## 5 Results
### Regression Analysis Results  
Regression analysis is a statistical method by which we estimate the value of some random variable called the dependent variable, also called the target variable, regressand, or explanatory variable, based on knowledge of other independent variables, regressors, covariates, or explanatory variables. A typical mathematical method used to solve problems of this type is Pearson linear discriminant analysis, which is used in this case study. Pearson correlation coefficient and relevant significance levels were calculated in Minitab 17 for the most important measurement variables. These experimental variables were motor oil level, amplitude related to the oil level, amplitude start value (determined manually), amplitude slope (shape of the level signal envelope), UART amplitude start value, pulse, and \(\Delta t\). The correlation matrix revealed that there are very strong correlations among the following pairs: pulse-UART amplitude start, pulse-amplitude start, slope-amplitude level, pulse-level, and UART amplitude start-level. There are also statistically significant correlations between the pairs: amplitude level-level, slope-level, and UART amplitude start-amplitude level; see Table 2.  
For the descriptive statistics, we chose 18 data rows (each of them representing 30 fractional measurements). Low-temperature raw data were eliminated from the statistics due to outliers (90 mm; \(-0.1\)\({}^{\circ}\)C) and/or incomplete data ensembles (58.5 mm and 80 mm; \(-5\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C). The descriptive statistics revealed that the average value of \(\Delta\)t is \(1.5\pm 0.27\)\(\upmu\)s. In terms of oil level measurement accuracy, it was \(\pm 0.41\) mm; see Table 3.  
Another step was multiple linear regression analysis. This type of analysis allowed us to determine the functional dependency of the crucial variables on motor oil level and \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Name** & **Unit** & **Level** & **AmplitudeLev** & **AmpStart** & **Slope** & **UARTAmpStart** & **Pulse** \\ \hline AmplitudeLev & mm & \(-0.736/0.000\) & - & - & - & - & - \\ AmpStart & \(\upmu\)s & \(0.967/0.000\) & \(-0.622/0.006\) & - & - & - & - \\ Slope & mV/\(\upmu\)s & \(-0.596/0.000\) & \(0.914/0.000\) & \(-0.444/0.065\) & - & - & - \\ UARTAmpStart & \(\upmu\)s & \(0.967/0.000\) & \(-0.620/0.006\) & \(1.000/0.000\) & \(-0.444/0.065\) & - & - \\ Pulse & \(\upmu\)s & \(0.967/0.000\) & \(-0.622/0.006\) & \(1.000/0.000\) & \(-0.447/0.063\) & \(0.999/0.000\) & - \\ dt & \(\upmu\)s & \(-0.183/0.468\) & \(0.036/0.887\) & \(-0.128/0.613\) & \(-0.266/0.286\) & \(-0.131/0.603\) & \(-0.121/0.632\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Correlation matrix of the most important experimental variables. All values in bold are significant (\(p<0.05\)). Figure 5: Dependence of amplitude from reference and oil surface on temperature. (**a**) Oil surface: 30 mm. (**b**) Oil surface: 60 mm. (**c**) Oil surface: 90 mm. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Variable** & **Unit** & **Total Count** & **Mean** & **StDev** & **Minimum** & **Median** & **Maximum** & **Range** \\ \hline dt & \(\upmu\)s & 18 & 1.5072 & 0.2763 & 1.0100 & 1.5400 & 1.9500 & 0.9400 \\ Offset & mV & 18 & 0.00683 & 0.000786 & 0.00500 & 0.00700 & 0.00800 & 0.00300 \\ Slope & mV/\(\upmu\)s & 18 & 0.03366 & 0.01444 & 0.00455 & 0.03647 & 0.06776 & 0.06322 \\ AmplitudeLev & mV & 18 & 0.05789 & 0.02838 & 0.00700 & 0.05550 & 0.10300 & 0.09600 \\ AmplitudeRef & mV & 18 & 0.01283 & 0.001383 & 0.01100 & 0.01100 & 0.01600 & 0.00500 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Descriptive statistics summary of the most important variables. temperature. The multiple linear regression model consists of the following information: statistical significance, the percentage of variation that might be explained by the model, and the final model equation.