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Introduction to Swarm IntelligenceOrigins, Observations of Nature

Core Concepts and PrinciplesSI based Algorithms

Ant Colony OptimizationParticle Swarm Optimization

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Swarm Intelligence:An Introduction

Nathan Bell

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What is Swarm Intelligence?Why is Swarm Intelligence Interesting?

Introduction to Swarm Intelligence

This presentation will cover the following topics:What is Swarm Intelligence?Origins of Swarm IntelligenceCore ConceptsApplications and SI based algorithms

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What is Swarm Intelligence?

What is “Swarm Intelligence” (SI)?http://www.youtube.com/watch?v=jEGV4ZSP22A

“The collective behaviour of a decentralized, self-organizedsystem”What is “Decentralization”?What is “Self-Organization”?

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http://www.youtube.com/watch?v=jEGV4ZSP22A




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What is Swarm Intelligence?

What does this definition mean to us?System consisting of a population of “agents”Simple Interactions between agentsLeading to complex high-level behaviour

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Why is Swarm Intelligence Interesting?

Why is SI interesting to us?Framework for decentralized and scalable problem solvingUnique perspective for addressing many problems

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SI naturally lends itself to alternative computational models:Distributed modelsMassively parallel modelsFlexible, dynamic modelsRoboticsetc...

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Useful in simulating many real-life systemsBehaviour of crowdsTraffic simulationSpread of diseaseetc...

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OriginsAnt ColoniesExample: Ant Foraging BehaviourTrail Optimization

Origins, Observations of Nature

Where did the idea of SI originate?Inspired by the success of swarming creatures in natureIn particular: ants, termites and bees

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Ant Colonies

What’s so interesting about ants?One of the most successful species on the planetColonies can range from tens, to millions, of antsIndividual ants are extremely basic creaturesColony displays a complex structure and behaviourHow do they achieve this success?

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Colony Behaviour

A basic look at ant colony behaviour:Coordinated among specialized workersLabour tasks include:

DefenceFood CollectionBrood CareNest CleaningNest Construction

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Colony Behaviour

Ants achieve complex behaviours:Division of labour, adaptive task allocationPath finding and optimizationClustering and sortingStructure formationRecruitment for foraging and collective transport of food

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Colony Behaviour

How does the colony address these tasks?Ants, as individuals, are extremely basicNo single ant could plan these complex tasksThese tasks are easily completed by the ant colonyHow is this possible?

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Colony Behaviour

Colony behaviour:Labour within ant colonies is decentralizedNo central planning or controlLocal interactions between antsLaying of chemical pheromonesComplex behaviour arisesA swarm intelligence is displayed

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Ant Foraging

Ant Foraging:Ants effectively locate and exploit food sourcesFood sources are discovered by random explorationAnts discover efficient paths to known food sourcesAnts have no high-level knowledge of the situationHow do ants form these efficient paths?

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Ant Foraging

The key is pheromoneWhile searching for food, ant movement is largely random

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Ant Foraging

Ants will return to the colony with foodAnts with food will lay a trail of pheromone along their path

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Ant Foraging

Ants are sensitive to pheromoneThe pheromone deposited attracts other antsAnts encountering a pheromone trail will tend to follow itThese ants are more likely to reach the food source

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Ant Foraging

Over Time:More ants discover the food sourceMore pheromone is depositedExisting trails to the food source are strengthened

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Trail Optimization

At the high level:Ants will seem to choose more optimal pathsHow is this achieved?Combination of:

Randomness of movementNature of pheromone

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Trail Optimization

Properties of pheromone:Pheromones are chemicals subject to:

EvaporationDiffusionOther environmental effects

Pheromone weakens over timePheromone trails will dissipate and spread out

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Trail Optimization

Example:A food source has been discovered by two antsOne ant happens to take a more efficient path

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Trail Optimization

Each trail initially attracts a roughly equal number of antsAnts on the more efficient path arrive earlier

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Trail Optimization

Ants return, depositing additional pheromonePheromone trail is strengthened on return

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Trail Optimization

Ants following the more efficient trail have shorter tripsTrips are completed with a higher frequencyHigher frequency leads to stronger pheromone

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Trail Optimization

Ants will favour the stronger pheromone trailLess efficient trail slowly dissipatesThe colony has chosen the more efficient pathNo knowledge of the global situation!

