SPE-192002-MS
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GPR for PCP RunlifeThe runlife of a PCP is the time between start of operation and failure of the PCP. Regression models of failure
time in maintainable systems, such as pumps, compressors,
and heavy mobile equipment, have been extensively studied over second half of last century by statisticians and reliability engineers (Meeker and
Escobar, Until the start of this century, the failure time regression models were predominantly parametric, requiring an assumption or understanding of the function between the variables of the system and failure time. The work of the statistician and engineer was finding the parameters that fit the function to the data. Parametric models are useful for relatively simple systems with a few variables (e.g. fatigue life of a plastic under different levels of stress
amplitude and temperature, the parameters of the function are independent of the variables, and failure time data is collected in a controlled laboratory environment.
Parametric models have limitations for most maintainable systems in industrial environments. The failure times of most systems are affected by numerous variables, and numerous failure modes can exist within the system.
Inmost cases, engineers do not know with certainty which variables, and which interactions of variables, affect the failure time. Parametric models also rely on smooth and continuous relationship between
variables and failure time, which is often not satisfied with numerous variables and interactions.
Additionally, the data collected from maintainable systems in industry are not controlled experiments where a sufficient quantity of data fora set of variables is acquired to obtain statistical significance in the prediction.
Regression models based on Gaussian process are non-parametric. A Gaussian process (GP) is a generalisation of the Gaussian (or normal) probability distribution (Rasmussen and Williams, p, and represents a collection of random variables. Typically, a probability distribution describes a scalar variable or vector for multivariate distributions. For the application of Gaussian
processes to regression, the random variables are functions. The functions are not defined algebraically, but as vectors "one can loosely think of a function as a very long vector, each entry in the vector specifying the function value f(x) at a particular input x".
The collection of random variables (i.e. the functions of the GP) are completely specified by mean function and covariance function of the GP.
Algebraically, f(x)
GP(m(x),k(x,x')), where f(x) is distributed according to the gaussian process, with m(x) as the mean function and k(x,x') as the covariance function.
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