Prediction of Brake Pad Wear of Trucks Transporting Oversize Loads Based on the Number of Drivers’ Braking and the Load Level of the Trucks—Multiple Regression Models

: Brake pad wear forecasting, due to its complex nature, is very di ﬃ cult to describe using engineering formulas. Therefore, the aim of this publication is to create high-quality brake pad wear forecasts based on three stochastic quantitative models based on multiple regression models (linear model, inverted linear model, and power model). The matrix of explanatory variables was extracted from the controllers of 29 vehicles: A—the driver’s style of using the brake pedal speci ﬁ ed on a 4-point scale and B—the number of vehicle load ranges speci ﬁ ed on a 5-point scale. Methodology: A matrix of explanatory variables was obtained over a 2-year period from trucks carrying oversize loads via OBD2 socket. The trucks operated under similar operating conditions. The created models were veri ﬁ ed in terms of their ﬁ t to the source data and by analyzing the residuals of the models. It should be emphasized that only the linear model met all the required criteria. The inverted linear and power-law models were rejected. Results: The veri ﬁ ed linear model is characterized by very small MAPE errors. The model was validated on 4 trucks and the brake pad wear prediction errors ranged from − 0.39% to 7.03%.


Introduction
The basic task of the braking system is to stop the vehicle, reduce its speed, and prevent the vehicle from rolling away when stationary [1].Friction disc brakes are a common solution today due to the effectiveness of the braking process, low cost, and high reliability [2].Despite the enormous progress in the development of the automotive industry, brake pads are commonly used actuators of braking systems [3].Currently, vehicles are becoming heavier and faster, so their operational efficiency must increase [4].The wear of pads used in real conditions is characterized by a number of non-linear, multi-dimensional factors [5].When braking, the vehicle generates friction force in the actuator systems, which results in a reduction of its kinetic energy.The change in kinetic energy leads to an increase in thermal energy occurring mainly on brake discs and pads [6].
The mechanisms of wear and degradation of brake discs and pads have received considerable attention in the literature, highlighting the importance of the materials used in production.D.K. Kolluri et al. [7] examined the effect of graphite particle size on the heating of brake discs.They observed that in composites, the use of small graphite particles compared to large particles improves the thermal properties of discs.The use of a coppermetal matrix for brake pads shows better tribological properties [8,9].New materials are also used.The authors of Ref. [10] used the newly developed DB-1 material and compared it to the materials currently used.In laboratory tests, they simulated the loads of highspeed trains.The new material has a very good coefficient of friction and is characterized by a lower wear rate due to the fine particles.There are publications in the literature regarding the use of natural material admixtures in effective braking systems.The authors [11] replaced synthetic fibers with hemp fibers.The use of natural hemp fibers reduced the specific wear rate while obtaining a consistent coefficient of friction for brake pads.
The aspect of accumulating significant heat energy in the braking system can prove very dangerous and shorten the life of brake pads and discs.The overheating phenomenon can occur due to the design of brake discs or pads made of materials with low thermal conductivity.
The paper [12] compared brake pads made from three asbestos-free composites.These composites contained different proportions of steel fibers 30, 35, and 41% and synthetic materials 11, 6, and 0%.The composite with 41% and 0% showed the best thermal stability and thermal conductivity.The effect of prolonged thermal loads on hardened steel causes a loss of mechanical properties [13].As the authors point out, this is due to the material's susceptibility to tempering after heating.Improving the resistance to tempering is to increase the amount of such chemical elements as molybdenum and vanadium.
Articles [14] describe the problem of overuse of the braking system, which is caused by an incorrect driving style.Prolonged use of the brake while driving downhill can cause this and contribute to complete pad wear [15].This problem is particularly dangerous for trucks with trailers or semi-trailers [16].As the authors point out [17] (p. 1) "… when using the lowest-priced brake discs and brake pads, a substantial reduction in their efficiency can occur if braking intensively or over a long period".Overheating of the brake system significantly reduces the friction coefficient of the brake pad against the brake disc.This forces an increase in braking force to achieve the same braking torque and results in accelerated wear [18,19].Corrosion of the braking system also adversely affects the vehicle's braking behavior.The primary corrosion factor is the composition of the brake pad and disc materials [20].The formation of corrosion intensifies during frequent changes in humidity and temperature difference between the brake disc and pad, which promotes a reduction in the friction coefficient [21,22].
Precision in the installation of new brake pads is also of great importance.When replacing new brake pads, a caliper that has not been cleaned of corrosion can result in faulty seating of the friction element and faster wear and vibration [23].The research topics presented above show the complexity of the braking phenomenon and the presence of many factors that affect brake pad wear.It should be noted that most of the research conducted was carried out in the laboratory and not on vehicles in real operating conditions.In the case of managing a fleet of multiple vehicles, the ability to estimate brake pad wear would make it possible to optimize the replacement schedule and identify the causes of rapid brake system wear.Currently, the most commonly used predictor is the vehicle distance traveled [24].
In a very interesting study [25], the authors measured the brake pad and disc wear on real objects.The study lasted 2 years, and 20 cars were analyzed.The influence of the type of traffic (urban-urban) and calendar month on the wear of the above-mentioned elements was determined.The study concluded that the wear and tear of the studied brake system components are influenced by the type of vehicle traffic and the season and are significantly statistical.However, the above work did not take into account many operational factors such as the driver's driving style, kilometers traveled, and vehicle load.
Some researchers have based their brake pad wear estimation results on machine learning methods [26,27].Good results were obtained for XGBoost + Logistic Regression and XGBoost + Deep Recurrent Neural Network-accuracy of 70% and 85%.The disadvantages of these methods are the need to collect a very large number of data and the selection of optimal configurations of processing methods.In the study [28], frictional thermal energy and car braking analysis were used to determine wear.The results showed that the decisive factor in pad wear is the vehicle's initial speed.
The Archard equation is a popular method for estimating brake pad wear.Kenneth Ma et al. tried to estimate brake pad wear based on the Archard equation.The estimation error on a real-world car proved to be very large, and as the authors stated "…without access to the associated usage data, accurate validation of the prediction cannot be carried out" [24] (p.12).Trucks carrying oversize loads experience frequent changes in the load carried.The weight of the load can vary up to 300% of the vehicle's weight.Therefore, in this case, estimating brake pad wear is important and should be based on reliable data.A key factor for fleet managers is the use of data stored in the vehicle's controllers.This allows verification of the driver's driving style and monitoring of the truck's operating data.OBD2 On-board vehicle diagnostics installed by manufacturers in each vehicle were used to download the data.Chunyu Yu et al. stated that "It is difficult to describe wear by using general formulas fully in engineering, because the wear characterization is related to several factors that are complex, nonlinear and multidimensional" [5] (p.2).
Therefore, the purpose of this publication is to create highly qualitative predictions of brake pad wear based on three stochastic quantitative models based on multiple regression.The matrix of explanatory variables based on real data takes into account the number of vehicle load ranges determined on a 5-point scale and the driver's style of using the brake pedal determined on a 4-point scale.In this work, "the driver's style of using the brake pedal" is understood as the number and intensity of pressing the brake pedal.

