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Klaas Faber (faber)
Senior Member
Username: faber

Post Number: 33
Registered: 9-2003
Posted on Thursday, March 13, 2008 - 4:14 am:   

Compression using wavelets can improve prediction errors. This has been proved theoretically in:

Boaz Nadler and Ronald R.Coifman, The prediction error in CLS and PLS: the importance of feature selection prior to multivariate calibration, J. Chemometrics 2005; 19: 107�118

N.B. These authors received the 2005 Kowalski prize for best theoretical paper for a related work.

Kind regards,

Klaas Faber
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Klaas Faber (faber)
Senior Member
Username: faber

Post Number: 32
Registered: 9-2003
Posted on Thursday, March 13, 2008 - 4:05 am:   

If the data is multivariate, then figures of merit such as SNR should be defined in a multivariate way also. For a good discussion of the rationale behind multivariate figures of merit, see:

K.S. Booksh and B.R. Kowalski
Theory of analytical chemistry
Analytical Chemistry, 66 (1994) 782A-791A

More often than not, a working model can be build for some properties, but not for all. That's something that can (often) be understood by looking at the multivariate SNR, which is calculated for each individual property. The current discussion focusses on the (unfortunately still) common multiple univariate SNRs, which characterize the instrument (rather than the model that produces the results). However, so-called munivariate numbers need not be informative at all for modeling results because it is the interplay of noise and overlap (among others) that determines whether a working model can be build.

For multiway data, the same principle applies: use a consistent generalization of the univariate concept.

Kind regards,

Klaas Faber
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venkatarman (venkynir)
Senior Member
Username: venkynir

Post Number: 56
Registered: 3-2004
Posted on Thursday, March 13, 2008 - 12:11 am:   

Dear Halamrk;
Thanks for your nice words.
Why we can't apply wavelet in the case of noise ?
It has denoising property .
I did feasibility study for pattern recognition and the results are encouraging .
Oh ! data compression may kill your vital informaiton ?.
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Howard Mark (hlmark)
Senior Member
Username: hlmark

Post Number: 184
Registered: 9-2001
Posted on Wednesday, March 12, 2008 - 8:15 am:   

Venky - I used Ohm's law for an analogy. Of course, no analogy is perfect, but the point is that under ordinary conditions, you can rely on the derived relationships between the engineer's definition of S/N and the effect on spectra, as long as the specified restrictions are adhered to.

In fact, if you read the discussion I referred to, you'll find that it's devided between "low noise" and "high noise" cases. "Low noise" essentially means "low enough that the mathematical approximations we ordinarily rely on (e.g., using only a limited number of terms from the expansion) can be relied on".

As you might expect, the "high noise" case is cosiderably more complicated.

Part of the derivations in the book was also published in Appl. Spect., 56(5), p.633-639 (2002) and was therefore formally reviewed, and passed muster. The rest was informally reviewed by readers of the "Chemometrics in Spectroscopy" column and I received several comments and corrections, which I included in the book, so all of it has essentially been reviewed one way or another.

As for the log(x) issue, consider that x represents the deviation from unity for a transmission (or reflection - I'm not going to repeat this every time I use the term) measurement; unity therefore represents the actual transmission of a sample with no absorption.

You might want to think of it this way:

ln(1+x) = (0+x) + ...
or
Log(1+x) = [conversion factor] * (0+x) + ...

since the absorbance of a sample of unity transmittance is zero.

But if you're still unhappy with the approximations, then I can only recommend you read the full discussion, where they are discussed at length and finally, formulas not using the approximations, are derived.

\o/
/_\
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venkatarman (venkynir)
Senior Member
Username: venkynir

Post Number: 55
Registered: 3-2004
Posted on Tuesday, March 11, 2008 - 11:27 pm:   

Dear Halamrk;
Fine.I read your view.
Are you sure that Ohms 'law applicaiton in hold good this case .
I understood that certian condition ohm's law fails to meet it definition .
More over there is mathamtically Log(1+x) and Log(x) are differ is it.
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Howard Mark (hlmark)
Senior Member
Username: hlmark

Post Number: 183
Registered: 9-2001
Posted on Tuesday, March 11, 2008 - 8:37 am:   

Tony - Now I'm awake again!!

The power series expansion for ln(x) (truncated to one term) is:

ln (1+x) = x + ...

Therefore, as long as x is close to unity (i.e., the noise is small), the noise on ln (x) equals the noise of x itself. So at zero absorbance (which corresponds to a transmittance/reflectance of unity) a good approximation to the noise of an absorbance spectrum is the noise of a transmission or reflection spectrum times the conversion factor to log(x), which is 0.4343.

