Curve fitting exponential

Fig. 10. Concentration curves fitted by exponential regression for the reaction of l,2,4,5r tetrabromobenzene (A) with surface bound chloride, a consecutive four-stage reaction with three intermediates (B,C,D) and the product E.

The calibration curve is generated by plotting the peak area of each analyte in a calibration standard against its concentration. Least-squares estimates of the data points are used to define the calibration curve. Linear, exponential, or quadratic calibration curves may be used, but the analyte levels for all the samples from the same protocol must be analyzed with the same curve fit. In the event that analyte responses exceed the upper range of the standard calibration curve by more than 20%, the samples must be reanalyzed with extended standards or diluted into the existing calibration range. 

Weighting of the calibration curve, 1 /x or 1 /x, is expected to provide better curve fit at the lower concentration levels. Alternative calculations, such as exponential or quadratic curve fits, are acceptable if they provide improved precision and/or accuracy. 

Figure 3.23. Plots of the Raman band integration of the strong 1550-1600 cm feature associated with the first species in the 342nm ps-KTR spectra (open squares and circles) and the strong 1630 cm region feature associated with the second species in the 400 nm ps-KTR spectra (solid squares and circles). Data are shown for ps-KTR spectra obtained in 25% water/75% acetonitrile (squares) and 50% water/50% acetonitrile solvent (circles) systems. The lines present best-fit exponential decay and growth curves to the data. (Reprinted with permission from reference [25]. Copyright (2004) American Chemical Society.)...

Fig. 2.5.5 A study examining the conformational changes of the protein ubiquitin, showing the population ratio of the A-state to the native-state as a function of time, (a) The reaction from 0 to 120 s. (b) The reaction for the first 40 s, including curves fit to a single exponential. Reprinted with permission from Ref. [37]. Copyright (2003) American Chemical Society.

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

Figure 4-21. The logarithmic transform of the exponential data set used in Figure 4-4. The fitted exponential curve appears as a straight line.

Fig. 3.16 The ALIS-MS responses from a dissociation rate experiment for a mixture of Zap-70 ligands using staurosporine as the quench reagent. See text for details. (A) The raw data and its fit curve for NGD-6367, one of the compounds in the mixture. (B) The exponential decay curves fit to normalized...

Figures 2 and 3 show typical test results for flux decline in laminar flow where the pressure and temperature are varied and the Reynolds number is held fixed. Similar behaviors are found with variations in Reynolds number and for turbulent flow. The important feature of the data is that the flux decline is exponential with time and an asymptotic equilibrium value is reached. Each solid curve drawn through the experimental points is a least-square fit exponential curve defined by Eq. (19). It is interesting to note that Merten et al ( ) in 1966 had observed an exponential flux decay in their reverse osmosis experiments. However, Thomas and his co-workers in their later experiments reported an algebraic flux decay with time (4,5).

In Figures 8 and 9 are shown the data for the dependence of the characteristic film buildup time t on Apg and U. In accord with the model, t is found to be independent of U, with only a very weak dependence on Apg indicated. This latter result could in part be a function of experimental inaccuracy. The data reduction for t introduces no assumptions beyond that needed to draw the exponential flux decline curves such as those shown in Figures 2 and 3. However, an error analysis shows that the maximum errors relative to the exponential curve fits occur at the earlier times of the experiment. This is seen in the typical error curve plotted in Figure 10. The error analysis indicates that during the early fouling stage the relatively crude experimental procedure used is not sufficiently accurate or possibly that the assumed flux decline behavior is not exponential at the early times. In any case, it follows that the accuracy of the determination of 6f is greater than that for t. 

Figure 10. Typical error curve for exponential fit of data...

Figure 1-5 Determination of the order of hypothetical reactions with respect to species A. (a) The initial reaction rate method is used. The initial rate versus the initial concentration of A is plotted on a log-log diagram. The slope 2 is the order of the reaction with respect to A. The intercept is related to k. (b) The concentration evolution method is used. Because the exponential function (dashed curve) does not fit the data (points) well, the order is not 1. The solution for the second-order reaction equation (solid curve) fits the data well. Hence, the order of the reaction is 2.

The increment in mechanical properties (tensile strength, 300% modulus, and Young s modulus) as a function of SAF is plotted in Fig. 39. In general, the higher level of SAF, which in turn indicates better exfoliation, results in high level of property enhancement. However, the level of increment with the increase in SAF is different in all three cases and follows a typical exponential growth pattern. The apparent nonlinear curve fitting of the experimental values presented in Fig. 39 is a measure of the dependence of mechanical properties on the proposed SAF function. 

