UCI ICS IGB SISL NKS Washington DC 06/15/06 Towards a Searchable Space of Dynamical System Models...

NKS Washington DC 06/15/06

UCI ICS IGB SISL

Towards a Searchable Space of Dynamical System Models

Eric MjolsnessScientific Inference Systems Laboratory (SISL)

University of California, Irvine

www.ics.uci.edu/~emj

In collaboration with: Guy Yosiphon

NKS June 2006


UCI ICS IGB SISL

Motivations shared with NKS

• Objective exploration of properties of “simple” computational systems

• Relation of such to the sciences

• Example: bit string lexical ordering of cellular automata rules; reducibility relationships; applications to fluid flow


UCI ICS IGB SISL

Criteria for a space of simple formal systems

• C1: Demonstrated expressive power in scientific modeling

• C2: Representation as discrete labeled graph structure– that can be searched and explored computationally– E.g. Bayes nets, Markov Random Fields

• roughly in order of increasing size - with index nodes (DD’s)

• C3: Self-applicability– useful transformations and searches of such dynamical

systems should be expressible• … as discrete-time dynamical systems that compute• So major changes of representation during learning are not excluded.


UCI ICS IGB SISL

C1: Demonstration of expressive power in scientific modeling


UCI ICS IGB SISL

Elementary Processes

• A(x) B(y) + C(z) with f (x, y, z)• B(y) + C(z) A(x) with r (y, z, x)• Examples

– Chemical reaction networks w/o params– .

– XXX from paper

• Effective conservation laws– E.g. ∫ NA(x) dx + ∫ NB(y) dy ,

∫ NA(x) dx + ∫ NC(z) dz


UCI ICS IGB SISL

Amino Acid Syntheses

Asp

Thr

KB

Pyr

Leu

Val

Ile

GlycolysisGlucose TCA cycleAla

+

tRNA-Leu

tRNA-Val

tRNA-Ile

tRNA-Ala

tRNA-ThrLys Met

Kmech: Yang, et al. Bioinformatics 21: 774-780, 2005Amino acid synthesis: Yang et al., J. Biological Chemistry, 280(12):11224-32, , Mar 25 2005. GMWC modeling: Najdi et al., J. Bioinformatics and Comp. Biol., to appear 2006.


UCI ICS IGB SISLExample: Anabaena Prusinkiewicz et al. model

G. Yosiphon,SISL, UCI


UCI ICS IGB SISL

Example: Galaxy Morphology

G. Yosiphon, SISL, UCI


UCI ICS IGB SISL

Example: Arabidopsis Shoot Apical Meristem (SAM)


UCI ICS IGB SISL

Co-visualization of raw and extracted nuclei data

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Quantification of growth


UCI ICS IGB SISL

QuickTime™ and aYUV420 codec decompressor


PIN1-GFP expression

Time-lapse imaging over 40 hrs

(Marcus

Heisler,

Caltech)


UCI ICS IGB SISL

Dynamic Phyllotactic Model

H. Jönnson, M. Heisler, B. Shapiro, E. Meyerowitz, E. Mjolsness - Proc. Nat’l Acad. Sci. 1/06

QuickTime™ and a decompressor


Emergence of new extended, interacting objects: floral meristem primordia.

DG’s at ≥ 3 scales: - molecular; - cellular; - multicellular.

QuickTime™ and a decompressor



UCI ICS IGB SISL

QuickTime™ and aMPEG-4 Video decompressor


Model simulation on growing template


UCI ICS IGB SISL

Spatial Dynamics in Biological Development

• Reimplemented weak spring model in 1 page

• Applying to 1D stem cell niches with diffusion, in plant and animal tissues


UCI ICS IGB SISL

Ecology: predator-prey models

with Elaine Wong, UCI


UCI ICS IGB SISL

Example: Hierarchical Clustering


UCI ICS IGB SISL

ML example: Hierarchical Clustering


UCI ICS IGB SISL

Logic Programming

• E.g. Horn clauses

• Rules

• Operators

• Project to fixed-point semantics


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An Operator Algebra for Processes

• Composition is by independent parallelism • Create elementary processes from yet more

elementary “Basis operators”– Term creation/annihilation operators: for each parm value,

– Obeying Heisenberg algebra

[ai, cj] = i j or

– Yet classical, not quantum, probabilities


UCI ICS IGB SISL

Basic Operator Algebra Composition Operations: +, *

Operator algebra

• H1 + H2

• H1 * H2

(noncommutative)

Informal meaning• independent,

parallel occurrence• instantaneous,

serial

co-occurrence

Syntax• parallel rules

• Multiple terms

on LHS, RHS


UCI ICS IGB SISL

Time Evolution Operators

• Master equation:d p(t) / dt = H p(t)

• where 1·H = 0, e.g.H = (H’) = H’ - 1· diag(1·H’ )

• H = time evolution operator– can be infinite-dimensional

• Formal solution: p(t) = exp(t H) p(0)


UCI ICS IGB SISL

Discrete-Time Semantics of Stochastic Parameterized Grammars

This formulation can also be used as a programming language, expressing algorithms.


