# A Two-Step Method for Missing Spatio-Temporal Data Reconstruction

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}covariance and the correlation coefficient need to be calculated for each missing data point. In addition, when data are missing continuously, interpolation accuracy is low, and it may even be impossible to obtain a final interpolation result.

## 2. Materials and Methods

#### 2.1. Problem Definitions

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Method Framework

- First, because the ST-2SMR method uses a neural network model to combine the interpolation results in spatial and temporal dimensions, the space-time missing dataset was divided, with 80% selected for parameter training and 20% used as a test dataset to evaluate model performance, to avoid overfitting and to improve generalization ability.
- Second, to avoid the influence of continuous missing data on fine-grained interpolation, coarse-grained interpolation for missing datasets was applied.
- Based on the results of coarse-grained interpolation, spatial and temporal heterogeneity was used to perform fine-grained interpolation. During this process, it was necessary to calculate the correlation coefficient and covariance between time and space sequences to fit the parameters. If whole time series or spatial sequences are involved in interpolation, redundant sample data increase computational complexity; therefore, to improve the accuracy and speed of interpolation, a suitable sliding window was introduced to ensure the strongest correlation between sample data and missing data. Next, a heterogeneous covariance function was constructed for the space-time dimension. The best unbiased estimate of missing data can then be obtained by maximizing the objective function.
- After interpolation of temporal and spatial dimensions, estimated values without missing data were chosen as training samples for the neural network, which is used to mine nonlinear relationships in spatio-temporal data. Estimated values of missing data were then obtained by inputting the spatio-temporal interpolation results of missing data into the trained neural network model.
- Finally, the performance of the model was evaluated using the test dataset.

## 3. Detailed Design of ST-2SMR

#### 3.1. Coarse-Grained Interpolation

Algorithm 1: Coarse-Grained interpolation. |

Input: Original Missing Matrix ${V}_{m\times n}$ |

Temporal Threshold $wc$ |

Parameter of IDW $\alpha $ |

Parameter of SES $\beta $ |

Output: Coarse-Grained Imputing Matrix $C\_{M}_{m\times n}$ |

1 $C\_{M}_{m\times n}\leftarrow Initialization\left({V}_{m\times n}\right)$ |

2 For $i$ = 1 to $m$ |

3 $j$ = 1 Forto $n$ |

4 ${V}_{ij}$ is missing value Ifthen |

5 ${v}_{c\_spatial}\leftarrow 0$ |

6 ${v}_{c\_temporal}\leftarrow 0$ |

7 ${v}_{c\_spatial}\leftarrow IDW\left({V}_{ij},\alpha \right)$ |

8 $\text{}{v}_{c\_temporal}\leftarrow SES\left({V}_{ij},\beta ,wc\right)$ |

9 ${v}_{c\_spatial}$ If, ${v}_{c\_temporal}$ are not missing value then |

10 $C\_{M}_{ij}\leftarrow \left({v}_{c\_spatial}+{v}_{c\_temporal}\right)/2$ |

11 ${v}_{c\_spatial}$ is not missing value Else ifthen |

12 $C\_{M}_{ij}\leftarrow {v}_{c\_spatial}$ |

13 ${v}_{c\_temporal}$ is not missing value Else ifthen |

14 $C\_{M}_{ij}\leftarrow {v}_{c\_temporal}$ |

15 Else |

16 $C\_{M}_{ij}\leftarrow \varnothing $ |

17 End for |

18 End for |

#### 3.2. Fine-Grained Interpolation

#### 3.2.1. Sliding Window

Algorithm 2: Selected optimal window (SOM). |

Input: Missing Spatial Series ${t}_{n}$ |

Output: Start of Window $w\_begin$ |

End of Window $w\_end$ |

1 $R\_begin\leftarrow 0$ |

2 $R\_end\leftarrow 0$ |

3 For $j=n-1$ to 1 |

4 $R\_last\leftarrow R\_begin$ |

5 $R\_begin\leftarrow \left(R\_begin+Corr\left({t}_{n},{t}_{j}\right)\right)/\left(n-j\right)$ |

