Local ordered weighted averaging in GIS-based multicriteria analysis

Figures & data

Table 1. Example: computing OWA.

k	Criterion values vk(ai)	Criterion weights wk	Ordered criterion values zik	Reordered criterion weights uk	Order weights λk	λk uk	λk uk zik
1	0.1	0.2	1.0	0.3	0.25	0.075	0.075	0.30
2	0.8	0.4	0.8	0.4	0.25	0.100	0.080	0.32
3	1.0	0.3	0.4	0.1	0.25	0.025	0.010	0.04
4	0.4	0.1	0.1	0.2	0.25	0.050	0.005	0.02
Σ						0.250		0.68

Table 2. Optimal order weights (λ_k) for selected values of the α parameter and the number of criteria n = 6.

α	0.0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
λ1	0.000	0.003	0.021	0.054	0.003	0.167	0.497	0.347	0.478	0.664	1.000
λ2	0.000	0.009	0.039	0.079	0.075	0.167	0.241	0.240	0.255	0.224	0.000
λ3	0.000	0.025	0.072	0.114	0.075	0.167	0.109	0.165	0.136	0.075	0.000
λ4	0.000	0.075	0.136	0.165	0.109	0.167	0.075	0.114	0.072	0.025	0.000
λ5	0.000	0.224	0.255	0.240	0.241	0.167	0.075	0.079	0.039	0.009	0.000
λ6	1.000	0.664	0.478	0.347	0.497	0.167	0.003	0.054	0.021	0.003	0.000
φ	0.000	0.532	0.763	0.901	0.970	1.000	0.970	0.901	0.763	0.532	0.000

Figure 1. Evaluating socio-economic status of neighbourhoods in the City of London, Ontario: spatial patterns of standardized scores the average-value-of-dwelling criterion generated by (a) global value function, v₄(a_i), and (b) local value function, v₄(a_iq).

Figure 2. Evaluating socio-economic status of neighbourhoods in the City of London, Ontario: spatial patterns of local criterion weights. (a) Median household income, w_1q, (b) incidence of low income, w_2q, (c) employment rate, w_3q, (d) average value of dwelling, w_4q, (e) university education, w_5q, and (f) residential burglary, w_6q.

Table 3. Evaluating socio-economic status of neighbourhoods in the City of London, Ontario: the global trade-offs and the average values of local trade-offs.

Criterion	OWA model	ai1	ai2	ai3	ai4	ai5	ai6
ai1	Global	1.00	1.39	1.67	1.79	1.92	1.79
	Local	1.00	1.48	1.45	1.90	2.05	2.22
ai2	Global	0.72	1.00	1.20	1.29	1.38	1.29
	Local	0.76	1.00	1.04	1.40	1.59	1.46
ai3	Global	0.60	0.83	1.00	1.07	1.15	1.07
	Local	0.75	1.05	1.00	1.39	1.80	1.53
ai4	Global	0.56	0.78	0.93	1.00	1.08	1.00
	Local	0.60	0.86	0.84	1.00	1.27	1.30
ai5	Global	0.52	0.72	0.87	0.93	1.00	0.93
	Local	0.31	0.48	0.46	0.55	1.00	0.74
ai6	Global	0.56	0.78	0.93	1.00	1.08	1.00
	Local	0.59	0.76	0.79	1.14	1.57	1.00

Note: a_i1 = median household income, a_i2 = incidence of low income, a_i3 = employment rate, a_i4 = average value of dwelling, a_i5 = university education, and a_i6 = residential burglary.

Figure 3. Evaluating socio-economic status of neighbourhoods in the City of London, Ontario: spatial patterns local criterion ranges of: (a) median income, r_1q (b) employment rate, r_3q, and (c) the local trade-offs between the median-income and employment criteria (w_1q/w_3q).

Figure 4. Evaluating socio-economic status of neighbourhoods in the City of London, Ontario: spatial patterns the overall global and local OWA values for: (a) and (d) α = 0.0; (b) and (e) α = 0.5; (c) and (f) α = 1.0.

Table 4. Evaluating socio-economic status of neighbourhoods in the City of London, Ontario: the Moran’s I coefficients of the spatial patterns of global and local OWA scores for selected values of α (see Figure 4).

α	Moran’s I
	Global OWA	Local OWA
0.0	0.502	0.301
0.1	0.495	0.297
0.2	0.482	0.293
0.3	0.469	0.289
0.4	0.456	0.285
0.5	0.444	0.283
0.6	0.402	0.241
0.7	0.334	0.205
0.8	0.295	0.157
0.9	0.271	0.138
1.0	0.254	0.115

Note: All coefficients are statistically significant at p < 0.000.