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Core Concepts and PrinciplesDecentralizationSelf-OrganizationEmergent Behaviour

Core Concepts and Principles

What differentiates Swarm Intelligence from otherPopulation-based methods?

We must understand the core components:DecentralizationSelf-OrganizationEmergent Behaviour

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Decentralization and Self-Organization

Decentralization:Population of roughly homogeneous agentsControl is fully distributed among the populationNo central “brain” controlling the agentsEach agent has roughly the same level of influence

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Decentralization and Self-Organization

Self-Organization:Agents each act according to their own behaviourAgents communicate and interactCommunications can include:

Direct contactLocal exchange of informationLocal broadcastsStigmergyetc...

Global behaviour arises from the interactions of the agents

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Emergent Behaviour

“Emergent Behaviour” is the heart of SIA High-level behaviour of a population based systemMust arise from the local interactions within the populationSelf-organization of a decentralized populationTwo categories: Weak and Strong

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Emergent Behaviour

Weak emergent behaviour of a population:can be reduced to the behaviour of a single individual

Strong emergent behaviour of a population:can not be reduced to the behaviour of a single individual

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Emergent Behaviour

Weak Emergent Behaviour:Extremely commonCan be easily predicted by looking at a single individualOften simple to engineerFor example, if individuals simply “move forward”, thepopulation will, as a whole, move forward

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Emergent Behaviour

Strong Emergent Behaviour:Heart of SI and very interesting phenomena itselfHard to predict from the behaviour of an individualMay seem to “transcend” the capabilities of the individual“The whole is greater than the sum of its parts”It is often quite difficult to engineerFor example, path optimization of foraging ants

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SI and Artificial LifeSI for Problem Solving

SI and Artificial Life

SI and Artificial Life:SI was first explored in the field of artificial lifeSimulations of swarming creatures“Boids” algorithmMotivation: learning more about emergent behaviours

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SI and Artificial LifeSI for Problem Solving

SI based Algorithms

SI for problem solving:Relatively new fieldActive field of researchSI algorithms have been used to address many problemsMost commonly, optimization problems

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Ant Colony OptimizationHow Does Ant Colony Optimization Work?Generic Ant Colony Optimization AlgorithmExample: Ant Colony Optimization for Travelling SalesmanExample Application: ACO for Classification

Ant Colony Optimization

Ant Colony Optimization (ACO):Meta-heuristic method for combinatorial optimizationOne of the first SI algorithms for optimizationModelled after the foraging behaviour of antsACO finds good paths through graphsApplicable to a wide variety of problems

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How ACO works

How does ACO work?ACO simulates the way ants communicate via pheromoneIteratively generates paths through graphsPheromone “deposited” on graph edgesPheromone levels effect edge choicesGood paths will emerge over time

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How ACO works

Generic ACO Process:Generic ACO consists of a two phase loop

Edge selection phasePheromone update phase

Loop iterates until reaching some termination criteria

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Generic Ant Colony Optimization

Generic edge selection phase:Population of ants placed into the graphAnts move randomly through the graphMovement influenced by edge weights and pheromonePhase ends when all ants have created some kind of path

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Generic ant movement:For current node x with set of neighbours YChoose edge xy , y ∈ Y with probability pxy

pxy =(τxy )(ηxy )∑

yi∈Y (τxyi )(ηxyi )

ηxy - Contribution of edge weightτxy - Contribution of pheromone

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Generic pheromone update phase:Existing pheromone levels are reduced via "evaporation"

τxy = (1− ρ)τxyρ - Evaporation coefficient parameter

Ants return along the path taken through the graphAt each edge, ants deposit pheromonePheromone deposited according to: ∆τ k

