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Attempt at a Determination of the Role of Legal Measures in the Orientation of Urban Growth
ATTEMPT AT A DETERMINATION OF THE ROLE OF LEGAL MEASURES IN THE ORIENTATION OF URBAN GROWTH* J.P. Cheylan* F. Desbordes-Cheylan* and L. Farinas Del Cerro** *Groupe d'Etudes pour l'Application M.ethodes Sci~ntlfiques a l'Arc~itecture et Z'AppZica#on de M.€thodes l'Urbanisme. Pedagoglque de MarseIlle - Lummy - 13009 MarseIlle - Franc. e Z'Urbanisme. Unite P€dagoglque **Laboratoire d'Informatique pour les Zes Sciences de l'Homme. Z'Homme. C.R.N.S. 31, Chemm Joseph Aiguier - 13274 Marseille Cedex 2 - France
ABSTRACT By means of formalized observations, which enable us to characterize a certain type of urban growth, we wish to determine precisely the true role played by some orientative measures in urban growth. The method of analysis employed to determine this role is based on the utilization of stochastic processes.
1 - Introduction The results of the research work described in this paper constitute a methodological step in the development of a more general program. This leads to the implementation of a process of analysis of urban growth mechanisms, which is aimed at producing an explicit understanding of these phenomena. The elaboration of formal and mathematical tools assisting in this step has been effected experimentally for a field of study - the community of Martigues (in the south of France).
In Martigues, these plans have not yet been adopted. The community however disposes of a plan (Preliminary to the POS), for the land occupation coefficients, which has been in operation since 1970, and which conforms to the same definition. These measures are put together to form an integral part of a municipal strategy, for the management of town space involving amongst other elements the previous state of urbanisation, the different opinions of the town council concerning the orientation of urban growth and the pressures of social groups and agents.
2 - Peri - urban growth
The measures may be expressed in two forms: forms : (1) A spatial ised form for the orientations, wh ich have.been selected, (2) A tactical form for concealed or true aims, which is analysable in a certain number of texts.
We have studied the role played by orientative legal measures in the peri-urban growth in Martigues, a medium-sized town (40,000 inhabitants) in the south of France, which is undergoing rapid expansion. This is partly due to its own dynamism promoted by the industrial employment in the petrochemical and associated industries, and to its role of regional center and also partly due to the influence of an industrial complex of national importance in Fos, the anticipated effects of which have been overestimated. Martigues is sufficiently distant from Marseille, the regional capital, so as not to be influenced. Furthermore, it is governed by a municipality asserting a policy of regional and urban planning. In France, plans for the occupation of land «Plan d'occupation des sols» - (POS), indicate the building rules appli cable in every part of the town area. These plans and the accompanying set of rules define, for homogenous districts, the minimum building surface areas, the maximum building density and for certain areas the restrictions on the authorized (e. g. industrial zones, regions of urban usuages of land (e. priority) .
Peri-urban growth, in the region studied, is marked on the one hand by the creation of estates in rural areas, occupied by town-dwellers and on the other by the appearance, in agricultural spaces, of individual constructions other than estates occupied by town-dwellers. This mode of growth spreads throughout the agricultural space without creating any urban complexes. This phenomenon shall be called «scattering». «scattering» . There exist several developmental modes of periurban growth, which may be characterized by : the nature of construction observed, their temporal mode of appearance, their previous agricultural usuages before construction. Moreover, these characterisations bear a relation with significant regularity to both the preferential localisation of certain types of construction and at the same time to the socio-professional characteristics of the people seeking planning permission. The municipal ity has expressed, by its discourse and by means of plans which are a spatial representation of classical orientation, a desire to bring a halt to the
* This research is supported by D.G.R.S. T., under contract number 75 7 0459
37
38
J. P. Chelyan
«scattering» mode of development. with a view to channelling it towards a zone of development based around two existing villages and hence combatting spatial segregation. measures. we To determine the role played by legal measures, are led to «compare» the observed evolution after implementation of the orientative measures for the proposed development with that of before.
