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?