Path of each ant is evaluated using heuristic∆τ k ∝ heuristic path value of ant k

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Ant Colony Optimization for Travelling Salesman

Travelling Salesman Problem (TSP):Given a set of citiesVisit each city exactly onceReturn to the originFind the tour with shortest total length

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Travelling Salesman Problem (TSP):NP-HardModelled as a complete graphEach node represents a cityEdges have weight equal to the distance between citiesACO is applied to find good solutions in polynomial time

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ACO for TSP:Convert edge distance values into attractiveness value ηFor each pair of cities x , y :

ηxy = P/dxyP - Initial attractiveness parameterdxy - Distance between cities x and y

Each ant will form a valid tour during edge selectionPheromone deposited according to tour values

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Edge selection for TSP:Each ant randomly assigned some city as their originAnts travel through the graph visiting all citiesAnts blacklist cities they have previously visitedBlacklisted cities do not contribute to pxy valuesAfter visiting all cities, ants return to their origin

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Selecting the next city in the tour:Current city xSet of unvisited cities YMove to city y ∈ Y with probability pxy

pxy =(τxy )(nxy )∑

yi∈Y (τxyi )(nxyi )

If Y is empty, move to origin then stop

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For example, consider the following situation:

A

B

C

D

E

F

Gd=8.00

d=16.12

d=14.14

d=14.56

T=3.21

T=1.10

T=0.81

T=2.98

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X d η τ pA – - – – - – – - – – - –B – - – – - – – - – – - –C 14.56 2.98D – - – – - – – - – – - –E 14.14 0.81F 16.12 1.10G 8.00 3.21

Calculate η for each move:ηxy = P/dxy

P = 100 for example

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X d η τ pA – - – – - – – - – – - –B – - – – - – – - – – - –C 14.56 6.87 2.98D – - – – - – – - – – - –E 14.14 7.07 0.81F 16.12 6.20 1.10G 8.00 12.5 3.21

Calculate p for each move:

pxy =(τxy )(ηxy )∑

yi∈Y (τxyi )(ηxyi )

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X d η τ pA – - – – - – – - – 0B – - – – - – – - – 0C 14.56 6.87 2.98 0.28D – - – – - – – - – 0E 14.14 7.07 0.81 0.08F 16.12 6.20 1.10 0.09G 8.00 12.5 3.21 0.55

Choose the next moverandomly according to p:

C is chosen

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Move to C and repeat:

A

B

C

D

E

F

G

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When all cities have been visited, a valid tour has been created

A

B

C

D

E

F

G

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Pheromone Update Phase:Each ant calculates the cost of their tourAnts travel along their tours depositing pheromoneFor each ant k :

∆τ k = Q/Lk

Lk - Value of ant k ’s tourQ - Pheromone attractiveness parameter

For each edge xy :τxy = (1− ρ)τxy +

∑k ∆τ k

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Consider this tour:

A

B

C

D

E

F

G

d=20

d=10.77

d=14.56

d=14.56

d=23.32d=10

d=18.44

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Pheromone deposit is calculated:

L = dAB + dBD + dDC + dCF + dFE + dEG + dGA

= 10.77 + 14.56 + 14.56 + 23.32 + 10 + 18.44 + 20= 111.65

Q = 100 for example

∆τ = Q/L= 100/111.65= 0.89

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Algorithm:For each edge xy :

Initialize ηxy = P/dxyInitialize τxy = 0

Run simulation loop until reaching termination criteriaEdge SelectionPheromone Update

Return best tour found as output

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Behaviour in early iterations:Little to no pheromoneAnts generate tours in a greedy wayClosest cities chosen with high probabilityHigh variety in TSP tours generated

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Behaviour in later iterations:Pheromone levels build up on popular edgesEdges used by “good” tours will have more pheromonePheromone levels begin to have greater influenceAnts generate similar tours using these “good” edges

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Early iterations:

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Later iterations:

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Convergence:

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ACO for Classification

ACO finds applications in a wide variety of fieldsFor example: data miningRule-based classification

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Rule Based Classification

Given an object with a set of attributes:Assign a predefined class to the object

Based on a set of rulesFind a rule matching the attributesAssign a class using this rule

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Rule Based Classification

Classification Rule:A rule assigns a class with the form:

IF < conditions > THEN < class >IF A and B THEN 0IF (A and C) or D THEN 1etc...

