Determination of Optimal Measurement Points for Calibration Equations—Examples by RH Sensors

The calibration points for sensors must be selected carefully. This study uses accuracy and precision as the criteria to evaluate the required numbers of calibration points required. Two types of electric relative humidity (RH) sensors were used to illustrate the method and the standard RH environments were maintained using different saturated salt solutions. The best calibration equation is determined according to the t-value for the highest-order parameter and using the residual plots. Then, the estimated standard errors for the regression equation are used to determine the accuracy of the sensors. The combined uncertainties from the calibration equations for different calibration points for the different saturated salt solutions were then used to evaluate the precision of the sensors. The accuracy of the calibration equations is 0.8% RH for a resistive humidity sensor using 7 calibration points and 0.7% RH for a capacitance humidity sensor using 5 calibration points. The precision is less than 1.0% RH for a resistive sensor and less than 0.9% RH for a capacitive sensor. The method that this study proposed for the selection of calibration points can be applied to other sensors.


Introduction
The performance of sensors is key for modern industries. Accuracy and precision are the most important characteristics. Calibration ensures sensors' performance. When a sensor is calibrated, the reference materials or reference environments must be specified. For a balance calibration, a standard scale is the reference materials. For temperature calibration, the triple point of ice-water or boiling matter is used to maintain the reference environment.
The number and the location of the calibration points.
The standard references and their uncertainties.
Betta [1] adopted minimizing the standard deviations for the regression curve coefficients or the standard deviation for the entire calibration curve to design an experiment to determine the number of calibration points, the number of repetitions, and the location of calibration points. Three types of sensor were used to demo the linear, quadratic and cubic calibration equations: a pressure transmitter, a platinum thermometer and E-Type thermocouple wires. The estimated confidence interval values were used to determine the validity of the regression equation. This method was extended to address calibration for complex measurement chains [2]. Lake [27] Lu and Chen [17] calculated the uncertainty for humidity sensors that were calibrated using 10 saturated salt solutions for two types of humidity sensors. The study showed that a second-order polynomial calibration equation gave better performance than a linear equation. The measurement uncertainty is used as the criterion to determine the precision performance of sensors [38].
The number of standard relative humidity values for fixed-point humidity systems is limited by the number and type of salt solutions. The number of salt solutions that must be used to specify the calibration points for the calibration of RH sensors is a moot point. More salt solutions allow more calibration points for the calibration of RH sensors. However, using more salt solutions is time-consuming. This study determined the effect of the number and type of salt solutions on the calibration equations for two types of humidity sensors. The accuracy and precision were determined in order to verify the method for the choice of the optimal calibration points for sensor calibration.

Relative Humidity (RH) and Temperature Sensors
Resistive sensor (Shinyei THT-B141 sensor, Shinyei Kaisha Technology, Kobe, Japan) and capacitive sensor (Vaisala HMP-143A sensor, Vaisala Oyj, Helsinki, Finland) were used in this study. The specification of the sensors is listed in Table 2.

Saturated Salt Solutions
Eleven saturated salt solutions were used to maintain the relative humidity environment. These salt solutions are listed in Table 3.

Calibration of Sensors
The humidity probes for the resistive and capacitive sensors were calibrated using saturated salt solutions. A hydrostatic solution was produced in accordance with OIML R121 [19]. The salt was dissolved in pure water in a ratio such that 40-75% of the weighted sample remained in the solid state. These salt solutions were stored in containers.
The containers were placed in a temperature controller at an air temperature of 25 ± 0.2 • C. During the calibration process, humidity and temperature probes were placed within the container above the salt solutions. The preliminary study showed that an equilibrium state is established in 12 h so the calibration lasted 12 h to ensure that the humidity of the internal air had reached an equilibrium state. Experiments for each RH environment were repeated three times. The temperature was recorded and the standard humidity of the salt solutions was calculated using Greenspan's equation [18].

