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ENVIRONMENTAL MODELLING PROBABILISTIC MODELS

ENVIRONMENTAL MODELLING PROBABILISTIC MODELS. Dr Claire H. Jarvis, chj2@le.ac.uk. The next lectures …. Focusing on ‘bottom-up’ models: Types of bottom up models For cellular automata in particular Theoretical concerns Practical applications

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ENVIRONMENTAL MODELLING PROBABILISTIC MODELS

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  1. ENVIRONMENTAL MODELLINGPROBABILISTIC MODELS Dr Claire H. Jarvis, chj2@le.ac.uk

  2. The next lectures … • Focusing on ‘bottom-up’ models: • Types of bottom up models • For cellular automata in particular • Theoretical concerns • Practical applications • Advantages & disadvantages of cellular automata for environmental modelling • A glimpse beyond cellular automata • Practical work • Cellular automata model for land use change in Bolivia

  3. Categories of environmental model

  4. Categories of environmental model • Environmental models may be categorised in a variety of ways (e.g. Burrough et al 1996, Heuvelink 1998, Steyaert 1993). Complexity of form Time Dependent Independent Physical Complex conceptual Simple conceptual Resolution Cell Block Global Important axes include stochastic/ deterministic, inductive/ deductive, bottom-up/ top-down

  5. Model types Bottom-up models are particularly, but not exclusively or necessarily, used for stochastic modelling. We shall indeed approach stochasticity, but for now consider the following categorisation: Top-down models (equations) Bottom-up models (A-life models) Cellular automata, a primitive class of A-Life (artificial life) model, are always developed from the ‘bottom-up’

  6. Why consider model taxonomies? • To provide a framework within which you can place the different techniques • Useful to weigh up the strengths and weaknesses of the various model types. • No one model type will be best for all applications • Transfer knowledge between different application areas of the environmental sciences (*Skidmore 2002, p8) • Reminder of the many uncertainties surrounding the modelling process • You might build a model with which you are well pleased, but others might approach the subject from a different but equally valid perspective *Skidmore, A. (2002) Taxonomy of environmental models in the spatial sciences, In Environmental Modelling with GIS and Remote Sensing, Taylor & Francis, London, p8-25.

  7. Cellular automataEquations vs CA models • This module deliberately introduces both top-down environmental models rooted in equations and bottom-up cellular automata models • This part of the module focuses on the possibilities of cellular automata. You need to consider what these models are, when these models might be most appropriately used and how. Neither equations nor artificial life models are a universal modellers’ panacea! • Viewing geographical problems and models from different perspectives is healthy and cross-comparisons provide further stimulation of research and debate

  8. Cellular automataWhy seek an alternative to equations? (1) • In many models it is common to refer to the derivative of a variable with respect to population size N. It assumes that N is large, not good when modelling small population effects • Equations are generally poor at dealing with highly non-linear effects such as thresholding or if-then-else conditionals, which arise very frequently in the description of animal and human behaviour • It would take tens to hundreds of lines of equations to express even a simple model of many geographical processes as a function of the many genetic, memory and environmental variables that affect their behaviour (After Taylor & Jefferson, 1996, p5-6)

  9. Cellular automataWhy seek an alternative to equations? (2) • Most differential equations, especially of the ‘real’ world, have no closed form solution • One of the well known CA-scientists, Tommaso Toffoli, contributed this problematic (Toffoli, 1984): "... in modeling physics with the traditional approach, we start for historical reasons ... with mathematical machinery that probably has much more than we need, and we have to spend much effort disabling these 'advanced features' so that we can get our job done in spite of them." (After A. Schalten, http://www.ifs.tuwien.ac.at/~aschatt/info/ca/Behav_Univers, Accessed 21 October 2003)

  10. Models based on the principles of artificial life

  11. ARTIFICIAL LIFE MODELS “Artificial life is the study of how to create man-made systems that behave as if they are alive”. Rucker (1993 p5, In Openshaw & Openshaw 1997 p239) “We can describe artificial life as the study of artificially created systems that embody at least some of the behaviours of ‘real’ life” Prata (1993 p1, In Openshaw & Openshaw 1997 p239) Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  12. Potential significance of A-Life technologies • They may lead to a better understanding of nature’s self-organising laws and the nature of life, with enormous philosophical and practical consequences; • They may lead to self-replicating factories; • They may carry life to the next level of evolution; • A rich source of inspiration for new types of research on artificial intelligence; • Creations of artificial life may be useful in their own right. (Prata 1993, In Openshaw & Openshaw 1997 p240)

  13. Artificial life models • Examples of this type of model, rooted in biological principles, include: • Cellular automata (Behaviour of simple cells) • Genetic algorithms (Complex swapping of alleles) • Neural networks (Inspired by brain cell activity) These three lectures focus on the simplest category of these a-life models – the cellular automata

