Exploring the advantages of the maximum entropy model in calibrating cellular automata for urban growth simulation: a comparative study of four methods

Figures & data

Figure 1. The classification of transition rules involved in this paper.

Figure 2. The urban growth of Beijing, Tianjin, and Wuhan during 2000–2020. The central zones of Beijing, Tianjin, and Wuhan have been enlarged to show details.

Figure 3. The spatial variables used to model the urban growth of Beijing, including dis2CC, dis2DC, dis2TC, dis2TRW, dis2PRW, dis2SEW, dis2TEW, dis2RST, dis2EEM, dis2WAB, Eleva, and Slope. Labels and units of spatial variables are explained in .Table 1

Figure 4. The spatial variables used to model the urban growth of Tianjin, including dis2CC, dis2DC, dis2TC, dis2TRW, dis2PRW, dis2SEW, dis2TEW, dis2RST, dis2EEM, dis2WAB, Eleva, and Slope. Labels and units of spatial variables are explained in .Table 1

Figure 5. The spatial variables used to model the urban growth of Wuhan, including dis2CC, dis2DC, dis2TC, dis2TRW, dis2PRW, dis2SEW, dis2TEW, dis2RST, dis2EEM, dis2WAB, Eleva, and Slope. Labels and units of spatial variables are explained in .Table 1

Table 1. The details of spatial factors and corresponding spatial variables

Dataset	Year	File type	Variable	Unit	Meaning	Source
City center	2020	Shapefile	dis2CC	Meter	Distance to city center	OpenStreetMap
District centers	2020	Shapefile	dis2DC	Meter	Distance to district centers	OpenStreetMap
Township centers	2020	Shapefile	dis2TC	Meter	Distance to township centers	OpenStreetMap
Trunk ways	2020	Shapefile	dis2TRW	Meter	Distance to trunk ways	OpenStreetMap
Primary ways	2020	Shapefile	dis2PRW	Meter	Distance to primary ways	OpenStreetMap
Secondary ways	2020	Shapefile	dis2SEW	Meter	Distance to secondary ways	OpenStreetMap
Tertiary ways	2020	Shapefile	dis2TEW	Meter	Distance to tertiary ways	OpenStreetMap
Railway & subway stations	2020	Shapefile	dis2RST	Meter	Distance to railway & subway stations	OpenStreetMap
Entrances & exits of motorways	2020	Shapefile	dis2EEM	Meter	Distance to entrances & exits of motorways	OpenStreetMap
Water bodies	2000	GRID	dis2WAB	Meter	Distance to water bodies	GlobeLand30
Elevation	2019	GRID	Eleva	Meter	The elevation of the terrain	EarthData
Slope	2019	GRID	Slope	Degree	The slope of the terrain	EarthData

Figure 6. The diagrams of forward hinge feature (left) and reverse hinge feature (right).

Figure 7. The architecture of the ANN model used in this study. Labels of spatial variables are explained in .Table 1

Table 2. The training and testing AUC values of the LR, ANN, SVM, and MaxEnt models, and the RMSE values of their projected TPMs

Study area	Method	AUC of training	AUC of testing	Deviation	RMSE
Beijing	LR	0.8389	0.8367	0.0022	1.3434
	ANN	0.8839	0.8709	0.0130	1.1806
	SVM	0.8380	0.8346	0.0034	1.3771
	MaxEnt	0.8772	0.8700	0.0072	0.8971
Tianjin	LR	0.7573	0.7609	0.0036	1.4551
	ANN	0.8309	0.8161	0.0148	1.2177
	SVM	0.7540	0.7562	0.0022	1.4286
	MaxEnt	0.8067	0.7986	0.0081	1.0564
Wuhan	LR	0.8347	0.8363	0.0016	1.2800
	ANN	0.9154	0.8980	0.0174	1.1611
	SVM	0.8304	0.8341	0.0037	1.2748
	MaxEnt	0.8869	0.8878	0.0009	0.8662

Note: Deviation is the absolute value of the difference between AUC values of training and testing.

Figure 8. Transition potential maps of Beijing projected by LR, ANN, SVM, and MaxEnt models. The central zone of Beijing has been enlarged to show spatial details.