3 - Steps and methods involved The method of analysis for urban growth elaborated has for objective the production of an understanding of growth mechanisms and conditions. We would like to show in particular the temporal interactions between the various phenomena studied and variations of the elements assumed to be explanatoI n order to constitute this ry of these phenomena. In data will be comprehension of urban phenomena. made up from observations on actual situations. The steps in this comprehension process may be follows : described as follows: problematic : study of the me(a) definition of a problematic: chanisms of peri - urban growth and in particular mechanisms for the appearance of buildings relating to the previous agricultural land usuage. study : (b) constitution of a field of study: i.e. the community of - geographical make-up Le. Martigues, Martigues. - constitution of the objects required for the i.e. appl ications for planning permisanalysis. Le. sion. representations of buildings on aerial photographs. (c) constitution of a formal representation of the field of study by the regular description of the objects involved with the help of a descriptive code. (d) transformation of this field of study to an experimental field (the experimental field is based on a stock of computerised data containing the representations of the phenomenon studied on which may be performed any calculation procedures. const itution of an interpreting system origi(e) constitution appl icable nating from the problematic and applicable field . This interpreting to the experimental field. system takes the form of a group of three factors : legal framework. natural milieu. relative localisation of the objects. It is the interdependance of these factors which enables in particular their temporal interactions to be expressed. (f) definition of mathematical and data-processing tools appropriate to the nature of information available and to the adopted problematic (for MAR KOV process is used to the moment. the MARKOV enable us to translate the evolution of construction in terms of the states of the explanatory
al. et al
factors.) ,• (g) organisation of hypotheses which are structured in terms of the explanatory factors and decomposable into groups of sub-hypotheses which are interdependant and may be handled by the tools created (we are working presently on a group of sub-hypotheses aimed at isolating. in space . the places. moments and the community space. modes of determination of construction attriframework . Another group butable to the legal framework. of sub - hypotheses aims at discovering the relationships between certain types of constructions and the previous agricultural usuage of the land - parcel on which they stand. (h) the interpretation of the results: this interpretation is carried out by direct reference to the relationship between the hypothesis considered ana the corresponding tool used in the implehypothesis. mentation of this hypothesis. For an example, we shall develop a group of subhypotheses originating from the following hypothesis : if the orientative measures governing consthesis: truction in operation since a given date. play a role in the appearance of consctructions. then different evolutionary mechanisms must be apparent for both observation periods, periods. before and after this date If the evolutions for latter observation period conform to those wh ich the orientative measures aimed at producing then it should be possible to recognise differences in mechanism. implied by the measures taken. by means of the differences in evolution : found for the two periods.
4 - Foreword on stationary Markov Processes with a finite set of states A stationary Markov Process with a finite number E = ( A. B. C•... C •.. . } is defined by a semigroup of of states. E". matrices. (pt: (p t : t ~O J which is in turn detransition matrices, fined by : 00 Pt,", Pt $ exp tG [. ~ Gn G n (1) n:O n=O n! (A.B)1 is a matrix satisfying the condicondi where G = [G (A.B)] tions : G A:P B if A:F G (A.B) ;, " 0 0 and G (A.B) = 0 for A to E (2)
L.
L
=
B G is called the infinitesimal operator of the semigroup : UOJ. UO). (pt It must be noted that: -L pt (A. B) :: G (A. B) for B ,;aA lim ...L ~A (3) t~ 0 tt 0 (A.B) dt describes the evolution of the Thus. G (A,B) dt) . process in the time interval (0. dt).
t"
=
The phenomenological description of the Markov follows : process with infinitesimal operator. G. is as follows: Knowing that the process is in the state. A. at the instant O. it follows that the distribution of its future sojourn time in A is a negative exponential of parameter G (A, (A. A). A) . The distribution for the following state. s. is discrete and defined by :
THE ORIENTATION OF URBAN GROWTH Prob
{S: B)" {s: Bl"'
39
- G (A, (A. B) / G (AA), B 1: A A (4)
5 - Description of the model The evolution of land - usuage is investigated for a group of land - parcels situated in a given region. It is known that the laws governing the evolution of these usuages depend on a certain number of factors. F. G. H. which are defined for each parcel. and are considered as fixed variables varying with time. We wish to use Markov Processes in order to model these evolutions. This is only possible over a time interval during which constant . The land usuage process. X. these factors are constant. wi will11 therefore be obta iined ned by pool iing ng together the d istinct Markov processes. Xo Xo ' X1 ..... •.. .. X, defined for the
..... .....
intervals [to t 1] . (t [tg tg.r] respectively ]. (t11 ~1 where for each i (1 ~ i ~ g). ttii is the first instant for which at least one of the factors succeeding ttii _ 1 changes value in anyone parcel.
(1.
states. where S • (1. 2. 3 ... mt m) Let Sn be the set of states. 3... defines the set of possible usuages for each parcel and m is the number of parcels. Each process Xi will therefore be given by : n k Xi ~1 xi (5) where the ',,;c life are independant stationary Markov processes. The distribution of processes. depends only on th e factors for the parcel k during the the value of the interval [ti tMJ. t i'1).
xr
6 - Statistical inference problems There are essentially two problems of statistical inference relevant to our model: - to test whether a process is a stationary Markov process. hypothesiss that two stationary - to test the hypothesi Markov processes have the same parameters.
process. 6-1 - Test of the markovian character of the process. Let us consider an interval of time (0. t) during which the factors remain constant for a set of parcels I- { i 1 ~ •...• •. ... i k 1 •
1
{i
I . denote the subset of I formed by the parLet I. ime zero. zero . The cels whicH are in the j - th state at ttime well known distribution of sojourn times
i .,
C C{
-et!
r:r:~) ~)"-.)-et! "')
of these parcels in the j - th state is a negative expot. nential with unknown parameter. truncated at t. The Markov hypothesis can thus be verified by ng. for eac~ j. ~ test qf goodness of fit of perform iing. the observations C' C') C' I"" "'J t' r.' . 2 distribution . ~ to this distribution. test ~all j,all be used to achieve 1) . this (see appendix 1).
