348
wavenumbers w i , and obtained by linear interpolation of the original data points.
In the first step of the algorithm we look for a template spectrum most similar to the
measured one and accept it as a component spectrum. then, we try to fit the linear
combinations of the spectra of the accepted and the next potential components to the
experimentally measured mixture spectrum.
Now, let us assume that we have already accepted some number of component spectra forming a reconstructed spectrum and we want to find out if another template spectrum can be included in it. to this end, we consider a series
of two-dimensional optimization problems by looking for the coefficients of the
linear combination of the spectrum reconstructed in the previous step treated as
a whole, and that of the next potential component. the algorithm proceeds until
the stop condition is fulfilled, which means that the difference in the value of the
correlation function for the best and worst fit divided by the value for the best fit,
or the value of the scaling coefficient for the next best potential component fall
below some threshold levels. All optimization tasks were performed under the assumption of non-negativity of components: c l ≥ 0 and
c l
l L
1 , which implies
that the scaling coefficients cannot be equal to zero simultaneously. otherwise
the denominator in the formula above would be zero. the correlation function is
non-concave and, consequently, local maxima may be present. hence, standard
gradient methods for finding the optimum solution may not be suitable. We chose
a more general stochastic method for finding the global extremum named differential Evolution [74], which is a representative of a wider class of genetic algorithms. the method is simple and competitive to other methods from this class,
though it cannot guarantee the best solution, due to its heuristic nature. Alternatively, we could have considered a linear combination of all 20 template spectra
identified by table 12.1, which means searching for 20 scaling coefficients at a
time. using such approach, the computing time would be long and the danger of
reaching a local maximum would be increased. therefore it is much safer to consider a series of one- and two-dimensional problems.
the method was tested using twenty mixtures, each containing between 1 and 8
amino acids; the qualitative compositions are presented in table 12.4. We used solid
powders and mixed them in approximately equal volumes. Naturally, this does not
mean that all components contribute equally to the mixture spectrum, as different
substances are different Raman scatterers. the results of the analysis are presented
in table 12.5. All mixtures were analysed using the same stop condition in the algorithm, assuming that the difference in the value of the correlation function (12.1) for
the best and the worst matching template spectrum was less than 1 % of the value
for the best match, or, alternatively, the value of the scaling coefficient for the best
matching template in the current step is less than 0.03.
the components identified but not present in the mixture are false positives,
whereas those not identified but present in the mixtures are false negatives (see
table 12.5). the analysis resulted in 4 false positives and 11 false negatives.
T. Roliński et al.
wavenumbers w i , and obtained by linear interpolation of the original data points.
In the first step of the algorithm we look for a template spectrum most similar to the
measured one and accept it as a component spectrum. then, we try to fit the linear
combinations of the spectra of the accepted and the next potential components to the
experimentally measured mixture spectrum.
Now, let us assume that we have already accepted some number of component spectra forming a reconstructed spectrum and we want to find out if another template spectrum can be included in it. to this end, we consider a series
of two-dimensional optimization problems by looking for the coefficients of the
linear combination of the spectrum reconstructed in the previous step treated as
a whole, and that of the next potential component. the algorithm proceeds until
the stop condition is fulfilled, which means that the difference in the value of the
correlation function for the best and worst fit divided by the value for the best fit,
or the value of the scaling coefficient for the next best potential component fall
below some threshold levels. All optimization tasks were performed under the assumption of non-negativity of components: c l ≥ 0 and
c l
l L
1 , which implies
that the scaling coefficients cannot be equal to zero simultaneously. otherwise
the denominator in the formula above would be zero. the correlation function is
non-concave and, consequently, local maxima may be present. hence, standard
gradient methods for finding the optimum solution may not be suitable. We chose
a more general stochastic method for finding the global extremum named differential Evolution [74], which is a representative of a wider class of genetic algorithms. the method is simple and competitive to other methods from this class,
though it cannot guarantee the best solution, due to its heuristic nature. Alternatively, we could have considered a linear combination of all 20 template spectra
identified by table 12.1, which means searching for 20 scaling coefficients at a
time. using such approach, the computing time would be long and the danger of
reaching a local maximum would be increased. therefore it is much safer to consider a series of one- and two-dimensional problems.
the method was tested using twenty mixtures, each containing between 1 and 8
amino acids; the qualitative compositions are presented in table 12.4. We used solid
powders and mixed them in approximately equal volumes. Naturally, this does not
mean that all components contribute equally to the mixture spectrum, as different
substances are different Raman scatterers. the results of the analysis are presented
in table 12.5. All mixtures were analysed using the same stop condition in the algorithm, assuming that the difference in the value of the correlation function (12.1) for
the best and the worst matching template spectrum was less than 1 % of the value
for the best match, or, alternatively, the value of the scaling coefficient for the best
matching template in the current step is less than 0.03.
the components identified but not present in the mixture are false positives,
whereas those not identified but present in the mixtures are false negatives (see
table 12.5). the analysis resulted in 4 false positives and 11 false negatives.
T. Roliński et al.