In the actual sound environmental systems, it seems to be essentially difficult to exactly evaluate a whole probability distribution form of its response fluctuation, owing to various types of natural, social and human factors. We have reported a unified probability density expression in the standard expansion form of Hermite type orthonormal series taking a well-known Gaussian probability density function (abbr. p.d.f.) as the basis for generally evaluating non-Gaussian, non-linear correlation and/or non-stationary properties of the fluctuation phenomenon. However, in the real sound environment, there still remain many actual problems on the necessity of improving the above standard type probability expression for practical use. First, a central point in this paper is focused on how to find a new probabilistic theory of practically evaluating the variety and complexity of the actual random fluctuations, especially through newly introducing an equvivalence transformation toward the standard type probability expression mentioned above in the expansion form of Hermite type orthonormal series. Then, the effectiveness of the proposed theory has been confirmed experimentally too by applying it to the actual problems on the response probability evaluation of various sound insulation systems in an acoustic room.

A specific signal in most of actual environmental systems fluctuates complicatedly in a non-Gaussian distribution form, owing to various kinds of factors. The nonlinearity of the system makes it more difficult to evaluate the objective system from the viewpoint of internal physical mechanism. Furthermore, it is very often that the reliable observation value can be obtained only within a definite domain of fluctuating amplitude, because many of measuring equipment have their proper dynamic range and the original random wave form is unreliable at the end of amplitude fluctuation. It becomes very important to establish a new signal processing or an evaluation method applicable to such an actually complicated system even from a functional viewpoint. This paper describes a new trial for the signal processing along the same line of the extended regression analysis based on the Bayes' theorem. This method enables us to estimate the response probability property of a complicated system in an actual situation, when observation values of the output response are saturated due to the dynamic range of measuring equipment. This method utilizes the series expansion form of the Bayes' theorem, which is applicable to the non-Gaussian property of the fluctuations and various kinds of correlation information between the input and output fluctuations. The proposed method is newly derived especially by paying our attention to the statistical information of the input-output data without the saturation operation instead of that on the resultantly saturated observation, differing from the well-known regression analysis and its improvement. Then, the output probability distribution for another kind of input is predicted by using the estimated regression relationship. Finally, the effectiveness of the proposed method is experimentally confirmed too by applying it to the actual data observed for indoor and outdoor sound environments.