CSC 599: Computational Scientific Discovery

Lecture 3: Data, Data Structures, and Reasoning

in Science

Outline

Computers Helping ScientistsData in Science

Data Catalogs

Data structures for scientific knowledge Equations Symbolic knowledge

Simulations and scientific reasoning Do simulations Architectures for sci. reasoning QSIM

Next time: Probabilistic reason and graphical models

Computers cooperating with Scientists

Play to the strengths of computers Fast Accurate (we hope!) Don't get bored Good for simulations of complex systems!

Avoiding their weaknesses No “common sense”

Garbage In, Garbage out Calculation bugs

Time to develop/maintain code Use someone else's program, if exists

Don't Just Code-It-Up!

Do you really need a computer? Analytical solutions are cooler Integrate! (If you can) Ordinary differential equation

System of eqns & derivatives w.r.t. one independ var Eg. time

Laplace and z-transformations Perturbation expansions and discrete time eqns

Partial differential equations Another variable can change

For fluids: pressure, temperature, etc. Separation of variables

See Gershenfeld “The Nature of Mathematical Modeling”

Data in Science

You make a measurement, should record? Value (numeric or conceptual) Precision

For measured numbers: Mean and std deviation For concepts: set identity

“car” vs. “Toyota” vs. “Prius” Domain

System constraints Range definition limits System extent limits Saturation limits

Instruments constraints Range definition limits Detection limits Reliable limits

Data constraints Observed min/max w/in system extent and detect limits

Data in Science (2)

You make a measurement, should record? Units and Dimensions When and Where

Speed of light expected to be observer-independent Count of robins? First bloom of dandelions?

With what equipment Eg. camera model and film type

By whom? Implicit paradigm Yesterday's Chinese “guest stars” are today's

supernovas Why?

What are you trying to look for? Might hint at what your data do not show

Laboratories: Where the Data Comes From

Places where observations are made Astronomical Observatories Particle Physics Accelerators

CERN FermiLab

Biology Labs Human Genome Project

Issues Distributed observations

Human Genome Project Collaborative observations

Catalogs

What is (and is not) Recorded

Distributed Nature of Catalogs

Synonyms in Catalogs

Missing and Error Values in Catalogs

Permanence of Catalogs

What is (and is not) recorded

Not record “outside of bounds” Space

Outside of area of interest Time (maybe Money!)

Budget ran out, no more money for sensing Magnitude (strength)

“Outside of bounds” may be a “soft” concept USGS records earthquake

All Mw > 7

Most Mw > 6

Has Mw ≈ 5 and below close to seismometers

Q: Is it true that the only small earthquakes are close to seismometers?

Distributed Nature of Catalogs

Distributed databases Hopefully designed properly

Redundancy Reliably online up-time

Central repository Centralizes data available from other dbs

Example: some biochemical dbs May use

inconsistent terms, different missing value conventions, etc

May be out-of-date relative to source dbs Format may be “least common denominator” of

more specialized formats

Synonyms and Different Meanings for a Given Word

Concepts ASCII or UNICODE string

Consider the term “robin” In N. America:

“American robin” “Turdus migratorius

In Europe “European robin” “Erithacus rubecula”

What about these terms? “thrush family bird” “songbird” “bird”

Ontologies can help (if used)

Missing and Errors Values

Missing values often coded as sentinel values There might be more than one! Esp. for “required” fields

Clerks were required to get customer telephone # Most common tele number? (111) 111-1111

Errors values Probably a bigger problem before computers Computers can solve some problems:

Sanity rules Parity, checksum (corruption), nonce (security)

Computers can add problems too NASA launched satellite to look at atmospheric ozone Satellite reported very low O

3 levels by south pole

Data was not believed, BUT WAS TRUE (Ozone hole)

Permanence of Catalog

How long does the media last? