Chemometrics
Calibration allows the user to relate instrumental measurements to the sample
of interest. Multivariate calibration allows for the analysis of several
measurements from several samples or specimens. This compares to univariate
calibration, which involves the use of a single instrumental measurement to
determine a single analyte. Either method may contribute to the two-step
procedure where 1) data is calibrated and 2) predictions based on the
calibration are made.
In calibration, indirect measurements are made from samples where the amount
of the analyte has been pre-determined, usually by an independent assay or
technique. These measurements, along with the pre-determined analyte levels,
comprise a group known as the calibration set. This set is used to develop a
model that relates the amount of sample to the measurements by the instrument.
In some cases, the construction of the model is simple due to a certain
relationship, such as Beer's Law in the application
of UV spectroscopy. Unlike spectroscopy, other cases can be much more complex,
and it is in these cases where construction of the model is the time-consuming
step. Once the model is constructed, it can predict analyte levels based on
measurements of new samples.
Another advantage of multivariate calibration (other than abiltiy to include
multiple measurements and samples) is that it can be used to separate samples
from interferences without the need of highly selective measurements for the
analyte (Thomas, 1994)
. In the case of HPLC, certain overlapping or anomalous peaks can be
systematically separated or deleted from the data set based on certain linear
combinations of measurements derived from one of several multivariate
calibration techniques.
As alluded to earlier, the multivariate calibration set contains multiple
measurements from multiple sources of samples and pre-determined analyte
amounts. Next would be the prediction step for new sample levels, and this
uses a model that provides the basis for the evaluation of a linear
combination of the measurements (Equation1).
Calibration techniques (used in the calibration step) differ in determining
coefficient values for the preceding equation (or a similar equations).
Three of these methods to be discussed are multiple linear regression,
principle components regression, and partial least squares.
Multiple linear regression (MLR): As stated earlier,
models constructed from spectroscopy are relatively simple due to linear
combinations of the instrumental measurements which makes the model a
correlation-based model. Models for a broader range of conditions (i.e.,
measurements from several wavelengths) have been constructed in order to
separate overlapping peaks elicited from the analyte plus other unknown
components or conditions. These methods for separating "outliers" are
based upon the following equation:
xi = b0 + b1 * yi1 +
b2 * yi2 + ... + bq * yiq
+ ei
where xi = analyte level of the i th specimen, yij
= j th instrumental measurement with the i th specimen, b =
model parameters, and ei = error associated with yi.
Next, using Equation1 above, the analyte levels
of new specimens can be predicted when estimated bj is
substituted for aj.
MLR does not require knowledge of the amount of interference or other samples
in the calibration set. But the maximum amount of measurements (q) used in
this method is restricted to
approximately 2 to 10 in most cases; therefore, selecting an appropriate set
of instrumental measurements is paramount (Thomas, 1994). For example, in critical
care environments, use of Vis-Near-IR spectroscopy noninvasively monitors
oxygen saturation in arterial blood (Mendelson, 1983). One wavelength in the Vis spectrum monitors
blood pulsatile volume and oxygen saturation; the other in the Near-IR
measures pulsatile blood volume only. Since the interference in this case is
the blood volume, the information from the two wavelengths can be used to
filter out the interference and provide a measurement of the oxygen content.
Principle Components Regression (PCR) and Partial Least Squares (PLS):
These methods belong to a class of statistical methods known as continuum
regression. Models for contiuum regression techniques are described by:
xi = b0 + b1 * ti1 +
b2 * ti2 + ... + bh * tiq
+ ei
where tik is the k th score affiliated with the i th
specimen. Every score has a linear combination of the instrumental
measurements, such that
where gammakj are coefficients. The calibration set data
determines these, along with the estimated model parameters (b) and the
metaparameter, h, which is the model size and is usually much smaller than
number of measurements, q
(Thomas, 1994). The prediction step follows where estimated x and
estimated b are correlated in the following equation:
x = b0 + b1 * t1 +
b2 * t2 + ... + bh * th
where
The prediction step equation is similar to Equation1, where estimated
x is simply a linear combination of the measurements affiliated with the new
specimen.
The difference between PCR and PLS is the coefficients and how they are
obtained: in PCR, only the calibration set measurements are used to acquire
the coefficients, while in PLS, the measurements and the analyte values are
used (Thomas, 1994)
. For information on how to calculate and obtain these coefficients,
detailed references are provided (Martens, 1989; Hoskuldsson, 1988).
Briefly, in is important to mention that before methods such as PCR or PLS
are utilized, data pretreatment is usually necessary. The most prevalent
and easiest type of transformation is centering the data. This method is
executed for both the analyte values and the instrumental measurements. It
tends to make subsequent calculations less prone to round-off and overflow
problems, and it can reduce the size of the model by one factor (Thomas, 1994). Another
data manipulation method is where the instrumental measurements are
differentially weighted. Each centered measurement's importance is
determined by a certain measurement-dependent weight. The centered and/or
weighted data is then used to construct the calibration model. More
information on these techniques is available in the "Thomas, 1994" article
below.
Thomas, E.V.
Analytical Chemistry, 1994, Vol. 66, pp. 797.
Thomas, p. 799.
Mendelson, Y. Ph.D.
Dissertation, Case Western University, Cleveland, OH, 1983.
Thomas, p. 799.
Thomas, p. 800.
Martens, H.; Naes, T.;
Multivariate Calibration; Wiley: Chichester, England, 1989.
Hoskuldsson, A.
J. Chemom., 1988, Vol. 2, pp. 211-228.
Thomas, p. 800.
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