Establish and Validate the Calibration Equation
The experimental design and flow chart for the data analysis is shown in Figure 1. The relationship between the standard humidity and the sensor reading values was established as the calibration equation.
This study used the inverse method. The standard humidity is the dependent (y i ) and the sensor reading values are the independent variables (x i ) [17].
The form of the linear regression equation is: where b 0 and b 1 are constants. The form of the higher-order polynomial equation is: Y = c 0 + c 1 X + c 2 X 2 + c 3 X 3 + . . . +c k X k (2) where c 0 , c 1 to c k are constants.  The relationship between the standard humidity and the sensor reading values was established as the calibration equation.
This study used the inverse method. The standard humidity is the dependent (yi) and the sensor reading values are the independent variables (xi) [17].
The form of the linear regression equation is: where b0 and b1 are constants. The form of the higher-order polynomial equation is: where c0, c1 to ck are constants.

Different Calibration Points
To model the calibration equations, the data for four different salt solutions was used, as listed in Table 3.
Case 1: The data set is for 11 salt solutions and 11 calibration points Case 2: The data set is for 9 salt solutions and 9 calibration points Case 3: The data set is for 7 salt solutions and 7 calibration points

Different Calibration Points
To model the calibration equations, the data for four different salt solutions was used, as listed in Table 3.
Case 1: The data set is for 11 salt solutions and 11 calibration points Case 2: The data set is for 9 salt solutions and 9 calibration points Case 3: The data set is for 7 salt solutions and 7 calibration points Case 4: The data set is for 5 salt solutions and 5 calibration points For each sensor, four calibration equations were derived using four different calibration points.

Data Analysis
The software, Sigma plot ver.12.2, was used to determine the parameters for the different orders of polynomial equations. The criteria to assess the fit of the calibration equations are the coefficient of determination R 2 , the estimated standard error of regression s and the residual plots.
The coefficient of determination, R 2 is used to evaluate the fit of a calibration equation. However, no standard criterion has been specified [15,16].
The single parameter coefficient was tested using the t-test to evaluate the order of polynomial regression equation. The hypotheses are: The t-value is: where b k is the value of the parameter for the polynomial regression equation of the highest order, and se(b k ) is the standard error of b k .

The Estimated Standard Error of Regression
The estimated standard error of regression s is calculated as follows: whereŷ i is the predicted valued of the response,ŷ i is the response, n 1 is the number of data and p is the number of parameters. The s value is the criterion that is used to determine the accuracy of a calibration equations [38]. It is used to assess the accuracy of two types of RH sensors that are calibrated using different saturated salt solutions.

Residual Plots
Residual plots is the quantitative criterion that is used to evaluate the fit of a regression equation. If the regression model is adequate, the data distribution for the residual plot should tend to a horizontal band and is centered at zero. If the regression equation is not accepted, the residual plots exhibit a clear pattern.
For the calibration equation, tests on a single regression coefficient and the residual plots are used to determine the suitability of a calibration equation for RH sensors that are calibrated using different saturated salt solutions. The estimated standard error of the regression equations is then used to determine the accuracy of the calibration equations.

Measurement Uncertainty for Humidity Sensors
The measurement uncertainty for RH sensors using different salt solutions was calculated using International Organization for Standardization, Guide to the Expression of Uncertainty in Measurement (ISO, GUM) [12,13,17]. u c 2 = u 2 x pred + u 2 temp + u 2 non + u 2 res + u 2 sta (7) where u c is the combined standard uncertainty, ux pred is the uncertainty for the calibration equation, u temp is the uncertainty due to temperature variation, u non is the uncertainty due to nonlinearity, u res is the uncertainty due to resolution, and u sta is the uncertainty of the reference standard for the saturated salt solution.
The uncertainty of x pred is calculated as follows [38]: where y is the average value of the response. The uncertainty in the value of u ref for the saturated salt solutions is determined using the reference standard for the salt solution. The scale and the uncertainty of these saturated salt solutions are listed in Table 3 that are taken from Greenspan [18] and the Organisation Internationale De Metrologies Legale (OIML) R121 [19]: where u ri is the uncertainty in the humidity for each saturated salt solution and N 2 is the number of saturated salt solutions that are used for calibration. The calibration equations use different numbers of saturated salt solutions had its uncertainty. This criterion is used to evaluate the precision of RH sensors.
The accuracy and precision of RH sensors that are calibrated using different saturated salt solutions was determined using the s and u c values. By Equations (7)- (9), the contrast between the number of saturated salt solutions is considered. The greater the number of data points that are used, the smaller is the s value that is calculated by Equation (6). However, this requires more experimental time and cost and the value of u ref may be increased. The uncertainty of each calibration point is different because different saturated salt solutions are used. The optimal number of calibration points were evaluated by accuracy and precision.