At different values of transmission or reflection, you can apply the formulas in the reference I gave to "Chemometrics in Spectroscopy".

You can even work back to the "engineering" value of S/N, although it's harder to solve for the S/N given the transmittance noise than vice versa, although you could always do it by applying successive approximations to the formulas given, if nothing else.

But it's far simpler to work with the case where both sample and reference signals are the same, i.e., zero absorbance. In that case, the single-beam noise is the square root of the noise of the transmission spectrum, and you can always calculate what the transmittance noise would be at any other value of sample transmission. The equations to do that are not subject to "statistical variability", they're as exact as, say, Ohm's law, even if the data have error.

The hardest part is getting a good value for the transmittance noise in the first place.

Doing it over only a few wavelengths doesn't give a good statistical estimate. Doing it over the whole spectrum doesn't characterize the performance for any particular portion of the spectrum. And performance will vary considerably, we used to see that routinely at Technicon: the absorbance noise, which mirrored the underlying S/N, was minimum at 1600 nm, corresponding to the maximum lamp output.

Any sample with absorbance bands will show decreased S/N at those bands. Water vapor in the air may or may not be a problem, however, depending on the instrument's resolution, since the water vapor bands are fairly sharp and likely won't show up.

Even my suggested method isn't perfect: at the microabsorbance levels we enjoy, even very tiny amounts of drift can affect the noise measurement, and long measurement times invariably allow some drift to occur. Another effect we would see is the absorption and desorption of water on the ceramic disks: the noise plot would show the characteristic spectrum of liquid water if the test was allowed to run for more than a couple of hours.

\o/
/_\
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Tony Davies (td)
Moderator
Username: td

Post Number: 170
Registered: 1-2001
Posted on Tuesday, March 11, 2008 - 6:10 am:   

Dear Christian,

Howard is right (he usually is!) S/N is tricky.

I think we should come back to it later but at the moment what you are really interested is the N. In most spectrometers one can only get measurements in terms of log(1/R) often called microabsorbance units, �Au, (log(1/R)* 1,000,000). Of course you can convert back to a reflectance or transmittance measurement in the computer but you really need something you can do quickly.

The amazingly (well it was amazing in 1980) low noise of NIR instruments is one of the most important reasons that we are able to use them for quantitative analysis but people have got so use to it that it is not discussed that often. Spectrometer manufacturers even forget to include any software to measure it. But instruments develop faults and we should be able to check that the spectrometer is functioning within spec.

When I saw the question last night I knew that I had only a few hours before Howard would have posted the answer but I could not find the reference I wanted and went to bed! (Now Howard is sleeping)
Fred McClure showed how he measured noise in his spectrometers and, as far as I know, no one has published any disagreement. Fred�s is in the two editions of the Williams and Norris monograph (p123 in the 2nd). He took two spectra of a Halon reference and subtracted them. This produces a �noise� spectrum that varies around zero �Au. He then calculated the RMS of the 1,700 points in the spectrum across the wavelength range of 900 -2,600 nm. I believe that the original NSAS operating system did much the same but it excluded the ends of the spectrum and also the �NOISE� program would automatically run a number of scans and report averaged data at each wavelength. It has been suggested that the regions of water vapour absorption should also be excluded if your laboratory suffers from rapidly varying levels of water vapour. I�m not sure that this is wise. If you suffer from this problem you need to know about it!

Karl Norris wrote in NIR news [3(3), 4-6 (1992)] about his results from an idea given to him by Dave Hopkins. This was to smooth a spectrum with a Savitsky-Golay smoothing function (they don�t just do derivatives!) and then subtract the smoothed spectrum from the original. The difference is the noise spectrum. I�m not sure if we should expect this to give a similar value to the Fred method but it has the great advantage that it can be done on any spectrum (and at any time). Like so many of Karl�s ideas I think this one has been largely ignored.

So to try to answer your questions:
We can define what we mean by �noise� and it is relatively easy to measure.
We do not know what the �signal� is so actually measuring S/N is difficult for the user BUT the user needs only to know about noise relative to his instrument.

As for your references, both Faber and Valderrama appear to agree with Howard and have a value at each wavelength. While Short is measuring noise in the analytical results, which includes many other sources of noise.
I am not convinced that there is any justification for believing that noise is wavelength dependant and you have to collect a lot of data to get the figures. We do see apparently higher noise at extreme wavelengths but we also tend to have high absorbance at high wavelengths. Also life is a bit short for frequent measurements of multiple noise spectra. A single figure from a single spectrum will warn the operator that the spectrometer has a problem.

