Transcript Computational Discovery of Communicable Knowledge

Computational Discovery of
Explanatory Process Models
Pat Langley
Computational Learning Laboratory
Center for the Study of Language and Information
Stanford University, Stanford, California
http://cll.stanford.edu/~langley
[email protected]
Thanks to N. Asgharbeygi, K. Arrigo, S. Bay, S. Dzeroski, A. Pohorille, J. Sanchez,
K. Saito, Oren Shiran, J. Shrager, and L. Todorovski for their contributions to this
research, which is funded by a grant from the National Science Foundation.
Adbuctive Model Construction
Most mature sciences focus their efforts not on discovering laws
or forming theories, but on constructing models that:
 build upon known laws and theoretical principles;
 adapt this knowledge to a particular scientific setting;
 augment the knowledge with auxiliary assumptions;
 use the resulting model to explain observed phenomena.
This task involves abduction of explanatory models from domain
knowledge, though it may also have inductive aspects.
In this talk, I examine the construction of explanatory models for
dynamical systems that change over time.
Time Series from the Ross Sea Ecosystem
Inductive Process Modeling
Our approach is to design and implement computational methods
for inductive process modeling, which:
 represent scientific models as sets of quantitative processes;
 use these models to predict and explain observational data;
 search a space of process models to find good candidates;
 utilize background knowledge to constrain this search.
This framework has great potential both for modeling scientific
reasoning and aiding practicing scientists.
A Process Model for an Aquatic Ecosystem
model AquaticEcosystem
variables: phyto, zoo, nitro, residue
observables: phyto, nitro
process phyto_loss
equations: d[phyto,t,1] =  0.307  phyto
d[residue,t,1] = 0.307  phyto
process zoo_loss
equations: d[zoo,t,1] =  0.251  zoo
d[residue,t,1] = 0.251
process zoo_phyto_grazing
equations: d[zoo,t,1] = 0.615  0.495  zoo
d[residue,t,1] = 0.385  0.495  zoo
d[phyto,t,1] =  0.495  zoo
process nitro_uptake
conditions: nitro > 0
equations: d[phyto,t,1] = 0.411  phyto
d[nitro,t,1] =  0.098  0.411  phyto
process nitro_remineralization;
equations: d[nitro,t,1] = 0.005  residue
d[residue,t,1 ] =  0.005  residue
Advantages of Quantitative Process Models
Process models offer scientists a promising framework because:
 they embed quantitative relations within qualitative structure;
 that refer to notations and mechanisms familiar to experts;
 they provide dynamical predictions of changes over time;
 they offer causal and explanatory accounts of phenomena;
 while retaining the modularity needed for induction/abduction.
Quantitative process models provide an important alternative to
formalisms typically used in scientific modeling.
Generic Processes as Background Knowledge
We cast background knowledge as generic processes that specify:
 the variables involved in a process and their types;
 the parameters appearing in a process and their ranges;
 the forms of conditions on the process; and
 the forms of associated equations and their parameters.
Generic processes are building blocks from which one can compose
a specific process model.
