retail planning in future cities a stochastic dynamical ... · pdf fileretail planning in...

Retail Planning in Future CitiesA Stochastic Dynamical Singly Constrained Spatial Interaction Model

Mark Girolami

Department of Mathematics, Imperial College LondonThe Alan Turing Institute

Lloyds Register Foundation Programme on Data-Centric Engineering

https://iconicmath.org/

Imperial College LondonStatistical Data Science Conference

03-07-2017

Mark Girolami (Imperial, ATI) Stochastic formulation of SIM 03-07-2017 1 / 42


Louis Ellam Mark Girolami Greg Pavliotis Sir Alan WilsonICL & Turing ICL & Turing ICL UCL & Turing


Motivation

Question: “What will cities and regions look like in the future?”


Motivation

Question: “What will demand for products be in the future?”


Motivation

120° W

110° W

100° W

90° W

80° W

70° W

30 ° N

40 ° N

50 ° N

Relative Purchase Level Volumes in USA Cities for B000HJBKMQ

‘Historical demand for a single product?”


Urban Analytics

How will cities and regions develop as time progresses?

Observational data from Twitter, Android location tracking, camera monitoring, travelticketing, demographics, etc.

Simple statistical summaries and models yielding insights the hallmark of UrbanAnalytics.

Mathematical modelling long history in urban and regional analysis and planning.

Build upon these mathematical formalisms with this new found data.


Complex Urban Systems

“We’re uncertain!”

Urban and regional systems are complex.Actions of individuals give rise to an emergent behaviour:

Phase transitions;Path dependence; andMultiple equilibria.

Mechanistic models of complex systems =⇒ model error.

Uncertainty should be addressed.

This talk: Improving insights into urban and regional development by addressing uncertaintiesarising in the modelling process.


The London Retail System

Figure: London retail structure for 2008 (left) and 2012 (right). The locations of retail zones andresidential zones are red and blue, respectively. Sizes are in proportion to floorspace and spendingpower, respectively. N = 625 and M = 201.


The London Retail System - Schoolboy Statistics

“Stratford and Shepherds Bush have risen up the rankings as a result of the opening ofWestfield Stratford City and Westfield London.”

“Knightsbridge appears to have experienced a significant reduction in town centrefloorspace. Ilford and Harrow also experienced declines in town centre floorspace.”

“The West End remains the largest centre in London.”

“Croydon, Kingston, Romford, Canary Wharf, Camden Town, Kings Road East and Angelshowed strong growth in total town centre floorspace over the 2007-2012 period.”

Source: 2013 London Town Centre Health Check Analysis Report.


Airports in England

Figure: Airports by number of passengers for 2007 (left) and 2016 (right). The locations of airports andresidential zones are red and blue, respectively. N = 33 and M = 25.


Urban and Regional Modelling


Spatial Interaction Model

Assume N origin zones and M destination zones.

Origin quantities {Oi}Ni=1.

Destination quantities {Dj}Mj=1.

{Tij}N,Mi ,j=1 denote the flows from zone i to j , respectively.


Assumptions

Flows from origin zones:

Oi =M∑j=1

Tij , i = 1, . . . ,N.

Flows to destination zones:

Dj =N∑i=1

Tij , j = 1, . . . ,M.

Known as a singly-constrained or production-constrained system since {Oi}Ni=1 is fixed and{Dj}Mj=1 is undetermined.


Assumptions

Fixed transport benefit:N∑i=1

M∑j=1

Tij lnWj = X ,

where Wj is the size, and Xj := lnWj is the attractiveness, of zone j .

Fixed transport cost:N∑i=1

M∑j=1

Tijcij = C ,

where cij is the cost of transporting a unit from zone i to zone j .


A Singly-Constrained Spatial Interaction Model

Maximization of an entropy function.

The destination flows are given by

Dj =N∑i=1

Oiexp

(α lnWj − βcij

)∑Mk=1 exp

(α lnWk − βcik

) .Xj := lnWj is the attractiveness of j .

cij is the inconvenience of transporting from zone i to j .

α is the attractiveness scaling parameter.

β is the cost scaling parameter.

αXj − βcij is the net utility from transporting from zone i to j .


