# Enhancing Indoor Air Quality Estimation: A Spatially Aware Interpolation Scheme

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}concentration, and particulate matter (PM), is critical to optimizing energy use in these environments as well as maintaining occupant health and comfort [14]. The application of spatial interpolation techniques has important implications for IAQ management because this can inform the optimization of ventilation and air conditioning systems, reduce energy consumption, and maintain a healthy indoor environment for occupants. Therefore, identifying important IAQ parameters in indoor spaces and developing accurate measurement techniques and data analysis methods have become active research areas.

_{2}concentrations using a limited number of sensors in a three-story smart building in Japan. Zhou [19] proposed a cross-sample learning algorithm to obtain a spatial graph model of sensors based on the horizontal and vertical effects of gravity on humidity and used it to learn the coefficient elements of labeled locations to predict the state of unlabeled locations. Machine-learning-based methods are relatively data-intensive and have a high calculation cost [20]. Choi [14] developed an accurate IAQ distribution map for large spaces using spatial interpolation methods. In their study, 18 sensors were installed in a library’s reading room, with 14 for data collection. Their study identified the optimal spatial interpolation method for each IAQ factor, determined the ideal number and layout of sensors, and confirmed the map’s effectiveness. In Huang [21], a study was conducted to select the optimal sensor installation location under the constraints of an indoor space. Their study compared two sampling methods in indoor air distribution measurement: the gridded method and the slope-based method. The data collected through each method were interpolated using the usual kriging method. As a result, the slope-based sampling method had a smaller interpolation error than the gridded method, and the authors recommended the slope-based sampling method for indoor air distribution measurement.

## 2. Related Works

## 3. Basic Concepts

_{1}, …, p

_{n}, we define the group distance GD(g) of group g as follows in Equation (1).

## 4. Indoor Spatial Interpolation Scheme

#### 4.1. Group Clustering

_{1}, …, p

_{n}are given and that each data point p

_{i}has m data values, y

_{i}

^{1}, … y

_{i}

^{m}for i = 1, …, n. The MSD for two data points p

_{i}and p

_{j}is defined as follows in Equation (3).

#### 4.2. Group Assignment

Algorithm 1. Assign a group to an unmeasured point |

procedure Group Assignment (q: unmeasured point)Let p _{1}, …, p_{n} be all the data points in an indoor space$p=\underset{{p}_{i}}{\mathit{argmin}}d({p}_{i},q),i=1,\dots ,n$ G(q) = G(p) end procedure |

#### 4.3. Group-Preferred K-Nearest Neighbor (GPKNN)

Algorithm 2. Find group-preferred K nearest neighbors |

procedure GPKNN (q: query point, K: integer)Let DPSet = {p _{1}, …, p_{n}} be a set of all the data points in an indoor spaceTSet = DPSet KSet = {} while size(KSet) != K and TSet != {}$\hspace{1em}\hspace{1em}p=\underset{{p}_{i}}{\mathit{argmin}}VD({p}_{i},q),{p}_{i}\in TSet\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}KSet=KSet\cup \{p\}\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}TSet=TSet-\{p\}$ end whilereturn KSet end procedure |

#### 4.4. Spatial Interpolation

#### 4.4.1. Spatial Structure IDW (SSI) Method

_{i}is the data point that is the i-th nearest data point to q, y(p

_{i}) is the measured value at p

_{i}, ω

_{i}is the weight value assigned to y(p

_{i}), and ŷ(q) is the estimated value at q. This method selects neighboring data points close to the query point and gives greater weight to the measured values of the points closer to the query point. Let λ(p, q) be the inverse of d(p, q) as in Equation (5) [17].