-X
6-2 - Test of homogeneity Let us consider two independant stationary Markov processes which have the same finite set of states (1.2 ... .. m).For 1 ~ j '" 4it m let
yi:[yi1 .y~ ... .. y~} yi} and Zi. {z~ • z£ .... z~.} denote yi:[yi 1 samples. of sojou1n sOjourn times in the state j with filed sizes nj and m .ar .arising ising from the corresponding processes. and likewise'
Li=fli li ji _ li li .,. li } ll'' 2 'I 'i-1 ~ " m t
exit from j. where:
o number of jumps from state j If n first sample. f If: (6) i
to state i in the
a
and
number of jumps from state j to state i in the ki i . kl. =m. (7) second sample. L.. k'. i I , We shall divide the problem of testing whether or not these two processes have the same law. into parts : two parts: (a) verification. for 1 ~ j ~ m. that the parameters (al of the exponential distributions. from wh ich originate. are equal. To solve this ..; and problem we employ the Neyman-Pearson statistics (see appendix 2).
!
(b)
7f 7!
verific~tion. for 1 6 j ~ m. that the samples verific~tion. and K' belong to the same population. This may be achieved. via classical X 2 tests. tests. Here again the distinct statistics are independant and can be summed to give an overall test of homogeneity . homogeneity.
t!.
7 - Example of a hypothesis Attention must be drawn firstly to the nature and role of hypotheses used in cognitive processes. It would seem important to mention that the lack of a global theory for urban growth operating as a network of explicit concepts. necessitates strategic choices. choices. These choices consist of postulating that any understanding of urban growth mechanisms may only be arrived at. if it is studied for a group of ach ievements of mechanism. and then only if this mechanism is this mechanism, itself of a sufficiently simple nature to be described. in theory . These considerations lead spite of this lack of theory. us to the implementation of apparently simple hypotheses but whose importance may be found within hypotheses concerning several growth modes.
J,P. Chelyan et al aZ
40
A
·
·~~............!i.. .• . ----------~ ~• B: B: •~ ·..._.... .. ,.-, -,.
A
Fig. 1 - pas regions Fig. 1·
is is is is
to is the date of the first observation t pas is the date of implementation of the pas the set of parcels homogenous w. w. r. t. the pas in the region A, not «built on» On!) at the instant tpos On!) at the instant tpos the set of parcels homogenous w. r. t. the pas in the region B, not «built on» the set of parcels homogenous w. r. t. the pas in the region A, not «built on» at the instant to On!) at the instant to the set of parcels homogenous w. w. r. t. the pas in the region B, not «built on»
PA
T
PB
T
PAp same laws as PB? p P
10-
L-
p'---
TRUE
NOT
TRUE
The POS has no effect on the evolutionary mechanism
(1) PAp same laws as PA pl ?
1? (2) PB p same laws as PB ppl?
I
TRUE NOT
TRUE Comparison of (1) and / or (2) with the objectives arising from the POS (expressed in terms of evolutionary mechanism for each zone).
concerning : Differences within (1) and / or (2) concerning: - acceleration of evolutionary mechanisms, or - nature of the states represented
Comparison of these differences with the objectives arising from the POS (expressed in terms of evolutionary mechanisms for each zone)
Fig. 2 .- Relationship between
pas and evolution of land land·- usuage
PA p is the model for the evolution of land-usuage for A PB p is the model for the evolution of land-usuage for B ~ PA p,;s the model for the evolution of land-usuage for Apt PB ptis the model for the evolution of land-usuage for B pt
THE ORIENTATION OF URBAN GROWTH
The hypothesis considered in(3)will be defined fo~ the following elements: - A, the space defined for a homogenous zone of the pas, POS, - B, the space defined for another homogenous zone of the pas, POS, - PA 1 .,. a mechanism for the diffusion of construction withi: withit the time interval (t o t pas l.), in the agriculpos tural space (<
41
The utilisation of the Markov Process presented, consisted of estimating the changes between two successive evolutionary periods in order to test for any effect of an event which divides these two periods. The same tools may be used to test for the similarities or differences between contemporary processes dealing with different classes of objects. It may therefore be verified that either different constructions are brought about by different mechanisms or that different evolutions occur after different types of starting states. From the point of view of comprehension, the analysis of the role played by orientative legal measures in urban growth has enabled us to show the nature and complexity of the relationships between these measures and the evolutions actually observed. In the example studied, it appears that the intentions contained in the official documents are only partially fulfilled. In from · their I n cases where these intentions differ from· preceding mechanisms, only the incitements to develop construction are fully achieved (the attempts at stopping an existing development running up against the inertia of this development). From the analysis point of view, certain of the difficulties, linked on the one hand to the development reasoni ng and on the other to of research on scientific reasoning the constitution of a stock of theoretical knowledge of urban phenomena will only be raised progressively by various and not always conscious contributions. We believe however that at present it is possible to contribute a few elements whose explanatory content may be relatively limited but whose conditions of explicit and validable constitution indicate however that they may be included in a cumulative process of comprehension acquired. acquired . This process might go towards the heuristic elaboration of a theoretical corpus of urban knowledge.
