Archive for the ‘source code’ Category
Fractals are a fascinating toy, one can easily spend an afternoon lost in Mandelbrot or Julia sets. Mathematicians were aware of fractals as early as the 1700s but it wasn’t until we had computers to do the calculations that we really discovered fractals.
Benoit B. Mandelbrot doing research at IBM was revisiting Gaston Julia’s work with fractals (1917) when he discovered the Mandelbrot set. Fractals are simple equations that are recursively computed. These simple equations create complex shapes.
The Mandelbrot function is z = z^2 + c. z and c are complex numbers, z is set to zero, c is the position on the x ( x, yi ) plane. You recursively compute this function to obtain the Mandelbrot fractal. Black is for the numbers that do not escape to infinity, the other colors represent how many loops it takes to escape.
Fractals have found some use in artificial intelligence. In the world of computer games, fractals create plant life, clouds, mountains and other scenery that would not be possible in such detail. Parkinson’s patients are diagnosed by their gait. In 2004 a sensor was developed that measures the patient’s gait, and analyzes the gait using fractals. 2002 fractals were put to use to help predict natural disasters and better model hurricanes. More recently fractal patterns have been found in solar wind. It is hoped this information will allow us to better predict solar storms.
Fractals have been found in Jackson Pollacks paintings and are being used to try to identify real paintings from fakes. They are also being used in image compression. A more fun way to play with fractals is to use them to predict the stock and commodity markets.
Genetic Algorithms create agents who compete to find the best solution to a problem using evolution.
Agents are initially created either using the best chromosomes from a previous run or a random mixture of chromosomes. After each round we evaluate each agent to see how well he has done. We then mix up the genes by switching among our best agents and or adding in some mutations. We repeat this for a set number of rounds or until some goal is reached.
If only gene switching is done and no mutations are entered into the population the agents will often only find local maximums not global maximums. If the number of agents is too small a solution may not be found.
Simple Genetic Program – you have predators, prey and food for the prey. When either a predator or prey reaches reproduction energy 2 of 3 genes are passed directly to the child, one is random.
There are knowledge based agents and expert systems that reason using rules of logic. These systems that do what an expert in a given field might do, tax consulting, medical diagnosis etc. They do well at the type of problem solving that people go to a university to learn. Usually predicate calculus is used to work through a given problem. This type of problem solving is known as ‘system inference’. The program should be able to infer relationships, functions between sets, some type of grammar, and some basic logic skills. The system needs to have three major properties: soundness, confidence that a conclusion is true; completeness, the system has the knowledge to be able to reach a conclusion; and tractability, it is realistic that a conclusion can be reached.
Reasoning is commonly done with if-then rules in expert systems. Rules are easily manipulated, forward chaining can produce new facts and backward chaining can check statements accuracy. The newer expert systems are set up so that users, who are not programmers, can add rules and objects and alter existing rules and objects. This provides a system that can remain current and useful with out having to have a full time programmer working on it.
There are three main parts to the expert system: knowledge base, a set of if-then rules; working memory, a database of facts; inference engine, the reasoning logic to create rules and data.
The knowledge base is composed of sentences. Each sentence is a representation of a fact or facts about the world the agent exists in or facts the expert system will use to make determinations. The sentences are in a language known as the knowledge representation language.
Rule learning for knowledge based and expert systems is done with either inductive or deductive reasoning. Inductive learning creates new rules, that are not derivable from previous rules about a domain. Deductive learning creates new rules from existing rules and facts.
Rules are made of antecedent clauses (if), conjunctions (and, or) and consequent clauses (then). A rule in which all antecedent clauses are true is ready to fire or triggered. Rules are generally named for ease of use and usually have a confidence index. The confidence index (certainty factor) shows how true something is, i.e. 100\% a car has four wheels, 50\% a car has four doors. Sometimes sensors are also part of the system. They may monitor states in the computer or environment. The Rete algorithm is the most efficient of the forward chaining algorithms.
Reasoning can be done using ‘Horn Clauses’, these are first-order predicate calculus statements that have, at most, one true literal. Horn Clauses have linear order time algorithms and this allows for a faster method of reasoning through lots of information. This is usually done with PROLOG or lisp. Clauses are ordered as such: goal, facts, rules. Rules have one or more negative literals and one positive literal that can be strung together in conjunctions that imply a true literal. A fact is a rule that has no negative literals. A list of positive literals with out a consequent are a goal. The program loops checking the list in order, when a resolution is performed a new loop is begun with that resolution. If the program resolves its goal the proof can be given in tree form, ‘and/or tree’.
Nonmonotomic reasoning is used to fix problems created by a change in information over time. More information coming in negates a previous conclusion and a new one needs to be drawn.
A conflict resolution process must be put in place as well to deal with conflicting information. This can be done by: first come, first serve; most specific rule is kept; most recently changed data rule triggered; once rule is resolved take it out of the conflict resolution set.
Forward chaining takes the available facts and rules and deduces new facts which it then uses to deduce more new facts, or invoke actions. Forward chaining can also be done by simply the application of if-then statements: The RETE algorithm is the most efficient at doing forward chaining right now, it compiles the rules into a network that it traverses efficiently. This is similar to the blackboard systems.