Materials and Methods
Modeling of the brake pads' wear system is understood as a list of procedures, leading to the above-stated goal (Figure 1).

Method of Collecting Real-World Data to Determine Model Parameters
Operational data were collected from a transportation company that owns 34 trucks adapted for the transportation of oversized cargo.The selected vehicles were a homogeneous group moving in a closed area (trucks without trailers or semi-trailers).The gross vehicle weight (GVW) of the trucks was 50,000 kg, 10 × 6 (11 units) and 10 × 4 (24 units) configuration.All trucks had 4 torsion axles and disc brake systems on non-driving axles.A drum brake system was installed on the drive axles.All trucks had the same tire size installed.
The tests were conducted from February 2020 to November 2021 and began after the brake pads were replaced on a particular truck.The brake pads used were from the same manufacturer.Completion of the study was defined as the thickness of the brake pads outside the serviceable range designated by the manufacturer (new 30 mm thick, worn < 8 mm thick).During the study, 5 trucks were eliminated from the analysis (3 trucks had mechanical or thermal damage to the brake pads, and 2 had driver changes).
The source data were read via diagnostic equipment from Knorr-Bremse (Munchen, Germany), Wabco (Friedrichshafen, Germany) and specialized CL2000 CAN 2.0A software from CSS Electronics (Aabyhoej, Denmark).Air brake system pressure signals and vehicle load were recorded via the CAN-BUS (J1939 protocol, OBD2).The recording of variable data was grouped in the recorder at 1000 km intervals.During brake pad replacement, the number of kilometers was read and rounded up to 100 km.The collected source data were contained in two matrices A = [129 × 88] and B = [145 × 88], where A-brake system pressure data matrix and B-vehicle load range data matrix.The collected source data were clustered according to the established algorithm (Table 1).Due to the large number of data, the source material was converted to statistically significant information [29,30].In this study, the value of the coefficient of variation related to a range of 1000 km was used for each explanatory variable.The critical value of the coefficient of variation (Var-co.) was set at <10% [31].The explanatory variable Y determines the brake pad wear period in kilometers for individual trucks Ti = {T1, T2, …, T29}.The explanatory variables Xi = {X1, …, X4} represent the number of brake pedals used within a certain range of brake system operating pressures per 1000 km.The explanatory variables Xi = {X5, …, X9} represent the number of individual truck load ranges per 1000 km with respect to GVW.

