Transcript The following presentation

Mathematics, Mathematical
Modeling, and Decision Making
Jennifer Slimowitz
Board on Mathematical Sciences and Their Applications
The National Academies
August 7, 2004
Outline
• National Academies and mathematics
• 3 examples of current project areas
– Environmental modeling
– Molecular biology and genomics
– Forensic science
August 7, 2004
The National Academies
• Provides independent, objective scientific advice
to the nation. Convenes scientific workshops and
panels of experts who work pro-bono to generate
recommendations.
• Chartered by Lincoln in 1863
• Comprised of the
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National Academy of Sciences
National Academy of Engineering
Institute of Medicine
National Research Council
August 7, 2004
Board on Mathematical Sciences and
Their Applications (BMSA)
• Mission: to support and promote the quality and
health of the mathematical sciences and their
benefits to the nation. To do this, we
– Conduct studies and meetings and disseminate reports
and technical assessments on mathematical sciences
topics of national interest; and
– Inform and cooperate with various governmental,
technical, and public bodies on technical and policy
matters relating to mathematical sciences.
August 7, 2004
National Academies
• Scientific boards deal with real-world issues
• “Pure Science” is paired with policy, law,
and other practical constraints to help
address issues
• Scientists play a role in decision-making,
but not the only role
August 7, 2004
3 Examples of Current Studies
• Mathematical models used by the EPA
• Molecular biology and genomics
• Forensic science
August 7, 2004
Mathematical Modeling in the EPA
EPA Mission: to protect human health and the
environment
Uses mathematical models to develop regulatory
standards (e.g. for air and water quality) and to
determine if proposed construction will be in
compliance with regulations
Differential equations, statistics, computational
science
August 7, 2004
Some Questions on the Use of
Environmental Models
• What are the underlying assumptions of a given
model?
• How do you aggregate models together?
• How do the limitations of the aggregate model
compare with those of its components?
• How do you responsibly convey the assumptions,
limitations, and degree of testing of models?
• How do you know of your model is applicable in a
certain circumstance?
August 7, 2004
Example: Community Multi-scale
Air Quality (CMAQ) Model
• Approaches air quality as a whole
– Ground level ozone
– Fine particles
– Visibility degradation
• Multiple pollutants and multiple spatial
scales
• Combines chemical, transport, and
meteorology models (technically hard!)
August 7, 2004
CMAQ (continued)
• Used regularly for research and regulatory
purposes (mostly ground level ozone)
• Involves ODEs and PDEs
• Inputs include grid size, temporal and spatial wind
vectors, and emissions of pollutants (often coming
as outputs from other models)
• Outputs include spatial and temporal
concentrations of pollutants
• Huge and complicated!
August 7, 2004
What steps could the EPA follow to
evaluate if a model is suitable?
• Peer review? Compare to data? Proprietary?
Need some kind of quality assurance.
• Council for Regulatory Environmental
Modeling (CREM) is developing
“knowledge data base” to elucidate models
that EPA uses. Motto: turn the black box to
Plexiglas.
• Why is this important to EPA?
August 7, 2004
National Academies study in progress:
“Environmental Decision Making:
Principles and Criteria for Models”
Statement of Task includes
• provide advice concerning the development of
guidelines and a vision of the selection and use of
models at the agency
• provide a report that will serve as a fundamental
guide for the selection and use of models in the
regulatory process at the EPA.
August 7, 2004
Who sits on the committee to give
this type of advice to EPA?
Experts in
• environmental, chemical, and civil engineering
• decision science and risk analysis and assessment
• biology
• mathematical ecology
• transportation and planning
• history of science
• public policy and law
• biostatistics and spatial statistics
August 7, 2004
Molecular Biology and Genomics
Big Question
Given the genome of a given organism, how
can you predict behavior?
• In humans, we can pinpoint the gene which
indicates cystic fibrosis. How comes kids
with CF get sick from microbial infections?
• Can a good model help us determine how
certain genetic mutations affect function?
August 7, 2004
Why is this important to policy
makers?