I would guess that �a blank� sample (in reflectance spectrometry) is the reference standard (in your case the ceramic reference). So you have done what I would recommend you (and all other operators) should do. I am not sure what you signify by �mean variances� but your result of 28 �Au is good and within spec.
I also recommend that we should all start following Karl�s recommendation of using Dave�s method.

I have questioned measuring noise at �zero� absorbance. Manufactures like it because it gives the lowest noise result but we cannot make useful measurements with spectra of zero absorbance so would it not be better to quote noise at a given absorbance?

I think there is quite a bit more to be said about this topic. It might be a good idea to try to reach a consensus that could be published in NIR news.

Best wishes,

Tony
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Howard Mark (hlmark)
Senior Member
Username: hlmark

Post Number: 181
Registered: 9-2001
Posted on Monday, March 10, 2008 - 7:07 pm:   

Christian - S/N is actually a lot more complicated than it first appears. To design engineers, it is literally that: the signal strength for a spectrometer at a given wavelength, divided by the magnitude of the random noise at that wavelength. This makes it simple, because both quantities are measured in the same units.

On the other hand, for starters this S/N is a function of wavelength, because the optical signal varies with wavelength (due to natural variations in the radiation intensity of the source, sensitivity of detectors, etc., with wavelength.)

But most spectroscopists don't have the means to measure S/N that way, and it's not directly pertinent to their concerns even though it determines all other measures of S/N. The spectroscopist is interested in the noise of the measured spectrum, and the relation to the spectral signal. Since most spectra from modern instruments or either dual-beam or ratioed, the definition of the "signal" is tricky, the measured transmittance would be unity and the measured absorbance zero, no matter what the actual underlying optical (or electrical, out of the detector) signal is. So typically, the magnitude of the random variations of the spectrum is compared to that value. This shows up very clearly in (mid-)FTIR, since a "100% line" shows greater variations at the ends of the spectrum where the optical signal is weak and less in the middle of the spectral range, where the signal is typically much stronger.

The mathematics relating the engineer's measurement of S/N to the spectroscopist's interests are fairly straightforward. If you're interested in this math (it's fairly simple, actually), it's laid out in "Chemometrics in Spectroscopy", Elsevier (2007), chapters 41-53.

Actually neasuring it is a little trickier. Typically, FTIR instrument manufacturers specify a value as a relative amount of variation (e.g., 0.5%). That value will often represent the smallest value acheived over the spectral range, and is normally either the range or the standard deviation of the values of the "100% line" at or near the middle of the range. Which one is used? You'll have to ask the manufacturer of your instrument what their specification means.

Similarly, NIR manufacturers specify the noise in "microabsorbance"; there's somewhat more consistency here, in that it's typically the standard deviation of many readings at the same wavelength, but to be sure, you should also ask he manufacturer of your instrument. Here, however, it may be quoted over a range of wavelengths, but is typically measured separately at each wavelength, by automatically rerunning a stable sample many times, one spectrum after another, then calculating the standard deviation of the readings at each wavelength.

Any value purporting to represent the instrument's behavior over the entire spectral range is a bogus value; due to the effects mentioned above, no single value can characterize the instrument's performance over more than a small wavelength region.

In any case, all manufacturers quote the lowest value they can get from their instruments, at the wavelengths where it is naturally the least.

To measure it yourself, you need to reproduce the measurement conditions: have a stable sample, allow the instrument to warm up thoroughly (24 hours as a minimum is not too much) so it doesn't drift, and allow the instrument to measure the sample multiple times, without any external disturbances.

Some guidance on these matters is available from the ASTM, which has committees that have formulated standard methods to perform these sorts of tests.

\o/
/_\
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Christian Mora (cmora)
Intermediate Member
Username: cmora

Post Number: 19
Registered: 2-2007
Posted on Monday, March 10, 2008 - 4:54 pm:   

Dear list members;

As you well know, in order to calculate the S/N, we need an estimate of the "instrumental" error. However, after reviewing some articles that report this figure, it's still not totally clear to me how to get it.

For example, Faber (1999) mentions that "standard deviation of the measurement error in the NIR absorbances was obtained by replication", Valderrama et al (2007) mention that "was obtained by replicate measurements of a blank sample and it was estimated as the square root of the mean variance in each wavelength", while Short et al. (2007) state that "the error was estimated as the mean standard deviation of the predicted concentration of 4 tablets varying in constituent concentrations...".

Does anyone know what exactly a "blank" sample is? Can anyone suggest, in simple words for somebody like me :-), how to calculate this error?

I've collected the ceramic reference of my spectrometer (FOSS model 5000) 30 times, and the square root of the mean variances at each wavelength is 2.788950E-5. Can I use this value as the estimate that I'm looking for?

Thanks in advance for your comments

CM

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