9th Annual North American Mee?ngs of the Regional Science Associa?on Interna?onal NARSC – RSAI November 7-‐10, 2012, ODawa, Canada Title: Entropic varia,ons of urban dynamics at different spa,o-‐temporal scales: geocomputa,onal perspec,ves Authors: Didier G. Leibovici (corresponding author) Nick Malleson & Mark Birkin Affilia?on: University of Leeds, UK Conference ID: P16142 Abstract: Popula,on movements monitoring and their spa,al paPern evolu,ons are fundamental components for urban management and policy decision-‐making. Societal issues such as health, transport or crime, to name a few, may lead to explore, understand and model the urban dynamics of popula,on characteris,cs at specific scales. Nonetheless, it is oUen the case that the models at different scales (dependent on the planning horizon) will complement each other within a bigger picture [1]. Spa,al microsimula,on can be more suited to a spa,al analysis focusing on evolu,ons of local to medium areal units on a yearly basis over a few decades whilst Agent Based Modelling may be able to model individual behaviours with intra-‐day moves [2]. These geosimula,on methods have nonetheless proven to be flexible enough, even to be combined for hybrid approaches (see Birkin and Wu in [2]). Using and genera,ng data at different aggregated/dis-‐aggregated scales [3], it is therefore desired to be able to analyse and compare the spa,o-‐temporal varia,ons simulated at a range of scale in order to bePer understand the model behaviour, its emerging proper,es in an aPempt to validate the results of the geosimula,on. The paper proposes to use mul,variate exploratory geocomputa,onal sta,s,cal mining in order to extract the emerging features of the spa,al and dynamic simula,ons. Mul,way methods [4] are adapted to allowing nested-‐inter-‐scale analysis, and methods based on the k-‐spa,al entropy framework for point processes [5][6], extended to take into account the various geometries, are used in this spa,o-‐temporal context to explore the entropic varia,ons in geographic and ,me scales. Besides the discrepancy of the modelling process, the uncertainty aPached to each data sources and their adequacy to the model along with the mul,ple run varia,ons, influences the spa,o-‐temporal varia,ons simulated. The challenge of integra,ng these uncertain,es within the k-‐spa,al entropy exploratory methods will be also discussed. keywords: geosimula,on, urban dynamics, popula,on microsimula,on, spa,al entropy, scales, spa,o-‐temporal data, exploratory methods, goecomputa,onal sta,s,cs [1] S,llwell, J and Clarke, M (2011) Popula(on Dynamics and Projec(on Methods. Springer [2] Heppenstall, A (2011) Agent-‐Based Models of Geographical Systems. Springer [3] Birkin, M Malleson, N Hudson-‐Smith, A Gray, S and Milton, R (2011) Calibra,on of a spa,al simula,on model with volunteered geographical informa,on. Interna,onal Journal of Geographical Informa,on Science, 25(8): 1221–1239 [4] Leibovici, D.G (2010) Spa,o-‐temporal Mul,way Decomposi,on using Principal Tensor Analysis on k-‐modes: the R package PTAk. Journal of Sta,s,cal SoUware, 34(10): 1-‐34 [5] Leibovici, D.G (2009) Defining Spa,al Entropy from Mul,variate Distribu,ons of Co-‐Occurrences. COSIT'09 Conference On Spa,al Informa,on Theory, Aber Wrac'h, France, September 21-‐25, 2009,LNCS: Volume 5756/2009, 392-‐404 [6] Leibovici, D.G Bas,n, L and Jackson, M (2011) Higher Order Cooccurrences in Point PaPern Analysis and Decision Tree Clustering. Computers & Geosciences, 37(3): 382-‐389
NARSC – RSAI November 7-‐10, 2012
Entropic varia?ons of urban dynamics at different spa?o-‐ temporal scales: geocomputa?onal perspec?ves Didier G. Leibovici & Mark Birkin University of Leeds, UK
geosTapial datA anaLysIS and siMulA,oN http://www.geotalisman.org/!
