Spatial sensitivity analysis for urban land use prediction withphysics-constrained conditional generative adversarial networks

Adrian Albert∗Lawrence Berkeley National Laboratory

Berkeley, CAaalbert@lbl.gov

Emanuele StranoMindEarth

Biel/Bienne, Switzerlandemanuele.strano@mindearth.org

Jasleen KaurPhillips Lighting Research

Cambridge, MAjasleen.kaur1@philips.com

Marta C. GonzalezUniversity of California, Berkeley

Berkeley, CAmartag@berkeley.edu

ABSTRACTAccurately forecasting urban development and its environmentaland climate impacts critically depends on realistic models of thespatial structure of the built environment, and of its dependence onkey factors such as population and economic development. Scenariosimulation and sensitivity analysis, i.e., predicting how changes inunderlying factors at a given location affect urbanization outcomesat other locations, is currently not achievable at a large scale withtraditional urban growth models, which are either too simplistic, ordepend on detailed locally-collected socioeconomic data that is notavailable in most places. Here we develop a framework to estimate,purely from globally-available remote-sensing data and withoutparametric assumptions, the spatial sensitivity of the (static) rateof change of urban sprawl to key macroeconomic development in-dicators. We formulate this spatial regression problem as an image-to-image translation task using conditional generative adversarialnetworks (GANs), where the gradients necessary for comparativestatic analysis are provided by the backpropagation algorithm usedto train the model. This framework allows to naturally incorporatephysical constraints, e.g., the inability to build over water bodies. Tovalidate the spatial structure of model-generated built environmentdistributions, we use spatial statistics commonly used in urban formanalysis. We apply our method to a novel dataset comprising oflayers on the built environment, nightlighs measurements (a proxyfor economic development and energy use), and population densityfor the world’s most populous 15,000 cities.

CCS CONCEPTS• Computing methodologies→ Artificial intelligence; Com-puter vision; Machine learning algorithms; • Applied com-puting → Environmental sciences; Economics.

∗Corresponding author. This research was performed while the author was with theMassachusetts Institute of Technology (MIT) in Cambridge, MA.

Permission to make digital or hard copies of part or all of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).KDD’19, August 2019, Anchorage, AK© 2019 Copyright held by the owner/author(s).ACM ISBN 123-4567-24-567/08/06.https://doi.org/10.475/123_4

KEYWORDSPhysics-Informed Machine Learning, Generative Adversarial Net-works, Urbanization Modeling, Spatial Sensitivity Analysis

ACM Reference Format:Adrian Albert, Emanuele Strano, Jasleen Kaur, and Marta C. Gonzalez. 2019.Spatial sensitivity analysis for urban land use prediction with physics-constrained conditional generative adversarial networks. In Proceedingsof ACM KDD Conference (KDD’19). ACM, New York, NY, USA, Article 4,8 pages. https://doi.org/10.475/123_4

1 INTRODUCTIONThe overwhelming impact of the built environment on a wide rangeof global issues such as climate change, energy use, and economicdevelopment and activity, is hard to overstate. Buildings consume60% of the world’s energy, more than 50% of the world’s populationlives in cities, which are responsible for 70% of GHG emissions and80% of economic output globally [20]. As such, realistic models ofthe evolution of the spatial distribution of the urban built environ-ment, and its dependence on key socio-economic factors, are ofhigh relevance for urban planning, energy management, and infras-tructure investments. However, the tools to study urban form at aglobal scale have to date been largely based on simplified, bottom-up models [22] that fail to capture the complexity observed in realdata and to produce realistic predictions of the spatio-temporaldynamics of land use. Further key limitations of traditional urbandevelopment models are i) their inability to effectively leverage thevast, ever-increasing amount of observational data on urban builtareas, and ii) their dependence on detailed, local socio-economicdata that are not available in the vast majority of cities.

