Adaptable Multivariate Calibration Models For Spectral .

1* .Adaptable MultivariateCalibration Models for Spectral ApplicationsEdward V. ThomasSandia National LaboratoriesAlbuquerque, NM 87185-0829b &EpGEl EG’EEC 7 w!!AbstractMultivariate calibration techniques have been used in a wide variety of spectroscopicsituations.In many of these situations spectral variation can be partitioned intomeaningful classes. For example, suppose that multiple spectra are obtained from eachof a number of different objects wherein the level of the analyte of interest varies withineach object over time. In such situations the total spectral variation observed across allmeasurementshas two distinct general sources of variation: intra-object and inter-object.One might want to develop a global multivariate calibration model that predicts theanalyte of interest accurately both within and across objects, including new objects notinvolved in developing the calibration model. However, this goal might be hard torealize if the inter-object spectral variation is complex and difficult to model. If the intraobject spectral variation is consistent across objects, an effective alternative approachmight be to develop a generic intra-object model that can be adapted to each obj ectseparately.This paper contains recommendationsanalysis in such situations.noninvasive measurementfor experimental protocols and dataThe approach is illustrated with an example involving theof glucose using near-infrared reflectance spectroscopy.Extensions to calibration maintenance and calibration transfer are discussed.OSTI

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.IntroductionMultivariateparticularlyin spectroscopy.on multivariatecontextcalibration has received significant attention in analytical chemistry,calibration.are associatedmanufacturingother contexts.withindustries,Examplesof multivariateapplicationscalibrationin the agriculturalmedical sciences4, the pharmaceuticalin a spectroscopicand foodindustries,industries,and in manySee the review by Lavine6 for more examples.The developmentmeasurementMartens and Naesl provide an excellent general referenceof a multivariate calibration model in spectroscopyinvolves theof n specimens at q different wavelengths comprising the calibration set. Inthe case of a linear multivariatecalibration model, the resultant predictive model can beexpressed in the form,j bo bl.x1 b2. x2 . bxq,q,(1)where is the predicted value of the analyte of interest for a new specimen given thespectral measurements(xi’s) associated with the new specimen and the model parameters(hi’s). Common methods for obtaining the model parameters include partial least-squaresregression (PLS) and principal components regression (PCR)l.The efficacy of thepredictive model depends on the applicability of the model form and how well thecalibration set represents the compositionsmodel will be applied.of the new specimens on which the predictiveThe efficacy of the predictive model also depends on how wellthe calibration set spans instrumental/environmentalconditions expected in the future.This is particularly critical when spectral effects induced by instrumental variation (e.g.,source fluctuations)and the instrument’s local environment (e.g., temperature)nontrivial when compared to effects introduced by varying the compositional2arefactors7.

. ,In many situations, it would be desirable to apply the predictive model to spectraobtained across and within different objects. It is assumed that the objects are dynamic inthe sense that the level of the analyte of interest can vary from spectrum to spectrumwithin an object. In such situations, the nature of the spectral variation across objectsmight make it difficult to develop a single model of the form in Equation 1 that is validacross objects.However, if the intra-object spectral variation is similar across objectsand amenable to a model of the form in Equation 1, there is a viable alternative approach.This approach, which is the topic of this paper, is to develop a generic model (operablewithin an object) that can be adapted to each object separately.recommendationsThis paper containsfor experimental protocols and data analysis in such situations.approach is illustrated with an example involving the noninvasive measurementTheofglucose using near-infrared reflectance spectroscopy where the objects are differentindividuals.Extensions of this approach to calibration maintenance and transfer arediscussed.TheoryConsider q-dimensionalspectral measurementsthat are obtained within andacross a population of objects. Let xv [xtil, Xvz,., xu ] denote thejt spectrumassociated with the ith object. This spectrum can be represented byxo p- ai zq,where P [,U1, P2,., ] is the average spectrum across the population,ai [ail, ai2,., aiq] is the specific effect of the ith object on the spectrum, and&ti [&i,, &i2,., &i ] is the specific effect of thej spectrum of the ith object. It is3(2)

