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Pat Langley Center for the Study of Language and Information Stanford University, Stanford, California http://cll.stanford.edu/~langley langley@csli.stanford.edu Computational Discovery of Explanatory Process Models Thanks to N. Asgharbeygi, K. Arrigo, S. Bay, A. Pohorille, J. Sanchez, K. Saito, and J. Shrager for their contributions to this research.

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Data Mining vs. Scientific Discovery induce predictive models from large, often business, data sets; induce predictive models from large, often business, data sets; cast models as decision trees, logical rules, or other notations invented by AI researchers. cast models as decision trees, logical rules, or other notations invented by AI researchers. There exist two computational paradigms for discovering explicit knowledge from data. The data mining movement develops computational methods that: Both approaches draw on heuristic search to find regularities in data, but they differ considerably in their emphases. constructing models from (often small) scientific data sets; constructing models from (often small) scientific data sets; stated in formalisms invented by scientists and engineers. stated in formalisms invented by scientists and engineers. In contrast, computational scientific discovery focuses on:

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In Memoriam Herbert A. Simon (1916 – 2001) Herbert A. Simon (1916 – 2001) Jan M. Zytkow (1945 – 2001) Jan M. Zytkow (1945 – 2001) Three years ago, computational scientific discovery lost two of its founding fathers: Both contributed to the field in many ways: posing new problems, inventing methods, training students, and organizing meetings. Moreover, both were interdisciplinary researchers who contributed to computer science, psychology, philosophy, and statistics. Herb Simon and Jan Zytkow were excellent role models who we should all aim to emulate.

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Time Line for Research on Computational Scientific Discovery 1989199019791980198119821983198419851986198719881991199219931994199519961997199819992000 Bacon.1–Bacon.5 Abacus, Coper Fahrehneit, E*, Tetrad, IDS N Hume, ARC DST, GP N LaGrange SDS SSF, RF5, LaGramge Dalton, Stahl RL, Progol Gell-Mann BR-3, Mendel Pauli Stahlp, Revolver Dendral AM GlauberNGlauber IDS Q, Live IE Coast, Phineas, AbE, Kekada Mechem, CDP Astra, GP M HR BR-4 Numeric lawsQualitative lawsStructural modelsProcess models Legend

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Successes of Computational Scientific Discovery Over the past decade, systems of this type have helped discover new knowledge in many scientific fields: qualitative chemical factors in mutagenesis (King et al., 1996) qualitative chemical factors in mutagenesis (King et al., 1996) quantitative laws of metallic behavior (Sleeman et al., 1997) quantitative laws of metallic behavior (Sleeman et al., 1997) qualitative conjectures in number theory (Colton et al., 2000) qualitative conjectures in number theory (Colton et al., 2000) temporal laws of ecological behavior (Todorovski et al., 2000) temporal laws of ecological behavior (Todorovski et al., 2000) reaction pathways in catalytic chemistry (Valdes-Perez, 1994) reaction pathways in catalytic chemistry (Valdes-Perez, 1994) Each has led to publications in the refereed scientific literature (e.g., Langley, 2000), but they did not focus on systems science.

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The Nature of Systems Science focus on synthesis rather than analysis in their operation; focus on synthesis rather than analysis in their operation; rely on computer modeling as one of their central methods; rely on computer modeling as one of their central methods; develop system-level models with many variables and relations; develop system-level models with many variables and relations; evaluate their models on observational, not experimental, data. evaluate their models on observational, not experimental, data. Disciplines like Earth science and computational biology differ from traditional fields in that they: Developing and testing such models are complex tasks that would benefit from computational aids. However, existing methods for computational scientific discovery were not designed with systems science in mind.

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Observations from the Ross Sea

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Inductive Process Modeling Our response is to design, construct, and evaluate computational methods for inductive process modeling, which: represent scientific models as sets of quantitative processes; represent scientific models as sets of quantitative processes; use these models to predict and explain observational data; use these models to predict and explain observational data; search a space of process models to find good candidates; search a space of process models to find good candidates; utilize background knowledge to constrain this search. utilize background knowledge to constrain this search. This framework has great potential for aiding systems science, but it raises new computational challenges.