A Dynamical Urban and Regional Model

Harris and Wilson Model

The sizes of the destination zones satisfy the (modified) competitive Lotka-Volterra equations

dWj

dt= ε(Dj − κWj

)Wj , j = 1, . . . ,M,

for some initial condition W (0) = w .

Definitions:

κ > 0 is the cost rate per unit size; and

ε > 0 is the responsiveness parameter.


Stochastic Urban and Regional Modelling


Potential Energy Function

In what proceeds, it is more natural to work in terms of attractiveness Xj := lnWj .

The ‘energy’ of a configuration X = (X1, . . . ,XM)T is given by an interaction potentialU(X ).

The ‘force’ acting on each site is given by a conservative vector field −∇U(X ).

Interaction Potential

γ−1U(X ) = κ

M∑j=1

exp(Xj)︸︷︷︸Running cost

−α−1N∑i=1

Oi lnM∑j=1

exp(αXj − βcij)︸︷︷︸Net utility

− δ

M∑j=1

Xj︸︷︷︸Economic Stimulus

.


More General Modelling Requirements

We show that:

1 There exists a potential function U ∈ C 2(RM ,R) such that

−γ−1∇U(X ) =∑

jRj(X )− κ exp(Xj) + δj(X ).

2 The drift coefficients −∇U(X ) are locally Lipshitz and satisfy the one-sided growthcondition in that there exist K1,K2 > 0 such that

−∇U(X ) · X ≤ K1 + K2‖X‖2, ∀X ∈ RM .

3 U satisfies the super-linear growth condition in that there exists K1,K2 > 0 such that

U(X ) ≥ K1 + K2‖X‖, ∀X ∈ RM .

4 The initial condition X (0) = x has finite variance E‖x‖2 <∞.


Key Results under the Model Structure

Urban dynamics, in terms of attractiveness, satisfies the overdamped Langevin dynamics

dX

dt= −∇U(X ) +

√2γ−1 ◦ dB

dt, X (0) = x .

The SDE has a unique solution X ∈ C ([0,T ];RM) that does not explode on [0,T ], almostsurely.

Since δj = δ > 0 and U ∈ C 2 with a growth condition, X (t) is ergodic with respect to awell-defined Boltzmann distribution, whose density w.r.t. Lesbgue Measure is given by

π∞(X ) =1

Zexp

(− γU(X )

), Z :=

∫ ∞−∞

exp(− γU(X )

)dX <∞.


A Stochastic Formulation of the Spatial Interaction Model

With our specification of U(X ), overdamped Langevin dynamics give the Harris andWilson model, plus multiplicative uncertainty.

Stochastic Urban Retail Model

Floorspace dynamics is a stochastic process that satisfies the following Stratonovich SDE.

dWj

dt= εWj

(Dj − κWj + δ

)+ σWj ◦

dBj

dt,

where(B1, . . . ,BM

)Tis standard M-dimensional Brownian motion and σ =

√2/εγ > 0 is the

volatility parameter.

Fluctuations (missing dynamics) are modelled as noise that we interpret in theStratonovich sense.

The extra parameter (or function) δ to represents local economic stimulus to preventzones from collapsing (needed for ergodicity).


Metropolised Time-Stepping methods

Metropolis-Adjusted Langevin Truncated Algorithm (MALTA)

Metropolis Hastings algorithm with an Euler-Maruyama proposal, with truncated drift:

X ∗ := Xn −τ∇U(Xn)

1 ∨ τ‖∇U(Xn)‖+√

2τγ−1ξn, ξn ∼ N (0, I ).

Truncated drift:

‖∇U(X )‖ ≤ τ gives an Euler-Maruyama proposal; and‖∇U(X )‖ > τ gives an Euler-Maruyama proposal, but with unit drift.

Resulting Markov chain is geometrically ergodic with respect to π∞.

Converges strongly to the Markov process {X (t) : t ∈ [0,T ]} on finite time intervals.


Illustrative Stochastic Urban Retail Model

M = 2 and N = 20:

Figure: Sample path of prototype example with parameter values α = 1.8, β = 1, γ = 1, δ = 0.5 andκ = 1. Under this parameter regime the two sites do not coexist and there are frequent phasetransitions.


