#
Assessment of the Equivalence of Low-Cost Sensors with the Reference Method in Measuring PM_{10} Concentration Using Selected Correction Functions

^{1}

^{2}

^{*}

## Abstract

**:**

_{10}concentrations obtained using low-cost devices. Such devices apply the optical method to values comparable with those obtained using the reference gravimetric method. An additional goal is to show that the results corrected in this way can be used to carry out the procedure for testing equivalence of these methods. The study used generalized regression models (GRMs) to construct corrective functions. The constructed models were assessed using the coefficients of determination and the methodology of calculating the measurement uncertainty of the device. Measurement data from the two tested devices and the reference method were used to estimate model parameters. The measurement data were collected on a daily basis from 1 February to 30 June 2018 in Nowy Sącz. Regression allowed building multiple models with various functional forms and very promising statistical properties as well as good ability to describe the variability of reference measurements. These models also had very low values of measurement uncertainty. Of all the models constructed, a linear model using the original PM

_{10}concentrations from the tested devices, air humidity, and wind speed was chosen as the most accurate and simplest model. Apart from the coefficient of determination, expanded relative uncertainty served as the measure of quality of the obtained model. Its small value, much lower than 25%, indicates that after correcting the results it is possible to carry out the equivalence testing procedure for the low-cost devices and confirm the equivalence of the tested method with the reference method.

## 1. Introduction

_{10}, PM

_{2.5}, and PM

_{1}, (i.e., fractions containing particles with a diameter smaller than 10, 2.5, and 1 m). The fraction of the largest particles is currently studied most often and most fully, while the fraction of the smallest particles is still relatively rarely studied due to technical difficulties. However, it is suspected of having the most destructive impact on health, therefore the number of analyses of this group of dusts is increasing [1,2,3,4].

_{2.5}and PM

_{1}dust penetrate deep into the lungs and even into the circulatory system, spreading pollution throughout the body. Research also shows the possibility of transmitting diseases by large dust particles in the air. The presence of viruses, including coronaviruses, has been observed on particulate matter, and the possibility of infection in this way is not excluded [4,5,6,7,8,9,10].

_{1}

_{0}concentration measuring device to values comparable with the measurements derived from the reference method, to prove that the equivalence of these measurement methods can be tested. This function can be used by the manufacturer to change the device software or, if such a change is not possible (even if no other measurements are available), to correct the measurement values at their recipients.

_{10}electronic meters, but it is only appropriate for the device used in the study. In addition, measurements from electronic sensors are usually corrected by correction factors or simple linear models (using only measurements from electronic devices) [11,19,20]. This leads in many cases to a situation in which the measurement results are correct within a certain range of values of factors affecting the functioning of the device (temperature, concentration of pollutants). Outside this range of values, the measurement results often deviate significantly from the actual values. In this study, the authors want to demonstrate the possibility of using non-linear models and models using weather factors for correction. The correction functions obtained in this way should be more flexible, and the obtained measurement errors less susceptible to changes in the measurement conditions.

## 2. Materials and Methods

_{10}) concentrations from low-cost electronic measuring devices using optical sensors. These devices are a new product and are just being launched. Their manufacturer did not allow the device name to be disclosed. They use an optical method for measurements. The device draws a specific amount of air into the reactor, which is illuminated by means of a laser beam of a certain length. Sensors installed in the reactor count the number of light reflections and its parameters and on this basis the concentration of pollutants in the air is calculated. Information on this subject is processed by electronic systems located in the device with a frequency of one measurement per minute. The obtained results are then sent to the recipient via a mobile network. The device allows the measurement of concentrations of various pollutants in the air, including PM

_{10}, PM

_{2.5}, and PM

_{1}. This also gives the opportunity to take measurements of many weather parameters. The design of the devices ensures their high mobility and measurement of pollutant concentrations virtually anywhere. This device, like many others, has already been described many times, and the results of its measurements have been analyzed in many aspects [10,11,12,13,16,17,21].

