Abstract
Interferometric optical systems have been proposed for implementing dual-parameter optical sensors. For this type of sensors, the sensitivity matrix equation is generally used to determine the parameters to be measured based on the sensitivity of each parameter to one particular feature of the output reflective spectrum of the interferometric system. One of the disadvantages of this method is that the measurement ranges will be very short if the sensitivities are not linear or if these present cross-sensitivity. In this work, a multiple regression model for the simultaneous detection of refractive index and temperature based on an interferometric optical sensor is proposed. Here, the mathematical model is a weighted sum of features used to estimate the values of two response variables. These features are functions of an initial set of 27 explanatory variables whose values were extracted of the output reflective spectrum of the interferometric system. Besides, in order to sustenance the model application, the sensor was modeled and experimentally carried out. Three cases were studied: the estimation of temperature at different refractive indices, the estimation of temperature when refractive index is equal to one, and the estimation of refractive index at different temperatures. For each one of these cases, an optimal basis of functions was founded with the algorithm proposed and used to estimate the values of the response variables. Besides, a technique to reduce the initial set of variables was implemented. Finally, for the experimental data, For each one of these cases, an optimal basis of functions was founded with the algorithm 1 proposed and used to estimate the values of the response variables.
1. Introduction
Optical sensors have attracted a lot of attention among researches due to their considerable applications and advantages over other types of sensors. These advantages relate to small size, immunity to electromagnetic interference, high sensitivities, high measurement ranges, and low cost. Specifically, interferometric optical sensors have been used to measure different parameters, as for example, temperature, refractive index, curvature, and strain [1–3]. Many of these sensing structures have been proposed as two-parameter optical sensors, for which one of the parameters is temperature; this is because the main fiber optic material, silica, has thermal properties. For example, Zhao et al. [4] reported a Michelson interferometer based on a waist-enlarged fiber bitaper for measuring temperature and refractive index in the measurement ranges between 11–98°C and 1.351–1.402 RIU, respectively. In that work, the changes of both parameters were reflected in variations in wavelength and reflected intensity of one interference peak. Moreover, the linear sensitivities were estimated and used to calculate the sensor’s parameters values. Another example is the temperature and salinity water in seawater sensor based on a tapered fiber Mach-Zehnder interferometer reported by Liu et al. [5]. These authors reported the temperature and refractive index sensitivities for the two variables, which were obtained from the spectral shifting of two peaks of the fringe pattern of the interferometer. For that sensor, the measurement ranges were 18.9–39.6°C and 0–49%. Due to that the thermooptic and thermal-expansion coefficients of silica are small, this type of sensors presents very low temperature sensitivities. Besides, in this type of sensors, the sensitivity matrix equation can be used to determine the values of the two variables since it can help to minimize errors in the calculation of the values of the measured parameters but requires that the sensitivities must be linear in the measurement ranges with a quasinegligible cross-sensitivity. Additionally, dual-parameter interferometric optical sensors based on materials with considerable thermal properties have been reported [6–8]. For example, Tan et al. [7] reported a UV-curable polymer microhemisphere-based fiber-optic Fabry-Perot interferometer for simultaneous measurement of refractive index and temperature in the measurement which ranges from 1.38 to 1.42 RIU and from 25 to 65°C, respectively. Here, the thermal properties of the polymer were considerably higher than those of silica. The authors reported that the fringe contrast was mainly sensible to refractive index changes, and wavelength shifting was sensible to temperature changes. For this optical sensor, the refractive index sensitivity is not linear, and authors mentioned that due to the cross-sensitivity of the sensor to temperature and refractive index, a measurement error of refractive index is induced.
On the other hand, in many optical sensors based on interferometric systems, the shift of the peak of one of the interference fringes is tracked. Here, the maximum measurement range is limited by the free spectral range () of the interference spectrum of the sensor due to the 2 ambiguity. In this sense, in order to expand the measurement range or to counteract the cross-sensitivity, other techniques different to the sensitive matrix equation have been reported. For example, the empirical mode decomposition algorithm [8], the method of intensity-modulated [9], and the use of neural network algorithms [10, 11] have been reported. In these last works, the mathematical models are based on different features of the output signal of the optical sensor utilized to predict the correct value of the measured parameter. Besides, it is important to highlight that nowadays, the application of different mathematical models that relate one independent variable with several explanatory variables is being proposed in different areas [12–15]. As an example in the medical area, Rajput et al. [13] applied different machine learning models as logistic regression and support vector machine, among others to develop a reference model in assisting rural people suffering from diabetes. These models were used to list the risks factors and correlation that exist among these. Another interesting example, in the communications networks area, Lakshmanna et al. [12] designed a model to achieve maximum energy utilization and lifetime in a wireless sensor network. This model was based on the teaching-learning-based optimization algorithm-based multihop routing technique which was applied to optimize the selection of routes of destinations.
