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Bayesian Networks II:Dynamic Networks and Markov Chains
By Peter Woolf ([email protected])University of Michigan
Michigan Chemical Process Dynamics and Controls Open Textbook
version 1.0
Creative commons
Existing plant measurementsPhysics, chemistry, and
chemical engineering knowledge & intuition Bayesian network models to
establish connections
Patterns of likely causes & influences
Efficient experimental design to test combinations of
causes
ANOVA & probabilistic models to eliminate irrelevant or uninteresting
relationships
Process optimization (e.g. controllers, architecture, unit
optimization, sequencing, and utilization)
Dynamical process modeling
From http://www.norsys.com/netlib/car_diagnosis_2.htm
Static Bayesian Network Example 1: Car failure diagnosis network
Static Bayesian Network Example 2:ALARM network: A Logical Alarm Reduction MechanismA medical diagnostic system for patient monitoring with 8 diagnoses, 16 findings, and 13 intermediate values
From Beinlich, Ingo, H. J. Suermondt, R. M. Chavez, and G. F. Cooper (1989) "The ALARM monitoring system: A case study with two probabilistic inference techniques for belief networks" in Proc. of the Second European Conf. on Artificial Intelligence in Medicine (London, Aug.), 38, 247-256. Also Tech. Report KSL-88-84, Knowledge Systems Laboratory, Medical Computer Science, Stanford Univ., CA.
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Yesterday(ti-1)
Today(ti)
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Unrolled Network
Dynamic Bayesian Networks
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These are both examples of Dynamic Bayesian Networks (DBNs)
OR
Collapsed Network
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ObservedPredicted
Predicts future responses
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ti-1 titi-2ti-3ti+1
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Model derived from past data
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Dynamic Bayesian Networks:Predict to explore alternatives
DBNs provide a suitable environment for MPC!
DBN: Thermostat example
From http://www.norsys.com/networklibrary.html#
N: fluctuations
Time(t)
ti ti+1
H: Heater
T: Temperature
G: Temp Set Pt.
S: Switch
V: Value/Cost
N: fluctuations
H: Heater
T: Temperature
G: Temp Set Pt.
S: Switch
V: Value/Cost
Unrolled networkCollapsed network
N: fluctuations
Time(t)
ti ti+1
H: Heater
T: Temperature
N: fluctuations
H: Heater
T: Temperature
Simplified DBN
A Dynamic Bayesian Network can be recast as a Markov Network
Assume each variable is binary (has states 1 or 0), thus any configuration could be written as {010} meaning N=0, H=1, T=0
A Markov network describes how a system will transition from system state to state
{000}
{001}
{010}
{110}
{011}{111}
{101}
{100}
A Dynamic Bayesian Network can be recast as a Markov Network
A Markov network describes how a system will transition from system state to state
{000}
{001}
{010}
{110}
{011}{111}
{101}
{100}
Each edge has a probability associated with it.
€
to
{000} {001} {010} {100} {101} {110} {011} {111}
from
{000}
{001}
{010}
{100}
{101}
{110}
{011}
{111}
0 p1 0 0 0 p2 0 0
p3 p4 0 0 0 0 0 0
0 0 0 p5 0 0 0 0
0 0 0 0 0 p6 p7 0
0 0 p8 p9 p10 0 0 0
0 0 0 0 0 p11 0 0
0 0 0 0 p12 0 0 p13
0 0 0 0 0 0 p14 p15
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Note: All rows must sum to 1P1+P2=1P5=1 etc.
Case Study: Synthetic StudySituation: Imagine that we are exploring the effect of a DNA damaging drug and UV light on the expression of 4 genes.
GFPGene AGene BGene C
Case Study 1: Synthetic Study
GFPGene AGene BGene C
IdealizedData
Case Study 1: Synthetic Study
GFPGene AGene BGene C
Noisy data
Case Study 1: Synthetic StudyNoisy data
Idealized Data
Given idealized or noisy data, can we find any relationships between the drug, UV exposure, GFP, and the gene expression profiles?
See miniTUBA.demodata.xls
Case Study 1: Synthetic Study
Google “miniTUBA” or go to http://ncibi.minituba.org
Case study 1: synthetic data
• Observations:– Stronger relationships require fewer
observations to identify– Noise in measurements are okay– Moderate binning errors are forgivable– Uncontrolled experiments can be your
friend in model learning
Take Home Messages
• Noisy, time varying processes can be modeled as a Dynamic Bayesian Network (DBN)
• A DBN can be recast as a Markov model of a stochastic system
• DBNs can be learned directly from data using tools such as miniTUBA
