyzer of its kind) on the vibration transducer in an effort to see if there were bearing problems. My math degree kicked in and a light-bulb went off in my head. I borrowed the HP 5420, and hooked it up to the flow transducer and pressure transducer. The signal was coupled through a capacitor so that only the AC signal and not the DC voltage were going to the HP. Within 15 – 20 minutes I had determined that one of the volutes on the pump was causing minor, but enough, cavitation to create the difference in flow from this pump. I later used this to tune the feed-water control system to such a degree that we were able to get about 2% more power out of the plant. Two percent of 1000 megawatts is 20 megawatts more power! I went on to solve several other problems with the analyzer, The HP allowed me to solve problems in a few hours instead of days or weeks. The HP analyzer became my best friend, I used it more than my new fangled pocket calculator. Many of the new “intelligent” flow, level, and temperature control systems have a bare-bones FFT analyzer in their electronics today. If the AGW crowd really wants to know the truth they will start analyzing the data correctly, not massaging to fill their pocket book.

cd

Richard
Firstly, whoa!
That’s a lot to take in, would it not have been easier just to refer to the Wiener–Khinchin theorem – would it not have cut down your article by half. Perhaps I’ve missed the point (or completely off track) but it did seem like hard going until I got to the best bits.
Secondly, do you not think you are taking a sledgehammer to crack a nut. Surely simple regression analysis will do for most bivariate issues. I agree that the t-stat may be a little to primitive but it could be vetted with other tests such as the Durbin Watson stat and simple cross-correlation. Don’t get me wrong, it seems really good to look at every scenario, but everyone seems data processing nuts in this field.
Thirdly, while I’m not questioning anything you say (you’re obviously the expert), you fleetingly mention phase shift as a result of certain types of filtering. In one of the examples above you show what looks like a Butterworth filter; this, as you allude to, will introduce a phase shift, surely this will affect any subsequent analysis. Perhaps a more elaborate discussion of this with result from a single study would make for another good article.

cd

Joe Born
Well bully for you. If the article is below you then no need to comment unless you want a brownie point for being so knowledgeable.
Yes the article is long winded and probably a referral to some good online articles would’ve saved a lot of screen space, but he at least has put something out there and does attempt, I admit unsuccessfully, to communicate some commonly applied techniques in a less technical fashion. In short, he hasn’t condescended the reader…unlike you I hasten to add.
I think a good follow up would be to show why such approaches are better than just using more common approaches or we even need worry about such things in this (or any) field.
You don’t have to please all the people all the time; that’s why it’s good for websites to host all kinds of articles. Just thought you might want to know what other posters think.

otsar

This is a good introductory article on data processing and should be a must read and comprehend for scientists interpreting data.
Someone that has the time and energy should write an article on the acquisition of the data. The the data from the point of the measurement: Pt RTD, TC, PH probe etc, to the point where the data is stored on disk in a useful format for data processing.
This is one of the areas I work in. Very often I run across situations where the investigator is not purely measuring what they believe they are measuring.
The article should address the problems of the actual measurement process: linear, non-linear sensors, S&H aperture time, sample interval choice, A/D conversion problems, driver problems, operating system problems (not real time vs real time), choice of floating point or binary, etc, etc.
The way I see it, this the point where the starting materials for the sausages are made.

Great post Richard!
Do you mind if I keep a copy for reference; not for publicizing?

Reed Coray

I want to second (third, fourth, ….) Richard’s comment to the effect that access to a good “tool box” of processing algorithms (e.g., the R suite of tools) isn’t sufficient to employ properly used those tools. My example is the man who is an expert in the use of every tool know to man. He is put in him in a room where any tool ever made is available to him and he knows how to properly use every tool. He is asked to build a car. His first question will likely be: “What’s a car?” I, too, have experience with the improper use of signal processing tools–especially digital signal processing tools.
I would like to add that for signal processing that can be implemented to a high degree of fidelity convolving the signal with a finite duration impulse response (e.g., lowpass, bandpass, differentiation, Hilbert Transform, etc.), the Discrete Fourier Transform (DFT) can be used (and often is used because it is computationally efficient) to filter a uniformly sampled time waveform. Although the magnitude of the DFT of a finite duration segment of an input signal provides a measure of the spectral content of that segment of the signal and thus provides a way to visualize the filtering process in the frequency domain, the spectral properties of the DFT are secondary to the filtering process. The DFT (and other transform procedures whose outputs give little if any information about the frequency content of the signal) can be used to efficiently implement finite impulse response convolution, The DFT is one of several digital signal processing algorithms having the property that the circular convolution of two equal length sampled waveforms is, within a scale factor, equal to the inverse DFT of the product of the DFTs of the two waveforms. The fact that the DFT of a finite duration segment of a waveform provides information about the spectral content of the waveform is a side benefit of the process.

JM VanWinkle

William C Rostron, thank you for raising the central point of physical sanity of “world temperature,” measured to a tenth of a degree C. Only someone who has lived their life in books would think that has real world meaning. One might just as well ask how many feathers are there on an average bird 100, 1000, 10,000 years ago, calculated to the nearest feather.

Reed Coray

Ooops. In my immediately foregoing comment, replace employ properly used with properly employ.