Specific examples of applications where urbanization forecastsare needed, but on-the-ground data is not available or is difficult toobtain that we have identified with partners such as theWorld Bankinclude i) the management and inclusion of large refugees campswithin the urbanization plans of cities in Africa; ii) the economicevaluation and forecasting of necessary future urban investmentsin central Africa; and iii) informing carbon emission, energy use,and water consumption forecasting for Asian and African regions.

In this paper, our goal is to build a machine learning simulationmodel of the spatial structure of the urban built land use that i)produces predictions (in a comparative statics sense, of the nexttime step) that are realistic, both qualitatively, and quantitatively,

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ii) can naturally incorporate physical constraints, and iii) can easilybe used to perform sensitivity analysis, at a large scale, of keyunderlying socio-economic factors affecting urbanization such aspopulation density and (proxies for) economic activity. The modelwould generate simulations, or“synthetic" cities, in a controlledway, to be used in scenario analyses and urbanization forecasts.

For this, we developed our earlier work [3] and propose a con-ditional generative adversarial network (GAN) [13] model thatformulates our task - a spatial regression - as an image-to-imagetranslation problem, inspired by [15]. We develop our GAN modelto predict the spatial distribution of built-up urbanized land start-ing from either i) random noise and input maps encoding physicalconstraints such as water areas (where nothing can be built), and,in addition, ii) maps of population density and luminosity levels(a rough proxy for economic development [24]). This formulationallows to naturally incorporate constraints such as the inability todevelop over water bodies by adding appropriate penalty termsin the GAN loss function. To measure the quality of generatedmaps, we compare real and generated built-up patterns maps usingdomain-inspired spatial statistics routinely used in urban analysis,economic geography, and “urban science" literature [17], such asbuilt-up area density and fractal dimension of built-up patterns.We note that the spatial sensitivity of urban built land use densityto underlying factors such as population density and economicactivity are practically spatial derivatives obtained as a by-productof the backpropagation algorithm used for training.

To train our model, we use a novel, global-scale dataset of remote-sensing data products as detailed in [2]. This dataset includes aunique data layer on built-up areas at global scale obtained fromthe German Aerospace Center (DLR) [9]. Further, the dataset in-corporates remote-sensing data on population density from theLandScan project [19], and on nightlights as proxy for local eco-nomic activity, e.g., [14], and of energy use [10] from NASA’s VIIRSmission [18] (see Section 4).

The paper is organized as follows: in Section 2 we review relatedliterature and current urban modeling approaches. Section 3 de-scribes the GAN model. Section 4 discusses our experimental setupand data. We discuss results in Section 5.1, and conclude in Section6. All the code and data will be made available shortly.

2 RELATED LITERATURE2.1 Modeling urbanization and built land useTraditional spatial explicit urban evolutionary models can be cat-egorized in two main classes: agent based modeling (ABM) andcomplex systems modeling (CSM) [5]. Both of them, as reported re-peatedly in the urban growth modeling literature, are not effectivetools for decision making [16]. The reason of their failure can besummarized as follows.

The ABM approach is based on a probabilistic model that assignsto a non-urban area the probability to become urban after a periodof time. This probability is calibrated with a wide and detailedvariety of spatial co-variants such as population density, price ofthe land, mobility, population demographic profiles, etc. Collectingspatial co-variants (typically through on-the-ground surveys) is apainstaking, long, difficult and expensive process and most of thetime, by the end of the survey process, the area of interest is already

changed. Moreover, in the very the areas where cities will growthmore in next decades, such as African cities, this kind of spatialand in-situ information, critical to inform such models, are minimalor even absent. Thus, the ABM approaches need highly-granulardata on urban development that are either very expensive or simplyimpossible to collect at scale. More agile and adaptable models arenecessary to impact decision making processes.

