Model Parameter Estimation - cs.bham.ac.ukszh/teaching/matlabmodeling/Lecture19... · Model...

Model Parameter Estimation


Shan He

School for Computational ScienceUniversity of Birmingham

Module 06-23836: Computational Modelling with MATLAB


Outline

Outline of Topics

Concepts about model parameter estimation

Parameter estimation using Nelder-Mead Simplex Method

Parameter estimation using Particle Swarm Optimiser

Parameter estimation for Agent-based models

Assignments



What is model parameter estimation?

I After building the model, we need to determine the modelparameters.

I Usually only a fraction parameters can be measured byexperiments.

I Many parameters are hard, expensive, time consuming, oreven impossible to measure.

I Model parameter estimation: to indirectly determine unknownparameters from measurements of experimental data.

I Challenging because:I Experimental data is noisy and sparse, e.g., only a few number

of time points.I Many models are not mathmatically well defined, e.g.,

Agent-based models



Methods for parameter estimation

For equation-based models, we have:I Derivative approximation methods: essentially approximate

derivatives by finite differences, e.g., Euler’s method, and thenfit the parameters by linear regression.

I Pros: the computation time is typically very fastI Cons: derivative approximations lead to inaccurate parameters.

I Bayesian methods: essentially use Bayesian methods to inferparameters from data

I Pros: can handle noisy or uncertain data; can also infer thewhole probability distributions of the parameters rather than apoint estimate.

I Cons: Computational time is very slow because of the need tosolve high-dimensional integration problems – Markov ChainMonte Carlo

I Optimisation methods.

http://www.mathworks.co.uk/matlabcentral/fileexchange/35725-monte-carlo-markov-chain-for-inferring-parameters-for-an-ordinary-differential-equation-model

http://www.mathworks.co.uk/matlabcentral/fileexchange/35725-monte-carlo-markov-chain-for-inferring-parameters-for-an-ordinary-differential-equation-model



Parameters estimation as an optimisation problem

I Suppose we have a system of ODE:

dy

dt= f (t, y;p), y ∈ Rn, f ∈ Rn,

where p ∈ Rm is the vector of parameters, and a collection ofk measurements of experimental data:(t1, y1), (t2, y2), . . . , (tk , yk).

I We aim to minimise the following objective function, which isthe mean square error between the model output andexperimental data:

obj(p) =k∑

j=1

|y(tj ;p)− yj|2,

where | · | denotes standard Euclidean vector norm.



Optimisation algorithms

To solve the above optimisation problem, a variety optimisationalgorithms can be chosen:

I Gradient-based, e.g., Levenberg-Marquardt.I Requirements:

I Mathematically well definedI Smooth and differentiable

I Derivative-free optimisation algorithms:I Direct search algorithms, e.g., pattern search, Nelder-Mead

methodI Metaheuristic algorithms, e.g., Evolutionary Computation and

Particle Swarm Optimiser



Example 1: A simple model

I Observation:

0 1 2 3 4 5 6−20

0

20

40

60

80

100

120

data1data2

I Suppose we have constructed a simple ODE model:{dxdt = −ay + x + t2 + 6t + bdydt = bx + ay − (a + b)(1− t2)

I Task: Estimate parameters a and b from observation



Nelder-Mead Simplex Method

I A well-established direct search algorithm

I A heuristic search method, no guarantee to find optimalsolutions

I Based on the concept of a simplex, which is a special polytopeof N + 1 vertices in N dimensions

I Derivative-free: does not use numerical or analytic gradients

I Implemented in MATLAB as a function: [x,fval] =

fminsearch(fun,x0), where fun is the objective function tobe minimised and x0 is the starting point



Pros and Cons of Nelder-Mead Simplex Method

I Pros: Very fast

I Cons: only suitable for low dimensional and unimodal cases.Performance sensitive to starting point x0



Example 2: Parameter estimation of LV model

I The fur data was collected by Hudson Bay Company morethan 100 years ago

I Theoretical model for predator-prey interaction:Lotka-Volterra (LV) Model



Example: Lotka-Volterra model

{dxdt = x(α− βy)dydt = −y(γ − δx)

I α: prey population growth rate

I β: prey population decline rate

I γ: predator population decline rate

I δ: predator population growth rate



Example: Parameter estimation of LV model

I Objective: to estimate parameters of LV model from theHudson Bay Company fur data

I We select the period between 1908-1935

I We estimate parameters by metaheuristic algorithms



Evolutionary Computation (EC)

I Umbrella term for algorithms inspired by evolutionary systems

I Algorithms are known as evolutionary algorithms (EAs):Genetic Algorithms (GAs), Evolutionary Programming (EP)and Evolution Strategy (ES), etc.

I Also includes closely related nature-inspired approaches:Particle Swarm Optimiser (PSO) and Ant Colony Optimiser(ACO)



Evolution is amazing!