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Challenges of Inductive Process Modeling process models characterize behavior of dynamical systems; process models characterize behavior of dynamical systems; variables are continuous but can have discontinuous behavior; variables are continuous but can have discontinuous behavior; observations are not independently and identically distributed; observations are not independently and identically distributed; models may contain unobservable processes and variables; models may contain unobservable processes and variables; multiple processes can interact to produce complex behavior. multiple processes can interact to produce complex behavior. Process model induction differs from typical learning tasks in that: Compensating factors include a focus on deterministic systems and the availability of background knowledge.

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Issue 1: Representing Scientific Models address observational rather than experimental data; address observational rather than experimental data; deal with dynamic systems that change over time; deal with dynamic systems that change over time; have an explanatory rather than a descriptive character; have an explanatory rather than a descriptive character; are causal in that they describe chains of effects; are causal in that they describe chains of effects; contain quantitative relations and qualitative structure. contain quantitative relations and qualitative structure. To assist system scientists modeling efforts, we must first encode candidate models that: We need some formal way to represent such models that can be interpreted computationally.

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Why Are Existing Formalisms Inadequate? d[ice_mass,t] = (18 heat) / 6.02 d[water_mass,t] = (18 heat) / 6.02 systems of equations B>6 C>0 C>4 14.318.711.516.9 regression trees gcd(X,X,X). gcd(X,Y,D) :- X<Y,Z is Y–X,gcd(X,Z,D). gcd(X,Y,D) :- Y<X,gcd(Y,X,D). Horn clause programs x =12, x =1 x =12, x =1 y =18, x =2 y =18, x =2 x =12, x =1 x =12, x =1 y =10, x =2 y =10, x =2 x =16, x =2 x =16, x =2 y =13, x =1 y =13, x =1 x =19, x =1 x =19, x =1 y =11, x =2 y =11, x =2 0.3 0.7 1.0 1.0 hidden Markov models

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A Process Model for an Aquatic Ecosystem model Ross_Sea_Ecosystem variables: phyto, nitro, residue, light, growth_rate, effective_light, ice_factor observables: phyto, nitro, light, ice_factor process phyto_loss equations:d[phyto,t,1] = 0.1 phyto equations:d[phyto,t,1] = 0.1 phyto d[residue,t,1] = 0.1 phyto process phyto_growth equations:d[phyto,t,1] = growth_rate phyto equations:d[phyto,t,1] = growth_rate phyto process phyto_uptakes_nitro conditions:nitro > 0 conditions:nitro > 0 equations:d[nitro,t,1] = 1 0.204 growth_rate phyto equations:d[nitro,t,1] = 1 0.204 growth_rate phyto process growth_limitation equations:growth_rate = 0.23 min(nitrate_rate, light_rate) equations:growth_rate = 0.23 min(nitrate_rate, light_rate) process nitrate_availability equations:nitrate_rate = nitrate / (nitrate + 5) equations:nitrate_rate = nitrate / (nitrate + 5) process light_availability equations:light_rate = effective_light / (effective_light + 50) equations:light_rate = effective_light / (effective_light + 50) process light_attenuation equations:effective_light = light ice_factor equations:effective_light = light ice_factor

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Advantages of Quantitative Process Models they embed quantitative relations within qualitative structure; they embed quantitative relations within qualitative structure; that refer to notations and mechanisms familiar to scientists; that refer to notations and mechanisms familiar to scientists; they provide dynamical predictions of changes over time; they provide dynamical predictions of changes over time; they offer causal and explanatory accounts of phenomena; they offer causal and explanatory accounts of phenomena; while retaining the modularity needed to support induction. while retaining the modularity needed to support induction. Process models are a good target for discovery systems because: Quantitative process models provide an important alternative to formalisms used currently in computational discovery.