In this work, a multiple regression model was applied to simultaneously measure temperature and refractive index (RI) over wide measurement ranges with a dual-parameter sensor based on an interferometric system. Here, the values of the model variables depend on the wavelength and the amplitude of fringes of the interference pattern of the optical sensor. Besides, these values were transformed by using a basis of functions. With experimental and theoretical data obtained using a sensor model based on three layers, three case studies based on an initial set of variables were analyzed. In addition, these cases were analyzed with a reduced set of variables selected by means a variable reduction technique. Finally, the efficiency of the regression equation obtained by applying the proposed model was evaluated with the root mean square error and the multiple determination coefficient.
2. Experimental Setup and Interferometric Sensor Model
In Figure 1(a), it showed the dual-parameter optical sensor model utilized to measure simultaneously refractive index and temperature. This sensor is based on an interferometric arrangement of three layers (L1, L2, and L3) deposited at the tip of an optical fiber. The details of the sensor head element fabrication are explained in [16]. The measured reflected spectra were obtained with the experimental setup shown in Figure 1(b). Here, the circulator was used to guide the light of a broadband source through its ports. The fiber-coupled interferometric arrangement was spliced to the port 2, and its reflected signal is monitored by an optical spectrum analyzer (OSA) at the port 3. Finally, a digital hot plate was used to control the temperature, and a digital refractometer and a calibrated sensor were used to corroborate the values of refractive index and temperature.
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Regarding the theoretical results, the relative reflected intensity spectra were simulated by using a mathematical model that takes into account the main rays reflected between layers [16]. In this model, the refractive index (), the thickness (), the thermoexpansion coefficient (), and the thermooptic coefficient () are considered. The characteristics of the layers of our interferometer are as follows: for L1, μm, 0, and 0 K-1; for L2, , 198 μm/mK, and K-1; and for L3, , where is the wavelength, is the temperature, μm, , and .
The reflection spectrum caused by one layer is a pattern of periodic fringes with an (separation between fringes) that depends on the refractive index and the thickness of the layer. Here, as the filter is formed by a set of layers, the resulting reflected intensity spectrum is a fringe pattern formed by the superposition of the different generated spectra. In this way, due to the different s of each superimposed spectra, the overall spectrum will present a modulated periodic fringe pattern, showing a variation in amplitudes of the fringes. For instance, the spectrum of the fabricated filter presented two , one of 3.5 nm and another of 25 nm, which are due to the silicon layer (L2) and the polymer layer (L3), respectively. In Figure 2(a), it can be seen both, the experimental and the simulated, reflected spectra at different external RIs () and at the same temperature. Here, it can be observed that the maximum amplitude of the fringes (A) decreases when the external RI increases while their wavelength positions remains almost constant. Now, the behaviour of the spectrum when the temperature increases and the refractive index is fixed is shown in Figure 2(b). Here, the wavelength is shifted toward the right, and the amplitude is changed. This behaviour was described in detail in [8].
The sensitivities that present our interferometric sensor are the refractive index sensitivities to the amplitude and to the wavelength position , the temperature sensitivities to the amplitude , and the wavelength position . Here, if the amplitude and the wavelength position of a fringe peak are used to determine changes in temperature and refractive index, then 0, , is linear in the whole measurement range, and is quasilinear in the measurement range needed to spectrally shift the fringe peak over 1 =3.5 nm. This range is limited by the 2 ambiguity that the interferometric sensors present. Moreover, is nonlinear in the measurement range limited by 1 , and there exists linearity only within small measurement ranges. Besides, there exists a strong cross-sensitivity to and since two of the layers are sensitive to temperature provoking changes in the amplitude of the fringes. This implies that the dual-parameter sensing matrix [4, 5], , cannot be used to determine the values of the parameters without errors for broad measurement ranges. Hence, we implemented a method with which the response variable depends linearly on several features that are explanatory variables transformed with a basis of functions. Here, a set of explanatory variables will be formed with different features of the reflection spectrum taking into account changes in wavelength and amplitude. Finally, and will be taken as the response variables.