A condition has the form:Attribute, Operator, Value

Colour = BlueWidth < 2etc...

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ACO for Rule Based Classification

ACO can discover rules from a data set:These rules are of the form:

IF < condition AND condition AND ... > THEN < class >These conditions are restricted to “categorical” attributes

Attribute = Value

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Rules as Paths

Rules can be interpreted as paths through a graph:

Attribute A

Value 1

Value 2

Value 3

Attribute B

Value 1

Value 2

Value 3Attribute C

Value 1

Value 2

Value 3

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Rules as Paths

Vertices represent conditions:Attribute-value pair“Attribute = Value”

Edges represent “AND” relations between conditions

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Rules as Paths

IF A=1 AND B=2 AND C=3:

Attribute A

Value 1

Value 2

Value 3

Attribute B

Value 1

Value 2

Value 3Attribute C

Value 1

Value 2

Value 3

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Rules as Paths

IF B=1 AND C=1:

Attribute A

Value 1

Value 2

Value 3

Attribute B

Value 1

Value 2

Value 3Attribute C

Value 1

Value 2

Value 3

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Applying ACO

To apply ACO to this graph representation, we require:Initial edge weightsHeuristic for depositing pheromone

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Initial Edge Weights

Edge weights are assigned using the training set:Edges leading to an Attribute-Value pair are assignedweightings based on that Attribute-Value pairWeights are determined according to the normalizedShannon Entropy of that Attribute-Value pair within thetraining set

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Initial Edge Weights

Shannon Entropy:

H(W |Aa = Vav ) = −∑

w∈W

(P(w |Aa = Vav ) ∗ log2 P(w |Aa = Vav ))

W : Set of possible classesAa: Attribute a ∈ AVav : Value v ∈ Va

Represents the information gained by observing a givenAttribute-Value pair according to the training set

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Edge Weights

For edges leading to value v ∈ Va of attribute a ∈ A:

η =log2 |W | − H(W |Aa = Vav )∑

i∈A∑

j∈Vi

(log2 |W | − H(W |Ai = Vij)

)A normalized and inverted Shannon Entropy

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Pheromone Heuristic

New rules are evaluated using the “quality” measure:

Q =TP

TP + FN∗ TN

FR + TN

TP : True PositivesTN : True Negatives

FP : False PositivesFN : False Negatives

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ACO Rule Generation Algorithm

Until termination:Initialize edge weights using the uncovered training setUntil convergence:

Generate a new rule as a pathDeposit and evaporate pheromone

Add the best-found rule to the final set of rulesDiscard training cases covered by the new rule

Output: A list of classification rules

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Convergence and Termination

Convergence occurs when either:The same rule is generated a specified number of timesThe maximum number of ants per iteration is reached

Termination occurs when:The number of uncovered cases reaches a threshold

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Result

ACO has successfully been applied to generate a set ofclassification rules from a training set.These rules can now be applied to perform classification.

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Particle Swarm OptimizationReal-valued Black-box OptimizationHow Particle Swarm Optimization WorksEffects of ParametersAlgorithmSwarm BehaviourBenefits of Particle Swarm OptimizationExample Application: Broadcast Tower Coverage

Particle Swarm Optimization

Particle Swarm Optimization (PSO):SI algorithm for real-valued, black-box optimizationDiscovered accidentallyOriginally a simulation of “social flocking” behaviourPopulation made of “particles” in the solution spaceCollision free “bird flocking” behaviour

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Real-valued Black-box Optimization

Real-valued optimization problems:Find the optimal input to some objective functionEach parameter is a real number

Solution space:Parameters define a “solution space”Each parameter corresponds to a dimensionA point in this space represents a function input