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Issue 2: 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; specify initial values for input variables and time step size; on each time step, determine which processes are active; on each time step, determine which processes are active; solve active algebraic/differential equations with known values; solve active algebraic/differential equations with known values; propagate values and recursively solve other active equations; propagate values and recursively solve other active equations; when multiple processes influence the same variable, assume their effects are additive. when multiple processes influence the same variable, assume their effects are additive. This performance method makes specific predictions that we can compare to observations.

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Issue 3: Encoding Background Knowledge Horn clause programs (e.g., Towell & Shavlik, 1990) Horn clause programs (e.g., Towell & Shavlik, 1990) context-free grammars (e.g., Dzeroski & Todorovski, 1997) context-free grammars (e.g., Dzeroski & Todorovski, 1997) prior probability distributions (e.g., Friedman et al., 2000) prior probability distributions (e.g., Friedman et al., 2000) To constrain candidate models, we can utilize available backround knowledge about the domain. Previous work has encoded background knowledge in terms of: However, none of these notations are familiar to domain scientists, which suggests the need for another approach.

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Generic Processes as Background Knowledge the variables involved in a process and their types; the variables involved in a process and their types; the parameters appearing in a process and their ranges; the parameters appearing in a process and their ranges; the forms of conditions on the process; and the forms of conditions on the process; and the forms of associated equations and their parameters. the forms of associated equations and their parameters. Our framework casts background knowledge as generic processes that specify: Generic processes are building blocks from which one can compose a specific process model.

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Generic Processes for Aquatic Ecosystems generic process exponential_lossgeneric process remineralization variables: S{species}, D{detritus} variables: N{nutrient}, D{detritus} variables: S{species}, D{detritus} variables: N{nutrient}, D{detritus} parameters: [0, 1] parameters: [0, 1] parameters: [0, 1] parameters: [0, 1] equations:d[S,t,1] = 1 S equations:d[N, t,1] = D equations:d[S,t,1] = 1 S equations:d[N, t,1] = D d[D,t,1] = Sd[D, t,1] = 1 D generic process grazinggeneric process constant_inflow variables: S1{species}, S2{species}, D{detritus} variables: N{nutrient} variables: S1{species}, S2{species}, D{detritus} variables: N{nutrient} parameters: [0, 1], [0, 1] parameters: [0, 1] parameters: [0, 1], [0, 1] parameters: [0, 1] equations:d[S1,t,1] = S1 equations:d[N,t,1] = equations:d[S1,t,1] = S1 equations:d[N,t,1] = d[D,t,1] = (1 ) S1 d[S2,t,1] = 1 S1 generic process nutrient_uptake variables: S{species}, N{nutrient} variables: S{species}, N{nutrient} parameters: [0, ], [0, 1], [0, 1] parameters: [0, ], [0, 1], [0, 1] conditions:N > conditions:N > equations:d[S,t,1] = S equations:d[S,t,1] = S d[N,t,1] = 1 S