The CSM framework employs a physics-like approach that needsno in-situ spatial and economic co-variants apart of urban footprintsat a reference point in time. Then it assigns new urbanized areasbased on simple assumptions such as bigger cities are more attrac-tive than small ones (gravity law). This approach is much fasterand simple than ABM, however, while it reproduces macroscopicurban characteristics such fractality [6] and scale-free behaviour[11], it is not able to reproduce realistic urban forms, i.e., the com-plex spatial patterns of land cover observed in cities. In a way, sinceCSM searches for universal laws of urbanization, its level of ab-stractions reproduces urban prototypes that are are too abstractand unrealistic and cannot be used in real-world scenarios.

In short, both types of traditional models (ABM and CSM) areimpractical for forecasting real-world urbanization in a low-dataregime such as the African continent. As such, there is a pronounceddisconnect between the current urban science academic literatureand the practical needs of policymakers who have to allocate re-sources based on urbanization forecasts. Moreover, the forecastingmodels work at very low resolution (500m to 1km), thus being un-able to provide enough granularity to be useful for on-the-groundoperational decision making.

2.2 GANs for urban science applicationsSince their introduction in 2014, GANs [13] have proven to be veryeffective at addressing long-standing challenges in computer vision(e.g., natural image generation, image super-resolution), speechsynthesis, and language translation, among others. Overviews ofearly techniques used for assessing model quality of different ar-chitectural choices are given e.g., in [23]. GANs have been recentlyshown to excel at generating data across a variety of disciplines, in-cluding, close to our topic of interest, land use modeling [3], remotesensing data processing, and climate modeling [25]. As such, thereis now a consistent body of evidence that GANs excel at implicitlysampling from highly complex, analytically-unknown distributionsin a great number of contexts.

In previous research [3], we have shown the effectiveness of aunconstrained, unconditional GAN model at generating realisticbuilt land use maps. There, we show that model-generated “cities”(that is, maps of built land use) display a high degree of similarityon several statistics used in the urban modeling literature withreal cities. The conditional GAN approach we develop in this pa-per builds upon the work in [3], in addition incorporating (soft)physical constraints such as water areas and the ability to predictland use maps from underlying socio-economic factors. Our modelis superior to traditional models in several key aspects. First, it istrained on satellite data products available at a large scale and inmost cases globally, making the data collection much faster, objec-tive and scalable than extensive in-situ surveys. Second, it allowsan easy, principled way to perform sensitivity analysis and scenario

Spatial sensitivity analysis for urban land use prediction with physics-constrained conditional generative adversarial networksKDD’19, August 2019, Anchorage, AK

simulation. Third, it casts urbanizationmodeling as a machine learn-ing problem, which allows us to leverage the tremendous advancesin AI models, software frameworks (e.g., PyTorch and Tensorflow),and hardware (GPUs and TPUs).

3 MODEL3.1 Comparative statics and sensitivity analysisConsider the spatial process of urbanization at a given instant intime xB as a function of several inter-related variables xA, amongwhich population density and the amount of “economic develop-ment" (of which nighttime luminosity is a noisy proxy [14]). Assumethat the urban built land use change process can be expressed inclosed form as 0 = F (t ,xB (t),xA(t)). For small changes in both tand xA, it is easy to see that the change of xB (its total derivative)can then be modeled as a linear combination of a time-dependentpart (with no spatial variation) and a space-dependent part (with notemporal variation). Here we focus only on the short-term spatialvariation at a given time instant t . In econometrics, this procedure ofanalyzing how endogenous factors change in response to changesin dependent variables is termed comparative statics, because it isagnostic of the time-path of the change process, and only focuseson before-and-after comparisons. A typical spatial regression (e.g.,[4]) would model this as:

xB = G(xA)+z = xpopγpop +xlumγlum +z (linear regression), (1)

where xA = (xpop ,xlum ), the pop and lum subscripts refer to night-time luminance and population density, γ refers to regression coef-ficients, i.e., γ = ∂xB

∂xA=(

∂xB∂xpop

, ∂xB∂xlum

), and z is the noise term.

