Calibration and measurement control based on Bayes statistics

The Bayesian methodology described in this paper has the inherent capability of choosing, from calibration-type curves, candidates which are plausible with respect to measured data, expert knowledge and theoretical models (including the nature of the measurement errors). The basic steps of Bayesian calibration are reviewed and possible applications of the results are described in this paper. A calibration related to head-space gas chromatographic data is used as an example of the proposed method. The linear calibration case has been treated with a log-normal distributed measurement error. Such a treatment of noise stresses the importance of modelling the random constituents of any problem.


Introduction
The large number of papers dealing with problems of calibration (for example [1, 2 and 10]) shows the importance of calibration to any method of analytical measurement. The published literature addresses a variety of aspects of this complex and so far unsolved problem.
The Bayes methodology applied in this paper proved to be an efficient tool for handling problems which are similar to those found in calibration. The method allows candidates which are plausible with respect to measured data, expert experience and theoretical models (including the nature ofthe measurement errors) to be highlighted in calibration curves. The potential of the Bayesian framework was recognized by the editorial board of the journal Technometrics, who started a discussion about its relevance to the calibration problem [3]. The discussion was, however, devoted mainly to the theoretical aspects of Bayesian calibration and only linear calibration with normally distributed measurement error was considered in depth.
The authors attempt to show here how the Bayesian approach can be used and to illustrate that it is a practical and powerful tool for solving calibration problems (see also [11]). This paper is complementary to Kirn) and Hangos (1989, [9]).
The calibration aims to estimate the mutual relation of measurements Y and concentrations X by evaluating measurements Yc for known values of the so-called control samples Xc. The information content of the calibration data is often substantially reduced due to generic non-linearity of the performed data reduction. Thus the random measurement errors are transformed into a systematic error, cf.
the analysis in reference [4].
The Bayesian methodology relies on the assumption that random and uncertain quantities have a probabilistic structure: they can be described by tools of probability theory. The relationship of measurements Y and concentrations X is expressed in the form of mutual probability density function (the term 'probability density funcion' will be abbreviated to p.d.f.) p(Y,X  [7]). (In the remainder of this paper notation is often simplified by dropping the subscript c. The proper meaning will be clear from the context.) The rather complex and general description above helps to combine all the available information (expert knowledge, theoretical models of deterministic relations, as well as of measurement errors and measured data) when solving different aspects ofthe calibration. The aim ofthis paper is to demonstrate how the principal term p(Y,XI[5 of the joint p.d.f, can be constructed and why it is generally necessary to extend parameters 0 to [5. The problems involved in quantifying in the expert knowledge by p([5) is beyond the scope of this paper (for a discussion of this topic see [8]). The reader interested in further aspects of Bayesian statistics should consult [7].
It will help to start with a simple example. Consequently, only p(IX,) has to be constructed. The theoretical calibration curve is assumed to be described by a deterministic function F(x,O) and the measurement errors ei of the particular measurements are independent and normally distributed with zero mean, i.e. only the random measurement error component is present. The variances of the random measurement errors 8i are concentration-dependent and this dependence is described by a known non-negative parametrized function of the form: var(8i) s(xi, Q)R, with Q and R unknown.
Under these assumptions the required c.p.d.f, is uniquely determined by the formulae p( Y,X[[5) p( IX,f)6(x X) where parameter [3 is the triple (0,Q,R). Note that the conventional calibration curve is the (conditional) mean of the c.p.d.f. (8). Remarks (1) As the example shows, the parameters of the measurement error distribution enter into the set of calibration parameters. This supports the idea that conventional calibration results in information reduction: the neglected information about the character of the measurement error distribution is clearly relevant for optimum data handling. For instance, the detailed stochastic characteristics of the measured data which are lost due to the reduction are significant to any measurement diagnostics.
(2) If the measurement errors E also contain systematic error components (i.e. they have non-zero expectation), then the deviation of the conditional expectation ofy conditioned on 13, taken as a function of x from the theoretical calibration curve F(x,), is just this systematic error. The calibration can compensate for this difference if the measurement conditions, i.e. the properties of the measurement error, are kept constant during the calibration and measurement phases.
(3) Low dimensionality of the parameter [3 is achieved by postulating the form (for example, Gaussian) of the measurement error distribution. This is, of course, restrictive. Including the discussed form in unknown quantities (so-called 'non-parametric estimation') would be preferable: it needs substantially more calibration measurements but provides more information about measurement errors.
(4) It is generally assumed that the measurement error has a Gaussian distribution. However, other distributions commonly found in practice [12] and other distributions will appear if there are any faults or failures in the measurement system or conditions.
(5) If it can be assumed that measurement errors are independent then the required c.p.d.f, modelling that measurement process has the following product form (this is known as a chain rule) N II p(flilXi,) This case is often acceptable for correctly controlled analytical measurements so it is used for the rest of this paper. Also, concentrations and parameters are assumed to be independent, i.e. [3 characterizes the measurement process only.
The general form of the chain rule [6] takes into account the fact that the result of a measurement can depend on the whole past measurement history. It has to be used when some dependency is encountered.
Clearly, there is a practical need to test whether the autocorrelated measurement errors are present both in the preliminary stages of analytical measurement and during routine measurements. A method for detecting this type of measurement error component is covered in reference [5].
(6) The generic term of the product (9) is where Fy(m,v) denotes Gaussian probability density function in argument y, with a given mean m and variance v.

