Maximum-likelihood method

When there is significant random error in all the variables, as in this example, the maximum-likelihood method can lead to better parameter estimates than those obtained by other methods. When Barker s method was used to estimate the van Laar parameters for the acetone-methanol system from these data, it was estimated that = 0.960 and A j = 0.633, compared with A 2 0.857 and A2- = 0.681 using the method of maximum likelihood. Barker s method uses only the P-T-x data and assumes that the T and x measurements are error free. 

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. 

The maximum-likelihood method is not limited to phase equilibrium data. It is applicable to any type of data for which a model can be postulated and for which there are known random measurement errors in the variables. P-V-T data, enthalpy data, solid-liquid adsorption data, etc., can all be reduced by this method. The advantages indicated here for vapor-liquid equilibrium data apply also to other data. 

The maximum-likelihood method, like any statistical tool, is useful for correlating and critically examining experimental information. However, it can never be a substitute for that information. While a statistical tool is useful for minimizing the required experimental effort, reliable calculated phase equilibria can only be obtained if at least some pertinent and reliable experimental data are at hand. 

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. 

Parameter Estimation. WeibuU parameters can be estimated using the usual statistical procedures however, a computer is needed to solve readily the equations. A computer program based on the maximum likelihood method is presented in Reference 22. Graphical estimation can be made on WeibuU paper without the aid of a computer however, the results caimot be expected to be as accurate and consistent. 

Maximum likelihood methods used in classical statistics are not valid to estimate the 6 s or the q s. Bayesian methods have only become possible with the development of Gibbs sampling methods described above, because to form the likelihood for a full data set entails the product of many sums of the form of Eq. (24) ... 

Maximum likelihood methods are commonly used to estimate parameters from noisy data. Such methods can be applied to image restoration, possibly with additional constraints (e.g., positivity). Maximum likelihood methods are however not appropriate for solving ill-conditioned inverse problems as will be shown in this section. 

While it is perfectly permissible to estimate a and b on this basis, the calculation can only be done in an iterative fashion, that is, both a and b are varied in increasingly smaller steps (see Optimization Techniques, Section 3.5) and each time the squared residuals are calculated and summed. The combination of a and b that yields the smallest of such sums represents the solution. Despite digital computers, Adcock s solution, a special case of the maximum likelihood method, is not widely used the additional computational effort and the more complicated software are not justified by the improved (a debatable notion) results, and the process is not at all transparent, i.e., not amenable to manual verification. 

Murshudov GN, Vagin AA, Oodson EJ. Refinement of macromolecular structures by the maximum-likelihood method. Acta Cryst 1997 053 240-55. 

The parameter values found by the two methods differ slightly owing to the different criteria used which were the least squares method for ESL and the maximum-likelihood method for SIMUSOLV and because the T=10 data point was included with the ESL run. The output curve is very similar and the parameters agree within the expected standard deviation. The quality of parameter estimation can also be judged from a contour plot as given in Fig. 2.41. 

Shirts, M. R. Bair, E. Hooker, G. Pande, V. S., Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods, Phys. Rev. Lett. 2003, 91,140601... 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

Maximum likelihood method The estimate of a parameter 9, based on a random sample Xi, X2, , Xn, is that value of 9 which maximizes the likelihood function L(Xi, X2, , Xn, 9) which is defined as... 

Everitt Hand (1981) recommended use of a sample of 200 or more to conduct a valid analysis of mixtures of populations. Even the maximum likelihood method, the... 

The method of maximum likelihood provides estimators which are usually quite satisfactory. They have the desirable properties of being consistent, asymptotically normal, and asymptotically efficient for large samples under quite general conditions. They are often biased, but the bias is frequently removable by a simple adjustment. Other methods of obtaining estimators are also available, but the maximum likelihood method is the most frequently used. 

These maximum likelihood methods can be used to obtain point estimates of a parameter, but we must remember that a point estimator is a random variable distributed in some way around the true value of the parameter. The true parameter value may be higher or lower than our estimate, ft is often useftd therefore to obtain an interval within which we are reasonably confident the true value will he, and the generally accepted method is to construct what are known as confidence limits. 

Strimmer K., and von Haeseler, A. (1996). Quartet puzzling a quartet maximum-likelihood method for reconstructing tree topologies. Mol. Biol. Evol. 13, 964—969. 

Note that there is a strong similarity to LDA (Section 5.2.1), because it can be shown that also for LDA the log-ratio of the posterior probabilities is modeled by a linear function of the x-variables. However, for LR, we make no assumption for the data distribution, and the parameters are estimated differently. The estimation of the coefficients b0, b, ..., bm is done by the maximum likelihood method which leads to an iteratively reweighted least squares (IRLS) algorithm (Hastie et al. 2001). 

In addition to these three major methods mentioned, several other computational approaches can also be used to deal with population stratification. For example, ADMIXMAP (22-26) is a model-based method that estimates the individual history of admixture. It can be applied to an admixed population with two or more ancestral populations. It also tests the association of a trait with ancestry at a marker locus with control for population structure. Wu et al. developed a software package in R (PSMIX) for the inference of population stratification and admixture (27). PSMIX is based on the maximum likelihood method. It performs as well as model-based methods such as STRUCTURE and is more computationally efficient. 

Wu, B., Liu, N., and Zhao, H. (2006) PSMIX an R package for population structure inference via maximum likelihood method. BMC Bioinformatics. 7, 317. Available at http //bioinfor-matics.med.yale.edu/PSMIX/. 

BIOMINERALIZATION MATRIX ISOLATION MAXIMUM LIKELIHOOD METHOD Maximum velocity,... 

W = AG. Of course, this relation can be tested only in the region of work values along the work axis where both distributions (forward and reverse) overlap. An overlap between the forward and reverse distributions is hardly observed if the molecules are pulled too fast or if the number of pulls is too small. In such cases, other statistical methods (Rennet s acceptance ratio or maximum likelihood methods. Section IV.B.3) can be applied to get reliable estimates of AG. The validity of the CFT has been tested in the case of the RNA hairpin CD4 previously mentioned and the three-way junction RNA molecule as well. Figure 9c,d and Fig. 10c show results for these two molecules. 

Chemical Element Balances Maximum Likelihood Method... 

Two adjustable parameters of fhe equafions can be found by an optimization technique using Marquardt s or Rosenbrock s maximum likelihood method of minimizafion... 

Figure 13.4. Phylogenetic analysis with WebPhylip. The phylogenetic analysis of DNA sequences encoding lysozyme precursors is performed with WebPhylip by maximum likelihood method.

Swift, M., J. Cohen, and R. Pinkham. A maximum-likelihood method for estimating the disease predisposition of heterozygotes. Am. j. Hum. Genet. 26 304-317, 1974. 

The objective function value is up to a constant, equal to minus twice the log-likelihood of the fit. Thus, a minimum objective function value reflects the maximum likelihood of the model parameters to describe the data best. The standard errors of the parameter estimates are also calculated by the maximum likelihood method. 

---