Multi Component Analysis (MCA):
Have you already understood the General Beer-Lambert law
in it's full extend?
If not, then please study it first, otherwise you will have only a very few success
by applying MCA to your work !
You find the “General Beer-Lambert law”[1]
on the Spectroscopy-Page.
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The Basics:
The absolute Basis for Multi Component Analysis in almost every Spectroscopy is the “General Beer-Lambert law”!
Let us start with some small discussions about the “General Beer-Lambert law”, it guaranteed you the later success using MCA:
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1. Discussion: For what is the product
 standing?
— You're right, it represents the actual number of molecule of the compound (n) in
the light-beam of your spectrometer! It's very wise, to never forget it!
Now let's make an experiment: First we will use a cuvette with a small path size, so we can use solutions
with very high concentrations without leaving the linear range of the spectrophotometer. As long as we are using
week solutions we find a very nice “Linearity” for our pure component, but what happens, when we use
stronger and stronger solutions of the same component. — Now you are wrong! — Has the
number of molecules changed, but where are there gone?
Or is
 not longer a constant
number? It's all wrong, — you forgot a parameter of the “General Beer-Lambert law”.
It's the parameter, which makes the law so “general”. We are talking about the important parameter
 .
You ask me, what is so important on m as m is equal one as we are talking about a pure component?
You are right and you are wrong at the same moment.
As long as we measured a week solution m was really equal one, but by the real stronger solutions m changed to at least two! —
First of all, what you are normally measuring in the UV/Vis region is the
 system of your compound.
Second, by high concentrations, molecules are not longer free, there form dimer till oligomer clusters.
Third, when they form this type of clusters, what is holding them together?:
— The system! —
 And that means, they must have an other spectrum as the monomers, because —
the systems will now influence each other!
Fourth, can you tell me, why we still see an "analog" response[2],
even when the parameter is not
able to change continuous from one to two?
It is a discrete (countable, "not analog"), maybe we could also say quantised parameter. |
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| There is an old, nice experiment to demonstrate this Effect in real if you only got the allowance to handle a small
amount of Benzene. Dilute it in Hexane, already by quite week solutions, you will “leave” the “linear”
range. What happened? Already in week solutions, Benzene will cluster and two Benzene rings form a "sandwich" and now its clear that both
system must recognize each other. |
| There is an other real, practical example on this site! You may have already read it whiteout thinking about it.
It's an example, where we also explained the solution of the problem in that kind, that it is possible to determine the correct concentration
over a wide range with a standard deviation far better then 0.3% rel. You find the example in our paper about our Experiences in the
NNIR Spectroscopy for quantitative and qualitative Quality Control, and it is called
tert-Butanol and Propanol-1. |
Let's make a 'gedanken experiment':
As you have just seen, that your measurement went wrong, because YOU did not obey the General Beer-Lambert
law with the required respect for other concentrations, let's continue the experiment we have started in our mind.
What's the next step happened after we have build in our cuvette more and more dimer of our basic species? Yes, it will
start to build the next higher, possible oligomer of our basic species, just to minimize the total energy of the system
(Gibb's Energy[3]).
And now my question: Where will you end up, if you repeat and repeat our fictive experiment in our fictive cuvette
(its pathsize has to decrise, to stay all time within the linear range of our spectrometer).
Why is the answer such a surprise for you - and others? |
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2. Discussion: What are you doing, if you have to measure a solution
containing two components? YES, of course, you are using an alternative method like HPLC, GC, or so. And what are
you doing, if you must analyse it with your spectrophotometer? Oh NO, why has it to be, — ok, you write down twice
the “Special Beer-Lambert” and double it for two wavelengths. Then you measure both pure substances at
two wavelengths, hoping you select for each substance the “main” wavelength. Finally you measure the sample at
the same wavelengths and then, O Dear, the whole calculation starts following the philosophy: “Two equations with two
unknowns is solvable!” — Well done! - for the first moment! But what are you doing, if both components are
building for a certain amount a cluster as in our discussion 1? Let's be told, that's more often as you belief!
I understand you, if you definitely drop the pen in this case. You need some tools to solve this and similarly problems. You
will find all necessary (and some more) tools on this site.