Figure 9. Transition potential maps of Tianjin projected by LR, ANN, SVM, and MaxEnt models. The central zone of Tianjin has been enlarged to show spatial details.

Figure 10. Transition potential maps of Wuhan projected by LR, ANN, SVM, and MaxEnt models. The central zone of Wuhan has been enlarged to show spatial details.

Table 3. The training and projecting time (second) for the LR, ANN, SVM, and MaxEnt models in Beijing, Tianjin, and Wuhan

Study area	Time\Method	LR	ANN	SVM	MaxEnt
Beijing	Training time	0.2780	3.2113	14.9110	13.1280
	Projecting time	0.3040	19.3607	5051.5970	156.4850
	Total time	0.5820	22.5720	5066.5970	169.6130
Tianjin	Training time	0.2160	3.1787	19.1640	13.0050
	Projecting time	0.2210	12.5205	4927.0870	120.4990
	Total time	0.4370	15.6992	4946.2510	133.5040
Wuhan	Training time	0.1760	4.6216	16.5870	13.0620
	Projecting time	0.2020	12.7564	2954.3230	101.3100
	Total time	0.3780	17.3780	2970.9100	114.3720

Figure 11. The observed urban growth in Beijing from 2000 to 2020 and the urban growth simulated by LR-CA, ANN-CA, SVM-CA and MaxEnt-CA models.

Figure 12. The observed urban growth in Tianjin from 2000 to 2020 and the urban growth simulated by LR-CA, ANN-CA, SVM-CA and MaxEnt-CA models.

Figure 13. The observed urban growth in Wuhan from 2000 to 2020 and the urban growth simulated by LR-CA, ANN-CA, SVM-CA and MaxEnt-CA models.

Table 4. The simulation accuracies of LR-CA, ANN-CA, SVM-CA, and MaxEnt-CA models in Beijing, Tianjin, and Wuhan

Study area	CA model	Kappa	Improvement (%)	FoM	Improvement (%)
Beijing	LR-CA	0.6903	–	0.3450	–
	ANN-CA	0.7070	1.67	0.3693	2.43
	SVM-CA	0.6870	−0.33	0.3404	−0.46
	MaxEnt-CA	0.7323	4.20	0.4076	6.26
Tianjin	LR-CA	0.6394	–	0.2418	–
	ANN-CA	0.6514	1.20	0.2577	1.59
	SVM-CA	0.6441	0.47	0.2480	0.62
	MaxEnt-CA	0.6732	3.38	0.2876	4.58
Wuhan	LR-CA	0.6786	–	0.3552	–
	ANN-CA	0.6972	1.86	0.3810	2.58
	SVM-CA	0.6878	0.92	0.3679	1.27
	MaxEnt-CA	0.7373	5.87	0.4401	8.49

Table 5. The landscape metrics of the urban landscape simulated by LR-CA, ANN-CA, SVM-CA, and MaxEnt-CA models in Beijing, Tianjin, and Wuhan

Study area	Model	LPI	LSI
Beijing	LR-CA	0.0899	44.2720
	ANN-CA	0.0978	42.2888
	SVM-CA	0.0884	44.6563
	MaxEnt-CA	0.1296	36.9353
Tianjin	LR-CA	0.0805	50.0072
	ANN-CA	0.0694	51.7027
	SVM-CA	0.0829	49.7773
	MaxEnt-CA	0.0937	49.4785
Wuhan	LR-CA	0.0386	42.0907
	ANN-CA	0.0399	41.3345
	SVM-CA	0.0392	40.8573
	MaxEnt-CA	0.0613	33.3524

Table 6. Partial selected features and corresponding parameters used in the MaxEnt modeling of the urban growth of Beijing. Labels of spatial variables are explained in Table 1

Feature type	Instance	Lambda-value	Min	Max
Linear	dis2CC	0	0.0086	0.9966
Quadratic	Eleva^2	0.4828	0.0051	0.9989
Product	dis2CC*dis2TRW	−1.8975	0.0009	0.9702
Forward hinge	‘dis2CC	0.5504	0.6852	0.9966
Reverse hinge	`dis2TRW	−0.2853	0.0397	0.4486

