Building W Matrices Using Selected Geostatistical Tools: Empirical Examination and Application
Abstract
:1. Introduction
2. Selected Geospatial Tools in Weight Matrix Construction and Examination: Proposal
2.1. Semivariograms
2.2. Standard Deviational Ellipse
2.3. Trend Surface Analysis
2.4. ESDA and Spatial Models
3. Results
3.1. Spatial Weight Matrices
3.1.1. Geographical Distance
3.1.2. Directional Matrix
3.1.3. Random Weight Matrix
3.2. ESDA
3.3. Spatial Modelling
- Substantive and formal correctness of conclusions; therefore, the designed matrices are correct;
- Consistency and stability of modelling results (i.e., statistically significant similarity of parameter estimates);
- Differences in the values of estimates of parameters reflecting the existing spatial processes;
- Selection of a spatial weight matrix should be dictated by the research objective, and the application of different matrices may alter the conclusions concerning spatial processes.
4. Discussion
- How can a directional matrix be built based on the semivariogram? Is there a possibility of attributing values to weights in the matrix based on this geostatistical measure?
- Is the construction of the matrix for the particular periods of research justified and essential (should one include a matrix designed for each period in the model?), and how can it be done (for example, in spatial panel models)?
- Is the construction of other or specific matrices essential for external variables compared to internal variables in SLX models?
- What are the possibilities to use other geospatial methods (e.g., geographically weighted or autoregressive regression) to investigate the effects of spatial endogenous weighting matrices on the modelling results in terms of the heterogeneous relationships between variables?
5. Conclusions
Funding
Conflicts of Interest
Appendix A
Statistics | Waste | Revenues | dWaste | dRevenues |
---|---|---|---|---|
Mean | 205.9 | 2915.2 | 1.02 | 1.08 |
Max | 609 | 8506 | 8.1 | 2.7 |
St. Deviation | 69.1 | 944.6 | 0.30 | 0.11 |
Coefficient of Variation | 33% | 32% | 30% | 11% |
Median | 206 | 2748 | 0.99 | 1.07 |
Min | 6 | 1267 | −0.05 | 0.48 |
Waste | Revenues | dRevenues | dWaste | Waste | Revenues | dRevenues | dWaste | |
---|---|---|---|---|---|---|---|---|
Year | 2005 | 2006 | ||||||
Mean | 214.6 | 1969.6 | 1.12 | 1.03 | 218.4 | 2223.0 | 1.13 | 1.03 |
St. Deviation | 70.7 | 473.7 | 0.05 | 0.38 | 69.4 | 606.3 | 0.10 | 0.32 |
Coefficient of Variation | 33% | 24% | 4.3% | 36.6% | 32% | 27% | 9.3% | 31.1% |
Max | 440 | 4143 | 1.47 | 4.88 | 464 | 5975 | 1.73 | 4.85 |
Min | 19 | 1267 | 0.99 | 0.003 | 19 | 1397 | 0.80 | 0.29 |
Median | 212 | 1808 | 1.12 | 1.004 | 214 | 2033 | 1.12 | 0.98 |
Year | 2007 | 2008 | ||||||