3. Implemented Model Theory
The multiple regression expresses a relationship between more than one explanatory variable () and one response variable (). Here, when the response data are not well fitted with a conventional linear model [13], these can be modeled by including additional features via a function applied to the independent variables, being expressed as the model as follows: where is the regression coefficient associated to the -th explanatory variable, is the coefficients number, and is a constant of regression. As the model is linear with respect to the parameters , therefore, the vector , that contains the values of the regression coefficients, can be found by using a ridge regression with regularization of the weights, which can be expressed by the following equation: where is the matrix for which each column has the transformed values of each one of the explanatory variables except for the first column that have ones, is a vector that contains the observed values of the response variable, is an identity matrix, and is a penalty term. Here, it is important to mention that the values of the explanatory variables can be transformed with a function formed by a basis of functions . When more than one function is used, the transformed data matrix is expressed as
The basis of functions that were used are , , , and , where is a constant and is a variable. Besides, it can be observed in Eq. (3) that the dimensions of matrix increases when more than one function is used in the data transformation. Here, when , it is convenient to use the matrix inversion lemma (Eq. (4)) in order to perform the inverse of Eq. (2) in the smallest space, the dimension of the number of data cases.
Finally, the regression equation to estimate the values of the response variable, , can be expressed as where is the estimation error. In this case, the response variable is a linear weighted sum of functions of each variable to model the outcome.
Additionally, the -fold crossvalidation technique [17] was implemented to find the value of that optimizes the regression equation obtained with the proposed model. Here, in general, first, a set of values of was proposed, and for each value of , values were calculated and averaged to obtain a value. In this way, the optimal value of was the one that generated the smallest . Finally, in order to measure how well the proposed model predicts the values of the response variable, we used the coefficient of multiple determination and the root mean square error , where is the estimated value of the response variable and is the mean of the values of .
3.1. Selection of the Basis of Functions
The regression equation obtained with the proposed model is based on a basis of five functions that are applied to the set of explanatory variables. These functions are , , , and . Now, in order to find the functions that make the regression equation efficient, the algorithm 1 was implemented.
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3.2. Selection of the Variables and Reduction of the Size of the Variables Set
As the spectra of the used optical filter are dependent from temperature and refractive index, the explanatory variables were obtained from features affected by these two parameters. Here, valleys (), peaks (), and spectral windows () (see Figure 2(c)) were considered in order to define the criterions of the proposed explanatory variables (see Table 1).
Later, a technique to reduce the number of explanatory variables used in the proposed model was implemented [17]. Briefly, this technique initiates with an equation based on the functions obtained after applying the algorithm 1 and taking into account all the explanatory variables. The variables were eliminated one by one, and at each time that a variable was removed, the new regression equation was used to calculate the value. Here, it is important to mention that the algorithm initiates with a transformed data matrix expressed as , and the transformed data matrix of the optimal regression equation is expressed as , where is the number of explanatory variables removed.
4. Results
A model of an interferometric optical sensor based on thermal sensitive layers was used to obtain simulated data. These data were obtained by simulating changes in the temperature and in the external refractive index. Besides, these data were used to apply the proposed mathematical model. Reflected spectra at different external refractive indices and temperatures were simulated. Here, the temperature range was from 0 to 100°C, and the refractive index range was from 1.0 to 2.0 RIU with steps of 1°C and 0.02 RIU, respectively. All the variables () listed in Table 1 were obtained for each simulated spectrum. Here, it is important to mention that the temperature sensor and the external refractive index were the response variables. The mathematical model was implemented taken into account three sensing cases. For the first and second cases, the estimation of temperature at different refractive indices and when the refractive index was equal to one were considered as different cases. The third case was the estimation of refractive index at different temperatures.
4.1. Estimation of Temperature at Different External Refractive Indexes
The mathematical model was applied to a synthetic dataset () that was formed with features of 5151 simulated spectra. In this way, the dataset is a matrix with 5151 rows (observations) and 27 columns (variables) that contains the values of the explanatory variables. This dataset was transformed with a basis of functions in order to find an optimal regression equation (Eq. (5)). In this way, the stages of the algorithm 1 were implemented (see Figure 3).