6 If $R\_begin<R\_last$ |

7 Return $j$ |

8 End if |

9 End for |

10 For $i=n+1$ to end of the timestamp |

11 $R\_last\leftarrow R\_end$ |

12 $R\_end\leftarrow \left(R\_end+Corr\left({t}_{n},{t}_{i}\right)\right)/\left(i-n\right)$ |

13 If $R\_end<R\_last$ |

14 Return $i$ |

15 End if |

16 End for |

17 $w\_begin\leftarrow j$ |

18 $w\_end\leftarrow i$ |

#### 3.2.2. Fine-Grained Spatial Dimension Interpolation

Algorithm 3: Fine-Grained spatial interpolation. |

Input: Coarse-Grained Matrix $C\_{M}_{m\times n}$ |

Number of Spatial Neighbors $ms$ |

Output: Fine-Grained Spatial Matrix $F\_{S}_{m\times n}$ |

1 For $i=1$ to $m$ |

2 For $j=1$ to $n$ |

3 $ws\leftarrow SOM\left(C\_{M}_{m\times n},C\_{M}_{ij}\right)$ |

4 $Rss\leftarrow Corrcoef\left(M\_{W}_{m\times ws},column\right)$ |

5 $S\_Correlate\leftarrow MaxMsCorrelate\left(Rss,S\_Missin{g}_{i},ms\right)$ |

6 $C\_S\leftarrow Cov\left(S\_Correlate\right)$ |

7 For each ${S}_{k}\in S\_Correlate$ |

8 ${C}_{k}\leftarrow Cov\left({S}_{k},S\_Missin{g}_{i}\right)$ |

9 ${b}_{k}\leftarrow Mean\left({S}_{k}\right)/Mean\left(S\_Missin{g}_{i}\right)$ |

10 $C\_Matrix\_Lef{t}_{\left(ms+1\right)\times \left(ms+1\right)}\leftarrow Combine\left(C\_S,b\right)$ |

11 $C\_Matrix\_Righ{t}_{\left(ms+1\right)\times 1}\leftarrow Combine\left(C,1\right)$ |

12 $w\leftarrow C\_Matrix\_Lef{t}_{\left(ms+1\right)\times \left(ms+1\right)}C\_Matrix\_Righ{t}_{\left(ms+1\right)\times 1}$ |

13 $F\_{S}_{ij}\leftarrow DotProduct\left(S\_Correlat{e}_{i},w\right)$ |

14 End for |

15 End for |

#### 3.2.3. Fine-Grained Temporal Dimension Interpolation

Algorithm 4: Fine-Grained temporal interpolation. |

Input: Coarse-Grained Matrix $C\_{M}_{m\times n}$ |

Number of Temporal Neighbors $nt$ |

Output: Fine-Grained Temporal Matrix $F\_{T}_{m\times n}$ |

1 For $i=1$ to $m$ |

2 For $j=1$ to $n$ |

3 $wt\leftarrow SOM\left(C\_{M}_{m\times n},C\_{M}_{ij}\right)$ |

4 $Rtt\leftarrow Corrcoef\left(M\_{W}_{m\times wt},row\right)$ |

5 $T\_Correlate\leftarrow MaxNtCorrelate\left(Rtt,T\_Missin{g}_{j},nt\right)$ |

6 $C\_T\leftarrow Cov\left(T\_Correlate\right)$ |

7 For each ${T}_{k}\in T\_Correlate$ |

8 ${C}_{k}\leftarrow Cov\left({T}_{k},T\_Missin{g}_{j}\right)$ |

9 ${b}_{k}\leftarrow Mean\left({T}_{k}\right)/Mean\left(T\_Missin{g}_{j}\right)$ |

