ESTIMATION OF PREDICTION ERROR FOR SAMPLES WITHIN THE CALIBRATION RANGE

A modification of a technique proposed by Lorber and Kowalski for the estimation of prediction errorsis presented.The method is applied to five data sets.The results show that for some data sets theestimated prediction errors are close to the actual prediction errors for samples within the calibrationrange,while samples outside the calibration range must be background corrected before quantificationof the prediction error.     

THE EFFECT OF MEAN CENTERING ON PREDICTION IN MULTIVARIATE CALIBRATION

Traditionally,one form of preprocessing in multivariate calibration methods such as principal componentregression and partial least squares is mean centering the independent variables(responses)and thedependent variables(concentrations).However,upon examination of the statistical issue of errorpropagation in multivariate calibration,it was found that mean centering is not advised for some datastructures.In this paper it is shown that for response data which(i)vary linearly with concentration,(ii)have no baseline(when there is a component with a non-zero response that does not change inconcentration)and(iii)have no closure in the concentrations(for each sample the concentrations of allcomponents add to a constant,e.g.100%)it is better not to mean center the calibration data.That is,the prediction errors as evaluated by a root mean square error statistic will be smaller for a model madewith the raw data than a model made with mean-centered data.With simulated data relativeimprovements ranging from 1% to 13% were observed depending on the amount of error in thecalibration concentrations and responses.     

MULTIVARIATE CALIBRATION WHEN DATA ARE SPLIT INTO SUBSETS

A calibration situation is considered where the calibration data are split into subsets with good linearrelationship between y and x within each group.Different strategies for good prediction in this case areproposed.Modifications for collinear data are considered and a simple simulated data set is used forillustration.     

GENERALIZED SIMULATED ANNEALING FOR CALIBRATION SAMPLE SELECTION FROM AN EXISTING SET AND ORTHOGONALIZATION OF UNDESIGNED EXPERIMENTS

ealibrared.The diseussion in this PaPer foeuses on near一infrared(NIR)sPeetroseoPy as the examPle instrument.However,the Proeedures Presented are aPPlieable tomost methods of instrumental analysis.Essentially,ealibration eonsists of assembling a seriesof samPles eontaining the analyte or analytes at     

THREE-WAY METHODS FOR THE CALIBRATION OF CHROMATOGRAPHIC SYSTEMS:COMPARING PARAFAC AND THREE-WAY PLS

For the calibration of chromatographic systems,different methods can be used.One class of methodsutilizes three-way approaches.The calibration problem is stated in such a way that the decompositionof a three-way array can serve for the prediction of retention on new stationary phases.Two three-way approaches are presented:the Unfold-PCA and PARAFAC models.The theory ofboth methods is presented and the differences are highlighted,the main difference being that PARAFACis a trilinear decomposition whereas Unfold-PCA is not.Both three-way methods are evaluated on asmall data set consisting of retention measurements of eight solutes at six mobile phase compositions onsix stationary phases.The differences in performance of the two models are minor,For calibration purposes,two variants of the methods are discussed:three-way PLS and an extensionof PARAFAC.Again the theory and differences between the two methods are explained.The predictiveperformance of the two methods is compared using the same data set as earlier.The differences inpredictive performance,however,are minor.Both methods are capable of predicting 98% of thevariation in the test sets.Yet,there are other considerations when comparing methods than predictiveperformance,e.g.the quality of the predictions.     