Generic Processes for Aquatic Ecosystems
generic process exponential_loss
variables: S{species}, D{detritus}
parameters:  [0, 1]
equations: d[S,t,1] = 1    S
d[D,t,1] =   S
generic process remineralization
variables: N{nutrient}, D{detritus}
parameters:  [0, 1]
equations: d[N, t,1] =   D
d[D, t,1] = 1    D
generic process grazing
variables: S1{species}, S2{species}, D{detritus}
parameters:  [0, 1],  [0, 1]
equations: d[S1,t,1] =     S1
d[D,t,1] = (1  )    S1
d[S2,t,1] = 1    S1
generic process constant_inflow
variables: N{nutrient}
parameters:  [0, 1]
equations: d[N,t,1] = 
generic process nutrient_uptake
variables: S{species}, N{nutrient}
parameters:  [0, ],  [0, 1],  [0, 1]
conditions: N > 
equations: d[S,t,1] =   S
d[N,t,1] = 1      S
Constructing Process Models
training data
process model
model AquaticEcosystem
variables: nitro, phyto, zoo, nutrient_nitro, nutrient_phyto
observables: nitro, phyto, zoo
process phyto_exponential_growth
equations: d[phyto,t] = 0.1  phyto
process zoo_logistic_growth
equations: d[zoo,t] = 0.1  zoo / (1  zoo / 1.5)
Induction
Abduction
process exponential_growth
variables: P {population}
equations: d[P,t] = [0, 1,]  P
process logistic_growth
variables: P {population}
equations: d[P,t] = [0, 1, ]  P  (1  P / [0, 1, ])
process constant_inflow
variables: I {inorganic_nutrient}
equations: d[I,t] = [0, 1, ]
process consumption
variables: P1 {population}, P2 {population}, nutrient_P2
equations: d[P1,t] = [0, 1, ]  P1  nutrient_P2,
d[P2,t] =  [0, 1, ]  P1  nutrient_P2
process no_saturation
variables: P {number}, nutrient_P {number}
equations: nutrient_P = P
process saturation
variables: P {number}, nutrient_P {number}
equations: nutrient_P = P / (P + [0, 1, ])
generic processes
process phyto_nitro_consumption
equations: d[nitro,t] = 1  phyto  nutrient_nitro,
d[phyto,t] = 1  phyto  nutrient_nitro
process phyto_nitro_no_saturation
equations: nutrient_nitro = nitro
process zoo_phyto_consumption
equations: d[phyto,t] = 1  zoo  nutrient_phyto,
d[zoo,t] = 1  zoo  nutrient_phyto
process zoo_phyto_saturation
equations: nutrient_phyto = phyto / (phyto + 0.5)
A Method for Process Model Construction
The IPM algorithm constructs explanatory models from generic
elements components in four stages:
1. Find all ways to instantiate known generic processes with
specific variables, subject to type constraints;
2. Combine instantiated processes into candidate generic models
subject to additional constraints (e.g., number of processes);
3. For each generic model, carry out search through parameter
space to find good coefficients;
4. Return the parameterized model with the best overall score.
Our typical evaluation metric is squared error, but we have also
explored other measures of explanatory adequacy.
Estimating Parameters in Process Models
To estimate the parameters for each generic model structure, the
IPM algorithm:
1. Selects random initial values that fall within ranges specified
in the generic processes;
2. Improves these parameters using the Levenberg-Marquardt
method until it reaches a local optimum;
3. Generates new candidate values through random jumps along
dimensions of the parameter vector and continue search;
4. If no improvement occurs after N jumps, it restarts the search
from a new random initial point.
This multi-level method gives reasonable fits to time-series data
from a number of domains, but it is computationally intensive.
Uses of Inductive Process Modeling
population dynamics
aquatic ecosystems
hydrology
biochemical kinetics
Intellectual Influences
Our approach to explanatory model construction draws on ideas
from many traditions:
 computational scientific discovery (e.g., Todorovski, 2003);
 methods for causal model abduction (e.g., Zupan et al., 2001);
 qualitative physics and simulation (e.g., Forbus, 1984);
 languages for scientific simulation (e.g., STELLA, MATLAB).
Our work combines these ideas in novel ways to support abduction
of models that explain the behavior of dynamical systems.
Some Recent Extensions
In recent work, we have extended our approach to incorporate:
 heuristic beam search through the space of process models;
 hierarchical generic processes that further constrain search;
 an ensemble-like method that mitigates overfitting effects;
 metrics for explanatory adequacy based on trajectory shapes.
We have also embedded our algorithms in an interactive software
environment for model construction and revision.
End of Presentation
Backup Slides
Generating Predictions and Explanations
To utilize or evaluate a given process model, we must simulate its
behavior over time:
 specify initial values for input variables and time step size;
 on each time step, determine which processes are active;
 solve active algebraic/differential equations with known values;
 propagate values and recursively solve other active equations;
 when multiple processes influence the same variable, assume
their effects are additive.
This performance method makes specific predictions that we can
compare to observations.
Results on the Ross Sea Ecosystem
Results on Protist Predator-Prey System
Results on the Rinkobing Fjord
Results on Biochemical Kinetics
observed trajectories
predicted trajectories