# A Computational Platform for Automatic Signal Processing for Bender Element Sensors +
## 4 Validation Using the International Parallel Test Benchmarks
### Statistical Analysis of the Results  
A statistical analysis is performed on the 16 output signals from the two sets of eight tests conducted at void ratios of 0.690 and 0.770. The signals are interpreted to extract the shear modulus using the time and frequency techniques outlined in Section 4.2 and utilizing the GeoHyTE platform. The presentation of the shear modulus readings is followed by a statistical assessment of the correlation between different interpretation methods and a comparison of the results with those from the International Parallel Test. The shear modulus readings for each of the eight test repetitions are detailed in Table 3 for \(e=0.690\) and Table 4 for \(e=0.770\). Additionally, the tables provide the mean, standard deviation, standard error of the mean, and the lower and upper limits of the 95% confidence interval for each interpretation method. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Void Ratio**} & \multirow{2}{*}{**Test No**} & \multicolumn{6}{c}{**\(\mathbf{G_{0}}\) [MPa]**} \\ \cline{3-10}  & & **Cross-Corr.** & **Peak-to-Peak** & **GeoHyTE** & **First Sharp** & **First Bump** & **Zero Crossing** & **Cross-Spectrum** \\ \hline \multirow{6}{*}{0.69} & 1 & 48.71 & 46.88 & 53.30 & 64.14 & 49.66 & 45.66 & 35.19 \\ \cline{2-10}  & 2 & 43.53 & 42.43 & 48.40 & 54.65 & 45.83 & 42.13 & 23.28 \\ \cline{2-10}  & 3 & 42.13 & 41.09 & 46.60 & 52.91 & 44.33 & 40.59 & 28.32 \\ \cline{2-10}  & 4 & 42.89 & 41.83 & 46.50 & 52.07 & 44.08 & 40.30 & 31.23 \\ \cline{2-10}  & 5 & 46.62 & 45.07 & 51.80 & 57.79 & 48.06 & 44.086 & 43.05 \\ \cline{2-10}  & 6 & 50.34 & 48.43 & 56.30 & 66.57 & 52.17 & 46.53 & 39.34 \\ \cline{2-10}  & 7 & 50.25 & 48.24 & 55.37 & 60.69 & 51.14 & 46.53 & 34.62 \\ \cline{2-10}  & 8 & 47.96 & 46.35 & 53.50 & 60.11 & 49.76 & 45.41 & 28.15 \\ \hline \multicolumn{1}{c}{Mean} & 46.55 & 45.04 & 51.47 & 58.62 & 48.13 & 43.90 & 32.90 \\ \hline \multicolumn{1}{c}{Std. deviation} & 3.311 & 2.916 & 3.854 & 5.241 & 3.081 & 2.571 & 6.455 \\ \hline \multicolumn{1}{c}{Std. error of the mean} & 1.171 & 1.031 & 1.363 & 1.853 & 1.089 & 0.9091 & 2.282 \\ \hline \multicolumn{1}{c}{LL 95\% CI} & 43.79 & 42.60 & 48.25 & 54.23 & 45.55 & 41.76 & 27.50 \\ \hline \multicolumn{1}{c}{UL 95\% CI} & 49.32 & 47.48 & 54.69 & 63.00 & 50.71 & 46.05 & 38.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Shear modulus values and statistical data for the tests performed on the sample with \(e=0.690\)[19]. For the void ratio of 0.690, the shear moduli range from 23.28 MPa (cross-spectrum method on Test 2) to 66.57 MPa (first sharp method on Test 6). GeoHyTE predictions fall between 46.50 MPa (Test 4) and 56.30 MPa (Test 6). The cross-spectrum method exhibits the widest spread of readings with a standard deviation (6.455 MPa). The first sharp method has a similar standard deviation (5.241 MPa), while the first bump, cross-correlation, and GeoHyTE methods show standard deviations ranging between 3 and 4 MPa. The peak-to-peak and zero-crossing methods have standard deviations ranging from 2 to 3 MPa. For the 0.770 void ratio, the shear moduli range from 24.24 MPa (cross-spectrum method on Test 5) to 48.68 MPa (first sharp method on Test 1). The GeoHyTE predictions range from 38.90 MPa (Test 5) to 42.00 MPa (Test 3). The spreading of the readings is more reduced compared to the previous case with standard deviations of 3.685 MPa and 1.999 MPa for the cross-spectrum and first sharp methods. Cross-correlation, peak-to-peak, GeoHyTE, first bump, and the zero crossing methods feature standard deviations between 0.5 and 1.20 MPa. In order to assess the correlation between shear modulus readings acquired from the seven interpretation techniques outlined in Tables 3 and 4, two statistical approaches are employed: the one-way analysis of variance (ANOVA) test and the correlation test. ANOVA tests are utilised to determine whether the means of two distributions are statistically similar. A lower ANOVA metric suggests a lower likelihood of the similarity hypothesis being true.