Shannon Information Retrieved from Overlapped Chromatographic Peaks with Kaiman Filter

Abstract

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...

Optimization of HPLC-ECD Conditions for Determination of Catechins with Precision and Efficiency Based on the FUMI Theory

“…15,16,20 The θ-optimal is the most efficient (rapid) condition and was expressed with a maximum of information flow θ (the amount of information φ transmitted in a unit time). 20 Hereafter, φ and θ were called precision and efficiency, respectively.…”

“…14 The precision φ-optimal is defined as the most precise condition among all the examined conditions and is characterized by the maximum of the precision φ. 15,16 The efficiency θ-optimal is the most efficient (rapid) condition and is described by the maximum of the efficiency θ. But the aforementioned parameters have never been applied to HPLC with electrochemical detection (HPLC-ECD).…”

Optimization of Detection Wavelelengths in Liquid Chromatography on the Basis of Information Theory and Kaiman Filter

“…4 For a peak present in a multi-component chromatogram, FUMI ~; of the j-th peak takes the simple form: KtO) ..,…”

“…'-3 Most recently, Hayashi et al proposed the function FUMI which describes the precision (variance) of chromatographic data as Shannon mutual information. 4 With the aid of FUMI, the optimal condition can be clearly defined as one that can transmit the maximal amount of mutual information, i.e. one that can provide the most precise measurements for a quantitative chromatographic analysis.…”

“…4 The total information P has been shown to be a good criterion by itself for chromatographic optimization, especially in selection of binary mobile phase composition.' In this situation, no other criteria were needed because of the experimental schemes designed (see below).…”