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process exponential_growth variables: P {population} variables: P {population} equations: d[P,t] = [0, 1, ] P equations: d[P,t] = [0, 1, ] P process logistic_growth variables: P {population} variables: P {population} equations: d[P,t] = [0, 1, ] P (1 P / [0, 1, ]) equations: d[P,t] = [0, 1, ] P (1 P / [0, 1, ]) process constant_inflow variables: I {inorganic_nutrient} variables: I {inorganic_nutrient} equations: d[I,t] = [0, 1, ] equations: d[I,t] = [0, 1, ] process consumption variables: P1 {population}, P2 {population}, nutrient_P2 variables: P1 {population}, P2 {population}, nutrient_P2 equations: d[P1,t] = [0, 1, ] P1 nutrient_P2, equations: d[P1,t] = [0, 1, ] P1 nutrient_P2, d[P2,t] = [0, 1, ] P1 nutrient_P2 d[P2,t] = [0, 1, ] P1 nutrient_P2 process no_saturation variables: P {number}, nutrient_P {number} variables: P {number}, nutrient_P {number} equations: nutrient_P = P equations: nutrient_P = P process saturation variables: P {number}, nutrient_P {number} variables: P {number}, nutrient_P {number} equations: nutrient_P = P / (P + [0, 1, ]) equations: nutrient_P = P / (P + [0, 1, ]) Issue 4: Inducing Process Models 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 equations: d[phyto,t] = 0.1 phyto process zoo_logistic_growth equations: d[zoo,t] = 0.1 zoo / (1 zoo / 1.5) equations: d[zoo,t] = 0.1 zoo / (1 zoo / 1.5) process phyto_nitro_consumption equations: d[nitro,t] = 1 phyto nutrient_nitro, equations: d[nitro,t] = 1 phyto nutrient_nitro, d[phyto,t] = 1 phyto nutrient_nitro d[phyto,t] = 1 phyto nutrient_nitro process phyto_nitro_no_saturation equations: nutrient_nitro = nitro equations: nutrient_nitro = nitro process zoo_phyto_consumption equations: d[phyto,t] = 1 zoo nutrient_phyto, equations: d[phyto,t] = 1 zoo nutrient_phyto, d[zoo,t] = 1 zoo nutrient_phyto d[zoo,t] = 1 zoo nutrient_phyto process zoo_phyto_saturation equations: nutrient_phyto = phyto / (phyto + 0.5) equations: nutrient_phyto = phyto / (phyto + 0.5) Induction training data generic processes process model

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A Method for Process Model Induction 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. We have implemented the IPM algorithm, which induces process models from generic components in four stages: The evaluation metric can be squared error or description length (e.g., M D = (M V + M C ) log (n) + n log (M E ).

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Estimating Parameters in Process Models 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. To estimate the parameters for each generic model structure, the IPM algorithm: This multi-level method gives reasonable fits to time-series data from a number of domains, but it is computationally intensive.

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identifying conditions on component processes identifying conditions on component processes inferring initial values of unobservable variables inferring initial values of unobservable variables keeping the structural search space tractable keeping the structural search space tractable reducing variance to mitigate overfitting effects reducing variance to mitigate overfitting effects Inductive process modeling raises a number of issues that have clear analogues in other paradigms: We have demonstrated promising responses to these problems within the IPM framework. More Issues in Process Model Induction

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Evaluation of the IPM Algorithm 1. We used the aquatic ecosystem model to generate data sets over 100 time steps for the variables nitro and phyto; 2. We replaced each true value x with x (1 + r n), where r followed a Gaussian distribution ( = 0, = 1) and n > 0; 3. We ran IPM on these noisy data, giving it type constraints and generic processes as background knowledge. To demonstrate IPM's ability to induce process models, we ran it on synthetic data for a known system: In two experiments, we let IPM determine the initial values and thresholds given the correct structure; in a third study, we let it search through a space of 256 generic model structures.

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Experimental Results with IPM The main results of our studies with IPM on synthetic data were: 1. The system infers accurate estimates for the initial values of unobservable variables like zoo and residue; 2. The system induces estimates of condition thresholds on nitro that are close to the target values; and 3. The MDL criterion selects the correct model structure in all runs with 5% noise, but only 40% of runs with 10% noise. These suggest that the basic approach is sound, but that we should consider more MDL schemes and other responses to overfitting.

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Observations from the Ross Sea

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Results on Training Data from Ross Sea

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Results on Test Data from Ross Sea

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Collecting Data on Photosynthetic Processes External stimuli (e.g., light) Adaptation Period Sampling mRNA/cDNA Equlibrium Period MicroarrayTrace Continuous Culture (Chemostat) /wwwscience.murdoch.edu.au/teach www.affymetrix.com/ Health of Culture Time

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Gene Expressions for Cyanobacteria

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Generic Processes for Photosynthesis Regulation generic process translationgeneric process transcription variables: P{protein}, M{mRNA} variables: M{mRNA}, R{rate} variables: P{protein}, M{mRNA} variables: M{mRNA}, R{rate} parameters: [0, 1] parameters: parameters: [0, 1] parameters: equations:d[P,t,1] = M equations:d[M,t,1] = R equations:d[P,t,1] = M equations:d[M,t,1] = R generic process regulate_onegeneric process regulate_two variables: R{rate}, S{signal} variables: R{rate}, S{signal} variables: R{rate}, S{signal} variables: R{rate}, S{signal} parameters: [ 1, 1] parameters: [ 1, 1], [0, 1] parameters: [ 1, 1] parameters: [ 1, 1], [0, 1] equations:R = S equations:R = S equations:R = S equations:R = S d[S, t,1] = 1 S generic process automatic_degradationgeneric process controlled_degradation variables: C{concentration} variables: D{concentration}, E{concentration} variables: C{concentration} variables: D{concentration}, E{concentration} conditions:C > 0 conditions:D > 0, E > 0 conditions:C > 0 conditions:D > 0, E > 0 parameters: [0, 1] parameters: [0, 1] parameters: [0, 1] parameters: [0, 1] equations:d[C,t,1] = 1 C equations:d[D,t,1] = 1 E equations:d[C,t,1] = 1 C equations:d[D,t,1] = 1 E d[E,t,1] = 1 E generic process photosynthesis variables: L{light}, P{protein}, R{redox}, S{ROS} variables: L{light}, P{protein}, R{redox}, S{ROS} parameters: [0, 1], [0, 1] parameters: [0, 1], [0, 1] equations:d[R,t,1] = L P equations:d[R,t,1] = L P d[S,t,1] = L P

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A Process Model for Photosynthetic Regulation model photo_regulation variables: light, mRNA_protein, ROS, redox, transcription_rate observables: light, mRNA process photosynthesis; equations:d[redox,t,1] = 0.0155 light protein equations:d[redox,t,1] = 0.0155 light protein d[ROS,t,1] = 0.019 light protein process protein_translationprocess mRNA_transcription equations:d[protein,t,1] = 7.54 mRNA equations:d[mRNA,t,1] = transcription_rate equations:d[protein,t,1] = 7.54 mRNA equations:d[mRNA,t,1] = transcription_rate process regulate_one_1process regulate_two_2 equations: transcription_rate = 0.99 light equations:transcription_rate = 1.203 redox equations: transcription_rate = 0.99 light equations:transcription_rate = 1.203 redox d[redox,t,1] = 0.0002 redox process automatic_degradation_1process controlled_degradation_1 conditions:protein > 0 conditions:redox > 0, ROS > 0 conditions:protein > 0 conditions:redox > 0, ROS > 0 equations:d[protein,t,1] = 1.91 protein equations:d[redox,t,1] = 0.0003 ROS equations:d[protein,t,1] = 1.91 protein equations:d[redox,t,1] = 0.0003 ROS d[ROS,t,1] = 0.0003 ROS

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Predictions from Best Parameterized Model

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Electric Power on the International Space Station

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Results on Battery Test Data

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Results on Data from Rinkobing Fjord

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specify a quantitative process model of the target system; specify a quantitative process model of the target system; display and edit the models structure and details graphically; display and edit the models structure and details graphically; simulate the models behavior over time and situations; simulate the models behavior over time and situations; compare the models predicted behavior to observations; compare the models predicted behavior to observations; invoke a revision module in response to detected anomalies. invoke a revision module in response to detected anomalies. Because few scientists want to be replaced, we are developing an interactive environment that lets users: The environment offers computational assistance in forming and evaluating models but lets the user retain control. Issue 5: Interfacing with Scientists

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Viewing and Editing a Process Model

