Competing Spatial Optimisation using the k-spatial entropy Didier G. Leibovici2, Konstantinos Daras1 Andy G.D. Turner1 1

University of Nottingham, U.K. [email protected] 1 Univerisity of Leeds, U.K. {K.Daras,A.G.D.Tuner}@leeds.ac.uk

1. Introduction Aggregating geographical data spatially is useful for visualisation, analysis and modelling to support policy formation and decision-making. 2001 and 2011 UK Census Data are available at Output Area (OA), Middle Layer Super Output Area (MSOA) and Ward level. Zoning system optimisation (Daras, 2006, Haynes et al. 2007) becomes useful in optimising the delineation of new aggregating areas to report specific data. Using higher resolution areal units, such as OAs, and optimising heterogeneity or homogeneity of particular descriptors across or within the looked for aggregated zones, similar in scale to MSOAs or Wards, define the problem. Spatial homogeneity of a categorical variable can be measured using the k-spatial entropy framework for point data or areal data (Leibovici et al. 2011, Leibovici and Birkin 2014), so that minimum spatial entropy ensures maximum heterogeneity and vice versa. This paper details the following optimisation procedures to aggregate areal units into zones: • minimum k-spatial entropy across the zones and maximum k-spatial entropy within each zone (minAmaxW) • finding a zoning system with maximum k-spatial entropy across the zones and minimum k-spatial entropy within (maxAminW). A minAmaxW optimised zoning system will have most homogeneous zones in terms of attribute spatial distribution but with heterogeneous population size, whilst a maxAminW optimied zoning will have regions most similar in total population but with very disparate attribute structuring. Policy-making can potentially use both types. Examples using a microsimulation of the evolution of the population in Leeds between 2001and 2031, are shown. These data are an output from the MoSeS project (Birkin et al. 2009).

2. Zoning and entropy

For a set of zones 𝑍 aggregating the distribution of a categorical variable 𝐶 over the sub-zones: a set of proportions 𝑝!" with !,! 𝑝!" = 1 representing the distribution of cases by category and by zone, 𝑝!" = 𝑛!" /𝑁, with 𝑁 as the total population count, one can use the property of the conditional entropy to get: 𝐻 𝐶, 𝑍 =𝑑𝑒𝑓 − !,! 𝑝!" 𝑙𝑜𝑔(𝑝!" )       (1) = − ! 𝑝.! 𝑙𝑜𝑔 𝑝.! − ! 𝑝.! ( ! 𝑝!/! 𝑙𝑜𝑔(𝑝!/! )       = 𝐻 𝑍 + 𝐻(𝐶/𝑍) =  𝐻 𝐶 + 𝐻(𝑍/𝐶) with 𝐻 . the Shannon entropy and where 𝑝!/! = 𝑝!" /𝑝.! with 𝑝.! = ! 𝑝!"  is the conditional probability of the category 𝑐 from the categorical variable 𝐶 given the zone 𝑍 = 𝑧. In other words (1) termed the entropy decomposition theorem (Theil 1972, Leibovici and Birkin 2014) insures that the entropy of a categorical variable disaggregated over a zoning is the entropy of the zoning plus the conditional entropy of the variable given the zoning. Moreover one has: 0 ≤ 𝐻(𝐶/𝑍) ≤ 𝐻 𝐶 (2)

reaching the lower bound when 𝐶 is completely determined by 𝑍 and the upper bound when 𝐶 and 𝑍 are two independent random variables. In regional sciences, a zoning system explaining most of a categorical variable distribution can facilitate policy implementations but working with a zoning system independent of the studied variable facilitates global policy-making expecting to impact equally in each area.

