CALIBRATION

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CALIBRATION Prof.Dr.Cevdet Demir cevdet_at_uludag.
edu.tr
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• LINKING TWO SETS OF DATA TOGETHER
• Peak height to concentration
• Spectra to concentrations
• Taste to chemical constituents
• Biological activity to structure
• Biological classification to chromatographic
  peak areas

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NORMALLY WE ARE INTERESTED IN SOME FUNDAMENTAL
PARAMETER e.g. concentration or biological
classification WE TAKE SOME MEASUREMENTS e.g.
spectra or chromatograms WE WANT TO USE THESE
MEASUREMENTS TO GIVE US A PREDICTION OF THE
FUNDAMENTAL PARAMETER
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UNIVARIATE CALIBRATION One measurement e.g. a
peak height MULTIVARIATE CALIBRATION Several
measurements e.g. spectra
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NOTATION x block is measured data e.g. spectra,
chromatograms, GCMS of biological extract,
structural parameters c block is what we are
trying to predict e.g. concentration, species,
acceptability of a product, taste
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c x
c X
C X
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• MULTIVARIATE CALIBRATION IN ANALYTICAL CHEMISTRY
• Single component.
• Example, concentration of chlorophyll a by
  uv/vis spectra.
• Mixture of components, all compounds known.
• Example, mixture of pharmaceuticals, all pure
  compounds known.

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• Mixture of components, only some compounds known.
• Example, coal tar pitch volatiles in industrial
  waste studied by spectroscopy, only some known.
• Statistical parameters.
• Example, protein in wheat by NIR spectroscopy.

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UNIVARIATE CALIBRATION x and c blocks
consist of single measurements. Traditional
analytical chemistry CLASSICAL CALIBRATION x ?
c . s Unknown s s ? c . x where c is the
pseudo-inverse
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TREATMENT OF ERRORS IN CLASSICAL CALIBRATION
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PROBLEMS 1. Modern lab dilution and sample
preparation errors (in c) are probably bigger
than spectroscopic errors (in x). Spectra are
more reproducible. Differs to classical
statistics. 2. Want to predict concentration
from spectra etc. not vice versa. Most classical
textbooks in analytical chemistry and most
spreadsheets incorrectly recommend classical
calibration.
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INVERSE CALIBRATION c ? x . b Unknown b b ?
c . x
c
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x
c
b


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COMPARING FORWARD AND INVERSE CALIBRATION
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INCLUDING THE INTERCEPT first column of x is
1s c ? b0 b1x c ? X . b b ? X . c
c
b
X


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• HOW WELL IS THE MODEL PREDICTED?
• Huge number of approaches
• Root mean square error (divide by degrees of
  freedom number of samples 1 or 2 according to
  parameters in the model).
• Often express as percentage either of the mean
  measurement or the standard deviation of the
  measurements

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• Correlation coefficient of predicted versus true
  has problems if the number of samples is small.
• ANOVA and replicates analysis using lack-of-fit
  error, as discussed in the experimental design
  lectures.
• Leaving samples out and predicting them
  cross-validation and testing will be discussed
  later.

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• PROBLEMS
• Outliers can be a major difficulty. Graphical
  ways of looking for outliers big area.
• Undue influence on least square models.

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• MULTIWAVELENGTH
•  
• Example four compounds, four wavelengths.
• MULTIPLE LINEAR REGRESSION (MLR)
• X C. B 
• Know
• X a series of spectra
• C concentrations

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• WAYS OF PERFORMING THE CALIBRATION
• Producing a series of mixture spectra of known
  concentrations by weighing different amounts and
  adding together
• Taking a series of spectra and calibrating
  against and independent method e.g. HPLC.

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EXAMPLE UV/VIS OF PAHs AT 4 WAVELENGTHS, NO
WAVELENGTH IS UNIQUE
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B X . C
estimated pyrene -3.870 A330 8.609 A335
5.098 A340 1.848 A345
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Can also use classical methods
This can be done by knowledge of the pure
spectra. Different to calibration where a series
of mixtures recorded
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• MULTIPLE LINEAR REGRESSION
• Why use only 4 wavelengths?
• Why not 10 or 100 wavelengths?
• More information not arbitrary choice of
  wavelengths.
• Number of wavelengths can be greater than number
  of compounds.

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• Example
• 25 spectra
• 10 compounds
• 100 wavelengths

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• B X . C
• In this case
• B is a matrix of coefficients, 100 ? 10
• X is a spectral matrix, 25 ? 100
• C is a concentration matrix, 25 ? 10
• Some technical problems using inverse calibration
  in this case, and often it does not work.

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• Better approach
• 1. First predict the spectra S.
• Either they are known from the calibration of the
  pure standards
• Or they can be predicted from the mixture spectra
• S ? C. X
• 2. Then use these predictions in a model (e.g. of
  unknowns)
• C ? X. S

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MLR effectively models a spectrum as a sum of
spectra of the components, e.g. for a 3 component
model Observed spectrum conc A ? spectrum A
conc B ? spectrum B conc C ? spectrum C
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• ENHANCEMENTS
• Selecting only certain variables, not all the
  wavelengths.
• Weighting of variables.