• Medical: enable the development of new
drugs targeted to people with specific
mutations
• Energy and Environment: engineer or find
organisms to aid in carbon sequestration,
bioremediation, or production of clean
energy
August 7, 2004
Role of Mathematics in Molecular
Biology and Genomics
To identify emergent properties, generate nonobvious hypotheses, and identify missing
information by
• Developing good models (dynamical systems,
differential equations, geometry)
• Learning how to analyze massive data sets
(statistics, pattern recognition algorithms)
August 7, 2004
Example: Statistics involved in gene
expression analysis
Traditionally, statistics deals with a few
independent observations taken from many
samples. In gene expression analysis, we
have many dependent observations taken
from only a few samples. Requires new
techniques!
Is it a vanity plate or just a coincidence?
August 7, 2004
National Academies Study in
Progress: “Mathematical Sciences
Research for DOE’s Computational
Biology”
Statement of Task: “recommend mathematical
sciences research activities to the Department of
Energy that will enable science to make effective
use of the large amount of existing genomic
information and the much larger and more diverse
collections of structural and functional genomic
information that are being created”
August 7, 2004
Who sits on the committee to give
this type of advice?
Experts in
• genetics, population genetics, and plant
biology
• statistics, computational science,
mathematical ecology, mathematical
physiology, and bioinformatics
• neuroscience
• chemical engineering and chemical physics
August 7, 2004
Quote from a committee member:
“Relationships between phenotype and
genotype are inherently difficult because
of the subtlety involved. If mechanisms
were big and obvious, organisms with
mutations would not survive .”
August 7, 2004
Forensic Science
• DNA forensic testing (success story!)
• Bullet lead analysis
False positives vs. False negatives – which are
worse?
August 7, 2004
DNA Forensic Testing
• DNA can be found in samples of saliva,
blood, semen, skin, hair or tears left at
crime scenes
• Sample is tested at 13 sites which are
known to vary greatly from individual to
individual, where there are different
numbers of “short tandem repeat” (STR)
units
August 7, 2004
DNA Forensic Testing (cont)
• Chance of any two individuals having same
number of STRs at any one of the 13 sites is
1/10
• Chance of two individuals matching at all
13 sites is (1/10) ^ 13 = less than 1 in a
trillion
• Typically, close relatives (siblings,
parent/child) will match at 4 or 5 sites.
August 7, 2004
Results
• At least 138 individuals have been
exonerated as a result of DNA forensic
testing
• In addition to letting the “innocent go free,”
DNA forensic testing helps to find the true
criminal
• CODIS (Combined DNA Index Systems) is
the FBI database of DNA samples of felons
August 7, 2004
Where is mathematics involved?
• Statistics – how do we know that the
probability of any two people having the
same number of STR units at any one site is
1/10?
• Computational Science – given a blood
sample, what algorithms do we use to
determine the DNA contained in it?
August 7, 2004
Compositional Analysis of Bullet
Lead
• Used to produce circumstantial evidence to
“match” a crime scene bullet to a bullet
from a suspect
• The FBI measures the concentrations of 7
elements (arsenic, antimony, tin, copper,
bismuth, silver, and cadmium) in the bullet
lead alloy
August 7, 2004
Where is mathematics involved?
Given concentrations of a bullet (or set of
bullets) found at crime scene and bullet (or
set of bullets) found on suspect, how does
the FBI determine if they match?
August 7, 2004
National Academies Study (2004):
Forensic Analysis: Weighing Bullet
Lead Evidence
Statement of Task: “…to assess the validity
of the scientific basis for the use of
elemental composition determination to
compare lead alloy-based items of evidence.
The following three areas will be addressed:
Analytical Method…Statistics for
Comparison…Interpretation Issues”
August 7, 2004
Who sits on the committee to give
this type of advice?
Experts in
• chemistry
• physics and materials science
• criminal justice, law, and forensic science
• statistics
• metallurgical engineering
August 7, 2004
Major concern of Committee
“Variations among and within lead bullet
manufacturers make any modeling of the
general manufacturing process unreliable
and potentially misleading in CABL
comparisons.”
August 7, 2004
Issues coming to the forefront:
Mathematics / Operations Research
in Voting
• Touch-screen voting – is it as accurate as voting
on a paper ballot? What algorithms will ensure
accountability while protecting anonymity?
• Gerrymandering – what constitutes fairness in
districting? How could lines be drawn to achieve
fairness? Could the current system of one
representative for each district be improved?
August 7, 2004