Mean | 218.8 | 2497.8 | 1.13 | 1.02 | 208.9 | 2663.2 | 1.08 | 1.02 |
St. Deviation | 68.4 | 702.2 | 0.14 | 0.24 | 77.5 | 658.7 | 0.10 | 0.22 |
Coefficient of Variation | 31% | 28% | 12.3% | 23.3% | 37% | 25% | 9.5% | 21.7% |
Max | 461 | 6703 | 2.68 | 2.84 | 609 | 7212 | 1.61 | 1.98 |
Min | 18 | 1604 | 0.67 | 0.29 | 1 | 1635 | 0.48 | 0.39 |
Median | 218 | 2257 | 1.12 | 1.00 | 207 | 2426 | 1.08 | 0.99 |
Year | 2009 | 2010 | ||||||
Mean | 210.2 | 2713.9 | 1.02 | 1.03 | 205.9 | 3002.1 | 1.11 | 1.02 |
St. Deviation | 76.0 | 633.7 | 0.09 | 0.28 | 76.4 | 841.0 | 0.15 | 0.22 |
Coefficient of Variation | 36% | 23% | 9.2% | 27.5% | 37% | 28% | 13.9% | 21.1% |
Max | 528 | 6200 | 1.35 | 2.57 | 521 | 8506 | 2.47 | 2.48 |
Min | 15 | 1763 | 0.73 | 0.19 | 10 | 1971 | 0.69 | 0.49 |
Median | 213 | 2491 | 1.02 | 1.00 | 204 | 2725 | 1.08 | 1.00 |
Year | 2011 | 2012 | ||||||
Mean | 204.0 | 3144.6 | 1.06 | 1.03 | 197.9 | 3232.8 | 1.03 | 1.00 |
St. Deviation | 69.7 | 845.2 | 0.13 | 0.26 | 65.2 | 894.3 | 0.11 | 0.24 |
Coefficient of Variation | 34% | 27% | 12.3% | 25.6% | 33% | 28% | 11.0% | 23.6% |
Max | 580 | 7246 | 1.81 | 2.83 | 423 | 7073 | 1.46 | 2.66 |
Min | 10 | 1967 | 0.55 | 0.34 | 14 | 1971 | 0.60 | 0.32 |
Median | 206 | 2865 | 1.04 | 1.00 | 200 | 2915 | 1.03 | 0.99 |
Year | 2013 | 2014 | ||||||
Mean | 193.7 | 3361.1 | 1.05 | 1.01 | 194.5 | 3572.4 | 1.07 | 1.03 |
St. Deviation | 58.0 | 913.1 | 0.08 | 0.22 | 57.0 | 936.3 | 0.08 | 0.32 |
Coefficient of Variation | 30% | 27% | 8.1% | 21.5% | 29% | 26% | 7.5% | 30.9% |
Max | 432 | 8305 | 1.31 | 2.29 | 388 | 7204 | 1.61 | 4.62 |
Min | 36 | 2194 | 0.69 | 0.13 | 14 | 2249 | 0.84 | 0.03 |
Median | 195 | 2989 | 1.05 | 1.00 | 201 | 3225 | 1.06 | 0.99 |
Year | 2015 | |||||||
Mean | 197.4 | 3686.6 | 1.03 | 1.06 | Waste—volume of collected mixed municipal waste in kg per capita, dWaste—dynamics of change of the Waste in the time period from 2005 to 2015, Revenues—revenues of cities in PLN per capita, dRevenues—dynamics of change of the revenues in the time period from 2005 to 2015, n = 279, T = 11, N = 3069 | |||
St. Deviation | 64.7 | 972.3 | 0.07 | 0.52 | ||||
Coefficient of Variation | 33% | 26% | 7.2% | 49.4% | ||||
Max | 597 | 7934 | 1.41 | 8.07 | ||||
Min | 6 | 2249 | 0.76 | −0.05 | ||||
Median | 200 | 3288 | 1.03 | 1.00 |
Tests | Statistics |
---|---|
Wilks’ lambda | 0.55 *** |
Pillai’s trace | 0.47 *** |
Lawley–Hotelling trace | 0.77 *** |
Roy’s largest root | 0.72 *** |
Up to 57 km: close (W1) | Up to 263 km: medium (W2) |
Number of regions: 279 | Number of regions: 279 |
Number of nonzero links: 10,470 | Number of nonzero links: 13,892 |
Percentage nonzero weights: 13.5 | Percentage nonzero weights: 17.9 |
Average number of links: 38. | Average number of links: 50 |
