First, let us introduce a random variable y with two exclusive values of 0 and 1 corresponding to normal situation without fault of machines and a failure situation with the fault. Furthermore, two kinds of variables on sound and vibration are expressed as and. In the case of paying attention to the variables, , and y, all the information on mutual correlations among, , and y is included in the conditional probability distribution. By using the well-known Bayes’ theorem:

the probability of fault occurrence can be predicted in an expansion expression, as follows:

where, denotes the averaging operation with respect to the random variables. As the fundamental probability density function, the generalized binomial distribution is adopted, and the orthonormal functions can be determined as

with

; the maximum value of y,; the minimum value of y,

; the level difference interval of y.

By use of a computer for the observed time series data on sound and vibration, on-line signal processing can easily be carried out.

Since the specific correlation relationship between sound and vibration emitted from an identical machine exists, it is possible to diagnose faults of the machine by detecting the change of correlation characteristic between sound and vibration. As examples of multiple faults of machine, and express the state variables corresponding to fault 1 and fault 2. In this section, when a specific failure occurs among multiple faults established in advance, a diagnosis method able to detect the machine part with the failure is proposed.

First, the joint probability distributions and are expanded into an orthonormal polynomial series.

where, correlation information among variables with lower and higher orders is reflected in the expansion coefficient. Here, and are orthonormal polynomials with two weighing functions, , respectively. As the fundamental probability density functions, , Gaussian distribution is adopted.

Thus, from Equation (8), the orthonormal functions can be determined as the Hermite polynomial [11] .

Furthermore, a trinomial distribution is adopted as the standard distribution expressing two kinds faults states.

Two variables and correspond to the following states.

Normal:, Fault 1:, Fault 2:,

where, p is probability of fault 1 and q is probability of fault 2. Thus, from Equation (10), the orthonormal function [11] can be determined as

Therefore, the arbitrary constants are determined as, and is a factorial function [11] of th order defined as

,; discrete level interval of y.

The coefficients are decided by the normalized condition for the probability.

Furthermore, in the case of considering more than three kinds of failures, a multinomial distribution can be adopted as the standard distribution of Equation (10). It is possible to derive the fault diagnosis in the same manners as the present study by calculating orthogonal polynomial based on the preciously published method [11] .

The following expression can be obtained from Equations ((5), (6)), using Bayes’ theorem on conditional probability distribution.

Therefore, fault occurrence probability is predicted from observation data of sound and vibration by considering the conditional expectation of Equation (13).

Since the specific correlation relationship between sound and vibration emitted from an identical machine exists, it is possible to diagnose faults of the machine by detecting the change of correlation characteristic between sound and vibration.

The fault diagnosis methods proposed in 2.1 and 2.2 require prior information in both situations before and after a failure occurs. However, only the prior information before the failure occurs can be really obtained. In this section, a faults diagnosis method is proposed based on only prior information in normal situation. The variables, are sound and vibration emitted from rotational machine, respectively. The faults occurrence can be diagnosed from changing information of the correlation of between sound and vibration.

First, the joint probability distribution is obtained in an orthonormal polynomial series.

Gaussian distribution is used as the standard distributions, and the Hermite polynomial is determined as the orthonormal polynomials. Then, the probability distribution for sound can be predicted from the vibration data, as follows:

Since the correlation information between and change when a fault occurs, the prediction of becomes difficult from Equation (18). Therefore, it is possible to diagnose the faults occurrence by evaluating the prediction error.

In general, cumulative distribution is more suitable than probability distribution for evaluating the prediction error.

There, by using Equation (18) cumulative distribution is given by the following equation.

with

Furthermore, the faults diagnosis based on the prediction error when evaluating the probability distribution of vibration from sound data is also possible by exchanging the variable for in Equations ((18)-(22)).

First, two different time series data sets (Data Set 1 and Data Set 2) for sound and vibration in three different time intervals were successively measured, in cases with and without fault occurrence. Using the 1500 data points of Data Set 1, which contained both cases of faults occurrence (500 data points for each fault) and fault-free cases (500 data points) as the learning data, the expansion coefficients of Equation (7) were first evaluated. Next, after dividing each data set into 30 sub-data sets consisting of 500 data points with 450 overlapping points, the probability of fault occurrence was predicted by use of Equation (14) based on the 500 sampled data in each sub-data set. The relationship between each Data Set and the sub-data sets is illustrated in Figure 3.

The results of prediction of the probability for normal situation, failure situations with the fault 1 and 2 are shown in Figure 4 for Data Set 1 and in Figure 5 for Data Set 2. From these results, we can find that the fault probability clearly exhibits large values in the sub-data sets corresponding to the failure situation. More precise prediction of the fault probability is possible by considering correlation information of lower and higher orders, because the correlation characteristics (of lower and higher orders) between sound and vibration change after failure. Thus, the proposed method to detect precisely the change of correlation information is useful in the detection of machine failures.

The fault diagnosis method with less prior information proposed in 2.3 was applied to real observation data. First, the expansion coefficients in Equation (15) were calculated by using 1000 data points in normal situation. Next, sub-data sets with 1000 data points were made from total 15,000 data points consisting of each 5000 data points in normal, fault 1 and fault 2 situations. Furthermore, probability distribution of vibration (or sound) was predicted by observing the data of sound (or vibration) by using sub-data sets. One of the prediction results is shown in Figure 5 in a form of cumulative distribution. The differences between experimental values and predicted values are shown in Table 1. Prediction errors at the 50% level and average of the prediction errors evaluated at every 10% level from the 10% level value to the 90% level value are shown in Table 1. The prediction errors were calculated from experimental values and predicted values at 9 points, from the 10% level to the 90% level by using the following relationships.

When fault 1 and fault 2 occur, large prediction errors are obtained for the probability distribution of vibration based on the observation of sound. As shown in Figure 6, the experimental values could not be predicted and there are great discrepancy between the experimental values and the theoretical values. Table 1, a prediction error is small for normal situation of data point from 1 to 5000. On the other hand, a prediction error becomes big for abnormality situation including a fault of data point from 5001 to 15,000. The validity of the fault diagnosis method based on the prediction error of the cumulative distribution has been confirmed numerically from the experimental result.