Two complexities and their models A friendly environment to simulate urban dynamics Arnaldo “Bibo” Cecchini

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Two complexities and their models The difficulty in dealing with urban systems’ complexity and the related difficulty to analyse and forecast is twofold: one kind of difficulty lies in the complexity of the system itself, and the other is due to the actions of actors, which are “acts of freedom”. We are firmly convinced that the planning in its strict sense (meaning by definition the production of “plans”) is absolutely necessary, today just as always, if not even more. And the production of plans, almost by definition implies substantially a set of rules, restrictions and incentives. But it is evident that the adequate combination of these three elements has to be determined in every concrete situation, and that the mix can vary.

GOAL – Why we do need “special models” for planning Give to the actors involved in planning processes the capability to read, understand, forecast the different systems that represent the “battleground” of their actions. Models must be useful for each party involved in the planning process, so they have to deal with both complexities.

WAYS TO THE GOAL We need modular models; they must be –friendly –flexible –multi-level –Unexpensive and must be linked to communication tools and must be useful for –decision –negotiation –consensus building –Evaluation These models must achieve this goal being “part” of the most sophisticated and hard ones used by experts.

Cities as Complex Systems: which models? To deal with this two types of complexities (linked together!) we need appropriate and interrelated models. For the first complexity we consider gaming simulations, role plays, scenarios techiniques, … For the second complexity could be interesting - for our purposes - the set of models with a bottom – up approach: neural networks, genetic algorithms, multi agent models, cellular automata each one appropriate for different tasks. Our experience is mainly related to Cellular Automata that seem efficient and effective to simulate urban growth, urban sprawl, location of functions, traffic flow, real estate values, land use transformation, …

Cellular Automata 1 discrete dynamic system A cellular automaton is a discrete dynamic system containing a great number of cells in the space (1-D, 2-D, o 3-D). evolution rule Each cell interacts only with its neighbour cells and there is a cellular automaton evolution rule that changes simultaneously the state of all cells within discrete temporal steps. simulate the behaviour of a complex system based on the interaction of a great number of cells that follow simple rules, The founding idea: to simulate the behaviour of a complex system based on the interaction of a great number of cells that follow simple rules, and not to describe the global behaviour through complex equations

Cellular Automata 2 Bi-dimensional grids and classical neighbourhoods

Cellular Automata 3 discrete dynamic systems continuous dynamic systems Cellular automata (CA) are discrete dynamic systems, but often represent an efficacious alternative to systems of diffeerential equations for the simulation of continuous dynamic systems In general, CA offer a model to study systems composed of numerous parts (cells) with no central control but only local interactions simulate the behaviour of a complex system based on the interaction of a great number of cells that follow simple rules, The founding idea is to simulate the behaviour of a complex system based on the interaction of a great number of cells that follow simple rules, and not to describe the global behaviour through complex equations

{ S N } { S A }  aa cc bb 11 Space: regularity State set: uniformity Neighborhood: stationarity Transition function: universality Transition function: invariance Time: regularity System closure: closed nonregularity time variance nonuniversality nonstationarity nonuniformity irregularity open Classical CA and limits for UM 1

Basic properties of CA (Couclelis 1985) The principles that Couclelis calls "leading conventions" are: definition of an infinite plane (space) spatial stationarity of neighborhoods regularity spatial homogeneity spatial and temporal invariance of transition rules closure towards external events. The analysis of urban systems need to “relax” almost all kind of classical constraints for CA, but leaving unchanged the CA “dogma”: locality of actions effects. Our approach is to build a general environment (CAGE) to develop AC with the possibility to tune ad libitum the constraints realizing the “right” AC for the actual task. Classical CA and limits for UM 2

Spatial and Temporal Stationariety of Neighbourhoods In CAGE the neighbourhood can vary with time and is defined as relations, non necessarily geometrical, among objects of the model. Regularity in spatial discretisation In CAGE the geometrical object associated to cells is considered an attribute not necessarily subject to spatial or temporal limitations. Stationariety and homogenity of the transition functions In CAGE the cells’ transition functions can also depend on parameters varying with time, local to the cell or global. Limited number of possible cells’ states In CAGE there is no a priori limitation of the number of states and CAGE makes available different types of sub-states (integer, real, char) Basics CAs for Urban Models 1