Typical spatial statistics methods make assumptions about thestructure of the noise term z, e.g., modeling it as a Gaussian processwith a covariance that depends on features computed locally [8].Moreover, the optimization typically is formulated via local lossterms, which average over statistics computed around a given loca-tion. Here, we make no assumptions about the form of z or abouthow xA and xB might interact, only that these (i.e., G(·)) can bemodeled by a neural network.

3.2 GANs for image-to-image translationGenerative adversarial networks (GANs) [13] learn unsupervisedrepresentations of input data by training two networks (a generatorand a discriminator) against each other. In the original formula-tion [13], the generator G receives as input a random noise vectorz, which it transforms in a deterministic way (e.g., by passing itthrough successive deconvolutional layers if G is a deep CNN) tooutput a sample xfake = G(z). WhenG is optimal, xfake is implicitlysampled from the data distribution that G seeks to emulate. Thediscriminator D takes an input x (which can be either real, froman empirical dataset, or synthetically generated by G), and outputsthe source probability P(s |x) = D(x) that x is either sampled fromthe real distribution (s = real), or produced by G (s = fake).

The noise input z is unstructured for basic GANs, which makesdifficult controlling the generation process for more complex taskssuch as simulating urban form with specific properties. In the con-text of image-to-image translation problems it has been proposed[15] to replace z and c with structured input (images). The taskhere is to transform an image from domain A (here, population

Figure 1: Model architecture for spatial regression withphysics-informed conditional GANs. We modify the image-to-image translation architecture in [15] by adding an ad-ditional constraint-enforcing model C, whose purpose is toguide training towards physically-sound solutions.

and luminosity data layers) to domain B (here, maps of built areasdensity). GAN-based methods that have recently been proposedachieve this via paired samples from the two domains {(x iA,x

iB )}

Ni=1

[15]. For paired data, the optimization objective function is of theform [15]:

Lpix2pix = LcGAN (G,D) + λLL1 (G)(λ is a trade-off parameter)LcGAN (G,D) = E(xA,xB )∼p(xA,xB )[log D(xA,xB )]+

ExA∼p(xA),z∼pzz [log [1 − D(xA,G(z;xA))]LL1 = E(xA,xB )∼p(xA,xB ),z∼pzz | |xB −G(z;xA)| |1.

LcGAN is the typical GAN objective ensuring that the generatedimages are realistic (i.e., sampled from the real image manifold). Theterm LL1 is a reconstruction error that enforces visual similarityof the generated image xB = G(z;xA) with the real image xB .

3.3 Conditional physics-constrained GANsThe structure of our final model architecture is shown in Figure1. Our first modeling goal is to control the generation process asto simulate realistic scenarios of the effect of changes in the inputmaps xA (nightlights, population density) to the output map of builtareas xB . The “pix2pix" architecture [15] that we build on trains ageneratorG that can produce realistic renderings of samples frominput domain A into domain B. As in [15], the form of noise z thatwe allow is via dropout (with p = 0.2).

Our second modeling goal is to incorporate physical constraintsinto GAN model training. For this, we modify the pix2pix archi-tecture to include a constraint-enforcing module C . This moduleitself can be a complex model such as a neural network perform-ing a more complicated task. However, here we opt for a simplecomputation of a error term ρB ≡ 1{xB > 0 & xW > 0} that

KDD’19, August 2019, Anchorage, AK A. Albert et al.

penalizes the model whenever it generates urbanized land patchesover water areas through . Information on water areas is encodedin a water mask xW that is passed both to the generator G as wellas the discriminator D and the constraint-enforcing model C . Thisis simple way to impose a soft constraint that does not guaranteethe desired solutions by design; however, in practice, we found itis very effective at enforcing the basic physical constraint we areinterested in (water areas). Our final optimization objective is:

L = Lpix2pix + Lconstr, with (2)Lconstr = αExB∼p(xB )[ρB ]

= αExB∼p(xB )[1{xB > 0 & xW > 0}], (3)

where α is a penalty term that we chose to have a large value (here,α = 100) as to ensure the model has an incentive not to generateurbanized land over water areas.