Bayesian calibration
The usual steps of Bayesian calibration The complete form of a joint p.d.f, for calibration can be chosen from the theoretical models, and from knowledge and experience of experts in the field. However, it could be useful to sketch a common way of constructing a joint p.d.f. The usual steps are: (1) The form of calibration curve.} F(x,O) is chosen; this specifies the dependence of the expected value of a measurement in a parametrized way, and assumes that the measured concentration is x.
(2) The distribution of the measurement errors is either guessed or estimated in a preliminary stage. The errors are assumed to be independent and zero mean (the systematic error is included in F(x,O) mostly as an additive constant). (3) When calibrating, a balance has to be found between cost of the calibration and possible losses caused by imprecisely determined values of measured concentration. Clearly, any compromise must be based on information about the uncertainties of the treated quantities. The unique property of the Bayesian framework is the ability to supply this information, regardless of the amount of data available. Measurements 0.2-ml volume of the control sample and 0"2 ml of the stock solution were measured into a 5-ml sampling vial and the vial was sealed. The samples in vials were equilibrated at 50 C in the water-bath of the HS-250 for at least 20 min. Then a 0-8-ml gas sample was injected into the gas chromatograph by the HS-250 equipped with injection syringe 1001 LTSN (1 ml) (Hamilton, Bonaduz, Switzerland). The measurement was performed in sequences three times a day (at 9 a.m., 11 a.m. and p.m.). In the sequence there were five samples of 0"5, 0"8, 1"0, 2"0 and 3"0 g/1 ethanol content, all of which were measured in random order.
The quantitative evaluation was performed on the basis of the peak area and height ratio of ethanol and propan-1-ol.

Models for calibration
The models for calibration are derived using Bayesian calibration (see the section on the usual steps of Bayesian calibration). The necessary assumptions and models for head-space gas chromatography of liquid samples are as follows: (1) The possible form of calibration curve is a line: (13) where a and b are unknown parameters forming the vector 0.
(2) The measurement error of the ith measurement i is assumed to have a log-normal distribution with concentration dependent variance, i.e. the measured values are Yi (a + bxi)i (14) or equivalently log(y/) log(a + bxi) + ei (15) where Ei has Gaussian distribution with zero mean and unknown variance s>0. The measurement errors are assumed to have a genuinely random nature, i.e. the components 1,2, are identically distributed and mutually independent.
The advantage of this model (14) includes (a) the higher the expected value of observations then the higher the variance is assigned to them by log-normal distribution. This feature matches all practical experience with the described case; (b) log-normal distribution models the observations with outliers so the calibration relying on it will be sound with respect to outlying data; (c) a linearized form of the (probabilistic) model (15) is easily tractable computationally [7]. x) and the calibration results can be computed according to equations (7) and (10).
(5) The prior distribution is chosen to be in the self-reproducing Gauss-Wishart form [7] characterized by a priori independent values of [ (A,B,s) with prior expectation (0,0,10-2). This choice is equivalent to the assumption that there will be zero offset and unit slope in (13) and measurement errors of the order 10 -1 The variances were chosen to assign about 95% ofprior probability to the assumption that the offset is in the range (-0"5, 0"5) and the slope is in the range (0-8, 1"2).

Calibration results
Typical sequences ofmeasured data are shown in tables and 2. Other series with outliers are shown in tables 3 and 4.  Table 3. Typical data of peak area ratio obtained for three different sequences of control samples with outliers.
Ethanol concentration/gl-   (11)]. Thus taking its section at a fixed measured value y the probability of concentrations which could cause this value can be seen.

Conclusion
The Bayesian calibration method has been described with emphasis on practical aspects of its application. In order to encourage its wider use, the basic steps of Bayesian calibration are reviewed and possible applications of the results are given. The calibration related to head-space gas chromatographic data was described to show the advantages of Bayesian calibration. The linear calibration case has been treated with log-normally distributed measurement noise. This treatment of noise stresses the (often overlooked) importance of modelling random constituents in any problem.
The authors hope that the paper has demonstrated that Bayesian calibration is especially advantageous when used for measurement control because information about uncertainty in the results after calibration is supplied.