Regression trace analysis

If data of the real system is available, the developed simulation model can be tested for similarity with the real system in a qnantitative way (bottom-right cell in Table 4.8). For this purpose, a lot of statistical procedures can be applied depending on the specific object to be tested. Typically, regression techniqnes, distribution tests, or time series analysis methods are used. A reliable qnantitative approach is to generate a forecast of the near future by means of the simulation model which is then compared with the real systems behaviour after the forecast period has expired. This is called predictive validation A mixture of trace analysis and fixed-value test is the trace-driven simulation where a historical situation is simulated. The model s output is compared with the historical records then. 

Millard BJ (1982) Sources of error in quantitative mass spectrometry. In Facchetti S (ed) Applications of mass spectrometry to trace analysis lectures of a course held at the Joint Research Centre, 29 Sept-3 Oct 1980, Ispra, Italy. Elsevier, Amsterdam, pp 163-179 Miller JN (1991) Basic statistical metiiods for analytical chemistry. Part 2. Calibration and regression methods. Analyst 116 3-14... 

Conventional XRF analysis uses calibration by regression, which is quite feasible for known matrices. Both single and multi-element standards are in use, prepared for example by vacuum evaporation of elements or compounds on a thin Mylar film. Comparing the X-ray intensities of the sample with those of a standard, allows quantitative analysis. Depending on the degree of similarity between sample and standard, a small or large correction for matrix effects is required. Calibration standards and samples must be carefully prepared standards must be checked frequently because of polymer degradation from continued exposure to X-rays. For trace-element determination, a standard very close in composition to the sample is required. This may be a certified reference material or a sample analysed by a primary technique (e.g. NAA). Standard reference material for rubber samples is not commercially available. Use can also be made of an internal standard,... 

Jin Q., Wang Z., Shan X., Tu Q., Wen B., Chen B. Evaluation of plant availability of soil trace metals by chemical fractionation and multiple regression analysis. Environ Pollut 1996 91 309-315. 

Vukjovic et al.199 recently proposed a simple, fast, sensitive, and low-cost procedure based on solid phase spectrophotometric (SPS) and multicomponent analysis by multiple linear regression (MA) to determine traces of heavy metals in pharmaceuticals. Other spectroscopic techniques employed for high-throughput pharmaceutical analysis include laser-induced breakdown spectroscopy (LIBS),200 201 fluorescence spectroscopy,202 204 diffusive reflectance spectroscopy,205 laser-based nephelometry,206 automated polarized light microscopy,207 and laser diffraction and image analysis.208... 

The samples were analyzed for trace metals and sulfate as well as for three fractions of particulate organic matter (POM) using sequential extraction with cyclohexane (CYC), dichloromethane (DCM) and acetone (ACE). Factor analysis was used to identify the principal types of emission sources and select source tracers. Using the selected source tracers, models were developed of the form POM = a(V) + b(Pb) + - - -, where a and b are regression coefficients determined from ambient data adjusted to constant dispersion conditions. The models for CYC and ACE together, which constitute 90% of the POM, indicate that 40% (3.0 pg/m ) of the mass was associated with oil-burning, 19% (1.4 pg/m ) was from automotive and related sources and 15% (1.1 pg/m ) was associated with soil-like particles. 

In developing a multiple regression model for apportioning sources of TSP in New York City, Kleinman, et al.(2) selected Pb, Mn, Cu, V and SO, as tracers for automotive sources, soil-related sources, incineration, oil-burning and secondary particulate matter, respectively. These were chosen on the basis of the results of factor analysis and a qualitative knowledge of the principal types of sources in New York City and the trace metals present in emissions from these types of sources. Secondary TSP, automotive sources and soil resuspension were found to be the principal sources of TSP in 1974 and 1975 ( ). 

In organic compound analysis, the instrument response is expressed as a response factor (RF), which is the ratio of the concentration (or the mass) of the analyte in a standard to the area of the chromatographic peak. Conversely, a calibration factor (CF) is the ratio of the peak area to the concentration (or the mass) of the analyte. Equation 1, Appendix 22, shows the calculation of RF and CF. In trace element and inorganic compound analyses, the calibration curve is usually defined with a linear regression equation, and response (calibration) factors are not used for quantitation. 

