"Universal" Fitting Function for Complex Systems: Case of the Short Samplings

The authors suggest an effective scheme for quantitative description of complex systems, when the number of measurements is relatively small. It has a great importance for quantitative description of expensive and rare experiments when the volume of the sampling is small. They proposed a simple theory that is based on the previous results associated with conception of the intermediate model (IM). The previous results can be generalized and applicable for description of complex systems with short samplings when the influence of the uncontrollable factors becomes significant. As an example, we consider the description of acoustic signals recorded from turbine bearings. It can be proved that the real signals have self-similar (fractal) properties. It helps to compress the length of the initial files (number of data points N = 44100) at least in 88 times and reduced essentially the number of the fitting parameters. The obtained results can be used for diagnosis of different defects during the process of technical exploitation. Each failure has own acoustic “picture” i.e. the amplitude-frequency response (AFR) expressed in terms of the generalized Prony spectrum (GPS). This AFR can be used as a “specific” fingerprint for identification of the unexpected failure and preventing a possible breakdown.