Parameter Estimation for SSystems using Evolutionary MultiObjective Optimization

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Parameter Estimation for S-Systems using
Evolutionary Multi-Objective Optimization

• Pros Naval
• Computer Vision Machine Intelligence Group
• Department of Computer Science
• University of the Philippines-Diliman

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Basic Facts about Yeast Fermentation

• Yeast Cells undergo alcohol fermentation.
• pathway results in the formation of ethanol and
  carbon dioxide.
• Net 2 ATP are gained by this pathway from a
  glucose molecule.
• Common Uses of Yeast
• 1. Bread
• Carbon dioxide released from the yeast cells
  causing dough to rise. Alcohol evaporates
  during baking.
• 2. Alcohol production
• Sparkling wines wines where the carbon dioxide
  is trapped.
• Other wines carbon dioxide is allowed to escape.

3
The Fermentation Pathway

• External glucose is transported into cell
  Transport is inhibited by Glucose-6-phosphate
• Hexokinase phosphorylates internal glucose into
  glucose-6-phosphate
• In vivo glucose-6-phosphate can flow towards
  ethanol, glycogen, trehalose, oxidative and
  non-oxidative pentose pathways. Only two are
  considered here.
• Glucose-6-phosphate and fructose-6-phosphate are
  at equilibrium are very readily converted into
  each other.
• 3. Phosphofructokinase phosphorylates
  fructose-6-phosphate
• 4. Fructose-1,6-diphosphate can be used to
  produce glycerol or phosphoenolpyruvate.
• 5. Pyruvate kinase catalyses the production of
  ethanol upon activation of fructose-1,6-diphosphat
  e.

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The 5 Dependent Variables
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The 9 Independent Variables
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GMA Test System

• 5 Differential Equations, 9 Independent Variables
• 21 unknowns Chromosome has 21 floating-point
  genes

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Constraints

• Precursor-Product Relations
• V1- V2 V51-
• a2 ß1 ß51
• g21 h11 h51,1
• g25 h15 h55,1
• g27 h17 h57,1
• etc.

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Time Course Data 40 Points
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Parameter Estimation

• Goal Estimate the values 21 parameters of the
  system
• from the given time course data.
• Use Evolutionary Multiobjective Optimization

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What are Genetic Algorithms ?

• Genetic Algorithms (GAs) are search algorithms
  based on the mechanics of natural selection and
  natural genetics.
• GAs combine survival of the fittest among string
  structures with a structured yet randomized
  information exchange to form a search algorithm
  with some of the innovative flair of human
  search.
• GAs effectively exploit historical information to
  speculate on new search points with expected
  improved performance.
• GAs were developed by John Holland and colleagues
  at the University of Michigan.

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Why GA ?

• Robustness balance between efficiency and
  efficacy necessary for survival in many different
  environments.
• GAs are not fundamentally limited by restrictive
  assumptions about the search space.
• Continuity
• Existence of derivatives
• Unimodality (one-peaked search spaces)

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Genetic Algorithms

• Simple Genetic Algorithm
• initialize population
• evaluate population
• while termination criterion not reached
• select solutions for next population
• perform crossover and mutation
• evaluate population

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How GAs differ from Traditional Methods

• 1. GAs work with a coding of the parameter set,
  not the parameters themselves.
• 2. GAs search from a population of points, not a
  single point
• 3. GAs use pay-off information (objective
  function) information, not derivatives or other
  auxiliary knowledge.
• 4. GAs use probabilistic transition rules, not
  deterministic rules

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Other GA Features

• Most optimization methods move from a single
  decision point to the next using some transition
  rule to determine the next point.
• point-to-point method lead to false peaks in
  multimodal (many-peaked) search spaces
• GAs work from a population of points
  simultaneously climbing peaks in parallel

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Other GA Features

• Many search techniques require much auxiliary
  information in order to work properly.
• E.g. gradient techniques need derivatives
• GAs do not need auxiliary information. GAs only
  require pay-off values

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Other GA Features

• GAs use probabilistic transition rules to guide
  their search towards regions of the search space
  with likely improvement.

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Desirable Properties of GAs

• Simplicity
• Parallelizability
• Robust
• Generality

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A Simple Genetic Algorithm

• Problem Maximize f(x) x2 on interval 0, 31
• 1. Encode x as a 5 bit binary string
• 2. Form Initial Population
• 01101
• 11000
• 01000
• 10011
• randomized through 20 successive flips
  of an unbiased coin
• 3. Define Operators
• Reproduction Operator
• Crossover Operator
• Mutation Operator

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Reproduction

• Reproduction process in which individual
  strings are copied according to their objective
  function values.
• String with a higher value have a higher
  probability of contributing one or more offspring
  in the next generation (Darwinian survival of the
  fittest principle)

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Reproduction

• To reproduce simply spin the wheel 4 times
• Once a string has been selected for reproduction,
  an exact replica of the string is made. This
  string is then entered into a mating pool for
  further genetic operator action.