The Boltzmann Distribution - A Prior Measure on Retail Configurations

Insights

Probability distribution that encodes our knowledge of spatial interactions.

Configurations are confined to a metastable state (path dependence).

As t →∞, system moves towards lowest energy configurations.

Sampling Challenges

π∞ is high-dimensional, anisotropic, multi-modal and involves rare events.

α = 1.8, β = 1, δ = 0.5 α = 1.8, β = 0, δ = 0.1 α = 1.1, β = 10, δ = 0.2


Sampling from the Boltzmann Distribution

A ‘Simplified’ Initial Distribution

Define the density π0(X ) ∝ exp(−γ0U0(X )) with

U0(X ) =M∑j=1

{κ exp(Xj)︸︷︷︸Running cost

−(δ +

1

M

M∑j=1

Oi

)Xj︸︷︷︸

Equal net utility

},

for some γ0 < γ. We can sample from π0 exactly and easily.

Bridging Distributions

Define a temperature schedule 0 = t0 < t1 < · · · < tT = 1 with bridging distributions

πj =1

Zjπ1−tj0 π

tj∞, where Zj :=

∫πjdX , for j = 1, . . . ,T .


Bridging distributions

Figure: {πj}T=9j=1 from left to right, then top to bottom.


The London Retail System: Invariant Distribution

Figure: A sample from the invariant distribution showing behaviour in the limit t →∞ (most stableconfigurations). The largest zone is Kensington with 12% of total floorspace and the West Endcontinues to be significant with 5% of total floorspace.


A Statistical Model of Urban Retail Structure


Hierarchical Modelling

Given observation data Y = (Y1, . . . ,YM), of log-sizes, infer the parameter valuesθ = (α, β)T ∈ R2

+ and corresponding latent variables X ∈ RM .

Assumption (Data generating process)

Assume that each observation Y1, . . . ,YM is an independent and identical realization of thefollowing hierarchical model:

Y1, . . . ,YM |X , σ ∼ N (X , σ2I ),

X |θ ∼ π(X |θ) ∝ exp(−U(X ; θ)),

θ ∼ π(θ).


Joint Posterior Distribution

The joint posterior is given by

π(X , θ|Y ) =π(Y |X , θ)π(X , θ)

π(Y ), π(Y ) =

∫π(Y |X , θ)π(X , θ)dXdθ.

We have a hierarchical prior given by

π(X , θ) =π(θ) exp(−U(X ; θ))

Z (θ), Z (θ) =

∫exp(−U(X ; θ))dX .

The joint posterior is doubly-intractable

π(X , θ|Y ) =π(Y |X , θ) exp(−U(X ; θ))π(θ)

π(Y )Z (θ).


Russian Roulette

Figure: On Russian Roulette Estimates for Bayesian Inference with Doubly-Intractable Likelihoods. AMLyne, M Girolami, Y Atchade, H Strathmann, D Simpson. Statistical Science, 30 (4), 443-467


Metropolis-within-Gibbs with Block Updates

We are interested in low-order summary statistics of the form

EX ,θ|Y [h(X , θ)] =

∫h(X , θ)π(X , θ|Y )dXdθ.

X and θ are highly coupled, so we use Metropolis-within-Gibbs with block updates.

X -update. Propose X ′ ∼ QX and accept with probability

min

{1,π(Y |X ′, θ) exp(−U(X ′; θ)) q(X |X ′)π(Y |X , θ) exp(−U(X ; θ)) q(X ′|X )

}.

θ-update. Propose θ′ ∼ Qθ and accept with probability

min

{1,π(Y |X , θ′)Z (θ) exp(−U(X ; θ′)) q(θ|θ′)π(Y |X , θ)Z (θ′) exp(−U(X ; θ)) q(θ′|θ)

}.

Unfortunately the ratio Z (θ)/Z (θ′) ratio is intractable!


Unbiased Estimates of the Partition Function

We can use the Pseudo-Marginal MCMC framework if we have an unbiased, positiveestimate of π(X |θ), denoted π(X |θ, u), satisfying

π(X |θ) =

∫π(X |θ, u)π(u|θ)du.

The Forward Coupling estimator (FCE)1 gives an unbiased estimate of 1/Z :

E [S ] = 1/Z .