Algorithm 3. Spatial Structure IDW (SSI) Method |

procedure SSI(q: query point, K: integer)Let DPSet = {p _{1}, …, p_{n}} be a set of all the data points in an indoor spaceLet y(p _{i}) be the data value of p_{i} for i = 1, …, n.{q _{1}, …, q_{K}} = GPKNN(q, K) where ${q}_{i}\in DPSet,i=1,\dots ,K$ $\hspace{1em}\mu \left({q}_{i},q\right)=\frac{1}{VD({q}_{i},q)}$ $\hspace{1em}{w}_{i}=\frac{\mu \left(q,{q}_{i}\right)}{{\sum}_{j=1}^{K}\mu \left(q,{q}_{j}\right)},i=1,\dots ,K,$ $\hspace{1em}\widehat{y}\left(q\right)={\displaystyle \sum _{i=1}^{K}}{w}_{i}y\left({q}_{i}\right),$ return ŷ(q) end procedure |

#### 4.4.2. Spatial Structure Kriging (SSK) Method

_{i}) is the expected value of y(p

_{i}) and where ω

_{i}is the kriging weight that is determined in a way that minimizes the variance of the error, ŷ(p

_{0}) − ŷ(p

_{0}). y(p) is a random field over a point p consisting of a trend m(p) and residual R(p), with the residual as a random field with a zero mean. The covariance of the residuals that is used to determine the weights of the method is assumed to be isotropic, which means that the covariance between two points depends only on their distance as in Equation (10) [27].

_{R}(h) is the isotropic covariance that depends only on h. Various models, such as the spherical model, exponential model, and wave model, can be used for calculating the isotropic covariance C

_{R}(h). There are three main kriging variants, (i) simple, (ii) ordinary, and (iii) kriging with a trend, which depend on the treatment of the trend component m(p).

Algorithm 4. Spatial Structure Kriging (SSK) Method |

procedure SSK (q: query point, K: integer)Let DPSet = {p _{1}, …, p_{n}} be a set of all the data points in an indoor spaceLet y(p _{i}) be the data value of p_{i} for i = 1, …, n.{q _{1}, …, q_{K}} = GPKNN(q, K) where ${q}_{i}\in DPSet,i=1,\dots ,K$$\hspace{1em}\widehat{y}\left(q\right)=kriging\left(q,\left\{{q}_{1},\dots ,{q}_{K}\right\}\right)withcov\left(R\left(p\right),R\left(q\right)\right)={C}_{R}\left(VD\right(p,q\left)\right)$ return ŷ(q) end procedure |

## 5. Experimental Results and Discussion

^{2}as shown in Equation (12), Equation (13), Equation (14), and Equation (15), respectively [49].

_{i}is the actual value of the i-th point, $\overline{{y}_{i}}$ is the mean of the true values, and ŷ

_{i}is the estimated value of the i-th point. A spherical covariance model is used in the kriging and SSK methods. In this paper, to evaluate the performance of the dataset, each data point in the dataset is considered to be an unmeasured point. The estimation of the unmeasured point is calculated using other data points within the dataset, and the error value of the unmeasured point is computed as the difference between the estimated value and the actual value. By utilizing the obtained error values, the final RMSE (root mean square error) is calculated.

#### 5.1. Experimental Results on an Office Dataset

_{2}and temperature sensors used in the office space.

_{2}concentration and temperature data every minute over a 5-day period from 29 June 2020 to 3 July 2020. For each of the 14 data points, we collected an average of 530 data points per day, totaling 37,086 data points for the CO

_{2}concentration and temperature data, respectively. Using the June 29 data, we varied N from 2 to 7 and K between 3, 6, 9, 12, and 14 to find the N and K values with the minimum RMSE. Using the found N and K values, we performed interpolation experiments on the data from 30 June to 3 July to verify the performance.

#### 5.1.1. Experimental Results for CO_{2} Data

_{2}data varies from two to seven. The RBF method is excluded from Figure 5 and the subsequent figures due to its large RMSE value compared to the other methods.

#### 5.1.2. Experimental Results for Temperature Data

_{2}values for IAQ01, IAQ02, and IAQ03 measured on 29 June.

_{2}is highly correlated within an independent room separated by walls. Figure 13 shows the result of dividing the sensors into three groups based on the CO

_{2}data using the group allocation and group assignment algorithms proposed in this paper. As shown in Figure 11, IAQ02 belongs to Group 1, the same group as IAQ01, while IAQ03 belongs to Group 3, a different group from IAQ01 and IAQ02.

