APPLICATION OF BAYESIAN APPROACH OF LOGISTIC REGRESSION FORMATION WHILE PROCESSING THE TESTS RESULTS ON THE DURABILITY OF AIRCRAFTCONSTRUCTIONS ELEMENTS

When solving the problems of predicting the durability of aircraft structures elements in case of a bird strike, in a collision with large hail, foreign objects, it is necessary to know the dependence of the probability of the barrier penetration on the characteristics of the impactor and the impact speed. The results of the experiments have the form of a dichotomous variable having only two possible values (the barrier is punched or not), so the required dependence can be represented in the form of a logistic regression given by the integral function of the logistic distribution. To evaluate the parameters of logistic regression with a small number of experiments, it is advisable to use the Bayesian approach. One of the components of the Bayesian parameter estimation is the calculation of the likelihood function. The resulting expressions for the likelihood function and its logarithm were used to estimate the logistic regression parameters using the maximum likelihood method, which was realized by Newton method. The calculated iterative scheme of the maximum likelihood method is given. Based on the results of experimental data, estimates are obtained that are used to compare with the results obtained by the Bayesian approach. A special case is considered – the Bayesian estimation of the shift parameter of the logistic curve for a given value of the scale parameter, and also a general case – the estimation of the shift and scale parameters. A normal distribution is used as a priori distribution for the shift parameter, and a classical (two-parameter) gamma distribution is used for the scale parameter. Point estimation of logistic regression parameters was carried out by the modal value of a posteriori distribution. Using the modal rather than the mean value, made it possible to significantly reduce the amount of calculations due to the fact that in this case there is no need to calculate the integral probability in the denominator of the Bayesian formula. 

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