Real Data Collection Method for Model Testing
Based on the created models, brake pad wear was predicted for four example trucks K1, K2, K3, and K4.
Trucks K1 and K2 are trucks that were not subject to testing, and their parameters are the same as T1 to T29.Trucks K3 and K4 belonged to another company specializing in bulk material transportation (2 trucks, 8 × 4 system, brake pads on 2 front non-drive axles, GVW-32,000 kg).The same equipment and methodology were used to acquire data as in the main study described in Section 2.1.

Model Class Selection
Finding the key relationships between the phenomena under consideration is the goal of the presented statistical model.Understanding of cause-effect relationships was realized using the linear model, inverted linear model, and power model [29,30,32].An important aspect is to carry out calculations for all forms of models.The following models were used in this study: Linear model (1): The power model described by Equation (3) was linearized using the natural logarithm.The structure of the quasi-linear model is shown in Equation ( 4): where Y-dependent variable [km], X1,2,…,n-explanatory variables, α0, α1, ..., αn-unknown parameters of the model, and ε-randomness component of the model.The models were recognized as high-quality models after meeting the following criteria: - Valve of adjusted determination coefficient R > 90%, -full analysis of the random components of the residuals of the models-correctly verified statistically.

Methods for Estimating Structural Parameters of Models
The general structural form of the model was chosen to provide the best possible fit [29,30,32].For this purpose, the parameters of the linear model αi (i = 0, 1, 2, …, n) were estimated using the classical method of least squares (5).
where yi-the actual value of the explanatory variable and ӯi-the value of the explanatory variable determined from the model.The coefficient of convergence φ 2 (6) describes what part of the data is not explained by the statistical model.
Coefficient of determination ( 7): Adjusted determination coefficient (8): where n-number of observations and k-degrees of freedom.
In the process of verifying an econometric model, it is first necessary to check whether there is a linear relationship between the explanatory variable Y and any of the explanatory variables Xi of the model.We test the significance of the determined regression coefficients and formulate hypotheses (9): We verify the set of hypotheses with statistics, Formula (10): In a valid econometric model, the explanatory variable Y must significantly depend on each of the explanatory variables Xi of the model.For each coefficient of the regression model, we pose hypotheses (11): We verify the set of hypotheses with statistics (12): where ai-the estimator of the coefficient αi and S(αi)-the estimator of the dispersion of the coefficient αi.

Methods for Analyzing the Random Components of the Model
In the method of least squares, in order for the obtained estimators of the coefficients αi (i = 0, 1, 2,…,n) to be effective, the Gauss-Markov assumptions [30,32,33] must be met: - The values of the explanatory variables are fixed (they are not random).- The randomness of the values of the explanatory variable y follows from the randomness of the component ε.

-
The random components ε for the individual values of the explanatory variables have a normal (or very close to normal) distribution with an expected value of zero and a constant variance: N(0, δε).

-
The random components are not correlated with each other.
Fulfillment of the Gauss-Markov assumptions was verified using the appropriate statistical tests and relationships presented below.

The Hypothesis of Normality of the Random Components of the Residual
The normality of residuals was assumed a priori when deriving all test statistics.If random errors in small samples are not normally distributed, the distributions of the test statistics differ from the values resulting from the normality of the distribution of residuals.
Hypothesis H0 was set: The random components have a normal distribution.Hypothesis verification was performed using the Shapiro-Wilk test.The value of the test statistic was determined by Formula (13): where an,i-Shapiro-Wilk coefficients, e1…en-values of the model residuals, and e -the mean value of the model residuals.
If W > Wα value, there is no basis for rejecting hypothesis H0.

The Hypothesis of Autocorrelation of the Random Components of the Residuals
Autocorrelation is the interdependence of random components and is clearly undesirable.The hypothesis about the lack of autocorrelation of random components was verified using the Durbin-Watson test.Hypotheses (14) were formulated as follows: where -autocorrelation coefficient of random components of order one.The empirical value of the Durbin-Watson statistic was determined by Formula (15).

