fuzzy logic and neural networks full report
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Fuzzy Logic Philosophical approach Decisions based on degree of truth Is not a method for reasoning under uncertainty that probability Crisp Facts distinct boundaries Fuzzy Facts imprecise boundaries Probability  incomplete facts Example €œ Scout reporting an enemy Two tanks at grid NV 54 (Crisp) A few tanks at grid NV 54 (Fuzzy) There might be 2 tanks at grid NV 54 (Probabilistic) Apply to Computer Games Can have different characteristics of players Strength: strong, medium, weak Aggressiveness: meek, medium, nasty If meek and attacked, run away fast If medium and attacked, run away slowly If nasty and strong and attacked, attack back Control of a vehicle Should slow down when close to car in front Should speed up when far behind car in front Provides smoother transitions €œ not a sharp boundary Fuzzy Sets Provides a way to write symbolic rules with terms like medium but evaluate them in a quantified way Classical set theory: An object is either in or not in the set Can „¢t talk about nonsharp distinctions Fuzzy sets have a smooth boundary Not completely in or out €œ somebody 6 is 80% in the tall set tall Fuzzy set theory An object is in a set by matter of degree 1.0 => in the set 0.0 => not in the set 0.0 < object < 1.0 => partially in the set Example Fuzzy Variable Fuzzy Set Operations: Complement Fuzzy Set Ops: Intersection (AND) If you have x degree of faith in statement A, and y degree of faith in statement B, how much faith do you have in the statement A and B Eg: How much faith in that person is about 6 „¢ high and tall Does it make sense to attribute more truth than you have in one of A or B Fuzzy Set Ops: Intersection (AND) Assumption: Membership in one set does not affect membership in another Take the min of your beliefs in each individual statement Also works if statements are about different variables Dangerous and injured  belief is the min of the degree to which you believe they are dangerous, and the degree to which you think they are injured Fuzzy Set Ops: Union (OR) If you have x degree of faith in statement A, and y degree of faith in statement B, how much faith do you have in the statement A or B Eg: How much faith in that person is about 6 „¢ high or tall Does it make sense to attribute less truth than you have in one of A or B Fuzzy Set Ops: Union (OR) Take the max of your beliefs in each individual statement Also works if statements are about different variables Fuzzy Rules If our distance to the car in front is small, and the distance is decreasing slowly, then decelerate quite hard Fuzzy variables in blue Fuzzy sets in red Conditions are on membership in fuzzy sets Actions place an output variable (decelerate) in a fuzzy set (the quite hard deceleration set) We have a certain belief in the truth of the condition, and hence a certain strength of desire for the outcome Multiple rules may match to some degree, so we require a means to arbitrate and choose a particular goal  defuzzification Fuzzy Rules Example (from Game Programming Gems) Rules for controlling a car: Variables are distance to car in front and how fast it is changing, delta, and acceleration to apply Sets are: Very small, small, perfect, big, very big  for distance Shrinking fast, shrinking, stable, growing, growing fast for delta Brake hard, slow down, none, speed up, floor it for acceleration Rules for every combination of distance and delta sets, defining an acceleration set Assume we have a particular numerical value for distance and delta, and we need to set a numerical value for acceleration Extension: Allow fuzzy values for input variables (degree to which we believe the value is correct) Set Definitions for Example Instance for Example Matching for Example Relevant rules are: If distance is small and delta is growing, maintain speed If distance is small and delta is stable, slow down If distance is perfect and delta is growing, speed up If distance is perfect and delta is stable, maintain speed For first rule, distance is small has 0.75 truth, and delta is growing has 0.3 truth So the truth of the and is 0.3 Other rule strengths are 0.6, 0.1 and 0.1 Fuzzy Inference for Example Convert our belief into action For each rule, clip action fuzzy set by belief in rule Defuzzification Example Three actions (sets) we have reason to believe we should take, and each action covers a range of values (accelerations) Two options in going from current state to a single value: Mean of Max: Take the rule we believe most strongly, and take the (weighted) average of its possible