Black-box optimization problems:Objective function provided as a “black-box”Nothing is known about the functionMust learn about the function through evaluations

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How Particle Swarm Optimization Works

Particles:Particles exist as points in the solution spaceThese points represent input values to objective functionAssigned values by evaluating the objective function“Fly” through solution spaceCommunicate via broadcast

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Each particle is simply made up of:Position ~XVelocity ~VPersonal Best Point ~Pi

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Particle behaviour:Particles move according to their velocityVelocity is updated through accelerationsInertial weighting is applied to prevent explosive speedsAcceleration towards the particle’s best found pointAcceleration towards the swarm’s best found point

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0x1

-10 10

-10

0x2

10

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Acceleration towards personal best:Self influenceInfluence of the particle’s own knowledge

Acceleration towards global best:Social influenceInfluence of the swarm’s collective knowledgeGlobal best points are communicated via broadcast

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Accelerations:Randomly weighted according to parameters

φp - Personal influence parameterφg - Social influence parameter

Separate random weighting per dimensionMore “explorative”

Constant deceleration is appliedω - Inertial weighting parameterAllows stronger accelerations without explosive speedsPromotes convergence

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Effects of Parameters

Low personal influence, high social influence:Faster convergenceMore susceptible to local optimaExploitation > Exploration

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High personal influence, low social influence:Slower convergenceLess susceptible to local optimaExploitation < Exploration

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NO personal influence, high social influence:“Social only” swarmAll particles immediately converge to the best found pointBehaviour degrades to simple repeated local search

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High personal influence, NO social influence:“Self only” swarmEach particle acts completely independentlyBehaviour degrades to multiple local searchNo “swarm intelligence” present

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Parameter values:Matter of exploration vs. exploitationExploration required for more “difficult” problemsExploitation required for convergence and efficiency

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Parameters:Typically, both φp and φg are set to 2ω set to around 0.7Must be tweaked for each problem to get best resultsThis tweaking is often unintuitiveRequires knowledge of the solution space

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Particle Swarm Optimization Algorithm

Velocity and position update equations:For each dimension d :

Vd = ωVd + U[0, φp](Pid − Xd ) + U[0, φg](Pgd − Xd )

Xd = Xd + Vd

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Example Particle Update

Consider a particle with:X = [5,9]

V = [7,−5]

P = [0,2]

And global best found point:Pg = [10,11]

With PSO Parameters:ω = 0.7φp = 2, φg = 2

1

2

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Inertial weighting is applied:

V1 = ωV1

= 0.7 ∗ 7= 4.9

V2 = ωV2

= 0.7 ∗ −5= −3.5

1

2

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Example Particle UpdateRandom acceleration towards P:

V1 = V1 + U[0, φp](P1 − X1)

= 4.9 + U[0,2](0− 5)

= 4.9 + (1.1 ∗ −5)

= −0.6

V2 = V2 + U[0, φp](P2 − X2)

= −3.5 + U[0,2](2− 9)

= −3.5 + (0.2 ∗ −7)

= −4.9

1

2

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Example Particle UpdateRandom acceleration towards Pg :

V1 = V1 + U[0, φg](P1 − X1)

= −0.6 + U[0,2](10− 5)

= 4.9 + (0.4 ∗ 5)

= 6.9

V2 = V2 + U[0, φg](P2 − X2)

= −4.9 + U[0,2](11− 9)

= −4.9 + (0.9 ∗ 2)

= −3.1

1

2

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Update position:

X1 = X1 + V1

= 5 + 6.9= 11.9

X2 = X2 + V2

= 9− 3.1= 5.9

1

2

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Particle Swarm Optimization Algorithm

Algorithm:Initialize particles:

Random positionRandom velocityEvaluate initial positions

While termination criteria not met:Update Pg via communicationFor each particle i :

Update velocityUpdate positionEvaluate new positionUpdate Pi

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Swarm Behaviour

Behaviour in early stages:Particles will fly through the solution spaceOvershoot the global and personal best pointsArc and loop around these pointsCoarse-grained searchSwarm finds “good” areasParticles slow over time, swarm begins to converge

