PROCEDURE FOR THE EVALUATION OF MEASURED DATA IN TERMS OF VIBRATION DIAGNOSTICS BY APPLICATION OF A MULTIDIMENSIONAL STATISTICAL MODEL Summary

The evaluation process of measured data in terms of vibration diagnosis is problematic for timeline constructors. The complexity of such an evaluation is compounded by the fact that it is a process involving a large amount of disparate measurement data. One of the most effective analytical approaches when dealing with large amounts of data is to engage in a process using multidimensional statistical methods, which can provide a picture of the current status of the flexibility of the machinery. The more methods that are used, the more precise the statistical analysis of measurement data, making it possible to obtain a better picture of the current condition of the machinery.


INTRODUCTION
From the perspective of understanding the context of the statistical evaluation of measured data regarding the amplitude and frequency of its engine, the Honda GX25 may experience irregularities during the application of the statistical model.This raises a number of important questions.Is the liner regression model suitable?What percentage of the measured data for the statistical model can be explained by this model?What percentage of the model is strong?Are the measured data affected by the multicollinearity, autocorrelation or heteroscedasticity?How much of this is determined by the normality of residues?What are the possible errors of the model and how can they be identified?These questions are very resolutely answered by the methodology described in this work.In the process, these questions will also evaluate the effectiveness of the correct application of the linear regression model with two variables using RStudio software.

CALCULATION OF THE REGRESSION COEFFICIENTS Β
The first step in the progressive realization of the linear regression model with two variables is to calculate the regression coefficients as follows: In view of Equation ( 1), it is clear that the RStudio software must calculate an inverse matrix by multiplying the transposed matrix of matrix X by XT, then multiplying it by the transposed matrix XT and matrix y.The problem with Equation ( 1) is easy to define in the program environment of RStudio when using the following command: This operation assures linear model parameter estimation, which will take the shape of a vector.The output from RStudio is a vector of estimation parameters in the following form:

RANDOM DISTURBANCES AND THEIR DISPERSION IN THE FORM OF A VARIANCE-COVARIANACE MATRIX
The next step is to define the problems of random disturbances in the environment program of RStudio.The difficulty of this step is the fact that the variance must be expressed in the variance-covariance matrix.From this point of view, it is necessary to define the problem expressed by Equations ( 3) and (4):   In order for Equation (3) to be expressed by the RStudio software, it is necessary to calculate the variance of random values according to Equation (4).This equation refers to the proportion by which the numerator is the product of the transposed matrix of the vector of residual components of the ordinary matrix vector eT of residuals e.In the denominator, the number of explanatory variables needs to increase by one.Then the coefficient is subtracted from the number of observations n.This problem can be defined in the RStudio software in the following form: In this case, I used the generic function resid (), which comes with a standard offer RStudio program environment.This feature was chosen because of the purpose in calculating the residual component of the regression model itself.In addition to the generic function resid (), I used another generic function, i.e., dim (), in this step [1].This function determines the number of rows of a matrix with dimension n, while the number of columns is provided by using the dimension in relation to k+1.The outcome of the RStudio program is based on the defined command to estate the variance of random failures, as presented in (5).In this case, its value equates to 7.019.After successfully estimating the variance of random failures, I then defined the problem of S2 (3) using the RStudio environment program.Since this software understands the calculated variance matrix, a scalar variable is not so necessary in this step in order to express this matrix in scalar form.This transformation takes place within the defined RStudio program, using the function as.vector (), as follows: After the expression of the scalar site of dispersion, it is necessary to estimate the variance-covariance matrix using another generic function.This time, I used the function vcov, which is found in an environment of the RStudio software in this form: The outcome from RStudio is a variance-covariance matrix of the dispersion folder of the measured values of the time series.

QUANTIFICATION OF THE COEFFICIENT OF DETERMINATION
The third and very important step in the application of a linear model with two variables is the quantification of the coefficient of determination, which helps to answer the question about how much, in percentage terms, my algorithms were taken into account in relation to the measured data.I have defined this problem in RStudio.In the first place, it was necessary to recall the fact that the coefficient of determination works with the total sum of three types of squares: the total sum of squares, the residual sum of squares and explained sum of squares.These squares are takes into consideration in the RStudio program by using the function r. squared.This function concerns the quantification issue relating to the coefficient of determination, whose function in RStudio takes this form: In this case, after applying the defining process in RStudio I calculated the value to be 98.9%.This value defines the following: approximately 98.9% of the total variability values of the dependent variable is explained by Equation (14).

