Details of Horwitz values.
\r\n\t
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He has edited four books and collaborated on ten others as well as seventeen patents and in the organization of three international conferences. He is a\nreviewer for ninety-eight journals.",institutionString:"Nanjing Medical University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"4",totalChapterViews:"0",totalEditedBooks:"4",institution:{name:"Nanjing Medical University",institutionURL:null,country:{name:"China"}}}],coeditorOne:{id:"184798",title:"Ms.",name:"Qiuqin",middleName:null,surname:"Tang",slug:"qiuqin-tang",fullName:"Qiuqin Tang",profilePictureURL:"https://mts.intechopen.com/storage/users/184798/images/13334_n.jpg",biography:"Qiuqin Tang is an attending doctor of The Women’s Hospital of Nanjing Medical University (Nanjing Maternity and Child Health Care Hospital). Her research interests include genetic and epigenetic risk factors of reproductive and developmental health. 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He is also the headmaster of the Family Planning Centre and Gynecological Cytology\nLaboratory at the same university. Dr. Tsikouras is a fellow of the\nInternational Academy of Clinical and Applied Thrombosis/Hemostasis. His scientific activities focus on paediatric and adolescence medicine, gynecological oncology, high-risk pregnancies. He is a reviewer for several international journals and has numerous scientific publications to his credit, including papers and book chapters. 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He\nis currently a professor in the Gynecology and Obstetrics Faculty\nof Medicine, University of Kiel, Germany, and honorary doctor\nat the Democritus University of Thrace, Alexandroupoli University Hospital He previously served as chief of the Department\nof Gynecology and Obstetrics at University Hospital RWTH Aachen,\nGermany. Dr. Rath is a reviewer for numerous journals and chief editor of Geburtshilfe und Frauenheilkunde (GebFra). 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From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"57909",title:"Validation of Analytical Methods",doi:"10.5772/intechopen.72087",slug:"validation-of-analytical-methods",body:'Analytical method validation is an essential requirement to perform the chemical evaluation [1, 2, 3]. Method validation is a procedure of performing numerous assessments designed to verify that an analytical test system is suitable for its intended reason and is capable of providing beneficial and legitimate analytical data [4, 5, 6, 7, 8]. A validation examine includes testing multiple attributes of a method to determine that it may provide useful and valid facts whilst used robotically [9, 10, 11]. To accurately investigate method parameters, the validation test ought to consist of normal test conditions, which includes product excipients [11, 12, 13, 14]. Therefore, a method validation examine is product-specific.
Selectivity/Specificity
Precision
Accuracy
Linearity
Range
Stability
Limit of Detection (LOD) and Limit of Quantitation (LOQ)
Selectivity of an analytical method is its ability to measure accurately an analyte in the presence of interferences that may be expected to be present in the sample matrix.
Selectivity is checked by examining chromatographic blanks (from a sample that is known to contain no analyte) in the expected time window of the analyte peak. And the raw data for selectivity will be recorded in the raw data in approved formats.
Precision of a method is the degree of agreement among individual test results when the procedure is applied repeatedly to multiple samplings.
Precision is measured by injecting a series of standards or analyzing series of samples from multiple samplings from a homogeneous lot. From the measured standard deviation (SD) and Mean values, precision as relative standard deviation (% rsd) is calculated.
The raw data for precision will be recorded in the approved format and the acceptance criteria for precision will be given in the respective study plan or amendment to the study plan.
Precision can be also calculated by using Horwitz equation:
The acceptable percent of relative standard deviation results for precision may be based on the Horwitz equation, an exponential relationship between the among-laboratory relative standard deviation (RSDR) and Concentration (C): [15]
For estimation of repeatability (RSDr), is modified to:
The Horwitz curve has been empirically derived and has been proven to be more or less independent of analyte, matrix and method of evaluation over the concentration range C = 1 (100%) to C = 10−9 by the evaluation of vast numbers of method precision studies. The modified Horwitz values for repeatability CV given under may be used for guidance. If measured repeatability is outside those values, suggested explanation must be submitted for consideration. The details were presented in Table 1.
Percent of analyte | Proposed acceptable % RSDr (Horwitz value × 0.67) |
---|---|
100.00 | 1.340 |
50.00 | 1.490 |
20.00 | 1.710 |
10.00 | 1.900 |
5.00 | 2.100 |
2.00 | 2.410 |
1.00 | 2.680 |
0.25 | 3.300 |
Details of Horwitz values.
Note: The unmodified Horwitz equation is used as a criterion of acceptability for methods collaboratively tested by CIPAC.