10 $C\_Matrix\_Lef{t}_{\left(nt+1\right)\times \left(nt+1\right)}\leftarrow Combine\left(C\_T,b\right)$ |

11 $C\_Matrix\_Righ{t}_{\left(nt+1\right)\times 1}\leftarrow Combine\left(C,1\right)$ |

12 $\phi \leftarrow C\_Matrix\_Lef{t}_{\left(nt+1\right)\times \left(nt+1\right)}C\_Matrix\_Righ{t}_{\left(nt+1\right)\times 1}$ |

13 $F\_{T}_{ij}\leftarrow DotProduct\left(T\_Correlat{e}_{j},\phi \right)$ |

14 End for |

15 End for |

#### 3.3. Spatio-Temporal Integration

Algorithm 5: Combining spatial and temporal. |

Input: Fine-Grained Spatial Matrix $F\_{S}_{m\times n}$ |

Fine-Grained Temporal Matrix $F\_{T}_{m\times n}$ |

Coarse-Grained Matrix $C\_{M}_{m\times n}$ |

Number of Spatial Neighbors $ms$ |

Number of Temporal neighbors $nt$ |

Output: Test Estimated Matrix $S{T}_{m\times n}$ |

Missing Estimated Matrix $M\_\mathrm{S}{T}_{m\times n}$ |

1 For $i=1$ to $m$ |

2 For $j=1$ to $n$ |

3 If $F\_{S}_{ij}$ $F\_{T}_{ij}C\_{M}_{ij}$ are not missing values then |

4 $Sample\leftarrow Sample\left(F\_{T}_{ij},F\_{S}_{ij},C\_{M}_{ij}\right)$ |

5 End if |

6 End for |

7 End for |

8 $Training\_Spl\leftarrow Divide\left(Sample,0.8\right)$ |

9 $Testing\_Spl\leftarrow Divide\left(Sample,0.1\right)$ |

10 $CrossValidation\_Spl\leftarrow Divide\left(Sample,0.1\right)$ |

11 $Net\leftarrow Train\left(Training\_Spl\right)$ $\nabla NeuralNetworkTraining$ |

12 For $i=1$ to $m$ |

13 For $j=1$ to $n$ |

14 If $C\_{M}_{ij}$ is missing value then |

15 $M\_S{T}_{ij}\leftarrow Sim\left(Net,F\_{T}_{ij},F\_{S}_{ij}\right)$ |

16 End if |

17 End for |

18 End for |

19 For $i=1$ to $m$ |

20 For $j=1$ to $n$ |

21 If $\{F\_{S}_{ij},F\_{T}_{ij}\}\in Testing\_Spl$ then |

22 $S{T}_{ij}\leftarrow Sim\left(Net,F\_{T}_{ij},F\_{S}_{ij}\right)$ |

23 End if |

24 End for |

25 End for |

## 4. Results

#### 4.1. Datasets

_{2.5}, CO, SO

_{3}, O

_{3}, NO

_{2}and other attributes, each of which was collected at 36 air quality monitoring stations at hourly intervals, as depicted in Figure 8. The dataset contained a total of 8759 records [30] (Table 1).

_{2.5}contains a complete case. For the other attributes, we used combination analysis to explore the patterns of missing spatio-temporal missing data (Figure 9).

_{2.5}dataset as an example, we set $wt$ = 48 as the time sliding window, which took 48 h of data to explore the missing pattern. The missing numbers of st

_{19}, st

_{27}, st

_{35}and st

_{36}at the same time were eight (i.e., the missing pattern in Figure 1c). Data for st

_{19}, st

_{27}, st

_{36}were completely missing (Figure 1d). Data for {st

_{18}, st

_{19}}, {st

_{35}, st

_{36}} showed random block loss (Figure 1b). Finally, a large number of patterns showed random missing data (Figure 1a). These patterns show that if the interpolation process was performed directly on the original dataset (i.e., without coarse-grained interpolation to eliminate the effect of successive missing data), it would be difficult to obtain accurate evaluation.