3.4 Conditional GANs as regression modelsIn effect, this formulation amounts to a regression of the outputbuilt map xB = G(xA) + z, with additional regularization providedbyLcGAN andLconstr. The GAN termLcGAN forces the generatorto output samples from the manifold of empirically-observed builtmaps xB , which ensures the realism of the regression predictionsxB . The penalty term Lconstr ensures compliance with the imposedphysical constraint. These two loss terms are structured, as theypenalize either the whole output map (in the case of LcGAN ) orglobal geometric properties of the output (in the case of Lconstr). Inaddition to these two key points, this framework allows to modelcritical spatial regression components:

Coefficients and standard errors. The deterministic input-outputtransformations are much simpler in standard regressions formula-tions (e.g., a linear transformation). This allows for straightforwardinterpretability and statistical tests on the properties of coefficients(gradients of dependent variable with respect to the input covari-ates). Here the dependence of the output to the input is modeled viaa highly-complex neural network. However, this transformation be-ing a composition of simple, differentiable functions, the gradient isreadily available as a by-product of the backpropagation algorithmused to train the network. Standard errors on the coefficients maythen be estimated via the inverse of the Hessian, which is availablereadily in certain computational frameworks.

Structure of noise term. Instead of the typical spatial regressionformulation in 1, where the structure in the noise term z is explicitlyand analytically modelled, our formulation incorporates noise im-plicitly through dropout operations and leaky ReLU activation func-tions. Note that if one would wish to impose an explicitly-modelledstructure on the noise z = z(θ ), with θ a vector of parameters, itwould be possible to estimate θ in the same end-to-end process viabackpropagation, as long as z(·) is differentiable in θ and z is partof the computational graph of the model.

4 EXPERIMENTAL SETUP4.1 Global remote-sensing urban data layersData layers.We observe spatial maps xi for i = 1, ...,N cities, withxi ∈ RW ×W ×S , where S is the number of data sources, as well ascorresponding binary masks x iW ∈ RW ×W , with x iW (x ,y) = 1 if

Figure 2: Geographical distribution of the world-wide urbandata on urban areas used in this paper.

Figure 3: Samples from the CityNet dataset [2] (cities oncolumns, data sources on rows). Example major cities, withwater areas in white and city bounds gray-shaded.

the land from city i at location (x ,y) can be developed (here, waterareas), and bi ∈ RW ×W , representing best-available administrativeboundaries for each city i , with bi (x ,y) = 1 if location (x ,y) iswithin the city boundary. S refers to the number of data sources,which are assumed distinct, if not independent, in the sense thateach brings additional, interpretable information. Here, we repre-sent a city by S = 3 data sources: population density xpop , nighttimeluminosity xlum , and building density xbld . We thus represent acity by the data layers x ≡ [xpop , xlum , xbld ,xW ,b].

The “CityNet” world cities dataset. We used a dataset thatwas recently introduced in [2] that contains the spatial maps asdefined above on world-wide cities with at least 10, 000 inhabitants,i.e., ∼ 32, 000 world-wide. For each of these cities, the dataset in[2] contains sample spatial maps as square windows of widthW =100km around a city center1 from four sources:

Built land areas density. The “Global Urban Footprint” (GUF)[9]is a novel dataset produced by the German Aerospace Center (DLR)1The maximum distance from the city center that most people would be willing tocommute for work (a one-hour commute driving at 50 kmph) [7]. Fixing a spatialscale ofW = 100km results in different image sizes (in pixels) for cities at differentlatitudes on Earth. For simplicity, we resize all images to 128 × 128 pixels.

Spatial sensitivity analysis for urban land use prediction with physics-constrained conditional generative adversarial networksKDD’19, August 2019, Anchorage, AK

thatmaps the distribution of human settlements for the entire planetat an unprecedented ∼ 12m/px (here we aggregate to 750m/px ).