_{10}concentrations (candidate method, CM). This period included both cold days in winter and early spring as well as very warm days in summer. This provided a cross-section of the various operating conditions of the devices. The measuring devices were placed in the immediate vicinity of the air pollution measuring station belonging to the Wojewódzki Inspektorat Ochrony Środowiska (Voivodship Inspectorate for Environmental Protection, WIOS), so that they could examine the composition of the same air. The use of two devices using the candidate method is intended not only to better assess equivalence with the reference method, but also to demonstrate the repeatability of measurements carried out by this method [15]. Electronic devices tested PM

_{10}concentrations every minute, then these data were aggregated into daily averages. Unreliable measurements were removed from the data series using the Grubbs test. In this way two series of PM

_{10}concentrations were obtained from each of the electronic devices used in the study (CM1 and CM2 expressed in μg/m

^{3}). These data were supplemented with daily PM

_{10}concentrations from the WIOS station, using the gravimetric method for measurements, as a reference method (RM) [22] and with meteorological data such as wind speed (WV, in m/s), relative humidity (humid, in %), and air temperature (temp, in °C). The data collected in this way were used to assess the equivalence of the test method with the reference method.

_{10}concentration values obtained with both tested devices and with the reference method. In the winter months, the concentrations obtained using all the methods are high with a large dispersion of results. In warm months, PM

_{10}concentrations clearly decrease. The dispersion of concentration values also decreases.

_{10}provided by both candidate devices and by the reference method, as well as the values of the characteristics for the other variables used in the study are presented in Table 2.

_{BS}value was 0.39 μg/m

^{3}, where ${u}_{BS}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left(CM{1}_{i}-CM{2}_{i}\right)}^{2}}{2n}}$. We can accept devices for which the values of this measure are less than 2.5 μg/m

^{3}[15].

_{10}concentration from the reference method as the dependent variable, PM

_{10}concentration from candidate method (separate modeling was carried out for each of the devices), and the factors that may significantly affect the behavior of the device (i.e., temperature, humidity, and wind speed as independent variables). Due to possible non-linear relationships between independent variables and the dependent variable, models with non-linearly transformed variables were also used. For this purpose, natural logarithm (ln X

_{i}), exponential transformation (e

^{Xi}), and second-degree polynomial function were used. In this way eight groups (clusters) of models were obtained which were subject to separate estimation procedures and selection of the best models:

- linear models,
- linear models with variable interactions,
- models based on second-degree polynomials,
- models based on second-degree polynomials with variable interactions,
- models using independent variables with their logarithms,
- models using independent variables along with their logarithms and variable interactions,
- models using independent variables with their exponential transformations,
- models using independent variables along with their exponential transformations and variable interactions.

_{0}+ b

_{1}X

_{0}) not significantly different from 0 and a slope value (b

_{1}) not significantly different from 1. Compliance with these requirements leads to the conclusion that the calibration function is an identity and the differences between the variables are in fact random. Both significances were tested using the t-value test, while in the second case the significance of the slope value was examined after subtracting one from it. The number of random errors was measured using adjusted ${\overline{R}}^{2}$ [15].

_{10}concentration measurement methods described in the Guide to the Demonstration [15]. These uncertainties are: uncertainty (inaccuracy) of the reference method—u

^{2}(x

_{i}); uncertainty caused by the regression model combining both methods ${S}_{e}^{2}$, caused by the imperfection of estimating the parameters of this model; and the uncertainty of the calibration function (1). LV is the maximum allowed concentration of PM

_{10}which is 50 μg/m

^{3}. Combined uncertainty can be written as:

^{2}

_{CR}is used to calculate the expanded relative uncertainty W

_{CM}:

_{10}concentrations. Values close to 0 mean that the candidate method gives a satisfactory approximation of the results obtained with the reference method. The limit of acceptance of the method is 25% for extended relative uncertainty [15,30].