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Results of Revising the NPP Model Initial model: E = 0.56 · T1 · T2 · W E = 0.56 · T1 · T2 · W T2 = 1.18 / [(1 + e 0.2 · (Topt – Tempc – 10) ) · (1 + e 0.3 · (Tempc – Topt – 10) )] T2 = 1.18 / [(1 + e 0.2 · (Topt – Tempc – 10) ) · (1 + e 0.3 · (Tempc – Topt – 10) )] PET = 1.6 · (10 · Tempc / AHI) A · PET-TW-M PET = 1.6 · (10 · Tempc / AHI) A · PET-TW-M SR {3.06, 4.35, 4.35, 4.05, 5.09, 3.06, 4.05, 4.05, 4.05, 5.09, 4.05} SR {3.06, 4.35, 4.35, 4.05, 5.09, 3.06, 4.05, 4.05, 4.05, 5.09, 4.05} RMSE on training data = 465.212 and r 2 = 0.799 Revised model: E = 0.353 · T1 0.00 · T2 0.08 · W 0.00 E = 0.353 · T1 0.00 · T2 0.08 · W 0.00 T2 = 0.83 / [(1 + e 1.0 · (Topt – Tempc – 6.34) ) · (1 + e 1.0 · (Tempc – Topt – 11.52) )] T2 = 0.83 / [(1 + e 1.0 · (Topt – Tempc – 6.34) ) · (1 + e 1.0 · (Tempc – Topt – 11.52) )] PET = 1.6 · (10 · Tempc / AHI) A · PET-TW-M PET = 1.6 · (10 · Tempc / AHI) A · PET-TW-M SR {0.61, 3.99, 2.44, 10.0, 2.21, 2.13, 2.04, 0.43, 1.35, 1.85, 1.61} SR {0.61, 3.99, 2.44, 10.0, 2.21, 2.13, 2.04, 0.43, 1.35, 1.85, 1.61} Cross-validated RMSE = 397.306 and r 2 = 0.853 [ 15 % reduction ]

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computational scientific discovery (e.g., Langley et al., 1983); computational scientific discovery (e.g., Langley et al., 1983); theory revision in machine learning (e.g., Towell, 1991); theory revision in machine learning (e.g., Towell, 1991); qualitative physics and simulation (e.g., Forbus, 1984); qualitative physics and simulation (e.g., Forbus, 1984); languages for scientific simulation (e.g., STELLA, MATLAB ); languages for scientific simulation (e.g., STELLA, MATLAB ); interactive tools for data analysis (e.g., Schneiderman, 2001). interactive tools for data analysis (e.g., Schneiderman, 2001). Intellectual Influences Our approach to computational discovery incorporates ideas from many traditions: Our work combines, in novel ways, insights from machine learning, AI, programming languages, and human-computer interaction.

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Contributions of the Research a new formalism for representing scientific process models; a new formalism for representing scientific process models; a computational method for simulating these models behavior; a computational method for simulating these models behavior; an encoding for background knowledge as generic processes; an encoding for background knowledge as generic processes; an algorithm for inducing process models from time-series data; an algorithm for inducing process models from time-series data; an interactive environment for model construction/utilization. an interactive environment for model construction/utilization. In summary, our work on computational scientific discovery has, in responding to various challenges, produced: We have demonstrated this approach to model creation on domains from Earth science, microbiology, and engineering.

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Directions for Future Research produce additional results on other scientific data sets produce additional results on other scientific data sets develop improved methods for fitting model parameters develop improved methods for fitting model parameters extend the approach to handle data sets with missing values extend the approach to handle data sets with missing values implement heuristic methods for searching the structure space implement heuristic methods for searching the structure space utilize knowledge of subsystems to further constrain search utilize knowledge of subsystems to further constrain search augment the modeling environment to make it more usable augment the modeling environment to make it more usable Despite our progress to date, we need further work in order to: Inductive process modeling has great potential to speed progress in systems science and engineering.

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Pat Langley Institute for the Study of Learning and Expertise and Center for the Study of Language and Information Stanford University, Stanford, California.

Pat Langley Institute for the Study of Learning and Expertise and Center for the Study of Language and Information Stanford University, Stanford, California.

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