_{2}, temperature, relative humidity, and light intensity, have distinct physics, and IAQ parameters are influenced not only by the layout of indoor spaces but also by these underlying physical properties. We assume that even if the physics of the IAQ parameters are different, the physics also would be reflected in the collected data. Therefore, we believe that the sensor grouping algorithm proposed in this paper partially reflects the spatial constraints on IAQ parameters.

#### 5.2. Experimental Results Based on the Intel Lab Dataset

#### 5.2.1. Experimental Results for Temperature Data

#### 5.2.2. Experimental Results for Humidity Data

#### 5.2.3. Experimental Results for Light Data

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 8.**Color-coded representation of groups for temperature, with the number of groups being 2 to 7.

**Figure 13.**Color-coded representation of groups for CO

_{2}with a group number of 3. Group 1 includes IAQ01, IAQ02, IAQ06, IAQ07, and IAQ09; Group 2 includes IAQ04, IAQ05, IAQ11, IAQ13, and IAQ14; and Group 3 includes IAQ03, IAQ10, IAQ08, and IAQ12.

**Figure 15.**Color-coded representation of groups for temperature with a group number of 3. Group 1 includes IAQ01, IAQ02, IAQ03, IAQ04, IAQ11, IAQ12, IAQ13, and IAQ14; Group 2 includes IAQ05 and IAQ06; and Group 3 includes IAQ07, IAQ08, IAQ09, and IAQ10.

