Reduced-order Likelihood Function

Use the vector Y / to denote the zero-mean random vector comprised of the response measurements from time n At to n At (n n ) in a time-descending order  

the joint PDF p(yi, y2,.. , yNp 0, C) follows an NgNp-vaiisite Gaussian distribution with zero mean and covariance matrix  

To compute this likelihood function, it involves only the solution of the linear algebraic equation = Ni jjp and the determinant of the matrix Xy, jvp which is NoNp x NoNp only. 

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Then, in the same manner, the reduced-order joint PDF p y, y2, , yA -i, C) can be further expanded. By continuing this process, the likelihood function can be factorized as a product of a reduced-order likelihood function and a number of conditional PDFs ... 

Therefore, the reduced-order likelihood function and the conditional PDFs in the approximated expansion are available and the procedure can be summarized as follows ... 

The reduced-order likelihood function p(yi, y2, yjVpl<, C) can be calculated using Equation (4.42) along with Equation (4.40) and (4.41). 

This approximation resolves the computational difficulty encountered in the direct exact formulation that requires repeated computations of the solution of linear simultaneous algebraic equations and determinants of the matrices with huge dimensions. The efficiency in the approximated expansion is gained by the appreciation that the conditioning information can be truncated within one period of the system only. For linear systems, the expressions for the reduced-order likelihood function p(yi, yj, - - -, yNp W, C) and the conditional PDFs p(.yn 0, yn-Np, yn-Np+1, , y -i, C) are available since they are Gaussian and the correlation functions are known in closed forms regardless of the stationarity of the response. For stationary response, the method is very efficient in the sense that evaluation of all the conditional PDFs p(ynW, yn-Np,yn-Np+i,, y -i, C) requires the inverse and determinant of two relatively small matrices only. 

The optimize command maximizes a statistical "likelihood function". The higher this function, the more likely is the parameter to be the correct one. In the figure below, the symbols represent points calculated by the program Topaz (the full model), and the solid lines are the values calculated from the reduced-order model using the parameters determined by the program. 

Involves all the tasks carried out on a piece of equipment in order to reduce the likelihood of a functional failure. The aim of the preventive maintenance is to prevent functional failure and, consequently, to ensure the possibility of using the equipment for a given time. 

The new expert s estimates can now be updated using general Bayes model. The mean of this posterior (jx), as the distribution marker, is compared with the true value (jxlvi), in order to determine if and how much the formulated likelihood function has been able to reduced the error of estimates. 

SILs are order of magnitude bands of PFDavg, which also reflects the amount of risk reduction of a preventive safety instrumented function. Non-SlS Mainly two parameters, namely, consequence and likelihood, which affect risk, are considered. The consequence is the potential severity of the hazard. The likelihood is the frequency of occurrence. Risk graphs/risk matrices are used for these purposes. The inherent risk can be reduced by non-SlS risk reduction. To assess the risk, one is required to know and evaluate the effectiveness of all non-SIS risk reduction measures to ensure that the risk is reduced to as low as possible before application of any SIS. In other words, it is required to assess whether an SIS is necessary to further reduce the risk. Non-SIS risk reduction methods could be consequence reductions such as a dike, whereas blast walls or blast-resistant control buildings could reduce likelihood. 

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