8 - Conclusion
Acknowledgments
The study presented in this paper may be interpreted as being the development of tools in an experimentation phase aimed at producing a validation. This validation draws on the one hand on the steps of analysis considered and on the the other on the mathematical tools required for this analysis and relating to the problems tackled. This work carried out on the community area of Martigues has enabled us to reach an understanding of the relationships existing between urban growth and the legal apparatus of its management.
We would like to express our thanks to Mr Fernandez de la Vega, for the helpful guidance in the completion of this work, and Mr J. D. Lee for the translation of the French text.
The use of Markov Processes as the tool for the representation of temporal phenomena has firstly been able to be carried out in accordance with the mathematical hypotheses implied by this type of tool and secondly has enabled the treatment of temporal data having complex structure.
J.P. Chelyan et aZ
42
Appendix 1.
References
We choose an integer p and a partition of the 11 ,12 ",, "" lp' lp, and put: put : interval (0, t) in p intervals 1"1
AMEDO D., GOLLEDGE R. G. G. - An introduction to scientific reasoning in geography. geography . J. Wiley and Sons, 1975.
X. :: ~ x.::;~
[n [ni-n i - nii (SF
l~i~p+l
,I
where: n is, for 1 ~ i i in the interval I ..
n ii (§) n (6) (0 ~
p, the number of observations lying
I
BILLlNGSLEY D., D., - Statistical inference for Markov Processes. I. M. M . S. statistical research monographs, The University of Chicago Press, 1961.
n p+ p.1 the number of parcels wh ich are still in state j at time t
"
p ... l, is the exp!cted value of ni ' n ii (<'),16 (<'), 1 ~ i ~ p1'"l, relatively to the estimated value ~ of the unknown parameter . parameter.
Xi is, under the null hypothesis, distributed as a chifreedom . square with n degrees of freedom. We can then sum the statistics corresponding to the various initial states i. j. 1 ~ j ~ m, since these statistics are clearly independent, to obtain again a"1? statistic with m, degrees of freedom which can be used as an overall test for the markovian character of process . the process.
Appendix 2. The Neyman-Pearson statistics is in this case: case :
(!')~max L . .E e i ~. e y v Zl Z'
S=2 \max L .. (') + max L
he v'
V
M . et Mme, FARINAS DEL CERRO L. CHEYLAN M. L. Essais de constitution de modeles expl icatifs partiels de I'evolution des utilisations des sols, march 1977 Bulletin . Ecole d'Architecture Marseille GAMSAU Bulletin. Luminy Luminy..
M ., 1975 - A computer DAJANI J., REINHARDT M., Comput . & Urban Soc. Soc. simulation of Urban Growth, Comput. vol 1 pp 159 - 168 - Pergamon Press. Press.
. (9) \ .(9)
where: -L. (e);..nlog (e);.-nlog
CHEYLAN J.P., P., DONATI P., J .P., CHIAVARI P., DESBORDES - CHEYLAN F. - Vers une explicitation et une formal isation des references spatiales et tempod' informations urbains. urbains. Commurelies dans les systemes d'informations Symposium : urban data managenication. VI European Symposium: ed . Dr G. G. Deprez, Liege - 18 - 22 april 1977. ment, ed.
DOOB - Stochastics Processes, Wiley New York, 1953
e (~)-1/.2:. (~)-1/'2:. y.--o't/ y.--o't/e
y/yll y/y" is the loglikelihood of the observations of the sample relatively to the negative exponential distribution Yi with parameter ~ ,truncated at tt..
L . - Power of some tests for FARINAS DEL CERRO L. exponential distribution against an alternative of mixappear) . ture (to appear).
- t is the total time ot observation - n the number of observation in the sample Y i falling into the interval (O,t) - n' the number of observations in the sample falling outside the interval (O,t)
,
Yi
,
( 9) have analogous exL. L. ((&) &) and L . " Zl yl vv zi Z Y Z pressions and are respectively the logl i kelohood of the sample Y yii and of the pooled sample V Vu\J Zi Zi
- the Neyman-Peason statistics S is asymptotically distributed as a chi-square with two degrees of freedom
GUENOCHE A. - Presentation d'un systeme de traitements documentaires et statistiques adapte au calcul en Sciences Humaines. Panorama de la Nouveaute Informatique en France - Tome 1 - pp 181 - 189, AFCET Nov 1966.