Spiny Rainforest Katydid Sand grasshopper

Walking Stick insectWalking leaf insect

http://www.environmentalgraffiti.com/featured/amazing-insect-camouflage/14128



Evolutionary Computation (EC)

Two main properties:

I Algorithms are usually population based: maintain a set ofpotential solutions at any one time;

I Algorithms are stochastic (non-deterministic): randomelements help obtain good solutions.



General design

In order to apply an EC to a problem, you need:

I A suitable representation of solutions to the problem;

I A way to evaluate solutions;

I A way to explore the space of solutions (variation operators);

I A way to select better solution to guide the search (exploit).



EC terminology

I Population: set of solutions

I Individual: a member in the population:

I Parents: reproducing individuals:

I Fitness function: a function to summarise how close anindividual (solution) is to achieving the set aims



Standard EC procedure

I Generate the initial population P(0) at random, and set t = 1I repeat

I Evaluate the fitness of each individual in P(t).I Select parents from P(t) based on their fitness.I Obtain population P(t + 1) by

I applying variation operators to parents

I Set t = t + 1.

I until termination criterion satisfied



Particle Swarm Optimiser (PSO)

I Invented by Kennedy and Eberhart 1995

I Inspired by bird flocking and fish schooling, more precisely,BIOD

I Simple rules for searching global optima

I Primarily for real-valued optimisation problems

I Simpler but sometimes better than GAs



PSO: detailed algorithm

I Can be seen as a swarm of particles flying in the search spaceto find the optimal solution.

I The variation operator consists of only two equations:

V k+1i = ωV k

i + c1r1(Pki − X k

i ) + c2r1(Pkg − X k

i )

X k+1i = X k

i + V k+1i

where X ki and V k

i are current position and velocity of the ithparticle, respectively; Pi is the best previous position of the ithparticle; Pg is the global best position of the swarm; ω isinertia weigh, typically in the range of (0, 1]; c1 and c2 areconstants, or so-called learning factors; r1 and r2 are randomnumber in the range of (0, 1)



PSO: algorithm illustration

I The search direction of PSO isdetermined by:

I The autobiographical memory,which remembers the bestprevious position of eachindividual Pi in the swarm

I The publicized knowledge,which is the best solution Pg

currently found by thepopulation



PSO: practical issues

I One key problem faced by EC researcher is how to choose theparameter

I PSO: Plug-and-play optimisation algorithm

I Only 3 parameters, all not very sensitive

I ω can be a linearly decreasing value - better local search



EC: further readings and MATLAB toolboxes

I My tutorial slides on Evolutionary Computation

I Evolutionary Computation Online tutorial.

I MATLAB Global Optimization Toolbox

I Genetic Algorithm Optimization Toolbox (GAOT)

http://www.cs.bham.ac.uk/~szh/teaching/matlabmodeling/EC_tutorial.pdf

http://www.lcc.uma.es/~ccottap/semEC/

http://www.mathworks.co.uk/products/global-optimization/

http://www.daimi.au.dk/~pmn/Matlab/dochome/toolbox/GAOT/gaotindex.html




I Estimating parameters by minimising the mean square error ofmodel output and experimental data.

obj(p) =k∑

j=1

|y(tj ;p)− yj|2,

where y denotes the agent-based model.

I Because of the stochastic nature of agent-based model, weneed to run the model many times to get the average output:

obj(p) =k∑

j=1

| 1m

m∑i=1

y(tij ;p)− yj|2,

I Problem: Computationally very demanding!I Solution: parallel computing or GPU to speed up simulation.

http://www.mathworks.co.uk/products/parallel-computing/

http://www.mathworks.co.uk/discovery/matlab-gpu.html



Future direction: automated model construction from data

I Genetic Programming (GP) is a power tools for machinecreativity: Genetic Programming for reinventing patentedinventions.

I GP is useful for constructing model (with parameters) fromexperiment data

I Distilling Free-Form Natural Laws from Experimental Data.

I Automated reverse engineering of nonlinear dynamicalsystems.

I Dialogue for Reverse Engineering Assessments and Methods(DREAM) .

http://www.genetic-programming.com/inventionmachine.html

http://www.genetic-programming.com/inventionmachine.html

http://creativemachines.cornell.edu/sites/default/files/Science09_Schmidt.pdf

http://creativemachines.cornell.edu/papers/PNAS07_Bongard.pdf

http://creativemachines.cornell.edu/papers/PNAS07_Bongard.pdf

http://www.the-dream-project.org/challenges

http://www.the-dream-project.org/challenges


Assignments

Assignments

Download my MATLAB code and data here, please:

I 1. use GAOT toolbox to estimate parameters of LV modelusing the the Hudson Bay Company fur data from year 1860to 1880;

I 2. try PSO to find the starting point for Nelder-Mead SimplexMethod.

http://www.cs.bham.ac.uk/~szh/teaching/matlabmodeling/code/ParameterEstimation.zip

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