The Hypothesis of Randomness of the Components of the Residuals
Verification of the hypothesis of the randomness of the distribution of deviations of the model's residuals is aimed at assessing the appropriateness of the choice of the analytical form of the model.To check the randomness of the residuals, the number of series tests (16)  where nε-the number of residuals of the same signs (even or odd).Hypothesis H0 was set: The error of the model residuals is random.

The Hypothesis of the Symmetry of the Random Components of the Residuals
The random components should have a normal distribution, which is a symmetric distribution.The test checks the number of residuals in plus ρ+ and in minus ρ−.We pose hypotheses (17) as follows: where +-the number of residuals in plus.
To test the hypotheses, the symmetry statistics of the random components were used in the form (18): where m-the number of residuals in plus.
The statistic, with the null hypothesis being true, has a t-Student's distribution with (n − 1) degrees of freedom.The critical area of the test is two-sided.

The Hypothesis of Homoskedasticity of the Random Components of the Residuals
The Breusch-Pagan test (19) was used to determine the presence of equality of variance of random components (homoskedasticity).

A Method for Evaluating the Use of Models
MAPE (Mean Absolute Percentage Error) was used to compare model results and actual values [29,30].MAPE reports the average magnitude of forecast errors for the test period expressed as a percentage.The MAPE value allows comparing the accuracy of forecasts of different models and was calculated by Formula (20): where  -predicted value.The primary predictive criterion for evaluating models is the minimization of MAPE error.The acceptable error must not exceed <15%.Scaling of the correctness of the models was performed according to the following criteria: MAPE < 5%-excellent, MAPE < 10%very good, and MAPE < 15%-good.

Results of the Initial Grouping of Source Data
The first part of sorting the data was to group them according to the assumptions shown in Table 1.The results of grouping the data according to the adopted algorithm are shown in Table 2.A preliminary analysis of the grouped data is shown in Table 3, which involves calculating the coefficient of variation value as a measure of dispersion [29,31].The results obtained in the range of less than Var-co.< 10% were eliminated [32].Analysis of the results presented in Table 3 postulates the elimination of the X9 variable.Therefore, after the first stage of selection, the variables Xi = {X1, X2,…, X8} were used for further model construction.

Linear Model Estimation and Verification Results
The constructed linear model, Formula (1) for Xi = {X1, X2,…, X8}, was verified at the significance level α = 0.05.The model's coefficient of determination is R = 0.926 (coefficient of convergence φ 2 = 5.3%).The model explains 92.6% of the variability of the studied trait, this indicates a good fit of the model to the empirical data (Table 4).The F statistic, given the truth of the null hypothesis, has an F Snedecor distribution with 8 degrees of freedom of the numerator and 20 degrees of freedom of the denominator.The empirical value of the statistic is F = 44.56,and the corresponding critical level of significance F = 4.413 × 10 −11 , which is less than the accepted significance level α = 0.05.We therefore reject hypothesis H0 in favor of H1.There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X1, X2,…, X8}.We test the significance of the individual regression coefficients.
The statistic at the truth of the null hypotheses has a Student's t-distribution with 20 degrees of freedom.The empirical values of the t-Student's statistic and the corresponding values of the critical level of significance (p-value) are shown in Table 4.There are no grounds for rejecting the hypothesis that the model constants ɑi = {α0,…,α8}\{α2,α5} are insignificant, i.e., equal to zero (the values of the critical level of significance for these coefficients are greater than the accepted level of significance α = 0.05).There is no basis for rejecting the hypothesis that the variables Xi = {X1,…,X8}\{X2,X5} are insignificant.The current model structure is flawed.

Re-Selection of the Linear Model Class
For the new linear model, Formula (1) for Xi = {X2,X5}, model fitting was carried out at a significance level of ɑ = 0.05.The coefficient of the model is R = 0.927 (coefficient of convergence φ 2 = 6.8%).The model explains 92.7% of the variation in the studied trait.This shows a good fit of the model to the empirical data (Table 5).The F statistic, with the null hypothesis being true, has a Snedecor F distribution with 2 degrees of freedom of the numerator and 26 degrees of freedom of the denominator.The empirical value of the statistic is F = 178.177.The critical level of significance F = 6.65 × 10 - 16 is less than the accepted level of significance α = 0.05.We therefore reject hypothesis H0 in favor of H1.There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X2,X5}.For each coefficient of the regression model αi, we test the hypothesis of its significance.We verify it with a statistic of Student's t-distribution with 26 degrees of freedom.The empirical values of the t-Student's statistic and the corresponding values of the critical level of significance (p-value) are shown in Table 5.All the coefficients of the model are significantly different from zero (the values of the critical level of significance are less than the accepted level of significance α = 0.05).There are no grounds for rejecting the hypothesis that all coefficients of the tested model are significantly different from zero.The current structure of the model is correct and is represented by the relation ( 21) and geometric interpretation in Figure 2