DRIED GRASS SILAGE ANALYSIS BY NIR REFLECTANCE SPECTROSCOPY-A COMPARISON OF STEPWISE MULTIPLE LINEAR AND PRINCIPAL COMPONENT TECHNIQUES FOR CALIBRATION DEVELOPMENT ON RAW AND TRANSFORMED SPECTRAL DATA

Calibrations to predict crude protein (CP) and in vitro dry matter digestibility (IVDMD) in dried grasssilage from reflectance data collected at 19 wavelengths on an InfraAlyzer 400R have been developedusing stepwise multiple linear (SML) and principal component (PC) regression techniques. A directcomparison of the efficacy of each multivariate technique in this application has been possible by usingidentical calibration development and evaluation sample sets. The effect of two data transformation stepsprior to PC regression was also investigated. PC regression of raw reflectance data yielded no significantimprovement in the standard errors of prediction (SEP) for CP and IVDMD over those obtained bySMLR, viz. 0.61 vs 0.63 and 2.9 vs 3.0 respectively. Computation time for development and evaluation ofthe PC regression equation was less than for selection of the best SMLR equation, and PCR equationsmay be more robust. Data transformation to reduce granularity effects prior to PCR did not produce anyimprovement in predictive accuracy for either IVDMD or CP.     

RECENT DEVELOPMENTS IN MULTIVARIATE CALIBRATION

With the goal of understanding global chemical processes,environmental chemists have some of the mostcomplex sample analysis problems.Multivariate calibration is a tool that can be applied successfully inmany situations where traditional univariate analyses cannot.The purpose of this paper is to reviewmultivariate calibration,with an emphasis being placed on the developments in recent years.The inverseand classical models are discussed briefly,with the main emphasis on the biased calibration methods.Principal component regression(PCR)and partial least squares(PLS)are discussed,along with methodsfor quantitative and qualitative validation of the calibration models.Non-linear PCR,non-linear PLSand locally weighted regression are presented as calibration methods for non-linear data.Finally,calibration techniques using a matrix of data per sample(second-order calibration)are discussed briefly.     

MIXTURE ANGLYSIS USING FACTOR ANGLYSIS I: CALIBRATION AND QUANTITATION

One of the major application areas of factor analysis, multivariate calibration and quantitation, is coveredin this review. The algorithms, methodologies and applications covered include principal componentregression, target transformation factor analysis, singular value decomposition and rank annihilationfactor analysis. Many important areas of research having relevance to multivariate calibration andquantitation problems are also covered in this review, including background correction, measurementerror, rank determination, cross-validation, figures of merit, detection of invalid samples, experimentaldesign, sample selection, statistical inference and wavelength selection.     

ADVANCES IN SECOND-ORDER CALIBRATION

Several multivariate methods are now available for the calibration of second-order or hyphenatedinstruments(e.g.GC/MS).When applied to bilinear data,it has been shown that calibration can beperformed in the presence of unknown interferences-a significant advantage over first-order calibration.In this paper,non-bilinear rank annihilation(NBRA),a method which has the potential of handling,second-order non-bi-linear data,is studied through theoretical analysis and computer simulation.It isfound that the second-order advantage can be carried over to non-bilinear data if a property defined asnet analyte rank(NAR)holds for the analyte of interest.The net analyte signal(NAS)is definedaccordingly for second-order calibration and the analogy to and difference from lower-order calibrationare discussed.With NAS,some analytical figures of merit such as signal-to noise ratio,selectivity,sensitivity and limit of determination can be calculated for second order calibration.An application toMS/MS data is also given.     

SPLITTING OF CALIBRATION DATA BY CLUSTER ANALYSIS

from eorresponding valuesof x.The most eommon aPProaeh to this Problem 15 linear regression(or ealibration),but1 inear methods are usually best suited for quite limited regions alld are not generallyaPPlieable.If a linear fit 15 not satisfactory,alternative aPProaehes are non一linear regression,non一Parametrie regression,transformations and sPlitting of the data into subgrouPs     

THE PERFORMANCE OF LEAST SQUARES AND ROBUST REGRESSION IN THE CALIBRATION OF ANALYTICAL METHODS UNDER NON-NORMAL NOISE DISTRIBUTIONS

By means of Monte Carlo simulations a comparison has been made between ordinary least squaresregression and robust regression. The robust regression procedure is based on the Huber estimate and iscomputed by means of the iteratively reweighted least squares algorithm. The performance of bothprocedures has been evaluated for estimation of the parameters of a calibration function and fordetermination of the concentration of unknown samples. The influence of the distributionalcharacteristics skewness and kurtosis has been studied, and the number of measurements used forconstructing the calibration curve has also been taken into account, Under certain conditions robustregression offers an advantage over least squares regression.     

