Calibration coefficients

We now use CLS to generate calibrations from our two training sets, A1 and A2. For each training set, we will get matrices, Kl and K2, respectively, containing the best least-squares estimates for the spectra of pure components 1-3, and matrices, Kl i and K2cnl, each containing 3 rows of calibration coefficients, one row for each of the 3 components we will predict. First, we will compare the estimated pure component spectra to the actual spectra we started with. Next, we will see how well each calibration matrix is able to predict the concentrations of the samples that were used to generate that calibration. Finally, we will see how well each calibration is able to predict the... 

Figure 21. Plots of the CLS calibration coefficients calculated for each component with each training set.

We perform CLS on A6 to produce 2 calibrations. K6 and K6, are the matrices holding the pure component spectra and calibration coefficients, respectively, for CLS with zero intercepts. K6a and K6aMl are the corresponding matrices for CLS with nonzero intercepts. 

The calibration matrix, Fc t has exactly the same format as K, the calibration matrix for CLS. It has one row for each component being predicted. Each row has one calibration coefficient for each wavelength in the spectrum. We can now use F , to predict the concentrations in an unknown sample from its measured spectrum. First, we place the spectrum into a new absorbance matrix, A,. We can now use equation [64] to produce a new concentration matrix, CHah, containing the predicted concentration values for the unknown sample. 

PCR calibration coefficients from Al/Cl PCR calibration coefficients from A2/C2... 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

Regression coefficient (calibration coefficient) intercept (general and regression in case of errors in both variables)... 

The different statistical character of the three variables becomes most clear in the different uncertainties of the calibration and evaluation lines. Notwithstanding the fundamental differences between xstandard and xsampiey the calculation of the calibration coefficients is carried out by regression calculus. 

As a rule the calibration coefficients do not alter significantly when WLS is applied instead of OLS. On the other hand, the uncertainty in the lower range of calibration is reduced in a remarkable way. The estimate of the residual standard deviation is... 

The EBV model has to be applied if both the measured values y and the analytical value x are error-affected quantities. The calibration coefficients of the model at Eq. (6.39) cannot be determined directly for the general case, but only according to certain assumptions or by approximations (Danzer et al. [1995]) from which three will be given here ... 

Direct calibration can be applied when the calibration coefficients are known, otherwise - in case of indirect calibration - the calibration coefficients are computed by means of experimentally estimated spectra-concentrations relations. 

Instead of the symbol A and the term sensitivity matrix also the symbol K (matrix of calibration coefficients, matrix of linear response constants etc) is used. Because of the direct metrological and analytical meaning of the sensitivities aj - in the A-matrix the term sensitivity matrix is preferred. 

The calibration coefficients are elements of the matrix P which can be estimated by... 

In the case that the original variables, the measured values y, are used for inverse calibration, there are no significant advantages of the procedure apart from the fact that no second matrix inversion has to be carried out in the analysis step see Eq. (6.87). On the contrary, it is disadvantageous that the calibration coefficients (elements of the P-matrix) do not have any physical meaning because they do not reflect the spectra of the single species. In addition, multicollinearities may appear which can make inversion of the T-matrix difficult see Eq. (6.86). 

In PCR the calibration coefficients (B-matrix) are estimated column by column according to... 

In PLS both the matrices of measured values Y and analytical values X are decomposed according to Eqs. (6.89) and (6.90) Y = TPr + EY and X = TQt + Ex and thus relations between spectra and concentrations are considered from the outset. The B-matrix of calibration coefficients is estimated by... 

In case of experimental calibration, the uncertainty of both the blank and the calibration coefficient, U (JBL, fi), have to be consider, e.g. according to... 

The accuracy of the measurement of radon concentrations with bare track detectors was found to be unsatisfactory due mainly to the changes of the deposition rate of radon progeny onto the detector as a result of air turbulence. In this work, therefore, a method was developed which can correct the contributions of the deposition to the track densities by classifying the etched tracks according to their appearance, i.e. round or wedge shaped. Using this method, about 30% improvement in the error of measurements was achieved. The calibration coefficient ob tained by experiment was 0.00424 tracks/cm /h/(Bq/m ), which agreed well with the calculated value. Comparison was also made of the present method with other passive methods, charcoal and Terradex, as to their performance under the same atmosphere. 

Kdf = calibration coefficient for activity of unattached daughters deposited on the detector,... 

Figure 3. Variation of calibration coefficients, Kc and K, derived from the calculation.

However, as regards the conventional method, a linearity exists only in the case when the fan was off, and the track densities T in the turbulent atmosphere were higher than those in the still atmosphere. The calibration coefficients K and K were calculated with a multi-step method of linear regression analysis (Skinner and Nyberg,... 

According to the experimental data, the calibration coefficients K and K were 0.00424 and 0.0172 tracks/cm /h/(Bq/m ), respectively. The mean value of K determined from the experimental data is higher by 13% than that previously derived purely on the basis of calculation. 

Calibration Coefficients. Two mean values of the calibration coefficient K are presented. 2 e mean value of K obtained by calculation was 6.00375 tracks/cin /h/(Bq/in ), and tha obtained by experiment was 0.00424. Taking account of the ambiguities in the assumptions for the calculation and the experimental errors, there is no significant difference between the two values of K. If the ambiguities and the experimental errors are negligible, the difference is due to the variation of the ratio of thoron concentration to radon. When the ratio is assumed equal to 0.14, the mean value of K obtained by calculation agrees with the experimental result. 

Using the present method, the bare CR-39 detector becomes useful for the long-term measurements of radon concentrations in various dwellings. The present method is useful for laboratories which wish to measure radon concentrations at low cost. The practical calibration coefficient Kc equals to 0.00424 tracks/cm /h/(Bq/m ). 

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