The accuracy of an analytical method is the degree of agreement of test results generated by the method to the true value.
Accuracy is measured by spiking the sample matrix of interest with a known concentration of analyte standard and analyzing the sample using the “method being validated.” The procedure and calculation for Accuracy (as% recovery) will be varied from matrix to matrix and it will be given in respective study plan or amendment to the study plan.
The linearity of an analytical method is its capability to elicit check consequences which might be at once, or with the aid of well described mathematical adjustments, proportional to the concentration of analytes in within a given range.
Linearity is determined by injecting a series of standards of stock solution/diluted stock solution using the solvent/mobile phase, at a minimum of five different concentrations in the range of 50–150% of the expected working range. The linearity graph will be plotted manually/using Microsoft Excel or software of the computer (Concentration vs. Peak Area Response) and which will be attached to respective study files.
The range of an analytical method is the interval between the upper and lower levels that have been demonstrated to be determined with precision, accuracy and linearity using the set method. This range will be the concentration range in which the Linearity test is done.
Many analytes readily decompose prior to chromatography investigations, for example during the preparation of the sample solutions, during extraction, clean-up, phase transfer, and during storage of prepared vials. Under these circumstances, method development should investigate the stability of the analyte. Accuracy test takes care of stability. It is required to mention in the method how long a sample after extraction can be stored before final analysis, based on the duration taken for accuracy test.
The term LOD is defined as the lowest concentration at which the instrument is able to detect but not quantify and the noise to signal ratio for LOD should be 1:3. The term LOQ is defined as the lowest concentration at which the instrument is able to detect and quantify. The noise to signal ratio for LOQ should be 1:10.
Determination of Limit of Detection (LOD) and Limit of Quantitation (LOQ) from Detector Linearity experiments (applicable to only instrument sensitivity).
LOD and LOQ values are calculated manually by taking Noise to signal ratio of a lowest/known concentration of linearity samples and it will be expressed in μg/ml or ppm. To calculate in %, values of LOD and LOQ will be multiplied by 100/lowest or known concentration of test item (mg/L) taken for analysis of that particular a.i. or impurity analysis.
Prepare a series of standard solutions (minimum five concentrations covering working concentrations used for routine analysis) and analyze each solution minimum twice and record the instruments response.
Using the concentrations and corresponding instrument response, LOD and LOQ can be calculated as follows:
Let the linear regression equation be
Where, X and Y are the variables (data of two parameters). Generally, X is called the independent variable and Y, the dependent variable.
Take concentration on X-axis and instrument response on Y-axis.
“a” and “b” are the regression constants. Further, “a” is known as the intercept and “b,” the slope of the line.
Let (X1, Y1), (X2, Y2), (X3, Y3)…(Xn, Yn) be the set of values required to be fit in the linear equation.
a. Method of arriving at “a” and “b”
Tabulate as given below:
X1 Y1
X2 Y2
. .
. .
. .
Xn Yn
____________________________________
____________________________________
ii. Calculate the following parameters:
iii. Calculate the slope “b,” and intercept “a” as given below:
b. Method of calculation r (correlation coefficient)
c. Method of calculation standard deviation for “a” and “b”
The standard deviation of the individual deviations of measured values in Y, above and below the linear line (fitted line) is:
From this, the standard deviation for “a” and “b” are calculated.
Standard deviation
for “a,” represented =
as Sa
Standard deviation.
For “b,” represented =
as Sb
When Sa is obtained for a linear calibration line, then it provides a clear information on the standard deviation of the “Blank” (or Control) response from the instruments.
The LOD and LOQ can be worked out, as given below:
The above calculations can be programmed in a computer but before every use, the computer program must be validated using the example given in section
The above procedure can also be used for obtaining LOD and LOQ of the method from recovery test results by taking fortified concentration on X-axis and obtained concentrations on Y-axis.
In this example, the linear regression equation is employed to find out the extent of linear response of an Detector to a reference analytical standard in the concentration range of about 0.2–3.0 ppm.
Each of these working standards is injected thrice (1 μl per injection), and the peak area counts corresponding to the active ingredient peak are given below.
From the peak areas corresponding to each concentration level, the mean, standard deviation (SD) and coefficient of variation (%CV) are also calculated. The details were presented in Table 2.