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Swarm Behaviour

Behaviour in later stages:Stagnation of global best point leads to convergenceParticles gather around this point, slowing over timeFine-grained search of the “good” areaSwarm finds the best point within the “good” area

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Benefits of Particle Swarm Optimization

Benefits of PSO:Very simple to implementNo costly computationsEasily extendable to problems of any dimensionEffective on difficult, noisy problemsCan be applied to any black-box real-valued functionCan even be used in meta-optimizationPSO finding optimal parameters for PSO on some problem

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Example Application

PSO can be applied to solve a problem if:A solution to the problem can be represented as a numberof real-valued parametersA solution’s quality can be represented by a single valueA function is provided which evaluates any given solution

As an example, consider the problem of maximizing thecoverage of a number of broadcast towers.

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Simple Broadcast Coverage Problem

Given a set number of broadcast towers:Assign each tower an x , y coordinateAssign each an amount of power

In order to:Maximize coverageMinimize power costs

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Simple Broadcast Coverage Problem

In order to apply PSO we must:Form a real-valued solution spaceRepresent the problem as a function which:

Assigns values to points in the solution spaceEvaluates the quality of a given solution point

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Tower Representation

Each tower has 3 parametersx coordinatey coordinatepower r

(x,y) r

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Solution Space

Multiple towers can be respresented:[Tower1,Tower2, ...,Towern]

[x1, y1, r1, x2, y2, r2, ..., xn, yn, rn]

A solution space with 3n dimensionsA particle’s position in this space describes:

The positions and power values of n towersA candidate solution to the tower coverage problem

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Example Point

For example, X = [3,3,5,8,11,3,13,4,2] corresponds to:

(3,3) 5

(8,11) 3

(13,4)

2

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Evaluating a Solution

PSO requires a function to evaluate these solutions:Total coverage of the towersPenalize for power costs

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Coverage

For simplicity:Houses withinbroadcast rangeEach house is worthsome set valueFor example:

10 per house

: Covered House

: Uncovered House

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Power Cost

For simplicity, the cost of a tower is:A flat initial cost, for example 100

If r <= 0, the tower is considered unusedNo initial cost in this case

Power cost scaling exponentially with rFor example: r2

Cost of n towers =∑n

i=0

{0, ri <= 0100 + r2

i , ri > 0

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Final Evaluation Function

The final evaluation function is then:

f (...) = HouseCoverage − PowerCosts

= 10 ∗ |Covered | −n∑

i=0

{0, ri <= 0100 + r2

i , ri > 0

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Example

f (7,7,7,21,10,6)

=10 ∗ |Covered |

−n∑

i=0

{0, ri <= 0100 + r2

i , ri > 0

=10 ∗ 34

−(

72 + 100)

−(

62 + 100)

=55

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Example

f (13,10,13,21,11,0)

=10 ∗ |Covered |

−n∑

i=0

{0, ri <= 0100 + r2

i , ri > 0

=10 ∗ 49

−(

132 + 100)

− (0)

=121

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Applying PSO

Applying PSO:Plug in the evaluation functionProvide appropriate parameter ranges

PSO will search for the optimal solution:According to the provided evaluation function

Accuracy of this function is essentialPSO will exploit errors in this function

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Thank You

Questions?

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IntroductionBare-Bones PSOAbstracting the BBPSO AlgorithmParticle Field OptimizationFurther Developing PFOFinal PFO Algorithm

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My research:Particle Field Optimization (PFO)Based on the PSO algorithmAbstraction of the “Bare-Bones” PSO model

Exploring:New high-level conceptNew behaviourNew avenues for development

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Bare-Bones PSO

Bare-Bones Particle Swarm (BBPSO):Abstraction of the core PSOSimplifies the particle update processRemoval of velocity and accelerationRetains roughly the same behaviour

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Bare-Bones PSO

From PSO to BBPSO:Observations of a single particle during swarm stagnationEach dimension shows a distinct bell-curve histogramCan a similar histogram be achieved in a simpler way?

