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### **An Improved Wavelet‐Based Multivariable Fault Detection Scheme**

Fouzi Harrou, Ying Sun and Muddu Madakyaru

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68947

#### Abstract

Data observed from environmental and engineering processes are usually noisy and correlated in time, which makes the fault detection more difficult as the presence of noise degrades fault detection quality. Multiscale representation of data using wavelets is a powerful feature extraction tool that is well suited to denoising and decorrelating time series data. In this chapter, we combine the advantages of multiscale partial least squares (MSPLSs) modeling with those of the univariate EWMA (exponentially weighted moving average) monitoring chart, which results in an improved fault detection system, especially for detecting small faults in highly correlated, multivariate data. Toward this end, we applied EWMA chart to the output residuals obtained from MSPLS model. It is shown through simulated distillation column data the significant improvement in fault detection can be obtained by using the proposed methods as compared to the use of the conventional partial least square (PLS)-based Q and EWMA methods and MSPLS-based Q method.

Keywords: data uncertainty, multiscale representation, fault detection, data-driven approaches, statistical monitoring schemes

#### 1. Introduction

Monitoring chemical and environmental processes has increasingly attracted greater attention of researchers and practitioners for improving the quality of products and enhancing process safety. For example, detecting anomalies in chemical or environmental plants is expected to reflect not only on the productivity and profitability of these plants, but also on the safety of people [1, 2]. To enhance process operation, we should monitor the process in an efficient manner and correctly detect abnormality events that may result in any degradation of

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Figure 1. Scheme of fault detection and diagnosis.

product quality, operation reliability, and profitability, in order that we can respond accordingly by making any necessary correction to the process. Fault detection and diagnosis represent two vital components of process monitoring (see Figure 1), during which abnormal events are first identified and then isolated to ensure that they can be appropriately handled [2, 3]. Generally, faults in modern automatic processes are difficult to avoid and may result in serious process degradations. Even small deviations in process parameters can result in lost time, and catastrophic failure can bring devastating health, safety, and financial consequences. Because of this, engineers must keep tweaking and improving the reliability of their processes, watching carefully for signs of anomalies that could lead to disaster. Therefore, it is crucial to be able to detect and identify any possible faults or failures in the system as early as possible [2, 4, 5].

Keeping an automated process running smoothly and safely and producing the desired results remains a major challenge in many sectors. Various fault detection techniques have been developed for the safe operation of systems or processes. There are two main types of these techniques: process history-based approaches and model-based approaches, as shown in Figure 2. Model-based approaches compare analytically computed outputs with measured values and signal an alarm when large differences are detected [2, 6, 7]. Unfortunately, the effectiveness of model-based fault-detection approaches relies on the accuracy of the models used. When there is no process model, model-free or process-history-based methods were successfully used in process monitoring because they can effectively deal with highly correlated process variables [8, 9]. Such methods require a minimal a prior knowledge about

Figure 2. Fault detection methods.

product quality, operation reliability, and profitability, in order that we can respond accordingly by making any necessary correction to the process. Fault detection and diagnosis represent two vital components of process monitoring (see Figure 1), during which abnormal events are first identified and then isolated to ensure that they can be appropriately handled [2, 3]. Generally, faults in modern automatic processes are difficult to avoid and may result in serious process degradations. Even small deviations in process parameters can result in lost time, and catastrophic failure can bring devastating health, safety, and financial consequences. Because of this, engineers must keep tweaking and improving the reliability of their processes, watching carefully for signs of anomalies that could lead to disaster. Therefore, it is crucial to be able to detect and identify any possible faults or failures in the

Keeping an automated process running smoothly and safely and producing the desired results remains a major challenge in many sectors. Various fault detection techniques have been developed for the safe operation of systems or processes. There are two main types of these techniques: process history-based approaches and model-based approaches, as shown in Figure 2. Model-based approaches compare analytically computed outputs with measured values and signal an alarm when large differences are detected [2, 6, 7]. Unfortunately, the effectiveness of model-based fault-detection approaches relies on the accuracy of the models used. When there is no process model, model-free or process-history-based methods were successfully used in process monitoring because they can effectively deal with highly correlated process variables [8, 9]. Such methods require a minimal a prior knowledge about

system as early as possible [2, 4, 5].

Figure 1. Scheme of fault detection and diagnosis.

176 Uncertainty Quantification and Model Calibration

process physics, but depends on the availability of quality input data. Process-history-based methods use implicit empirical models derived from analysis of available data and rely on computational intelligence and machine learning methods [10–12]. In the last four decades, process-history-based methods such as principal component analysis (PCA) and partial least squares (PLSs) have become more and more important in statistical process monitoring. They have been extensively applied in the field of chemometrics [5, 13, 14]. In contrast to the classical univariate statistical process monitoring tools, these approaches take the correlations between variables into account and monitor a set of correlated variables simultaneously. Moreover, by projecting the original measurements into a latent sub space, latent variables (LVs) are monitored in a reduced dimensional space. A PCA or PLS model is built on good historical data of normal or process operation [15, 16]. This model can then be used to monitor or predict the future behavior of the process [17].

However, most of the processes are in dynamic state, with various events occurring such as abrupt process changes, slow drifts, bad measurements due to sensor failures, and human errors. Data from these processes are not only cross-correlated, but also autocorrelated. Applying conventional latent variable regression (LVR) methods directly to dynamic systems results in false alarms, making it insensitive to detect and discriminate different kinds of events. In addition, noisy data and model uncertainties negatively affect the performance of fault detection methods. In fact, wavelet-based multiscale representation of data has been shown to provide effective noise-feature separation in the data, to approximately decorrelate autocorrelated data, and to transform the data to better follow the Gaussian distribution [18]. Multiscale representation of data using wavelets has been widely used for data denoising, compression, and for process monitoring [18–21].

The detection of incipient faults is crucial for maintaining the normal operations of a system by providing early fault warnings. The problem is that incipient anomalies are often too weak to be detected by conventional monitoring methods. The objective of this chapter is to extend the fault detection techniques developed to take into account the uncertainty of the data. To this end, multiscale data representation, a powerful feature extraction tool, will be used to reduce false alarms by improving noise-feature data separation and decorrelation of autocorrelated measurement errors. To do so, multiscale partial least square (MSPLS)-based exponentially weighted moving average (EWMA) fault detection techniques will be developed. The overarching goal of this work is to tackle multivariate challenges in process monitoring by merging the advantages of EWMA chart and multiscale-PLS modeling to enhance their performance. It is shown through simulated distillation column data that significant improvement in detecting small fault can be obtained using the MSPLS-EWMA approach as compared to the PLS-EWMA fault detection approach.

The remainder of this chapter is organized as follows. Section 2 gives a brief overview of the PLS and the multiscale PLS approach. In Section 3, we present the proposed MSPLS-EWMA fault-detection procedure. In Section 4, EWMA chart is briefly presented. Section 5 applies the proposed fault-detection procedure to a simulated distillation column process. Finally, Section 6 concludes the chapter.

#### 2. Preliminary materials

#### 2.1. Partial least squares (PLS)-based charts

The objective of PLS models is to find relations between input and output data blocks by relating their latent variables. A detailed description of the PLS technique is given in Ref. [22]. This data-driven empirical statistical model approach is extremely useful under the situation where either a first principal model or analytical model is difficult to obtain or the measured variables are highly correlated (collinear) to each other. The PLS methods have been extensively researched and applied in the chemometrics field.

Consider an input data matrix X∈Rn�m and an output data matrix Y∈Rn�p , where n is the number of samples or observations, m and p are the number of input and output variables, respectively. The objective of PLS is to maximize the covariance matrix between linear combinations of X and Y. A PLS model is given by the inner model and the outer model [15, 23] (see Figure 3). The input and output matrices can be related to LVs as follows via the outer model [23]:

$$\begin{cases} \mathbf{X} = \widehat{\mathbf{X}} + \mathbf{E} = \sum\_{i=1}^{l} \mathbf{t}\_i \mathbf{P}\_i^T + \mathbf{E} = \mathbf{T} \mathbf{P}^T + \mathbf{E} \\\ \mathbf{Y} = \widehat{\mathbf{Y}} + \mathbf{F} = \sum\_{i=1}^{l} \mathbf{u}\_i \mathbf{q}\_i^T + \mathbf{F} = \mathbf{U} \mathbf{Q}^T + \mathbf{F} \end{cases} \tag{1}$$

Figure 3. Principle of PLS.

of data using wavelets has been widely used for data denoising, compression, and for process

The detection of incipient faults is crucial for maintaining the normal operations of a system by providing early fault warnings. The problem is that incipient anomalies are often too weak to be detected by conventional monitoring methods. The objective of this chapter is to extend the fault detection techniques developed to take into account the uncertainty of the data. To this end, multiscale data representation, a powerful feature extraction tool, will be used to reduce false alarms by improving noise-feature data separation and decorrelation of autocorrelated measurement errors. To do so, multiscale partial least square (MSPLS)-based exponentially weighted moving average (EWMA) fault detection techniques will be developed. The overarching goal of this work is to tackle multivariate challenges in process monitoring by merging the advantages of EWMA chart and multiscale-PLS modeling to enhance their performance. It is shown through simulated distillation column data that significant improvement in detecting small fault can be obtained using the MSPLS-EWMA approach as compared to the PLS-EWMA

The remainder of this chapter is organized as follows. Section 2 gives a brief overview of the PLS and the multiscale PLS approach. In Section 3, we present the proposed MSPLS-EWMA fault-detection procedure. In Section 4, EWMA chart is briefly presented. Section 5 applies the proposed fault-detection procedure to a simulated distillation column process. Finally, Section

The objective of PLS models is to find relations between input and output data blocks by relating their latent variables. A detailed description of the PLS technique is given in Ref. [22]. This data-driven empirical statistical model approach is extremely useful under the situation where either a first principal model or analytical model is difficult to obtain or the measured variables are highly correlated (collinear) to each other. The PLS methods have been exten-

number of samples or observations, m and p are the number of input and output variables, respectively. The objective of PLS is to maximize the covariance matrix between linear combinations of X and Y. A PLS model is given by the inner model and the outer model [15, 23] (see Figure 3). The input and output matrices can be related to LVs as follows via the outer model [23]:

i þ E ¼ TPT þ E

i þ F ¼ UQT þ F

l

i¼1 tiPT

l

uiqT

i¼1

, where n is the

ð1Þ

monitoring [18–21].

178 Uncertainty Quantification and Model Calibration

fault detection approach.

6 concludes the chapter.

2. Preliminary materials

2.1. Partial least squares (PLS)-based charts

sively researched and applied in the chemometrics field.

8 >>>><

>>>>:

Consider an input data matrix X∈Rn�m and an output data matrix Y∈Rn�p

X ¼ Xb þ E ¼ X

Y ¼ Yb þ F ¼ X

where Xb and Yb are approximated data matrices of X and Y, respectively, the matrices T∈Rn�l and U ∈Rn�q consist of l retained LVs of the input and output data, respectively. E∈Rn�m and F∈Rn�p represent the residuals matrices that were the unexplained variance of the input and output data, respectively, P∈Rm�l and Q ∈ Rp�q are the loading of matrices X and Y, respectively. In practice, how to choose a proper number l for LVs is an important step in PLS modeling. If all LVs are used in modeling, the model may fit the noise and therefore reduce the predictive ability of the model. Here, the cross-validation method can be used to determine a proper number of LVs [24]. The inner model can be computed as

$$\mathbf{U} = \mathbf{T}\mathbf{B} + \mathbf{H},\tag{2}$$

where B is a regression matrix and H is a residual matrix. The information in Y can be expressed as

$$\mathbf{Y} = \mathbf{T} \mathbf{B} \mathbf{Q}^T + \mathbf{F}^\* \tag{3}$$

where matrix F� was the residue that presented the unexplained variance.

#### 2.2. Wavelet transform

Most engineering processes generate data with multiscale properties, signifying that they include both useful information and noise at different times and frequencies. The majority of fall detection approaches are based on time-domain data (operates on a single time scale) that do not take multiscale characteristics of the data into consideration. Wavelet analysis has been show to represent data with multiscale properties, efficiently separating deterministic and stochastic features [18].

Multiresolution time series decomposition was initially applied by Mallat, who used orthogonal wavelet bases during data compression for image decoding [25]. Wavelets represent a family of basis functions that can be expressed as the following localized in both time and frequency [18]:

$$
\psi\_{a,b}(t) = \frac{1}{\sqrt{a}} \psi \left( \frac{t-b}{a} \right),
\tag{4}
$$

where a represents the dilation parameter, b is the translation parameter [26] and ψðtÞ is the mother wavelet. Both these parameters are commonly discretized dyadically as a¼ 2m, b¼ 2mk, ðm;kÞ∈ Z2 , and the family of wavelets can represented as ψmnðtÞ ¼ 2 �m 2 ψð2�mt � mÞ.  Different families of basis functions are created based on their convolution with different filters, such as the Haar scaling function and the Daubechies filters [26, 27]. Parameters that are discretized dyadically force downsampling reduce the number of parameters dyadically with every decomposition. However, dyadically discretized wavelet force samples at nondyadic locations to become decomposed only after a certain time delay.