Analysis of variance (ANOVA and MANOVA) has been used to investigate the influence of location on forms of metals in roadside soil (Nowak, 1995). Multiple regression analysis has proved valuable in processing sequential extraction data to obtain information on plant availability of trace metals in soils (Qian et al, 1996 ... 

An alternative method was also studied. This involved ratioing the intensity of the 698 cm 1 styrene band to the intensity of the 2921 cm-1 C-H stretching vibration. Since oils and other additives would interfere with this approach they were extracted with acetone. Vacuum oven drying was then necessary to remove all traces of acetone prior to PA analysis. Otherwise the PA spectrum would be that of acetone rather than that of the rubber since the gas phase spectrum of the acetone would overwhelm that of the solid phase rubber. This technique allowed both solution and emulsion SBR to be analysed by a common method. The results can be expressed by a least squares linear regression equation over the range of 5%-40% styrene in SBR. 

The stationary case is given by the description shown below in Fig. 2.2. It is a sketch of the way by which complex formation constants are traced back to and correlated with E (L) using the most simple approach of linear regression analysis (applications in Figs. 2.2 and 2.15). There are two terms in Eq. 2.4 one of which (c) describes an intrinsic binding stability... 

These mathematical models arc generally used for determining minor clement and trace element concentrations. In the case of major elements, a variation in the concentration level of these elements induces massive matrix variations and. in this case, it is preferable to use. where possible, methods that attenuate the matrix effects prior to using analysis by regression. 

Whether this is considered a problem may depend on the situation. In the context of assessing the ability to trace analytical methods from regression analysis of paired patient sample measurements, an error of 1% solely because of the applied regression procedure would indeed constitute a problem (see Traceability and Measurement Uncertainty later in this chapter). 

For the first set of materials, and with the aim of assessing the dispatch conditions, a short-term stability study was conducted at 40°C. The layout chosen for the stability study was the so-called isochronous scheme samples were taken from the bulk, placed at 40° C and then moved back to the reference temperature (4°C), after 1 and 2 weeks. Then, at the same time, the samples were analysed for major components and trace elements. The results, 3 time-points (0, 1, 2 weeks) and 2 units analysed per time-point, were evaluated by one-way analysis of variance ANOVA. As some parameters (especially As, Cd, Cu, and to a minor extent also Mn, pH) showed a statistically significant slope of the regression line, it was decided to assure the dispatch of the samples at 4°C (with cooling elements). 

Figure 9.9 shows the distribution of lutetium (using trace concentrations of "Lu, (jS y ti 6.71 d) between an aqueous solution and benzene containing an organic complex former (HA = acetylacetone). Eqn. (9.23.a) has been Etted to the experimental points by regression analysis to yield the equilibrium constants, by which the solid curves have been calculated. 

NB Apparent pKa values were determined by titration in eitiier aqueous DMSO (30-80 wt%) solutions or in aqueous metiianol (10-50 wt%) solutions. Ionic strength effects were corrected with Davies modification of the Debye-Huckel equation. Wei t percent compositions were converted to mole fraction and plots (often exhibiting traces of curvature) of apparent pX were extrapolated to 100% water by linear regression analysis. The following values were reported for tiie pJCa value in water (pX )-... 

The system coefficients of the skin/medium [c, r, s, a, b, v] can be obtained by multiple linear regression analysis of the LEER equation matrix (Equation 5.2). These system coefficients are properties of the skin/medium systan. They will not change with minor or trace chemicals in composition or proportion. 

Using a portable beam stability-controlled XRF spectrometer, Romano et al. (2005) have determined the concentrations of Rb, Sr, Y, Zr, and Nb in 50 fine potsherds from the votive deposit of San Francesco in Catania (Italy) by using a multilinear regression method in their bid for quantitative nondestructive determination of trace elements in archaeological pottery. A small portion of a few potsherds was even powdered in order to test the homogeneity of the material composing the fine pottery samples and the XRF data were compared with those obtained by chemical analysis of the powdered samples. 

The application of statistical methods to research dealing with foods has included determinations of the significance of differences, making of confidence statements concerning estimates, and tracing trends through the use of analysis of variance and/or regression techniques. At the present time a considerable portion of the literature on foods contains some use of statistical methods. The possibilities of more extensive use of statistical methods in food research are vast. However, still wider use is needed if we are to make the most of this valuable tool. 

---