December 5, 2017, 715004753 Rev. B
Page 63
Finally, note that the length of the vectors is proportional to concentration.
5.2.2
Vectors derived from multiple wavelengths
When absorbance ratios are limited to two wavelengths, the chance that two different spectra
share an absorbance ratio is greater than if comparison is made using absorbance ratios at many
wavelengths. Therefore, the spectral contrast algorithm uses absorbances from multiple
wavelengths to form a vector in an
n
-dimensional vector space, where
n
is the number of
wavelengths from the spectrum.
To compare two spectra, the spectral contrast algorithm forms a vector for each spectrum in an
n
-dimensional space. The two spectral vectors are compared mathematically to compute the
spectral contrast angle.
As with the two-wavelength comparison, a spectral contrast angle of zero in
n
-dimensional space
means that all ratios of absorbances at corresponding wavelengths match. Conversely, if any
comparison of ratios does not match, the corresponding vectors point in different directions.
5.3
Spectral contrast angles
Spectra of identical shape have vectors that point in the same direction. Spectra of varying shapes
have vectors that point in different directions. The angle between the two vectors of any two
spectra, the spectral contrast angle, expresses the difference in direction between the spectral
vectors of two spectra.
A spectral contrast angle can vary from 0
to 90
. A spectral contrast angle approaching 0°
indicates little shape difference between the compared spectra. Matching a spectrum to itself
produces a spectral contrast angle of exactly 0
. The maximum spectral contrast angle, 90
,
indicates that the two spectra do not overlap at any wavelength.
To illustrate the relationship between the spectral contrast angle and spectral shape differences,
consider the pairs of spectra shown in the next three figures.
5.3.1
Spectra with different shapes
In the following figure, the absorbance spectra of two compounds, A and B, are distinctly different.
They therefore produce a large spectral contrast angle (62.3
).
Summary of Contents for 2998
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