RCSaumarez

Thanks for the comments.
This post was in response to an argument about filtering data and then cross correlating them – is it valid?
1) @ Joe Born and others. I agree that if you are familiar with DSP, this is a very simple exposition. My goal was to show that filtering and correlation are related and this may not be realised by some readers. If done incorrectly, correlation will be affected by filtering and this has been the subject of a number of posts and disputes – Eschenbach, Mckintyre, Briggs et al.
2) @Otsar. Thanks, but I would need to write a textbook!
3) @cd. The purpose of the last section was explain, in simple terms, that autocorrelation affects statistical degrees of freedom and so needs to be taken into account when testing hypotheses statistically. As regards filtering, if one is using a physical filter with a phase shift on the input and output, the point of cross-correlation is that it eliminates the phase response of the filter, provided the filters have the same characteristics (and generally one should use Bessel filters that linear phase characteristics). If one uses a digital zero-phase filter for both, you won’t screw up the phase response at least.
4) @Useless grad student. Yes you are basically correct. One has to careful in defining the sign of phase, otherwise the signal comes out backwards! Also, the magnitude in most DFTs needs care because it represents HALF the spectrum. If you remember that cos(wt)=(exp(jwt)+exp(-jwt))/2, the spectrum is represented in both positive and negative frequencies (with sine, imaginary, component negative) you will calculate amplitude/2 if you use a real transform.
5) @David in Texas. A point rotating in unit radius circle will trace out cos(wt) on the x axis and sin(wt) on the y axis, i.e. they will be 90 degrees out of phase. I had some difficulty in drawing figure 3, and the solution to your problem depends on when you consider time to zero. If I start measuring a cosine wave at t=0, the phase will be zero but if I start 1/4 of a period later, the phase will 90 degrees. Phase is related to delay and this depends in a given signal on when you started and finished recording it. The important point in signals is not the absolute phase but the way phase changes with frequency – the phase spectrum.
A more general point is this. The type of analysis I have described is related to LINEAR systems, that obey 1) the output is proportional to the input, 2) the output of the sum of the two different input functions is equal to the sum of the outputs of the responses of the individual functions (superposition) and 3) the process does not vary with time (stationarity)..
In the brutal real world, as opposed to technological signal processing, systems are rarely truly linear or stationary and outputs may depend on the product of the inputs; signal processing in these systems is much more difficult (tiger country!). If you start with a basic proposition of, say, the oceans absorb heat and release it, they may act as a simple mixer and one can treat their behaviour through linear methods. In practice, mixing between levels, currents etc (I’m no expert) may make this invalid. However, for small fluctautions, over short periods they may be sufficiently linear for this not to matter – but this depends on the problem that you are trying to model and why you are doing so .
The importance of this idea is signal signal processing is not a “dark art” but a tool to gain insight into the behaviour of physical systems, especially when one has noise, or other signals mixed in. Therefore one has to start with an idea of what one is trying to establish rather than using a battery of techniques blindly (and incorrectly).

Bart

Jeff L says:
April 9, 2013 at 8:01 am
“Now we need to assume stationarity & linearity (and those assumptions ultimately would need to be tested) but in theory, using the convolutional model, we can calculate h(t) “
Actually, a valid model over the past 55 years is not stationary, but it has stationary increments. The relationship is essentially
dCO2/dt = k*(T – To)
dCO2/dt = rate of change of CO2
k = coupling factor in ppm/degC/unit-of-time
T = global temperature anomaly
To = equilibrium level of temperature
This highly representative model shows that temperature is necessarily driving CO2, and not the reverse. The arrow of causality is necessarily from temperature to CO2, as it would be absurd to argue the converse, that temperatures are being driven by the rate of change of CO2 – once the CO2 reached a plateau, no matter what its level, the temperature would return to the equilibrium value of To.
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.
I know it is hard to believe, but the climate modelers have got even the most basic relationship backwards and wrong.

Bart

RCSaumarez says:
April 9, 2013 at 11:04 am
“Therefore one has to start with an idea of what one is trying to establish rather than using a battery of techniques blindly (and incorrectly).”
So true. The climate community should have started by employing identification techniques, such as PSD estimation, to identify how things were actually working, then fit their models to the observations. Instead, they have attempted to work up models, and then fit the data to them. Doyle’s famous detective warned against such methodology over a century ago:
“It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.” – Sherlock Holmes

Bart

Bart says:
April 9, 2013 at 11:10 am
On
dCO2/dt = k*(T – To)
This type of relationship is to be expected in a continuous transport system where CO2 is constantly entering and exiting the surface system, and the differential rate at which it does so is modulated by surface temperature.
It’s a done deal. I’m just waiting for others to recognize it.

Greg Goodman says:
April 9, 2013 at 9:15 am
…..
Hi Greg
It makes sense (the CET temperatures correlate with AO in winter months, the Antarctic’s summer). For correlation I use Excel, but it requires same ‘scale resolution’ for both variables. As a test I suggest use the average of monthly AO’s data ( for the ice melting months) tabulated as annual data. Hope someone else may know of a better method.

commieBob

There is a serious caveat when it comes to applying this stuff. It works only for linear time-invariant systems. If we understand a system well enough, we can work it so our analysis is LTI (or close enough). In the case of the climate, that probably isn’t the case.

Greg Goodman

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.”
It will or it could do ?
Call be skeptical, but I’d like to see at least a simple demonstration that is the case. I have no reason to think you are wrong but that is not enough for me to accept such a bold statement without question.

Greg Goodman

Bob: ” In the case of the climate, that probably isn’t the case.”
Aw, com’on Bob. Climate is CO2 plus stochastic variation. I know cos Tamino told me so.

Bart

commieBob says:
April 9, 2013 at 12:31 pm
“In the case of the climate, that probably isn’t the case.”
Actually, it probably is the case. The inapplicability of these methods is more the exception than the rule. And, as I demonstrate above, this system is clearly amenable to such an approach.

Die Zauberflotist

OK, you lost me after “Correlations”, but I love the way you fill out an x-y axis, so I trust you. What’s your opinion of the 5 second rule? And, is it safe to use a meat-tenderizing hammer on a ganglion cyst?