Estimation and Verification Results of Inverted Linear Model
For the inverted linear model, Formula (2) for Xi = {X1,…,X8}, the results of parameter estimation are presented in Table 6.The constructed linear inverse model was verified at the significance level α = 0.05.The model's coefficient of determination is R 0.961 (coefficient of convergence φ 2 = 2.8%).The model explains 96.1% of the variation in the studied trait, this indicates a good fit of the model to the empirical data.The significance of the regression coefficients was tested, and we formulated hypotheses for a linear model.The F statistic, with the null hypothesis being true, has a Snedecor F distribution with 8 degrees of freedom of the numerator and 20 degrees of freedom of the denominator.The empirical value of the statistic is F = 86.598,and the corresponding critical level of significance F = 8.01 × 10 −14 .The level is less than the accepted significance level α = 0.05.We reject hypothesis H0 in favor of H1.There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X1, X2,…,X8}.
For each coefficient of the regression model, we hypothesize as for a linear model.The empirical values of the t-Student's statistic and the corresponding values of the critical level of significance (p-value) are shown in Table 6.No basis for rejecting the hypothesis that the model constants α = {α0,…,α8}\{α2,α3} are insignificant, i.e., equal to zero; incorrect model structure.

New Inverse Linear Model
For the new inverted linear model, Formula (2) for Xi = {X2,X3}, model fitting was carried out at a significance level of α = 0.05.Based on the verified data, we determine the parameter values of the new inverted linear model (Table 7).The model fit was verified at a significance level of α = 0.05.The coefficient of the model is R = 0.962 (coefficient of convergence φ 2 = 3.5%).Conclusion: The model explains 96.2% of the variability of the studied trait.This shows a very good fit of the model to the empirical data.We hypothesize that the coefficients of the regression model are not significant.The F statistic, with the null hypothesis being true, has a Snedecor F distribution with 2 degrees of freedom of the numerator and 26 degrees of freedom of the denominator.The empirical value of the statistic is F = 355.962,and the corresponding critical significance level F = 1.92 × 10 −19 .The level is less than the accepted significance level α = 0.05.We therefore reject hypothesis H0 in favor of H1.There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X2,X3}.
For each coefficient of the regression model αi, we test the hypothesis of its significance.We verify it with a statistic with a Student's t-distribution with 26 degrees of freedom.The empirical values of the t-Student's statistic and the corresponding values of the critical level of significance (p-value) are shown in Table 7.All the coefficients of the model are significantly different from zero (the values of the critical level of significance are less than the accepted level of significance α = 0.05).There is no basis for rejecting the hypothesis that all model coefficients are significantly different from zero.The current structure of the inverse linear model is correct and is represented by the relation (22) and geometric interpretation in Figure 3:

Results of Estimation and Verification of the Power Model
For the power (quasi-linear) model specified by Formula ( 14), the parameter values are shown in Table 8.We will verify the constructed model at the significance level α = 0.05.The model's coefficient of determination is R = 0.941 (coefficient of convergence φ 2 = 4.1% φ 2 = 4.2%).The model explains 94.1% of the variation of the studied trait, this is a good fit of the model to the empirical data.The significance of the regression coefficients was checked.We put hypotheses as for previous models.The F statistic, with the null hypothesis being true, has an F Snedecor distribution with 8 degrees of freedom of the numerator and 20 degrees of freedom of the denominator.The empirical value of the statistic is F = 56.656,and the corresponding critical level of significance F = 4.62 × 10 −12 .The level is less than the accepted significance level α = 0.05.We reject hypothesis H0 in favor of H1.There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables Xi = {X1, X2,…,X8}.
For each coefficient of the regression model, we pose hypotheses as for previous models.The empirical values of the Student's t-statistic and the corresponding values of the critical level of significance (p-value) are shown in Table 8.There are no grounds for rejecting the hypothesis that the model constants ɑ = {α0,…,α8}\{α2} are insignificant, i.e., equal to zero (the values of the critical level of significance for these coefficients are greater than the accepted level of significance α = 0.05).There are no grounds for rejecting the hypothesis that the variables Xi = {X1,…,X8}\{X2} are insignificant.The current model structure is incorrect.