**Figure 16.**Arrangement of sensors in Intel Lab. The numbers 1 through 54 indicate where each sensor is installed.

Sensor | CO_{2} | Temperature |
---|---|---|

Model | E + E | Sensirion |

Range | 0~2000 ppm | −4~125 °C |

Accuracy | <±50 ppm + 2% | ±0.3 °C ± 2% |

Interface | I2C | I2C |

Country of manufacture | Austria | Switzerland |

Data Point | X Location (cm) | Y Location (cm) |
---|---|---|

IAQ01 | 100 | 243 |

IAQ02 | 126 | 354 |

IAQ03 | 187 | 335 |

IAQ04 | 265 | 249 |

IAQ05 | 392 | 335 |

IAQ06 | 511 | 283 |

IAQ07 | 637 | 384 |

IAQ08 | 387 | 178 |

IAQ09 | 507 | 111 |

IAQ10 | 603 | 176 |

IAQ11 | 325 | 8 |

IAQ12 | 386 | 15 |

IAQ13 | 591 | 19 |

IAQ14 | 62 | 354 |

N | K | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|---|

2 | 3 | 43.26 | 41.42 | 59.62 | 241.94 | 31.04 | 29.96 |

6 | 39.80 | 38.50 | 41.66 | 177.38 | 29.71 | 30.47 | |

9 | 38.93 | 39.91 | 41.86 | 159.08 | 30.07 | 32.06 | |

12 | 38.11 | 38.36 | 47.18 | 156.94 | 30.69 | 32.83 | |

14 | 38.00 | 39.10 | 47.18 | 155.20 | 31.05 | 34.74 | |

3 | 3 | 43.26 | 41.42 | 59.62 | 241.94 | 25.79 | 26.54 |

6 | 39.80 | 38.50 | 41.66 | 177.38 | 26.37 | 27.22 | |

9 | 38.93 | 39.91 | 41.86 | 159.08 | 26.97 | 29.65 | |

12 | 38.11 | 38.36 | 47.18 | 156.94 | 28.56 | 33.79 | |

14 | 38.00 | 39.10 | 47.18 | 155.20 | 28.87 | 38.17 | |

4 | 3 | 43.26 | 41.42 | 59.62 | 241.94 | 26.03 | 24.00 |

6 | 39.80 | 38.50 | 41.66 | 177.38 | 26.54 | 26.12 | |

9 | 38.93 | 39.91 | 41.86 | 159.08 | 26.80 | 33.52 | |

12 | 38.11 | 38.36 | 47.18 | 156.94 | 28.30 | 40.42 | |

14 | 38.00 | 39.10 | 47.18 | 155.20 | 28.60 | 43.37 | |

5 | 3 | 43.26 | 41.42 | 59.62 | 241.94 | 26.59 | 24.36 |

6 | 39.80 | 38.50 | 41.66 | 177.38 | 28.32 | 28.94 | |

9 | 38.93 | 39.91 | 41.86 | 159.08 | 29.12 | 34.86 | |

12 | 38.11 | 38.36 | 47.18 | 156.94 | 29.56 | 42.54 | |

14 | 38.00 | 39.10 | 47.18 | 155.20 | 29.93 | 46.72 | |

6 | 3 | 43.26 | 41.42 | 59.62 | 241.94 | 22.82 | 22.51 |

6 | 39.80 | 38.50 | 41.66 | 177.38 | 23.97 | 20.90 | |

9 | 38.93 | 39.91 | 41.86 | 159.08 | 25.85 | 21.13 | |

12 | 38.11 | 38.36 | 47.18 | 156.94 | 27.09 | 21.11 | |

14 | 38.00 | 39.10 | 47.18 | 155.20 | 27.91 | 21.43 | |

7 | 3 | 43.26 | 41.42 | 59.62 | 241.94 | 25.79 | 25.54 |

6 | 39.80 | 38.50 | 41.66 | 177.38 | 25.44 | 23.41 | |

9 | 38.93 | 39.91 | 41.86 | 159.08 | 26.86 | 22.86 | |

12 | 38.11 | 38.36 | 47.18 | 156.94 | 27.71 | 22.52 | |

14 | 38.00 | 39.10 | 47.18 | 155.20 | 28.55 | 22.96 |

**Table 4.**Performance metrics for each method implemented with optimal N and K for CO

_{2}data collected from 30 June to 3 July.