The discrete wavelet transform (DWT) analyzes the signal at different scales (or over different frequency bands) by decomposing the signal at each scale into a coarse approximation (low frequency information), A, and detail information (high frequency information), D. DWT employs two sets of functions: the scaling functions φj;kðtÞ ¼ ffiffiffiffiffiffi 2�j p φð2�j t � kÞ;k∈ Z and wavelet functions ψj;kðtÞ ¼ ffiffiffiffiffiffi 2�j p ψð2�j t � kÞ;j ¼ 1;…;J;k∈ Z, which are associated with low pass filter H and high pass filter G, respectively. Where the coarsest scale J usually termed the decomposition level. Any signal can be represented by a summation of all scaled and detailed signals as follows [26]:

$$\overbrace{\mathbf{x}(t) = \overbrace{\sum\_{k=1}^{n2^{-l}} a\_{lk} \phi\_{jk}(t)}^{A\_l(t)} + \sum\_{j=1}^{l} \sum\_{k=1}^{n2^{-j}} d\_{jk} \psi\_{jk}(t)} \tag{5}$$

where j, k, J, and n represent the dilation parameter, translation parameter, number of scales, and number observations in the original signal, respectively [28, 29]. djk and aJk are respectively the scaling and the wavelet coefficients, and AJðtÞ and DjðtÞ;ð Þ j ¼ 1; 2;…;J represent the approximated signal and the detail signals, respectively. Of course, by passing a series of high and low pass wavelets filters, it is decomposed into signals at different scales as shown in Figure 4.

In the next section, we highlight the advantages of multiscale.

Figure 4. Principle of multiscale representation based on wavelet transform.

#### 2.3. Advantages of multiscale representation

do not take multiscale characteristics of the data into consideration. Wavelet analysis has been show to represent data with multiscale properties, efficiently separating deterministic and

Multiresolution time series decomposition was initially applied by Mallat, who used orthogonal wavelet bases during data compression for image decoding [25]. Wavelets represent a family of basis functions that can be expressed as the following localized in both time and

> ffiffi a p ψ t � b a � �

where a represents the dilation parameter, b is the translation parameter [26] and ψðtÞ is the mother wavelet.  Different families of basis functions are created based on their convolution with different filters, such as the Haar scaling function and the Daubechies filters [26, 27]. Parameters that are discretized dyadically force downsampling reduce the number of parameters dyadically with every decomposition. However, dyadically discretized wavelet force samples at nondyadic locations to become decomposed only after a certain time

The discrete wavelet transform (DWT) analyzes the signal at different scales (or over different frequency bands) by decomposing the signal at each scale into a coarse approximation (low frequency information), A, and detail information (high frequency information), D. DWT

and high pass filter G, respectively. Where the coarsest scale J usually termed the decomposition level. Any signal can be represented by a summation of all scaled and detailed signals as

> þ X J

where j, k, J, and n represent the dilation parameter, translation parameter, number of scales, and number observations in the original signal, respectively [28, 29]. djk and aJk are respectively the scaling and the wavelet coefficients, and AJðtÞ and DjðtÞ;ð Þ j ¼ 1; 2;…;J represent the approximated signal and the detail signals, respectively. Of course, by passing a series of high and low pass wavelets filters, it is decomposed into signals at different scales as shown in Figure 4.

j¼1

Xn2�j k¼1

2�j p φð2�j

t � kÞ;j ¼ 1;…;J;k∈ Z, which are associated with low pass filter H

djkψjkðtÞ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ DjðtÞ:

employs two sets of functions: the scaling functions φj;kðtÞ ¼ ffiffiffiffiffiffi

xðtÞ ¼ Xn2�J

In the next section, we highlight the advantages of multiscale.

k¼1

aJkφJkðtÞ zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ AJðtÞ

2�j p ψð2�j

, and the family of wavelets can represented as ψmnðtÞ ¼ 2

; ð4Þ

�m

t � kÞ;k∈ Z and wavelet

ð5Þ

2 ψð2�mt � mÞ.

ψa;bðtÞ ¼ 1

stochastic features [18].

180 Uncertainty Quantification and Model Calibration

frequency [18]:

b¼ 2mk, ðm;kÞ∈ Z2

functions ψj;kðtÞ ¼ ffiffiffiffiffiffi

follows [26]:

delay.

Conventional methods are referred to as time-domain analysis methods. These methods are more sensitive toward impulsive oscillations and are unable to extract frequencies and patterns in the data that may be hidden. Before the introduction of multiscale wavelet analysis, mathematical tools such as Fourier transform analysis, coherence function analysis, and power spectral density analysis were used. However, these tools would only allow the signal to imitate the tool being used for analysis. For example, the use of Fourier transform analysis would decompose the signal into a sum of cosine and sine functions. Multiscale helps overcome this problem as it helps simultaneously examine both the time and frequency domains, while Fourier transform is only capable of shifting between the time and frequency domain.

Ganesan et al. in a literature review of multiscale statistical process monitoring state the following advantages of using wavelet coefficients in Multiscale statistical process control (MSSPC) over conventional Statistical process control (SPC) methods [20]:

#### 2.4. Separating noise feature

Two important applications, data compression and data denoising, can be achieved through wavelet multiscale decomposition. One of the biggest advantages of multiscale representation is its capacity of distinguishing measurement noise from useful data features, by applying low and high pass filters to the data during multiscale decomposition. This allows the separation of features at different resolutions or frequencies, which makes multiscale representation a better tool for filtering or denoising noisy data than traditional linear filters, like the mean filter and the EWMA filter. Despite their popularity, linear filters rely on defining a frequency threshold above where all features are treated as measurement noise. The ability of multiscale representation to separate noise has been used not only to improve data filtering, but also to improve the prediction accuracy of several empirical modeling methods and the accuracy of state estimators.

A noisy signal is filtered by a three-step method [30]:

#### 2.5. Multiscale PLS modeling

Data observed from environmental and engineering processes are usually noisy and correlated in time, which makes the fault detection more difficult as the presence of noise degrades fault detection quality, and most methods are developed for independent observations. Multiscale representation of data using wavelets is a powerful feature extraction tool that is well suited to denoising and decorrelating time series data.

The integrated multiscale PLS (MSPLS) modeling approach is to take advantage of the both latent variable regression and denoising ability of the multiscale decomposition using wavelets. Thus, improve in prediction ability of the model, which in term improves the fault detection methods. The given input variable data matrix X and response variable matrix y are decomposed at different scales using multiscale basis function called wavelets. Let the decomposed data at each scale ðjÞ be Xj and yj . Then, the MSPLS model is developed using decomposed data, can be expressed as

$$\mathbf{y}\_{\circ} = \mathbf{T}\_{\circ} \mathbf{B}\_{\circ} \mathbf{Q}\_{\circ}^{T} - \mathbf{F}\_{\circ}, \tag{6}$$

where Xj ∈ Rn�m is the filtered input data matrix at scale ðjÞ, yj ∈ Rn�1 is the response output vector at scale ðjÞ. F∈ Rm�p is the MSPLS model residual at jth decomposition scale.

However, denoising the input and output variables a prior to developing model results in poor prediction ability of the MSPLS model due to removal of features which may be important to model. Therefore, in the proposed integrated MSPLS modeling approach, the selection of optimum decomposition depth based on the prediction ability of the developed MSPLS model is used. The integrated MSPLS modeling algorithm is summarized next [8].

Once an MSPLS model based on past normal operation is obtained, it can be used to monitor future deviation from normality. Two monitoring statistics, the T2 and Q statistics, are usually utilized for fault detection purposes [31]. First, the Hoteling T2 statistics indicates the variation within the process model in the LVs subspace. Second, the Q statistic, also known as the squared prediction error (SPE), monitors how well the data conforms to the model (see Figure 5).

The T2 statistic based on the number of retained LVs, l, is defined as [31]

$$T^2 = \sum\_{i=1}^{l} \frac{t\_i^2}{\lambda\_i},\tag{7}$$

where λi is eigenvalue of the covariance matrix of X. The T2 statistic measures the variation in the LVs only. A large change in the PC subspace is observed if some points exceed the confidence

Figure 5. (a) Hoteling T2 statistic and (b) Q statistic.

2.4. Separating noise feature

182 Uncertainty Quantification and Model Calibration

signal.

2.5. Multiscale PLS modeling

each scale ðjÞ be Xj and yj

expressed as

denoising and decorrelating time series data.

A noisy signal is filtered by a three-step method [30]:

Two important applications, data compression and data denoising, can be achieved through wavelet multiscale decomposition. One of the biggest advantages of multiscale representation is its capacity of distinguishing measurement noise from useful data features, by applying low and high pass filters to the data during multiscale decomposition. This allows the separation of features at different resolutions or frequencies, which makes multiscale representation a better tool for filtering or denoising noisy data than traditional linear filters, like the mean filter and the EWMA filter. Despite their popularity, linear filters rely on defining a frequency threshold above where all features are treated as measurement noise. The ability of multiscale representation to separate noise has been used not only to improve data filtering, but also to improve the prediction accuracy of several empirical modeling methods and the accuracy of state estimators.

• Apply wavelet transform to decompose the noisy signal into the time-frequency domain.

• Transform back into the original domain the threshold coefficients to obtain a filtered

Data observed from environmental and engineering processes are usually noisy and correlated in time, which makes the fault detection more difficult as the presence of noise degrades fault detection quality, and most methods are developed for independent observations. Multiscale representation of data using wavelets is a powerful feature extraction tool that is well suited to

The integrated multiscale PLS (MSPLS) modeling approach is to take advantage of the both latent variable regression and denoising ability of the multiscale decomposition using wavelets. Thus, improve in prediction ability of the model, which in term improves the fault detection methods. The given input variable data matrix X and response variable matrix y are decomposed at different scales using multiscale basis function called wavelets. Let the decomposed data at

yj ¼ TjBjQT

vector at scale ðjÞ. F∈ Rm�p is the MSPLS model residual at jth decomposition scale.

where Xj ∈ Rn�m is the filtered input data matrix at scale ðjÞ, yj ∈ Rn�1 is the response output

However, denoising the input and output variables a prior to developing model results in poor prediction ability of the MSPLS model due to removal of features which may be important to model. Therefore, in the proposed integrated MSPLS modeling approach, the selection of

. Then, the MSPLS model is developed using decomposed data, can be

j � Fj; ð6Þ

• Threshold the detail coefficient and remove coefficients a selected threshold.

limit of the T2 chart, indicating a big deviation in the monitored system. Confidence limits for T2 at level (1 � α) relate to the Fisher distribution, F, as follows [31]:

$$T\_{l,n,\alpha}^2 = \frac{l(n-1)}{n-l} F\_{l,n-l,\alpha},\tag{8}$$

where Fl;n�l;α is the upper 100α% critical point of F with l and n � l degrees of freedom.

The squared prediction error (SPE) or Q statistic, which is defined as [31]

$$Q = \mathbf{e}^T \mathbf{e},\tag{9}$$

captures the changes in the residual subspace. e ¼ x � x^ represents the residuals vector, which is the difference between the new observation, x, and its prediction, x^, via the MSPLS model. Eq. (9) provides a direct mean of the Q statistic in terms of the total sum of measured variation in the residual vector e. The SPE can be considered a measure of the system-model mismatch. The confidence limits for SPE are given in Ref. [32]. This test suggests the existence of an abnormal condition when Q > Qα, where Qα is defined as

$$Q\_{\alpha} = q\_1 \left[ \frac{h\_0 c\_{\alpha} \sqrt{2 \rho\_2}}{\rho\_1} + 1 + \frac{q\_2 h\_0 (h\_0 - 1)}{\rho\_1^2} \right],\tag{10}$$

where

$$\varphi\_i = \sum\_{j=l+1}^{m} \lambda\_j^i, \text{for } i = 1, 2, 3,\tag{11}$$

$$h\_0 = 1 - \frac{2\varphi\_1 \varphi\_3}{3\varphi\_2^2}.\tag{12}$$

cα is the confidence limits for the 1 � α percentile in a normal distribution.

However, the MSPLS-based T2 and Q approaches fail to detect small faults [9]. Here, we use only the Q-based chart as a benchmark for fault detection with PLS and MSPLS. Motivated by the power of the EWMA chart, which are widely used univariate control chart, is proposed as improved alternatives for fault detection. The objective is to tackle MSPLS challenges in process monitoring by merging the advantages of the EWMA and MSPLS approaches to enhance their performance and widen their practical applicability.

