Linear least squares

The data are also represented in Fig. 39.5a and have been replotted semi-logarithmically in Fig. 39.5b. Least squares linear regression of log Cp with respect to time t has been performed on the first nine data points. The last three points have been discarded as the corresponding concentration values are assumed to be close to the quantitation limit of the detection system and, hence, are endowed with a large relative error. We obtained the values of 1.701 and 0.005117 for the intercept log B and slope Sp, respectively. From these we derive the following pharmacokinetic quantities ... 

A least squares linear regression has been applied to the data pertaining to the p-phase, yielding the values of 1.745 and 0.005166 for the intercept log B and slope Sp, respectively. Using these results, we can compute the extrapolated plasma concentrations between 0 and 20 minutes. From the latter, we subtract the observed concentrations C which yields the concentrations of the a-phase C ... 

The residual a-phase concentrations C are shown in the semilogarithmic plot of Fig. 39.13b. Least-squares linear regression of log C upon time produced 1.524 and -0.02408 for the intercept log and the slope respectively. 

Concentrations of terbacil and its Metabolites A, B and C are calculated from a calibration curve for each analyte run concurrently with each sample set. The equation of the line based on the peak height of the standard versus nanograms injected is generated by least-squares linear regression analysis performed using Microsoft Excel. 

Ordinary least squares Linear projection Fixed shape, linear a, maximum squared correlation between projected inputs and output 0, minimum output prediction error... 

Least-squares linear regression analysis was performed on the data. 

Where AG is free energy, R is gas constant (1.987 cal/deg K mole-1), T is degrees Kelvin, and AS is entropy. Kd is the distribution constant of the herbicide between the solution phase and the adsorbed phase (equation 4). Thus, least squares linear regression analysis of ln(Kd) vs. 1/T yielded values for heats of adsorption (AH) for the herbicides in Keeton soil. 

As shown in Figure 6.21, excellent linearity was obtained, as represented by the high coefficient of correlation obtained for the least square linear regression. Similar results were obtained for the evaluation of autosampler accuracy when other analytes (propyl paraben and rhodamine 110 chloride) were employed in the determinations. Liu et al.9 conducted similar evaluations for the samples employed in the evaluation of the drug release rate profile of OROS with similar results to those discussed above. 

Fig. 5.6. Calibration curve for permanganate standards. Line is a least-squares linear regression for the data. Graphical interpolation is illustrated for an unknown with an absorbance of 0.443.

Fig. 18 Top CD spectra. Oligomer 15 (solid line) oligomer 15 in the presence of 100 equiv of (-)-a-pinene (dotted line) and 100 equiv of (+)-a-pinene (dashed line) in a mixed solvent of 40% water in acetonitrile (by volume) at 295 K. [15]=4.2 pmol. Bottom Plot of IniCn for 15 against the solvent composition. The solid line is the least-squares linear fit (correlation coefficient=0.9987), and the dotted line is the extrapolation to 100% water. Error bars are from the nonlinear fitting of the data to...

Fig. 28 Plot of A 3i4 vs mol % chiral octadecamer 45 (n= 18) for solutions of varying amounts of chiral and achiral 18 (n=18) octadecamers in different concentrations of water/acetoni-trile (v/v). All spectra were recorded in solutions with a total oligomer concentration of 3.3 pM. The dotted lines are the expected signals that should arise upon dilution of a sample containing only chiral octadecamer. The solid line is the least-squares linear fit of the chiral octadecamer dilution data (top left, correlation coefficient=0.998...

If a calibration function is used with coefficients obtained by fitting the response of an instrument to the model in known concentrations of calibration standards, then the uncertainty of this procedure must be taken into account. A classical least squares linear regression, the default regression... 

On these longer time scales, least square linear fits show an increase in UV-B of about 1% per year in the 90s at 305nm which is associated with a 0.45% per year ozone decline. This long-term change seems to be constant during the period and it is not a result of the period of sampling or of the improvement of air quality and changes in environmental conditions. The later were taken into account with a careful analysis of... 

The data fitting of the linear interval is made with a least squares linear regression. The obtained calibration lines were... 

Figure 18.5 Most atomic anions are not detectable on common ICS-AES instruments. Addition of detectable anions can be used as proxies for more abundant anions. For a DNA duplex it was shown that cacodylate serves as a good substitute. (A) Excluded anions can be measured in a regime that is independent of pH for a 24-bp DNA duplex (I 20 mM Na+ with 10 mM cacodylate and II 1 mM Mg + 20 mM Na+ with 10 mM cacodylate). (B) The calculated number of excluded anions (see Eq. (18.2)) does not depend on the fractional abundance of cacodylate for three ionic conditions (O) 20 mM Na+, ( ) 100 mM Na+, and (0) 1 mM Mg and 20 mMNaf Lines are least-squares linear fits. Total excluded anions per DNA are calculated according to Eq. (18.2) and are plotted above the main plot with the same symbols. Reprinted from Bai et al. (2007).

Comparison of V/Al atomic ratios for A120, and V/(Si+Al) atomic ratios for V-loaded gels are presented rrrFig. 8. Overall, and somewhat independent of whether or not the samples have been subjected to H, reduction, these ratios for the gel are approximately 30 s smaller than the values observed for V on Al-O,. The least squares linear regression fits to the V/A120, data extrapolates near the origin whereas the corresponding aaxa for V/gel does not, Fig. 8. Hence, V dispersion on the gel surface, as monitored by XPS, is lower than on Al-O,. The hydrothermal instability of V loaded gel (Table 1) is prooably responsible for the lower V dispersion and for the lower V/Al+Si ratios seen in Fig. 8. 

Least squares linear regression analyses of the data over the range of 20%-50% ACN gave the following equations ... 

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. 

These methods have been reviewed by Silas, Yates and Thornton [55]. In a PA-FTIR method Parker and Waddell also used the intensity of the 910 cm"1 butadiene band and ratioed it to the intensity of the 1450 cm 1 C-H bending band as a function of the % vinyl-butadiene [51]. The results can be expressed by a least squares linear regression equation over a range of 10%-60% vinyl-butadiene. 

The 1378 cm"1 band is from the CH3 symmetric bending and the 1156 cm 1 band is a complex skeletal vibration involving the CH3 branch of propylene. The 722 cm"1 band represents the CH2 rock and the 1462 cm"1 band is a combination of the CH2 scissor and the asymmetric CH3 bend. In the photoacoustic spectra the 1378 and 1462 bands are strong while the 1154 and 722 cm"1 bands are weak. Least squares linear regression... 

Linearity A calibration curve was obtained using the eight calibration standards that were described above. A 1/x2 weighted least squares linear regression using the area ratios of analyte/intemal standard against the nominal concentration was performed. A regression line was obtained from these data, which was used for back calculation of the concentration for unknowns and quality controls. 

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