,.assumed that the ZYterm describes random spectral effects that, in a sense, have acommon distributionacross objects. That is, these effects are just as likely to beassociated with one object as well as another. In contrast, the ai effects are specific to anobject. It is also assumed that the level of the associated analyte of interest varies acrossthe measurementsof each object (e.g., the analyte of interest could be changing in time).Furthermore it is assumed that a unit-change in the analyte of interest induces an identicalspectral effect across objects.The usual strategy for developing a global model (one that is applicable acrossobjects) is to use a modeling method such as PLS or PCR in conjunction with anappropriate calibration set containing spectra from multiple objects. The calibration setconsists of Xand Y, where X [X1,X2, . . . X1], Y 1, y2, . . . JYIl,xi [Xii,%2,. . . xinJisthe nix q matrix (i.e. number of spectra by number of pixels) of the calibration spectrafrom the iti of I objects, and i [yiI, i2,,.,yin) is the ni x 1 vector of the associatedanalyte reference values. The dimensions of X and Y are Nxq and Nxl respectively,where N ni . A linear calibration model is obtained by using a method such as PLSisl(or PCR) withXandY. The resulting predictor of the analyte level associated with thej’ spectrum of the i kobject (denoted by yy) can be expressed as(3)The set of coefficientscoefficients.{bk}k.1,. are sometimes referred to as final regressionNote that Equation 3 can be expanded (via Equation 2) as9 j bo bk”(pk aik sgkk l4) (4)

rIn the usual strategy for developing a global model, the inter-object spectral effects andintra-object spectral effects are not considered independently.That is, the model isdeveloped in view of the combined inter- and intra-object effects ( ai and Zv). As longas both the inter- and intra-object effects are amenable to modeling, this strategy can beeffective.On the other hand, if the inter-object spectral effects are for some reasondifficult to model (e.g., the calibration data do not sufficiently span the inter-objecteffects), then this approach may be ineffective.An alternative approach involves modeling the intra-object spectral effectsseparately and then adapting the model to a target object. In many cases, the intra-objecteffects are small relative to the inter-object effects and are distributed similarly acrossobjects. The latter condition is assumed here. In this approach the calibration data aremean-centered by object. That is Xm [Xlm, Xzm,., XIm ] and [ylm, y ,.,y ],Xpwhere [xzy , x;Y/w‘[J’’il-Yi7Yi2yi —. 1n] J dJi x”ZyJ.ni zj l1i, . xinim l [xil‘Yi ””” Yi.,‘zi xi2‘Xi .-- xini‘xi](5)(6)‘Yil?i’liThe mean-centeringoperation on the spectrajnlremoves the average spectrum @) and the object-specificspectral effect ( i) from theoriginal calibration spectra leaving only the specific effect of thej h spectrum of the itkobject (see Equation 1). That is(7)W’ and ’ will heretofore be referred to as generic calibration data.5

.Standard multivariatethe generic calibrationanalyteof interestcalibration techniques like PLS and PCR can be applied todata to develop a calibration model that relates variation of theto intra-objectoperations ensure that both spectraleffects.Note that the mean-centeringand c have a mean of zero. Thus it is unnecessarytoinclude an intercept in these modeling activities, resulting in a predictive model of theform,(8)This model is referred to as a generic model as it is applicable to intra-object spectralvariation from all objects. The model coefficients ({g )bl,,,., J are differentiatedfromthose in Equation 3. Because the generic model is developed with regard to intra-objectspectral variation only, these model coefficients will not generally be the same as thoseassociated with the global model ({b } l,,,., J. Note that the quality of the modeldepends on the level of the intra-object variation of the analyte of interest (i.e. thevariation in c).For maximum benefit, it is important to observe each object in thecalibration set at a relatively wide range of analyte levels.In order to be a viable predictor of the analyte of interest, the generic model mustbe adapted to each new target object. The global model (in Equation 4) does thisadaptation implicitly through the fitted model coefficients that are developed inrecognition of inter-obj ect spectral effects as well as the intra-object spectral effects. InEquation 4, bk “pk is a constant that does not depend on i orj.Furthermore,k la bk “ ik is an object-specificconstant.Thus, Equation 4 can be rewritten ask l6

jv di bk”& k.(9)k lCast in this formulation,it is clear that di bk “a ik provides an adaptation to the it k lobject through the interaction of the regression coefficients ({bk}) and the object-specificspectral effects (at). Hence the adaptation is linear with respect to the object-specificspectral effects.In the case of the generic model, the proposed adaptation takes a similar form.That is, for the P* prediction sample associated with the target object (where the Atarget object is not normally represented in the calibration set), we seek to find anappropriate value of c such that9 tp ct (lo)gk”&tpkk lis a good predictor of yp. Equivalently,we want to find a value offi such that9ip(11)‘ft gk”xtpkk lis a good predictor ofy@ given the spectral measurement, xp . A direct way toaccomplish this is to use one or more representativespectra and associated referencevalues from the tt target object. Let Xt [x ,x z,.,Xtii] and yt ,M2,.representative.Ytit]denote%spectra and associated reference values from the t fitarget object.Adaptation to the t target object is provided by(12)The predictor based onfi given in Equation 12 is constructed so that, over the nt7