1 least connected region: 27 with 5 links, | 1 least connected region: 27 with 6 links; |
1 most connected region: 194 with 69 links; | 1 most connected region: 184 with 87 links |
Up to 545 km: far (W3) | Directional: (W4) |
Number of regions: 279 | Number of regions: 279 |
Number of nonzero links: 37,750 | Number of nonzero links: 14,793 |
Percentage nonzero weights: 48.5 | Percentage nonzero weights: 29 |
Average number of links: 135 | Average number of links: 50 |
1 least connected region: 27 with 32 links; | 1 least connected region: 27 with 6 links; |
1 most connected region: 142 with 215 links | 1 most connected region:184 with 87 links |
Random: (W5) | |
Number of regions: 279 Number of nonzero links: 2232 Percentage nonzero weights: 2.867384 Average number of links: 8 Non-symmetric neighbours list | W1, W2, W3—Geographical Distance Matrices W4—Directional Matrix W5—Exogenous (Random) Matrix |
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Class | Distance | Semivariogram | Class | Distance | Semivariogram |
---|---|---|---|---|---|
Wasteav | Wasteav | ||||
1 | 7075 | 0.168 | 21 | 290,095 | 0.140 |
2 | 21,226 | 0.109 | 22 | 304,246 | 0.167 |
3 | 35,377 | 0.165 | 23 | 318,397 | 0.165 |
4 | 49,528 | 0.168 | 24 | 332,548 | 0.163 |
5 | 56,831 | 0.132 | 25 | 346,699 | 0.147 |
6 | 77,830 | 0.132 | 26 | 360,850 | 0.155 |
7 | 91,981 | 0.136 | 27 | 375,001 | 0.166 |
8 | 106,132 | 0.141 | 28 | 389,152 | 0.211 |
9 | 120,283 | 0.212 | 29 | 403,303 | 0.209 |
10 | 134,434 | 0.217 | 30 | 417,454 | 0.187 |
11 | 148,585 | 0.177 | 31 | 431,605 | 0.201 |
12 | 162,736 | 0.176 | 32 | 445,756 | 0.197 |
13 | 176,887 | 0.161 | 33 | 459,907 | 0.171 |
14 | 191,038 | 0.179 | 34 | 474,057 | 0.192 |
15 | 205,189 | 0.206 | 35 | 488,208 | 0.224 |
16 | 219,340 | 0.214 | 36 | 502,359 | 0.176 |
17 | 233,491 | 0.194 | 37 | 516,510 | 0.184 |
18 | 247,642 | 0.154 | 38 | 530,661 | 0.173 |
19 | 262,793 | 0.148 | 39 | 544,812 | 0.127 |
20 | 275,944 | 0.149 | 40 | 558,963 | 0.172 |
Moran’s I | Wasteav. | Waste2005 | Waste2010 | Waste2015 |
---|---|---|---|---|
Up to 57 km: close (W1) | 0.18 *** | 0.09 *** | 0.17 *** | 0.09 *** |
Up to 263 km: medium (W2) | 0.16 *** | 0.08 *** | 0.15 *** | 0.08 *** |
Up to 545 km: far (W3) | 0.07 *** | 0.05 *** | 0.08 *** | 0.04 *** |
Directional: (W4) | 0.15 *** | 0.20 *** | 0.20 *** | 0.10 *** |
Random: (W5) | 0.19 *** | 0.15 *** | 0.20 *** | 0.15 *** |
Model | Spatial Weight Matrix | α0 | α1 | λ |
---|---|---|---|---|
Up to 57 km: close (W1) | 2.25 *** | 0.58 *** | 0.37 *** | |
Up to 263 km: medium (W2) | 2.25 *** | 0.58 *** | 0.41 *** | |
Up to 545 km: far (W3) | 2.25 *** | 0.58 *** | 0.44 *** | |
Directional: (W4) | 2.25 *** | 0.58 *** | 0.10 *** | |
Random: (W5) | 2.25 *** | 0.57 *** | 0.19 *** |