Please keep patience — we will discuss it all in time! |
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3. Discussion: Have you realised it? All our discussions
started from the point of your knowledge (Thinking in single wavelength) and incorporate more and more my knowledge (Thinking
in spectra) and so we found at least the causes of different problems and for a few we found already a solution or at
least the way how we could solve it without missing the respect to the “General Beer-Lambert law”
Finally I hope you have realised, that a lot of problems can only be understood and solved if you analyse the problems
from the view (and thinking) inside the space of spectra. |
Now it's time to start with the
“Real MCA”:
Why I call it “Real” is easily told, when I explain you, that
there are quite a number of different MCA algorithms often with some side conditions.
But be told, even if hundreds of so called 'specialists' are telling others,
this simple algorithm you will now learn is the only one that holds and is in the
right consistency with the “General Beer-Lambert law”
After you have learned how the MCA algorithm works and how you are able to adjudge
that the results of the MCA are valid or only lucky numbers, I tell you several rules
you have to obey to have the same success with this spectroscopy as we have. |
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Our starting point is — of course, the “General
Beer-Lambert law”:
Is it looking a little bit different to you? — Not really! It's exactly
the same equation as in the beginning of this page.
But we have a small number of points to discuss! First, as we are measuring
the sample and the standards (Yes, so are usually the references (n) called in the process
of the MCA) in the same cuvette, so we can isolate the parameter d on both side of the
equal sign and drop it , without having made a real change to the equation. Of course,
that's only true for our MCA calculation. |
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Now a very important step is following, - we tried it
already several times before. — We have to go from a single wavelength to full
spectra's! To make it a little easier for you, I try to visualise it in the following
picture:
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(1)[4]
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Are you able to find the “General Beer-Lambert law” in this picture? How large
is m in this picture? You're right, m is four in this picture!
You're asking what the red spectrum, which is labeled “Resid.” (= residual) has to do in this expression of the
“General Beer-Lambert law”? Please keep patience, we will discuss it in a short moment
Let's give a few hints for all of us who are not familiar with Matrixes and Vectors:
(But keep in mind, I can not give you several lessons in Mathematics not at this place and not in this
moment!)
You may have already realised, that this four bars on the left side are symbolising four standard spectra's, each going from
top to the bottom. Each spectrum written in this kind is called a vector, a column vector to be precise. If column vectors
are put together as our four standards spectra's in our pictured equation, then we are talking from a Matrix, with other words
our four standard spectra's form a matrix which is mathematically symbolised with the capital letter A in the equation (1).
The four selected concentrations form also a column vector, but of course a very short one. As you can see in equation (1),
column vector are symbolised by a lower case letter (c) with an arrow on its top. Now as you can see in our pictured equation,
if you multiply a matrix like our standard spectra matrix with a column vector as our selected concentrations you get as result
exactly the same, as when you multiply each standard spectrum with its selected concentration and summarise all multiplied
spectra's together. With other words - you have build a mixture spectrum using the standard spectra's in there selected,
individual concentrations. The resulting mixture spectrum forms of course also a column vector, which is symbolised in our
equation (1) with the small case letter (b) with an arrow on its top.
If you put all what you have now learnt together, you realise, that equation (1) is nothing else then the
“General Beer-Lambert law” for spectra's, only extended by a residual spectrum!
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Now let's come to this “residual spectrum” additionally labeled with a exclamation
mark:
First — When you read the equation (1) from the left to the right, it meens with known
concentrations, you are synthesising a mixture spectrum and that can be done mathematically absolute correct. With other
words our residual spectrum will contain at every wavelength the value 0.0. So we see, adding this residual spectrum is a
valid extension to the “General Beer-Lambert law”
Second — When you read the equation (1)
from the right to the left, it meens with unknown concentrations, you are analysing a measured mixture spectrum
with a set of measured standards, and that cannot be mathematically correct solved anymore.
(You're telling, it can be solved mathematically correct, when we use the same number of wavelengths as standards we have!
Then I tell you, YOU are a good mathematician but a very bad spectroscopist and a very bad analyst, too!
Why? Read ahead and you will find it by yourself!)
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Have you realized it? - You are already in middle of what is called “Real MCA”.