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The results of the stage 1 are shown in Figure 3(a). Here, the values, as a function of obtained at each time that a power function was added, can be seen. The set of values of were from 1 to 100 with steps of 1 for all the functions of this stage. The solid circle indicates the value. Moreover, in this stage, the matrix was transformed four times, and the optimal values of of the power functions were 1, 3, 6, and 10. Furthermore, as can be observed, the values decrease each time that a power function is added. Here, the values were 7.650°C, 3.225°C, 2.488°C, and 1.956°C which correspond to the power functions , , , and , respectively. This means that for the first power function, , the smallest value () was for ; for the second added function, , the was 3.225°C for , and so on for the next two added functions. Here, it is important to mention that all the functions are potential as was mentioned in the algorithm, and no more of four functions were added because there was no significant decrement in the value. For this stage, they selected the first three functions () because of the change in (0.5325°C) which is small when the fourth function is added. Finally, the transformed matrix in this stage was .
Regarding step (d) of the stage 1 of the algorithm 1, the -fold crossvalidation technique was implemented to find the optimal value of used to find the regression coefficients of the regression equations. Here, each time the matrix was transformed with an additional function and standardized, it was divided in 5 groups; thus, each training and test group had 80% and 20% of data. Here, it is important to mention that for the added function, each time a value of was used, the matrix was divided. The set of values of were with steps of . The optimal values of were 0, , and , which correspond to the functions , , and , respectively.
In Figure 3(b), it showed the results obtained with the stage 2 of the algorithm 1. Here, was set as , with steps of 0.1, for the sine, the cosine, and the logarithm functions. Additionally, it was set with steps of 0.01 for the exponential function. At this stage, the values were 1.2452°C, 1.0053°C, 1.6684°C, and 1.6558°C for the functions , , , and , respectively. Here, the cosine was selected as the added function, since the regression equation based on presented the lowest value. Afterwards, for the stage 3 for which the results are shown in Figure 3(c), the power and cosine functions were not considered because these functions were selected in the previous stages. Besides, for the stage 3, the set of values of were defined within for the sine and the logarithm functions; moreover, it was set as for the exponential function. In this way, the obtained values were 0.8626°C, 0.9623°C, and 1.0093°C for the functions , , and , respectively. Based on these results, the proposed function at this stage was . Finally, the results of stage 4 are shown in Figure 3(d). In this stage, the set of values of were the same that were used in stage 3. Here, the values were 0.9135°C for and 1.3584°C for . Here, as the change in (0.0509°C) was not significant with respect to that obtained in the previous stage (0.8626°C), no function of this stage was added. In this way, the final basis of functions to estimate temperature was , and the transformed matrix was . A summary of the optimal values of and of the mentioned stages are listed in Table 2.
Regarding step (d) of the stages 2–4 of the algorithm 1, the set of values of were also with steps of . The optimal values of were , , and which correspond to the functions , and , respectively. Finally, the regression equation based on was used to calculate the estimated temperature values. These were compared with the real temperature values, as shown in Figure 4(a). Here, the values of each diagonal correspond to a refractive index value between 1.00 and 2.00 RIU. It is observed that the estimated values presents a high fit to the real values. Based on these results, the values of and achieved were 0.8626°C and 0.9991, respectively.
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After finding the functions with their coefficients with which the best estimates were obtained, the technique to reduce the size of the explanatory variables set was implemented. In Figure 4(b), the evolution of as a function of the number of eliminated variables is presented. The numbers listed in the figure show the number and order in which each one of the variables was eliminated. Here, the dash means that the first is calculated by using all the explanatory variables. Later, it showed the obtained when the first variable () was removed. In this way, the is evolving as another variable is removed until these are finished; in our case, the last removed variable was . Here, it can be observed that the increases as more explanatory variables are removed. The criterion to choose which variables to eliminate was that the value obtained with the reduced variables subset was with respect to the value obtained with all the variables set. In this way, we removed the first 7 variables listed in Figure 4(b), and the size of the new subset of explanatory variables was . With this variables subset the resulting transformed matrix was . The real temperature values and estimated temperature values with the regression equation, based on 20 variables and the functions , are shown in Figure 4(c). It is observed that the fit is practically the same as that observed in Figure 4(a) that was obtained considering 27 variables. Moreover, for this later case in which a reduced number of variables was used, the reached and were 0.9868°C and 0.9989, respectively.
The proposed mathematical model was also evaluated using experimental data. Here, the reduced subset of variables and the function were used, and the optimal value of was . The experimental data were obtained from 96 reflected spectra, which were recorded with the OSA when the interferometric sensor was submerged in liquids at different temperatures and refractive indices. In our experiments, temperature was varied between 22°C and 26°C, and liquids with refractive indices between 1.000 RIU and 1.497 RIU were used. The comparison between measured and estimated temperature values is shown in Figure 4(d). The and values were 0.0887°C and 0.9958, respectively.
