Optimal designs with information theory in least-squares problems

Thijssen, P.C.; Kateman, G.; Smit, H.C.

doi:10.1016/s0003-2670(00)83610-7

Cited by 25 publications

(3 citation statements)

References 10 publications

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“…Various criteria have already been developed to allow for automatic wavelength selection. − Among the proposed methods, the determinant and the condition number of calibration matrix are the most preferred criteria for prediction of the best wavelength combination. − Nevertheless, these two criteria are principally designed for exactly determined systems, although some modifications have been made to deal with overdetermined systems. [Most applications use RMSEP to characterize the performance of principal component regression (PCR) procedure, carried out under optimized conditions as obtained by the above criteria.]…”

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confidence: 99%

Wavelength Selection for Simultaneous Spectroscopic Analysis. Experimental and Theoretical Study

1996

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Simultaneous multicomponent analysis is usually carried out by multivariate calibration models (such as principal component regression) that utilize the full spectrum. We demonstrate, by both experimental and theoretical considerations, that better results can be obtained by a proper selection of the spectral range to be included in calculations. We develop the theory that models the analytical uncertainty in multicomponent analysis and show the conditions where wavelength selection is essential (for example, when considerable spectral overlapping exists). An error indicator function is developed to predict the analytical performance under given experimental conditions, using a certain spectral range. This function is applied for allocation of the most informative spectral ranges to be utilized in multicomponent analysis. Selection of spectral ranges by this method is shown to ensure optimal results that considerably improve analytical performance in some cases. The similarity between the results obtained by this function and actual experimental results prove the validity of the proposed error indicator for wavelength selection. In addition to the experimental examples, extensive computer simulations have been carried out in order to study the validity of the theory over a wide range of the relevant parameters.Multicomponent spectral analysis is now gaining popularity. Numerous spectral data can be collected by modern instruments (such as photodiode arrays and CCD detectors), and several mathematical approaches have been designed to deal with these over-determined systems. [1][2][3][4][5][6] Early applications used complete spectra for determination of all components in a mixture. Ross and Pardue showed, however, that accuracy can be improved by a careful selection of wavelength ranges, which results in a collinearity or spectral overlap reduction. 7 The decision of how many data points and which wavelengths should be included in the analytical process is not trivial and is usually an empirical choice. 8 Various criteria have already been developed to allow for automatic wavelength selection. 8-25 Among the proposed meth-ods, the determinant and the condition number of calibration matrix are the most preferred criteria for prediction of the best wavelength combination. [8][9][10][11][12] Nevertheless, these two criteria are principally designed for exactly determined systems, although some modifications have been made to deal with overdetermined systems. [Most applications use RMSEP to characterize the performance of principal component regression (PCR) procedure, carried out under optimized conditions as obtained by the above criteria.] The major disadvantage of the current criteria is that all components are examined together with the same matrix, while in most cases, the optimal conditions for determination of each component is different and should be optimized separately. Recently, two stochastic search heuristics, namely, simulated annealing and genetic algorithms, have been suggested for wavelength selectio...

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Wavelength Selection for Simultaneous Spectroscopic Analysis. Experimental and Theoretical Study

1996

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“…The exact form of the filtering error Pk can be derived by the mathematical induction applied to the recursive correspondent (see Appendix (i) and also refs. 7,8). Importantly, the approximate Pkt takes a definite chemometric meaning: clearly, (kPkt)-1 is equal to S/ N S/ N = (ZF,2) / ( Wit)…”

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confidence: 99%

Shannon Information Retrieved from Overlapped Chromatographic Peaks with Kaiman Filter

1989

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A function (FUMI) which describes the Shannon mutual information in chromatography is derived. FUMI covers two subjects in one formula: (i) the degree of peak overlap and the noise level; (ii) the mathematical formalism of peak deconvolution and quantitation based on the one-dimensional Kalman filter for peak resolution. Computer simulation demonstrates that a sufficient amount of mutual information can be retrieved from overlapped peaks (Gaussian) through (i) observation and (ii) data processing. A practical advantage of FUMI is logical evaluation and optimization of chromatographic experiments without resort to experience: the optimal can be defined as the chromatogram which can transmit the maximal amount of mutual information in a unit time. KeywordsInformation theory, Kalman filter, mutual information, chromatography, optimizationAnalytical systems provide us with information in various forms and ways.An important aim in analytical chemistry is to elaborate a method through which more information can be transmitted from samples of interest. Logical evaluation of an overall analytical system, therefore, should be founded on the quantitative description of the information.As an example, high performance liquid chromatography (HPLC) is taken here. We focus our discussion on quantitative analysis. We can easily guess that superior chromatographic separation of target materials will yield more information.However, the amount of useful information is not only regulated by the elution pattern: reliability of the information also depends on analytical powers of data processing because of the inevitable noise-contamination, baseline drift, etc., in the measurement process.The Shannon mutual information will serve for the quantitative evaluation of the system (see below), but should cover at least two subjects: (i) the degree of peak overlap in a noise level; (ii) mathematical formalism of a data processing.We review the fundamental concepts of information theory to clarify our reasoning. Let 1[Q] be the amount of information concerning an unknown quantity Q to be estimated. If the probability of Q is known or welldefined, I[SO] can be calculated according to the Shannon equation.' We cannot know any details of the quantity Q before measurement of Q, but the complete or undistorted transmission of the information I[Q] through analytical media is always thwarted by various error sources such as calibration, interference, drift, etc. In general, the useful information which we can actually collect from an ensemble !P of the measure- If ambiguity is absent (1[Q/ !PJ 0), then we can retrieve from the measurements !V all of the original information on the unknown quantity Q. In other words, we can establish the estimation for Q with 100% certainty. If I[Q/ VI takes the largest value (=1[Q]), then nothing can be known concerning the information source Q. Chromatography achieves a decrease in the ambiguity I[Q/ !P] by separating peaks of interest, increasing the sensitivity of detection systems, etc. Data processing is also...

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“…Making a number of assumptions, the usual Kalman filter algorithms can be used resulting in [ 4 , 5 , 6 ]

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Accuracy in trace analysis. II. Considerations of the measurement process. Quality assurance and reference materials for trace analysis

Kateman¹

1988

J. RES. NATL. BUR. STAN.

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Most analytical measurements are not absolute but depend on the correlation between physical phenomena and some intrinsic property, e.g., concentration. Therefore, calibration is an indispensable part of analytical chemistry. Unfortunately, calibrations are not free from interference by the environment. This disturbing environment can be the micro-environment, components in the sample that influence the calibration line. As a rule this interference is usually constant, though not always (e.g., separation processes).The macro-environment, however, changes continuously. Temperature, pressure, chemicals, and man are stationary only during a short time. These influences will be seen as random fluctuations or, when autocorrelated, as drift. One approach is to monitor the properties of the calibration system internally by incorporating a calibration system and a measuring system. By monitoring the calibration system, the results of the unknown can be corrected. Kalivas and Kowalski [1] described the solution for the multicomponent situation, using the generalized standard addition method (GSAM). By treating drift as a time dependent component they obtain the equation A powerful method which uses all available information is "state estimation" or Kalman filtering.

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Optimal designs with information theory in least-squares problems

Cited by 25 publications

References 10 publications

Wavelength Selection for Simultaneous Spectroscopic Analysis. Experimental and Theoretical Study

Wavelength Selection for Simultaneous Spectroscopic Analysis. Experimental and Theoretical Study

Shannon Information Retrieved from Overlapped Chromatographic Peaks with Kaiman Filter

Accuracy in trace analysis. II. Considerations of the measurement process. Quality assurance and reference materials for trace analysis

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