- (1)
- selection of factors affecting PM
_{10}concentration measurement in the tested device (i.e., potential independent variables of the correction function); - (2)
- selecting all relevant functional forms (groups) of correction functions;
- (3)
- construction of all models in each group and selection of the best model in the group (i.e., the model that meets all statistical estimation assumptions and gives the highest value of the coefficient of determination);
- (4)
- evaluation of selected models using a calibration function (i.e., checking for possible systematic errors in the corrected measurements);
- (5)
- evaluation of selected correction functions employing extended relative uncertainty (assessment of the amount of random errors) and selection of the final form of the correction function.

_{10}concentration in the device of the data recipient.

## 3. Results

#### 3.1. Linear Model

_{i}is an independent variable, and parameters a

_{i}are estimates of structural parameters.

_{10}was affected by the concentration obtained from the tested device and the relative humidity of the air. An increase in the humidity value caused a decrease in the corrected PM

_{10}value. In the case of air with high humidity, the tested low-cost device tends to overestimate the value of the concentration of pollution. Both models differ primarily in the significance of the influence of wind speed. In the model for the CM1 device, the wind speed is statistically significant, while in the model constructed based on the concentrations from CM2 this variable is statistically insignificant. The difference of significance of the parameter may result from even very small differences in the data. In the case of a parameter on the limit of significance, small differences in the data may cause rejection or non-rejection of the hypothesis about the significance of this parameter in the t-test.

_{10}concentrations from the reference device. The remainder of the variation is explained by the model.

_{1}

_{0}concentration values from the test part of the data. After calculating the value of the correction functions for these data, their identity with the PM

_{10}concentration measurements for the reference method was verified. To this end, calibration functions were built (i.e., linear regression models between corrected concentrations from the tested devices and concentrations from the reference method). The expected result is a linear model for which the slope is statistically insignificantly different from 1, while the intercept is statistically insignificantly different from 0. The estimation of parameters results for calibration function FC1 and FC2 are presented in Table 4.

_{0}) estimates were significantly indifferent from 0 and for this reason they are not included in Table 4. The slope estimates (a

_{1}) are 0.979 for both models and were significantly indifferent from 1. This is confirmed by values t (a

_{1}–1) statistics which are close to 0. It can be concluded that both calibration functions are in fact identity transformations, and the concentration values obtained from the correction models do not contain systematic errors.

#### 3.2. Linear Models with Variable Interactions

_{i}is an independent variable, X

_{i}× X

_{j}is the interaction of the i-th variable with the j-th variable, parameters a

_{i}are estimates of structural parameters, and l is the number of all interactions l = k(k − 1)/2.

_{10}concentrations, temperature, humidity, and wind speed), as well as the interactions of all possible pairs of these variables. The interactions describe the effect of the simultaneous influence of factors on a dependent variable. Table 5 presents the results of the estimation of the correction model.

_{10}concentration, and humidity, as well as wind speed and humidity. A positive estimate of the structural parameter for wind speed indicates that with stronger winds the PM

_{10}concentrations obtained by the CM1 are underestimated. In the F4 model, the CM2 concentration, temperature, interaction of PM

_{1}

_{0}concentration, and humidity as well as temperature and humidity are statistically significant for the CM2. A positive estimate of the structural parameter for temperature indicates that the increase in temperature causes a greater underestimation of the concentration values in the CM2 device. Negative and small values of structural parameters for all interactions indicate that these factors reduce PM

_{10}concentration. This impact is relatively small but statistically significant.

_{10}concentration values. Both models can be used to correct data from the tested devices.

#### 3.3. Models Using Independent Variables with Their Logarithms

_{i}is an independent variable, ln X

_{i}is the natural logarithm of the variable, and parameters a

_{i}are estimates of structural parameters.

_{10}concentration from the tested devices and the relative humidity of the air. Negative signs of the structural coefficients of models for humidity (−28.055 and −31.352) indicate, just as in the case of F1 and F2 linear models, overestimation of PM

_{10}concentrations in the tested devices caused by high humidity. In addition, the influence of logarithm of wind speed in the F5 model was significant. A positive sign of the structural coefficient shows that the increase in wind speed affects the reduction of PM

_{10}concentrations from the tested device in relation to the RM concentrations.