The idea is to find two sequences of consistent estimators {V(i)}, {V(i)}, each with thesame distribution, such that V(i) and V(i−1) are“coupled”.

1Markov Chain Truncation for Doubly-Intractable Inference, C. Wei, and I. Murray, AISTATS (2016)Mark Girolami (Imperial, ATI) Stochastic formulation of SIM 03-07-2017 33 / 42

Unbiased Estimates of the Partition Function

Requires N estimates of 1/Z using path sampling e.g. annealed importance sampling orthermodynamic integration, for a random stopping time N.

Coupling between V(i) and V(i−1) is introduced with a Markov chain that shares randomnumbers.

Variance reduction technique.

Then the unbiased estimate is given by

S := V(0) +N∑i=1

V(i) − V(i−1)

Pr(N ≥ i).


The Signed Measure Problem

S can be negative when V(i) < V(i−1) for many i . This is known as the ‘sign problem’.

Rejecting when S is negative would introduce a bias.

Instead, we note that

E[h(X , θ)] =1

π(Y )

∫h(X , θ)π(Y |X , θ)π(X |θ, u)π(θ)π(u)dudθdX ,

=

∫h(X , θ)σ(X |θ, u)π(X , θ, u|Y )dudθdX∫

σ(X |θ, u)π(X , θ, u|Y )dudθdX,

where σ is the sign function and we have defined

π(X , θ, u|Y ) =π(Y |X , θ)|π(X |θ, u)|π(θ)π(u)∫

π(Y |X , θ)|π(X |θ, u)|π(θ)π(u)dudθdX.


Pseudo-Marginal Markov Chain

We can sample from π(X , θ, u|Y ) using Metropolis-within-Gibbs with block updates.

X -update. Propose X ′ ∼ QX and accept with probability

min

{1,π(Y |X ′, θ) exp(−U(X ′; θ)) q(X |X ′)π(Y |X , θ) exp(−U(X ; θ)) q(X ′|X )

}.

(θ, u)-update. Propose (θ′, u′) ∼ Qθ,u and accept with probability

min

{1,π(Y |X , θ′)|S(θ, u)| exp(−U(X ; θ′))π(θ′)π(u′) q(θ|θ′)π(Y |X , θ)|S(θ′, u′)| exp(−U(X ; θ))π(θ)π(u)q(θ′|θ)

}.

Posterior expectations are estimated using

EX ,θ|Y [h(X , θ)] = limN→∞

∑Ni=1 h(Xi , θi )σ(Xi |θi , ui )∑N

i=1 σ(Xi |θi , ui ).


Illustration: Airports in England

Figure: Left: Trace plots of α and β. 10.1% of the σ values are negative. The posterior mean of α andβ are 0.97 and 0.94, respectively, indicating a moderate attraction towards to larger but fairly localairports. Right: 5th and 95th percentiles of the latent states X . There tends to be more variability inlarger airports such as Heathrow and Manchester.


Illustration: Retail Centres in London

Figure: LVisualization of the posterior-marginals for the latent variables {xj}Mj=1 over a map of London.The outer/inner rings (red) show the posterior mean of the sizes, w(xj) = exp(xj), plus/minus twostandard deviations. We show the residential areas (blue) with magnitude corresponding to number ofresidents. The retail sites and residential areas have been plotted at their latitude-longitudeco-ordinates.


Conclusions and Outlook


Further work and extensions

Investigation of new data assimilation methodologies to calibrate models to data availableat different scales. For example:

Population data;Cost matrix; orTime dependent parameters.

Deployment of new methodology to a global problem to provide new insights into urbanretail structure.

Improved mechanistic models of urban and regional phenomena.

Melding of data and models takes us beyond data analytics.


Summary

We have revisited the Harris and Wilson model to formally account for uncertainties inthe modelling process.

A stochastic extension of the model, under certain conditions, may be expressed as aLangevin diffusion.

The invariant distribution provides new insights into urban and regional dynamics, andsuggests the introduction of a new parameter in the model.

We presented a Bayesian hierarchical model for urban and regional systems. We havedemonstrated our approach using an example of airports in England and retail in London.


Acknowledgment

EPSRC Programme Grant - EP/P020720/1

Inference, Computation, and Numerics for Insights into Cities - ICONIC




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