The curve fitting programs cope better with fewer variables in the equations. Try to reduce the number of variables. For example, suppose you have to fit a multiphasic curve to three exponentials that are moderately separated in time. There are seven unknowns three rate constants three amplitudes and an endpoint. If the slowest phase is sufficiently separated from the second, first fit the tail of the slowest phase to a single exponential. Then fit the whole curve to a triple exponential equation in which the rate constant and the amplitude that were derived for the third phase are used as constants. Use a time window that focuses on the first two phases and not the whole time course. Similarly, if the first phase is much faster than the second and third, fit the tail of the process to two exponentials. Then fit the fast time region to a triple exponential in which the last two phases have fixed rate constants and amplitudes. 

Figure 8.16 Reaction curve for the dehydration of carbonic acid by conductometric detection. Lower dots, every tenth experimental point solid line, fitted exponential curve. Middle dots, logarithmic plot, every tenth point solid line, least-squares line. Upper residuals (G - Gcajcd). [From Ref. 20, reprinted with permission. Copyright 1978 American Chemical Society.]...

Real (viscoelastic) materials give an intermediate response that is an exponential curve. The exponential time constants associated with the curve are used to approximate the relaxation times of the material itself. Thus, the shape of the output curve is analyzed to give viscoelastic information, although this model fitting is only strictly legitimate in the linear viscoelastic region. Workers have shown that the mechanical parts of the models (springs and dashpots) can be associated with specific parts of a food s makeup. 

Fig. 3 Temporal trend of BB-153 concentrations in fishes from the Great Lakes. The best-fit exponential curves are shown the fitted lines have the following correlation coefficients (r2) Superior, 0.481 Huron, 0.739 Michigan, 0.236 Erie, 0.128 and Ontario, 0.527. Only the regression for Lake Huron is statistically significant. From Zhu and Hites [19]...

Typical kinetic profiles (hybridoma). (A) Cell concentration and viability (B) glucose consumption (GLC) and lactate production (LAC) (C) monoclonal antibody production (mAb) (D) glutamine consumption (GLN) and ammonium production (NH4+) (E) specific growth rate (px) (F) alanine (ALA) and glycine (GLY) production. Adapted from Lee (2003). Symbols correspond to the experimental data and the lines to the manual curve fitting. Vertical lines indicate the instant at which exponential growth phase ended (gx < Px.max)-... 

Fig. 4.29. Normalized integrated intensities (left) of substrate core levels in dependence on deposition time for the spectra shown in Fig. 4.26. The deposition rate is estimated to be 2nmmin 1. The lines in the left graph are obtained by curve fitting of the data to an exponential decay. The derived attenuation times are displayed in the right graph in dependence on electron kinetic energy together with theoretical energy-dependent escape depth calculated using the formula by Tanuma, Powell, and Penn [37] and using a y/ E law [38]...

Figure 6 (a) Decay ofIR absorption atl984 cm 1 with exponential curve fit (b) plots ofkobs v.v. [Mel] for oxidative addition reactions of [Polymer][Rh(CO)2I2] and Bu4N[Rh(CO)2I2] (25 °C)... 

The temporal behavior of the various fragments is presented in Fig. 21 and the rise and decay times obtained by either single or multiple exponential curve fitting procedures are presented in Table 5. These results can be summarized as follows ... 

The Freundlich equation is an exponential relation of gas loading with adsorbate gas pressure. The form of the equation used for curve fitting is given below ... 

When P > 3, exponential curve-fitting procedures for the WSGG spectral model become significantly more difficult for hand computation but are quite routine with the aid of a variety of readily available... 

Figure 3.49 Viscosity as a function of temperature for IM, 3M and SM aqueous di-methylsulfoxide (DMSO) solutions. Solid curves are curve fits to the data ba on the exponential expression tj = tj, exp [ACocf (IVT - INT )], where T is absolute temperature in Kelvin, R is the universal gas content, A o is the activation energy, Tp is the phase change temperature, and r)pi, is the viscosity of the solution at the phase-change temperature. The dashed curve is the exponential curve fit for water, and the dashed circles indicate the region of temperature where phase change occurred. (Reprinted with pennis-sion. See Ref. [72aJ. 1993 ASME International.)...

Figure 2 a) typical experimental decay of the 2F component of the signal (full line) and the corresponding fitted exponential curve (dashed line), b) plot of the inverse of the relaxation time of the exponential decay versus the square of the wave vector, the linear fit indicates that the process is diffusive and the slope of the line is equal to the diffusion coefficient. 

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