Model | Spatial Weight Matrix | Year | α0 n | α1 n | λnnn |
---|---|---|---|---|---|
W1 | 2005 | 2.31 *** | 0.12 ** | 0.14 | |
W2 | 2.31 *** | 0.14 ** | 0.16 | ||
W3 | 2.49 *** | 0.20 ** | −0.02 | ||
W4 | 2.28 *** | 0.15 ** | 0.18 * | ||
W5 | 2.52 *** | 0.02 ** | 0.16 ** | ||
W1 | 2006 | 2.31 *** | −0.03 | 0.23 * | |
W2 | 2.31 *** | −0.04 | 0.19 | ||
W3 | 2.35 *** | −0.04 | 0.07 | ||
W4 | 2.28 *** | −0.05 | 0.18 * | ||
W5 | 2.35 *** | −0.03 | 0.11 * | ||
W1 | 2007 | 2.31 *** | 0.008 * | 0.20 * | |
W2 | 2.31 *** | 0.005 * | 0.16 | ||
W3 | 2.27 *** | 0.04 * | −0.05 | ||
W4 | 2.28 *** | 0.01 * | 0.17 * | ||
W5 | 2.27 *** | 0.04 * | 0.05 | ||
W1 | 2008 | 2.27 *** | 0.09 * | −0.02 | |
W2 | 2.27 *** | 0.09 * | 0.003 | ||
W3 | 2.34 *** | 0.06 * | −0.09 | ||
W4 | 2.26 *** | 0.10 * | 0.17 ** | ||
W5 | 2.28 *** | 0.05 * | −0.008 | ||
W1 | 2009 | 2.29 *** | 0.09 * | 0.32 ** | |
W2 | 2.29 *** | 0.07 * | 0.30 ** | ||
W3 | 2.29 *** | 0.002 * | 0.47 ** | ||
W4 | 2.26 *** | 0.11 * | 0.18 * | ||
W5 | 2.29 *** | 0.01 * | 0.13 * | ||
W1 | 2010 | 2.28 *** | −0.02 | 0.28 ** | |
W2 | 2.28 *** | −0.02 | 0.29 ** | ||
W3 | 2.32 *** | −0.05 | 0.41 ** | ||
W4 | 2.24 *** | −0.03 | 0.18 * | ||
W5 | 2.32 *** | −0.04 | 0.16 ** | ||
W1 | 2011 | 2.28 *** | −0.004 | 0.21 ** | |
W2 | 2.28 *** | −0.006 | 0.23 * | ||
W3 | 2.31 *** | −0.03 | 0.37 ** | ||
W4 | 2.26 *** | −0.01 | 0.17 * | ||
W5 | 2.32 *** | −0.001 | 0.17 ** | ||
W1 | 2012 | 2.27 *** | 0.03 * | 0.21 * | |
W2 | 2.27 *** | 0.03 * | 0.20 * | ||
W3 | 2.30 *** | 0.03 * | 0.28 ** | ||
W4 | 2.23 *** | 0.01 * | 0.18 * | ||
W5 | 2.29 *** | 0.002 * | 0.11 * | ||
W1 | 2013 | 2.26 *** | 0.03 * | 0.39 *** | |
W2 | 2.26 *** | 0.02 * | 0.42 ** | ||
W3 | 2.37 *** | 0.01 * | 0.41 ** | ||
W4 | 2.23 *** | 0.01 * | 0.20 * | ||
W5 | 2.38 *** | 0.01 * | 0.30 *** | ||
W1 | 2014 | 2.26 *** | −0.01 | 0.40 *** | |
W2 | 2.26 *** | −0.007 | 0.40 *** | ||
W3 | 2.32 *** | −0.05 | 0.20 ** | ||
W4 | 2.24 *** | −0.04 | 0.24 * | ||
W5 | 2.32 *** | −0.06 | 0.25 *** | ||
W1 | 2015 | 2.26 *** | 0.04 * | 0.31 ** | |
W2 | 2.26 *** | 0.04 * | 0.32 ** | ||
W3 | 2.35 *** | 0.01 * | 0.31 ** | ||
W4 | 2.24 *** | 0.03 * | 0.20 * | ||
W5 | 2.35 *** | 0.09 * | 0.23 *** |
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Antczak, E. Building W Matrices Using Selected Geostatistical Tools: Empirical Examination and Application. Stats 2018, 1, 112-133. https://doi.org/10.3390/stats1010009
Antczak E. Building W Matrices Using Selected Geostatistical Tools: Empirical Examination and Application. Stats. 2018; 1(1):112-133. https://doi.org/10.3390/stats1010009
Chicago/Turabian StyleAntczak, Elżbieta. 2018. "Building W Matrices Using Selected Geostatistical Tools: Empirical Examination and Application" Stats 1, no. 1: 112-133. https://doi.org/10.3390/stats1010009