_{1}–1) statistics), while intercept values are insignificantly different from 0. It can be stated that PM

_{10}concentration values obtained as a result of the operation of the correction functions F5 and F6 do not contain systematic errors compared to the values derived from RM, and the differences between them are random.

#### 3.4. Models Using Independent Variables with Their Exponential Transformations

^{x}form has the following general form:

_{i}is an independent variable, e

^{Xi}is the value of the exponential transformation (exponent) of the variable, and parameters a

_{i}are estimates of structural parameters.

_{10}concentration values from the tested devices, air temperature, and wind speed with positive marks of structural coefficient estimates as well as air humidity and exponent from wind speed with negative signs. This means that in this model the increase in temperature and wind speed skews down the results obtained in relation to reference measurements, while the increase in humidity increases it slightly. It should be noted that both constructed models have a very similar structure.

_{10}concentrations from the tested devices gives a very good fit to the data derived from RM. The values of the determination coefficients indicate the greatest ability of these models to correct concentration measurements.

#### 3.5. Other Models

_{10}measurements is influenced by the original measurements coming from the tested device, air humidity, and wind speed. Estimates of structural parameters and the value of the coefficient of determination for this model are identical to those for the linear model.

_{10}concentration values are not related to the squares of any of the variables.

_{10}concentration was affected only by the original PM

_{10}concentration values and natural logarithm from relative humidity. Estimates of structural parameters lead to a model identical to the model built solely on variable logarithms. Therefore, it should be recognized that the interaction of variable logarithms does not have a significant impact on the correction function.

#### 3.6. Measurement Uncertainty of Built-Up Correction Functions

_{10}by the tested method. Low values of both uncertainty measures are considered better. In the case of expanded relative uncertainty, the method acceptance limit, the method equivalence limit, is 25%. To calculate both measurement uncertainties, the entire available data set (i.e., teaching part of data and verification part of data), were used. The results are presented in Table 11.

_{10}concentrations obtained from the tested devices. These models, however, differ in terms of the degree of complication (the remaining parameters of model evaluation are at a similar level). Therefore, the authors would be inclined to indicate the linear model F1 and F2 as the best for correcting data from the tested device.

## 4. Discussion

_{10}particulate matter to values comparable with measurements obtained using the reference gravimetric method. A further aim was to demonstrate the equivalence of both methods. For this purpose, groups of functions (functional forms) were selected that could be used to construct the appropriate corrective function. For each of them, all models were built that can be constructed for a selected set of variables. From among them, the best model was selected in each group (i.e., one that gave the largest value of the adjusted coefficient of determination ${\overline{R}}^{2}$ when all stochastic assumptions were met for linear econometric models).

_{10}concentrations performed by the tested devices were influenced by humidity and wind force. The air temperature had a very limited influence, although it seemed that it was the factor that should have the strongest effect. In the case of wind force, structural parameters were positive in all models. This means that strong wind causes a decrease in the observed concentration of PM

_{10}and the model must later correct this value upwards. In the case of humidity, the reverse is true. Negative values of structural parameters indicate that the increase in humidity level contributes to the apparent increase in PM

_{10}concentrations detected by the device.

_{10}concentration values produced by the reference method (determination coefficients) and in terms of the absence of systematic errors, it can be concluded that all models have similar properties in this respect. The values of the adjusted coefficients of determination calculated for both the data training set and the verification set are similar and very high. The values of adjusted ${\overline{R}}^{2}$ exceed 0.95. This means that the models are very well adapted to the empirical data (i.e., PM

_{10}concentrations from the reference method). All models also passed the systematic failure assessment. Calibration models for all selected correction functions have satisfactory properties, in that they do not differ statistically from the identity function. It can be considered that in terms of design all correction functions meet the assumptions and have similar ability to correctly adjust raw results.