Table 7. Product features used in the MaxEnt modeling of the urban growth of Beijing, Tianjin, and Wuhan. Labels of spatial variables are explained in Table 1

Study area	Product feature	Lambda-value	Min	Max
Beijing	dis2CC*dis2PRW	0.0089	0.0002	0.9864
	dis2CC*dis2RST	0.0190	0.0002	0.9895
	dis2CC*dis2TRW	−1.8975	0.0009	0.9702
	dis2DC*dis2TRW	0.1507	0.0069	0.9872
	dis2PRW*Slope	0.3711	0.0174	0.9975
	dis2RST*Eleva	0.7815	0.0040	0.9863
	dis2RST*Slope	0.3402	0.0066	0.9952
Tianjin	dis2CC*dis2DC	0.0099	0.0048	0.9517
	dis2CC*dis2TEW	0.3476	0.0079	0.9776
	dis2DC*dis2EEM	0.5271	0.0030	0.9489
	dis2DC*dis2TRW	0.1103	0.0289	0.9851
	dis2EEM*dis2RST	−0.1047	0.0166	0.9763
	dis2EEM*dis2TC	1.3685	0.0085	0.9744
	dis2PRW*dis2TRW	−0.9914	0.0228	0.9671
	dis2RST*dis2SEW	−0.2697	0.0446	0.9978
	dis2RST*dis2TRW	0.1302	0.0060	0.9938
	dis2TEW*dis2WAB	−1.8440	0.0105	0.9933
	dis2TEW*Eleva	0.0395	0.0532	0.9845
Wuhan	dis2DC*dis2EEM	−1.6721	0.0001	0.9071
	dis2DC*dis2TC	0.0864	0.0061	0.9800
	dis2EEM*dis2WAB	0.3513	0.0050	0.9908
	dis2RST*dis2TC	−1.6537	0.0046	0.9828

Figure 14. The response curves showing the relationship between urban growth of Beijing and spatial variables dis2CC, dis2DC, dis2TC, dis2TRW, dis2PRW, dis2SEW, dis2TEW, dis2RST, dis2EEM, dis2WAB, Eleva, and Slope. Labels of spatial variables are explained in .Table 1

Figure 15. The response curves showing the relationship between urban growth of Tianjin and spatial variables dis2CC, dis2DC, dis2TC, dis2TRW, dis2PRW, dis2SEW, dis2TEW, dis2RST, dis2EEM, dis2WAB, Eleva, and Slope. Labels of spatial variables are explained in .Table 1

Figure 16. The response curves showing the relationship between urban growth of Wuhan and spatial variables dis2CC, dis2DC, dis2TC, dis2TRW, dis2PRW, dis2SEW, dis2TEW, dis2RST, dis2EEM, dis2WAB, Eleva, and Slope. Labels of spatial variables are explained in Table 1.

Table 8. The percent contributions of spatial variables in the MaxEnt modeling of the urban growth of Beijing, Tianjin, and Wuhan. Labels of spatial variables are explained in Table 1

Rank	Beijing		Tianjin		Wuhan
	Variable	Percent contribution	Variable	Percent contribution	Variable	Percent contribution
1	Eleva	34.6	dis2TEW	36.1	dis2TEW	23.4
2	Slope	25.8	dis2RST	16.9	dis2CC	20.9
3	dis2CC	14.3	dis2WAB	14.9	dis2RST	14.6
4	dis2TEW	7.1	dis2DC	9.3	dis2WAB	11.6
5	dis2PRW	3.3	dis2CC	5.2	dis2PRW	8.8
6	dis2WAB	3.3	dis2PRW	4.1	dis2TRW	7.9
7	dis2SEW	2.8	Eleva	4.0	dis2SEW	6.2
8	dis2DC	2.5	dis2SEW	3.5	dis2DC	3.2
9	dis2TC	2.5	dis2TC	2.2	dis2EEM	2.2
10	dis2RST	1.9	dis2TRW	2.2	Eleva	1.0
11	dis2TRW	1	dis2EEM	1.1	Slope	0.1
12	dis2EEM	0.9	Slope	0.5	dis2TC	0.1