#### 4.2. Evaluation Criteria

#### 4.3. Experimental Results

#### 4.3.1. Overall Results

#### 4.3.2. Effect of Coarse-Grained Interpolation

#### 4.3.3. Effect of the Coarse-Grained Missing Data Rate

#### 4.3.4. Effect Sample Point Number

#### 4.3.5. Effect of Sliding Window

#### 4.3.6. Performance of Two- and Three-Step Interpolation

#### 4.3.7. Performance Comparison for Different Datasets

_{2}, CO, SO

_{3}and O

_{3}datasets (Figure 13). The results confirmed that the proposed method is superior to the other three methods in terms of accuracy. We found that only our new method can guarantee a complete reconstruction result and is able to maintain consistent stability across different datasets. For example, the P-BSHADE method performed better on the SO

_{3}dataset, but worse for other datasets. The ST-HC method performed better on the NO

_{2}dataset, but worse on the other datasets. This variable performance reflects the fact that different datasets have completely different missing data patterns, from which existing methods directly interpolate results (i.e., they do not eliminate the influence of missing patterns before interpolation).

#### 4.3.8. Evaluation of Computational Efficiency

## 5. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Wang, J.; Yong, G.E.; Lianfa, L.I. Spatiotemporal data analysis in geography. Acta Geogr. Sin.
**2014**, 69, 1326–1345. [Google Scholar] - Deng, M.; Fan, Z.; Liu, Q. A Hybrid Method for Interpolating Missing Data in Heterogeneous Spatio-Temporal Datasets. ISPRS Int. J. Geo-Inf.
**2016**, 5, 13. [Google Scholar] [CrossRef] - Gao, Z.; Cheng, W.; Qiu, X. A Missing Sensor Data Estimation Algorithm Based on Temporal and Spatial Correlation. Int. J. Distrib. Sens. Netw.
**2015**, 2015, 1–10. [Google Scholar] [CrossRef] - Galán, C.O.; Lasheras, F.S.; Juez, F.J.D.C.; Sánchez, A.B. Missing data imputation of questionnaires by means of genetic algorithms with different fitness functions. J. Comput. Appl. Math.
**2017**, 311, 704–717. [Google Scholar] [CrossRef] - Durán-Rosal, A.M.; Hervás-Martínez, C.; Tallón-Ballesteros, A.J. Massive missing data reconstruction in ocean buoys with evolutionary product unit neural networks. Ocean Eng.
**2016**, 117, 292–301. [Google Scholar] [CrossRef] - Tak, S.; Woo, S.; Yeo, H. Data-Driven Imputation Method for Traffic Data in Sectional Units of Road Links. IEEE Trans. Intell. Transp. Syst.
**2016**, 17, 1762–1771. [Google Scholar] [CrossRef] - Tonini, F.; Dillon, W.W.; Money, E.S. Spatio-temporal reconstruction of missing forest microclimate measurements. Agric. For. Meteorol.
**2016**, 218–219, 1–10. [Google Scholar] [CrossRef] - Londhe, S.; Dixit, P.; Shah, S. Infilling of missing daily rainfall records using artificial neural network. ISH J. Hydraul. Eng.
**2015**, 21, 255–264. [Google Scholar] [CrossRef] - Tipton, J.; Hooten, M.; Goring, S. Reconstruction of spatio-temporal temperature from sparse historical records using robust probabilistic principal component regression. Adv. Stat. Clim. Meteorol. Oceanogr.
**2017**, 3, 1–16. [Google Scholar] [CrossRef] - Ruan, W.; Xu, P.; Sheng, Q.Z. Recovering Missing Values from Corrupted Spatio-Temporal Sensory Data via Robust Low-Rank Tensor Completion; International Conference on Database Systems for Advanced Applications; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Lu, G.Y.; Wong, D.W. An adaptive inverse-distance weighting spatial interpolation technique. Comput. Geosci.
**2008**, 34, 1044–1055. [Google Scholar] [CrossRef] - Bartier, P.M.; Keller, C.P. Multivariate interpolation to incorporate thematic surface data using inverse distance weighting (IDW). Comput. Geosci.
**1996**, 22, 795–799. [Google Scholar] [CrossRef] - Pesquer, L.; Cortés, A.; Pons, X. Parallel ordinary kriging interpolation incorporating automatic variogram fitting. Comput. Geosci.
