Maximum-likelihood method processing data

It is worth mentioning that statistical and probabilistic techniques have had a significant impact in how heavy atoms are found and models are refined (e.g., SHARP, SOLVE, REFMAC). Baysian statistics and maximum likelihood methods are now used instead of least-squares methods. One may want to consider how various data collection strategies may affect the later steps in the process by keeping this in mind, i.e., high redundancy in the data makes for better statistics. 

The fission process is less well understood than the others, and there is not a good theoretical argument for a particular statistical distribution. Porter and Thomas 42) predict that it should be a fairly broad distribution corresponding to a small value of v (say, in the range 1 to 4) rather than a very high value as is the case with capture widths. One is forced to rely on the experimental data in spite of the poor statistics. Fischer 32) analyzed 12 resonances of and 19 resonances of Pu , both by the moments method (79) and the maximum likelihood method (87), with the results shown in Table V. 

Another class of methods such as Maximum Entropy, Maximum Likelihood and Least Squares Estimation, do not attempt to undo damage which is already in the data. The data themselves remain untouched. Instead, information in the data is reconstructed by repeatedly taking revised trial data fx) (e.g. a spectrum or chromatogram), which are damaged as they would have been measured by the original instrument. This requires that the damaging process which causes the broadening of the measured peaks is known. Thus an estimate g(x) is calculated from a trial spectrum fx) which is convoluted with a supposedly known point-spread function h(x). The residuals e(x) = g(x) - g(x) are inspected and compared with the noise n(x). Criteria to evaluate these residuals are Maximum Entropy (see Section 40.7.2) and Maximum Likelihood (Section 40.7.1). 

Assumptions may be made or models adopted (often by implication) about a system being measured that are not consistent with reality. The selection of the method of data reduction may be partly on the basis of the model adopted and partly on the basis of features such as computation time and simplicity. Kelly classified data processing methods as direct, graphical, minmax, least squares, maximum likelihood, and bayesian. Each method has rules by which computations are made, and each produces an estimate (or numerical result) of reality. 

Existing methods for data rectification with process models including, maximum likelihood and Bayesian methods, are inherently single-scale in nature, since they represent the data at the same resolution everywhere in time and frequency. The multiscale Bayesian data rectification method developed in this section combines the benefits of Bayesian rectification and multiscale filtering using orthonormal wavelets. 

Firstly, we process two kinds of crack initiation site data uniformly. In order to test the probability distribution of fatigue life, we assume that the fatigue life submits the lognormal distribution. To make the runout data (>10 ) available (if there is such data), maximum likelihood estimation method with curtailed time data can be used to estimate the mean and standard deviation. If n specimens are tested at a certain stress level, the curtailed time is T, the number of failure specimens is r and their lives are ... 

Similarly, the estimators of the mean and standard deviation using maximum likelihood estimation method for curtailed time data is fi= 19.4409 and(T = 1.3110. And the critical value r , is 0.5714 when significance level takes 0.1. The calculation process is also shown in Table 3. 

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