  14. Cellular automata

  15. Cellular AutomataHistory (1) • The THEORY of cellular automata was introduced in the late 1940´s by John von Neumann & and Stanislaw Ulam, working at the Los Alamos National Laboratory in the United States (von Neumann, 1966; Toffoli, 1987) • PRACTICAL aspects of CAs were developed in the late 1960´s when John Horton Conway developed the Game of Life (Gardner, 1970) and with the increased popularity of the computer Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  16. Game of life (Mathematician John Conway) • A cell becomes live if it is surrounded by three life neighbours • Cell becomes dead if surrounded by more than three (overcrowding) or less than two (isolation) • The number of life sites fluctuates over time according to a power law until steady state is reached

  17. Move 1 … • A cell becomes live if it is surrounded by three life neighbours • Cell becomes dead if surrounded by more than three (overcrowding) or less than two (isolation)

  18. After 10 iterations … Startingpoint

  19. Game of life: Potential patterns • The cell patterns resulting from the initial configuration of 0/1 with particular rule sets have been given all sorts of names: • Blinkers, ponds, gliders, shuttling bees, puffer trains. • Wolfram’s (2002) book discusses many of these phenomena in more detail. • There are a number of Internet sites where you can try creating your own universes. See http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life for applets demonstrating the principle emerging patterns. • The purpose of this modelling course is to explore & build custom-build universes that demonstrate geographical purposes. DEMO

  20. Cellular automataWhat are they? (1) Informally … ‘The cellular automaton consists of a line of cells, each colored either black or white. At every step there is then a definite rule that determines the color of a given cell from the color of that cell and its immediate left and right neighbors on the step before’. Stephen Wolfram (A New Kind of Science, 2002) Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  21. Cellular automataWhat are they? (2) More formally … Consider a space divided into cells, where each cell is repeatedly "updated" to a new state in an evolving sequence. A program of this nature is specifically called a cellular automaton when it is 1) parallel, 2) local, and 3) homogeneous. Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  22. Cellular automataWhat are they? (3) • Parallelism - the individual cell updates are performed independently of each other, i.e. all of the updates are done at once • Locality - when a cell is updated, its new value is based solely on the old values of the cell and of its nearest neighbours. • (3) Homogeneity - each cell is updated according to the same rules. Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  23. Cellular automataWhat are they? (4) • Importantly, using a CA, a complex system is not described using complex equations, but rather complexity emerges through the interaction of simple individuals (cells) following simple rules. A localised rather than centralised mindset is adopted. • “There is no ‘global blueprint’ but one emerges bottom-up” • (Openshaw & Openshaw 1997) Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  24. Major elements of cellular automata Cell space:The space is composed of individual cells. Theoretically, these cells may be in any geometric shape. Cell states:The states of each cell may represent any spatial variable, e.g. the various types of land use. Time steps: A CA will evolve at a sequence of discrete time steps. At each step, the cells will be updated simultaneously based on transition rules. Transition rules: A transition rule normally specifies the states of cell before and after updating based on its neighbourhood conditions. White, R., and G. Engelen, 2000, High-resolution integrated modeling of the spatial dynamics of urban and regional systems, Computer, Environment and Urban Systems24:383-400.

  25. Major elements of cellular automata (1)Cell space • Theoretically, cells may be in any geometric shape. • Most CA adopt regular grids to represent such space, which make CA very similar to a raster GIS. • Recent work in GIScience by Wu challenges this notion, and looks at x spaces

  26. Major elements of cellular automata (2)Cell states • The states of each cell may represent any spatial variable. • In your practical exercises, as for much of the geographical literature, the states that you will consider will be various types of land use. • Land use is a discrete state.

  27. Major elements of cellular automata (3)Time steps • As you saw in the ‘Game of Life’ example, a CA will evolve at a sequence of discrete time steps. • At each step, the cells will be updated simultaneously based on transition rules. • These transition rules may alter across time. • In the special case of the reversible cellular automata, at any time-step of the development the rules may to go forwards or backwards in time without losing any information.