CONDITION NUMBERS, ITERATIVE REFINEMENT AND ERROR BOUNDS

Concentration estimates of components present in a sample mixture can be obtained using matrixmathematics. In the past, the condition number of the calibration matrix has been used to give anamplification factor by which uncertainties in data can work through to errors in the concentrationestimates. This paper explores an additional interpretation of condition numbers with regards tosignificant figures and rounding errors. A procedure is suggested which will always give the most accurateconcentration estimates provided the calibration matrix is not too ill-conditioned. Condition numbershave also been used by analytical chemists to discuss the error bounds for concentration estimates.Unfortunately, only one representative error bound can be approximated for all the components. Thispaper will show how to compute bounds for individual concentration estimates obtained as solutions to asystem of m equations and n unknowns. The procedure is appropriate when calibration data and sampleresponses are inaccurate.     

COMPUTER-GENERATED MULTICOMPONENT CALIBRATION DESIGNS FOR OPTIMAL ANALYSIS SAMPLE PREDICTIONS

This paper utilizes variable step size generalized simulated annealing(VSGSA)to design multicomponentcalibration samples for spectroscopic data.VSGSA is an optimization procedure which is capable ofconverging to exact positions of global optima located on multidimensional continuous functions.On thebasis of analysis sample response vectors,optimally designed calibration concentration matrices areobtained assuming knowledge of components present.The complexity of response surfaces establishedby the optimization criteria is described.     

SHORT COMMUNICATION MULTIVARIATE CALIBRATION IN UV-VIS SPECTROSCOPY

A multivariate calibration procedure based on principal component analysis is proposed.UV-vis spectraof ternary mixtures have been used to check the applicability of the procedure.     

SHORT COMMUNICATION A PROGRAM FOR THE SIMULTANEOUS POTENTIOMETRIC DETERMINATION OF PROTONATION CONSTANTS AND ELECTRODE CALIBRATION PARAMETERS

A program for the potentiometric determination of the protonation constants of mononuclear polyproticsubstances is described.A maximum of twelve parameters can be determined simultaneously,includingup to six protonation constants,four electrode calibration parameters,the protolysis constant of thesolvent and the titrant concentration.Optimization is carried out by using the non-simplifiedNewton-Raphson method,which is potentiated by the Marquardt algorithm and a distance speeding-upcoefficient.A direct search method is also used to improve the initial set of values.Variances arecalculated very accurately,since the real Hessian function is used.Statistical weights and ionic strengthcorrections are also considered,The program has been tested by using simulated titration curves ofpolyprotic acids with close constants.     

THE RANK-SIZE MODEL: A NON-LOGARITHMIC CALIBRATION*

The rank-size model of city-size distributions is ordinarily calibrated by means of a logarithmic transformation. Because this procedure gives undue weight to the smallest cities, the regression results may be unrepresentative of the overall structure and regularity of the rank-size distribution. In order to avoid this problem a non-logarithmic calibration is proposed. Empirical tests indicate that the performance of the nonlog model is superior to that of the log model.     

Diatom-based water chemistry reconstructions from northern Sweden: a comparison of reconstruction techniques