Determination of the New Power Model
The new quasi-linear power model is described by Equation (23): Estimation of structural parameters for the presented model is provided in Table 9.We verify the fit of the model at a significance level of α = 0.05.The coefficient of the model is R 0.901 (convergence rate φ 2 = 9.6%).The model explains 90.1% of the variability of the studied trait.This indicates a good fit of the model to the empirical data.We hypothesize that the coefficients of the regression model are not significant.The F statistic, given the truth of the null hypothesis, has an F Snedecor distribution with 1 degree of freedom of the numerator and 27 degrees of freedom of the denominator.The empirical value of the statistic is F = 254.865,and the corresponding critical level of significance F= 2.821 × 10 −15 is less than the accepted significance level α = 0.05.We reject hypothesis H0 in favor of H1.There is no basis for rejecting the hypothesis that brake pad wear depends on at least one of the variables X2.
For the regression model coefficient α2, we test the hypothesis of its significance.We verify it with a statistic of t-Student's distribution with 27 degrees of freedom.The empirical values of the t-Student's statistic and the corresponding values of the critical level of significance (p-value) are shown in Table 9.The coefficients of the model are significantly different from zero (the values of the critical level of significance are less than the accepted level of significance α = 0.05).There are no grounds for rejecting the hypothesis that the coefficients of the tested model are significantly different from zero.The current structure of the quasi-linear power model is correct.After simple transformations, we obtain a power model in the form (24) and geometric interpretation in Figure 4:

Results of Random Component Analysis of the Models
In the least squares method, as mentioned earlier, the Gauss-Markov assumptions must be met.
The random components of the residuals of the obtained models ( 21), (22), and (24) were analyzed, and the results are shown in Table 10.An analysis of Table 11 shows that there is no basis for rejecting the hypothesis that the random components of the models have a normal distribution, for the N(0, 2583.179)linear model and the N(0, 0.048) power model.Note that for the inverse linear model N(0; 0.048) value W < Wα, and this means that there are grounds for rejecting hypothesis H0.The large difference between the distribution of the residuals and the normal distribution may disturb the assessment of the significance of the coefficients of the individual variables of the model.Therefore, the inverted linear model was rejected.There is no basis for rejecting hypothesis H0 about the lack of autocorrelation of random components of order one.13.The empirical value of the statistic does not fall into the critical area.There is no basis for rejecting the hypothesis H0 that the distribution of the components of the model residuals is random.

Results of the Hypothesis on the Symmetry of the Random Components of the Residuals
The critical area of the test is two-sided, and the results are shown in Table 14.

Figure 1 .
Figure 1.Procedure for determining the statistical model.

Figure 3 .
Figure 3. Geometric interpretation of the inverse linear model (22).Y2-Period of exploitation of the brake pads [km]; X2-Number of brake applications in the range of 0.15-0.24[MPa]; X3-Number of brake applications in the range of 0.25-0.40[MPa].

Figure 4 .
Figure 4. Geometric interpretation of the power model (24).Y3-Period of exploitation of the brake pads [km]; X2-Number of brake applications in the range of 0.15 to 0.24 [MPa].

4. 5 . 3 .
Results of the Hypothesis of Randomness of the Components of the Residuals Hypothesis H0 was set: The error of the model residuals is random.The data are shown in Table

Table 1 .
Algorithm for clustering source data.

Period of Exploitation of the Brake Pads [km] A = Brake System Pressure Range [MPa] B = Vehicle Load Range GVW [%]
was used.

Table 2 .
Results of source data grouping.

Table 3 .
Analysis of grouped data for the linear model.

Table 4 .
Multivariate regression results for the linear model.

Table 5 .
Multivariate regression results for the new linear model. :

Table 6 .
Multiple regression results for the inverted linear model.

Table 7 .
Multiple regression results for the new inverted linear model.

Table 8 .
Multivariate regression results for the power model.

Table 9 .
Multiple regression results for the new power model.

Table 11 .
Values of the Shapiro-Wilk statistic.Results of the Hypothesis of Autocorrelation of the Random Components of the Residuals Autocorrelation is the interdependence of random components and is clearly undesirable.The results are presented in Table12.

Table 13 .
Results of verification of the randomness of the distribution of the model residuals.

Table 14 .
Results of verification of the symmetry of the distribution of the residuals of the models.