Examples exponential curve fitting

We do not design our own algorithm here but use the fin Insearch. m function supplied by Matlab. It is based on the original Nelder, Mead simplex algorithm. As an example, we re-analyse our exponential decay data Data Decay. m (see p. 106], this time fitting both parameters, the rate constant and the amplitude. Compare the results with those from the linearisation of the exponential curve, followed by a linear least-squares fit, as performed in Linearisation of Non-Linear Problems, (p.127). 

When using the above algorithm as an on-line help, the operator must pay attention to the fact that, in general, the mono-exponential hypotheses need not hold. If, for example, the estimated fitting error is too large, the usual course of action would be to increase either the number of scans or the number of blocks. When, however, the apparent fitting errors are due to the fact that the relaxation curves are non-exponential, rather than insufficient data quality, improving the precision of the curve will not be to be of... 

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. 

Once the production potential of the producing wells is insufficient to maintain the plateau rate, the decline periodbegins. For an individual well in depletion drive, this commences as soon as production starts, and a plateau for the field can only be maintained by drilling more wells. Well performance during the decline period can be estimated by decline curve analysis which assumes that the decline can be described by a mathematical formula. Examples of this would be to assume an exponential decline with 10% decline per annum, or a straight line relationship between the cumulative oil production and the logarithm of the water cut. These assumptions become more robust when based on a fit to measured production data. 

As has long been recognized it is extremely difficult to accurately fit experimental decay curves to sums of exponentials, especially for relatively close lifetimes (< factor of 2). 55,56,59,60) That is, one can get good fits but with parameters that are physically meaningless. The same is true of many different types of models. However, the point is so important as to justify repeating. Earlier we gave several examples. 55,56 We... 

One consequence of this situation is that the FFC relaxation curves cannot be analyzed assuming any fixed starting or ending value. For example, under the mono-exponential hypothesis, the relaxation rate R (inverse of the relaxation time) must be estimated by fitting the three-parameter formula... 

An example of the goodness-of-fit attainable with these equations is given in Figure 8, where the decay kinetics for electron transfer from A,A-dimethylaniline to photoexcited octadecylrhodamine on the surface of CTAB micelles is compared with the theoretical fits for two different concentrations of the donor [82bj. The theoretical decay curves were obtained by numerical integration of Eq. 13, followed by ensemble averaging over the Poisson distribution of donors within the micelles an exponential factor was also added to account for the spontaneous decay of the excited molecules. The parameters AGf and 1 were calculated by assuming that the donor and acceptor molecules are on the surface of a sphere of low dielectric... 

In various sections above we have discussed the simple kinetic schemes, some of which have been applied to experimental data on annealing of recoil atoms. For example, it was thought at first (91) that the kinetics of cobaltic trisethylenediamine annealing obeyed a linear combination of twm unimolecular terms. Further analysis (53) has shown that this interpretation was probably in error, and also that the potassium chromate and bromate data could not be fitted by a small number of simple exponential terms. We have also attempted to fit the potassium bromate and cobaltic trisethylenediamine data by the error-function expression suggested by Fletcher and Brown (10, 27), see Section Bid above, and although the fit is reasonably good over portions of the isothermal curves, the point of inflection near the time-zero axis in the error-function expression is not observed experimentally. 

Fitting the data to this simple phenomenological model affords a substantial dimensionality reduction while preserving most of the original fidelity. Each data matrix (intensity vs. frequency and time) is reduced to two vectors, namely temperature and area versus time. A representative example of the temperature and area curves are presented in Fig 5. The standard error in the area was about ten times greater than temperature, and is periodically displayed in Fig 5. Initial temperatures were 1700-2000 K and often followed an exponential decay with rates between 0.91-1.24 s Examining Fig 2, we note that errors in the residuals due to temporal aliasing contribute less than 0.5% since the interferometer scanned at least 16 times faster than the temperature decay rate. 

BDT, HW, and BK models extended the Ho-Lee model to match a term structure volatility curve (for example the cap prices) in addition to the term structure. The BK model is a generalization of the BDT model and it overcomes the problem of negative interest rates assuming that the short rate r is the exponential of an Ornstein-Uhlenbeck process having time-dependent coefficients. It is popular with practitioners because it fits the swaption volatility surface well. Nevertheless, it does not have closed formulae for bonds or options on bonds. 

FIGURE 7.16 (a) Example of fluorescence decays of Pro4[M + H] + ions at 303 and 438 K and the instrument response function (light solid curve) (b) fits to the measured decays by a stretched exponential model (solid black curves) showing the fit decay constant Linear and logarithmic intensity scales are used in (a) and (b), respectively. (From lavarone, A.T. Duft, D. Parks, J.H. J. Phys. Chem. 2006,110, 12714-12727. With permission.)... 

Using a low-coverage surface, single R6G molecules were positioned within the near field at places of maximum fluorescence intensity, and data were collected until each molecule photobleached (yielding from 10 to 10 photocounts). Examples of fluorescence decay curves from individual R6G molecules are shown in Figs. ll(e-h). Each is well fitted by a single exponential, but they are all very different from the bulk decay. This occurs because the molecules had different orientations, and the position of maximum fluorescence corresponded to different positions under the tip. 

The temporal evolution of the OKE signals in solutions and films of the linear conjugated polynitriles with different m values was observed. As an example, the time-resolved transient optical Kerr signals of PBN with m = 11.8 are presented (Fig. 24). The signal profiles of all samples are approximately symmetric with respect to the delay time, which indicates a primarily pulse width-limited response. To obtain the relaxation time of the Kerr medium we can fit the experimental curve with an exponential function [45,57]. Because the time constants of all samples are all less than the pulse duration, we can only roughly determine that the relaxation time of all samples is shorter than the laser pulse width (165 fs). The ultrafast optical response may be caused by distortion of the ir-electron cloud occurring with the nonresonant excitation [58]. 

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