#### 3. EWMA monitoring charts

In this section, we briefly introduce the basic idea of the EWMA chart and its properties. For a more detailed discussion of EWMA charts, see Ref. [33]. EWMA is a statistic that gives less weight to old data, and more weight to new data. The EWMA charts are able to detect small shifts in the process mean, since the EWMA statistic is a time-weighted average of all previous observations. The EWMA control scheme was first introduced by Roberts [34], and is extensively used in time series analysis. The EWMA monitoring chart is an anomaly detection technique widely used by scientists and engineers in various disciplines [6, 33, 35]. Assume that {x1;x2;…;xn} are individual observations collected from a monitored process. The expression for the EWMA is [33]

limit of the T2 chart, indicating a big deviation in the monitored system. Confidence limits for T2

n � l Fl;n�l;α; ð8Þ

Q ¼ eTe; ð9Þ

; fori ¼ 1; 2; 3; ð11Þ

: ð12Þ

; ð10Þ

l;n;α ¼ lðn � 1Þ

where Fl;n�l;α is the upper 100α% critical point of F with l and n � l degrees of freedom.

captures the changes in the residual subspace. e ¼ x � x^ represents the residuals vector, which is the difference between the new observation, x, and its prediction, x^, via the MSPLS model. Eq. (9) provides a direct mean of the Q statistic in terms of the total sum of measured variation in the residual vector e. The SPE can be considered a measure of the system-model mismatch. The confidence limits for SPE are given in Ref. [32]. This test suggests the existence of an

at level (1 � α) relate to the Fisher distribution, F, as follows [31]:

184 Uncertainty Quantification and Model Calibration

abnormal condition when Q > Qα, where Qα is defined as

where

Qα ¼ ϕ1

h0cα

ϕi ¼ Xm j¼lþ1 λi j

cα is the confidence limits for the 1 � α percentile in a normal distribution.

enhance their performance and widen their practical applicability.

3. EWMA monitoring charts

ffiffiffiffiffiffiffiffi 2ϕ2 p ϕ1

h0 ¼ 1 � 2ϕ1ϕ3

However, the MSPLS-based T2 and Q approaches fail to detect small faults [9]. Here, we use only the Q-based chart as a benchmark for fault detection with PLS and MSPLS. Motivated by the power of the EWMA chart, which are widely used univariate control chart, is proposed as improved alternatives for fault detection. The objective is to tackle MSPLS challenges in process monitoring by merging the advantages of the EWMA and MSPLS approaches to

In this section, we briefly introduce the basic idea of the EWMA chart and its properties. For a more detailed discussion of EWMA charts, see Ref. [33]. EWMA is a statistic that gives less

3ϕ2 2

þ 1 þ ϕ2h0ðh0 � 1Þ ϕ2 1

" #

T2

The squared prediction error (SPE) or Q statistic, which is defined as [31]

$$\begin{cases} z\_t = & \lambda x\_t + (1 - \lambda) \ z\_{t-1} & \text{if } t > 0 \\\\ z\_0 = & \mu\_0, & \text{if } t = 0. \end{cases} \tag{13}$$

The starting value z0 is usually set to the mean of the fault-free data, μ0. Zt is the output of EWMA and xt is the observation from the monitored process at the current time. The forgetting parameter λ∈ ð0; 1� determines how fast EWMA forgets historical data. Equation (13) can also be written as

$$\mathbf{z}\_t = \lambda \sum\_{t=1}^n (1 - \lambda)^{n-t} \mathbf{x}\_t + (1 - \lambda)^n \boldsymbol{\mu}\_0,\tag{14}$$

where λð1 � λÞ n�t is the weight for xt, which falls off exponentially for past observations. We can see that if λ is small, then more weight is assigned to past observations. Thus, the chart is tuned to have efficiency for detecting small changes in the process mean. On the other hand, if λ is large, then more weight is assigned to the current observations, and the chart is more suitable for detecting large shifts [33]. In the special case, λ ¼ 1, the EWMA is equal to the most recent observation, xt, and provides the same results as Shewhart chart. As λ approaches zero, EWMA approximates the CUSUM criteria, which gives equal weights to the current and historical observations.

Under fault-free conditions, the standard deviation of zt is defined as

$$
\sigma\_{z\_\ell} = \sigma\_0 \sqrt{\frac{\lambda}{\left(2 - \lambda\right)} [1 - \left(1 - \lambda\right)^{2t}]},\tag{15}
$$

where σ0 is the standard deviation of the fault-free or preliminary data set. Therefore, in such cases, zt � N μ0;σ2 zt � �. However, in the presence of a mean shift at the time point 1 ≤ τ ≤ n, zt � N μ0 þ ½1 � ð1 � λÞ n�τþ1 �ðμ1 � μ0Þ;σ2 zt � �. The upper and lower control limits (UCL and LCL) of the EWMA chart for detecting a mean shift are UCL=LCL ¼μ0 � Lσzt , where L is a multiplier of the EWMA standard deviation σzt . The parameters L and λ need to be set carefully [33]. In practice, L is usually set to three, which corresponds to a false alarm rate of 0.27%. If zt is within the interval [LCL and UCL], then we conclude that the process is under control up to time point t. Otherwise, the process is considered out of control.

#### 4. Combining MSPLS model with EWMA chart: MSPLS-EWMA

In this chapter, we combine the advantages of MSPLS modeling with those of the univariate EWMA monitoring chart, which results in an improved fault detection system, especially for detecting small faults in highly correlated, multivariate data. Toward this end, we applied EWMA charts to the output residuals obtained from the MSPLS model (see Figure 6). Indeed, under normal operation with little noise and few errors, the residuals are close to zero, while they significantly deviate from zero in the presence of abnormal events. In this work, the output residuals from MSPLS are used as a fault indicator.

As given in Eq. (6), the output vector y can be written as the sum of a predicted vector y^ and a residual vector F, i.e.,

$$
\mathbf{y} = \hat{\mathbf{y}} + \mathbf{F}.\tag{16}
$$

The residual of the output variable, F ¼ f 1;…;f t ;…;f n , which is the difference between the observed value of the output variable, y, and the predicted value, y^, obtained from the MSPLS model, is a potential indicator for fault detection. The EWMA statistic based on the residuals of the response variable can be calculated as follows:

$$z\_t = \lambda f\_t + (1 - \lambda)z\_{t-1}t \in [1, n] \tag{17}$$

In this case, since the EWMA control scheme is applied on the residual data matrix, one EWMA decision function will be computed to monitor the process.

Figure 6. Principle of MSPLS-EWMA procedure.

### 5. Monitoring a simulated distillation column

4. Combining MSPLS model with EWMA chart: MSPLS-EWMA

output residuals from MSPLS are used as a fault indicator.

The residual of the output variable, F ¼ f 1;…;f t

the response variable can be calculated as follows:

Figure 6. Principle of MSPLS-EWMA procedure.

EWMA decision function will be computed to monitor the process.

residual vector F, i.e.,

186 Uncertainty Quantification and Model Calibration

In this chapter, we combine the advantages of MSPLS modeling with those of the univariate EWMA monitoring chart, which results in an improved fault detection system, especially for detecting small faults in highly correlated, multivariate data. Toward this end, we applied EWMA charts to the output residuals obtained from the MSPLS model (see Figure 6). Indeed, under normal operation with little noise and few errors, the residuals are close to zero, while they significantly deviate from zero in the presence of abnormal events. In this work, the

As given in Eq. (6), the output vector y can be written as the sum of a predicted vector y^ and a

observed value of the output variable, y, and the predicted value, y^, obtained from the MSPLS model, is a potential indicator for fault detection. The EWMA statistic based on the residuals of

In this case, since the EWMA control scheme is applied on the residual data matrix, one

;…;f n

y ¼ y^ þ F: ð16Þ

, which is the difference between the

zt ¼ λf t þ ð Þ 1 � λ zt�1t∈ ½1;n� ð17Þ

In this section, the ability of the proposed MSPLS-EWMA technique to detect faults is studied through simulation data and the results compared with those obtained using a traditional PLS-EWMA method. In all monitoring charts, the red-shaded area is the region where the fault is injected to the test data while the 95% control limits are plotted by the horizontal-dashed line.

#### 5.1. Description and data generation of the process

A distillation column is most commonly used unit operation in chemical process industries. The objective of the distillation operation is to separate the component from a mixture of component. The operation of distillation column is very energy expensive. Therefore, monitoring of such process plays very important role in bringing down the cost of the operation. The schematic diagram of the distillation column is shown in Figure 7.

The efficacy of the proposed fault detection strategy tested using simulated (using ASPEN simulation software [36]) distillation column. The input variables consist of temperature measurements at different location of the distillation column along with feed flow rate and reflux flow. The light distillate from reflux drum considered as the response variable. The operating conditions, nominal operating conditions, and detailed steps involved in the data generation can be found in Ref. [36]. The generated 1024 data samples are then corrupted with zero mean Gaussian white noise with signal-to-noise ratio (SNR) of 10 dB used for model development and testing the Fault detection (FD) strategy. Figure 8 shows dynamic data of the distillation column, i.e., variations of the light component for changes in the reflux and feed flow. The MSPLS model is developed from first 512 data samples and later part of the data points is used for testing purpose. The optimal LVs for the model are achieved through cross-validation methods and found to be three LVs for the MSPLS model.

A scatter plot of the measured and predicted data is presented in Figure 9. This plot indicates a reasonable performance of the selected models.

#### 5.2. Detection results

After a process model has been successfully identified, we can proceed with fault detection. Three types of faults in distillation columns will be considered here: abrupt, intermittent, and gradual faults.

To quantify the efficiency of the proposed strategies, we use two metrics: the false detection rate (FAR) and the miss detection rate (MDR) [37]. The FAR is the number of normal observations that are wrongly judged as faulty (false alarms) over the total number of fault-free data. The MDR is the number of faults that are wrongly classified as normal (missed detections) over the total number of faults.

#### 5.2.1. Case (A): abrupt fault detection

In this case study, an abrupt change is simulated by adding a small constant deviation which is 2% of the total variation in temperature Tc3, to the temperature sensor measurements Tc3,

Figure 7. Distillation column diagram.

between sample times 150 and 200. In the example, the testing data with low SNR, SNR ¼ 5, are generated for the purpose of evaluation of MSPLS-EWMA and PLS-Q monitoring performances. Results of the PLS-Q and MSPLS-Q statistics are demonstrated in Figure 10(a) and (c), respectively. It can be seen from Figure 10(a) and (c) that PLS-Q and MSPLS-Q cannot detect this small fault. Figure 10(b) shows that the PLS-EWMA chart is capable of detecting this simulated fault but with a lot missed detection (i.e., MDR ¼ 55% and FAR ¼ 0.96%). Figure 10(d) shows that although the MSPLS-EWMA chart clearly detected this abrupt faults without missed detection (i.e., MDR ¼ 0% and FAR ¼ 0.96%).

#### 5.2.2. Case (B): intermittent fault

In this case study, we introduce into the testing data a bias of amplitude 2% of the total variation in temperature Tc3 of between samples 50 and 100, and a bias of 10% between samples 350 and 450. Figure 11(a)–(d) shows the monitoring results of the PLS-based Q and EWMA charts, and MSPLS-based Q and EWMA charts. Figure 11(a) shows that the PLS-based

An Improved Wavelet‐Based Multivariable Fault Detection Scheme http://dx.doi.org/10.5772/intechopen.68947 189

Figure 8. Simulation of a distillation column: variation of input-output data with SNR ¼ 10 (Solid line: noise-free data; dots: noisy data).

between sample times 150 and 200. In the example, the testing data with low SNR, SNR ¼ 5, are generated for the purpose of evaluation of MSPLS-EWMA and PLS-Q monitoring performances. Results of the PLS-Q and MSPLS-Q statistics are demonstrated in Figure 10(a) and (c), respectively. It can be seen from Figure 10(a) and (c) that PLS-Q and MSPLS-Q cannot detect this small fault. Figure 10(b) shows that the PLS-EWMA chart is capable of detecting this simulated fault but with a lot missed detection (i.e., MDR ¼ 55% and FAR ¼ 0.96%). Figure 10(d) shows that although the MSPLS-EWMA chart clearly detected this abrupt faults without missed

In this case study, we introduce into the testing data a bias of amplitude 2% of the total variation in temperature Tc3 of between samples 50 and 100, and a bias of 10% between samples 350 and 450. Figure 11(a)–(d) shows the monitoring results of the PLS-based Q and EWMA charts, and MSPLS-based Q and EWMA charts. Figure 11(a) shows that the PLS-based

detection (i.e., MDR ¼ 0% and FAR ¼ 0.96%).

5.2.2. Case (B): intermittent fault

Figure 7. Distillation column diagram.

