Bayesian Networks II: Dynamic Networks and Markov Chains

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Nov 7, 2013 (4 years and 1 day ago)

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Bayesian Networks II:
Dynamic Networks and
Markov Chains
By Peter Woolf (pwoolf@umich.edu)
University of Michigan
Michigan Chemical Process
Dynamics and Controls
Open Textbook
version 1.0
Creative commons
Existing plant
measurements
Physics, 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 Mechanism
A 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
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Today
(t
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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
Predicts
future
responses
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t
i-1
t
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t
i-2
t
i-3
t
i+1
t
i+2
Model derived from past data
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Today
(t
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Tomorrow
(t
i+1
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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 network
Collapsed 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 1
P1+P2=1
P5=1 e
tc.
Case Study
: Synthetic Study
Situation:
Imagine that we are exploring the effect of a DNA
damaging drug and UV light on the expression of 4 genes.
GFP
Gene A
Gene B
Gene C
Case Study 1
: Synthetic Study
GFP
Gene A
Gene B
Gene C
Idealized
Data
Case Study 1
: Synthetic Study
GFP
Gene A
Gene B
Gene C
Noisy
data
Case Study 1
: Synthetic Study
Noisy 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