values Center of Mass: Take all the rules we partially believe, and take their weighted average In this example, we slow down either way, but we slow down more with Mean of Max Mean of max is cheaper, but center of mass exploits more information Evaluation of Fuzzy Logic Does not necessarily lead to nondeterminism Advantages Allows use of continuous valued actions while still writing crisp rules €œ can accelerate to different degrees Allows use of fuzzy concepts such as medium Biggest impact is for control problems Help avoid discontinuities in behavior In example problem strict rules would give discontinuous acceleration Disadvantages Sometimes results are unexpected and hard to debug Additional computational overhead There are other ways to get continuous acceleration References Nguyen, H. T. and Walker, E. A. A First Course in Fuzzy Logic, CRC Press, 1999. Rao, V. B. and Rao, H. Y. C++ Neural Networks and Fuzzy Logic, IGD Books Worldwide, 1995. McCuskey, M. Fuzzy Logic for Video Games, in Game Programming Gems, Ed. Deloura, Charles River Media, 2000, Section 3, pp. 319329. Neural Networks Inspired by natural decision making structures (real nervous systems and brains) If you connect lots of simple decision making pieces together, they can make more complex decisions Compose simple functions to produce complex functions Neural networks: Take multiple numeric input variables Produce multiple numeric output values Normally threshold outputs to turn them into discrete values Map discrete values onto classes, and you have a classifier! But, the only time I „¢ve used them is as approximation functions Simulated Neuron  Perceptron Inputs (aj) from other perceptrons with weights (Wi,j) Learning occurs by adjusting the weights Perceptron calculates weighted sum of inputs (ini) Threshold function calculates output (ai) Step function (if ini > t then ai = 1 else ai = 0) Sigmoid g(a) = 1/(1+ex) Output becomes input for next layer of perceptron Network Structure Single perceptron can represent AND or OR, but not XOR Combinations of perceptron are more powerful Perceptron are usually organized on layers Input layer: takes external input Hidden layer(s) Output layer: external output Feedforward vs. recurrent Feedforward: outputs only connect to later layers Learning is easier Recurrent: outputs can connect to earlier layers or same layer Internal state Neural network for Quake Four input perceptron One input for each condition Four perceptron hidden layer Fully connected Five output perceptron One output for each action Choose action with highest output Or, probabilistic action selection Choose at random weighted by output Learning Neural Networks Learning from examples Examples consist of input, t, and correct output, o Learn if network „¢s output doesn „¢t match correct output Adjust weights to reduce difference Only change weights a small amount () Basic perceptron learning Wi,j = Wi,j + (to)aj If output is too high (to) is negative so Wi,j will be reduced If output is too low (to) is positive so Wi,j will be increased If aj is negative the opposite happens Neural Net Example Single perceptron to represent OR Two inputs One output (1 if either inputs is 1) Step function (if weighted sum > 0.5 output a 1) Initial state (below) gives error on (1,0) input Training occurs Neural Net Example Wj = Wj + (to)aj W1 = 0.1 + 0.1(10)1 = 0.2 W2 = 0.6 + 0.1(10)0 = 0.6 After this step, try (0,1)1 example No error, so no training Neural Net Example Try (1,0)1 example Still an error, so training occurs W1 = 0.2 + 0.1(10)1 = 0.3 W2 = 0.6 + 0.1(10)0 = 0.6 And so on ¦ What is a network that works for OR What about AND Why not XOR Neural Networks Evaluation Advantages Handle errors well Graceful degradation Can learn novel solutions Disadvantages Neural networks are the second best way to do anything Can„t understand how or why the learned network works Examples must match real problems Need as many examples as possible Learning takes lots of processing Incremental so learning during play might be possible References Mitchell: Machine Learning, McGraw Hill, 1997. Russell and Norvig: Artificial Intelligence: A Modern Approach, Prentice Hall, 1995. Hertz, Krogh & Palmer: Introduction to the theory of neural computation, AddisonWesley, 1991. Cowan & Sharp: Neural nets and artificial intelligence, Daedalus 117:85121, 1988. Todo By Monday, Nov 3, Stage 3 demo Thurs Nov 6, Midterm Everything up to and including lecture 15 Use Search at http://topicideas.net/search.php wisely To Get Information About Project Topic and Seminar ideas with report/source code along pdf and ppt presenaion