Dynamic knowledge bases, known as truth maintenance systems, may be used. This uses a ‘spreadline’ which is similar to a spread sheet that will calculate missing and updated values as other values entered.
General algorithm forward chaining
load rule base into memory
load facts into memory
load initial data into memory
match rules to data and collect triggered rules
if conflict resolution done BREAK
use conflict resolution to resolve conflicts among rules
fire selected rules
Backward Chaining evaluates a goal and moves backward through the rules to see if true. An example is a medical diagnosis expert system, that takes in information from questions then returns a diagnoses. PROLOG systems are backward chaining.
General algorithm backward chaining
load rule base into memory
load facts into memory
load initial data
specify a goal
load rules specific to that goal onto a stack
if stack is empty BREAK
WHILE MORE ANTECEDENT CLAUSES
if antecedent is false pop stack and NEXT WHILE
if antecedent true fire rule and NEXT WHILE
if antecedent unknown PUSH onto stack (we may later ask user for more information about this antecedent.
The Rete Algorithm is considered to be the best algorithm for forward chaining expert systems. It is the fastest but also requires much memory. It uses temporal redundancy, rules only alter a few facts at a time, and structural similarity in the left hand side of rules to do so.
The Rete is a an acyclic graph that has a root node. The nodes are patterns and the paths are the left hand sides of the rules. The root node has one kind node attached to it for each kind of fact. Each kind node has one alpha node attached to it for each rule and pattern. Then the alpha nodes have associated memories which describe relationships. Each rule has two beta nodes. The left part is from alpha(i) and the right from alpha(i+1). Each beta node stores the JOIN relationships. Changes to rules are entered at the root and propagated through the graph.
Knowledge based agents loop through two main functions. One is to sense the world and TELL the knowledge base what it senses, two is to ASK what it should do about what it senses, which it then does. An agent can be constructed by giving it all the sentences it will need to perform its functions. An agent can also be constructed by a learning mechanism that takes perceptions about the environment and turns them into sentences that it adds to the knowledge base.
Particle Swarms take several particles move them randomly around an area looking for a minimum/maximum or some other condition. As particles test each location they land at the best score for the group and each particle is updated. The particles velocity is then adjusted to head more in the direction of each one’s personal best and in the direction of the group best. The velocities also contain a random variable to keep particles from swarming the first good location they find and ending up at a local min or max.
The algorithm is very simple:
For each particle:
test location to see if it is better than previous best
update velocity vectors
End for each particle
Test to see if any particle has beaten the group best so far and up date if so
The velocity vector is calculated:
OldVelocity += ConstantForGroup * randomNumberBetweenZeroAndOne * (GroupBestLocation – CurrentParticleLocation) + ConstantForParticles * randomNumberBetweenZeroAndOne * ( PersonalBestLocation – CurrentLocation)
The time step is decreased each cycle by multiplying dt by .99. Otherwise the particles start zooming around and fail to home in except by brute force.
The example takes six particles through a hundred loops and attempts to find the location of the maximum. The code is heavily commented and should be easy to understand and adjust.
Particle Swarm Optimization
Swarm Intelligence, Focus on Ant and Particle Swarm Optimization
Particle Swarm Optimization ( PSO ) in Python (source code available)
Swarm Behavior – National Geographic
The swarm is reporting for duty
Swarming birds, plasma, crowds and stock markets
Swarm intelligence reaches a new level
Are swarms chaotic
Swarm approach to photography
This AI process mimicks the physical process of annealing. Annealing is the process of heating and cooling a metal in a controlled manner. This gives us a stronger metal if the cooling part is done slowly. A higher temperature means more energy. So if I am trying to place a marble on a hilly landscape in the global minimum I may have to shake things up a great deal to bounce the marble out of a local minimum. So I start out with lots of energy hoping to locate the global minimum. Then reduce the energy hoping to zero in on the minimum better.
Start with a random solution or your best solution from a previous run. This is our current solution. The energy of our solution is then measured. For the eight queens problem the amount of energy is the amount of conflicts between our queens.
We then take a copy of the solution, bounce it around a bit and see if it has less energy than the previous solution. If so we use that as our working solution. If not we test it using P(dE) = exp( -dE/T). At higher temperatures we take worse solutions than we do when we get close to finding the correct solution.
Simulated Annealing is used in traveling salesman and scheduling problems and also it has been used to do some image clean up work.
Set up initial board
Copy initial solution to current, working and best solutions
Bounce the queens around a the board
See if this solution is better than last one ( uses less energy / has less conflicts )
if so use new solution
if not test again using P(dE) = exp( -dE/T )
if this gives a lower energy than a random number we pick, use this solution
Switch copy boards as needed
Bail from loop if problem solved ( energy < some number )
Simulated Annealing was developed by S. Kirkpatrick, CD Gelatt and MP Vecchi in 1983 and by Cerny 1985. It is a Monte Carlo type of solution to a problem.