188 Uncertainty Quantification and Model Calibration

Q chart has no power to detect this fault. From Figure 11(b), it can be seen that the MSPLS-Q chart can detect the intermittent faults but with several missed detections. Figure 11(c) shows that the PLS-EWMA chart can indeed detect this fault, but with some missed detections. On the other hand, the MSPLS-EWMA chart with λ ¼ 0:3 correctly detects this intermittent fault (see Figure 11(d)). In this case study, we can see that detection performance is much enhanced when using the MSPLS-EWMA chart compared to the others.

Figure 9. Scatter plots of predicted and observed training data.

Figure 10. Monitoring results of PLS-Q chart (a), PLS-EWMA chart (b), MSPLS-Q chart (c), and MSPLS-EWMA chart (d) in the presence of a bias anomaly in the temperature sensor measurements 'Tc3' with SNR ¼ 30, Case (A).

#### 5.2.3. Case (C): drift failure detection

A slow drift fault is simulated by adding a ramp change with a slope of 0.01 to the temperature sensor, Tc3, from sample 250 through the end of the testing data. Monitoring results of PLS and

An Improved Wavelet‐Based Multivariable Fault Detection Scheme http://dx.doi.org/10.5772/intechopen.68947 191

Figure 11. Monitoring results of PLS-Q chart (a), PLS-EWMA chart (b), MSPLS-Q chart (c), and MSPLS-EWMA chart (d) in the presence intermittent sensor fault in 'Tc3' with SNR ¼ 30, Case (B).

MSPLS-based Q and EWMA statistics are shown in Figure 12(a)–(d). Figure 12(a) shows the monitoring results of PLS-Q chart, in which we can see that a signal is first given at sample 313 with a significant false alarm rate (i.e., FAR ¼ 22.4%). Figure 12(b) shows that the PLS-EWMA chart first detects the fault at the 290th observation. The MSPLS-Q chart is shown in Figure 12(c), which first flags the fault at sample 323. Figure 12(d) shows that the MSPLS-EWMA chart first detects the fault at the 288th observation. Therefore, a fewer observations are needed for the MSPLS-EWMA chart to detect a fault compared to the other charts.

This case study testifies again to the superiority of the proposed approaches compared to conventional PLS-based fault detection. Of course, this chapter also demonstrates through

Figure 12. Monitoring results of PLS-Q chart (a), PLS-EWMA chart (b), MSPLS-Q chart (c), and MSPLS-EWMA chart (d) in the presence drift sensor anomaly in 'Tc3' with SNR ¼ 30, Case (C).

5.2.3. Case (C): drift failure detection

Figure 9. Scatter plots of predicted and observed training data.

190 Uncertainty Quantification and Model Calibration

A slow drift fault is simulated by adding a ramp change with a slope of 0.01 to the temperature sensor, Tc3, from sample 250 through the end of the testing data. Monitoring results of PLS and

Figure 10. Monitoring results of PLS-Q chart (a), PLS-EWMA chart (b), MSPLS-Q chart (c), and MSPLS-EWMA chart (d)

in the presence of a bias anomaly in the temperature sensor measurements 'Tc3' with SNR ¼ 30, Case (A).

simulated data that significant improvement in fault detection can be obtained by using the MSPLS model when combined with the EWMA chart.

#### 6. Conclusion

The objective of this chapter is to extend the PLS fault-detection methods to deal with uncertainty in the measurements. The developed approach merges the flexibility of multiscale PLS model and the greater sensitivity of the EWMA control chart to incipient changes. Specifically, in this approach, the multiscale PLS model has been constructed using the wavelet coefficients at different scales, and then EWMA monitoring chart was applied using this model to improve the fault detection abilities of this PLS fault detection method even further. Using a simulated distillation column, we demonstrate the effectiveness of MSPLS-EWMA to detect abrupt and drift faults. Results show that the MSPLS-EWMA can achieve better fault-detection efficiency than the PLS-EWMA, PLS-Q, and MSPLS-Q monitoring approaches.

#### Acknowledgements

This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No: OSR-2015- CRG4-2582.

#### Author details

Fouzi Harrou1 \*, Ying Sun1 and Muddu Madakyaru2

\*Address all correspondence to<EMAIL_ADDRESS>
1 Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

2 Department of Chemical Engineering, Manipal Institute of Technology, Manipal University, Manipal, India

#### References

[3] Ralston P, DePuy G, Graham J. Computer-based monitoring and fault diagnosis: A chemical process case study. ISA Transactions. 2001;40(1):85–98

simulated data that significant improvement in fault detection can be obtained by using the

The objective of this chapter is to extend the PLS fault-detection methods to deal with uncertainty in the measurements. The developed approach merges the flexibility of multiscale PLS model and the greater sensitivity of the EWMA control chart to incipient changes. Specifically, in this approach, the multiscale PLS model has been constructed using the wavelet coefficients at different scales, and then EWMA monitoring chart was applied using this model to improve the fault detection abilities of this PLS fault detection method even further. Using a simulated distillation column, we demonstrate the effectiveness of MSPLS-EWMA to detect abrupt and drift faults. Results show that the MSPLS-EWMA can achieve better fault-detection efficiency

This publication is based upon work supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No: OSR-2015-

1 Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King

2 Department of Chemical Engineering, Manipal Institute of Technology, Manipal University,

[1] Aldrich C, Auret L. Unsupervised process monitoring and fault diagnosis with machine

[2] Isermann R. Model-based fault-detection and diagnosis: Status and applications. Annual

Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia

MSPLS model when combined with the EWMA chart.

192 Uncertainty Quantification and Model Calibration

than the PLS-EWMA, PLS-Q, and MSPLS-Q monitoring approaches.

\*, Ying Sun1 and Muddu Madakyaru2

\*Address all correspondence to<EMAIL_ADDRESS>
learning methods. London: Springer; 2013

Reviews in Control. 2005;29:71–85

6. Conclusion

Acknowledgements

CRG4-2582.

Author details

Fouzi Harrou1

Manipal, India

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### **Practical Considerations on Indirect Calibration in Analytical Chemistry**

Antonio Gustavo González

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.68806

#### Abstract

Indirect or methodological calibration in chemical analysis is outlined. The establishment of calibration curves is introduced and discussed. Linear calibration is presented and considered in three scenarios commonly faced in chemical analysis: external calibration (EC) when there are no matrix effects in the sample analysis; standard addition calibration (SAC) when these effects are present and internal standard calibration (ISC) in cases of intrinsic variability of the analytical signal or possible losses of the analyte in stages prior to the measurement. In each kind of calibration, the uncertainty and confidence interval for the determined analyte concentration are given.

Keywords: external calibration, standard addition method, internal standard, uncertainty measurement

#### 1. Introduction

Direct absolute methods such as gravimetry, titrimetry or coulometry (among others) are directly traceable to SI units. Thus, traceability of contemporary instrumental methods is accomplished by applying indirect calibration procedures. In a direct calibration, the value of the standard (reference value) is expressed in the same quantity as the measurement of the equipment (for instance, the calibration of an analytical balance). In an indirect calibration, the value of the standard is expressed in a quantity different from the output one, that is, the measurement and the measurand are different. This is the most common kind of calibration in chemical analysis, for example, the calibration of a spectrophotometric method. Accordingly, the indirect calibration in analytical chemistry, also known as methodological calibration, is the operation that determines the functional relationship between measured values and analytical

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

quantities, characterizing types of analytes, and their amounts. In this chapter, the establishment and validation of the mathematical model for the calibration function will be studied and discussed as well as the habitual scenarios concerning interferences coming from the chemical environment (matrix effects) and physical/instrumental lack of control leading to signal modification (standard additions and internal standard methodology). Confidence intervals for the calculated analyte concentration will be outlined and discussed.

#### 2. The calibration in analytical chemistry

Calibration, as previously defined, can be assimilated to a mathematical function, Y ¼ f(x), where Y is the analytical signal or response corresponding to the analyte concentration x. The major analytical aim consists of finding this functionality. When applying absolute methods of analysis [1], where traceability is assured, such as gravimetry, titrimetry or coulometry, there is no need for indirect calibration. The analyte amount is evaluated from the analytical signal with the use of physicochemical constants (atomic mass, Faraday constant) and the concentration of the standardized titration solution in titrimetry, leading to a typical linear response model x ¼ KY:

• Coulometry: x ¼ Yðtotal chargeÞ nðelectrons transferredÞFðFaraday constantÞ

But in the field of relative methods (the majority of instrumental ones), traceability is reached just by performing an indirect calibration, that is by establishing the relationship between the analyte concentration and the analytical response. There are some theoretical relationships [2–5] verified for special analytical techniques as depicted in Table 1.

Nevertheless, in the common situations, the response function has to be empirically established by using standard analyte solutions. Many response functions exhibit linear zones, generally at low concentrations of analyte and other zones where a curvature appears, and in some cases,

Table 1. Theoretical response functions used in some analytical instrumental techniques.

i

i

there are regions where the response signal is independent of the analyte concentration [6]. Analysts are interested to the portion of the response function where the variation of the analytical signal with the analyte concentration contains useful analytical information. This portion of response function with analytical interest for calibration purposes is called the calibration curve. From the calibration curve, the amount of analyte in an unknown sample is evaluated from interpolation. The calibration step is of utmost importance within the realm of method validation.

quantities, characterizing types of analytes, and their amounts. In this chapter, the establishment and validation of the mathematical model for the calibration function will be studied and discussed as well as the habitual scenarios concerning interferences coming from the chemical environment (matrix effects) and physical/instrumental lack of control leading to signal modification (standard additions and internal standard methodology). Confidence intervals for the

Calibration, as previously defined, can be assimilated to a mathematical function, Y ¼ f(x), where Y is the analytical signal or response corresponding to the analyte concentration x. The major analytical aim consists of finding this functionality. When applying absolute methods of analysis [1], where traceability is assured, such as gravimetry, titrimetry or coulometry, there is no need for indirect calibration. The analyte amount is evaluated from the analytical signal with the use of physicochemical constants (atomic mass, Faraday constant) and the concentration of the standardized titration solution in titrimetry, leading to a typical linear response

calculated analyte concentration will be outlined and discussed.

• Gravimetry: x ¼ Gðgravimetric factorÞ � Yðmass of weighing formÞ

• Titrimetry: x ¼ pðstoichiometryÞ � Cðtitrant concentrationÞ � Yðtitrant volumeÞ

nðelectrons transferredÞFðFaraday constantÞ

But in the field of relative methods (the majority of instrumental ones), traceability is reached just by performing an indirect calibration, that is by establishing the relationship between the analyte concentration and the analytical response. There are some theoretical relationships [2–5]

Nevertheless, in the common situations, the response function has to be empirically established by using standard analyte solutions. Many response functions exhibit linear zones, generally at low concentrations of analyte and other zones where a curvature appears, and in some cases,

Response function Reference Analytical technique y ¼ A þ Bx Beer's law Absorption spectroscopy

B þ D Rodbard four parameter Logistic equation [5] Immunoassay

Table 1. Theoretical response functions used in some analytical instrumental techniques.

y ¼ AxB Scheibe-Lomaking [2] Atomic emission spectrometry y ¼ A þ Bx þ Cx2 Wagenaar et al. [3] Atomic absorption spectrometry

y ¼ A þ Blogx Nernst's equation Electrochemistry

2. The calibration in analytical chemistry

198 Uncertainty Quantification and Model Calibration

• Coulometry: x ¼ Yðtotal chargeÞ

y ¼ A þ B½1 � e�Cx� Andrews et al. [4]

y ¼ A�D 1þ x ð ÞB

verified for special analytical techniques as depicted in Table 1.

model x ¼ KY:

In many situations, the calibration curve is linear, and a calibration straight line is obtained. From the mathematical models applied for establishing the response function, the most straightforward, studied, and easy to handle is the linear one. Accordingly, the linear calibration model will be considered throughout this chapter.

In case of non-linear response, there are several alternatives. The use of linearizing transformations is a common tool [7], but when this procedure does not work, curve-fitting methods are chosen. The best procedure is to try with polynomials of degrees successively larger until the F-test of residual variances indicates that the systematic error due to the lack of fit is negligible. If the plot has "N" points, the major degree polynomial to be used is of degree N�1. But the blind use of high-order polynomial may lead to overfitting. This kind of fitting is solved by multilinear regression [8]. This technique sometimes fails because the coefficient matrix is nearly singular. To avoid this, we can use orthogonal polynomials. The use of these polynomials leads to a diagonal coefficient matrix, overcoming singularities, and simplifying calculations. The orthogonal polynomials commonly used in curve fittings are the Chebischev's polynomials [9] and the Forsythe ones [10].