Closure with regard to external events In CAGE is it possible that external phenomena influence, even on a local level, the evolution of the simulation. Locality of the evolution control In CAGE, in order to be able to obtain realistic simulations, there is also a mechanism of global control (steering) of the system’s evolution Difficulties in the interoperability between CA scenarios and GIS environments In CAGE specific functions for importing/exporting spatial and alphanumeric data available in existing GIS environments will be developed Environments usable only by expert programmers In CAGE, the modelling-simulation-analysis cycle is simplified by a specific user graphical interface including the functionality of semi-visual modelling of evolution rules Basics CAs for Urban Models 2

MODEL I MODEL V MODEL IV MODEL III MODEL II j i g ij t+  t g ij t g ij t+  t g ij t-2 g ij t-1 g ij t g ij t +  t u ij t v ij t w ij t g ij t +  t g ij t 1±p,1 ±q g ij t +  t Tobler 1979 The fifth one "hides" two kind of CA, one which we could call "synchronous" where it's: g i,j = F(g i ±p, j ±q ) and which corresponds to a sort of "filter" for interpolation - extrapolation, and one which we could call “diachronic” where it's: g i,j t +dt = F(g i,j t, n i,j t ) Basics Territorial models

Basics land use change CA model Final Land Use P ij P ik P jk Static Variables: technical infrastructure; social infrastructure; real state market; occupation density; etc. Dynamic Variables: type of land use- neighbouring cells; dist. to certain land uses Source: Adapted from Soares (1998). Transition probabilities Initial Land Use Calculates Amount of Transitions Meta rules Iterations Calculates Dynamic Variables Calculates Spatial Transition Probability

Example of CA Urban Simulation 1 t0t0 t1t1 t3t3 t4t4 CA Model calibration validation forecast Source: adapted from Almeida (2004).

Predominantly Deterministic Models of Land Use Change Spontaneous (random) new growth Organic growth Diffusive growth and spread of a new growth centre Road influenced growth (Clarke et al., 1997) Cell urbanised at previous step Growth moved to road, and spread Seed Cell Cell urbanised by this step Road Example of CA Urban Simulation 2 A

Evolution of Urban Growth - San Francisco Bay Area (USA) Clarke et al., (1997) topography, roads in 1920, roads in Decisive factors for urban growth: Predominantly Deterministic Models of Land Use Change Example of CA Urban Simulation 2 B

SIMLUCIA - Model with categories of urban land use, which incorporates regionalised variables. RIKS - Research Institute for Knowlege Systems, University of Maastricht, Holland (1999) Predominantly Stochastic Models of Land Use Change Global Transition Probabilities or Transition Rates (impacts on the system as a whole): Regionalized Economic/Demographic Models, Markov Chain, Multivariate Regressions, etc. Local Transition Probabilities or Cells Probabilities Weights of Evidence, Logistic Regression, Analytical Hierarchical Programming (AHP), Neural Networks, Decision Tree, etc. Example of CA Urban Simulation 3 A

P z = f(S z ) f (A z ).   (w z,y,d x I d,i ) +  z d i P z is the potential for transition to state z f(S z ) is the suitability of the cell for activity z (S z is a suitability coefficient) f(A z ) is the accessibility of the cell for activity z (A z is an accessibility coefficient) w z,y,d is the weighting parameter applied to cells with state y in distance zone d I d,i =1 if the state of cell i in distance zone d = y; I d,i =0 if the state of cell i in distance zone d ≠ y;  z is a stochastic disturbance term Cells Transition Probability: Predominantly Stochastic Models of Land Use Change Example of CA Urban Simulation 3 B