**2011**, 37, 464–473. [Google Scholar] [CrossRef] - Bhattacharjee, S.; Mitra, P.; Ghosh, S.K. Spatial Interpolation to Predict Missing Attributes in GIS Using Semantic Kriging. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 4771–4780. [Google Scholar] [CrossRef] - Dutilleul, P. Spatio-Temporal Heterogeneity: Concepts and Analyses; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Xu, C.; Wang, J.; Hu, M. Interpolation of Missing Temperature Data at Meteorological Stations Using P-BSHADE. J. Clim.
**2013**, 26, 7452–7463. [Google Scholar] [CrossRef] - Yozgatligil, C.; Aslan, S.; Iyigun, C. Comparison of missing value imputation methods in time series: The case of Turkish meteorological data. Theor. Appl. Climatol.
**2013**, 112, 143–167. [Google Scholar] [CrossRef] - Gardner, E.S., Jr. Exponential smoothing: The state of the art—Part II. Int. J. Forecast.
**2006**, 22, 637–666. [Google Scholar] [CrossRef] - Li, Y.; Li, Z. Efficient missing data imputing for traffic flow by considering temporal and spatial dependence. Transp. Res. Part C Emerg. Technol.
**2013**, 34, 108–120. [Google Scholar] [CrossRef] - Ran, B.; Tan, H.; Wu, Y. Tensor based missing traffic data completion with spatial-temporal correlation. Phys. A Stat. Mech. Appl.
**2016**, 446, 54–63. [Google Scholar] [CrossRef] - Qi, H.; Liu, M.; Wang, D. Spatial-Temporal Congestion Identification Based on Time Series Similarity Considering Missing Data. PLoS ONE
**2016**, 11, e162043. [Google Scholar] [CrossRef] [PubMed] - Holland, R.C.; Jones, G.; Benschop, J. Spatio-temporal modelling of disease incidence with missing covariate values. Epidemiol. Infect.
**2015**, 143, 1777–1788. [Google Scholar] [CrossRef] [PubMed] - Reynolds, K.M.; Madden, L.V. Analysis of epidemics using spatio-temporal autocorrelation. Phytopathology
**1988**, 78, 240–246. [Google Scholar] [CrossRef] - Li, D.; Deogun, J.; Spaulding, W. Towards Missing Data Imputation: A Study of Fuzzy K-means Clustering Method. Lect. Notes Comput. Sci.
**2004**, 3066, 573–579. [Google Scholar] - Qu, L.; Li, L.; Zhang, Y. PPCA-based missing data imputation for traffic flow volume: A systematical approach. IEEE Trans. Intell. Transp. Syst.
**2009**, 10, 512–522. [Google Scholar] - Kong, L.; Xia, M.; Liu, X.Y. Data Loss and Reconstruction in Sensor Networks. IEEE Infocom
**2013**, 25, 1654–1662. [Google Scholar] - Yi, X.; Zheng, Y.; Zhang, J. ST-MVL: Filling Missing Values in Geo-Sensory Time Series Data. In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016, New York, NY, USA, 9–15 July 2016; pp. 2704–2710. [Google Scholar]
- Karydas, C.G.; Gitas, I.Z.; Koutsogiannaki, E. Evaluation of spatial interpolation techniques for mapping agricultural topsoil properties in Crete. EARSeL eProceedings
**2009**, 8, 26–39. [Google Scholar] - Rumelhart, D.E.; Hinton, G.E.; Williams, R.J. Learning rep-resentation by back-propagating errors. Nature
**1986**, 323, 533–536. [Google Scholar] [CrossRef] - Zheng, Y.; Yi, X.; Li, M. Forecasting fine-grained air quality based on big data. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Sydney, NSW, Australia, 10–13 August 2015. [Google Scholar]
- Cesare, L.D.; Myers, D.E.; Posa, D. Estimating and modeling space-time correlation structures. Stat. Probab. Lett.
**2001**, 51, 9–14. [Google Scholar] [CrossRef] - Wang, J.F.; Li, X.H.; Christakos, G. Geographical Detectors-Based Health Risk Assessment and its Application in the Neural Tube Defects Study of the Heshun Region, China. Int. J. Geogr. Inf. Sci.
**2010**, 24, 107–127. [Google Scholar] [CrossRef] - Fan, Z.; Gong, G.; Liu, B. A Space-time Interpolation Method of Missing Data Based on Spatio-temporal Heterogeneity. Acta Geod. Cartogr. Sin.
**2016**, 45, 458–465. [Google Scholar] - Kondrashov, D.; Ghil, M. Spatio-temporal filling of missing points in geophysical data sets. Nonlinear Proc. Geophys.
**2006**, 13, 151–159. [Google Scholar] [CrossRef]