Population density data. The LandScan data [19] consists of popu-lation density estimates available worldwide at a 1km/px resolution.It has been produced yearly since 2000 at the Oak Ridge NationalLaboratory using remote-sensing imagery and census surveys.

Nighttime luminosity data. NASA’s Visible Infrared Imaging Ra-diometer Suite (VIIRS) [18] satellite mission provides data on rel-ative luminance values between 20:00 and 22:00 local time at a750m/px resolution and on a scale from 0 (no lights) to 180.

City boundaries. Data on boundaries at a municipality levelwas integrated from the Global Administrative Boundaries project(GADM) [21] (compiled from open source and census surveys).

We further removed those cities for which the amount of signal inthe built up layer (fraction of image pixels with non-zero values) wasbelow 2%, which resulted in a final training dataset of almost 15, 000world-wide cities. The largest, spatially uniformly-distributed 3, 000such cities are shown in Figure 2.

Several examples of our training set are presented in Figure 3.For each city (column), we present the data sources used, frombottom to top: built areas, luminosity, and population density (thelatter two on log-scale). In the left panel, we show several majorcities, where the white areas represent the water mask xW appliedto each map layer, and the regions outside official city boundaries(b(x ,y) = 0) are gray-shaded.

4.2 Model validation via spatial statisticsFollowing typical analysis in statistical geography and urban de-velopment [17], we compute several spatial statistics on the builtareas of all our samples, either real or synthetic. The simplest suchmeasure for a given map x is the percentage of built area a in afixedW ×W window (hereW = 200km) around the city center,a = 1

W 2∑i, j xi, j . Next, as a simple extension, we compute the

distribution of the sizes of of built agglomerations (contiguouspatches) for a given input image x2. In the top row in Figure 4 weillustrate this computation for Paris. The top-right panel shows thetop 20 such contiguous patches, including the large urban core. Thetop-left panel shows the log-log distribution of the patch size.

Another commonly-used statistic in “urban physics" is the fractaldimension f of a spatial distribution, which we illustrate in the tworightmost panels in Figure 4. To compute f , we use the classic “box-counting" algorithm [12]. This algorithm divides up a give inputimage into a successively finer grid (boxes of sizes 1

2 ,14 ,

18 , ...) and

calculates the number of boxes at each scale that cover at least athreshold amount of non-zero pixels. This procedure is illustratedin the bottom row in Figure 4 for “Sierpinski’s triangle", a classicfractal shape consisting of self-similar triangles. Then f is computedas the slope of the logarithm of the number of boxes containingenough non-zero pixels with the logarithm of the box size.

5 RESULTS5.1 Producing synthetic citiesWe trained our model from Section 3 using the data from Section 4.We validate the G and D components of the model as follows.2For this, we use the morphology.label algorithm in skimage package in Pythonthat extracts the connected components of a given image at the pixel level.

Figure 4: Spatial statistics for urban form analysis. From leftto right: top 20 patches for Paris at a 256px × 256px resolu-tion; log-log distribution of number of built patches withtheir sizes; illustration of the box counting algorithm [12]for computing fractal dimension f ; computing f for Paris.

Simulating urban forms using G. In Figure 7 (left panel) weshow examples of synthetic spatial maps generated by our modelthat sees only water masks xW as inputs. To enhance the visual-ization of the generated patterns, we display population and builtareas density maps as green and blue channels. For comparison, weshow real, randomly-selected cities on the left panel in the samefigure. Note that G is able to generate crisp, realistically-lookingurban forms that are virtually undistinguishable from the real cities.