LAROUCHE P., 1965 - The simulation of residential growth in the Montreal region, Quebec, Quebec Depart ment of roads. roads. 1965
(X~)· SI EGE L S. S. - Nonparametric statistics for the behavioral nternational Student Edition, Mc Grawraj sciences, IInternational Hill - New York. 1956.
ABSTRACT By means of formalized observations, which enable us to characterize a certain type of urban growth, we wish to determine precisely the true role played by some orientative measures in urban growth. The method of analysis employed to determine this role is based on the utilization of stochastic processes.
1 - Introduction The results of the research work described in this paper constitute a methodological step in the development of a more general program. This leads to the implementation of a process of analysis of urban growth mechanisms, which is aimed at producing an explicit understanding of these phenomena. The elaboration of formal and mathematical tools assisting in this step has been effected experimentally for a field of study - the community of Martigues (in the south of France).
In Martigues, these plans have not yet been adopted. The community however disposes of a plan (Preliminary to the POS), for the land occupation coefficients, which has been in operation since 1970, and which conforms to the same definition. These measures are put together to form an integral part of a municipal strategy, for the management of town space involving amongst other elements the previous state of urbanisation, the different opinions of the town council concerning the orientation of urban growth and the pressures of social groups and agents.
2 - Peri - urban growth
The measures may be expressed in two forms: forms : (1) A spatial ised form for the orientations, wh ich have.been selected, (2) A tactical form for concealed or true aims, which is analysable in a certain number of texts.
We have studied the role played by orientative legal measures in the peri-urban growth in Martigues, a medium-sized town (40,000 inhabitants) in the south of France, which is undergoing rapid expansion. This is partly due to its own dynamism promoted by the industrial employment in the petrochemical and associated industries, and to its role of regional center and also partly due to the influence of an industrial complex of national importance in Fos, the anticipated effects of which have been overestimated. Martigues is sufficiently distant from Marseille, the regional capital, so as not to be influenced. Furthermore, it is governed by a municipality asserting a policy of regional and urban planning. In France, plans for the occupation of land «Plan d'occupation des sols» - (POS), indicate the building rules appli cable in every part of the town area. These plans and the accompanying set of rules define, for homogenous districts, the minimum building surface areas, the maximum building density and for certain areas the restrictions on the authorized (e. g. industrial zones, regions of urban usuages of land (e. priority) .
Peri-urban growth, in the region studied, is marked on the one hand by the creation of estates in rural areas, occupied by town-dwellers and on the other by the appearance, in agricultural spaces, of individual constructions other than estates occupied by town-dwellers. This mode of growth spreads throughout the agricultural space without creating any urban complexes. This phenomenon shall be called «scattering». «scattering» . There exist several developmental modes of periurban growth, which may be characterized by : the nature of construction observed, their temporal mode of appearance, their previous agricultural usuages before construction. Moreover, these characterisations bear a relation with significant regularity to both the preferential localisation of certain types of construction and at the same time to the socio-professional characteristics of the people seeking planning permission. The municipal ity has expressed, by its discourse and by means of plans which are a spatial representation of classical orientation, a desire to bring a halt to the
* This research is supported by D.G.R.S. T., under contract number 75 7 0459
37
38
J. P. Chelyan
«scattering» mode of development. with a view to channelling it towards a zone of development based around two existing villages and hence combatting spatial segregation. measures. we To determine the role played by legal measures, are led to «compare» the observed evolution after implementation of the orientative measures for the proposed development with that of before.