Aside from the advantages and applications of orthogonal polynomials, they are not at all the ultimate weapon. Rice proposed rational polynomial functions of the type FðxÞ ¼ X aixi .X bixi

that present a higher flexibility than orthogonal polynomials for adjusting purposes [11]. Another approach is to fit the points to a curve consisting of several linked sections of different geometrical shapes. This is the basis of the spline functions. Cubic spline [12] is the most used. They approximate the data to a series of cubic equations. These cubic links overlap in p interpolation points called "knots," and it is essential that splines show continuity at such points. This continuity applies to the spline function and its first derivatives. A total cubic spline has p�1 links, with four coefficients (S ¼ a þ bx þ cx2 þ dx3). Thus, 4(p�1) coefficients have to be calculated. This technique has been successfully applied in radioimmunoassay, gas– liquid chromatography, and atomic absorption spectrometry [13].

The most usual technique for establishing a calibration straight line is the method of least squares. This consists of minimizing the function Q ¼ X Yi � Y^ i � �2 where Yi is the observed value of the response function at a xi analyte concentration, and Y^ i is the estimated response value according to the linear model Y ¼ a þ bx þ εðYÞ or Y^ ¼ a þ bx. The minimization ∂Q ∂a ¼ 0 and ∂Q ∂b ¼ 0 leads to the values of a, b as well as their variances and covariance [13].

Three main requisites must be fulfilled before using this method [14], namely:

In analytical calibrations, the analyte concentration is known with high accuracy and precision and, accordingly, the requirement (i) is accomplished. The condition (ii) is assumed by many researchers without a previous testing. There are several statistical assays for testing normality [13], and they should be performed before embarking in the fitting. Analysts have paid much more attention to the requirement (iii). In situations of heteroscedasticity (non-constant variance), the method of least squares can be applied but by using the so-called weighing factors [15], which are defined as wi ¼ 1 σ2ðYiÞ . Thus, the function to be minimized now is Q ¼ Xwi Yi � Y^ i � �2 leading to expressions similar to the one obtained in simple linear regression. This is the weighted regression [13].

Let us assume that we deal with a situation often found in routine analysis where the three mentioned requirements are fulfilled. In the following, we consider the different scenarios we can face.

#### 3. Metrological foundations on indirect calibration

Consider a new proposed analytical method which is applied to dissolved test portions of a given sample within the linear dynamic range of the linear analytical response (Y). This response may be expressed by the following linear relationship involving both analyte and matrix amounts [16]:

$$
\hat{Y} = A + B\mathbf{x} + \mathbf{C}z + \mathbf{D}\mathbf{x}z \tag{1}
$$

where Y^ is the estimated analytical response and A, B, C and D are constants.

A is a constant that does not change when the concentrations of the matrix, z, and/or the analyte, x, change. It is obviously related to the constant error blank correction. The blank must account for signals coming from reagents and solvents used in the assay as well as any bias resulting from interactions between the analyte and the sample's matrix. It is well known that the calibration blank and the reagent blank compensate for signals from reagents and solvents, but neither of them can correct for a bias resulting from an interaction between the analyte and the sample's matrix. The suitable blank must include both the sample's matrix and the analyte, and so it must be determined using the sample itself. The term A is called the true sample blank and can be estimated from the Youden sample plot, which is defined as the "sample response curve" [17]. Thus, by applying the selected analytical method to different test portions, namely m (a different mass taken from the test sample), different analytical responses Y are obtained. The plot of Y versus m is the Youden sample plot, and the intercept of the corresponding regression line is the so-called total Youden blank (TYB) which is the true

sample blank [17–19]. However, when a "matrix without analyte" is available, the term A can be determined by evaluating the system blank (calibration and reagent blank).

Bx is the essential term that justifies the analytical method because it directly deals with the sensitivity to the presence of analyte.

Cz refers to the signal contribution from the matrix, depending only on its amount, z. When this term occurs, the matrix is called interferent. This contribution must be absent, because a validated analytical method should be selective enough with respect to the potential interferences appearing in the samples where the analyte is determined. Accordingly, the majority of validated methods do not suffer from such a direct matrix interference.

Dxz is an interaction analyte/matrix term. This matrix effect occurs when the sensitivity of the instrument to the analyte is dependent on the presence of the other species (the matrix) in the sample [20]. For the sake of determining analytes, this effect may be overcome by using the method of standard additions as we consider later.

Thus, the calibration function remains as:

• The x variable is free from error εðxÞ ¼ 0.

200 Uncertainty Quantification and Model Calibration

factors [15], which are defined as wi ¼ 1

3. Metrological foundations on indirect calibration

sion. This is the weighted regression [13].

dasticity).

Q ¼ Xwi Yi � Y^ i

matrix amounts [16]:

can face.

• The error associated to Y variable, εðYÞ, is normally distributed, N(0,σ2

• The variance of response Y, σ2ðYÞ, remains uniform in the dynamic range of x (homosce-

In analytical calibrations, the analyte concentration is known with high accuracy and precision and, accordingly, the requirement (i) is accomplished. The condition (ii) is assumed by many researchers without a previous testing. There are several statistical assays for testing normality [13], and they should be performed before embarking in the fitting. Analysts have paid much more attention to the requirement (iii). In situations of heteroscedasticity (non-constant variance), the method of least squares can be applied but by using the so-called weighing

σ2ðYiÞ

Let us assume that we deal with a situation often found in routine analysis where the three mentioned requirements are fulfilled. In the following, we consider the different scenarios we

Consider a new proposed analytical method which is applied to dissolved test portions of a given sample within the linear dynamic range of the linear analytical response (Y). This response may be expressed by the following linear relationship involving both analyte and

A is a constant that does not change when the concentrations of the matrix, z, and/or the analyte, x, change. It is obviously related to the constant error blank correction. The blank must account for signals coming from reagents and solvents used in the assay as well as any bias resulting from interactions between the analyte and the sample's matrix. It is well known that the calibration blank and the reagent blank compensate for signals from reagents and solvents, but neither of them can correct for a bias resulting from an interaction between the analyte and the sample's matrix. The suitable blank must include both the sample's matrix and the analyte, and so it must be determined using the sample itself. The term A is called the true sample blank and can be estimated from the Youden sample plot, which is defined as the "sample response curve" [17]. Thus, by applying the selected analytical method to different test portions, namely m (a different mass taken from the test sample), different analytical responses Y are obtained. The plot of Y versus m is the Youden sample plot, and the intercept of the corresponding regression line is the so-called total Youden blank (TYB) which is the true

where Y^ is the estimated analytical response and A, B, C and D are constants.

� �2 leading to expressions similar to the one obtained in simple linear regres-

).

. Thus, the function to be minimized now is

Y^ ¼ A þ Bx þ Cz þ Dxz ð1Þ

$$
\hat{Y} = A + B\mathbf{x} + D\mathbf{x} \mathbf{z} \tag{2}
$$

This function has to be established by using standards and could be applied to samples according to different methodologies. Calibration standards are prepared from primary standards containing the analyte or a surrogate, that is, a pure substance equivalent to analyte in chemical composition, separation and measuring that is taken as representative of the native analyte. It must be absent in the sample. Commonly, a surrogate is used in an internal methodology and in this case is termed as internal standard (IS) [21].

Three different scenarios can be considered for establishing the calibration function in order to determine the analyte in the sample: the external calibration (EC) (applicable when there is no matrix effect); the standard addition calibration (SAC) (used when matrix effect is present); and the internal calibration (IC) (applied for compensate uncontrolled analytical signal variations). These methodologies are outlined in the following section.

#### 4. The external calibration

The external calibration (EC) is the most commonly used calibration methodology. It is named so because the calibration standards are not made up of the sample test portion. Instead, they are prepared and analysed separately from samples [21]. Accordingly, the signals recorded accounts for the analyte added as primary standard, reagents, solvents and other agents according to the analytical procedure, except the sample matrix. Accordingly, because EC is established in a free matrix environment, it can be applied for analyte determination only when sample matrix effects are absent. Thus, as a preliminary step within the method, validation to assess constant and proportional bias due to matrix effects has to be performed [22]. Being a matrix free calibration scenario, z ¼ 0, B is the slope of EC, bEC, and A can be taken as the system blank, aEC. In order to evaluate the goodness of the fit, the regression analysis of the analytical signal on the analyte concentrations established in the calibration set yields the calibration curve for the predicted responses. The simplest model is the linear one, very often found in analytical methodology, leading to predicted responses according to

$$
\ddot{Y} = a\_{\text{EC}} + b\_{\text{EC}} \mathbf{x} \tag{3}
$$

Eq. (3) must be checked for goodness of fit.

The correlation coefficient r ¼ Xðxi � xÞðYi�YÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xðxi � xÞ 2 XðYi � YÞ 2 q , although commonly used, espe-

cially in linear models, is not appropriate owing to the little value of this parameter for detecting curvature [23, 24]. In statistical theory, correlation is the measure of the association between two random variables, but in our case, x and Yare strongly related. Thus, there is no correlation in its mathematical sense. Values of r near þ1 or �1 provide an environment of respectability but not much else. Some authors apply statistical tests for significance of the correlation coefficient, for instance, the student t-test [13] t ¼ jrj ffiffiffiffiffiffiffiffiffiffiffiffi N � 2 p ffiffiffiffiffiffiffiffiffiffiffiffi 1 � r2 p or the Fisher transformation [9] z ¼ 1 2ln 1 þ r 1 � r � �, but they cannot ward off danger because the null hypothesis is that the variables are uncorrelated (zero correlation), and accordingly, a small r value can be considered significantly different from r ¼ 0. As Thompson [23] pointed out, "certainly it is true that, if the calibration points are tightly clustered around a straight line, the experimental value of r will be close to unity. But the converse is not true". Thus, some more suitable criteria should be considered. A very simple way to prove that the linear model suitably fits the experimental data and is right for searching possible calibration pathologies is the analysis of residuals [8, 13, 25]. So, if the model is suitable, the residuals should be normally distributed. This can be assessed by plotting them on a normal probability graph. The presence of curvature reveals a lack of fit due to non-linear behaviour. A residual segmented pattern may indicate heteroscedasticity in the data, and a weighted linear regression could be used.

Another parameter measuring the goodness of the fit is called the on-line linearity [26] and is a parameter that measures the dispersion of the points around the calibration straight line and is evaluated as the relative standard deviation of its slope: on-line linearity ¼ RSDbEC ¼ sbEC bEC . The typical critical threshold for considering a suitable linear model is RSDb ≤ 0:05.

Nevertheless, the best way to test the goodness of fit is by comparing the variance of the lack of fit against the pure error variance [27]. For an adequate assess of the lack of fit of the linear model, a suitable experimental design for performing the calibration is needed as indicated in the following [28]:

#### iii. Each calibration standard should be measured in triplicate to account for pure error variance.

From these data, we can test homoscedasticity. Accordingly, we have a triplicate of responses for each calibration standard and hence an estimation of the pure error variance of the response is available at each calibration point. We can apply the Cochran's assay because the number of observations is same for all concentration levels of analyte. Thus, if the number of calibration standards is N and they are replicated n times, the Cochran statistics is calculated as:

$$C = \frac{s\_{\text{max}}^2}{\sum\_{i=1}^N s\_i^2} \tag{4}$$

where s2 i is the response variance at the concentration level i and s2 max is the maximum variance. This value is compared against the critical tabulated value CtabðN, n, PÞ, P being the selected confidence level. If C≤ Ctab, then the response variances can be considered to be uniform across the range of analyte concentrations, and an estimated pooled sum of squares due to pure errors, SSPE, can be obtained:

$$SS\_{PE} = \sum\_{i=1}^{N} \sum\_{j=1}^{n} \left( Y\_{ij} - \overline{Y}\_i \right)^2 = \frac{n-1}{N} \sum\_{i=1}^{N} s\_i^2 \tag{5}$$

The residual sum of squares of the model SSR is given by

the system blank, aEC. In order to evaluate the goodness of the fit, the regression analysis of the analytical signal on the analyte concentrations established in the calibration set yields the calibration curve for the predicted responses. The simplest model is the linear one, very often

> Xðxi � xÞðYi�YÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2

cially in linear models, is not appropriate owing to the little value of this parameter for detecting curvature [23, 24]. In statistical theory, correlation is the measure of the association between two random variables, but in our case, x and Yare strongly related. Thus, there is no correlation in its mathematical sense. Values of r near þ1 or �1 provide an environment of respectability but not much else. Some authors apply statistical tests for significance of the

the variables are uncorrelated (zero correlation), and accordingly, a small r value can be considered significantly different from r ¼ 0. As Thompson [23] pointed out, "certainly it is true that, if the calibration points are tightly clustered around a straight line, the experimental value of r will be close to unity. But the converse is not true". Thus, some more suitable criteria should be considered. A very simple way to prove that the linear model suitably fits the experimental data and is right for searching possible calibration pathologies is the analysis of residuals [8, 13, 25]. So, if the model is suitable, the residuals should be normally distributed. This can be assessed by plotting them on a normal probability graph. The presence of curvature reveals a lack of fit due to non-linear behaviour. A residual segmented pattern may indicate heteroscedasticity in the data, and a weighted linear regression could

Another parameter measuring the goodness of the fit is called the on-line linearity [26] and is a parameter that measures the dispersion of the points around the calibration straight line and is evaluated as the relative standard deviation of its slope: on-line linearity ¼ RSDbEC ¼ sbEC

Nevertheless, the best way to test the goodness of fit is by comparing the variance of the lack of fit against the pure error variance [27]. For an adequate assess of the lack of fit of the linear model, a suitable experimental design for performing the calibration is needed as indicated in

i. At least six calibration points spaced over the concentration range of the method scope

ii. Calibration standards should be measured over 5 days for suitably covering the possible

typical critical threshold for considering a suitable linear model is RSDb ≤ 0:05.

are required for establishing the calibration straight line.