3.  Self-­‐k-­‐spatial  entropy     The decomposition in (1) provides a way to communicate with a map (see Figure 1), the variability of categorical data with the local entropies of each zone. Nonetheless the Shannon entropy reflects distributional homogeneity but not spatial homogeneity within each zone. A random permutation of the sub-zones 𝑅 where 𝐶 is recorded (the OA here) gives the same entropy. To take into account the spatial pattern, Leibovici (2009) introduced a spatial entropy index based on co-occurrences distributions: the k-spatial entropy. A co-occurrence is defined by vicinity, e.g., a maximum distance between k occurrences (k, the order of co-occurrence, being the number of events to be considered in one collocation). For a given categorical variable the cooccurrence distribution can be seen as multivariate multinomial distribution, k=3, giving a trivariate distribution. Leibovici (2011) introduced an univariate version, the self-k-spatial entropy (3), easier to understand and compute, looking only at co-occurrences of one category with itself: 𝑝!!! ,! = 𝑝!!!,! for example with k = 3, so only the hyper-diagonal of the co-occurrence table is used: ! 𝐻!" 𝐶, 𝑑 =𝑑𝑒𝑓 − 1 𝑙𝑜𝑔 𝑛 (3) ! 𝑝!!  …!,! 𝑙𝑜𝑔(𝑝!!…!,! ) !

The classical entropy is derived from the distribution of the occurrences whilst the self-kspatial entropy is derived from the spatial co-occurrences for each category. As the self-k-spatial entropy is the normalised Shannon entropy of the co-occurrence distribution, equations (1) with (3) holds with normalising weights coefficients. Nonetheless, in order to make sense of the conditional co-occurrence distribution, the co-occurrences is constrained by the zoning (4), i.e., only co-occurrences whitin a given zone are counted. Cross-boundary co-occurrences for C have to be missed out, so that co-occurrence distributional wise we still have 𝑝 𝐶, 𝑍 = 𝑝 𝐶/ 𝑍 𝑝(𝑍)=  𝑝 𝑍/𝐶 𝑝(𝐶). Note that Z being a spatial zoning containing the observations with C, the first equation 𝑝 𝐶, 𝑍 = 𝑝 𝐶/𝑍 𝑝(𝑍) is true for both constrained and and unconstrained cooccurrence distributions. Computationally the constrained version of the self-k-spatial entropy is faster as parallel evaluations can be done per zone. ! ! 𝐻!"# 𝐶, 𝑑 =𝑑𝑒𝑓 𝐻!" 𝐶, 𝑍, 𝑑 =𝑑𝑒𝑓 − 1 𝑙𝑜𝑔 𝑛 (4) ! 𝑝!!  …!, !,! 𝑙𝑜𝑔(𝑝!!…!, !,! ) !

In Figure 1 the variation obtained from using the co-occurrence distribution rather than the occurrence distribution with the Shannon entropy is seen with the decomposition (top panel) and the conditional entropies or local entropies (bottom panel). The centre of Leeds appears the least homogeneous in relation to social grades but the North-West and South-East areas of the district showing relatively homogeneous Wards. Can we find a zoning that accentuates this structuring, describing the optimality of the solution in reference to the initial Ward zoning?

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Figure 1: Entropy decompositions for social grades with Ward zoning in 2016 (top panel): Shannon and the constrained self-k-spatial zoning entropy, (bottom panel): local values of the within self-k-spatial entropies at 3000m.

4. Zoning optimisation A generic homogeneity function for zoning optimisation is the within variance for the grouping in 𝑁! zones (Daras 2006): 𝑍!"# = 𝑎𝑟𝑔𝑚𝑖𝑛     𝑡 𝑦 𝐼𝑑 − 𝑃! 𝑦 /(𝑛 − 𝑁! ) (5) ! !≻! !∈𝒵

where the numerator is just expressing using projectors the sum of squares of residuals from the

local mean for each zone of the attribute 𝑦, with the zoning  𝑍 aggregating the 𝑅 high resolution zones (noted 𝑍 ≻ 𝑅) belonging to a range of valid zoning 𝒵 (defined by a set of constraints such as the compactness of the shapes). The compactness constraint is operating in a competing way during the algorithm and pre-defines the order of testing for local optimum of the objective function. For categorical variables the zoning often deals only with the proportion of one category (or combined categories). The spatial-entropy index presented in (4) is a good candidate to take into account the whole set of categories to express spatial homogeneity or heterogeneity. Because of the decomposition (1) there is no real competing between maxA and minW in the maxAminW optimisation (6) which also influences the joint entropy: ! ! 𝑍!"#!$% = 𝑎𝑟𝑔min     𝛼/𝐻!"# 𝑍 + 𝛽𝐻!"# (𝐶/𝑍) (6) ! !≻! !∈𝒵

where the optimisation weights : (𝛼 + 𝛽) = 1, allow some flexibility along with the set of quality constraints fixed by the ensemble 𝒵 (e.g., number of zones, minimum number of population).