Conc. of standard solution (μg/ml) | Peak area | Mean | SD (n − 1) | %CV | ||
---|---|---|---|---|---|---|
1 | 2 | 3 | ||||
0.1956 | 32,827 | 33,299 | 32,731 | 32,952 | 304 | 0.923 |
0.4890 | 87,783 | 88,480 | 87,446 | 87,903 | 527 | 0.600 |
0.9780 | 176,037 | 174,675 | 177,203 | 175,972 | 1265 | 0.719 |
1.467 | 246,212 | 250,786 | 246,849 | 247,949 | 2477 | 0.999 |
1.956 | 319,143 | 319,615 | 315,316 | 318,025 | 2358 | 0.741 |
2.934 | 415,059 | 410,773 | 418,407 | 414,746 | 3827 | 0.923 |
Calculation details of mean, SD, and %CV.
%CV = SD × 100/Mean: The coefficient of variation (CV) shows that the Injection variation is less than 1%.
Let the equation be
Where, Y = Mean peak area counts and X = Concentration of standard solution, μg/ml.
The calculations were presented in Table 3.
Analytical methods for quantitation of major excipients and/or active ingredients, and preservatives in finished goods.
Analytical methods for determination of impurities or degradation compounds in finished goods. These methods include quantitative assays and limit tests, titrimetric and bacterial endotoxin tests.
Analytical methods for determination of performance characteristics, e.g., sterility testing, dissolution and drug release for pharmaceutical products.
Details of required validation parameters of assay presented in Table 4.
Sl. no. | Y | X |
---|---|---|
1. | 32952 | 0.1956 |
2. | 87903 | 0.4890 |
3. | 175972 | 0.9780 |
4. | 247949 | 1.4670 |
5. | 318025 | 1.9560 |
6. | 414746 | 2.9340 |
Calculation details of additional parameters.
Analytical validation data playing a fundamental role in pharmaceutical industry, pesticide industry for releasing the economic batch and long term stability information consequently, the records must be produced to suited regulatory authority requirements.
System identification is an important topic nowadays since it can be used in solving numerous problems [1]. The aim of system identification is to estimate an unknown model using the available and observed data, namely the input and output of the system. In this context, the well-known Wiener filter is a popular solution, along with the adaptive filters which can be derived starting from this approach.
In multilinear system identification, dealing with a large parameter space represents an important challenge [2, 3]. The huge length of the filter (hundreds or thousands of coefficients) is also a serious problem [4, 5]. The methods used for addressing these issues usually rely on tensor decomposition and modeling [2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], meaning that a high-dimension problem is rewritten as a combination of lower-dimension structures, using the Kronecker product decomposition [18].
In the context of multilinear forms identification, a few approaches were proposed recently, addressing the cases when the large system is decomposed into two or three smaller components (i.e., bilinear and trilinear forms, respectively) [10, 17, 18, 19, 20, 21, 22, 23]. The aforementioned solutions outperform their conventional counterparts, offering at the same time a lower computational complexity.
Motivated by the appealing performance of these previous developments, we extended the tensor decomposition technique to higher-order systems, and in this framework, this chapter presents a part of the work and results obtained recently by the authors in the context of multilinear system identification. An iterative Wiener filter and a family of LMS-based algorithms tailored for multilinear forms are presented. For more details on the results summarized here, the works [24, 25, 26] can be consulted.
Related to the work presented here, several other tensor-based solutions relying on the recursive least-squares (RLS) algorithm were also developed recently [27, 28]. Possible applications of such system identification frameworks can be encountered in topics such as big data [29], machine learning [14], but they may be also useful in nonlinear acoustic echo cancelation [30, 31], source separation [13, 32, 33], channel equalization [12, 34], array beamforming [16, 35], blind identification [36], object recognition [37, 38], and cardiac applications [39].
The rest of this chapter is organized in the following way. In Section 2, we introduce the system model for the multiple-input/single-output (MISO) system identification problem. In this context, Section 3 presents an iterative Wiener filter tailored for the identification of multilinear systems. Next, in Section 4, an LMS-based algorithm is presented, together with its normalized version, and then, in Section 5, the performance of these algorithms is proved through simulations. Finally, conclusions are drawn in Section 6.