CAGE Application Example: the Heraklion Model The data-sets used for modelling and calibration: –urban density variable for 1980 and 2000, expressed in persons/ha; –real-estate value for 1980 and 2000, expressed in Dr./ha; –a “social value” indicator for 1980 and 2000, expressed as a qualitative ordinal value; –effective land-use for ; –1998 master plan zoning and restrictions. several layers where distinctive dynamics are simulated. irregular cells representing city blocks and containing predominantly buildings, but also internal and capillary street network and undeveloped land; poly-lines representing the main street and road network, organised in main and secondary roads; points representing positions of urban services and special facilities. Example of CA Urban Simulation 4 A

CAGE Application Example: the Heraklion Model Example of CA Urban Simulation 4 B Initial ( observed) Final ( simulated)

CAGE Application Example: the Heraklion Model Observed 2000 data Example of CA Urban Simulation 4 C Real-estate values: simulated values after 20 steps The model parameters was calibrated trough genetic algorithms More than 60% of the total cell number (1671) take the correct real estate value class at the configuration of calibration

CAGE (Cellular Automata General Environment) Irregular cells allowed Typed states Layers Cells’ neighbourhoods as result of queries Non-homogeneous non-stationary neighbourhoods Access to georeferenced data Calibration tools Visualization Our Environment 1

GUI User-friendly GUI for visual modelling of cellular automata Cross-platformQT Cross-platform application based on QT libraries (Windows-Linux) Overcoming of many restrictions Overcoming of many restrictions typical of the CA-based environments Scenario structured in layers Scenario structured in layers (on different layers simulations of various relevant phenomena take place) library functions Some library functions are predefined and help in the definition of evolution rules Export/Import Export/Import of data and graphs Execution on remote computer TCP/IP Execution on remote computer via TCP/IP protocol Essential Characteristics of CAGE Our Environment 2

The CAGE’s GUI Structure modelling Graphical windows Our Environment 2

The cellular automata The cellular automata is defined by three elements: is a finite set of values of g-dimensional vector of global parameters: is a vector of global parameters’ update functions is a set of n cell layers. The CA Model Our Environment 3

The Layer cell layer A cell layer is defined by three elements: is a set of cells is a finite set of values of r-dimensional vector of layer’s parameters. is a vector of layer parameters’ update functions Our Environment 4

The Cell 1 cell The cell is defined by six elements: is a finite set of values of the q-dimensional vector of the cell’s state is a finite set of values of the r-dimensional vector of cell’s local parameters O i is a finite set of geometrical objects, possibly geo-referenced and characterised by an adequate vectorial description Our Environment 5

is a vector of n neighbourhood functions defined as: Horizontal neighbourhood Vertical neighbourhood The Cell 2 Our Environment 6

is a vector of local parameters’ update functions where: is the transition function of a generic layer’s cell that describes the transition of the cell’s state and is defined as: The Cell 3 Our Environment 7

The Update functions of layer’s parameters are defined as: The layer’s parameters can assume values depending on the current configuration of the entire layer, thus offering a mechanism for global control of the layer’s evolution. Layer Parameters Updating Our Environment 9

The value of the k-th global parameter can be calculated on the basis of the values assumed by other global parameters and by all layers’ parameters: A global parameter can be updated, even based on an external variable, by a generic calculum model evolving in parallel to the cellular automaton Global Parameters Updating Our Environment 10

The Architecture of CAGE TCP CAGE server CAGE client CAGE client CAGE client C++ compiler CA kernel PIPE Our Environment 11

Characteristics of CAGE: Structure Modelling Function generationChange of selected element’s attributes Tree-structure representation Flow-chart representation of the function Inserting: layers, parameters, constants, sub-states, vertical neighbourhoods Function source-code

Attributes of layers Per each layer it is possible to define: The label; The type of spatial discterisation: regular: the classical discretisation of the plane in hexagonal, rectangular, triangular cells. The regular discretisation allows to select one of the classical neighbourhoods (Moore, Von Neumann, ecc); non regular: the cells are constituted of graphical objects to be defined with the editing tools available in CAGE; The type of CA border: toroidal, limited, inactive (the border cells are used in the internal cells’ neighbourhoods, but do not get updated) Our Environment 13

