Transcript No Slide Title

BAYESIAN NETWORKS
IN MODEL AND DATA INTEGRATION AND
DECISION MAKING IN RIVER BASIN
MANAGEMENT USING
Consideration of opportunities for Bayes networks in predictive water quality
modelling
Olli Malve (M. Sc.)
Water Resources Research and Management
Citations from
Ames, D.P. & Neilson, B. (Utah Water resources Laboratory) 2001: Bayesian Decision Networks for total maximum daily load analysis;
East Canyon Creek Case Study (WWW-document).
Reckhow, K.H. (UNC Water Resources Research Institute, North Carolina State University, Raleigh, USA) 1999: Water quality prediction
and probability network models; probability network model for nitrogen enrichment and algal blooms in the Neuse River (Can.J. Aquat.
Sci. 56:1150-1158 (1999)).
see also:
http://www2.ncsu.edu/ncsu/CIL/WRRI/ken's_page.html (home page of K. Reckhov)
http://www.epa.gov/OWOW/tmdl/ (A Total Maximum Daily Load (TMDL) program)
Bayes network for discrete variables
implement with Hugin software
• Do not include real Bayesian update of parameters
with new data.
• There are several other statistical and
computational methods and software: one of the
best - OpenBugs for continuous variables was
used in hierarchical modeling of Finnish lakes.
• Resembles Structural equation models.
• They both belong to the family of Graphical
probibilistic models.
Hirarchical linear chlorophyll a model
DAG diagram
xijk
β
σ2
βi
σ2i
βij
τ
yijk
Structural equation model
LAKE
PYHÄJÄRVI
in SÄKYLÄ;
Planktiv
– Planktivorous
fish model
research
Z – zooplankton
(Crustacea)
A3- Cyanobacteria
TP – total phosphorus
TN – total nitrogen
PHYSICAL WAY OF THINKING
Hydraulic routing of ground and surface water
flow in drainage basin, in river channels, in
lakes and in estuaries.
Drainage basin, river, lake and estuary are
linked with hydraulic principles
High spatial and temporal resolution
STATISTICAL INFERENCE
Small-scale transport and transformation processes of
pollutants
in drainage basin
are summarized
with probabilistic expression
that characterize
the aggregate response of interest
to the decision makers.
Outcomes
expressed as probabilities
are
an acknowledgement of
the lack of precission in
predictive models
BAYES NETWORKS
Formally, BNTs are directed acyclic graphs in
which each node represents a random
variable, or uncertain quantity, whick can
take two or more possible values.
Each node represents a multi-valued variable,
comprising a collection of mutually exclusive
hypothesis (state of a lake: Oligotrophic,
Mesotrophic, Eutrophic)
or
observations (nutrient loading: Low, Medium,
High)
The arcs signify the existence of direct causal
influence between the linked variables, and
the strength of these influences are quantified
by conditional probabilities
Conditional probability (each direct link X->Y)
discrete variables is quantified by a fixed
conditional probability matrix M, in which the
(x,y) entry is given by
My|xP(y|x) P(Y=y |X=x)=
P(y1|x1) P(y2|x1) ... P(yn|x1)
P(y1|x2) P(y2|x2) ... P(yn|x2)
.
.
.
.
.
.
.
.
.
P(y1|xm) P(y2|xm) ... P(yn|xm)
QUANTIFYING THE LINKS
Bayes learning of Conditional Probability Matrix (CPM) from
1. Observational data
-simultaneus observations of each variable are tabulated, sorted by
the parent variables and converted into categories as prescribed in
node definitions.
-for every combination of states of parent nodes, the number of
occurences of states of the child is counted.
-probabilities are calculated as a number of occurences of a child
state divided by the total number of observations for the combination
of parent states
2.Parameter learning from Model simulations
(uncertainty analysis such as Monte Carlo simulations);
-varying the selected input variables about an appropriate distribution
and drawing random samples from model parameter distributions
->results of simulations at the selected output variables are tabulated
with their corresponding set of input variable conditions
->CPM is generated from this data tabulation using the same method
described above for observational data
3. Parameter learning from scientists, experts, stakeholders, cost
and benifits
If data is not available and typical models are not appropriate,
conditional probability tables can be generated by eliciting
information from experts and stakeholders.
-in the case of cost and benifit analysis for example the costs
assosiated with wastewater treatment plant upgrade will likely
need to be elicited from experts and through market inquiries
-benefits assosiated with water quality improvement (recreation,
biological habitat, esthetics and other environmental benefits) are
subjective in nature and are difficult to quantify without input from
local individuals, stakeholders and experts
The probabilistic relationships described here may be more difficult
to generate than those calculated from data and models.
DECISIONS AND UTILITY
A Bayesian Decision Network (BDN) is a specific form of a
Bayesian network that includes decision and utility nodes and is
used to model the relationship between decisions and outcomes.
Decision node contain descrete options instead of a probability
distribution across states. Decision node can only exist in one state
at a time, representing a decision or management option made
between multiple choices.