Let us consider a MISO system, whose output signal at the time index
where the individual channels are modeled by the vectors:
the superscript
where
where
where
and so on, where
where
with
Here, an observation can be made: the solution of the decomposition in Eq. (11) is not unique [17, 24]. Despite this, no scaling ambiguity occurs in the identification of the global impulse response,
Using Eqs. (9)–(11), we may rewrite
We aim to identify the global impulse response,
where
with
where
The optimization criterion is the minimization of the mean-squared error (MSE), which can be defined using Eq. (15):
where
Relation (17) provides the global impulse response. In order to obtain the
It can be easily checked that
where
Hence, the cost function given by Eq. (16) may be expressed in
where
If all coefficients except
The minimization of this convex cost function with respect to
Using this result, an iterative approach can be derived. A set of initial values
The minimization of the cost function yields
Using
where
The minimization of the cost function yields
All the other estimates
The limitations of the Wiener filter (e.g., matrix inversion, statistics estimation) can restrict the applicability of the previously presented approach in real-world situations (for example, in nonstationary conditions, or when real-time processing is needed). Therefore, a better approach may be represented by adaptive filters. In this context, the well-known least-mean-square (LMS) algorithm is among the most popular solutions, due to its simplicity. In the following, a family of LMS-based algorithms for multilinear forms identification is presented.
By using the estimated impulse responses
where
We can easily check that
where
The global filter estimate is obtained as
We may also identify the global impulse response using the classical LMS algorithm:
where
When choosing the constant values of the step-size parameters from Eqs. (39)–(41), we need to consider the compromise between convergence rate and steady-state misadjustment. In certain cases, it can be more useful to have variable step-size parameters. Hence, the update equations become
Then, the a posteriori error signals can be defined as
After replacing Eq. (39) in Eq. (50), Eq. (40) in Eq. (51), and Eq. (41) in Eq. (52), and then canceling the a posteriori error signals, we get
We assume that
In the numerators of Eqs. (56)–(58), the normalized step-size parameters
The initialization of the individual impulse responses can be done using Eqs. (42, 43). We may also identify the global impulse response using the regular NLMS algorithm:
where
The purpose of this section is to illustrate through simulations the improved performance of the proposed solutions for the identification of multilinear forms. We performed experiments involving MISO system identification. As input signals, we used white Gaussian noises and AR(1) processes, obtained by filtering white Gaussian noises through a first-order system with the transfer function
Impulse responses used in simulations of the multiple-input/single-output (MISO) system identification scenario (for
The measure of performance is the normalized misalignment (in dB) for the identification of the global impulse response, computed as
First, we aim to show comparatively the performances of the LMS-MF and LMS algorithms. When choosing the step-size parameter values, we need to take into account the theoretical upper bound, which for the conventional LMS is
Figure 2 shows the case when
Performance of the LMS-MF and LMS algorithms (using different step-size parameters), for the identification of the global impulse response
Next, in Figure 3,
Performance of the LMS-MF and LMS algorithms (using different step-size parameters), for the identification of the global impulse response
When the system order increases, the improvement in performance brought by the LMS-MF is even more apparent. This can be seen in Figure 4, where
Performance of the LMS-MF and LMS algorithms (using different step-size parameters), for the identification of the global impulse response
In the following, we aim to illustrate the performance of the NLMS-MF and NLMS algorithms in the identification of the global system. Since the step-size parameter does no longer have a constant value, the normalized algorithms can work better in nonstationary environments. The fastest-convergence bound for the value of the normalized step-size parameter of the conventional NLMS algorithm is 1 [40].
Figure 5 illustrates the case when the inputs are white Gaussian noises and
Performance of the NLMS-MF and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response
The improvement offered by the proposed approach is even more significant for correlated inputs. In Figure 6, the input signals are AR(1) processes. It is noticed that even when the NLMS-MF algorithm uses lower values for the normalized step-sizes, it can still outperform the NLMS algorithm working in the fastest convergence mode.
Performance of the NLMS-MF and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response
The same conclusion applies when the order
Performance of the NLMS-MF and NLMS algorithms (using different normalized step-size parameters), for the identification of the global impulse response
Next, we aim to show the influence of the normalized step-size values on the performance of the proposed algorithm. In Figure 8, the order of the system is
Performance of the NLMS-MF algorithm using different normalized step-size parameters (with equal values of
The last experiment involving the NLMS-MF algorithm aims to show the performance in the case when the normalized step-size parameters
Performance of the NLMS-MF algorithm using different normalized step-size parameters (with different values of
Due to the important improvement in performance brought by the adaptive tensor-based LMS algorithms, observed through experiments, these algorithms may represent appealing solutions for the identification of long-length separable system impulse responses.
In this chapter, we have presented a decomposition-based approach for dealing with the identification of high-dimension MISO systems. Unlike the conventional method, which is based on the identification of the global system impulse response, our solution focuses on regarding the system as an
This work was supported by a grant from the Romanian Ministry of Education and Research, CNCS–UEFISCDI, Project Number PN-III-P1-1.1-PD-2019-0340, within PNCDI III.
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