XðYi � YÞ

Y^ ¼ aEC þ bECx ð3Þ

N � 2 p ffiffiffiffiffiffiffiffiffiffiffiffi

1 � r2 p or the Fisher transfor-

bEC . The

2 q , although commonly used, espe-

, but they cannot ward off danger because the null hypothesis is that

found in analytical methodology, leading to predicted responses according to

Xðxi � xÞ

correlation coefficient, for instance, the student t-test [13] t ¼ jrj ffiffiffiffiffiffiffiffiffiffiffiffi

Eq. (3) must be checked for goodness of fit.

The correlation coefficient r ¼

202 Uncertainty Quantification and Model Calibration

2ln 1 þ r 1 � r � �

mation [9] z ¼ 1

be used.

the following [28]:

sources of uncertainty.

$$SS\_{\mathbb{R}} = \sum\_{i=1}^{N} \sum\_{j=1}^{n} \left( Y\_{i\circ} - \hat{Y}\_{\circ i} \right)^2 = \sum\_{i=1}^{N} n \left( \overline{Y}\_i - \hat{Y}\_i \right)^2 \tag{6}$$

where Yij is the recorded analytical signal of the calibration point i at the replication j and Y^ ij. This value can be split into two terms: the sum of squares corresponding to pure error (SSPE) and the sum of squares corresponding to the lack of fit (SSLOF):

$$\text{SS}\_{\text{LOF}} = \text{SS}\_{\text{R}} - \text{SS}\_{\text{PE}} \tag{7}$$

The pure error variance is SSPE/(n�1), and the variance of the lack of fit, by considering N�2 degrees of freedom for SSR, is SSLOF/(N�n�1). So, for assessing the adequacy of the model, the Fisher F-test is applied:

$$F = \frac{(\mathcal{SS}\_{\mathbb{R}} - \mathcal{SS}\_{\text{PE}})/(N - n - 1)}{\mathcal{SS}\_{\text{PE}}/(n - 1)}\tag{8}$$

The calibration model is considered suitable if less than the one-tailed tabulated value FtabðN � n � 1, n � 1, PÞ exists at a P given confidence level.

Once the model is adequate for application, analyte determination is carried out by interpolating the analytical signal of the sample in the calibration model. Typical statistical calculations for evaluating the variances of slope, intercept, its covariance as well as the uncertainty associated to the estimated analyte concentration can be found in several texts, for instance, Miller and Miller [13]. Thus, if Y0 is the response signal recorded by applying the analytical method on the sample, the concentration of native analyte, x0, is given by

$$
\hat{x}\_0 = \frac{Y\_0 - a\_{EC}}{b\_{EC}} \tag{9}
$$

In order to evaluate its standard deviation, and the corresponding expanded uncertainty, the theorem of variance propagation is applied. The propagation of variance is the common approach for evaluating the uncertainty of indirect measurements according to the current edition of the guide for the expression of uncertainty measurement (GUM). However, an essential limitation has to be taken into account. The non-linearity of the function (here the calibration function) must be negligible. This is fundamental because the function is expanded in a Taylor series, and then, it is truncated by neglecting second- and higherorder terms. To avoid this drawback, the propagation of distributions instead of the propagation of variance is a very suitable way for estimating the measurement uncertainty. The application of Monte-Carlo method to carry out the propagation of distributions is very effective [29].

Saying that brute-force Monte-Carlo (MC) methods are "very effective" may seem strange to some readers, as one major problem of MC is their methodological in-efficiency. It is due to large sampling variance of the relatively small samples acceptable in computationally demanding applications. In other words, any acceptable sample of 100 values may have a large random unknown error, generally different from any other sample of comparable size. To overcome this inefficiency, approximate simplified surrogate models are often used to allow for sampling a much as 106 times, just to reduce sampling variability. I would thus rather call MC methods 'general', 'useful', 'simple' and 'powerful' etc., as they apply to any parametric model and any distribution (if a random generator can be found), and can be utilized by anybody with very little statistical training.

But in our case, where the calibration function has been considered linear, the use of theorem of variance propagation can be applied without risks:

$$\begin{split} s\_{x\_0}^2 &= \left(\frac{\partial \mathbf{x}\_0}{\partial Y\_0}\right)^2 s\_{Y\_0}^2 + \left(\frac{\partial \mathbf{x}\_0}{\partial \mathbf{a}\_{\mathrm{EC}}}\right)^2 s\_{a\mathrm{EC}}^2 + \left(\frac{\partial \mathbf{x}\_0}{\partial b\_{\mathrm{EC}}}\right)^2 s\_{b\mathrm{EC}}^2 + 2\left(\frac{\partial \mathbf{x}\_0}{\partial \mathbf{a}\_{\mathrm{EC}}}\right)\left(\frac{\partial \mathbf{x}\_0}{\partial b\_{\mathrm{EC}}}\right) \mathrm{cov}(a\_{\mathrm{EC}}, b\_{\mathrm{EC}}) \\ &= \left(\frac{1}{b\_{\mathrm{EC}}}\right)^2 s\_{Y\_0}^2 + \left(-\frac{1}{b\_{\mathrm{EC}}}\right)^2 s\_{a\mathrm{EC}}^2 + \left(-\frac{Y\_0 - a\_{\mathrm{EC}}}{b\_{\mathrm{EC}}^2}\right)^2 s\_{Y\_0}^2 + 2\left(-\frac{1}{b\_{\mathrm{EC}}}\right)\left(-\frac{Y\_0 - a\_{\mathrm{EC}}}{b\_{\mathrm{EC}}^2}\right)\left(-\overline{\mathbf{x}}\_{\mathrm{b\_{\mathrm{EC}}}}^2\right) \end{split} (10)$$

Considering the following equivalences (see Ref. [13] for instance):

#### Practical Considerations on Indirect Calibration in Analytical Chemistry http://dx.doi.org/10.5772/intechopen.68806 205

$$\begin{aligned} s\_{Y\_0}^2 &= s\_R^2 = SS\_R/(N-2) \\ s\_{a\_{\rm{loc}}}^2 &= s\_R^2 \left(\frac{1}{N} + \frac{\overline{\pi}^2}{S\_{\rm{xx}}}\right) \\ s\_{b\_{\rm{loc}}}^2 &= \frac{s\_R^2}{S\_{\rm{xx}}} \\ S\_{\rm{xx}} &= \sum\_i \left(\mathbf{x}\_i - \overline{\mathbf{x}}\right)^2 \end{aligned} \tag{11}$$

After some algebraical manipulations, we get

Once the model is adequate for application, analyte determination is carried out by interpolating the analytical signal of the sample in the calibration model. Typical statistical calculations for evaluating the variances of slope, intercept, its covariance as well as the uncertainty associated to the estimated analyte concentration can be found in several texts, for instance, Miller and Miller [13]. Thus, if Y0 is the response signal recorded by applying the analytical method on the sample, the concentration of native analyte, x0, is

> x^0 ¼ Y0 � aEC bEC

In order to evaluate its standard deviation, and the corresponding expanded uncertainty, the theorem of variance propagation is applied. The propagation of variance is the common approach for evaluating the uncertainty of indirect measurements according to the current edition of the guide for the expression of uncertainty measurement (GUM). However, an essential limitation has to be taken into account. The non-linearity of the function (here the calibration function) must be negligible. This is fundamental because the function is expanded in a Taylor series, and then, it is truncated by neglecting second- and higherorder terms. To avoid this drawback, the propagation of distributions instead of the propagation of variance is a very suitable way for estimating the measurement uncertainty. The application of Monte-Carlo method to carry out the propagation of distributions is very

Saying that brute-force Monte-Carlo (MC) methods are "very effective" may seem strange to some readers, as one major problem of MC is their methodological in-efficiency. It is due to large sampling variance of the relatively small samples acceptable in computationally demanding applications. In other words, any acceptable sample of 100 values may have a large random unknown error, generally different from any other sample of comparable size. To overcome this inefficiency, approximate simplified surrogate models are often used to allow for sampling a much as 106 times, just to reduce sampling variability. I would thus rather call MC methods 'general', 'useful', 'simple' and 'powerful' etc., as they apply to any parametric model and any distribution (if a random generator can be found),

But in our case, where the calibration function has been considered linear, the use of theorem

bEC þ 2 ∂x0 ∂aEC � � ∂x0

s2

Y0 þ 2 � 1

∂bEC � �

bEC � � covðaEC, bECÞ

� Y0 � aEC b2 EC

�xs2 bEC � � ð10Þ

!

s2

Y0 � aEC b2 EC

!2

and can be utilized by anybody with very little statistical training.

∂x0 ∂bEC � �2

of variance propagation can be applied without risks:

s2 aEC þ

s2 aEC þ �

Considering the following equivalences (see Ref. [13] for instance):

∂x0 ∂aEC � �2

1 bEC � �2

s2 Y0 þ

s2 Y0 þ � ð9Þ

given by

204 Uncertainty Quantification and Model Calibration

effective [29].

s2 x0 ¼ ∂x0 ∂Y0 � �2

> ¼ 1 bEC � �2

$$s\_{\mathbf{x}\_0}^2 = \frac{s\_\mathcal{R}^2}{b\_{\mathrm{EC}}^2} \left[ 1 + \frac{1}{N} + \frac{\left( Y\_0 - \overline{Y} \right)^2}{b\_{\mathrm{EC}}^2 S\_{\mathrm{xx}}} \right] \tag{12}$$

If the signal Y0 is obtained as the average of m measurements, we have

$$s\_{x\_0}^2 = \frac{s\_R^2}{b\_{EC}^2} \left[ \frac{1}{m} + \frac{1}{N} + \frac{\left(Y\_0 - \overline{Y}\right)^2}{b\_{EC}^2 S\_{xx}} \right] \tag{13}$$

And the corresponding expanded uncertainty can be evaluated from the tabulated Student tstatistics or by assuming a Gaussian distribution and using the z score at a given confidence level (generally P ¼ 95%) and so:

$$\begin{aligned} \mathbf{U}\_{\mathbf{x}\_{0}} &= t\_{\text{table}} (\mathbf{N} - \mathbf{2}, \mathbf{95\%}) \mathbf{s}\_{\mathbf{x}\_{0}} \\ \mathbf{U}\_{\mathbf{x}\_{0}} &= \mathbf{z}\_{\mathbf{95\%}} \mathbf{s}\_{\mathbf{x}\_{0}} \approx \mathbf{2} \mathbf{s}\_{\mathbf{x}\_{0}} \\ \mathbf{C} &= t\_{\text{eq}} t\_{\text{1}} \mathbf{n} \mathbf{I} \mathbf{1} \quad \text{and} \quad \mathbf{1} \quad \text{and} \quad \mathbf{1} \end{aligned} \tag{14}$$

Confidence Interval : x0 � Ux0

EC is adequate for analytical procedures that could be considered as methods free from matrix effects, but it has the main limitation coming from the assumption that the different environments (matrices) of the calibration standards (solvent, buffer,…) and of the samples are equivalent, and they have no effect on the calibration function [21]. If this assumption is incorrect, additive and/or proportional systematic errors may appear. Accordingly, in a preliminary stage within the method validation, constant and proportional bias due to matrix effects must be investigated with the help of standard addition calibration and Youden plot [22].

#### 5. Standard addition calibration

The standard addition calibration (SAC) or standard addition method was originally proposed in 1937 by Hans Hohn in polarographic studies [30]. He used this strategy in order to avoid the matrix effects on the intensity of emission signal, and nowadays, it is widely used in chemical analysis. SAC can be applied with three fundamental goals [31]:

In essence, the calibration for the two first purposes comprises three steps [32]:

The SAC can be performed either at a final fixed volume or at a variable volume [19]. In this discussion, we only consider the first case by working at constant final volume.