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ERROR ANALYSIS This now becomes more
sophisticated. In addition to errors in the c
block (concentration errors), now also errors in
the x block (reconstruction of
spectra). Discuss later.
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• LIMITATIONS AND PROBLEMS WITH MLR
• Number of experiments and number of wavelengths
  must never be less than number of compounds
• All significant compounds must be known. If
  still unknowns, then these are mixed up with the
  knowns. Problems if no pure standards and no
  reliable reference method. THIS IS THE BIGGEST
  LIMITATION.
• Sometimes extra wavelengths can be bad ones e.g.
  noise or background.
• Assume that concentrations are perfectly known,
  errors in only one variable, using classical
  approach.

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However if information on all the significant
compounds is known then MLR is a simple an
effective method.
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PRINCIPAL COMPONENTS REGRESSION (PCR)
Do not need to know all components in advance,
simply "how many components", and the compounds
of interest. Overcomes a major limitation of MLR
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c ? T . r
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The first step is to perform PCA. Obtain a
scores matrix, retaining A components The value
of A may be a guess of the number of compounds in
the mixture. Then r T. c
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Can extend to more than one concentration C ?
T . R

T
R
C

?

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Example 25 spectra taken at 100 wavelengths We
know about and want to predict 4 compounds We
think there are around 10 compounds in the
mixture, 6 are unknown. T is a matrix of
dimensions 25 ? 10 C is a matrix of dimensions 25
? 4 R is a matrix of dimensions 10 ? 4
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Example of the calculation of the concentration
of pyrene in a set of 25 uv/vis spectra
containing 10 different PAHS. How many PCA
components to use? The prediction gets better the
more the number of components.
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ERRORS x block Simply as in PCA, look at
eigenvalues as more principal components are
calculated
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ERRORS c block Look at errors in calculation
of concentrations often different behaviour
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Predictions for pyrene concentration using 1, 5
and 10 principal components.

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Why not use a large number of PCA
components? Then one can get perfect
prediction? FALLACY the idea is to predict
unknowns, after the knowns have been modelled.
Later PCs often model noise. Choose no of PCs
equal to number of compounds in the mixture?
Methods for determining number of PCs described
later when this is unknown.
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• Advantage over MLR - only partial knowledge
  necessary.
•  
• Disadvantage assumption that all errors in the
  "x" block.
• Practical situation. 
• Modern instruments very reproducible.
• Volumetrics, measuring cylinders, syringes are
  inaccurate.

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PARTIAL LEAST SQUARES (PLS) This technique
assumes that errors in both x and c block are
equally significant.
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What does this mean? X T.P E c T.q f
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THERE IS A COMMON SCORES MATRIX FOR BOTH x AND
c BLOCKS. In PCR we calculate the scores just
for the x block and then use a separate step
for regression. A big difference between PCR and
PLS is that in PCR there is only one scores
matrix whereas for PLS (using 1 column) there are
different scores matrices according for each
compound. The vector q is analogous to loadings.
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• PLS components have some analogies to PC
  components.
• In PCA, each component consists of a
• scores vector
• loadings vector
• eigenvalue.

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• In PLS, each component consists of a
• scores vector
• x loadings vector (p)
• c loadings vector (q) a single number
• magnitude.

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• FOR THE TECHNICALLY MINDED.
• Unlike eigenvalues, the magnitudes of success PLS
  components do not necessarily decrease in size,
  although they do model the overall datasets.
• Unlike loadings for PCA, loadings in PLS are not
  orthogonal.
• In most cases PLS loadings are not normal.
• There are many algorithms for PLS and it can be
  confusing.

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ERROR ANALYSIS similar principles to PCR but
different curves for different compounds. Sometime
s different number of PLS components are used to
model different compounds in one mixture.
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• For a dataset consisting of 25 spectra observed
  at 27 wavelengths, for which 8 PLS components are
  calculated, there will be
• a T matrix of dimensions 25 ? 8,
• a P matrix of dimensions 8 ? 27,
• an E matrix of dimensions 25 ?27,
• a q vector of dimensions 8 ? 1 and
• an f vector of dimensions 25 ? 1.

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PLS2 when more than one c variable
.

P






E

X

T


.

Q




F



T

C
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• X T.P E
• C T.Q F
• Differences to PLS1
• C is now a matrix
• Q is also a matrix
• F is also a matrix
• Single scores for all compounds in the mixture.

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• Theoretically PLS2 should perform better than
  PLS1 but in practice it often performs worse.
• Computationally faster, important 10 years ago.
• Useful for non-linear problems such as QSAR where
  interactions, but not so useful in analytical
  chemistry which is very linear.

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• SUMMARY OF MAIN METHODS
• Univariate calibration
• Classical
• Inverse
• Multiple linear regression
• Principal components regression
• Partial least squares
• PLS1
• PLS2