Parameters and Sub-states Per each parameter it is possible to define: the label; the type: integer, real, char the updating method: constant or based on function Per each sub-state it is possible to specify the following properties: the label the type: integer, real, char Our Environment 14

Neighbourhoods Per each horizontal neighbourhood it is possible to define wether it is: of a classical type (Moore, Von Neumann, ecc. ), only in the case of a regular spatial discretisation based on a function (query) Per each vertical neighbourhood it should be specified: the reffering layer the neighbourhood’s updating function Our Environment 15

Functions Per each function (parameters, sub- states and neighbourhoods updating) it is necessary to define: the label the frequency of execution the probability of execution an execution subordination condition Our Environment 15

Function Generation Flow-chart creationProperties of the selected component Guided code input (variables and library functions) Our Environment 16

Flow-Chart Components If-then If-then-else If-then-else if Block Single Statement Free Instruction Flow-Chart End Our Environment 17

Logical Conditions Tree-like representation Example: (a=1 AND b=2) OR (c=0) AND OR a=1 b=2 c=0 Our Environment 18

Logical Propositions Creation of a composed logical proposition Our Environment 19

Library Functions Geometrical Functions dist(obj1, obj2)Returns the distance between object obj1 and object obj2 bari-centres centroidX(obj)Returns the X coordinate of the object obj bari-centre centroidY(obj)Returns the Y coordinate of the object obj bari-centre area(obj)Returns the area of the object obj perimeter(obj)Returns the perimeter of the object obj length(obj)Returns the length of the object obj Our Environment 20

Library Functions Mathematical Functions abs(num)Returns the absolute value of the number num max(num1, num2)Returns the maximum value between num1 e num2 min(num1, num2)Returns the minimum value between num1 e num2 odd(num)Returns true if num is an unpair number even(num)Returns true if num is a pair number prob(num, over)Returns true with the probability of num/over Example: prob(10,100) returns true in 10% of cases div(n1, n2)Returns true if n1 is a mutiplier of n2 Our Environment 21

Library Functions Layer Aggregation Functions Sum(var)Returns the sum of values of the variable var on the cells Min(var)Returns the minimum value of the variable var on the cells Max(var)Returns the maximum value of the variable var on the cells Average(var)Returns the arithmetical mean value of the variable var on the cells NEqVal(var, val)Returns the number of cells where the variable var assumes to value val Our Environment 22

Library Functions Neighbourhood Aggregation Functions NeighCell[lay].Sum(var)Returns the sum of values of the variable var on the cells belonging to the actual cell’s neighbourhood on the layer lay NeighCell[lay].Min(var)Returns the minimum value of the variable var on the cells belonging to the actual cell’s neighbourhood on the layer lay NeighCell[lay].Max(var)Returns the maximum value of the variable var on the cells belonging to the actual cell’s neighbourhood on the layer lay NeighCell[lay].Average(var)Returns the arithmetic mean value of the variable var on the cells belonging to the actual cell’s neighbourhood on the layer lay NeighCell[lay].NEqVal(var, val)Returns the number of cells belonging to the actual cell’s neighbourhood on the layer lay where the variable var assumes the value val Our Environment 1

CA Structure Layer cell state Result: source-code of the transition function Example: LIFE Per each variable (parameter or sub-state) represented by the identifier [Name] it is automatically defined the variable New[Name] The variable’s updating function must contain at least one assignment instruction like: New[Name] = expression Constants Our Environment 24

Function-based Neighbourhood Updating Query attributes Target cell inclusion condition (dist(Obj,Cell[Layer].Obj) 5000) Cell Neighbourhood at the first step dist(Obj, Cell[Layer].Obj)<100 Update QueryHorizontal neighbourhood ProbabilityFrequencyCondition Our Environment 25

A Non-regular Spatial Discretisation Editing Tools Our Environment 26

Objects Visualisation Control Layers Visualisation Control Variable Values Editing Other Useful Tools Colormap creation Our Environment 26

Colormap Our Environment 27