4.1.1. Estimation of Temperature when External Refractive Index Is Equal to One
Many optical temperature sensors are used to detect temperature when these are not submerged in liquids; for this reason, we decided to simulate and characterize our proposed interferometric system for the case when only . The mathematical model is applied using a synthetic dataset that is formed with features of 101 simulated spectra for which the external refractive index is equal to one. In this way, the dataset is a matrix with 101 rows (observations) and 27 columns (variables) that contains the values of the explanatory variables. This dataset was also transformed with a basis of functions in order to find an optimal regression equation (Eq. (5)). In this way, the stages of the algorithm 1 were also implemented.
The results of all the stages are presented in Figure 5 and summarized in Table 3. In the stage 1 (Figure 5(a)), the set of values of were from 1 to 100 with steps of 1 for all the functions. In this stage, the matrix was transformed five times, and the optimal values of of the power functions were 15, 29, 93, 73, and 3, resulting in the following functions , , , , and . The values obtained by using , , , , and were 1.5704°C, 0.3318°C, 0.1553°C, 0.0870°C, and 0.0644°C, respectively. For this stage, we decided the first four power functions (), because the variation in was also small (0.0226). Finally, for this stage, the transformed matrix is expressed as .
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The results obtained in the other stages are shown in Figures 5(b)–5(e). Here, the set of values of were with steps of 1 for the sine and cosine functions, with steps of 0.01 for the logarithm function, and with steps of 0.01 for the exponential function. In the stage 2 (Figures 5(b) and 5(c)), the optimal values of were 61, 76, -0.34, and 0.36 for the functions , , and , respectively. The values obtained with these functions were 0.0370°C, 0.0389°C, 0.0673°C, and 0.0559°C, respectively. Based on these results, the chosen function was . Now, for the stage 3 (Figures 5(d) and 5(e)), the implemented functions and the values were , , and and 0.0154°C, 0.0312°C, and 0.0311°C, respectively. In this stage, the chosen function was . And for the last stage (Figure 5(f)), the tested functions were and , and the resulting values were 0.0081°C and 0.0079°C, respectively. Finally, the functions proposed to estimate temperature were with which the transformed matrix was expressed as .
The process to obtain the optimal values of was the same. In this way, using the functions , , , , , , and , the optimal values of were , , , , , , and , respectively. The regression equation based on was used to calculate the estimated temperature values. The comparison between real and estimated temperature values is shown in Figure 6(a). It is observed that the estimated values present a high fit to the real values. Based on these data, the reached values of and were 0.0079°C and 0.9999, respectively. The algorithm to reduce the size of the set of explanatory variables was also implemented. The evolution of the as a function of the number of eliminated variables is shown in Figure 6(b). It is observed in the list that the first and last variables removed were and , respectively. The criterion to choose the variables was that the variation in was . In such manner, the first 12 variables listed were removed, leaving a subset of 15 explanatory variables. Here, the resulting transformed matrix was . The comparison between real and estimated temperature values, calculated with the regression equation based on the 15 chosen variables and , is shown in Figure 6(c). It is observed that the fit is practically the same as that observed in Figure 6(a). The and values were 0.0498°C and 0.9999, respectively.
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The proposed mathematical model was also evaluated using experimental data. Here, the reduced subset of variables and was used; the optimal value of was . The experimental reflected spectra were obtained when the interferometric sensor was no submerged in a liquid. Then, from the 96 experimental reflected spectra, there were considered only the corresponding to an external refractive index equal to one. Here, the temperature of the interferometric sensor was varied from 22°C to 26°C. The comparison between measured and estimated temperature values is shown in Figure 6(d). The and values were 0.0138°C and 0.9999, respectively.
4.2. Estimation of Refractive Index at Different Temperatures
The mathematical model was applied with the same synthetic dataset used to estimate refractive index at different temperatures (). The values of refractive index () were taken as values of the response variable. The dataset was also transformed with a basis of functions in order to find an optimal regression equation to estimate external refractive index without taking temperature into account. In this way, the stages of the algorithm 1 was implemented.
The results obtained in each one of the stages can be seen in Figure 7. For the stage 1, the set of values of was from 1 to 100 with steps of 1 for all the power functions. The matrix was transformed four times, and the optimal values of of each one of the added functions were 1, 3, 9, 13, and 87. In this way, the tested functions were , , , , and . In Figure 7(a), it showed the evolution of the as a function of for these four functions. The values obtained using , , , , and were 0.0063 RIU, 0.0031 RIU, 0.0020 RIU, 0.0016 RIU, and 0.0014 RIU, respectively. For this stage, we decided a basis of functions with the first three power functions (), resulting a transformed matrix as .
