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Mathematics, Mathematical Modeling, and Decision Making

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Mathematics, Mathematical Modeling, and Decision Making Jennifer Slimowitz Board on Mathematical Sciences and Their Applications The National Academies – PowerPoint PPT presentation

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Title: Mathematics, Mathematical Modeling, and Decision Making


1
Mathematics, Mathematical Modeling, and Decision
Making
  • Jennifer Slimowitz
  • Board on Mathematical Sciences and Their
    Applications
  • The National Academies

2
Outline
  • National Academies and mathematics
  • 3 examples of current project areas
  • Environmental modeling
  • Molecular biology and genomics
  • Forensic science

3
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
  • National Academy of Sciences
  • National Academy of Engineering
  • Institute of Medicine
  • National Research Council

4
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.

5
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

6
3 Examples of Current Studies
  • Mathematical models used by the EPA
  • Molecular biology and genomics
  • Forensic science

7
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

8
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?

9
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!)

10
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!

11
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?

12
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.

13
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

14
Molecular Biology and GenomicsBig 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?

15
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

16
Role of Mathematics in Molecular Biology and
Genomics
  • To identify emergent properties, generate
    non-obvious 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)

17
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?

18
National Academies Study in Progress
Mathematical Sciences Research for DOEs
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

19
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

20
  • 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 .

21
Forensic Science
  • DNA forensic testing (success story!)
  • Bullet lead analysis
  • False positives vs. False negatives which are
    worse?

22
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

23
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.

24
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

25
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?

26
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

27
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?

28
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
    MethodStatistics for ComparisonInterpretation
    Issues

29
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

30
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.

31
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?
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