**Figure 9.**Pattern of missing spatio-temporal PM

_{2.5}data. Red squares represent missing data, and values on the right ordinate are the number of missing data combinations.

**Figure 12.**Influence of the sliding window on the interpolation results. The static window refers to the selection of fixed size sliding window (i.e., the center of missing data, taking the first 24 h and last 24 h as the sample data for interpolation).

**Figure 13.**Performance of the ST-2SMR using different spatio-temporal datasets. (

**a**) The result of the experiment in NO

_{2}datasets; (

**b**) The result of the experiment in CO datasets; (

**c**) The result of the experiment in SO

_{3}datasets; (

**d**) The result of the experiment in O

_{3}datasets .

Data | Ratio of Missing | Complete Case | Number of Missing |
---|---|---|---|

PM2.5 | 13.25% | 29.11% | 41,771 |

CO | 15.10% | 0.00% | 47,604 |

SO_{3} | 15.24% | 0.00% | 48,041 |

O_{3} | 15.43% | 0.00% | 48,667 |

NO_{2} | 16.01% | 0.00% | 50,470 |

**Table 2.**Combination of different methods. P-BSHADE, point estimation model of biased hospital-based area disease estimation; ST, spatio-temporal; HC, heterogeneous covariance.

Method | Coarse-Grained | Window | Coarse-Grained + Window |
---|---|---|---|

ST-kriging | ST-kriging-C | ST-kriging-W | ST-kriging-C-W |

P-BSHADE | P-BSHADE-C | P-BSHADE-W | P-BSHADE-C-W |

ST-HC | ST-HC-C | ST-HC-W | ST-HC-C-W |

**Table 3.**Performance comparison of different methods. RC, ratio of construction; ST-2SMR, spatio-temporal missing data reconstruction.