3 - Steps and methods involved The method of analysis for urban growth elaborated has for objective the production of an understanding of growth mechanisms and conditions. We would like to show in particular the temporal interactions between the various phenomena studied and variations of the elements assumed to be explanatoI n order to constitute this ry of these phenomena. In data will be comprehension of urban phenomena. made up from observations on actual situations. The steps in this comprehension process may be follows : described as follows: problematic : study of the me(a) definition of a problematic: chanisms of peri - urban growth and in particular mechanisms for the appearance of buildings relating to the previous agricultural land usuage. study : (b) constitution of a field of study: i.e. the community of - geographical make-up Le. Martigues, Martigues. - constitution of the objects required for the i.e. appl ications for planning permisanalysis. Le. sion. representations of buildings on aerial photographs. (c) constitution of a formal representation of the field of study by the regular description of the objects involved with the help of a descriptive code. (d) transformation of this field of study to an experimental field (the experimental field is based on a stock of computerised data containing the representations of the phenomenon studied on which may be performed any calculation procedures. const itution of an interpreting system origi(e) constitution appl icable nating from the problematic and applicable field . This interpreting to the experimental field. system takes the form of a group of three factors : legal framework. natural milieu. relative localisation of the objects. It is the interdependance of these factors which enables in particular their temporal interactions to be expressed. (f) definition of mathematical and data-processing tools appropriate to the nature of information available and to the adopted problematic (for MAR KOV process is used to the moment. the MARKOV enable us to translate the evolution of construction in terms of the states of the explanatory
al. et al
factors.) ,• (g) organisation of hypotheses which are structured in terms of the explanatory factors and decomposable into groups of sub-hypotheses which are interdependant and may be handled by the tools created (we are working presently on a group of sub-hypotheses aimed at isolating. in space . the places. moments and the community space. modes of determination of construction attriframework . Another group butable to the legal framework. of sub - hypotheses aims at discovering the relationships between certain types of constructions and the previous agricultural usuage of the land - parcel on which they stand. (h) the interpretation of the results: this interpretation is carried out by direct reference to the relationship between the hypothesis considered ana the corresponding tool used in the implehypothesis. mentation of this hypothesis. For an example, we shall develop a group of subhypotheses originating from the following hypothesis : if the orientative measures governing consthesis: truction in operation since a given date. play a role in the appearance of consctructions. then different evolutionary mechanisms must be apparent for both observation periods, periods. before and after this date If the evolutions for latter observation period conform to those wh ich the orientative measures aimed at producing then it should be possible to recognise differences in mechanism. implied by the measures taken. by means of the differences in evolution : found for the two periods.
4 - Foreword on stationary Markov Processes with a finite set of states A stationary Markov Process with a finite number E = ( A. B. C•... C •.. . } is defined by a semigroup of of states. E". matrices. (pt: (p t : t ~O J which is in turn detransition matrices, fined by : 00 Pt,", Pt $ exp tG [. ~ Gn G n (1) n:O n=O n! (A.B)1 is a matrix satisfying the condicondi where G = [G (A.B)] tions : G A:P B if A:F G (A.B) ;, " 0 0 and G (A.B) = 0 for A to E (2)
L.
L
=
B G is called the infinitesimal operator of the semigroup : UOJ. UO). (pt It must be noted that: -L pt (A. B) :: G (A. B) for B ,;aA lim ...L ~A (3) t~ 0 tt 0 (A.B) dt describes the evolution of the Thus. G (A,B) dt) . process in the time interval (0. dt).
t"
=
The phenomenological description of the Markov follows : process with infinitesimal operator. G. is as follows: Knowing that the process is in the state. A. at the instant O. it follows that the distribution of its future sojourn time in A is a negative exponential of parameter G (A, (A. A). A) . The distribution for the following state. s. is discrete and defined by :
THE ORIENTATION OF URBAN GROWTH Prob
{S: B)" {s: Bl"'
39
- G (A, (A. B) / G (AA), B 1: A A (4)
5 - Description of the model The evolution of land - usuage is investigated for a group of land - parcels situated in a given region. It is known that the laws governing the evolution of these usuages depend on a certain number of factors. F. G. H. which are defined for each parcel. and are considered as fixed variables varying with time. We wish to use Markov Processes in order to model these evolutions. This is only possible over a time interval during which constant . The land usuage process. X. these factors are constant. wi will11 therefore be obta iined ned by pool iing ng together the d istinct Markov processes. Xo Xo ' X1 ..... •.. .. X, defined for the
..... .....
intervals [to t 1] . (t [tg tg.r] respectively ]. (t11 ~1 where for each i (1 ~ i ~ g). ttii is the first instant for which at least one of the factors succeeding ttii _ 1 changes value in anyone parcel.
(1.
states. where S • (1. 2. 3 ... mt m) Let Sn be the set of states. 3... defines the set of possible usuages for each parcel and m is the number of parcels. Each process Xi will therefore be given by : n k Xi ~1 xi (5) where the ',,;c life are independant stationary Markov processes. The distribution of processes. depends only on th e factors for the parcel k during the the value of the interval [ti tMJ. t i'1).
xr
6 - Statistical inference problems There are essentially two problems of statistical inference relevant to our model: - to test whether a process is a stationary Markov process. hypothesiss that two stationary - to test the hypothesi Markov processes have the same parameters.
process. 6-1 - Test of the markovian character of the process. Let us consider an interval of time (0. t) during which the factors remain constant for a set of parcels I- { i 1 ~ •...• •. ... i k 1 •
1
{i
I . denote the subset of I formed by the parLet I. ime zero. zero . The cels whicH are in the j - th state at ttime well known distribution of sojourn times
i .,
C C{
-et!
r:r:~) ~)"-.)-et! "')
of these parcels in the j - th state is a negative expot. nential with unknown parameter. truncated at t. The Markov hypothesis can thus be verified by ng. for eac~ j. ~ test qf goodness of fit of perform iing. the observations C' C') C' I"" "'J t' r.' . 2 distribution . ~ to this distribution. test ~all j,all be used to achieve 1) . this (see appendix 1).