THT-B121 Resistive Humidity Sensor
Calibration equations for resistive sensors using 11 salt solutions: The distribution of the relative humidity data for the reading values for a resistive sensor is plotted against the standard humidity values that are maintained using 11 saturated salt solutions in Figure 2. is the uncertainty due to resolution, and usta is the uncertainty of the reference standard for the saturated salt solution.
The uncertainty of xpred is calculated as follows [38]: where is the average value of the response. The uncertainty in the value of uref for the saturated salt solutions is determined using the reference standard for the salt solution. The scale and the uncertainty of these saturated salt solutions are listed in Table 3 that are taken from Greenspan [18] and the Organisation Internationale De Metrologies Legale (OIML) R121 [19]: where is the uncertainty in the humidity for each saturated salt solution and is the number of saturated salt solutions that are used for calibration.
The calibration equations use different numbers of saturated salt solutions had its uncertainty. This criterion is used to evaluate the precision of RH sensors.
The accuracy and precision of RH sensors that are calibrated using different saturated salt solutions was determined using the s and uc values. By equations [7][8][9], the contrast between the number of saturated salt solutions is considered. The greater the number of data points that are used, the smaller is the s value that is calculated by Equation [6]. However, this requires more experimental time and cost and the value of uref may be increased. The uncertainty of each calibration point is different because different saturated salt solutions are used. The optimal number of calibration points were evaluated by accuracy and precision.

THT-B121 Resistive Humidity Sensor
Calibration equations for resistive sensors using 11 salt solutions: The distribution of the relative humidity data for the reading values for a resistive sensor is plotted against the standard humidity values that are maintained using 11 saturated salt solutions in Figure 2. The estimated parameters and the evaluation criteria for regression analysis are listed in Table 4. The residual plots for the calibration equations for different orders of polynomial equations are shown in Figure 3. The estimated parameters and the evaluation criteria for regression analysis are listed in Table  4. The residual plots for the calibration equations for different orders of polynomial equations are shown in Figure 3.   The coefficient of determination, R 2 , for the linear, 2nd, 3rd and 4th order polynomial calibration equations are 0.9967, 0.9974, 0.9987 0.9993, respectively. High R 2 values do not give useful information for the specification of an appropriate calibration equation. The estimated values of standard deviation, s, is used to define the uncertainty for an inverse calibration equation [35]. The s values for the four calibration equations are 1.6098, 1.4612, 0.9820 and 0.7719, respectively. It is seen that an appropriate calibration equation gives a significant reduction in uncertainty.
Calibration equations for resistive sensor using 5 salt solutions: The estimated parameters and the evaluation criteria for the regression analysis for 5 calibration points for a resistive sensor are listed in Table 5. The residual plots for four calibration equations are shown in Supplementary Materials. Similarly to the regression results for 11 salt solutions, the linear, 2nd and 3rd order polynomial equations all employed a systematic distribution in the residuals plots. These equations are clearly not appropriate calibration equations. For a resistive sensor, the residual plots for the 4th order polynomial equations presented a random distribution.  The regression results for the 4th order polynomial equations using different calibration points in different salt solutions are listed in Table 6. The results for 9 and 7 calibration points are similar to those for 11 and 5 calibration points. Table 6. Estimated parameters and evaluation criteria for the 4th order polynomial equations for THT-B121 resistive sensors using four different calibration points.  The R 2 value is used b to evaluate the calibration equations [27,33]. Even the linear calibration equation for this study shows a high R 2 value. However, the estimated error was higher than that for other equations. The residual plots all exhibited a clear pattern distribution so the R 2 value cannot be used as the sole criterion to assess the calibration equation. Betta and Dell'Isola [1] mention R 2 , Chi-square and F-test to verify the accuracy of a model. This study used t-value for a parameter was used as the criterion. This method bases on statistical theory.