Condition | Method | MAE | MSE | RC |
---|---|---|---|---|

ST-Kriging | 18.3796 | 0.2211 | 96.20% | |

P-BSHADE | 18.2085 | 0.2190 | 96.20% | |

ST-HC | 26.4273 | 0.3066 | 65.94% | |

ST-2SMR | 15.5247 | 0.1319 | 65.94% | |

Window | ST-Kriging-W | 14.3281 | 0.1724 | 99.60% |

P-BSHADE-W | 14.6172 | 0.1758 | 99.29% | |

ST-HC-W | 11.2211 | 0.1349 | 93.69% | |

ST-2SMR | 9.5920 | 0.0822 | 93.69% | |

Coarse-Grained | ST-Kriging-C | 13.1726 | 0.1585 | 100% |

P-BSHADE-C | 12.9178 | 0.1554 | 100% | |

ST-HC-C | 8.7650 | 0.1054 | 100% | |

ST-2SMR | 7.4292 | 0.0470 | 100% | |

Coarse-Grained + Window | ST-Kriging-C-W | 12.9717 | 0.1560 | 100% |

P-BSHADE-C-W | 12.6669 | 0.1524 | 100% | |

ST-HC-C-W | 7.9196 | 0.0953 | 100% | |

ST-2SMR | 7.2285 | 0.0623 | 100% |

**Table 4.**Performance of different coarse-grained interpolation methods

^{1}. SES, simple exponential smoothing.

Method | IDW + SES | ST-Kriging | P-BSHADE | |||
---|---|---|---|---|---|---|

MAE | MRE | MAE | MRE | MAE | MRE | |

ST-kriging | 13.1726 | 0.1585 | 13.6027 | 0.1636 | 13.6010 | 0.1636 |

P-BSHADE | 12.9178 | 0.1554 | 13.5883 | 0.1635 | 13.6081 | 0.1637 |

ST-HC | 8.7650 | 0.1054 | 9.0637 | 0.1089 | 9.1601 | 0.1101 |

ST-2SMR | 7.4292 | 0.0470 | 7.4826 | 0.0475 | 7.6002 | 0.0484 |

^{1}The abscissa represents fine-grained interpolation; the ordinate represents coarse-grained interpolation.

Neighbor Station Number | MAE | MRE | RC | |
---|---|---|---|---|

Spatial | Temporal | |||

5 | 5 | 7.3300 | 0.0625 | 100% |

5 | 10 | 7.3276 | 0.0635 | 100% |

5 | 15 | 7.4736 | 0.0643 | 100% |

10 | 5 | 7.2787 | 0.0630 | 100% |

10 | 10 | 7.2285 | 0.0623 | 100% |

10 | 15 | 7.2761 | 0.0630 | 100% |

15 | 5 | 7.2952 | 0.0631 | 100% |

15 | 10 | 7.2892 | 0.0631 | 100% |

15 | 15 | 7.3332 | 0.0650 | 100% |

Method | Three-Step | Two-Step | ||||||
---|---|---|---|---|---|---|---|---|

ST-Kriging | P-BSHADE | ST-HC | IDW + SES | |||||

MAE | MRE | MAE | MRE | MAE | MRE | MAE | MRE | |

ST-Kriging | 13.0498 | 0.1570 | 13.0235 | 0.1567 | 13.0741 | 0.1573 | 13.1726 | 0.1585 |

P-BSHADE | 12.7898 | 0.1539 | 12.7964 | 0.1539 | 12.8304 | 0.1543 | 12.9178 | 0.1554 |

ST-HC | 8.7075 | 0.1048 | 8.7206 | 0.1049 | 8.6618 | 0.1042 | 8.7650 | 0.1054 |

ST-2SMR | 7.4143 | 0.0651 | 7.4215 | 0.0652 | 7.4080 | 0.0641 | 7.4292 | 0.0470 |

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**MDPI and ACS Style**

Cheng, S.; Lu, F. A Two-Step Method for Missing Spatio-Temporal Data Reconstruction. *ISPRS Int. J. Geo-Inf.* **2017**, *6*, 187.
https://doi.org/10.3390/ijgi6070187

**AMA Style**

Cheng S, Lu F. A Two-Step Method for Missing Spatio-Temporal Data Reconstruction. *ISPRS International Journal of Geo-Information*. 2017; 6(7):187.
https://doi.org/10.3390/ijgi6070187

**Chicago/Turabian Style**

Cheng, Shifen, and Feng Lu. 2017. "A Two-Step Method for Missing Spatio-Temporal Data Reconstruction" *ISPRS International Journal of Geo-Information* 6, no. 7: 187.
https://doi.org/10.3390/ijgi6070187