Method | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|

RMSE | 45.44 | 46.04 | 43.98 | 175.42 | 28.84 | 26.66 |

MAE | 38.84 | 37.96 | 38.78 | 166.43 | 23.35 | 21.71 |

MAPE | 10.21 | 9.94 | 10.97 | 39.12 | 10.13 | 8.00 |

R2 | 0.40 | 0.42 | 0.34 | 0.07 | 0.51 | 0.57 |

N | K | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|---|

2 | 3 | 0.98 | 0.99 | 0.96 | 12.86 | 0.88 | 0.94 |

6 | 0.99 | 1.02 | 0.95 | 10.21 | 0.83 | 0.88 | |

9 | 0.99 | 1.03 | 0.96 | 8.15 | 0.83 | 0.99 | |

12 | 1.02 | 1.03 | 0.95 | 7.90 | 0.85 | 0.95 | |

14 | 1.05 | 1.02 | 0.95 | 8.09 | 0.85 | 1.00 | |

3 | 3 | 0.98 | 0.99 | 0.96 | 12.86 | 0.78 | 0.86 |

6 | 0.99 | 1.02 | 0.95 | 10.21 | 0.75 | 0.81 | |

9 | 0.99 | 1.03 | 0.96 | 8.15 | 0.77 | 0.83 | |

12 | 1.02 | 1.03 | 0.95 | 7.90 | 0.84 | 0.81 | |

14 | 1.05 | 1.02 | 0.95 | 8.09 | 0.87 | 0.83 | |

4 | 3 | 0.98 | 0.99 | 0.96 | 12.86 | 0.81 | 0.88 |

6 | 0.99 | 1.02 | 0.95 | 10.21 | 0.77 | 0.81 | |

9 | 0.99 | 1.03 | 0.96 | 8.15 | 0.78 | 0.84 | |

12 | 1.02 | 1.03 | 0.95 | 7.90 | 0.85 | 0.82 | |

14 | 1.05 | 1.02 | 0.95 | 8.09 | 0.89 | 0.83 | |

5 | 3 | 0.98 | 0.99 | 0.96 | 12.86 | 0.91 | 0.99 |

6 | 0.99 | 1.02 | 0.95 | 10.21 | 0.83 | 0.90 | |

9 | 0.99 | 1.03 | 0.96 | 8.15 | 0.85 | 0.97 | |

12 | 1.02 | 1.03 | 0.95 | 7.90 | 0.94 | 0.90 | |

14 | 1.05 | 1.02 | 0.95 | 8.09 | 0.98 | 0.92 | |

6 | 3 | 0.98 | 0.99 | 0.96 | 12.86 | 0.69 | 0.73 |

6 | 0.99 | 1.02 | 0.95 | 10.21 | 0.66 | 0.72 | |

9 | 0.99 | 1.03 | 0.96 | 8.15 | 0.72 | 0.72 | |

12 | 1.02 | 1.03 | 0.95 | 7.90 | 0.81 | 0.71 | |

14 | 1.05 | 1.02 | 0.95 | 8.09 | 0.86 | 0.72 | |

7 | 3 | 0.98 | 0.99 | 0.96 | 12.86 | 0.71 | 0.74 |

6 | 0.99 | 1.02 | 0.95 | 10.21 | 0.71 | 0.71 | |

9 | 0.99 | 1.03 | 0.96 | 8.15 | 0.76 | 0.76 | |

12 | 1.02 | 1.03 | 0.95 | 7.90 | 0.88 | 0.76 | |

14 | 1.05 | 1.02 | 0.95 | 8.09 | 0.94 | 0.78 |

**Table 6.**Performance metrics for each method implemented with optimal N and K for temperature data collected from 30 June to 3 July.

Method | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|

RMSE | 1.07 | 1.09 | 1.06 | 7.64 | 0.81 | 0.88 |

MAE | 0.91 | 0.90 | 0.93 | 7.25 | 0.66 | 0.72 |

MAPE | 4.66 | 4.78 | 4.13 | 35.25 | 4.02 | 4.37 |

R2 | 0.36 | 0.35 | 0.36 | 0.02 | 0.43 | 0.41 |

**Table 7.**Performance metrics values for each method implemented with optimal N and K for Intel Lab temperature data collected from 29 February to 3 March.

Method | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|

RMSE | 2.38 | 2.45 | 2.44 | 5.90 | 1.86 | 2.90 |

MAE | 1.92 | 2.13 | 2.11 | 4.66 | 1.78 | 2.43 |

MAPE | 10.19 | 12.20 | 10.77 | 35.09 | 9.01 | 14.93 |

R2 | 0.72 | 0.67 | 0.74 | 0.19 | 0.77 | 0.73 |

**Table 8.**Performance metrics for each method implemented with optimal N and K for Intel Lab humidity data collected from 29 February to 3 March.

Method | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|

RMSE | 1.85 | 1.85 | 1.77 | 7.57 | 1.55 | 1.60 |

MAE | 1.52 | 1.39 | 1.43 | 5.57 | 1.23 | 1.26 |

MAPE | 6.21 | 6.13 | 4.72 | 12.83 | 4.19 | 4.25 |

R2 | 0.86 | 0.88 | 0.89 | 0.32 | 0.93 | 0.93 |

**Table 9.**Performance metrics for each method implemented with optimal N and K for Intel Lab light data collected from 29 February to 3 March.

Method | IDW | Kriging | Natural Neighbor | RBF | SSI | SSK |
---|---|---|---|---|---|---|

RMSE | 170.22 | 161.47 | 139.71 | 175.88 | 84.46 | 90.47 |

MAE | 137.10 | 122.95 | 112.15 | 132.32 | 64.72 | 68.41 |

MAPE | 17.08 | 15.75 | 14.65 | 18.24 | 8.64 | 9.35 |

R2 | 0.49 | 0.50 | 0.69 | 0.44 | 0.75 | 0.72 |

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## Share and Cite

**MDPI and ACS Style**

Jung, S.; Han, S.; Choi, H.
Enhancing Indoor Air Quality Estimation: A Spatially Aware Interpolation Scheme. *ISPRS Int. J. Geo-Inf.* **2023**, *12*, 347.
https://doi.org/10.3390/ijgi12080347

**AMA Style**

Jung S, Han S, Choi H.
Enhancing Indoor Air Quality Estimation: A Spatially Aware Interpolation Scheme. *ISPRS International Journal of Geo-Information*. 2023; 12(8):347.
https://doi.org/10.3390/ijgi12080347

**Chicago/Turabian Style**

Jung, Seungwoog, Seungwan Han, and Hoon Choi.
2023. "Enhancing Indoor Air Quality Estimation: A Spatially Aware Interpolation Scheme" *ISPRS International Journal of Geo-Information* 12, no. 8: 347.
https://doi.org/10.3390/ijgi12080347