HMP 140A Capacitive Humidity Sensor
Calibration equations for a capacitive sensors using 11 salt solutions The relationship between the reading values for a capacitive sensor and the standard humidity values that are maintained using 11 saturated salt solutions is shown in Figure 4.
as the criterion. This method bases on statistical theory.

HMP 140A Capacitive Humidity Sensor
Calibration equations for a capacitive sensors using 11 salt solutions The relationship between the reading values for a capacitive sensor and the standard humidity values that are maintained using 11 saturated salt solutions is shown in Figure 4.
The estimated parameters and the evaluation criteria for regression analysis are listed in Table 7.  The residual plots for the calibration equations for different orders of polynomial equations are shown in Figure 5.  The estimated parameters and the evaluation criteria for regression analysis are listed in Table 7. The residual plots for the calibration equations for different orders of polynomial equations are shown in Figure 5.

Calibration equations for a capacitive sensor using 5 salt solutions
The estimated parameters and the evaluation criteria for the regression analysis for 5 calibration points for a capacitance are listed in Table 8. The residual plots for four calibration equations are shown in Supplementary Materials. Similarly to the regression results for 11 salt solutions, residuals plots for the linear equation exhibit a systematic distribution. Residual plots for the 2nd order polynomial equations presented a random distribution. The  Table 9. The results of R 2 values for 5, 7, 9 and 11 calibration points are similar. However, the calibration equation for 11 calibration points gives the smallest s value. Table 9. Estimated parameters and evaluation criteria for the 2nd order polynomial equations for HMP 140A capacitive sensors using four different calibration points.

Evaluation of Accuracy
The distribution between the number of saturated salt solutions and the estimated standard error for the calibration equations of two types of RH sensors is in Figure 6. For a resistance sensor, the s values of 7, 9, 11 calibration points are <0.8% RH. For a capacitance sensor, the s values for four saturated salt solutions are <0.8% RH. The accuracy of these calibration equations is <0.8% for both types of RH sensors. In terms a practical application [20,21], the calibration equation can be established using 7 salt solutions for a resistance sensor and 5 salt solutions for a capacitance sensor.

The Measurement Uncertainty for the Two Humidity Sensors
The method that is used to calculate the measurement uncertainty is that of Lu and Chen [17]. Two Types "A" and "B" method are used to evaluate the measurement uncertainty. The Type A standard uncertainty is evaluated by statistical analysis of the experimental data. The Type B standard uncertainty is evaluated using other information that is related to the measurement.
The Type A standard uncertainty for the two types of humidity sensors used the uncertainty for the predicted values from the calibration equations. The Type B standard uncertainty for humidity sensors uses the reference standard, nonlinear and repeatability, resolution and temperature effect. The results for the Type B uncertainty analysis for resistive and capacitive sensors are respectively listed in Tables 10 and 11.