Utility node provide a simple mean for estimating expected values
of different outcomes. Expected value E of an uncertain outcome
with n states (i=1…n) is computed as:
E=Pi Bi ,
where a benifit Bi, associated with each state, and a probability, Pi,
of being in each state.
APPLICATION OF Bayes Decision Networks
1. Defining the problem
2. Integrating disparate data rources
3. Scenario generation and analysis
4. Building a Bayesian Decision Network (Influence diagram)
5. Obtaining Probability Distributions
Decision tree
• Bayesin networks can be transformed to
decision tree
Bayes net
Decision tree
0.7
Algal bloom
Go swimming
(yes/no)
(yes/no)
Algal
yes
bloom
Get ill
yes
0.3
no
Feeling
well
Go swimming
no
Get ill
Algal
bloom
(yes/no)
Hot
sunshine
0.1
yes
0.9
no
Get ill
Feeling
well
SUMMARY
Bayesian Decision Networks provide successful way to make
educated decisions.
BDN is simple for stakeholder involvement and understanding,
while still containing proven and defensible science.
BDN is a tool for communication between scientists, stakeholders
and decision makers.
Bayesian Decision Networks
1. provides a good conceptual framework for clear defining
relevant variables
2. etablishes the relationship between causes and effects in the
system
3. Integrates different sources of information into a single
analytic tool
4. Captures model responses for quick scenario generation and
investigation
5. Quantifies risk which can be used in establishing the marigin of
safety
A carefully devised and calibrated probabiltiy network model is
ideally designed to communicate at the interface between
scientists, stakeholders, and decision makers.
By acknowledging the sometimes-substantial uncertainty in
model predictions, we enhance, rather than diminish, the value
of predictive modelling by focusing on the model ability to
estimate risk.
Bayesian Decision
network (Influence
diagram) of Lake
Säkylän Pyhäjärvi
Management scenarion
Studying the effect of
zooplankton and TotP-load
Studying the effect of
management actions on the costs
and the attainment of water
quality standards
Conditional marginal distributions of costs, attainment of
water quality satndard and Cyanobacteria (BlueGmax)
summer maximum biomass with given Buffer Strip width
(21 – 36 m), wetland percentage (1.1 – 1.25 %),
forestation (25 –31 %) and fish catch ( 3, in a artificial
scale which will be replaced after expert judgement).
Water quality modelling and probability
network models
with reference to
Reckhow, K.H. Can. J. Fish. Aquat. Sci. 56:1150-1158 (1999).
Modelling for nitrogen enrichment and algal blooms in
Nuese River, Canada with Bayes nets - probabilistic
prediction of eutrophication
Initial forcing function ”Spring precipitation” is expressed as marginal probabilities assessed from statistics
on historic precipitation data in the watershed. Distribution was segmented into three eually likely
precipitation ranges (below average, average, above average).
The probabilities for ”precentage forested buffer” reflect a judgemental assessment of the total perennial
stream miles in the Neuese River watershed that would be required to have a maintained minimum width
buffer, based on the project outcome of proposed management plans. The resultant probability estimates are
given in the table.
Conditional probabilities were assessed for the four intermediate conditional probabilities. ”Precentage of
nitrogen load reduction” was conditional on only the ”precentage of forested buffer”. A scientific expert
was consulted for a probabilistic statement reflecting the expected reduction in nitrogen loading due to
buffers alone.
The ”nitrogen concentration” was expressed as a fuction of ”spring precipitation” and the ”nitrogen loading reduction”; in
the absence of data to fit a statistical model for these variables, nitrogen concentration was based on scientific judgement.
The relationship between ”summer precipitation” and ”summer streamflow” were based on the statistical model developed
from precipitaion and sreamflow data.
The conditional probabilities for the reponse variable ”algal bloom” were based in the scientific judgement
(for the effect of nitrogen concentration) and in part on the interpretation of chlorophyll a versus flow data.
Using the data, the chlorophyll levels were grouped to algal bloom categories, and flow data were grouped
into flow categories. The relative frequency of data points in each ”algal bloom” / ”flow” group determined
the initial probabilities; these probabilities were further decomposed, using judgement, to account for the
effect of ”nitrogen concentration”.
Conditional probabilities for ”anoxia” were based on judgement. These responce variable
conditional probabilitites are presented in the table below.
Probabilities expressed in earlier pages can be combined into a joint probability on all
variables, which when allows us to solve for a number of interesting variables. While all
marginal and conditional probabilities can be easily calculated using the estimates, computation
in larges problems is facilitaed with Bayes nets software.
From the probabilities expressed earlier the marginal probability of
anoxia is 0.30; in Bayesian terms, this calculation reflects only prior
information. If the implementation of management option could assure
that at least 95% of streams had the the required buffer (p(95-100% for
forested buffer) = 1.0), then anoxia probability drops slightly to 0.27. This
calculation, although hypothetical, is indicative of the types of policy
related questions that can be addressed with a complete probabiltiy
network model.