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Tham khảo tài liệu 'volume 17 - nondestructive evaluation and quality control part 19', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Fig. 2 Schematic of distribution of detection probabilities for cracks of fixed length Equation 1 implies that the POD a function is the curve through the averages of the individual density functions of the detection probabilities. This curve is the regression equation and provides the basis for testing assumptions about the applicability of various POD a models. In Ref 4 seven different functional forms were tested for applicability to available POD data and it was concluded that the log-logistics log odds function best modeled the data and provided an acceptable model for the data sets of the study. Note that the log odds model is commonly used in the analysis of binary hit miss data because of its analytical tractability and its close agreement with the cumulative log normal distribution Ref 8 . Two mathematically equivalent forms of the log odds model have subsequently been used. The earliest form is given by Eq 2 This parametrization can also be expressed as Eq 3 In the Eq 3 form the log of the odds of the probability of detection the left-hand side of Eq 3 is expressed as a linear function of ln a and is the source of the name of the log odds model. Note that given the results of a large number of independent inspections of a large number of cracks the parameters of the model can be fit with a regression analysis. As an example Fig. 3 shows Eq 3 fit to the data of Fig. 1. This regression approach will not be discussed further because the maximum likelihood estimates see the section Analysis of Hit Miss Data in this article can be applied to much smaller samples of inspection results and can give equivalent answers for large sample sizes. Fig. 3 Example linear relation between log odds of crack detection and log crack size Although the parametrizations of Eq 2 and 3 are sensible in terms of estimation through regression analyses 1 and Jare not easily interpretable in physical terms. A mathematically equivalent form of the log odds POD a model is given by Ref 8