The Measurement Uncertainty for the Two Humidity Sensors
The method that is used to calculate the measurement uncertainty is that of Lu and Chen [17]. Two Types "A" and "B" method are used to evaluate the measurement uncertainty. The Type A standard uncertainty is evaluated by statistical analysis of the experimental data. The Type B standard uncertainty is evaluated using other information that is related to the measurement.
The Type A standard uncertainty for the two types of humidity sensors used the uncertainty for the predicted values from the calibration equations. The Type B standard uncertainty for humidity sensors uses the reference standard, nonlinear and repeatability, resolution and temperature effect. The results for the Type B uncertainty analysis for resistive and capacitive sensors are respectively listed in Tables 10 and 11. The Type A standard uncertainty that are calculated using the predicted values for the 4th order polynomial equation for the resistive sensor and the 2nd order polynomial equation for a capacitive sensor are added to give a combined uncertainty using Equation (7). The combined uncertainty for three RH observations for the two humidity sensors using calibration equations that use different calibration points are in Figures 7 and 8. The Type A standard uncertainty that are calculated using the predicted values for the 4th order polynomial equation for the resistive sensor and the 2nd order polynomial equation for a capacitive sensor are added to give a combined uncertainty using Equation (7). The combined uncertainty for three RH observations for the two humidity sensors using calibration equations that use different calibration points are in Figures 7 and 8.

The Precision of the Two Types of RH sensors
The combined uncertainty is the criterion that is used to determine the precision of the sensors. The values for the combined uncertainty for the resistive sensor at a RH of 30%, 60% and 90% are 0.8618%, 0.8506% and 0.8647% for the calibration equation that uses 11 calibration points, and 1.1155%, 1.1040% and 1.1271% for the calibration equation that uses 5 calibration points. The calibration equation that uses 9 calibration points gives the smallest uc values. The combined uncertainty for 7, 9 and 11 calibration points is <1.0% RH.
The values for the combined uncertainty for a capacitive sensor at a RH of 30%, 60% and 90% are 0.7787%, 0.7690% and 0.7813% for the calibration equation that uses 11 calibration points and 0.8803%, 0.8717% and 0.8890% for the calibration equation that uses 5 calibration points. The  The Type A standard uncertainty that are calculated using the predicted values for the 4th order polynomial equation for the resistive sensor and the 2nd order polynomial equation for a capacitive sensor are added to give a combined uncertainty using Equation (7). The combined uncertainty for three RH observations for the two humidity sensors using calibration equations that use different calibration points are in Figures 7 and 8.

The Precision of the Two Types of RH sensors
The combined uncertainty is the criterion that is used to determine the precision of the sensors. The values for the combined uncertainty for the resistive sensor at a RH of 30%, 60% and 90% are 0.8618%, 0.8506% and 0.8647% for the calibration equation that uses 11 calibration points, and 1.1155%, 1.1040% and 1.1271% for the calibration equation that uses 5 calibration points. The calibration equation that uses 9 calibration points gives the smallest uc values. The combined uncertainty for 7, 9 and 11 calibration points is <1.0% RH.
The values for the combined uncertainty for a capacitive sensor at a RH of 30%, 60% and 90% are 0.7787%, 0.7690% and 0.7813% for the calibration equation that uses 11 calibration points and 0.8803%, 0.8717% and 0.8890% for the calibration equation that uses 5 calibration points. The

The Precision of the Two Types of RH Sensors
The combined uncertainty is the criterion that is used to determine the precision of the sensors. The values for the combined uncertainty for the resistive sensor at a RH of 30%, 60% and 90% are 0.8618%, 0.8506% and 0.8647% for the calibration equation that uses 11 calibration points, and 1.1155%, 1.1040% and 1.1271% for the calibration equation that uses 5 calibration points. The calibration equation that uses 9 calibration points gives the smallest u c values. The combined uncertainty for 7, 9 and 11 calibration points is <1.0% RH.
The values for the combined uncertainty for a capacitive sensor at a RH of 30%, 60% and 90% are 0.7787%, 0.7690% and 0.7813% for the calibration equation that uses 11 calibration points and 0.8803%, 0.8717% and 0.8890% for the calibration equation that uses 5 calibration points. The combined uncertainty for 5, 7, 9 and 11 calibration points is <0.9% RH. In terms of practical applications, this performance is sufficient for industrial applications [20,21].
The accuracy and precision are 0.80% and 0.90% RH for a resistance RH sensor that uses 7 calibration points and 0.70% and 0.90% RH for a capacitance RH sensors that uses 5 calibration points.