But we have still to solve our problem from before. The most important step we have already done, we have added the residual
spectrum. No that's' not making our solution mathematically correct. It gives us on the contrary an infinite number of
mathematically possible solutions. This situation can also not meet our goal. But you are wrong, it gives us at the same time
the tool to find the mathematically, the spectroscopic, and the analytical BEST solution, all at once! And with
a little statistics we are finally able to validate our estimated solution, too. Is there anything more you can wish from a
analytical method? — YES, Real MCA can give you even more, — more as you ever thought off!
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You may have already realised it, the residual spectrum is the absolute most
important part of our equation (1) for the whole process of the “Real MCA”:
Let's start with a few number of conditions which define us the three optimums we would like to have!
All required conditions are of course related with the residual spectrum: |
Conditions, which must hold for the residual:
(Don't mingle this Conditions for the residual spectrum with the
Side Conditions for MCA, from above!)
- ) All values of the residual spectrum should be 'normal'(Gaussian) distributed. This is quasi equivalent to the
condition, that the residual spectrum must only contain “White Noise” (= must be structureless).
- ) The Mean of residual values must be zero. That's equal to:
- ) The residual spectrum has to be as minimal as possible! As the residual spectrum must have positive and negative parts
(How can the sum be zero, otherwise?), we must
define the minimum a little different:
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This are already enough Conditions to solve our MCA problem. There is one other algorithm who
solves the same equation with a little different kind of optimisation, and that is called 'Maximum Likelihood'.
But all other algorithm, and there are quite a number, are solving the
problem not with enough, or not with the right respect to the “General Beer-Lambert law”. And specially, if
there is somebody who is telling you he has an even better algorithm, please test first, - no, - not how better his
algorithm will be, first test how his algorithm will respect the “General Beer-Lambert law”. |
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Now we have all together to solve our “Real MCA” Problem:
Starting point is our equation (3) above and of course the “General Beer-Lambert law” (1). If we insert
the “General Beer-Lambert law” (1) into equation (3) we result in:
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Now we have to multiply both parentheses in equation (4). That gives us the next equation:
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Using the basic rules for Matrix- and Vector- algebra gives us:
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Now we are ready to find the minimum. To do so we must, as you are knowing, build the first
derivative and set it equal zero. For our problem it is meaning, that we have to build the first partial derivative with
respect to concentration of each component and set the result to zero. Doing so with equation (6) looks as follow:
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If we execute this prescription with the respect of the mathematics for matrixes and vectors we result in:
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| The equation (9) is an equivalent statement to equation (8): |
 |
| And if we solve this equation (9) for our concentration-vector, we will get our final equation of the form: |
(OLS)[5] |
| This equation (10) is solving our
“Real MCA” problem by determining the unknown concentrations with respect to the condition to make the
residual spectrum as small as ever possible. |
| But this solution is by far not satisfying. We need more, - we
need additional tools! Why? Let's discuss equation (10) and you will realise it by yourself at the end. |
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Discussion of equation (10) (OLS)[5]:
As easily can be seen, the equation (10) is a straightforward solution of the problem. That's on one hand very nice and
convenient, but on the other hand it is a big problem, too. Into a straightforward equation you can put, in the figurative
sense, potatoes, elephants, trees, stars, mountains, cities, and what ever you want, and you will get a concentration result out.
| That cannot be satisfying and never be reliable at all!
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| The only way, how we can survive with our “Real MCA”,
is, when we qualify and validate our concentration results in a absolute reliable way: |
For you as a gentle reader, it may be clear, that the only term on equation (1) and equation (10)
we can use to qualify and to validate our concentration results is the residual spectrum
 .[6]
But how can we do that in a reliable way? There are a few vendors, who give you a parameter, with which you should be
able to test your MCA result, but this particular parameter is also straightforward and in this case absolutely meaningless.
This parameter is called “Fit Error” and can be found in approximately every statistic book. So I will
not talk about all its really big disadvantages.
There are other very excellent chemometrics Augurs and their programs, they calculate for you a disguised correlation
coefficient as a quality of estimation(validation). But also this hidden R2 between sample spectrum
und estimated spectrum is an absolute unreliable parameter.