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Crossover Operation

• After reproduction, a simple crossover may
  proceed in 2 steps
• Members of the newly reproduced strings in the
  mating pool are mated at random.
• Each pair of string undergoes crossing over as
  follows
• An integer position k along the string is
  selected uniformly at random between 1 and the
  string length less one 1, l-1. Two new strings
  are created by swapping all characters between
  positions k1 and l inclusively
• Example
• A1 01101 A1 01100
• A2 11000 A2 11001

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Mutation

• Mutation occasional (with small probability)
  random alteration of the value of a string
  position
• Changing a 1 to 0 or vice-versa
• Mutation rate typically one mutation per one
  thousand bits

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Computing the Second Generation
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The Second Generation
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Evolutionary Multiobjective Optimization (EMOO)

• Real-world optimization problems are most of the
  time multiobjective in nature
• Conflicting and competing objectives
• Trade-offs (cost and reliability, mass and
  stability, etc)
• Traditional Approach simplify multi-objective
  problem into a single objective problem
• Use penalty methods in which the different
  criteria are weighted and summed to obtain a
  scalar objective function which is to be
  minimized or maximized
• Multi-to-single objective transformation often
  changes the problem to a different one

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Domination

• A 100, 80, 90, 100
• B 90, 80, 90, 100
• A dominates B because A is better than B in all
  aspects.
• C 90, 80, 90, 100
• D 80, 100, 70, 100
• C and D are non-dominated with respect to each
  other.
• we are interested in non-dominated vectors too
• Very distinct solutions
• Single-objective optimization throws away many
  non-dominated vectors and forces us to accept
  just one or a few very similar optimized
  solutions.

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Multiobjective Optimization (MO)

• We want to find a vector x x1, x2, , xnT
  satisfying the m inequality constraints
• gi(x) 0 i 1, , m
• and the p equality constraints
• hi(x) 0 i 1, , p
• and which optimizes the vector function
• f(x) f1(x), f2(x), , f3(x) T
• where x x1, x2, , xnT is the vector of
  decision variables.

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Pareto Optimality

• x is Pareto-optimal if there is no feasible
  vector that would decrease one criterion without
  causing a simultaneous increase in some other
  criterion.
• Pareto-optimum is very often not a single
  solution but a set of solutions called
  non-dominated solutions.
• Pareto-optimal solutions are optimal in the
  wider sense that no other solutions in the search
  space are superior to them when all objectives
  are considered.

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The Pareto Front
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Why Evolutionary Algorithms ?

• Classical Approaches
• Several independent optimization runs are
  required to obtain an approximation of the
  Pareto-optimal set synergies are not exploited
  causing high computation overhead
• For large scale problems, hardly any real
  classical multi-objective optimization
  techniques had previously been available (Horn,
  1997)
• Weighting method may be sensitive to the shape of
  the Pareto-optimal front
• Problem knowledge may be required which may not
  be available
• Evolutionary Computation Approach
• Single optimization run
• Large search spaces can be handled
• Generality Non-differentiable, non-continuous
  objective functions
• scalability

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Why Evolutionary Algorithms ?

• Evolutionary Algorithms seem to be especially
  suited to Multi-objective optimization for two
  main reasons
• They capture multiple Pareto-optimal solutions in
  a single simulation run
• They exploit similarities of solutions by
  recombination

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EMO Algorithms

• 1. Penalty-Function-Based Selection
• VEGA (Schaefer, 1994)
• 2. Pareto-Based Selection
• NPGA (Horn et. al. 1994)
• MOGA (Fonseca Fleming, 1993)
• 3. Elitism-based Selection
• SPEA (Zitzler et. Al. 2000)
• NSGA-II (Deb et. al. 2000)
• PAES (Knowles Corne, 2000)
• PESA (Corne et. al. 2000)
• 4. With Local Search
• MOGLS (Ishibuchi Murata, 1996), Jaszkiewicz
  (2002)

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NSGA-II

• Basic Idea
• Generate a population of n children from n
  parents
• Population of parents and children are merged
• Non-dominated sort is performed to keep the best
  n individuals for the next generation.

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Evolutionary Computation Issues

• Population Size
• Choice of GA Operators
• Operator Probabilities
• Representation of Decision Variables
• Selection
• Others elitism, niching, etc.
• These parameters interact in complex ways.