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Bare-Bones PSO

New method of position update:Update particles according to Gaussian distributionDistribution constructed to mimic histogram bell-curveVelocity and accelerations discarded entirelyParticle movement abstracted to a random sampling

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Bare-Bones PSO

BBPSO particle update:For each particle i:

Pm =Pi+Pg

2For each dimension d :

Xid = N (Pmd , |Pid − Pgd |)

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Bare-Bones PSO

Result of BBPSO:Roughly the same high-level behaviourRoughly the same level of performanceParticles no longer “fly” through the spaceMuch simpler particle behaviourMore predictable, easier to analyze

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Abstracting the BBPSO Algorithm

Abstracting the BBPSO:It is possible to further abstract the BBPSORecall the components of the canonical PSO particle:

PositionVelocityPersonal best found pointCommunicated knowledge of global best

Each component is required to update the particle

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Abstracting the BBPSO:The BBPSO algorithm removes the velocity componentPosition is updated by random samplingSampled distribution is independent of current positionCommunication of global bests depend on personal bestsCurrent position is used only to update the personal best

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Particles without positions:Particles can be updated without storing a positionSample the random distributionEvaluate new pointUpdate the best found point if neededDiscard the new point

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Effects of removing particle position:Identical behaviour to BBPSODifferent conceptsParticles no longer exist as explicit pointsEach particle defined by its best found pointParticle’s construct and sample a random distributionProbability of that particle existing at a given point

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New Model

Moving forward with this concept:Individuals are no longer candidate solutionsEach represents a probability fieldSimple Gaussian distributionsNo longer “Particles”Instead, “Particle Fields”

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New Model

At the population level:Population forms a complex probability fieldSum of simple individual fieldsProbability field of all possible particle locations

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New Model

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

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Population probability field:Probability field updated as individuals are updatedShows how PSO explores the solution spaceDemonstrates how PSO “learns”Similarities to a limited Bayesian learning process

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Generating candidate solutions:Solutions have been separated from particlesMaintain a population of candidate solutionsSolutions generated by sampling complex populationdistributionFor each candidate solution:

Randomly select a particle field individualGenerate position from that individual’s distribution

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Updating the particle field individuals:Each solution chooses an individual during generationIndividual’s are updated using associated solutionsSimilar to the standard PSO methodAdapted to handle any number of associated solutions

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Moving forward again:Still roughly the same behaviour as the BBPSOSlightly more randomBiggest difference is the high-level conceptNew concept provides new avenues for development

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Modifying the population distribution:Now possible to modify the population probability fieldConsider complex population distributionSum of simple individual distributionsApply simple weighting schemeDrastic change to population distributionFocus search to better areas?

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Weighting Schemes

Effects of weighting schemes:Introduce new behaviourChanges the population probability fieldChanges how the swarm “learns”Incorporate different information into the search

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Effects of Weighting

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Population Probability Field

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Effects of Weighting

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Weighted Population Probability Field

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p1: 0.01p2: 0.4p3: 0.01p4: 0.1p5: 0.3p6: 0.08population

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Relative Population Sizes

Modifying relative population sizes:Model now consists of two separated populationsPossible to modify population sizes independentlyRelative sizes effects the high level behaviourMore particle field individuals:

May be more exploratoryLess particle field individuals:

Faster convergence

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Final PFO Algorithm

Final PFO Algorithm:Combining all these changes we have a distinct algorithmNew behaviour and perspectiveNew avenues for future development

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PFO Simulation Step

For each candidate solution to be generated:Sample complex distribution defined by PF populationRandomly choose a particle field individualAccording to weighting valuesSample that individual’s simple distribution

For each PF individual:Choose best associated solutionUpdate personal best point if neededCalculate new weighting value

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Results

Preliminary experimental results:Weighting scheme for individuals:

Average value of candidate solutions generated by eachBetter information than just the best found?

Tests done with different ratios of population sizesCompared to BBPSOBetter performance on all test problems

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Thank You

Questions?

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