Calibration Transfer, Part III: The Mathematical Aspects

Jun 01, 2013

This column is a continuation from our previous two columns on the subject of multivariate calibration transfer (or calibration transfer) for spectroscopy. As we noted in the previous columns, calibration transfer is a series of approaches or techniques used to attempt to apply a single spectral database, and the calibration model developed using that database, to two or more instruments. Those instruments may be of like or different technical design. In this installment, we review the mathematical approaches and issues related to the calibration transfer process.

As we have discussed in this series of column installments (1,2), calibration transfer involves several steps. Basic reviews on the subject briefly described here are found in the literature (3–5). The basic spectra are initially measured on at least one instrument (that is, the parent, primary, or master instrument) and combined with the corresponding reference chemical information (that is, actual or reference values) for the development of calibration models. These models are maintained on the original instrument over time, used to make the initial calibration, and transferred to other instruments (that is, child, secondary, or transfer instruments) to enable analysis using the child instruments with minimal intervention. We note that the issue of calibration transfer disappears if the instruments are precisely alike. If the instruments are the "same," then one sample placed on any of the instruments will predict precisely the "same" result using the same model. Because instruments are not alike, and in fact change over time, the use of calibration transfer mathematics is applied to produce the best attempt at model or data transfer. As mentioned in the first installment of this series (1), there are important issues of attempting to match calibrations using spectroscopy to the reference values. The main principle is that spectroscopy measures the volume fractions of the various components of a mixture. The reference values may be based on one of several physical or chemical properties that are only vaguely related to this measured volume fraction. These include the weight fraction of materials, the volume percent of composition with unequal densities, the physical or chemical residue after some processing or separation technique, the weight fraction of an element found in a larger molecule (such as total nitrogen vs. protein), and other measured or inferred properties. The nonlinearity caused by differences in the volume fraction measured by spectroscopy and the reported reference values must be compensated for by using specific mathematics for nonlinear fitting during calibration modeling. This compensation often involves additional factors when using partial least squares (PLS), or additional wavelengths when using multiple linear wavelength regression (MLR).

Multivariate calibration transfer, or simply calibration transfer, is a set of software algorithms and physical materials (or product standards) measured on multiple instruments, and is used to move calibrations from one instrument to another. All the techniques used to date involve measuring product samples on the parent instrument and child instrument and then applying a variety of algorithmic approaches to complete the transfer procedure. Traditionally this has been accomplished by measuring 10–40 or more product samples for each constituent on both the parent and child instruments, comparing the average near-infrared (NIR) predicted value from the parent instrument predictions to the average predicted value from the child instruments, and then biasing the child instrument to the same average value as the parent instrument. Note this procedure is carried out for each product and constituent combination! After this has been accomplished the user of the child instruments may also compare the average NIR predicted values to their corresponding laboratory reference values for each constituent, and then again adjust each constituent model with a new bias value, resulting in an extremely tedious and unsatisfying procedure. This entire wearisome process is exacerbated if different preprocessing and calibration algorithms are used for each constituent calibration or when one is attempting to transfer calibrations from spectrometers of different optical design.

The Mathematical Approaches to Calibration Transfer

A basic instrument correction must be applied to align the wavelength axis and photometric response for each instrument to make their measurement spectra somewhat alike. This process will create spectra and measurement characteristics that are most similar and repeatable. The correction (or internal calibration) procedure requires photometric and wavelength measurement reference materials that are stable over time and can be relied on to have accurate and repeatable characteristics. It is of paramount importance that the standards do not change appreciably over time. All instruments will change with time because of lamp color temperature drift, mechanical wear, electronic component and detector aging, and variations associated with the instrument operating environment, such as temperature, vibration, dust, and humidity.

In the process of transferring calibrations from a parent to a child instrument, one may take four different fundamental strategies for matching the predicted values across instruments. Each of these strategies varies in complexity and efficacy. One may adjust the calibration model (that is, the regression or b-vector), the instrument as it measures spectra (that is, the x and y axes), the spectra (using various spectral transformations, such as matching x and y axes and apparent lineshapes via smoothing), or the final predicted results (via bias or slope adjustments). All of these methods have been applied individually or in combination in an attempt to match the reported predicted results derived from parent and child instruments. Ideally, one would adjust all spectra to look alike across instruments, such that calibration equations all give the same results irrespective of the specific instrument used. This is the challenge for instrument designers and manufacturers, and a significant challenge it is.