## 5. Conclusions

_{10}concentrations in air. The purpose of the work is to demonstrate the possibility of building a correction function for the measurements from the tested device and to prove that the corrected data will provide the opportunity to perform testing for equivalence with the reference method. Several models meeting the requirements were constructed in the study. The best of them was a linear model (8) and (9), using PM

_{10}concentration values from the tested devices, wind speed and humidity. It approximated the reference method concentration values almost perfectly. The analysis using expanded relative uncertainty has shown that there is a good chance that after applying the correction it will be possible to demonstrate equivalence with the reference method. This will allow measurements from this device to be treated as equivalent to reference measurements. The obtained model can be used by the device manufacturer to improve the device’s functioning.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Average daily concentrations of PM

_{10}from the reference method (RM) and candidate method devices (CM1 and CM2) on 1 February–30 July 2018 in Nowy Sącz.

**Table 1.**Pearson’s linear correlation coefficients for PM

_{10}concentrations obtained by reference method (RM) and candidate methods (CM1 and CM2).

Variable | CM1 | CM2 |
---|---|---|

RM | 0.968 | 0.965 |

Variable | Mean | Median | Minimum | Maximum | St. Deviation |
---|---|---|---|---|---|

RM | 40.157 | 28.9 | 10.8 | 148.1 | 26.927 |

CM1 | 49.874 | 26.9 | 3.8 | 273.4 | 50.860 |

CM2 | 50.872 | 25.5 | 4.0 | 259.2 | 53.709 |

Temp | 11.697 | 15.3 | −12.6 | 26.0 | 9.771 |

Wind speed (WV) | 0.913 | 0.8 | 0.3 | 2.7 | 0.355 |

Humid | 79.531 | 78.4 | 51.5 | 99.9 | 11.136 |

**Table 3.**Estimates of structural parameters for the best models of linear correction function F1 and F2 of the candidate methods PM

_{10}concentration, the test statistics F value, and the adjusted ${\overline{R}}^{2}$.

Name | Device CM | Intercept | CM | WV | Humid | F value | Adjusted ${\overline{\mathit{R}}}^{2}$ |
---|---|---|---|---|---|---|---|

F1 | CM1 | 39.413 | 0.521 | 3.967 | −0.368 | 806.46 | 0.964 |

F2 | CM2 | 46.792 | 0.485 | −0.400 | 1012.31 | 0.958 |

**Table 4.**Verification of the linear correction function: parameter estimates, estimation errors, adjusted ${\overline{R}}^{2}$, and the value of t-test statistics for calibration functions F1 and F2.

Name | Device CM | Slope a_{1} | Estimation Error s(a_{1}) | Adjusted ${\overline{\mathit{R}}}^{2}$ | t (a_{1}–1) |
---|---|---|---|---|---|

FC1 | CM1 | 0.979 | 0.031 | 0.957 | −0.668 |

FC2 | CM2 | 0.979 | 0.032 | 0.957 | −0.669 |

**Table 5.**Estimates of structural parameters for the best linear models with interactions of the correction function F3 and F4, the value of the F-test statistics, and adjusted coefficient of determination ${\overline{R}}^{2}$.

Name | Device | Intercept | CM | Temp | WV | CM × Humid | Temp × Humid | WV × Humid | F Value | Adjusted ${\overline{\mathit{R}}}^{2}$ |
---|---|---|---|---|---|---|---|---|---|---|

F3 | CM1 | 9.087 | 0.880 | 23.659 | −0.004 | −0.243 | 620.787 | 0.965 | ||

F4 | CM2 | 13.540 | 0.859 | 1.367 | −0.004 | −0.017 | 498.528 | 0.957 |

**Table 6.**Verification of the linear correction function with interactions: parameter estimates, estimation errors, adjusted coefficient of determination values ${\overline{R}}^{2}$, and the value of t-test statistics for calibration functions FC3 and FC4.

Name | Device | Slope a_{1} | Estimation Error s(a_{1}) | Adjusted ${\overline{\mathit{R}}}^{2}$ | t (a_{1}–1) |
---|---|---|---|---|---|

FC3 | CM1 | 0.980 | 0.030 | 0.960 | −0.644 |

FC4 | CM2 | 0.979 | 0.031 | 0.958 | −0.663 |

**Table 7.**Estimate parameters for the best models of the correction function F5 and F6 using variable logarithms, the value of F-test statistics, and the adjusted coefficient of determination ${\overline{R}}^{2}$.