NARSC – RSAI November 7-‐10, 2012
geosimula,on & urban dynamics • purposes because data availability (scale, privacy) and forecas,ng demographic characteris,cs, behaviour models … “what if?” policy scenario
• spa,al support and models Cellular automata, microsimula,on, Agent-‐Based modelling
Birkin & Wu (2009) NARSC – RSAI November 7-‐10, 2012
MoSes models & our data example • hybrid models • microsimula,on dynamic evolu,ons
Wu, Birkin & Rees (2011)
NARSC – RSAI November 7-‐10, 2012
MoSes models & our data example census data 2001 + microsimula,on dynamic evolu,on SARS is “downscaled” to households in Leeds area (census) 700 000 individuals
NARSC – RSAI November 7-‐10, 2012
MoSes models & our data example • data at OA levels regioning 2439 spa,al units ⊂ LSOA 476 spa,al units ⊂MSOA 108 spa,al units ⊂ Wards 33 spa,al units 1
2
5
10
kilometers
1200
1000
800
600
400
200 110
Nb of popula,on (OA)
% to the uniform (Ward)
NARSC – RSAI November 7-‐10, 2012
Which entropic varia,ons? • system complexity, uncertainty, increasing randomness • paPern in the data vs uniformity • distribu,onal paPern, spa,al paPern, spa,o-‐temporal paPern
• geosimula,on results, data mining, variables associa,ons • dominant “traits”, rare events vs uncertainty • scale effects NARSC – RSAI November 7-‐10, 2012
Shannon entropy • distribu,onal globally, or, x spa,al support (Regions) R
“dec
si ( o p om
” m e r eo h t n o
*
*
*: H()*=1/log(nclasses)H() NARSC – RSAI November 7-‐10, 2012
*
Shannon entropy • distribu,onal globally, or, x spa,al support (Regions) R
local within entropies NARSC – RSAI November 7-‐10, 2012
k-‐spa,al entropy • co-‐occurrences distribu,on (spa,al distribu,on) (ababab)
(random)
(BAB)
(AB2)
…… distances on centroids ……..
NARSC – RSAI November 7-‐10, 2012
96 94 92 90
100 * self k-spatial entropy
98
k-‐spa,al entropy (1class /r)
2000
4000
6000
8000
10000
co-occurrence distance
1
2
5
10
kilometers
1200
1000
)
800
600
400
Figure$8:)FCAk)at)OA)levels)of)the)Contingency)Table:)2439)7)39;)the)variables)representing) 39)categories)evaluated)on)2439)OAs)for)7)time)period)–)(top)panel):)one)particular)tensor) within) the) decomposition) of) the) complete) lack) of) independence) in) the) 34way) contingency) table) ;) the) scatter) plot) of) the) component) of) the) years) is) overlaid) with) simply) the) zero) as) dotted) green) line.) –) (bottom) panel):) map) of) the) difference) in) population) over) the) full) period.(the)Ward)map)is)overlaid)with)their)number).))) ) The) simple) difference) of) population) counts) along) the) period) of) 30) years) is) shown) on) the) lower)panel)and)the)maps)are)in)agreement.)Brown)parts)of)the)top)map)are)associated)with) y01) and) y06,) corresponding) to) the) green) parts) of) the) difference) map) (decrease) in) population),) dark) blue) correlated) with) last) years) y26,) y31) and) corresponding) to) the) pale) beige)of)the)bottom)map)(increase)in)population).)) )
200 110
!
