Articulating Spatial Statistics and Spatial Optimization Relationships: Expanding the Relevance of Statistics
Abstract
:1. Introduction
2. Background
3. Methods and Results
3.1. The p = 1 Spatial Median Problem
3.1.1. The p = 1 Univariate Spatial Median Problem
3.1.2. The p = 1 Bivariate Spatial Median Problem
3.2. The p = 2 Spatial Median Problem
3.2.1. The p = 2 Univariate Spatial Median Problem
3.2.2. The p = 2 Bivariate Spatial Median Problem
3.3. The p = 3 Spatial Median Problem
3.3.1. The p = 3 Univariate Spatial Median Problem
3.3.2. The p = 3 Bivariate Spatial Median Problem
4. Discussion and Conclusions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Raju, C. Probability in ancient India. In Handbook of the Philosophy of Science; Bandyopadhyay, P., Forster, M., Eds.; Volume 7: Philosophy of Statistics; Elsevier: Amsterdam, The Netherlands, 2011; pp. 1175–1195. [Google Scholar]
- Stigler, S. The History of Statistics: The Measurement of Uncertainty before 1900; Belknap/Harvard: Cambridge, MA, USA, 1986. [Google Scholar]
- Smith, D. History of Mathematics; Dover: New York, NY, USA, 1986; Volume 1−2. [Google Scholar]
- Cliff, A.; Ord, J. Spatial Autocorrelation; Pion: London, UK, 1973. [Google Scholar]
- Paelinck, J.; Klaassen, L. Spatial Econometrics; Saxon House: Farnborough, UK, 1979. [Google Scholar]
- Cressie, N. Statistics for Spatial Data; Wiley: New York, NY, USA, 1991. [Google Scholar]
- Griffith, D. A family of correlated observations: From independent to strongly interrelated ones. Stats 2020, 3, 14. [Google Scholar] [CrossRef]
- Ghosh, A.; Rushton, G. Spatial Analysis and Location-Allocation Models; Van Nostrand Reinhold: New York, NY, USA, 1987. [Google Scholar]
- Church, R.; Murray, A. Location Covering Models: History, Applications and Advancements; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Fermat, P. Essai sur les Maximas et les Minimas, in Œuvres Fermat, Publiées par Lessoins de P. Tannery et C. Henry Sous les Auspices du Ministère de L’instruction Publique; Gauthier-Villars et Fils: Paris, France, 1629; pp. 1891–1912. [Google Scholar]
- Weber, A. Über den Standort der Industrien; J.C.B. Mohr: Tübingen, Germany, 1909; English translation: The Theory of the Location of Industries; Chicago University Press: Chicago, IL, USA, 1929. [Google Scholar]
- Small, C. A survey of multidimensional medians. Int. Stat. Rev. 1990, 58, 263–277. [Google Scholar] [CrossRef]
- Ninimaa, A. Bivariate generalizations of the median. In Multivariate Statistics and Matrices in Statistics, Proceedings of the 5th Tartu Conference, Tartu-Pühajärve, Estonia, 23–28 May, 1994; Tiit, E., Kollo, T., Niemi, H., Eds.; De Gruyter: Boston, MA, USA, 1995; pp. 163–180. [Google Scholar] [CrossRef]
- Eftelioglu, E. Geometric median. In Encyclopedia of GIS, 2nd ed.; Shekhar, S., Xiong, H., Zhou, X., Eds.; Springer: Cham, Switzerland, 2017; pp. 701–704. [Google Scholar]
- Kellerman, A. Centrographic Measures in Geography; GeoAbstracts, University of East Anglia: Norwich, UK, 1981. [Google Scholar]
- Kuhn, H.; Kuenne, R. An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. J. Reg. Sci. 1962, 4, 21–33. [Google Scholar]
- Griffith, D. Using estimated missing spatial data in obtaining single facility location-allocation solutions. l’Espace Géographique 1997, 26, 173–182. [Google Scholar] [CrossRef]
- Griffith, D. Using estimated missing spatial data with the 2-median model. Ann. Oper. Res. 2003, 122, 233–247. [Google Scholar] [CrossRef]
- Anselin, L. Local indicators of spatial association—LISA. Geogr. Anal. 1995, 27, 93–115. [Google Scholar] [CrossRef]
- Ord, J.; Getis, A. Local spatial autocorrelation statistics: Distributional issues and an application. Geogr. Anal. 1995, 27, 286–306. [Google Scholar] [CrossRef]
- Griffith, D.; Chun, Y. Spatial autocorrelation in spatial interactions models: Geographic scale and resolution implications for network resilience and vulnerability. Netw. Spat. Econ. 2015, 15, 337–365. [Google Scholar] [CrossRef]
