Download presentation

Presentation is loading. Please wait.

Published byJake McGinnis Modified over 3 years ago

1
UCI ICS IGB SISL NKS Washington DC 06/15/06 Towards a Searchable Space of Dynamical System Models Eric Mjolsness Scientific Inference Systems Laboratory (SISL) University of California, Irvine www.ics.uci.edu/~emj In collaboration with: Guy Yosiphon NKS June 2006

2
UCI ICS IGB SISL NKS Washington DC 06/15/06 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

3
UCI ICS IGB SISL NKS Washington DC 06/15/06 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 (DDs) 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.

4
UCI ICS IGB SISL NKS Washington DC 06/15/06 C1: Demonstration of expressive power in scientific modeling

5
UCI ICS IGB SISL NKS Washington DC 06/15/06 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. N A (x) dx + N B (y) dy, N A (x) dx + N C (z) dz

6
UCI ICS IGB SISL NKS Washington DC 06/15/06 Amino Acid Syntheses Kmech: Yang, et al. Bioinformatics 21: 774-780, 2005 Amino 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.

7
UCI ICS IGB SISL NKS Washington DC 06/15/06 Example: Anabaena Prusinkiewicz et al. model G. Yosiphon, SISL, UCI

8
UCI ICS IGB SISL NKS Washington DC 06/15/06 Example: Galaxy Morphology G. Yosiphon, SISL, UCI

9
UCI ICS IGB SISL NKS Washington DC 06/15/06 Example: Arabidopsis Shoot Apical Meristem (SAM)

10
UCI ICS IGB SISL NKS Washington DC 06/15/06 Co-visualization of raw and extracted nuclei data Quantification of growth

11
UCI ICS IGB SISL NKS Washington DC 06/15/06 PIN1-GFP expression Time- lapse imaging over 40 hrs (Marcus Heisler, Caltech)

12
UCI ICS IGB SISL NKS Washington DC 06/15/06 Dynamic Phyllotactic Model H. Jönnson, M. Heisler, B. Shapiro, E. Meyerowitz, E. Mjolsness - Proc. Natl Acad. Sci. 1/06 Emergence of new extended, interacting objects: floral meristem primordia. DGs at 3 scales: - molecular; - cellular; - multicellular.

13
UCI ICS IGB SISL NKS Washington DC 06/15/06 Model simulation on growing template

14
UCI ICS IGB SISL NKS Washington DC 06/15/06 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

15
UCI ICS IGB SISL NKS Washington DC 06/15/06 Ecology: predator-prey models with Elaine Wong, UCI

16
UCI ICS IGB SISL NKS Washington DC 06/15/06 Example: Hierarchical Clustering

17
UCI ICS IGB SISL NKS Washington DC 06/15/06 ML example: Hierarchical Clustering

18
UCI ICS IGB SISL NKS Washington DC 06/15/06 Logic Programming E.g. Horn clauses Rules Operators Project to fixed-point semantics

19
UCI ICS IGB SISL NKS Washington DC 06/15/06 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 [a i, c j ] = i j or –Yet classical, not quantum, probabilities

20
UCI ICS IGB SISL NKS Washington DC 06/15/06 Basic Operator Algebra Composition Operations: +, * Operator algebra H 1 + H 2 H 1 * H 2 (noncommutative) Informal meaning independent, parallel occurrence instantaneous, serial co-occurrence Syntax parallel rules Multiple terms on LHS, RHS

21
UCI ICS IGB SISL NKS Washington DC 06/15/06 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)

22
UCI ICS IGB SISL NKS Washington DC 06/15/06 Discrete-Time Semantics of Stochastic Parameterized Grammars This formulation can also be used as a programming language, expressing algorithms.

23
UCI ICS IGB SISL NKS Washington DC 06/15/06 Algorithm Derivation: Conceptual Map DG rules stochastic program (H, e t H ) (H´, H´ n /(1· H´ n ·p)) Eulers formula Heisenberg Picture Time Ordered Product Expansion CBH (c) (d) Operator Space (high dim) Functional Operator Space Trotter Product Formula

24
UCI ICS IGB SISL NKS Washington DC 06/15/06 C2: Representation as discrete labeled graph structure that can be searched and explored computationally

25
UCI ICS IGB SISL NKS Washington DC 06/15/06 Basic Syntax for a Modeling Language: Stochastic Parameterized Grammars (SPGs) = 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

26
UCI ICS IGB SISL NKS Washington DC 06/15/06 Graph Meta-Grammar = 1 = 2 = 3 = 1 = 2

27
UCI ICS IGB SISL NKS Washington DC 06/15/06 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

28
UCI ICS IGB SISL NKS Washington DC 06/15/06 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 SPGs 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?)

29
UCI ICS IGB SISL NKS Washington DC 06/15/06 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 DGs subsume: ODEs, SDEs, PDEs, SPDEs –Unification with SPGs too

30
UCI ICS IGB SISL NKS Washington DC 06/15/06 C3: Self-applicability -Arrow reversal -Arrow reversal graph grammar exercise -Machine learning by statistical inference -e.g. hierarchical clustering (reported) -? Equilibrium reaction networks for MRFs -Further possible applications …

31
UCI ICS IGB SISL NKS Washington DC 06/15/06 Template: A-Life Concisely expressed in SPGs Steady state condition: total influx into g = total outflow from g

32
UCI ICS IGB SISL NKS Washington DC 06/15/06 Applications to Dynamic Grammar Optimization and a Grammar Soup Map genones to grammars Map hazards to functionality tests Map reproduction to crossover or simulation

33
UCI ICS IGB SISL NKS Washington DC 06/15/06 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

34
UCI ICS IGB SISL NKS Washington DC 06/15/06 For More Information www.ics.uci.edu/~emj modeling frameworks

Similar presentations

OK

Some Probability Theory and Computational models A short overview.

Some Probability Theory and Computational models A short overview.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Slide show view ppt online Ppt on paintings and photographs related to colonial period dates Ppt on cooperative society Free ppt on layers of atmosphere Ppt on spiritual leadership Ppt on principles of object-oriented programming languages Ppt on latest technology in ece Maths ppt on exponents and powers Ppt on object recognition in dip Ppt on covering the environmental problems causes effects and solutions