Measurement errors Fitting

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

For complex reactions more than one dependent variable is measured. The fitting procedure should take all the observed variables into account. When each of the variables has a normally distributed error, all data are equally precise, and there is no correlation between the variables measured, parameters can be estimated by minimizing the following function ... 

However, an important question that needs to be answered is "what constitutes a satisfactory polynomial fit " An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. 

The MaxEnt method will always deflate deformation features by the (<80 ) ,1S corresponding to measurements error [39]. To obtain an empirical estimate of this intrinsic spread allowed by the noise, twenty noisy data sets were generated as in formula (31), and fitted with BUSTER using the fragment and NUP already described in the previous paragraph. 

The more the precision of the instrument, and the more the points for the time unit in the acquired profile, the better the result of the fitting of experimental data. For this reason instruments with a low measure error and connectable to a computer for the automatic and continous aquisition of data are very much prefered. The UV-Vis spectrophotometer is by far the most used instrument in chemical kinetics. It has a good sensitivity and a good control of the temperature. It is connected or easily connectable to a computer and is available nearly everywhere. The absorbance has a very low dependence on the temperature so that, in the used temperature range, its variation can be neglected during the VTK experiments. 

Since tar concentrations are also corrupted by measurement errors, and since we do not know which variable is more reliable, it is equally meaningful to fit the inverse model x = Ay + to the data, thereby regarding the nicotine concentration as independent variable. Show that the two regression... 

The probability of F is the probability of finding the observed ratio of mean squares given the null hypothesis that the data do indeed fit the calibration model. If this probability falls below an acceptable level (say, a = 0.05), then H0 can be rejected at the (1 — a) level. If H0 is not rejected, then both MSLOF and MSME are estimates of the measurement error and can be pooled to give a better estimate a2 = (SSLOF + SSME)/(N — 2) with N—2 degrees of freedom. The steps are given in table 8.4. 

Spreadsheet 8.2. Calculations for test of linear range of data shown in spreadsheet 8.1. Range tested, 3.1-50.0 nM N=24 data points number of concentrations, k, = 6. ME = measurement error, LOF = lack of fit, SS = sum of squares, MS = mean square. 

This is acceptable provided that the assumption is stated up front. The melt curves in the lower part of Fig. 17.3 were fit this way. Alternatively, the full model can be used, with additional steps taken to ensure the validity of the results. An example is shown in the upper part of Fig. 17.3. Here, the experiments were repeated multiple times (minimizing the measurement error), the data were fit simultaneously (using a global fitting algorithm), and the results were corroborated using a separate singular value decomposition analysis. 

In absorption spectrometry, <7i is usually fairly constant, and x1 fitting has no advantages. Typical examples of data with nonconstant and known standard deviations are encountered in emission spectroscopy, particularly if photon counting techniques are employed, which are used for the analysis of very fast luminescence decays [27], In such cases, measurement errors follow a Poisson distribution instead... 

Suppose that the variables BJ are to be determined by a least-squares fit of the relations, Eq. 16, to the measured values T exp (vector Yexp). Assume that the measurements Yexp are unbiased ( (Yexp) = Ytrue where E() represents the mean or expectation value) and that the measurement errors and their correlations are described by the positive-definite nxn variance-covariance matrix 0Y which can be written as the dyadic P... 

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