UCI ICS IGB SISL

Algorithm Derivation:Conceptual Map

DG rules

stochastic program

(H, et H)

(H´, H´n/(1· H´n ·p))

Euler’sformula

Heisenberg Picture

TimeOrderedProductExpansion

CBH

(c)

(d)

Operator Space (high dim)

FunctionalOperatorSpace

TrotterProductFormula


UCI ICS IGB SISL

C2: Representation as discrete labeled graph structure that can

be searched and explored computationally


UCI ICS IGB SISL

Basic Syntax for a Modeling Language: Stochastic Parameterized Grammars (SPG’s)• = set of rules• Each rule has:

– LHS RHS {keyword expression}*

– Parameterized term instances within LHS and/or RHS– LHS, RHS: sets (of such terms) with Variables

• LHS matches subsets of parameterized term instances in the Pool

– Keyword clauses specify probability rate, as a product

• Keyword: with– Algebraic sublanguage for probability rate functions

• rates are independent of # of other matches; oblivious.

• Rule/object : verb/noun : reaction/reactant bipartite graphs– … with complex labels


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Graph Meta-Grammar= 1

= 2

= 3

= 3

= 3

= 1

= 2

{ Aiterm

i,x

i, A i,

i I

Ajterm

j, x

j, A j ,

j I

with ; r 0,1

}


UCI ICS IGB SISL

“Plenum” SPG/DG implementation

• builds on Cellerator experience• [Shapiro et al., Bioinformatics 19(5):677-678 2003]

• computer algebra embedding provides – probability rate language– Symbolic transformations to executability

• includes mixed stochastic/continuous sims


UCI ICS IGB SISL

SPG/DG Expressiveness Subsumes …• Logic programming (w. Horn clauses)

– LHS RHS; all probability rates equal– Hence, any simulation or inference algorithms can in principle be

expressed as discrete-time SPG’s

• Chemical reaction networks– No parameters; stoichiometry = weighted labeled bipartite graph

• Context-free (stochastic) grammars– No parameters; 1 input term/rule– Formally “solvable” with generating functions

• Stochastic (finite) Markov processes– No parameters; 1 input/rule, 1 output/rule– “Solvable” with matrices (or queuing theory?)


UCI ICS IGB SISL

SPG/DG Expressiveness Subsumes …• Bayes Nets

– Each variable x gets one rule:

Unevaluated-term, {evaluated predecessors(y)} evaluated-term(x)

• MCMC dynamics– Inverse rule pairs satisfying detailed balance

– Each rule can itself have the power of a Boltzmann distribution

• Probabilistic Object Models– “Frameville”, PRM, …

• Petri Nets• Graph grammars

– Hence, meta-grammars and grammar transformations

• DG’s subsume: ODE’s, SDE’s, PDE’s, SPDE’s– Unification with SPG’s too


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C3: Self-applicability

-Arrow reversal

-Arrow reversal graph grammar exercise

-Machine learning by statistical inference

-e.g. hierarchical clustering (reported)

-? Equilibrium reaction networks for MRF’s

-Further possible applications …


UCI ICS IGB SISL

Template: A-Life

Concisely expressed in SPG’s

Steady state condition: total influx into g = total outflow from g


UCI ICS IGB SISLApplications to

Dynamic Grammar Optimizationand a “Grammar Soup”

• Map genones to grammars

• Map hazards to functionality tests

• Map reproduction to crossover or simulation


UCI ICS IGB SISL

Conclusions• Stochastic process operators as the semantics for a language

– A fundamental departure– Specializes to all other dynamics

• Deterministic, discrete-time, DE, computational, …• Graph grammars allow meta-processing

• Operator algebra leads to novel algorithms• Wide variety of examples at multiple scales

– Sciences• Cell, developmental biology; astronomy; geology• multiscale integrated models

– AI• Pattern Recognition• Machine learning

• Searchable space of simple dynamical system models including computations


UCI ICS IGB SISL

For More Information

• www.ics.uci.edu/~emj modeling frameworks

UCI ICS IGB SISL NKS Washington DC 06/15/06 Towards a Searchable Space of Dynamical System Models...

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