Discussion
The number of calibration points that are required for sensors represents a compromise between the ideal number of calibration points and the time and cost of the calibration. The criterion that Betta [1] used to determine the optimal number of points used the ratio of the standard deviation of the regression coefficients (s bj ) to the established standard error of regression (s).
Accuracy and precision are the most important criteria for sensors so this study uses both values. Using statistical theory, the best calibration equation is determined using the t-value for the highest-order parameter and the residual plots. The estimated standard errors for the regression equation are then used to determine the accuracy of the sensors. The combined uncertainty considered the uncertainty of reference materials, the uncertainty for the predicted values and other B type sources. The combined uncertainties for the calibration equations for different numbers of calibration points using different saturated salt solutions are the criteria that are used to evaluate the precision of sensors.
Two types of electric RH sensors were calibrated in this study. Some calibration works, such as those for temperature and pressure sensors, are calibrated by an equal spacing of calibration points. The RH reference environments are maintained using different saturated salt solutions.
It is seen that the optimum number of calibration points that is required to calibrate a resistive humidity sensors involves 7 saturated salt solutions (LiCl, MgCl 2 , K 2 CO 3 , NaBr, NaCl, KCI and K 2 SO 4 ), so seven points are specified. Five saturated salt solutions (LiCl, MgCl 2 , NaBr, NaCl and K 2 SO 4) are specified for a capacitive humidity sensor. Considering factors that influence the choice of salts, such as price, toxicity and rules for disposal, the choice of these salt solutions is suitable.
The calibration equations key to measurement performance. This study determines that te 4th order polynomial equation is the adequate equation for the resistive humidity sensor and the 2nd order polynomial equation is the optimum equation for the capacitive humidity sensor. The accuracy of the calibration equations is 0.8% RH for a resistive humidity sensor that uses 7 calibration points and 0.7% RH for a capacitance humidity sensor that uses 5 calibration points. The precision is less than 1.0% RH for the resistive sensor and less than 0.9% RH for the capacitive sensor.
The method that is used in this study applicable to other sensors.

Conclusions
In this study, two types of electric RH sensors were used to illustrate the method for the specification of the optimum number of calibration points. The standard RH environments are maintained using different saturated salt solutions. The theory of regression analysis is applied. The best calibration equation is determined in terms of the t-value of the highest-order parameter and the residual plots. The estimated standard errors for the regression equation are the criteria that are used to determine the accuracy of sensors. The combined uncertainty involves the uncertainty for the reference materials, the uncertainty in the predicted values and other B type sources. The combined uncertainties for the calibration equations for different number of calibration points using different saturated salt solutions are the criteria that are used to evaluate the precision of the sensors.
The calibration equations are key to good measurement performance. This study determines that the 4th order polynomial equation is the adequate equation for the resistive humidity sensor and the 2nd order polynomial equation is the best equation for the capacitive humidity sensor. The accuracy of the calibration equations is 0.8% RH for a resistive humidity sensor that uses 7 calibration points and 0.7% RH for a capacitance humidity sensor using 5 calibration points. The precision is less than 1.0% RH for the resistive sensor and less than 0.9% RH for the capacitive sensor.
The method to determine the number of the calibration points used in this study is applicable to other sensors.
Supplementary Materials: The following are available online at http://www.mdpi.com/1424-8220/19/5/1213/ s1. The residual plots for the calibration equations for different orders of polynomial equations for resistive humidity sensor using 5 saturated salt solutions (LiCl, MgCl 2 , NaBr, NaCl and K 2 SO 4 ). The residual plots for the calibration equations for different orders of polynomial equations for capacitance humidity sensor using 5 saturated salt solutions (LiCl, MgCl 2 , NaBr, NaCl and K 2 SO 4 ).
Author Contributions: H.-Y.C. drafted the proposal, executed the statistical analysis, interpreted the results and revised the manuscript. C.C. reviewed the proposal, performed some experiments, interpreted some results and criticized the manuscript and participated in its revision. All authors have read and approved the final manuscript.