-X
6-2 - Test of homogeneity Let us consider two independant stationary Markov processes which have the same finite set of states (1.2 ... .. m).For 1 ~ j '" 4it m let
yi:[yi1 .y~ ... .. y~} yi} and Zi. {z~ • z£ .... z~.} denote yi:[yi 1 samples. of sojou1n sOjourn times in the state j with filed sizes nj and m .ar .arising ising from the corresponding processes. and likewise'
Li=fli li ji _ li li .,. li } ll'' 2 'I 'i-1 ~ " m t
exit from j. where:
o number of jumps from state j If n first sample. f If: (6) i
to state i in the
a
and
number of jumps from state j to state i in the ki i . kl. =m. (7) second sample. L.. k'. i I , We shall divide the problem of testing whether or not these two processes have the same law. into parts : two parts: (a) verification. for 1 ~ j ~ m. that the parameters (al of the exponential distributions. from wh ich originate. are equal. To solve this ..; and problem we employ the Neyman-Pearson statistics (see appendix 2).
!
(b)
7f 7!
verific~tion. for 1 6 j ~ m. that the samples verific~tion. and K' belong to the same population. This may be achieved. via classical X 2 tests. tests. Here again the distinct statistics are independant and can be summed to give an overall test of homogeneity . homogeneity.
t!.
7 - Example of a hypothesis Attention must be drawn firstly to the nature and role of hypotheses used in cognitive processes. It would seem important to mention that the lack of a global theory for urban growth operating as a network of explicit concepts. necessitates strategic choices. choices. These choices consist of postulating that any understanding of urban growth mechanisms may only be arrived at. if it is studied for a group of ach ievements of mechanism. and then only if this mechanism is this mechanism, itself of a sufficiently simple nature to be described. in theory . These considerations lead spite of this lack of theory. us to the implementation of apparently simple hypotheses but whose importance may be found within hypotheses concerning several growth modes.
J,P. Chelyan et al aZ
40
A
·
·~~............!i.. .• . ----------~ ~• B: B: •~ ·..._.... .. ,.-, -,.
A
Fig. 1 - pas regions Fig. 1·
is is is is
to is the date of the first observation t pas is the date of implementation of the pas the set of parcels homogenous w. w. r. t. the pas in the region A, not «built on» On!) at the instant tpos On!) at the instant tpos the set of parcels homogenous w. r. t. the pas in the region B, not «built on» the set of parcels homogenous w. r. t. the pas in the region A, not «built on» at the instant to On!) at the instant to the set of parcels homogenous w. w. r. t. the pas in the region B, not «built on»
PA
T
PB
T
PAp same laws as PB? p P
10-
L-
p'---
TRUE
NOT
TRUE
The POS has no effect on the evolutionary mechanism
(1) PAp same laws as PA pl ?
1? (2) PB p same laws as PB ppl?
I
TRUE NOT
TRUE Comparison of (1) and / or (2) with the objectives arising from the POS (expressed in terms of evolutionary mechanism for each zone).
concerning : Differences within (1) and / or (2) concerning: - acceleration of evolutionary mechanisms, or - nature of the states represented
Comparison of these differences with the objectives arising from the POS (expressed in terms of evolutionary mechanisms for each zone)
Fig. 2 .- Relationship between
pas and evolution of land land·- usuage
PA p is the model for the evolution of land-usuage for A PB p is the model for the evolution of land-usuage for B ~ PA p,;s the model for the evolution of land-usuage for Apt PB ptis the model for the evolution of land-usuage for B pt
THE ORIENTATION OF URBAN GROWTH
The hypothesis considered in(3)will be defined fo~ the following elements: - A, the space defined for a homogenous zone of the pas, POS, - B, the space defined for another homogenous zone of the pas, POS, - PA 1 .,. a mechanism for the diffusion of construction withi: withit the time interval (t o t pas l.), in the agriculpos tural space (<
41
The utilisation of the Markov Process presented, consisted of estimating the changes between two successive evolutionary periods in order to test for any effect of an event which divides these two periods. The same tools may be used to test for the similarities or differences between contemporary processes dealing with different classes of objects. It may therefore be verified that either different constructions are brought about by different mechanisms or that different evolutions occur after different types of starting states. From the point of view of comprehension, the analysis of the role played by orientative legal measures in urban growth has enabled us to show the nature and complexity of the relationships between these measures and the evolutions actually observed. In the example studied, it appears that the intentions contained in the official documents are only partially fulfilled. In from · their I n cases where these intentions differ from· preceding mechanisms, only the incitements to develop construction are fully achieved (the attempts at stopping an existing development running up against the inertia of this development). From the analysis point of view, certain of the difficulties, linked on the one hand to the development reasoni ng and on the other to of research on scientific reasoning the constitution of a stock of theoretical knowledge of urban phenomena will only be raised progressively by various and not always conscious contributions. We believe however that at present it is possible to contribute a few elements whose explanatory content may be relatively limited but whose conditions of explicit and validable constitution indicate however that they may be included in a cumulative process of comprehension acquired. acquired . This process might go towards the heuristic elaboration of a theoretical corpus of urban knowledge.
