# Polarization Influence on Algorithms of TFBG Sensors Data Analysis for Bending Application
## 4 Partial Regression of the Transformed Spectrum Length Method  
The transmission spectrum of the TFBG grating can be treated as an analytical signal and methods such as multivariate calibration can be used.  
The imperfection in measuring instruments means that optical spectra contain components that do not carry information and make its extraction difficult. As with every measurement, there is noise, but the characteristic distortion of the spectra is the baseline. For the data presented in this article, an additional effect that makes interpretation difficult is the change in polarization. The calibration regression model looks for the relationship between the measurement vector (optical spectrum) and the determined radius (curvature).  
Methods such as PLS regression use the concept of the so-called latent variables. This method is used in many fields [23]. Even in cases of complex dependencies and additional impacts [24]. The partial regression method belongs to the group of chemometric methods, which are a specific application of statistics and mathematics to the analysis of multidimensional data. Cumulative contour spectra were used to build the model (Figure 11). The data set was divided into a training set consisting of 338 points and a test set of 169 points.  
Figure 10: Polarization influence on normalized wavelength: (**a**) normalized wavelength versus curvature; (**b**) normalized wavelength versus radius; (**c**) estimated radii versus real radii; (**d**) error of radii estimation.  
It could be seen in Figure 12a that the estimated radius reflects the value from the experiment. There are no points that strongly deviate from the trend determined by the characteristics. The error of the bending radius estimation is presented in Figure 12b and it could be seen that its value is only slightly changing for the different curvatures. The mean square error for the training data set was only 0.215 mm. However, for the test data set 0.223 mm. The method can handle the entire range of analyzed radii.  
Table 1 presents a summary of the calculations performed. The parameters of the root mean square error (RMS) and the maximum error were used as indicators for comparing individual methods. The parameters were calculated for two ranges of bending radii, the limit of which is 30 cm. In the range of 12-30 cm, the mean squared error value for classical methods used in TFBG demodulation differs by two to three times compared to the PLS method. The maximum error is three to six times larger. In the range of 31-50 cm, the amplitude methods are characterized by a much higher value of both RMS and maximum error. Envelope and contour methods in this area can only be used under strict polarization  
Figure 11: Cumulative contour length versus wavenumber for whole 39 radii.  
Figure 12: Predicted versus measured values of bending radii (**a**); prediction error for whole analyzed dataset (test and model) (**b**).  
control. The mean deviation from the local mean method fares only slightly better here. The wavelength method has a good RMS value of 0.87 cm. The wavelength method has a more even distribution of errors. By far the best parameters, respectively, 0.23 cm for RMS and 0.44 cm for maximum error, can be found with the newly developed method using the PLS model.