Fast Inverse Distance Weighting-Based Spatiotemporal Interpolation: A Web-Based Application of Interpolating Daily Fine Particulate Matter PM2.5 in the Contiguous U.S. Using Parallel Programming and k-d Tree
Abstract
:1. Introduction
1.1. Background
- IDW interpolation is simple and intuitive.
- IDW interpolation is fast to compute the interpolated values.
- The choice of IDW interpolation parameters are empirical (i.e., based on, concerned with or verifiable by observation or experience rather than theory or pure logic).
- The IDW interpolation is always exact (i.e., no smoothing).
- The IDW interpolation has sensitivity to outliers and sampling configuration (i.e., clustered and isolated points).
1.2. Literature Review on Interpolation in GIS
2. Methods
2.1. Experimental PM2.5 Data
2.2. IDW-Based Spatiotemporal Interpolation Method Using the Extension Approach
2.2.1. Original IDW-Based Spatiotemporal Interpolation Method Using the Extension Approach
Day | t | c * t |
---|---|---|
1 January 2009 | 1 | 0.1086 |
2 January 2009 | 2 | 0.2172 |
3 January 2009 | 3 | 0.3258 |
4 January 2009 | 4 | 0.4344 |
... | ... | ... |
31 December 2009 | 365 | 39.6390 |
2.2.2. Improved IDW-Based Spatiotemporal Interpolation Method Using the Extension Approach
2.2.3. Discussion of the Methods
2.3. Applying Parallel Computing Techniques
2.3.1. Motivation of Using Parallel Computing
2.3.2. Implementation of Parallel Computing
2.4. k-d Tree Data Structure
2.4.1. Motivation of Using k-d Tree
2.4.2. Properties of a k-d Tree
2.4.3. Constructing a k-d Tree
2.4.4. Searching a k-d Tree
2.4.5. Nearest Neighbor Search Algorithm using k-d Tree to Find One Nearest Neighbor
- Given a current best estimate of the node that may be the nearest neighbor, a candidate hyper-sphere can be constructed that is centered at the query point q(q0, q1, q2,...,qk−1) and running through the current best node point. The nearest neighbor to the query point must lie inside the hyper-sphere.
- If the hyper-sphere is fully to one side of a splitting hyper-plane, then all points on the other side of the splitting hyper-plane cannot be contained in the sphere and, thus, cannot be the nearest neighbor.
- To determine whether the candidate hyper-sphere crosses the splitting hyper-plane that compares coordinate at dimension i, check whether |qi − ai| < r.
2.4.6. Adapted Neighbor Search Algorithm Using a k-d Tree to Find Multiple Nearest Neighbors
Algorithm 1: getNearestNeighbors(k, value) k-nearest neighbor search in a k-d tree |
Algorithm 2: searchNode(value, curr, k, neighborList, examined), Part I moving up the k-d tree to look for better nearest neighbors |
Algorithm 3: searchNode(value, curr, k, neighborList, examined), Part II |
2.5. Cross-Validation
2.5.1. K-Fold Cross-Validation Method
2.5.2. Leave-One-Out Cross-Validation Method
2.5.3. Error Statistics
3. Results
3.1. Computational Performance Improvement by Using Parallel Computing
3.2. Computational Performance Improvement of by Using k-d Tree
3.3. Leave-One-Out Cross-Validation Results for the Original IDW-Based Method
Neighbors | Exponent | MARE |
---|---|---|
3 | 1.0 | 0.54797 |
3 | 1.5 | 0.54301 |
3 | 2.0 | 0.54012 |
3 | 2.5 | 0.53849 |
3 | 3.0 | 0.53768 |
3 | 3.5 | 0.53739 |
3 | 4.0 | 0.53742 |
3 | 4.5 | 0.53768 |
3 | 5.0 | 0.53807 |
3.4. Cross-Validation Results for the Improved IDW-Based Method
3.4.1. Leave-One-Out Cross-Validation Results
Exponent | MARE: n = 3 | MARE:n=4 | RMSPE:n=3 | RMSPE:n=4 |
---|---|---|---|---|
1.0 | 0.37027 | 0.38444 | 242.1428 | 279.8240 |
1.5 | 0.36641 | 0.37664 | 210.7435 | 241.7803 |
2.0 | 0.36423 | 0.37154 | 185.4772 | 208.6938 |
2.5 | 0.36302 | 0.36816 | 167.2276 | 182.7473 |
3.0 | 0.36240 | 0.36595 | 155.4234 | 164.5911 |
3.5 | 0.36214 | 0.36454 | 148.5401 | 153.2638 |
4.0 | 0.36209 | 0.36367 | 144.8794 | 146.9017 |
4.5 | 0.36219 | 0.36317 | 143.0899 | 143.6380 |
5.0 | 0.36237 | 0.36294 | 142.2915 | 142.0947 |
3.4.2. Ten-Fold Cross-Validation Results
Exponent | MARE: n = 7 | RMSPE: n = 7 |
---|---|---|
1.0 | 1.88664 | 4016.4020 |
1.5 | 1.64071 | 3626.1590 |
2.0 | 1.47143 | 3472.6450 |
2.5 | 1.36437 | 3472.7490 |
3.0 | 1.29870 | 3542.2100 |
3.5 | 1.25827 | 3627.0180 |
4.0 | 1.23285 | 3702.0150 |
4.5 | 1.21648 | 3759.8730 |
5.0 | 1.20577 | 3801.4410 |
3.4.3. Comparison of Leave-One-Out and 10-Fold Cross-Validation Results
3.5. A Web-Based Spatiotemporal IDW Interpolation Application
3.5.1. Interpolation and Cross-Validation
3.5.2. Visualization and Animation
4. Conclusions and Future Work
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Li, L.; Losser, T.; Yorke, C.; Piltner, R. Fast Inverse Distance Weighting-Based Spatiotemporal Interpolation: A Web-Based Application of Interpolating Daily Fine Particulate Matter PM2.5 in the Contiguous U.S. Using Parallel Programming and k-d Tree. Int. J. Environ. Res. Public Health 2014, 11, 9101-9141. https://doi.org/10.3390/ijerph110909101
Li L, Losser T, Yorke C, Piltner R. Fast Inverse Distance Weighting-Based Spatiotemporal Interpolation: A Web-Based Application of Interpolating Daily Fine Particulate Matter PM2.5 in the Contiguous U.S. Using Parallel Programming and k-d Tree. International Journal of Environmental Research and Public Health. 2014; 11(9):9101-9141. https://doi.org/10.3390/ijerph110909101
Chicago/Turabian StyleLi, Lixin, Travis Losser, Charles Yorke, and Reinhard Piltner. 2014. "Fast Inverse Distance Weighting-Based Spatiotemporal Interpolation: A Web-Based Application of Interpolating Daily Fine Particulate Matter PM2.5 in the Contiguous U.S. Using Parallel Programming and k-d Tree" International Journal of Environmental Research and Public Health 11, no. 9: 9101-9141. https://doi.org/10.3390/ijerph110909101