Popula,on counts (OA) NARSC – RSAI November 7-‐10, 2012
12!
k-‐spa,al entropy
(all classes /r)
• co-‐occurrences distribu,on (spa,al distribu,on) 1
OA profiles: eth: white
OA profiles: eth: Asian
0.0652173913043478 0.06
0.9
0.05 0.8
0.04 0.7
0.03
0.6
0.02
0.5
0.01
0.4 0.38
0 0.0470016207455429 OA profiles: eth: black
0.04
0.03
0.02
0.01
0
0.026 0.025 OA profiles: eth: other 0.02 0.015 0.01 0.005 0
NARSC – RSAI November 7-‐10, 2012
W
10
0
s
H
H
s H
tl e p
l t e p
l et p
y21
y26
y31
H tl e p
H tl
H tl
e p
e p
tl e p
y01
y06
y11
y16
c a
a c
s
s
H
H
H
s H
tl e p
l t e p
l te p
et p
y16
y21
y26
y31
H tl e p
H tl e p
tl e p
y01
y06
y11
x c a h
x c a h
x c h a
x h c a
x h
h x
c a
a c
s
s
s
70
s
25
s H
s
x c a h
55
s
100
h x
85
x h
s s
40
x h c a
s H
s
25
s
10
s
25
s
x c h a
l
10
55
40
s
h
x c a h
0
a c
x c a h
100 x k-sptial Entropy OA
c a
x c a
85
a
x h
4000
70
a
x h
55
x h c
40
x c h
0
x c a h
70
h
x c a h
100 x k-sptial Entropy OA
x c a
3000 100
2000
85
100
k-‐spa,al entropy ( all classes /r) W
H
H
H
H
H
tl e p
l t e p
l et p
l et p
y16
y21
y26
y31
H tl e p
H tl e p
tl e p
y01
y06
y11
NARSC – RSAI November 7-‐10, 2012
s
k-‐spa,al entropy ( all classes /r) W eth llt pro
hea soc trw
age car hlt sex a
e e
l
e e
l
e p
e p
l
l
H p t
t
t
H
H
t
t
s H s H
t
s s
pl y06
y11
y16
y21
y26
y31
y01
y06
eth llt pro
y11
y16
y21
y26
y31
y01
e
p e
el
e l
l
s
t
Ht
H s H
t
s s
p l y01
y11
y16
y21
y26
y31
y01
y06
y11
y16
y11
a
s y21
a
a
y16
y21
x c
y26
h
c c
x c
a
c
y31
a
a a c
h x
x
h
x a
a
s
st
h
y26
y31
y01
NARSC – RSAI November 7-‐10, 2012
h
h
h
x
t
H y06
x c
y06
H
t
a
age car hlt sex
H t
xc
x
h
s
H
p
a
h
s
100 x k-sptial Entropy OA
p
el
100 x k-sptial Entropy OA
4000m
p
h
h
a x
h
el
p
x a
hea soc trw
e l p
c
c
t
s
H
y01
h
xc
100 x k-sptial Entropy OA
2000m
100 x k-sptial Entropy OA
p
h
c
l
pl
e p
H
cx x c
y06
y11
y16
y21
y26
y31
spa,ally discriminant-‐ra,o entropy • distances between classes (spa,al distribu,on)
(ababab)
(random)
(BAB)
(AB2)
…… distances on centroids ……..
NARSC – RSAI November 7-‐10, 2012
spa,ally discriminant-‐ra,o entropy W
NARSC – RSAI November 7-‐10, 2012
9th$Annual$North$American$Meetings$of$the$Regional$Science$Association$International$$ e)decomposition)in)2001)of)the)entropy)Shannon)and)self4k4spatial)entropy)for) NARSC$–$RSAI$November$7=10,$2012,$Ottawa,$Canada$ ade)at)OAs)level)using)as)constraining)zoning)system)the)Ward)map)(with)a) ! 99)OAs)within)each)ward)) Table$2.)Scale)decomposition)in)2001)of)the)entropy)Shannon)and)self4k4spatial)entropy)for) the)Social)grade)at)OAs)level)using)as)constraining)zoning)system)the)Ward)map)(with)a) pro kEnt 3000 OA/Ward range)of)56499)OAs)within)each)ward)) 95
85
85
100 x self-k-spatial Zoning Entropy OA / Z= Ward
75 65 ShanZ
kZ
kZ.2km
kZ.3km
kZ.4km
kZ.5km
kZ.6km
Zoning and collocation
75 65
0.6
55 45
0.4
35 25
0.3 0.2 0.1
!