Consider now the application of the analytical procedure to a dissolved test portion of an unknown sample within the linear working range. The analyte concentration x is the sum of the fixed native concentration coming from the sample (volume of test portion V0) and the variable spiked concentration (spiked volume, Vspike) and keeping a final constant volume V. The amount of matrix in the test portion (z) is constant. Accordingly, the analytical response can be now modelled as:

$$\begin{aligned} \hat{Y} &= A + B\mathbf{x} + D\mathbf{x}\mathbf{z} = A + (B + D\mathbf{z})\mathbf{x} = A + (B + D\mathbf{z}) \left( \frac{V\_0 \mathbf{C}\_{\text{native}}^0 + V\_{\text{spike}} \mathbf{C}\_{\text{spike}}^0}{V} \right) = \\\ A + (B + D\mathbf{z})\mathbf{C}\_{\text{native}} + (B + D\mathbf{z})\mathbf{C}\_{\text{spike}} &= a\_{\text{SAC}} + b\_{\text{SAC}} \mathbf{C}\_{\text{spike}} \end{aligned} \tag{15}$$

where Cnative is the actual concentration of the analyte in the unspiked sample, Cspike the actual concentration of the spiked analyte and aSAC and bSAC are the intercept and the slope of the SAC calibration straight line. If we try to estimate the analyte concentration of a spiked sample by using the external calibration line, we obtain an estimation of the total observed analyte concentration:

$$
\hat{\mathbf{C}}\_{\text{obs}} = \frac{\hat{Y} - a\_{\text{EC}}}{b\_{\text{EC}}} = \frac{a\_{\text{SAC}} - a\_{\text{EC}} + b\_{\text{SAC}} \mathbf{C}\_{\text{spike}}}{b\_{\text{EC}}} \tag{16}
$$

For the unspiked sample, Cspike ¼ 0, we obtain

$$
\hat{C}\_{\text{native}} = \frac{a\_{\text{SAC}} - a\_{\text{EC}}}{b\_{\text{EC}}} \tag{17}
$$

According to Eqs. (16) and (17), the spiked concentration of the analyte is estimated from the external calibration as:

Practical Considerations on Indirect Calibration in Analytical Chemistry http://dx.doi.org/10.5772/intechopen.68806 207

$$
\hat{\mathbf{C}}\_{\text{spike}} = \hat{\mathbf{C}}\_{\text{obs}} - \hat{\mathbf{C}}\_{\text{native}} = \frac{b\_{\text{SAC}} \mathbf{C}\_{\text{spike}}}{b\_{\text{EC}}} \tag{18}
$$

From Eq. (18), an overall estimation of the overall consensus recovery is calculated as:

$$\mathbf{Rec} = \frac{\hat{\mathbf{C}}\_{spike}}{\mathbf{C}\_{spike}} \tag{19}$$

When proportional bias is absent, we have bSAC ¼ bEC, and that implies Rec ¼ 1. This must be tested for statistical significance by using the student t-test [22]:

$$t = \frac{|\text{Rec-1}|}{s\_{\text{Rec}}} \tag{20}$$

with the recovery standard deviation given by:

• To determine analytes in samples where the analyte-matrix interactions lead to inaccurate

• To determine analytes where the content in the sample is smaller than the quantitation

• To check the accuracy of an analytical result when no reference materials or reference

b. Spike the test solution with one or more amounts of analyte to get corresponding solu-

c. From the responses, calculate a straight-line fit of the experimental data and from that evaluate the concentration that produced the response obtained from the untreated test

The SAC can be performed either at a final fixed volume or at a variable volume [19]. In this

Consider now the application of the analytical procedure to a dissolved test portion of an unknown sample within the linear working range. The analyte concentration x is the sum of the fixed native concentration coming from the sample (volume of test portion V0) and the variable spiked concentration (spiked volume, Vspike) and keeping a final constant volume V. The amount of matrix in the test portion (z) is constant. Accordingly, the analytical response

where Cnative is the actual concentration of the analyte in the unspiked sample, Cspike the actual concentration of the spiked analyte and aSAC and bSAC are the intercept and the slope of the SAC calibration straight line. If we try to estimate the analyte concentration of a spiked sample by using the external calibration line, we obtain an estimation of the total observed analyte concentration:

C^ native ¼ aSAC � aEC

According to Eqs. (16) and (17), the spiked concentration of the analyte is estimated from the

bEC

¼ aSAC � aEC þ bSACCspike bEC

V0C0

native þ VspikeC0

V !

spike

¼

ð15Þ

ð16Þ

ð17Þ

In essence, the calibration for the two first purposes comprises three steps [32]:

discussion, we only consider the first case by working at constant final volume.

a. Measure the analytical response produced by the test solution.

Y^ ¼ A þ Bx þ Dxz ¼ A þ ðB þ DzÞx ¼ A þ ðB þ DzÞ

A þ ðB þ DzÞCnative þ ðB þ DzÞCspike ¼ aSAC þ bSACCspike

C^obs ¼ Y^ � aEC bEC

For the unspiked sample, Cspike ¼ 0, we obtain

results when the EC is used.

206 Uncertainty Quantification and Model Calibration

method is available (recovery assay).

tions and measure the new responses.

solution.

can be now modelled as:

external calibration as:

limit but within the range of analytical sensitivity.

$$\mathbf{s\_{Rec}} = \sqrt{\frac{s\_{b\_{\rm SAC}}^2}{b\_{\rm EC}^2} + \frac{b\_{\rm SAC}^2 s\_{b\_{\rm CE}}^2}{b\_{\rm EC}^4}} \tag{21}$$

Thus, if the degrees of freedom ν corresponding to the uncertainty of consensus recovery are known, student t-statistic is compared with the critical two-tailed tabulated value, ttab(ν,P), at P% confidence. If t ⩽ ttab, the consensus recovery is not significantly different from 1. Alternatively, instead of ttab, a coverage factor k taken as z score may be used for the comparison. Typical values are k ¼ 2 or k ¼ 3 for 95 or 99% confidence, respectively [22], so

Although recovery is sometimes considered a separate validation parameter, it should be established as a part of method validation because it is directly related to the trueness assessment [33]. Aside from the statistical testing considered above, the Association of Official Analytical Chemists (AOAC) has published tables of acceptable recovery percentages as a function of the level of analyte in the sample (see Table 1 of [22]). The relative uncertainty for proportional bias owing to matrix effects is taken as sRec Rec according to SAC.

The relationships between the analytical signal and the analyte concentration when amatrix effect is present are given by Eq. (15). The independent term"A" is the total Youden blank, which is included in the intercept of the SAC calibration (aSAC ¼ A þ bSACCnative). The Youden's plot [17–19] consists of plotting the analytical response (Y) against the amount of the test portion taken for analysis:

$$
\hat{Y} = A + b\_Y \overline{w}\_{sample} \tag{22}
$$

The intercept of this plot is an evaluation of the TYB, which is the sum of the system blank (SB) corresponding to the intercept of the EC (aEC) and the YB associated with the constant bias in the method [13]. Thus, we can equate TYB ¼ A, SB ¼ aEC and YB ¼ A – aEC. We can define the method constant bias as:

$$
\Theta = \frac{A - a\_{\text{EC}}}{b\_{\text{EC}}} \tag{23}
$$

The uncertainty of the constant bias can be obtained by the law of variance propagation [22]:

$$s\_{\theta} = \sqrt{\frac{s\_A^2}{b\_{\text{EC}}^2} + \frac{s\_{\text{EC}}^2}{b\_{\text{EC}}^2} + \frac{(A - a\_{\text{EC}})^2 s\_{b\_{\text{EC}}}^2}{b\_{\text{EC}}^4} + \frac{2(A - a\_{\text{EC}})}{b\_{\text{EC}}^3} \text{cov}(a\_{\text{EC}}, b\_{\text{EC}})} \tag{24}$$

The variances s2 aEC , s2 bEC and the covariance are obtained from the statistical parameters of the EC straight line and s2 A from the Youden's plot. Once sθ is calculated, the constant bias may be assessed for significance as in the case of recovery.

Accordingly, if after performing the assessment of proportional and constant bias matrix effects are present, the uncorrected result x0, found by EC, must be suitably corrected as

$$\chi\_0 = \frac{x\_0^{\text{uncorr}} - \theta}{\text{Rec}} \tag{25}$$

Another way of getting the correct result from the reading of analytical signal Y0 is

$$\mathbf{x}\_0 = \frac{Y\_0 - A}{b\_{\text{SAC}}} \tag{26}$$

On the other hand, when using the SAC for evaluating the analyte concentration x0 of a sample, its standard deviation can be obtained by applying the theorem of variance propagation to the function

$$x\_0 = \frac{a\_{\text{SAC}} - A}{b\_{\text{SAC}}} \tag{27}$$

leading to

$$\begin{split} s\_{x\_0}^2 &= \left(\frac{\partial x\_0}{\partial A}\right)^2 s\_A^2 + \left(\frac{\partial x\_0}{\partial a\_{SAC}}\right)^2 s\_{a\_{SAC}}^2 + \left(\frac{\partial x\_0}{\partial b\_{SAC}}\right)^2 s\_{b\_{SAC}}^2 \\ &+ 2\left(\frac{\partial x\_0}{\partial a\_{SAC}}\right) \left(\frac{\partial x\_0}{\partial b\_{SAC}}\right) \text{cov}(a\_{SAC}, b\_{SAC}) \\ &= \left(-\frac{1}{b\_{SAC}}\right)^2 s\_A^2 + \left(\frac{1}{b\_{SAC}}\right)^2 s\_{a\_{SAC}}^2 + \left(\frac{-(a\_{SAC} - A)}{b\_{SAC}^2}\right)^2 s\_{b\_{SAC}}^2 \\ &- 2\left(\frac{1}{b\_{SAC}}\right) \left(\frac{-(a\_{SAC} - A)}{b\_{SAC}^2}\right) \overline{\pi} s\_{b\_{SAC}}^2 \end{split} \tag{28}$$

After several algebraical manipulations, we obtain

the method [13]. Thus, we can equate TYB ¼ A, SB ¼ aEC and YB ¼ A – aEC. We can define the

θ ¼ A � aEC bEC

The uncertainty of the constant bias can be obtained by the law of variance propagation [22]:

2 s2 bEC

ð Þ A � aEC

sθ ≤ k, the constant bias is not significantly different from 0.

b4 EC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sθ > k, the constant bias is significantly different from 0, and the results have to be

0 � θ

Accordingly, if after performing the assessment of proportional and constant bias matrix effects are present, the uncorrected result x0, found by EC, must be suitably corrected as x0 ¼ xuncorr

> x0 ¼ Y0 � A bSAC

On the other hand, when using the SAC for evaluating the analyte concentration x0 of a sample, its standard deviation can be obtained by applying the theorem of variance propaga-

> x0 ¼ aSAC � A bSAC

> > s2 aSAC þ

covðaSAC, bSACÞ

s2

xs2 bSAC

∂x0 ∂bSAC � �2

aSAC þ �ðaSAC � AÞ b2 SAC

!2

s2 bSAC

> s2 bSAC

∂x0 ∂aSAC � �2

1 bSAC � �2

Another way of getting the correct result from the reading of analytical signal Y0 is

þ

2ð Þ A � aEC b3 EC

vuut ð24Þ

bEC and the covariance are obtained from the statistical parameters of the

A from the Youden's plot. Once sθ is calculated, the constant bias may be

covðaEC, bECÞ

Rec ð25Þ

ð23Þ

ð26Þ

ð27Þ

ð28Þ

method constant bias as:

208 Uncertainty Quantification and Model Calibration

The variances s2

• If jθj

• If jθj

EC straight line and s2

corrected by θ.

tion to the function

s2 x0 ¼ ∂x0 ∂A � �2

þ2 ∂x0 ∂aSAC � � ∂x0

¼ � 1 bSAC � �2

�2 1 bSAC s2 A þ

s2 A þ

� � �ðaSAC � AÞ b2 SAC

∂bSAC � �

!

leading to

sθ ¼

aEC , s2

s2 A b2 EC þ s2 aEC b2 EC þ

assessed for significance as in the case of recovery.

$$s\_{x\_0}^2 = \frac{s\_{y/x}^2}{b\_{SAC}^2} \left[ \frac{1}{N} + \frac{\overline{Y}^2}{b\_{SAC}^2 \mathbb{S}\_{\text{xx}}} \frac{+A(A - 2\overline{Y})}{b\_{SAC}^2 \mathbb{S}\_{\text{xx}}} \right] + \frac{s\_A^2}{b\_{SAC}^2} \tag{29}$$

But many workers apply the SAC without considering the true blank, that is, by setting A ¼ 0 and sA ¼ 0 leading to

$$s\_{x\_0}^2 = \frac{s\_{y/x}^2}{b\_{SAC}^2} \left[ \frac{1}{N} + \frac{\overline{Y}^2}{b\_{SAC}^2 \mathcal{S}\_{xx}} \right] \tag{30}$$

This expression is presented in several standard analytical textbooks, for instance [13, 19, 32, 34]. However, Ortiz et al. [35] pointed out that when extrapolating, the analyte concentration is obtained by setting Y0 ¼ 0 and calculating x0 ¼ �aSAC=bSAC, but even in this case, the uncertainty of the signal must be included in calculations, leading to

$$s\_{\mathbf{x}\_0}^2 = \frac{s\_{y/x}^2}{b\_{\rm SAT}^2} \left[ 1 + \frac{1}{N} + \frac{\overline{Y}^2}{b\_{\rm SAT}^2 \mathbf{S}\_{\rm xx}} \right] \tag{31}$$

The SAC, as it has been outlined, is considered as an extrapolation method but an interpolation approach is available [32, 36]. A plot of the data obtained from SAC and how the analyte concentration is predicted by extrapolation are depicted in Figure 1. Nevertheless, an interpolation alternative is also gathered there. The latter is discussed in the following.