Name | Device | Intercept | CM | ln (WV) | ln (Humid) | F Value | Adjusted ${\overline{\mathit{R}}}^{2}$ |
---|---|---|---|---|---|---|---|

F5 | CM1 | 136.661 | 0.528 | 4.975 | −28.055 | 823.406 | 0.965 |

F6 | CM2 | 151.775 | 0.487 | −31.352 | 1023.823 | 0.958 |

**Table 8.**Verification of the linear correction function with variable logarithms: parameter estimates, estimation errors, adjusted coefficient of determination values ${\overline{R}}^{2}$, and the value of t-test statistics for calibration functions FC5 and FC6.

Name | Device | Slope a_{1} | Estimation Error s(a_{1}) | Adjusted ${\overline{\mathit{R}}}^{2}$ | t (a_{1}–1) |
---|---|---|---|---|---|

FC5 | CM1 | 0.979 | 0.032 | 0.957 | −0.670 |

FC6 | CM2 | 0.979 | 0.032 | 0.957 | −0.675 |

**Table 9.**Estimate parameters for the best models of the correction function F7 and F8 using the exponential transformations of variables, the value of F-test statistics, and the adjusted coefficient of determination ${\overline{R}}^{2}$.

Name | Device | CM | Temp | WV | Humid | Exp (WV) | F Value | Adjusted ${\overline{\mathit{R}}}^{2}$ |
---|---|---|---|---|---|---|---|---|

F7 | CM1 | 0.579 | 0.262 | 22.715 | −0.080 | −2.526 | 999.366 | 0.982 |

F8 | CM2 | 0.565 | 0.414 | 21.589 | −0.093 | −2.392 | 908.694 | 0.981 |

**Table 10.**Verification of the correction function with exponential transformations: parameter estimates, estimation errors, adjusted coefficient of determination values ${\overline{R}}^{2}$, and the value of t-test statistics for calibration functions FC7 and FC8.

Name | Device | Slope a_{1} | Estimation Error s(a_{1}) | Adjusted ${\overline{\mathit{R}}}^{2}$ | t (a_{1}–1) |
---|---|---|---|---|---|

FC7 | CM1 | 0.969 | 0.038 | 0.937 | −0.818 |

FC8 | CM2 | 0.967 | 0.039 | 0.933 | −0.841 |

**Table 11.**Combined uncertainty and extended relative uncertainty for all best models of correction function.

Measure | Symbol | Model | |||||||
---|---|---|---|---|---|---|---|---|---|

F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | ||

Combined uncertainty | u^{2}_{CR} | 11.137 | 11.243 | 11.771 | 14.087 | 10.856 | 11.318 | 18.490 | 23.564 |

Extended relative uncertainty | W_{CR} | 13.3% | 13.4% | 13.7% | 15.0% | 13.2% | 13.5% | 17.2% | 19.4% |

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**MDPI and ACS Style**

Owczarek, T.; Rogulski, M.; Czechowski, P.O.
Assessment of the Equivalence of Low-Cost Sensors with the Reference Method in Measuring PM_{10} Concentration Using Selected Correction Functions. *Sustainability* **2020**, *12*, 5368.
https://doi.org/10.3390/su12135368

**AMA Style**

Owczarek T, Rogulski M, Czechowski PO.
Assessment of the Equivalence of Low-Cost Sensors with the Reference Method in Measuring PM_{10} Concentration Using Selected Correction Functions. *Sustainability*. 2020; 12(13):5368.
https://doi.org/10.3390/su12135368

**Chicago/Turabian Style**

Owczarek, Tomasz, Mariusz Rogulski, and Piotr O. Czechowski.
2020. "Assessment of the Equivalence of Low-Cost Sensors with the Reference Method in Measuring PM_{10} Concentration Using Selected Correction Functions" *Sustainability* 12, no. 13: 5368.
https://doi.org/10.3390/su12135368