The diatom composition in surface sediments from 119 northern Swedish lakes was studied to examine the relationship with lake-water pH, alkalinity, and colour. Diatom-based predictive models, using weighted-averaging (WA) regression and calibration, partial least squares (PLS) regression and calibration, and weighted-averaging partial least squares (WA-PLS) regression and calibration, were developed for inferences of water chemistry conditions. The non-linear response between the diatom assemblages and pH and alkalinity was best modelled by weighted-averaging methods. The lowest prediction error for pH was obtained using weighted averaging, with or without tolerance downweighting. For alkalinity there was still some information in the residual structure after extracting the first weighted-averaging component, which resulted in a slight improvement of predictions when using a two component WA-PLS model. The best colour predictions were obtained using a two component PLS model. Principal component analysis (PCA) of the prediction errors, with some characteristics of the training set included as passive variables, was performed to compare the results for the different alkalinity predictive models. The results indicate that calibration techniques utilizing more than one component (PLS and WA-PLS) can improve the predictions for lakes with diatom taxa that have broad tolerances. Furthermore, we show that WA-PLS performs best compared with the other techniques for those lakes that have a high relative abundance of the most dominant taxa and a corresponding low sample heterogeneity.     

THE CONSTRUCTION OF SIMULTANEOUS OPTIMAL EXPERIMENTAL DESIGNS FOR SEVERAL POLYNOMIALS IN THE CALIBRATION OF ANALYTICAL METHODS

In this paper a criterion is described for the construction of experimental designs for the evaluation ofcalibration models in analytical chemistry.The proposed criterion seeks a compromise between theD-optimal designs for estimating the parameters of different polynomial models.A computer algorithmis presented for a sequential construction of experimental designs using the optimality criterion.Theperformance of the optimality criterion and the computer algorithm is elaborated for the problem ofdiscrimination between a first-to a third-degree polynomial for the calibration of analytical methods.Anexperimental design consisting of replicate measurements at five distinct levels equally spaced over thecalibration range proved a good solution.     

Real-time genetic spatial optimization to improve forest fire spread forecasting in high-performance computing environments

Abstract

Forest fires are a kind of natural hazard with a high number of occurrences in southern European countries. To avoid major damages and to improve forest fire management, one can use forest fire spread simulators to predict fire behavior. When providing forest fire predictions, there are two main considerations: accuracy and computation time. In the context of natural hazards simulation, it is well known that part of the final forecast error comes from uncertainty in the input data. These data typically consist of a set of GIS files, which should be appropriately conflated. For this reason, several input data calibration methods have been developed by the scientific community. In this work, the Two-Stage calibration methodology, which has been shown to provide good results, is used. This calibration strategy is computationally intensive and time-consuming because it uses a Genetic Algorithm as a solution. Taking into account the aspect of urgency in forest fire spread prediction, it is necessary to maintain a balance between accuracy and the time needed to calibrate the input parameters. In order to take advantage of this technique, one must deal with the problem that some of the obtained solutions are impractical, since they involve simulation times that are too long, preventing the prediction system from being deployed at an operational level. A new method which finds the minimum resolution reduction for such long simulations, keeping accuracy loss to a known interval, is proposed. The proposed improvement is based on a time-aware core allocation policy that enables real-time forest fire spread forecasting. The final prediction system is a cyberinfrastructure, which enables forest fire spread prediction at real time.     

‘Diatoms and pH reconstruction’ (1990) revisited

The 167 sample lake-water pH-diatom calibration data-set created as part of the Palaeolimnology Programme within the Surface Water Acidification Project (SWAP) is re-analysed numerically using nine different numerical methods, six based on simple two-way weighted-averaging (WA), and the other three involving Gaussian logit regression (GLR) and maximum-likelihood (ML) calibration, the modern analogue technique, or weighted-averaging partial least-squares regression and calibration. Root mean squared error of prediction and maximum bias were estimated for all nine methods based on 10,000 internal and 10,000 external cross-validations involving a training-set, an optimisation-set, and a test-set. The results show that WA with a monotonic deshrinking spline equals or slightly outperforms WA with linear inverse deshrinking, especially in external cross-validation. Methods that employ tolerance downweighting generally have an inferior performance except when combined with monotonic deshrinking. It appears that simple two-way WA extensively used in SWAP cannot be significantly bettered. Thanks to increased computing power, better software, and more rigorous cross-validations, GLR shows good performance, especially in external cross-validation.