Other vendors give you even more worse parameters to solve this problem we still have.
(Appropos: Do you know the difference in significance of the “Fit Error” and
“SEP” ['apperent' or 'corrected'], and “SEC”, and “RMSEP”,
and “PRESS”, and “RSM”, and even “SE..”, and “....”.
Sorry, do you know other such parameters? - Please let me know. Thank you.) |
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From all we was talking till now, it should be clear, that we need this parameter, as
it is telling us, if it is allowed to use the concentration results from equation (10), or if we did a mistake, or not
observe the “General Beer-Lambert law” in it's full extend. With the possible ownership of this important
parameter the usability of this analytical method is given or lost.
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But be told, we have designed this important parameter in a way,
that is making it of such a high significance, that the parameter has already mutated to a significant quality parameter
for many products. This parameter is called “Relative Fit Error”(RFE).
But the most important questions about the RFE are:
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How relative ?
Relative to what ?
Is it making the parameter so significant as I told you ? |
AND NOTE:
For special situations (not analytical!) we have designed him a twin, called
“Normalised Relative Fit Error” (NRFE), too. |
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We will continue with the RFE, just after Discussion the 2nd, of equation (10). |
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Discussion the 2nd, of equation (10):
Invert Singular Matrixes, Collinearity in Matrix A, and other similar terms:!!!
( A lesson for Dummies like ...., and ...., and Mathematician, and Statisticians, and ....,
and so on!)
As soon as you met a Mathematician or a Statistician with your equation (10), he/she will arias his/her finger and higher
the voice and explains to you: "There are several problems with your equation: have you seen, that you have to invert the
matrix (A'A) ?? - That's maybe very dangerous !!" And you, very, very impressed and shocked too, will keep
away your finger from equation (10) for now and forever. But that's only very bad for you, and never bad for the
purse/wallet of Mathematician or Statistician, - Have you realized?
But let's discuss:
Singular Matrixes and Collinearity are like pretty siamese twins, because one is the cause for the other in our equation:
A Singular Matrix is a square Matrix, like our (A'A), from which you cannot calculate an inverse
- a real big disadvantage!
But that's NOT ANY problem at all for us, for "good modern spectroscopists". Why?
As the Matrix (A'A) is generated from our Matrix of Standardspectra A, the Collinearity must already be
contained in A, if ever possible!
'Mathematical Collinearity', as it is required, is anyway only a product of such "Sand-Box Tools"[7],
because (almost not only for spectra !):
Collinearity exist, if it is possible to modeling mathematically at least one Standardspectrum of our Matrix
A with the remaining Standardspectra of our Matrix A in any combination and concentration !!!!!!!
(Uffff... I hope it's enough exact for Mathematician?)
As formula: Collinearity
 .
[*] |
k = number of Spectra in Matrix A.
(For Mathematician we would also have to obey the higher order Terms! But is that true?)
You may have already realized, that this is only possible in a "sand-box" [7]
or in a not correct programmed Softwarepackage, because the real World looks different to "good modern spectroscopist's".
"Our (only?)" real world is never a "sand-box"[7]
and that's why the formula is a little more decorated in real:
Collinearity
 . |
Now you will realize, that the only case for collinearity is, if you have twice the MATHEMATICAL IDENTICAL spectrum in your
Matrix A of Standardspectra, and that's only possible in a "sand-box" [7]
or a bad programmed Spectrometer tool!
You see, too, that you are not able to generate collinearity, even when you measure twice the same cuvette without
replacing and/or refilling it, and insert both spectra into our Matrix A !!!
How much more security will you have, if you follow :
The First Golden MCA Rule to determine the minimal Number of Wavelengths:
Count your Standards in your Matrix A and increase it by one for the Baseline, and increase it also by one
for the measuring noise. This number multiplied by eight gives you the minimal Number of Wavelengths to do MCA.
As formula:
Minimal Number of Wavelengths = 8 * (# of Standards + 1(Baseline) + 1(Noise)) |
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Maybe you have just recognized from our discussion, that quite 'similar' is true also for the
other "Multi Variate Data analysis" (MVD/MVDA).
Of course, It's not all times so easy to see as with spectra.