8 - Conclusion
Acknowledgments
The study presented in this paper may be interpreted as being the development of tools in an experimentation phase aimed at producing a validation. This validation draws on the one hand on the steps of analysis considered and on the the other on the mathematical tools required for this analysis and relating to the problems tackled. This work carried out on the community area of Martigues has enabled us to reach an understanding of the relationships existing between urban growth and the legal apparatus of its management.
We would like to express our thanks to Mr Fernandez de la Vega, for the helpful guidance in the completion of this work, and Mr J. D. Lee for the translation of the French text.
The use of Markov Processes as the tool for the representation of temporal phenomena has firstly been able to be carried out in accordance with the mathematical hypotheses implied by this type of tool and secondly has enabled the treatment of temporal data having complex structure.
J.P. Chelyan et aZ
42
Appendix 1.
References
We choose an integer p and a partition of the 11 ,12 ",, "" lp' lp, and put: put : interval (0, t) in p intervals 1"1
AMEDO D., GOLLEDGE R. G. G. - An introduction to scientific reasoning in geography. geography . J. Wiley and Sons, 1975.
X. :: ~ x.::;~
[n [ni-n i - nii (SF
l~i~p+l
,I
where: n is, for 1 ~ i i in the interval I ..
n ii (§) n (6) (0 ~
p, the number of observations lying
I
BILLlNGSLEY D., D., - Statistical inference for Markov Processes. I. M. M . S. statistical research monographs, The University of Chicago Press, 1961.
n p+ p.1 the number of parcels wh ich are still in state j at time t
"
p ... l, is the exp!cted value of ni ' n ii (<'),16 (<'), 1 ~ i ~ p1'"l, relatively to the estimated value ~ of the unknown parameter . parameter.
Xi is, under the null hypothesis, distributed as a chifreedom . square with n degrees of freedom. We can then sum the statistics corresponding to the various initial states i. j. 1 ~ j ~ m, since these statistics are clearly independent, to obtain again a"1? statistic with m, degrees of freedom which can be used as an overall test for the markovian character of process . the process.
Appendix 2. The Neyman-Pearson statistics is in this case: case :
(!')~max L . .E e i ~. e y v Zl Z'
S=2 \max L .. (') + max L
he v'
V
M . et Mme, FARINAS DEL CERRO L. CHEYLAN M. L. Essais de constitution de modeles expl icatifs partiels de I'evolution des utilisations des sols, march 1977 Bulletin . Ecole d'Architecture Marseille GAMSAU Bulletin. Luminy Luminy..
M ., 1975 - A computer DAJANI J., REINHARDT M., Comput . & Urban Soc. Soc. simulation of Urban Growth, Comput. vol 1 pp 159 - 168 - Pergamon Press. Press.
. (9) \ .(9)
where: -L. (e);..nlog (e);.-nlog
CHEYLAN J.P., P., DONATI P., J .P., CHIAVARI P., DESBORDES - CHEYLAN F. - Vers une explicitation et une formal isation des references spatiales et tempod' informations urbains. urbains. Commurelies dans les systemes d'informations Symposium : urban data managenication. VI European Symposium: ed . Dr G. G. Deprez, Liege - 18 - 22 april 1977. ment, ed.
DOOB - Stochastics Processes, Wiley New York, 1953
e (~)-1/.2:. (~)-1/'2:. y.--o't/ y.--o't/e
y/yll y/y" is the loglikelihood of the observations of the sample relatively to the negative exponential distribution Yi with parameter ~ ,truncated at tt..
L . - Power of some tests for FARINAS DEL CERRO L. exponential distribution against an alternative of mixappear) . ture (to appear).
- t is the total time ot observation - n the number of observation in the sample Y i falling into the interval (O,t) - n' the number of observations in the sample falling outside the interval (O,t)
,
Yi
,
( 9) have analogous exL. L. ((&) &) and L . " Zl yl vv zi Z Y Z pressions and are respectively the logl i kelohood of the sample Y yii and of the pooled sample V Vu\J Zi Zi
- the Neyman-Peason statistics S is asymptotically distributed as a chi-square with two degrees of freedom
GUENOCHE A. - Presentation d'un systeme de traitements documentaires et statistiques adapte au calcul en Sciences Humaines. Panorama de la Nouveaute Informatique en France - Tome 1 - pp 181 - 189, AFCET Nov 1966.
LAROUCHE P., 1965 - The simulation of residential growth in the Montreal region, Quebec, Quebec Depart ment of roads. roads. 1965
(X~)· SI EGE L S. S. - Nonparametric statistics for the behavioral nternational Student Edition, Mc Grawraj sciences, IInternational Hill - New York. 1956.