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Chapter 8

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  1. Chapter 8 Geocomputation Part B: Artificial Neural Networks (ANNs) & Genetic Algorithms (GAs) www.spatialanalysisonline.com

  2. Geocomputation: ANNs In this presentation on geocomputation: ANNs discussed include • Multi-level perceptrons (MLPs) • Radial basis function neural networks (RBFNNs) • Self organising feature maps (SOFMs) ANNs are particularly concerned with • Function approximation and interpolation • Image analysis and classification • Spatial interaction modelling www.spatialanalysisonline.com

  3. Geocomputation: Evolutionary computing In this presentation on geocomputation: EC elements discussed include • Genetic algorithms (GAs) • Genetic programming (GP) EC is particularly concerned with • Complex problem solving using GAs • Model design using GP methods www.spatialanalysisonline.com

  4. Geocomputation • Artificial Neural Networks (ANNs) • A computational model based on emulating biological neural networks • A form of non-linear modelling tool • Often a 3-layer network structure is used: input, hidden, output • The output layer of such structures are typically modified weighted sums of intermediate layers, which are modified weighted sums of the input layer www.spatialanalysisonline.com

  5. Artificial Neural Networks Hence at each output node (hidden or final) a two-step process takes place: www.spatialanalysisonline.com

  6. Artificial Neural Networks • Simple 3-layer feedforward ANN • Fully inter-connected; each connection is given a weight, w • Known as a Multi-level perceptron (MLP) • In this case: 3 input nodes, 5 hidden nodes, 2 output nodes and 2 bias nodes (bias, B, is similar to the constant term in regression models) • At hidden node 1 we have: where the wij are weights to be determined, b1=1, and the xi are the observed input values www.spatialanalysisonline.com

  7. Artificial Neural Networks is simply a linear weighted sum of the inputs. To generate a non-linear output it must be modified by some (well behaved) non-linear function, g(), e.g. the logistic function: i.e. Sample activation functions www.spatialanalysisonline.com

  8. Artificial Neural Networks We can now compute the output layer values as the weighted sum Suppose we have known input values x1=1, x2=-3, x3=5, and known outputs of 0 and 1. Can we select the weights to ensure the inputs generate the known outputs? Suggestion: <build your own worked example & program here> www.spatialanalysisonline.com

  9. Artificial Neural Networks • Learning • Supervised learning • Split training/test data sets (control data) • Known inputs and output (target) values for training data • (Network output-Target output) = Error signal, e • Systematically adjust weights to minimise sum of e2 • Adjustment typically based on backpropagation and gradient descent • Used in many classification/pattern recognition applications and in function approximation • Unsupervised learning • No training data • Must create clusters by analysing dataset for patterns/clusters www.spatialanalysisonline.com

  10. Artificial Neural Networks • Some basic issues: • local vs global minimisation • Initialisation and selection • Data normalisation and coding • Momentum • Model design and over-fitting • Overtraining • Interpolation vs Extrapolation/Forecasting www.spatialanalysisonline.com

  11. Artificial Neural Networks • MLP: Example 1 function approximation source data fitted solution curve RMSE vs epochs www.spatialanalysisonline.com

  12. Artificial Neural Networks • MLP Example 2: LCM www.spatialanalysisonline.com

  13. Artificial Neural Networks • MLP Example 2: LCM www.spatialanalysisonline.com

  14. Artificial Neural Networks • MLP Example 2: LCM www.spatialanalysisonline.com

  15. Artificial Neural Networks • MLP Example 2: LCM weights matrix www.spatialanalysisonline.com

  16. Artificial Neural Networks • MLP Example 3: Spatial interaction model • Generalised model: Tij=f(Oi,Dj,dij) • Sample data format (log transformed): www.spatialanalysisonline.com

  17. Artificial Neural Networks • MLP Example 3: Spatial interaction model www.spatialanalysisonline.com

  18. Artificial Neural Networks • Radial Basis Function Networks Basic functional form: Gaussian RBF: www.spatialanalysisonline.com

  19. Artificial Neural Networks • Self organising function maps • SOM as an output space • Neighbourhood relations • Grid size, form and topology www.spatialanalysisonline.com

  20. Artificial Neural Networks • Self organising function maps • Dimensional reductions • Mapped output – similar vectors (units) are close to each other • Typically an unsupervised procedure • Spatial mapping of SOM can follow using simple assignment to best matching unit (BMU) www.spatialanalysisonline.com

  21. Artificial Neural Networks • Self organising function maps • Choose a grid size, form and topology • Train the network • Identify the best matching units • Modify the BMU and its neighbours (spatially biased learning rule) • Map the trained network www.spatialanalysisonline.com

  22. Artificial Neural Networks • Self organising function maps – some issues • Initialisation • Pre-processing • Normalisation • Missing data • Masking and weighting • Learning and tuning • Distance metrics • Neighbourhood functions (kernels) • Learning rate functions www.spatialanalysisonline.com

  23. Artificial Neural Networks • Self organising function maps – Idrisi www.spatialanalysisonline.com

  24. Artificial Neural Networks • Self organising function maps – Idrisi www.spatialanalysisonline.com

  25. Genetic Algorithms • Solutions are represented as individuals • Individuals are modelled as chromosomes • Chromosomes are comprised of genes • Genes have values known as alleles • Chromosomes have a measurable fitness • New chromosomes (children) are created by reproduction and mutation processes • The fittest individuals survive • The creation process is then iterated www.spatialanalysisonline.com

  26. Genetic Algorithms • GAs: Example 1 - TSP allele=12 (ID of town in TSP problem set) chromosome genes • Each chromosome contains complete list of towns • create a set of m randomly permuted strings and compute lengths, d • evaluate the fitness of each string (e.g. 1/d) • select random pairs of tours (biased by fitness) • combine pairs by crossover operation • evaluate fitness of offspring • apply replacement rule (fittest retained) and iterate till stable www.spatialanalysisonline.com

  27. Genetic Algorithms • GA components • Encoding or representation – binary, list, tree etc • Fitness function selection – use of rank transforms • Population initialisation • Selection: roulette, tournament, uniform random • Reproduction • Crossover e.g. A = [a b c d e f g h] B = [1 2 3 4 5 6 7 8] and the crossover point is 3, the following children are generated: child 1 = [a b c 4 5 6 7 8] child 2= [1 2 3 d e f g h] • Mutation • Local search • Termination www.spatialanalysisonline.com

  28. Genetic Algorithms • GAs: application areas • TSP (as above) • Clustering • Map labelling • Optimum location with capacity constraints • Concept can be extended to alleles that are expressions or program elements rather than numerical values  Genetic programming www.spatialanalysisonline.com