If we summaries, we can note, that it is almost never (at least for spectra) possible to interrupt/crash the estimation
calculation with real world data (not constructed in any way). That means, we will ALL time get (concentration-) results, independent what ever we do!
So we have some work to do for validation the algorithm in use and our estimated results, as you have learned in the first
discussion of equation (10):
Validating the used algorithm under the analytical condition with the RFE.
Validating our estimated results with the RFE.
Determining the presence and the resulting influence of "Collinearity" to our estimated results.
Soon we will discuss the last item in more details on another separate page, und you will soon
realize, that it is not necessary to really make a significant difference for your estimation to test for possible "Collinearity".
You will then also learn, How easy you can determine the presence of "Collinearity"
by only one keystroke during calibration, and what possibilities you have to avoid any "Collinearity", if you prefer!
- And finally, I will have a "small" surprise for you, too !! - |
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Relative Fit Error (RFE):
Don’t mingle this definition of “RELATIVE FIT ERROR”
with the one for the “NORMAL” Fit Error, which is reported almost in every mathematics and/or statistic book. This RFE has
an absolutely different significance level than the other.
It is absolutely true:
The “RELATIVE FIT ERROR” in its generalised form is more then only use full
for almost every multivariate Data analysis!
We would have Publications of much higher quality and reliability, in science, if the authors would be able
to use the RFE at the correct moment, and in the correct manner! Peter Forster
Independent if for: MCA, Factorial Analysis, PCA, PCR, PLS, ......, or every similar based Chemometrics —
The “RELATIVE FIT ERROR” IS the measure of YOUR quality! Peter Forster
The RFE played already a very important part in the report on our experiences made in the NNIR wavelength range,
[18], and is well accepted in routine work of our customers. The RFE is also playing a very
important part in our report about the
New System Suitability Test for Diode Array Spectrophotometers.
The RFE is the most important parameter of the Multi Component Analysis, and of course of any other MDA, as it
will tell you, if it is allowed to interpret the concentration results estimated by “Real MCA”
in any other way, as to erase them!!, or if you did not strictly follow a spectrometric rule. The RFE is calculated
from the residual spectrum
.[6]
The residual at each wavelength is weighed to the noise at this wavelength, squared and summed to give the RFE.
The mathematical formula is given in equation 1:
(1)
where n is the number of wavelengths,
 1to
n;
 abs(i) is
the residual at the wavelength i;
noise(i) is the noise at the wavelength i;
the number of components is equal to the number of standards.
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This means, that the value of the RFE of a calculated MCA should be as low as possible.
The best RFE with a value of 1.00 can be interpreted in the way, that the residual spectrum
(the part of the sample spectrum the MCA algorithm was not able to modeling)[6]
is from the same size AND kind as the noise that was present at the moment the sample spectrum was taken. If there
is only an absolute minor mistake (“You may think”), for example in preparation the sample, the value of the RFE
will explode.
Oh! — before I forget, as also the RFE can not completely fulfil all our wishes, we have designed a twin
beside him, called NRFE : "Normalised Relative Fit Error".
But that's coming next, others too!
Be told, too: The RFE may have "only the direction (?!?) in common" with the PRESS you know ???
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References
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[1] "General/Specific Beer-Lambert-Bouguer law":
What's the real differences?
www.p-forster.com/english/themes/spectroscopy/Spectroscopy.htm"
[2] "Analog Response:":
This is especially and absolutely the case for "One/Single Wavelenght Measurements"!!!!!
Back
[3] "Gibb's Energy: What a surprise: Molecules do, as soon as they earn !!!"
Wiki: http://en.wikipedia.org/wiki/Gibbs_Energy
[4] "Matrix Representations and Criteria for Selecting
Analytical Wavelengths for Multicomponent Spectroscopic Analysis",
C. W. Brown, P. F. Lynch, R. J. Obremski & D. S. Lavery, Anal.Chem., 1982, 54, p 1472-1479
[5] "OLS: Ordinary Least Squares"Carl F. Gauss This algorithm, designed
by C. F. Gauss is the best method to solve such and similar problems, it means problems, which can be expressed in
the Matrix form: b = A * c + r. The Vector r is often written
as Vector e for error. If you have to use other "'"more sophisticated"'"
algorithms, you are only declaring, that you have at least something to hide, and if it is only to hide, that
you don't know "anything" about your spectra. Quiet similar is true also for other fields of since.
