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Modelling dynamics of electrical responses in plantsSanmitra Ghosh
Supervisor: Dr Srinandan Dasmahapatra & Dr Koushik Maharatna
Electronics and Software Systems,School of Electronics & Computer Science
University of Southampton
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OutlineIntroductionBlack Box models (System Identification)Modelling plant responses as ODEsCalibration of Models (Parameter Estimation
in ODE using ABC-SMC)ABC-SMC using Gaussian processesFuture workReferences
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IntroductionExperiments
Models ???
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IntroductionTypical electrical responses
Light
Ozone (sprayed for 2 minutes)
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Black Box models
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1 1
( ) ( )( ) ( ) ( ) ( )( ) ( )B q C qA q y t u t e tF q D q
Generalized least-square estimator {A,B,F,C,D} are rational polynomials
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1
( )( )C qD q
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1
( )( )B qF q
1
1( )A q
( )e t
( )y t( )u t
Linear estimator
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Black Box models
Linear BlockInput Nonlinearity
Output Nonlinearity
fBF h
( )u t ( )w t ( )x t ( )y t
Nonlinear Hammerstein-Wiener model structure
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1
( )( ) ( ( ( )))( )pB qy t h f u tF q
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1
( ( ) ( ))N
N p mt
V y t y t
System output
Cost function
This cost function is minimized using optimization
Black Box models
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Modelling responses as ODEs
Proposed model:
(t) models a latent stimulus (v) is a chosen non-linear function of voltage V(t)
𝑑𝐼𝑑𝑡=−𝜇 𝐼
𝑑𝑉𝑑𝑡 =𝐼+ 𝑓 (𝑣)
2 4 6 8 10
4.05
4.10
4.15
4.20
time
Voltage
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Modelling responses as ODEs
(v) is chosen as Micheles-Menten (sigmoidal) and Fitzhugh-Nagumo (cubic) type non-linear function of v(t) (voltage)
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ABC
𝑝 (𝜃∨𝑦 )≈1( ∆( 𝑦 , 𝑥 ) ≤ 𝜀¿ 𝑓 ( x |𝜃 ) π ( θ )
Approximate Bayesian Computation
Prior
Likelihoodposterior
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ABC
ABC Rejection Sampler (Pickard, 1999)
1. Given , π(θ), (x|) 2. Sample a parameter ∗ from the prior
distribution π().
3. Simulate a dataset x from model (x| ∗) with parameter θ .∗
4. if ∆(, ) ≤ then
5. Accept ∗ otherwise reject.
Note: to generate data x from model (x| ∗) we have to solve the ODE
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ABC
ABC-Sequential Monte Carlo (Toni et al, 1999) …
Limitation: extremely slow due to large number of explicit ODE solving for generating simulated data
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ABC
Data
Gaussian Process
== + noise
𝑑 X̂ ( t )𝑑𝑡
≤
The Gaussian process trick
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ABC
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ABCPredator-Pray a b
generated 2.10
2.10
estimated (ABC-SMC) 2.10
2.07
estimated (ABC-SMC-GPDist)
2.10
2.06
Fitzhugh-Nagumo a b c
generated 0.20 0.20 3.00
estimated (ABC-SMC) 0.19 0.20 2.97
estimated (ABC-SMC-GPDist)
0.21 0.22 2.62
Mackay-Glass β n γ τ
generated 2.00 9.65 1.00 2.00
estimated (ABC-SMC) 2.07 9.42 1.03 2.01
estimated (ABC-SMC-GPDist)
2.04 9.16 1.00 2.04
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Future workModel needs to be extended to capture the
variability seen among different electrical responses.
More models are required to represent other stimuli.
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References1. J K Pritchard, M T Seielstad, a Perez-Lezaun, and M W Feldman. Population growth of human Y chromosomes: a study of Y chromosome microsatellites. Mol. Biol. Evol., 16(12):1791–8, December 1999.
2. T. Toni, D. Welch, N. Strelkowa, a. Ipsen, and M. P.H Stumpf. Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface, 6(31):187–202, February 2009.