DETECTION OF THE VECTOR OF PARAMETERS IN THE FORM OF A CONFIDENCE INTERVAL
After the evaluation of the coefficient of determination, it is necessary to go to the next step, namely, to detect the interval of estimation β.The statistics indicate that this interval is a confidence interval, which, with 95% probability, include parameters β.The progress of this detection is implemented in an RStudio environment program using another generic function: confint ().The function code () takes the following shape in the RStudio environment program: The outcome of this calculation reveals a 95% confidence interval for the estimation of the parameters β in the form of a two-sided vector.

VERIFICATION OF THE NORMAL HYPOTHESIS OF THE MODEL WITH TWO VARIABLES
The proper application of this model evolves out of the nature of the hypothesis that the model must meet.This is a hypothesis that states that the residual component in the time series must follow the normal distribution probability.For this reason, in order to verify this assumption, it is necessary to use the well-known statistical test known as the Jarque-Bera test of normality.Of course, this issue is dealt with in the defined RStudio program.The functions are determined by using the jbTest (), which comes with the standard R-studio package.This function enters the calculation process within RStudio with the following code: This test takes into consideration two possible hypotheses, which might occur during the evaluation.The zero hypothesis H0 states that the residual component shall enter into a normal probability distribution.The alternative hypothesis H1 states that this component shall be something other than a normal probability distribution.An important feature of this test is that involves a p-value.This value is an essential indicator of the acceptance or rejection of the null hypothesis H0.The aim of this test is to equate 0 with a p-value of 739.If the p-value is greater than the level of significance for α, the null hypothesis could not be rejected at 0.05.It is therefore possible that the hypothesis of a normal distribution of residual components can be regarded as satisfied.

DETECTION OF POSSIBLE DEFECTS IN THE MODEL (RAMSEY TEST)
The most important part of this work is to verify whether a linear regression model is defined correctly.In order to verify the accuracy of the model specifications for the purposes of this thesis, I used a famous statistical test known as the Ramsey regression equation specification error test.On the chosen level of significance, α = 0.05, as defined according to Ramsey test in R-studio.Using the generic function resettest () enables this program to assess very quickly whether the proposed model is defined correctly or not.If the generic resettest () function is used on the basis of model specifications, errors in the RStudio environment program occur as follows: specifications for the chosen level of significance's testing errors were determined in relation to the zero hypothesis H0, which states how the shape of this model is correctly defined in comparison to the alternative hypothesis H1, which considers the model to be incorrectly defined.As such, implementing the square expansion of the independent variable should take this form: After using this test, I failed to reject the zero hypothesis H0 at this stage.Given the level of significance α, the result of this test represents clear statistical proof that the proposed model meets all the statistical assumptions and may be considered as a model in the correct form.

QUANTIFICATION OF ANTICIPATED VALUES OF THE MODEL
One of the advantages of using a linear regression model in practice is the fact that this model offers the possibility of calculating the anticipated values.This issue is in the defined RStudio program whose purpose is to estimate the values that can inform future generations of the model.For this purpose, it is necessary to estimate the interval of values.To this end, this chapter makes use of the command predict (), located in the argument folder of the confidence interval's specified code, in which the value is calculated.After the inclusion of this interval, the command to predict () in RStudio takes the following form: The outcome from RStudio, after the anticipated data (13) are defined, is the estimation interval of the variables X.

CONCLUSION
The application of the RStudio program in order to measure data in terms of the diagnosis of internal combustion engines included measurement data on the amplitude and speed of the Honda GX25 engine.This can be expressed as a linear regression model in the following form: This equation enables the following interpretation: the change of the parameter y on one drive will change the value of the parameter X to 4.216.The parameter y in this case is the magnitude and parameter X is the speed of the Honda GX25 engine, which is used in the Shell Eco-marathons.
Equation ( 14) is not the only outcome following application of a linear model with two variables.This methodology also statistically confirmed the fact that the model relating to ( 14) is based on the Ramsey test, whose specifications were defined in the right form as errors.Meanwhile, as this equation is able to include 98.9% of all measured values, it clearly calculates the value of the coefficient of determination.