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• Crossover is useful for problems that rely on
  building-block exchange. Mutation may destroy
  already-found good information and may not be
  suitable for such problems.
• Use large crossover probability and small
  mutation probability

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Difficulties encountered by EC Algorithms

• Multimodality presence of false attractors
• Deception lower-order schema are led to
  sub-optimal solutions
• Isolation (e.g. needle in the haystack problem)
  no information is available to guide the search
  in any direction
• Collateral Noise noise hides the presence of a
  good sub-solution in a solution

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Population Size

• Too small or too large a population size is
  detrimental
• Both require a large number of generations to
  arrive at the optimum

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Chromosomal Representation of Problem

• The ai, ßi, gij, hij, parameters we want to
  estimate are encoded as real-valued genes in the
  chromosome.
• Initialization 0.01, 0.5
• Constraints 1.0e-9, 1.0
• Macro repairs the sign of the parameters

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The Vector-Valued Fitness Function

• Other objective functions
• values of the numerically calculated dependent
  variables
• values of the numerically calculated slopes

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Simulations

• Settings
• 1. Gaussian Probabilities
• Mean (0.0, 0.1)
• Sigma (0.5, 0.1)
• Pm 0.1
• 2. Crossover Single-Point, pc 0.8
• 3. Population Sizes (500, 300, 100)
• Very Preliminary Results
• Solution found for pop. Size 500
• Generation 2260 (after 38 hrs)
• 1 minute/generation

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3 Solutions Found

• Best Solution at Generation 2260
• x1' 0.403538 x2-0.109715 x6 -0.279625
  x10.344337 x50.483996 x7
• x2' 0.279625 x10.344337 x50.483996 x7
  -0.155649 x21e-09 x5-0.267273 x8 - 0.230363
  x20.417923 x11
• x3' 0.155649 x21e-09 x5-0.267273 x8
  -0.0276623 x30.378066 x50.37052 x9
  x14-0.000762412 -0.0909049 x30.0350299
  x40.785198 x5-0.302526 x12
• x4' 0.0797786 x30.378066 x50.37052 x9
  x14-0.000762412 -0.131994 x30.0350299
  x40.785198 x5-0.302526 x10
• x5' 0.0797786 x30.378066 x50.37052 x9
  x14-0.000762412 0.131994 x30.0350299
  x40.785198 x5-0.302526 x10 -0.155649 x21e-09
  x5-0.267273 x8
• - 0.392697 x5 x13

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• Best Solution
• Objective1 (sum of differences in x) minimum
• Sol 2
• Objective1 (sum of differences in x) minimum
• Sol 3
• Objective2 (sum of differences in xdot) minimum

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X1 (Internal Glucose) Estimate
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X1 (Internal Glucose) Estimate
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X1 (Internal Glucose) Estimate
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X2 (Glucose-6-Phosphate) Estimate
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X2 (Glucose-6-Phosphate) Errors
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X3 (Fructose-1,6-diphosphate) Estimate
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X3 (Fructose-1,6-diphosphate) Errors
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X4 (Phosphoenolpyruvate) Estimate
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X4 (Phosphoenolpyruvate) Errors
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X5 (ATP) Estimate
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X5 (ATP) Errors
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Errors of Best Estimate
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X-dot Based Objective Function
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Summary (Best)
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Using 2 Objective Functions for Fitness

• Other EA Parameters are the same as before.

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The Vector-Valued Fitness Function
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X-Xdot Pareto Fronts
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• Choose Generation 2000
• Computation Time 34 hours
• Pareto Set
• 189 Members
• Select min x, mid, min xdot

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X1 (Internal Glucose) Generation 2000 Errors
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X2 (Glucose-6-Phosphate) Generation 2000 Errors
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X3 (Fructose-1,6-diphosphate) Generation 2000
Errors
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X4 (Phosphoenolpyruvate) Generation 2000 Errors
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X4 (Phosphoenolpyruvate) Generation 2000 Errors
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X5 (ATP) Generation 2000 Errors
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Errors of Best Estimate (Gen 2000)
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Errors of Mid Estimate (Gen 2000)
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Errors of Min Xdot Estimate (Gen 2000)
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Summary Generation 2000 (Min X)
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Future Work

• Determine
• Population Size
• Mutation Rate, Gaussian Probability
• Appropriate Crossover Type, Crossover Probability
• Add noise to data
• Effect of data size on estimate
• Whats next ?
• EMO SA ?
• EMO Local Search ?
• EMO SA Local Search ?
• Needed lots of computational resources