Figure 2:History of the maxAminW optimisation (stopping rule: small improvement over last 100 iterations). As seen in Figure 2, in order to prevent local minima, the algorithm is set to behave alike a simulating annealing optimisation from allowing a little increase of the whole score.

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Figure 3: Entropy decompositions with the maxAminW optimised zoning (top panel): all for the Shannon and the self-k-spatial zoning entropy. (bottom panel ): local values of the within self-kspatial entropies at 3000m (with Wards overlaid).

5. Competing compactness In the optimisation (5) or (6) and the result in Figure 3, the compactness is a competing constraint but it can be included as a real competing optimisation:

! ! 𝑍!"#!$% = 𝑎𝑟𝑔min     𝛼/𝐻!"# 𝑍 + 𝛽𝐻!"# (𝐶/𝑍) + 𝛾𝐶𝑜𝑚!,! (𝑍) ! !≻! !∈𝒵!

(7)

where instead of being defined as a rule or threshold for selection as valid zoning in 𝒵′, the compactness is now a third component in the objective function : (𝛼 + 𝛽 + 𝛾) = 1. Because of the differences in variability and in order to insure fair competing, the compactness score has to ! be normalised against 𝐻!"# (𝐶/𝑍) for its range.

5. Discussion With basic compactness constraint, both optimisation paradigms (maxAminW and minAmaxW) “converge” quickly and provided useful zonings for this example. The unconstrained value of the ! self-k-spatial entropy at collocation distance 3000m is 𝐻!" 𝐶, 𝑑 = 0.358, so quite close to the ! zoning constrained value 𝐻!"# 𝐶, 𝑑 = 0.373 with the optimal zoning in Figure 3, and better ! than with the initial zoning of the wards 𝐻!"# 𝐶, 𝑑 = 0.529 in Figure 1. Removing crossborder co-occurrences have here a global “smoothing” effect (for both zonings) which is partially recovered locally within the zoning. The compatible findings reveal the spatial patterns associated with the categorical variable whilst loosing on the re-aggreagated statistic. For our example the different choices for compactness integration were not crucial for the interpretation but for policy-making and public communication results. The application for zoning optimisation opens up the choices of criteria for this type of spatial clustering where homogeneity and heterogeneity integrates a spatial constraint. The paper proposes using the distribution of co-occurrences and the k-spatial entropy framework; the optimality of the solutions are in reference to this and the objective funtion associated to the chosen optimisation paradigm (maxAminW or minAmaxW).

6. Acknowledgements This work has been funded partially by the ESRC TALISMAN (geosTapial datA anaLysIS and siMulAtioN) project http://www.geotalisman.org

References Birkin MH, Townend P, Turner AGD, Wu BM and Xu J, 2009, MoSeS: A Grid-enabled spatial decision support system. Social Science Computing Review, 27(4):493-508. Daras K, 2006, An information statistics approach to zone design in the geography of health outcomes and provision. Ph.D. Thesis, University of Newcastle, UK pp 206 Haynes R, Daras K, Reading R and Jones A 2007 Modifiable neighbourhood units, zone design and residents’ perceptions. Health & Place, (13):812–825. Leibovici DG, 2009, Defining Spatial Entropy from Multivariate Distributions of Co-Occurrences. Lecture Notes in Computer Sciences, (5756/2009): 392-404. Leibovici DG, Bastin L and Jackson M, 2011, Higher-Order Co-occurrences for Exploratory Point Pattern Analysis and Decision Tree Clustering on Spatial Data. Computers & Geosciences, 37(3):382-389. Leibovici DG and Birkin MH, 2014, On Geocomputational Determinants of Entropic Variations for Urban Dynamic Studies. Geographical Analysis (accepted) Theil H, 1972, Statistical Decomposition Analysis. Amsterdam: North Holland. Wu BM and Birkin MH, 2012, Agent-Based Extensions to a Spatial Microsimulation Model of Demogaphic Change. In: Heppenstall et al. (eds) Agent-Based Models of Grographical Systems, Springer Sciences Business Media, 347-360.