Next, we use our GAN-based spatial regression framework topredict built maps xB = G(xA) on the test set. To validate the globalstructure of the generated built density maps, we compare model-estimated spatial statistics (a, f ) with ground-truth statistics (a, f )as discussed in Section 4. We show the results of this comparison inFigure 9. Note that the synthetic built density maps are very closeto the real built density map, as indicated by values of Pearson’s R2of ∼ 0.75 (for a) and ∼ 0.85 (for f ). While these domain statisticsare relatively simple, more complex ones (e.g., the distribution ofthe urbanized areas in Figure 4) can be similarly incorporated.

Comparing cities using features learned by D. A second ex-periment we performed was to extract autoencoder bottleneck layerrepresentations ϕi = De (xi ) for all cities i = 1, ...,N . As in [1], webuild a simple classifier (a KD-Tree) that allows to efficiently com-pare cities by performing nearest-neighbor queries in feature space.This is illustrated in the right panel in Figure 7, where we showthe top 3 “most similar” cities to Paris, San Francisco, Boston, and

KDD’19, August 2019, Anchorage, AK A. Albert et al.

Figure 5: Real (left) and synthetic (right) cities. For visualization purposes we display population and built areas density mapsas green and blue channels. Note the complex, realistic-looking synthetic spatial patterns in the right panel and their qualita-tive similarity to the real urban patterns on the right.

Figure 6: Comparing the ground truth and estimated statis-tics: the fraction of built area (a vs a, left) and the fractaldimension (f vs f , right).

Lagos. For example, for San Francisco, the model identifies otherelongated, coastal towns with complex morphology as top neigh-bors, whereas for Paris the model returns other circular-shapedcities. This measure of similarity can provide a way to identify“classes” of cities by their urban development.

5.2 Scenario and sensitivity analysis via GANsWe present examples results in Figure 8 for Paris. The first threecolumns (from left to right) show input population and nightlightsmap xA (a multi-channel image), true built map xB , and predictedbuilt map xB = G(xA). The last two panels show the spatial gra-dient structure for the two components (population density andluminosity). As it is not feasible (or necessary) to compute a full

Figure 7: Validating D by using its bottleneck features ϕ toassess city similarity. Themost similar 3 cities to each querycity are displayed (xbld , xpop , xlum ) are stacked for display.

256 × 256 × 2 gradient matrix for every one of the 256 × 256 el-ements in xB , we compute the spatial gradient for a given localregion of interest R (highlighted in gray over the true built mapxB , second column in the figure) by averaging over all pixels inthat region. We simulate two scenarios: i) R is the main urban core,and ii) R consists of the top 3 secondary area patches. Note how

Spatial sensitivity analysis for urban land use prediction with physics-constrained conditional generative adversarial networksKDD’19, August 2019, Anchorage, AK

a “spillover" effect can be qualitatively observed - in particular forthe secondary urbanized areas - where non-zero gradient valuesappear outside of the areas over which the gradient is computed.This suggests that (static) local changes in the population density(or luminosity) propagate beyond the immediate locality. We nextset to characterize the spatial properties of this effect.

5.3 Spatial gradient structure and “spillover"Next, we study how a local change in factors related to urban form- population density and economic development (luminosity) - canpropagate in space and lead to changes in urban form xB potentiallyrelatively far away from where the change originated. Specifically,we compute the dependence of the magnitude of the gradient dueto a change in a given local region R with the distance from thatregion. We use the following sampling procedure (see Figure 9):(1) For a local region of interest R (circled with a red line in the

figure, the second-larges urban patch around Paris), generatem(herem = 50) sampling directions (rays) r uniformly at random;

(2) Compute the magnitude of the gradient |∂xB/∂xA |(d) as a func-tion of distance d from the center of R, by averaging along eachr over ∼ 7km bins, and then averaging over the rs.An example scenario of applying this sampling method is shown

in the rightmost panel in Figure 9. There, we characterize the “ac-tion at a distance" (spatial dependence of the gradient magnitude) ofa unit change in either population density (blue curve) or luminos-ity (green curve) in the second-largest urban region around Paris.Again, we observe that the “effects" of changes in these factors inthe distribution of built environment propagate spatially outward.Next, to quantify this effect at a global level, we apply the samesampling technique over the computed gradient maps with respectto local changes in luminosity and population density over the testdataset of ∼ 3000 largest metropolitan regions worldwide. Thisanalysis is shown in Figure 10. First, we compute the percentage ofgradient magnitude outside of the originating region (here we usedthe top 3 secondary contiguous urban regions outside of the urbancore) - what we term “gradient spillover". The left panel in thefigure shows this calculation for cities in four major geographicalregions in the case of luminosity. The distributions indicate thatspatial gradient “spillover", while relatively small, is still present.