[6] "Residual Spectrum:
"
It can not be stressed enough, that what follows is also absolute essential to most of all
other Multivariate Data Analysis:
We will calculate/"not estimate!" the residual spectrum in two separate steps to make it more transparent
to everybody:
| 1.) |
First we RECALCULTE our Sample spectrum by inserting our estimated concentrations (equation 10) into the
General Beer Lambert Law (equation 1):
A *
= 
==> A * (A'A)-1*A'
= 
REC
REC
is called the "Recalculated Sample (Spectrum)", a term you should memorise very well,
because you will met it quite often from now! This process is also called "Data Reproduction"! (Why?)
|
| 2.) |
Now we are ready to calculate (NOT Estimate!) our absolute important Residual Spectrum:
= –
REC Now the circle is closed. We are back on equation (3)!
Have you realised, that the residual spectrum is the absolute only term of all our estimation/calculations, who
is able to tell us, if we did something valid or only number crunching in its worst sense !!!
What's finally missing is a real adequate judgement to get all relevant information out
of the residual spectrum!
This continuous lack we have exactly filled with our "Relative Fit Error"
(RFE).
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This two steps, together with the RFE (adequate judgement) was in the history called "Short-Circuit Reproduction".
Please, don't ask me why such important knowledge could get lost, without any better substitute!
"Short-Circuit Reproduction": - what an appropriate headline for this process, as it also implicate, that the
fuse could be blown to protect from misuse/misinterpretation inadequate data/measurements!
Peter Forster
[7] "Sand-Box Tools" are Programs, which contains a more
or less complete collection of functions, methods, algorithm, etc. to manipulate numbers (also known as scalars),
vectors, matrixes, and so on, in any way you like or dislike.
Typically representatives are: .... .... .... .... .... - (NO!).
[18] "Near Near Infrared Spectroscopy for Quantitative
and Qualitative Quality Control”
Tamzin A. Lafford, Yvette Cornélis and Peter Forster*, ANALYST , Vol. 117, 1543ff 1992.
For References: see also under References on the page
“System Suitability Test (SST)”
[*] "You may read it as": Collinearity exists (=
TRUE), IF there exist any (more precise: at least one!) i (The Letter i, mostly: index
i) in the closed Interval (1:k), which evaluates the following Condition (==)
to TRUE!
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(To give you a helping hand is also a risen, why I opened this Site!)
Epilog to Mathematics
of "real" Multi Component Analysis:
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As you may have learned now, that “real”
MCA is nothing else as applying the “General Beer-Lambert law” to spectra!
But this means, that only optimisation algorithms are allowed, which also respect the
“General Beer-Lambert law” in its full extent, and that are in the
moment only this explained 'Gaussian Least Square' (OLS) and in certain cases the 'Maximum Likelihood'
algorithms!
Now it should be clear, that also for Multi Component Analysis the user has to complies
with all the spectroscopic requirements “General Beer-Lambert law”
sets!
That specially means, that m = complete must hold, and that's by almost
every other algorithm and procedure neglected and not even proved nor tested.
We will discuss several bad mathematical and spectroscopic practices during the explanation of designing and execution
of MCA.
The condition number 1 is not really used in the equations (2) till (10). But it
is used as a test criteria inside of the RFE. As soon as a structure appears in the residual
spectrum, larger then the measuring noise, the RFE starts to explode!
As the “real” MCA algorithm also holds for m = one, so the
whole spectroscopy for single wavelength analytic can be done with MCA.
This has the absolutely important advantage for the routine,
to have automatically incorporated a Sample-Method-Validation (SMV)
and a Sample-Reference-Identification (SRI). This benefits
costs only a very small effort during the method development, but it costs absolutely
nothing during the later routine analysis.
SMV and SRI are two terms a good modern spectroscopist should put for all times
alongside of the “General Beer-Lambert law” and includes all three
“in his daily prayer”!
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| Conclusion: |
Combined with the RFE, “real” MCA is
certainly one of the most powerful tools a
spectroscopist can use today. Peter Forster
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