  28. Major elements of cellular automata (4)Transition rules • Establishing transition rules is a key issue when building cellular automata • Transition parameters include: • The neighbourhood • The nature of the transition

  29. Transition rules: The neighbourhood (1) • What configuration of neighbourhood? • The neighbourhood need not be square, but may be circular or hexagonal or even triangular • The Margolus Neighbourhood takes a different approach & considers 2x2 cells of a lattice at once. Neumann Moore (Figures from A. Schalten, http://www.ifs.tuwien.ac.at/~aschatt/info/ca/Behav_Univers, Accessed 21 October 2003)

  30. Transition rules: The neighbourhood (2) • How broad is the neighbourhood? • Should this extend only to the neighbouring cells, or beyond them? • Should the neighbourhood be symmetrical? Moore Extended Moore (Figures from A. Schalten, http://www.ifs.tuwien.ac.at/~aschatt/info/ca/Behav_Univers, Accessed 21 October 2003)

  31. Transition rules: Nature of transition • Standard rules • Every group of states of the neighbourhood cells is related a state of the core cell • Totalistic Rules • The state of the next state core cell is only dependent upon the sum of the states of the neighbourhood cells and not their individual configurations. • Legal Rules • Related to totalistic rules, these cover the special case where all cells start with the same state. Essentially, these rules are the subset of totalistic rules that enforce a change of state where applied

  32. Classes of CA "...many (perhaps all) cellular automata fall into four basic behaviour classes.", Stephen Wolfram (Wolfram, 1984) Class 1 – Reaches a unique state from limited points Class 2 – Creates repeating patterns Class 3 - Creates chaotic, a-periodic patterns which re often self-similar (fractal) in nature Class 4 - More complex behaviour, irreversible, die after initially stable period (After A. Schalten, http://www.ifs.tuwien.ac.at/~aschatt/info/ca/Behav_Univers, Accessed 21 October 2003) Real problems need to hit here On the ‘edge of chaos’ (Langton) Too simple Too chaotic DEMO

  33. Cellular automataWhat relevance to geography? • CAs are inherently spatial, incorporating the intrinsically spatial concept of the neighbourhood. • Relationships can be seen between CAs and earlier work such as Tobler's (1979) cellular geography, raster GIS models and Hagerstrand's (1968) diffusion models. • The relevance of CAs to geography depends on the extent to which simple rules underlie the apparent complexity of geographical systems; particularly human systems • CAs may be seen as a simple way of integrating time into GIS/spatial data structures. Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  34. Cellular automataGeographical application areas (1) • The literature to date shows us that a world where each cell, or small region of space "updates" itself independently (parallelism), basing its new state on the appearance of its immediate surroundings (locality) and on some generally shared laws of change (homogeneity) can be relevant for • Physical phenomenon e.g. forest fires, invasive species, • Sociological phenomena e.g. process of urbanisation of territories by individuals. Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  35. Cellular automataGeographical application areas (2) • Dadson (1999) summarised the types of problem that can be approached using cellular structure and 'rules' as: • Spatially complex systems (e.g., landscape processes especially in relation to criticality e.g. earthquakes, river meanders, landslides) • Discrete entity modelling in space and time (e.g., ecological systems, population dynamics) • Emergent phenomena (e.g., evolution, earthquakes) Environmental Modelling GY7202: PROBABILISTIC MODELS (1)

  36. NEXT WEEK • Advantages & disadvantages of cellular automata – student-led reviews • Practical steps to consider whenbuilding a cellular automata model

  37. Group literature reviews (1) • Objectives: • Each member of the class to have read a paper on a CA application and to have discussed your understanding of it with in a small group; • As a class, to share your newly acquired knowledge on CA applications in summary form, so covering a wider range of the literature than might be achievable individually • To develop your ability to read and summarise journal papers in an efficient manner

  38. Group literature reviews (2) • In groups of two/three, please choose one application paper from the selection provided in the areas of wildfire spread or ecology/biogeography. • Please skim read the papers and extract the four fundamental characteristics of the CA models used (cell space, cell states, time steps & transition rules used) • Using the pro-forma provided, summarise your findings. • These summaries will be photocopied and distributed to all members class afterwards, so please make them succinct and legible. • Be ready to report your group summaries back to the class orally.

  39. Review: General properties of CAs • CA´s develop in both space & time, both of which are defined in discrete steps. • A CA is built up from cells, that for geographical work are generally arranged in what is termed a ‘two dimensional lattice’ • The number of states of each cell is finite and each state is discrete • All cells have identical neighbourhoods consisting of their immediate or near neighbours in the lattice • The future state of each cell depends only on the current state of the cell and the states of the cells in the neighbourhood • The development of each cell is defined by ‘rules’ that are applied consistently across all cells. These rules may be deterministic or stochastic.

  40. Final questions What do you think the role of CA models could be within geography?Can you think of some other applications, for example in geomorphology, where a CA modelling approach might be interesting?

  41. READING • Wolfram, S. (2002) A new kind of science: • ‘Two dimensions and beyond’, Chapter 5, pp169-221 • ‘Implications for everyday systems’, Chapter 8, pp 363-432. • Resnick, M. (1997) Turtles, Termites, and Traffic Jams, MIT Press, pp163. • Openshaw, S., Openshaw, C. (1997) Artificial intelligence in Geography • ‘Artificial Life’, Chapter 8, p236-266

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