What value of the analytical signal Y0 will correspond to a spiked x value that is equal to the concentration of the native analyte? That is:

$$\mathbf{Y}\_0 = a\_{\text{SAC}} + b\_{\text{SAC}} \\ \mathbf{x}\_0 = A + 2b\_{\text{SAC}} \\ \mathbf{x}\_0 = 2\mathbf{Y}\_{\text{unspick}} - A \tag{32}$$

Figure 1. The plot of extrapolation and interpolation for prediction of the native analytical concentration of a sample by using the SAC.

And if we disregard the true blank, we get

$$Y\_0 = a\_{\text{SAC}} + b\_{\text{SAC}} \mathbf{x}\_0 = 2b\_{\text{SAC}} \mathbf{x}\_0 = 2Y\_{unspikal} \tag{33}$$

Thus, the native analyte concentration can be obtained by interpolation by setting the analytical signal for the sample as the double of the signal, corresponding to the unspiked sample minus the true blank:

$$
\hat{\mathfrak{x}}\_0 = \frac{Y\_0 - a\_{\text{SAC}}}{b\_{\text{SAC}}} \tag{34}
$$

This leads to a variance for the native analyte

$$s\_{x\_0}^2 = \frac{s\_{y/x}^2}{b\_{SAC}^2} \left[ 1 + \frac{1}{N} + \frac{\left(Y\_0 - \overline{Y}\right)^2}{b\_{SAC}^2 S\_{xx}} \right] \tag{35}$$

According to Andrade et al. [32, 36], the use of extrapolation in the SAC is a risky practice because it may lead to biased prediction and uncertainties substantially different from interpolation. Confidence interval from extrapolation is always higher than those obtained by interpolation.

#### 6. The internal standard calibration

The method of internal standard calibration (ISC) was first applied in the 1950s in several analytical fields [37–41]. This method is especially useful when the analytical response varies slightly from run to run due to different causes, for instance:

The use of an internal standard is also needed for analytical methods where there are multiple sample preparation steps, especially when volumetric recovery at each step may vary (extraction with separation cartridges) or when involving chemical derivatizations with low or variable yields of reaction.

An internal standard is a substance different from the analyte but that has physicochemical properties very similar to the analyte. Evidently, the internal standard cannot be a component of the sample.

It is added to the sample, and the patterns in known amounts and the signal produced by both the analyte and the internal standard are measured. If in repeated measurements, there is signal oscillation, it will occur both in the analyte and in the internal standard, and the ratio of the signals of both will not change.

Thus, instead of the response Y, the ratio of responses Y/YIS is used in the calibration procedure. Assuming that in the instrumental method the signal is in direct proportion to the analyte and internal standard, we get:

$$\begin{aligned} Y &= kx \\ Y\_{IS} &= k\_{IS} \mathbf{x}\_{IS} \\ \left(\frac{Y}{Y\_{IS}}\right) &= F\left(\frac{x}{\mathbf{x}\_{IS}}\right) \end{aligned} \tag{36}$$

Here, Yis the analytical signal due to the analyte and YIS is the analytical signal corresponding to the internal standard. The calibration straight line is performed as in EC by preparing standards at several analyte concentrations and with the same concentration fixed for internal standard xIS. Thus, the calibration constant F is evaluated. Whereas the dispersion of the calibration straight line Y ¼ kx may be significant, the one obtained with the ISC is negligible.

The sample is then treated in the same way by spiking the internal standard at the same concentration in the standards. Thus, if the reading of the sample is Y0 Y0 IS � �

$$
\hat{\mathbf{x}}\_0 = \left(\frac{\mathbf{y}^0}{\mathbf{y}\_{IS}^0}\right) \frac{\mathbf{x}\_{IS}}{F} \tag{37}
$$

By applying the variance propagation law and considering negligible variance of xIS, we get

$$s\_{X\_0} = \frac{\mathbf{x}\_{ISR}}{F} \sqrt{1 + \frac{\mathbf{x}\_{IS}^2 \left(\frac{\mathbf{Y}^0}{Y\_{IS}^0}\right)^2}{F^2 \sum \mathbf{x}\_i^2}}\tag{38}$$

The main advantage of ISC is that this quantification method does not need a previous calibration because it is implicit in the quantification [21]. Accordingly, the use of onepoint calibration method can be used. It only requires the addition of known and equal amounts of internal standards to the standard analyte solution and to sample solution and measures the analytical signals of analyte and internal standard in the standard and in the sample. Evidently, the signals of analyte and internal standard must be distinguishable without overlapping.

Thus,

And if we disregard the true blank, we get

210 Uncertainty Quantification and Model Calibration

This leads to a variance for the native analyte

6. The internal standard calibration

resonance (NMR).

variable yields of reaction.

of the sample.

s 2 x0 ¼ s2 y=x b2 SAC

slightly from run to run due to different causes, for instance:

• Changes in the capillary characteristics in polarography.

• Temperature fluctuations in atomic emission spectrometry.

minus the true blank:

Y0 ¼ aSAC þ bSACx0 ¼ 2bSACx0 ¼ 2Yunspiked ð33Þ

Y0 � Y � �2 b2 SACSxx

" #

ð34Þ

ð35Þ

Thus, the native analyte concentration can be obtained by interpolation by setting the analytical signal for the sample as the double of the signal, corresponding to the unspiked sample

> x^0 ¼ Y0 � aSAC bSAC

> > 1 þ 1 N þ

According to Andrade et al. [32, 36], the use of extrapolation in the SAC is a risky practice because it may lead to biased prediction and uncertainties substantially different from interpolation. Confidence interval from extrapolation is always higher than those obtained by interpolation.

The method of internal standard calibration (ISC) was first applied in the 1950s in several analytical fields [37–41]. This method is especially useful when the analytical response varies

• Inhomogeneities in the effective magnetic field due to shielding effect in nuclear magnetic

The use of an internal standard is also needed for analytical methods where there are multiple sample preparation steps, especially when volumetric recovery at each step may vary (extraction with separation cartridges) or when involving chemical derivatizations with low or

An internal standard is a substance different from the analyte but that has physicochemical properties very similar to the analyte. Evidently, the internal standard cannot be a component

• Variability in the injection volume in gas chromatography (manual injection).

• Differences in the nature of particulate matter in the sample in X-ray fluorescence.

• Irreproducibility of automatic injectors in capillary electrophoresis.

$$\begin{aligned} \left(\frac{Y^{std}}{Y\_{IS}^{std}}\right) &= F \left(\frac{\mathbf{x}^{std}}{\mathbf{x}\_{IS}}\right) \quad \text{and} \quad \left(\frac{Y^0}{Y\_{IS}^0}\right) = F \left(\frac{\mathbf{x}^0}{\mathbf{x}\_{IS}}\right) \\ \dot{\mathbf{x}}\_0 &= \mathbf{x}\_{IS} \frac{\begin{pmatrix} Y^0 \\ Y\_{IS}^0 \end{pmatrix}}{\begin{pmatrix} Y^{std} \\ Y\_{IS}^{std} \end{pmatrix}} \end{aligned} \tag{39}$$

Another exclusive feature of ISC is the possibility of performing the quantification of several analytes of the same chemical family in the same test portion and in a unique internal calibration with a single internal standard. Consequently, it could be possible to evaluate the mass fraction of each analyte according to [21].

$$\% \text{x}\_{0i} = \frac{\text{x}\_{0i}}{\sum\_{j}^{\text{x}\_{0j}} \text{(x}\_{0j}\text{)}} 100 = \frac{\left(\frac{\text{x}\_{IS}}{F}\right) \left(\frac{Y\_i^0}{Y\_{IS}^0}\right)}{\sum\_{j} \left(\frac{\text{x}\_{IS}}{F}\right) \left(\frac{Y\_j^0}{Y\_{IS}^0}\right)} 100 = \frac{Y\_i^0}{\sum\_{j} Y\_j^0} 100 \tag{40}$$

Accordingly, ISC is a very powerful method for congener analysis (for instance in fat analysis, determination of waxes, sterols, aliphatic alcohol and so on) by using only a unique internal standard.

#### 7. Synthesis

Indirect calibration is a key concept for method validation. Instrumental analysis involving indirect calibration is a common feature in routine analysis, and three typical scenarios can be found depending on the analyte-matrix interaction and the uncontrolled variation of the analytical signal owing to intrinsic characteristics of the analytical process. Thus, when the interaction of the matrix of sample is negligible, the external calibration is the normal choice. Otherwise, the Standard Addition calibration together with the Youden plot have to be applied. In cases where there are non-random signal variations run to run or possible analyte losses due to sample preparation procedures or derivatization reactions, Internal Standard calibration must be considered. These three approaches have been outlined and discussed. Uncertainty values for the analyte concentration coming from the calibration step are considered and evaluated from the calibration data.

#### Author details

Antonio Gustavo González

Address all correspondence to<EMAIL_ADDRESS>
Department of Analytical Chemistry, Faculty of Chemistry, University of Seville, Seville, Spain

#### References

Ystd Ystd IS

x^0 ¼ xIS

fraction of each analyte according to [21].

212 Uncertainty Quantification and Model Calibration

7. Synthesis

Author details

Antonio Gustavo González

%x0i ¼ X

ered and evaluated from the calibration data.

Address all correspondence to<EMAIL_ADDRESS>
x0i

x0j � � 100 ¼

j

¼ F xstd xIS � �

> Y0 Y0 IS

Ystd Ystd IS

!

!

and Y0 Y0 IS

Another exclusive feature of ISC is the possibility of performing the quantification of several analytes of the same chemical family in the same test portion and in a unique internal calibration with a single internal standard. Consequently, it could be possible to evaluate the mass

> xIS F � � Y0 i Y0 IS

> > xIS F � � Y0 j Y0 IS

Accordingly, ISC is a very powerful method for congener analysis (for instance in fat analysis, determination of waxes, sterols, aliphatic alcohol and so on) by using only a unique internal standard.

Indirect calibration is a key concept for method validation. Instrumental analysis involving indirect calibration is a common feature in routine analysis, and three typical scenarios can be found depending on the analyte-matrix interaction and the uncontrolled variation of the analytical signal owing to intrinsic characteristics of the analytical process. Thus, when the interaction of the matrix of sample is negligible, the external calibration is the normal choice. Otherwise, the Standard Addition calibration together with the Youden plot have to be applied. In cases where there are non-random signal variations run to run or possible analyte losses due to sample preparation procedures or derivatization reactions, Internal Standard calibration must be considered. These three approaches have been outlined and discussed. Uncertainty values for the analyte concentration coming from the calibration step are consid-

Department of Analytical Chemistry, Faculty of Chemistry, University of Seville, Seville, Spain

X j

!

! 100 ¼ Y0

X i

100 ð40Þ

j Y0 j

!

¼ F x0 xIS � �

ð39Þ

!

editor. Basic Chemometric Techniques in Atomic Spectroscopy. 2nd ed. Milton Road, Cambridge, CB4 0WF, UK: The Royal Society of Chemistry; 2013. pp. 52–117

[33] Taverniers I, De Loose M, van Bockstaele E. Trends in quality in the analytical laboratory. II. Analytical method validation and quality assurance. Trends in Analytical Chemistry. 2004;23:535–552

[17] Cardone MJ. New technique in chemical assay calculations. 2. Correct solution of the model problem and related concepts. Analytical Chemistry. 1986;58:483–445

[18] Cardone MJ, Willavice SA, Lacy ME. Method validation revisited: A chemometric

[19] Harvey D. Analytical Chemistry 2.0. Electronic edition: http://www.asdlib.org/online Articles/ecourseware/Analytical Chemistry 2.0/Text\_Files.html. 2016. p. 1133

[20] Booksh KS, Kowalski BR. Theory of analytical chemistry. Analytical Chemistry. 1994;66: