Maximum Likelihood Fits

Bro R, Sidiropoulos ND, Smilde AK, Maximum likelihood fitting using simple least squares algorithms, Journal of Chemometrics, 2002, 16, 387 100. 

Fig. 6.7 distribution measured in data. Overlaid are the result of the maximum likelihood fit and the simulated template distributions. The dashed and the dotted line are the b- and cMiip-template, respectively. The filled circles correspond to the data distribution, while the solid line is the result of the fitting procedure... 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

One limitation of clique detection is that it needs to be run repeatedly with differei reference conformations and the run-time scales with the number of conformations pt molecule. The maximum likelihood method [Bamum et al. 1996] eliminates the need for reference conformation, effectively enabling every conformation of every molecule to a< as the reference. Despite this, the algorithm scales linearly with the number of conformatior per molecule, so enabling a larger number of conformations (up to a few hundred) to b handled. In addition, the method scores each of the possible pharmacophores based upo the extent to which it fits the set of input molecules and an estimate of its rarity. It is nc required that every molecule has to be able to match every feature for the pharmacophor to be considered. 

We thus get the values of a and h with maximum likelihood as well as the variances of a and h Using the value of yj for this a and h, we can also calculate the goodness of fit, P In addition, the linear correlation coefficient / is related by... 

Koninklijke Nederlands Akadamie van Wetenschappen 53 386-392, 521-525, 1397-1412 Titterington DM, Halliday AN (1979) On the fitting of parallel isochrons and the method of maximum likelihood. Chem Geol 26 183-195... 

Compute the phase behavior of the system and compare with the data. If the fit is excellent then proceed to maximum likelihood parameter estimation. 

Conversion of experimental dose/response data into a form suitable for extrapolation of human risk using least squares or, more usually, maximum likelihood curve fits. 

Fig. 21.8. Frequency histograms of single-channel conductances for (A) SENS and (B) LEVR parasites. Gaussian curves were fitted to each distribution using the maximum likelihood procedure. The peaks for the SENS isolate were 21.4 + 2.3 pS (8% area) labelled G25 33.0 + 4.8 pS (31% area) labelled G35 38.1 + 1.2 pS (19% area) labelled G40 and 44.3 + 2.2 pS (42% area) labelled G45. The peaks for the LEVR isolate were 25.2 4.5 pS (21% area) labelled G25 41.2 1.7 pS (49% area) labelled G40 and 46.7 1.1 pS (30% area) labelled G45.

When experimental data is to be fit with a mathematical model, it is necessary to allow for the fact 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 linear 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 linear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. See Press et al. (1986) for a description of maximum likelihood as it applies to both linear and nonlinear least squares. 

Finally, it is interesting to note that biases can be introduced by data fitting at low counts even with the use of ordinarily unbiased estimators like the maximum likelihood estimator [37],... 

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

The basis upon which this concept rests is the very fact that not all the data follows the same equation. Another way to express this is to note that an equation describes a line (or more generally, a plane or hyperplane if more than two dimensions are involved. In fact, anywhere in this discussion, when we talk about a calibration line, you should mentally add the phrase ... or plane, or hyperplane... ). Thus any point that fits the equation will fall exactly on the line. On the other hand, since the data points themselves do not fall on the line (recall that, by definition, the line is generated by applying some sort of [at this point undefined] averaging process), any given data point will not fall on the line described by the equation. The difference between these two points, the one on the line described by the equation and the one described by the data, is the error in the estimate of that data point by the equation. For each of the data points there is a corresponding point described by the equation, and therefore a corresponding error. The least square principle states that the sum of the squares of all these errors should have a minimum value and as we stated above, this will also provide the maximum likelihood equation. 

Mathematically, all this iirformation is used to calculate the best fit of the model to the experimental data. Two techniques are currently used, least squares and maximum hkelihood. Least-squares refinement is the same mathematical approach that is used to fit the best line through a number of points, so that the sum of the squares of the deviations from the line is at a minimum. Maximum likelihood is a more general approach that is the more common approach currently used. This method is based on the probability function that a certain model is correct for a given set of observations. This is done for each reflection, and the probabilities are then combined into a joint probability for the entire set of reflections. Both these approaches are performed over a number of cycles until the changes in the parameters become small. The refinement has then converged to a final set of parameters. 

In the BMD approach, a curve is fitted to discrete responses (binary, dichotomous/quantal data, i.e., yes/no) or to continuous mean effect values (a response such as weight that can assume any value in a range). The curve is usually fitted to data using the maximum likelihood approach. 

This model assumes that any dosage effect has the same mechanism as that which causes the background incidence. Low-dose linearity follows directly from this additive assumption, provided that any fraction of the background effect is additive no matter how small. A best fit curve is fitted to the data obtained from a long-term rodent cancer bioassay using computer programs. The estimates of the parameters in the polynomial are called Maximum Likelihood Estimates (MLE), based upon the statistical procedure used for fitting the curve, and can be considered as best fit estimates. Provided the fit of the model is satisfactory, the estimates of these parameters are used to extrapolate to low-dose exposures. 

The criterion for best fit is based on the maximum likelihood principle (Fisher 1922) where the best estimates of the model parameters should maximise the likelihood function, L, for the observation of N different experimental observations. 

Maximum likelihood (ML) is the approach most commonly used to fit a parametric distribution (Madgett 1998 Vose 2000). The idea is to choose the parameter values that maximize the probability of the data actually observed (for fitting discrete distributions) or the joint density of the data observed (for continuous distributions). Estimates or estimators based on the ML approach are termed maximum-likelihood estimates or estimators (MLEs). 

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