ShanZ
kZ
kZ.2km
95
H(Z)
0.3 0.2 H(C,Z)
Entropic Profiles
H(C/Z)
0.12 0.1
kZ.6km
!
0.08
85 75 55
65
95
0.06
45 35
0.1 0.0565668005373745
H(Z/C)
0.14
0.0048
25
25 15
H(C)
0.4
kZ.5km
0.16
eth kEnt 3000 OA/Ward
0.02
15
0.5
ShanZ kZ kZ.2km kZ.3km kZ.4km kZ.5km kZ.6km
kZ.4km
H(C) H(Z) H(C,Z) H(C/Z) H(Z/C)
0.04
0 5
35
C= soc
100 x self-k-spatial Zoning Entropy OA / Z= Ward
0.6
kZ.3km
0.00081
C= soc
Zoning and collocation
85 75 45
65 55
0.7
0 5
100 x self-k-spatial Zoning Entropy OA / Z= Ward
0.87 0.8
0.001
0.0326716079089894
See R code …and NARSC paper … liPle errors soc kEnt 3000 OA/Ward
0.0015
0.5
0 5
15
H(C) H(Z) H(C,Z) H(C/Z) H(Z/C)
Z W
0.75 0.7
15
55 45 35 25
C= soc
0.0025129408532653 0.0025
0.002
soc kEnt 3000 OA/Ward 2016
0 5
100 x self-k-spatial Zoning Entropy OA / Z= Ward
95
K-‐spa,al or discriminant-‐ra,o entropy and Zoning
C= soc ShanZ kZ kZ.2km kZ.3km kZ.4km kZ.5km kZ.6km
) H(C) H(Z) H(C,Z) H(C/Z) H(Z/C) ale)decomposition)in)2001)of)the)entropy)Shannon)and)self4k4spatial)entropy)for) Entropic Profiles ade)at)OAs)level)using)as)constraining)zoning)system)the)Ward)map)(Table)2)) ) Figure$9:)Scale)decomposition)in)2001)of)the)entropy)Shannon)and)self4k4spatial)entropy)for) NARSC – RSAI November 7-‐10, 2012 st)a)few)results)no)much)which)would)need)to)be)commented)and)particularly) the)Social)grade)at)OAs)level)using)as)constraining)zoning)system)the)Ward)map)(Table)2))
geocomputa,onal aspects
• • • •
areal data vs point data Weigh,ng distances for different geometries using Zoning as co-‐occurrence vicinity distance of co-‐occurrence within Zoning
• computa,onal costs • R package kOO • Zoning or upper scale (R’) in the decomposi,on
NARSC – RSAI November 7-‐10, 2012
first aPempt of a summary
NARSC – RSAI November 7-‐10, 2012
•
distributions)of)the)vari entropy) as) in) (1)) or) aPempt of a summary + other perspec,ves configuration,)proximiti is)of)interest.)Therefore probabilis,c framework useful (decomposi,on things)is)needed.)Notice theorem)
-‐but distance based measures surroga,ng a distribu,on can be useful )which)is)focusing)on)re -‐easier also for quality assessments the)variable)distribution (3),) • role of the density: BaPy’s entropy formula,on Δ!)is)the)size)of)the •
to) discard) the) areal) zo support!Δ!)can) be) tran uncertainty / quality of observed / simulated ddistribution)or)the)area4 ata
-‐error propaga,on -‐sampling “H1” (the envelope represen,ng H0)
• R, R’, Zoning etc… -‐,me as an extra crossing factor -‐as an “extended” spa,al selec,on NARSC – RSAI November 7-‐10, 2012
where)Δ! =
!" ! ∥ !! =
! Δ!
is)t
! !! log
from)the)prior)distribut brought)by)(or)needed) from)! .)Therefore)the) expressed)by)the)refere over)the)spatial)units)fo ) With)more)than)one)se