- Griffith, D.; Paelinck, J. Chapter 2.6: Relationships between spatial autocorrelation and solutions to location-allocation problems. In Morphisms for Quantitative Spatial Analysis; Advanced Studies in Theoretical and Applied Econometrics Series; Springer: Berlin, Germany, 2018; pp. 18–22. [Google Scholar]
- Hotelling, H. Stability in competition. Econ. J. 1929, 39, 41–57. [Google Scholar] [CrossRef]
- Meager, K.; Teo, E.; Xie, T. Socially-optimal locations of duopoly firms with non-uniform consumer densities. Theor. Econ. Lett. 2014, 4, 431–445. [Google Scholar] [CrossRef] [Green Version]
- Chen, L.-A.; Welsh, A. Distribution-function-based bivariate quantiles. J. Multivar. Anal. 2002, 83, 208–231. [Google Scholar] [CrossRef]
- Olkin, I.; Trikalinos, T. Constructions for a bivariate beta distribution. Stat. Probab. Lett. 2015, 96, 54–60. [Google Scholar] [CrossRef] [Green Version]
- Overton, S.; Stehman, S. Properties of designs for sampling continuous spatial resources from a triangular grid. Commun. Stat. 1993, 22, 251–264. [Google Scholar] [CrossRef]
- Grekousis, G. Chapter 3: Analyzing geographic distributions and point patterns. In Spatial Analysis Methods and Practice: Describe–Explore–Explain through GIS; Cambridge University Press: Cambridge, UK, 2020; pp. 147–206. [Google Scholar] [CrossRef]
- Wang, B.; Shi, W.; Miao, Z. Confidence analysis of standard deviational ellipse and its extension into higher dimensional Euclidean space. PLoS ONE 2015, 10, e0118537. [Google Scholar] [CrossRef] [PubMed]
- Cooper, L. Heuristic methods for location-allocation problems. SIAM Rev. 1964, 6, 37–53. [Google Scholar] [CrossRef]
- Ostresh, L. An efficient algorithm for solving the two center location-allocation problem. J. Reg. Sci. 1975, 15, 209–216. [Google Scholar] [CrossRef]
- Hyson, C.; Hyson, W. The economic law of market areas. Q. J. Econ. 1950, 64, 319–324. [Google Scholar] [CrossRef]
- Okabe, A.; Boots, B.; Sugihara, K.; Chiu, S. Spatial Tessellations: Concepts andApplications of Voronoi Diagrams, 2nd ed.; Wiley: Chichester, UK, 2000. [Google Scholar]
- Dacey, M. The geometry of central place theory. Geogr. Annaler. Ser. B Hum. Geogr. 1965, 47, 111–124. [Google Scholar] [CrossRef]
- Leepmeier, M. The Voronoi Cell in a saturated Circle Packing and an elementary proof of Thue’s theorem. arXiv 2019, arXiv:1905.05837. [Google Scholar]
Distribution of Demand | Relevant Integration Answers | Spatial Medial | Random Sample (n = 201; 10,000 Replications) |
---|---|---|---|
sinusoidal | × ) = 0.5 | 0.50000 | 0.49959 (0.05443) |
uniform | x = 0.5 | 0.50000 | 0.50003 (0.03525) |
bell-shaped | 70x9 − 315x8 + 540x7 − 420x6 + 126x5 = 0.5; only one real root | 0.50000 | 0.50020 (0.01431) |
left/negative skewed | 3060x19 − 12,920x18 + 20,520x17 − 14,535x16 + 3876x15 = 0.5; only one real root | 0.75846 | 0.75831 (0.00848) |
right/positive skewed | 3060x19 − 45,220x18 + 311,220x17 − 1,322,685x16 + 3,879,876x15 − 8,314,020x14 + 13,430,340x13 − 16,628,040x12 + 15,872,220x11 − 11,639,628x10 + 6,466,460x9 − 2,645,370x8 + 755,820x7 − 135,660x6 + 11,628x5 = 0.5; only one real root | 0.24154 | 0.24168 (0.00855) |
tri-density function additive mixture | Solved with Mathematica 12.1 assuming x R+ | 0.44045 | 0.46118 (0.03576) |
Demand Points Distribution | Spatial Median Coordinate | Map Pattern of Weights | |||
---|---|---|---|---|---|
Random | Linear Gradient | Quadratic Gradient | Periodic (i.e., SINE Function) | ||
uniform—Beta(1, 1) | theoretical U, V | 0.5, 0.5 | |||
complete data U | 0.50352 | 0.59413 | 0.65100 | 0.55542 | |
complete data V | 0.50119 | 0.59295 | 0.65240 | 0.61038 | |
sampled data U | 0.50517 (0.06348) | 0.59524 (0.05589) | 0.65026 (0.05813) | 0.55680 (0.05808) | |
sampled data V | 0.49924 (0.05947) | 0.5929 (0.05795) | 0.65114 (0.05694) | 0.60905 (0.05421) | |
skewed—Beta(9, 5) | theoretical U, V | 0.35, 0.35 | |||
