The multivariate analysis with using all variables is doubtful from both the calculation time and the expected effect, and in this study, it is decided to extract the data on the crank angle capable of sufficiently explaining the above-mentioned relationship from one cycle of the engine. As described before, the bow-shaped vibration pattern is characteristic and clearly changed with the running time of the engine, the data in the vicinity of the crank angle is examined. More specifically, the bow-shaped pattern (the slide vibration of No. 1 cylinder) is generated twice, i.e., when the piston is lowered and when the piston is elevated, and the data at the bow-shaped top point when the piston is lowered at the crank angle of 72° was adopted. At this angle, when 200 variables obtained in one frequency analysis are referred to as the data of one case, one case is constituted by allotting the spectrum values at 100Hz, 200Hz, ..., 20kHz to the explanatory variables of x₁, x₂, ..., x₂₀₀. 85 cases of the data under the same condition were prepared in the analysis, and 22 groups data were prepared for each different running time of the engine. When the observed value is explained using a plurality of variables x₁, x₂, ..., x_p, the expected value of y is expressed by p variables as follows.

E[y] = β₀+β₁x₁+β₂x₂+……+β_px_p　(1)

The probability fluctuation part ε is added thereto, and the observed value leads to as follows.

y = β₀+β₁x₁+β₂x₂+……+β_px_p+ε　(2)

The equation (2) is the multiple regression equation of y to x₁, x₂, ..., x_p, and the data is expressed as indicated in Table 1.

　

Table 1 Data constitution

　

The dependent variable of y is the running time (run) of the engine, and ε is the remainder of the estimated value subtracted from the observed value. In this case, the multiple regression models and the remainder are expressed by (3) and (4), respectively.

runh_j = β₀+β₁x_1,j+β₂x_2,j+……+β_px_p,j+ε_j　(3)

where, (j= 1, 2,..., N), N: total case number

ε_j = runh_j - (β₀+β₁x_1,j+β₂x_2,j+……+β_px_p,j)　(4)

The multiple regress analysis is the analytical technique to obtain the estimated values b₀, b₁, b₂, ..., b_p of β₁, β₂, ..., β_p so that the sum of squares of the remainder is minimum, and the multiple regression equation can be obtained as (5) below.

runh_j = b₀+b₁x_1,j+b₂x_2,j+……+b_px_p,j　(5)

The multiple correlation coefficient of the multiple regression equation obtained above is R=0.935, and the decision coefficient R²=0.873. It is indicated that the multiple regression equation obtained from this result applies well. The multiple correlation coefficient is the result using 200 variables, but it can not be always concluded that the analysis is achieved with only the variables affecting the dependent variables among 200 explanatory variables. Thus, to select the variables to be used for the analysis, the stepwise method is used. The analysis is achieved with the input probability of 0.05 and the removal probability of 0.1 as for the significant probability of the F-value in inputting or removing the explanatory variables in/from the multiple regression equation, and 60 variables are finally obtained. As a result of the analysis using the variables, the multiple correlation coefficient R=0.928, and the coefficient of decision R²=0.861. The rest of 140 variables except the selected 60 variables less affect the dependent variables, and it can be concluded that 140 variable may be excepted.

The average processing of the cases is taken in order to decrease the noise component contained in the data when the dependent variables are calculated using the obtained multiple regression equation. Fig, 10 is the figure to indicate the relationship between the average number and the error. As shown in the figure, there is not linear relationship between the average number and the error, and the error time is rapidly reduced in the range between 1 and 20, and then, gradually decreased. As shown in the result, the error is 330 at the average number of 100, and further reduction of the error can not be expected. In this study, it is desired to reduce the error as much as possible in obtaining the estimated value from the regression equation, but it is also necessary to select an appropriate average number when taking into consideration the calculation time. The average number where the error is approximately 400 hours was adopted here.

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