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For the other stages, the values of were with steps of 1 for the sine, cosine, and logarithm functions and with steps of 0.01 for the exponential function. The optimal values of obtained in the stage 2 (Figure 7(b)) were 11, 16, 76, and 0.34 for the functions , , and , respectively. The values obtained with these functions were 0.0015 RIU, 0.0014 RIU, 0.0016 RIU, and 0.0017 RIU, respectively. Then, the chosen function was . Now, for the stage 3 (Figure 7(c)), the implemented functions with their optimal value of were , , and . The values obtained with these functions were 0.0012 RIU, 0.0013 RIU, and 0.0016 RIU, respectively. In this stage, the chosen function was . Moreover, for the fourth and last stage (Figure 7(d)), the optimal values of were 80 and 0.1 for the functions and for which the resulting values were 0.0042 RIU and 0.0061 RIU, respectively. Here, as the values increased with any of the functions tested in this stage, it was decided not to add any function to . Thus, the reached values of and with function were 0.0012 RIU and 0.9999, respectively. Finally, the optimal values of and obtained at each one of the stages for each one of the functions are listed in Table 4.
The optimal values of were , , , , and for the functions , , , , and , respectively. In Figure 8(a), the real and estimated refractive index values obtained with the regression equation based on can be seen. Here, each vertical line of the figure corresponds to a temperature value, and the and values were 0.0012 RIU and 0.9999, respectively. Now, regarding to the process for reducing the number of variables, the evolution of is presented in Figure 8(b). For this case, the criterion was that the variation in was . Here, the first 15 variables listed in the figure were removed, leaving a subset of 12 explanatory variables. The and values with this subset of variables were 0.0020 RIU and 0.9999, respectively. Finally, the real and estimated refractive index values based on these 15 variables and are shown in Figure 8(c).
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As in the previous cases, the mathematical model was also evaluated by using experimental data in order to validate the regression equation obtained with simulated data. Here, the same experimental spectra used in the case to estimate temperature at different refractive indices were utilized. In this case, the response variable had the values of refractive indices of the liquids. Besides, the optimal value of λ used in the regression equation was , and the reached values of and were 0.0007 RIU and 0.9999, respectively. Finally, the comparison between measured and estimated temperature values is shown in Figure 8(d).
As can be seen in the results of the three cases, the simultaneous measurement of temperature and refractive index and their measurement ranges was not limited by the proposed algorithm, and all the values of the response variable were estimated with high precision for the measurement ranges used. These measurement ranges are larger than those achieved with the analysis of the output signals of interferometric sensors with conventional methods such as the use of the sensitivities matrix equation for the case of two-parameter sensors. In addition, these results were achieved because several characteristics of the output spectrum were taken into account and not just one or two as in conventional methods. The mathematical method that was proposed in this work relates a weighted sum of features, which are functions of an initial set of explanatory variables, with two response variables.
5. Conclusions
In this work, it was demonstrated that a multiple regression model based on explanatory variables, transformed by means a basis of functions, can be implemented to use a thermosensitive interferometric system for simultaneous measurement of temperature and refractive index. The model is based on variables that depend on the wavelength and the amplitude of peaks and fringes of the output signal of the interferometric sensor that change when the interferometer is subjected to changes in temperature and refractive index. Here, as not all sensor sensitivities are linear, therefore, it is not possible to use the sensitivity matrix equation to obtain simultaneously the temperature and refractive index values over wide measurement ranges. In this way, it was shown that transforming the variables of the model with a basis of functions (power, exponential, logarithmic, sine, and cosine), it is possible to ideally measure refractive index values between 1.00 and 2.00 RIU and temperature values between 0 and 100°C. Besides, with the proposed mathematical model, it was possible to overcome the ambiguity 2, being able to measure in a temperature measurement range greater than the typical limited by 1 . The proposed model was experimentally supported by using liquids with refractive index values between 1.00 and 1.49 RIU and temperature values between 22°C and 26°C achieving a root mean square error and a multiple determination coefficient of 0.0012 RIU/0.9999 and 0.0887°C/0.9958. Finally, it is important to note that since the estimation of the values of the response variables is based on multiple features, multiple machine learning models can be used for this type of applications, which will be aborded in future work.
Data Availability
The experimental and simulated data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
This research was partially funded by the Universidad de Guanajuato under projects CIIC 208/2022 and CIIC 144/2022.