Finally, in the right panel in Figure 10 we present the distancedependence of the average (log-scale, normalized) gradient magni-tude, again in the same scenario where a change occurs in the top3 secondary urban regions outside the urban core, for the case ofpopulation density. For each of the ∼ 7.5km intervals, we show thedistribution over the global city sample as box plots, broken downby major geographical region. While there are visible differencesacross regions, a trend is apparent in that changes propagate, onaverage, over more than 15km before they contribute less than 1%to the spatial gradient at that distance.

6 DISCUSSION AND CONCLUSIONSTo the best of our knowledge, this is the first study in both themachine learning and in the urban analysis literature on a model ofurban forms derived from data at a planetary scale. Prior literatureon urban analysis almost entirely focuses on cities in developedcountries, due to data availability. However, projections show thatmost of the urbanization in the next 30 years will take place in

developing countries, with more than a doubling of the amountof land used in cities in the absence of data-informed regulatorypolicies. However, in many such places, socio-demographic andeconomic data is either extremely limited, unreliably reported, ordifficult to collect. Together with the poor predictive performance oftraditional urban models, this calls for developing better predictivemethods, calibrated using a globally-consistent dataset, and is ofinterest to international planning agencies, real estate markets, andnational regulatory agencies.

A flexible generative model such as a GAN is needed becauseexisting models of urban form are not able to reproduce the rich-ness in the spatial distribution of key macroeconomic indicators(building density, population density, economic activity) observedin real data, and require extensive design and calibration to in-corporate heterogeneous data. This paper is the first applicationof a top-down approach to modeling urban spatial maps, wherelong-range connections between macroeconomic factors are im-plicitly modeled via a map-to-map regression, without the need ofhand-engineering of features. As such, we can uncover the type ofglobal dependencies that we highlight in our analysis of sensitivity(gradient) profiles in a completely non-parametric way. We see thisas key differentiator for the task of scenario analysis, and as firststep towards a realistic simulator of urban form.

Lastly, we caution that the opacity of GANs (and deep networksmore generally) is a common problem in applying these types ofmodels to real decision-making. In absence of ground truth labels ofany kind, interpretation is certainly very hard. Clearly, the simplespatial statistics that we use to validate model output do not takeinto account all of the richness in spatial maps; however they serveas first proof of concept of a GAN-based methodology. Learning dis-entangled, interpretable representations is certainly an importantstep towards interpretability and usability. We recognize that, incontrast with other problems typically studied in machine learning(e.g., image classification), in our application there are no straight-forward labels to guide training, or, in fact, to allow for validatingalgorithm output. In future work, we intend to add the capabilityof conditioning the generation process on any available ancillaryinformation, e.g., macro-economic and geographical regions andcity-level data of economic output, diversity etc.

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Figure 8: Left: GAN spatial regression example for Paris. The first three columns (from left to right) show the input mapsxA, ground truth built areas map xB , model-generated built map xB . The last two columns show the gradients ∂xB/∂xpop and∂xB/∂xlum. Right: fraction of gradient outside of a local region of interest R produced by a unit change in luminosity in R.

Figure 9: Left: sampling rays out of a local region of inter-est (red outline); right: example gradient dependence withdistance d from selected region for Paris.

Figure 10: Spatial propagation of gradients: (log-normalized)gradient ∂xB/∂xpop magnitude with distance from R.

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