UCI ICS IGB SISL NKS Washington DC 06/15/06 Towards a Searchable Space of Dynamical System Models...
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Transcript of 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
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
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.
NKS Washington DC 06/15/06
UCI ICS IGB SISL
C1: Demonstration of expressive power in scientific modeling
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
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.
NKS Washington DC 06/15/06
UCI ICS IGB SISLExample: Anabaena Prusinkiewicz et al. model
G. Yosiphon,SISL, UCI
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Example: Galaxy Morphology
G. Yosiphon, SISL, UCI
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Example: Arabidopsis Shoot Apical Meristem (SAM)
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
UCI ICS IGB SISL
QuickTime™ and aYUV420 codec decompressor
are needed to see this picture.
PIN1-GFP expression
Time-lapse imaging over 40 hrs
(Marcus
Heisler,
Caltech)
NKS Washington DC 06/15/06
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
are needed to see this picture.
Emergence of new extended, interacting objects: floral meristem primordia.
DG’s at ≥ 3 scales: - molecular; - cellular; - multicellular.
QuickTime™ and a decompressor
are needed to see this picture.
NKS Washington DC 06/15/06
UCI ICS IGB SISL
QuickTime™ and aMPEG-4 Video decompressor
are needed to see this picture.
Model simulation on growing template
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Ecology: predator-prey models
with Elaine Wong, UCI
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Example: Hierarchical Clustering
NKS Washington DC 06/15/06
UCI ICS IGB SISL
ML example: Hierarchical Clustering
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Logic Programming
• E.g. Horn clauses
• Rules
• Operators
• Project to fixed-point semantics
NKS Washington DC 06/15/06
UCI ICS IGB SISL
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
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
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)
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Discrete-Time Semantics of Stochastic Parameterized Grammars
This formulation can also be used as a programming language, expressing algorithms.
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
UCI ICS IGB SISL
C2: Representation as discrete labeled graph structure that can
be searched and explored computationally
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
UCI ICS IGB SISL
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
}
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
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?)
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
UCI ICS IGB SISL
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 …
NKS Washington DC 06/15/06
UCI ICS IGB SISL
Template: A-Life
Concisely expressed in SPG’s
Steady state condition: total influx into g = total outflow from g
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
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
NKS Washington DC 06/15/06
UCI ICS IGB SISL
For More Information
• www.ics.uci.edu/~emj modeling frameworks