complete data U | 0.35139 | 0.37292 | 0.38368 | 0.36325 | |
complete data V | 0.35265 | 0.37169 | 0.38448 | 0.37711 | |
sampled data U | 0.35228 (0.02154) | 0.37254 (0.02218) | 0.38397 (0.02151) | 0.36470 (0.02146) | |
sampled data V | 0.35359 (0.02395) | 0.37155 (0.02086) | 0.38406 (0.02247) | 0.37783 (0.02105) |
Demand Points Distribution | Spatial Median Coordinate | Map Pattern of Weights | ||||
---|---|---|---|---|---|---|
Random | Linear Gradient | Quadratic Gradient | Periodic (i.e., SINE Function) | |||
region #1 | theoretical U | triangle | 0.29289 | |||
rectangle | 0.25000 | |||||
theoretical V | triangle | 0.29289 | ||||
rectangle | 0.50000 | |||||
complete data U | 0.74921 | 0.50940 | 0.58642 | 0.29617 | ||
complete data V | 0.48371 | 0.78234 | 0.81068 | 0.68327 | ||
sampled data U | 0.74979 (0.02094) | 0.51026 (0.02161) | 0.58840 (0.04718) | 0.29731 (0.01607) | ||
sampled data V | 0.48451 (0.05633) | 0.78300 (0.01095) | 0.81223 (0.01116) | 0.69387 (0.02193) | ||
region #2 | theoretical U | triangle | 0.70711 | |||
rectangle | 0.75000 | |||||
theoretical V | triangle | 0.70711 | ||||
rectangle | 0.50000 | |||||
complete data U | 0.24599 | 0.70223 | 0.75412 | 0.77668 | ||
complete data V | 0.50031 | 0.32890 | 0.37663 | 0.55974 | ||
sampled data U | 0.24982 (0.02075) | 0.70190 (0.02322) | 0.74355 (0.03638) | 0.77080 (0.01317) | ||
sampled data V | 0.50255 (0.06146) | 0.32999 (0.01625) | 0.37797 (0.03768) | 0.54873 (0.02427) |
Iteration | Regional Demand | Regional Spatial Median | Objective Function | ||||
---|---|---|---|---|---|---|---|
Region #1 | Region #2 | Region #3 | Region #1 | Region #2 | Region #3 | ||
0 | 0.500 | 0.000 | 0.500 | 0.137 | 0.440 | 0.759 | 0.250 |
1 | 0.387 | 0.276 | 0.337 | 0.097 | 0.440 | 0.766 | 0.117 |
2 | 0.381 | 0.292 | 0.327 | 0.095 | 0.440 | 0.767 | 0.115 |
3 | 0.374 | 0.298 | 0.328 | 0.093 | 0.440 | 0.767 | 0.114 |
4 | 0.371 | 0.301 | 0.328 | 0.092 | 0.440 | 0.767 | 0.114 |
5 | 0.371 | 0.305 | 0.324 | 0.092 | 0.440 | 0.767 | 0.114 |
Geographic Distribution of Weights | p = 1 | p = 2 | p = 3 | |||
---|---|---|---|---|---|---|
Region #1 | Region #2 | Region #1 | Region #2 | Region #3 | ||
sinusoidal | 100 | 50 | 50 | 36.15 | 27.70 | 36.15 |
uniform | 100 | 50 | 50 | 33.33 | 33.33 | 33.33 |
bell-shaped | 100 | 50 | 50 | 30.92 | 38.16 | 30.92 |
negatively skewed | 100 | 50 | 50 | 31.95 | 38.72 | 29.33 |
positively skewed | 100 | 50 | 50 | 29.33 | 38.72 | 31.95 |
irregular multimodal | 100 | 50 | 50 | 37.11 | 30.47 | 32.42 |
Scheme | Solution Type | Sampling + ALTERN | Random Initiation of ALTERN | ||
---|---|---|---|---|---|
Frequency | Objective Function | Frequency | Objective Function | ||
random | exact | 1 | 80.5101 | 1 | 80.5101 |
best sample | 43 | 80.5101 | 92 | 80.5101 | |
2nd ranked sample | 219 | 82.1280 | 123 | 81.1721 | |
linear gradient | exact | 1 | 147.9170 | 1 | 147.9170 |
best sample | 608 | 147.9170 | 0 | 147.9170 | |
2nd ranked sample | 282 | 148.4167 | 873 | 148.4207 | |
quadratic gradient | exact | 1 | 98.6672 | 1 | 98.6672 |
best sample | 725 | 98.6672 | 0 | 98.6672 | |
2nd ranked sample | 252 | 99.4786 | 669 | 98.6812 | |
periodic geographic distribution | exact | 1 | 154.0689 | 1 | 154.0689 |
best sample | 176 | 154.0689 | 47 | 154.0689 | |
2nd ranked sample | 176 | 154.8659 | 141 | 154.8659 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Griffith, D.A. Articulating Spatial Statistics and Spatial Optimization Relationships: Expanding the Relevance of Statistics. Stats 2021, 4, 850-867. https://doi.org/10.3390/stats4040050
Griffith DA. Articulating Spatial Statistics and Spatial Optimization Relationships: Expanding the Relevance of Statistics. Stats. 2021; 4(4):850-867. https://doi.org/10.3390/stats4040050
Chicago/Turabian StyleGriffith, Daniel A. 2021. "Articulating Spatial Statistics and Spatial Optimization Relationships: Expanding the Relevance of Statistics" Stats 4, no. 4: 850-867. https://doi.org/10.3390/stats4040050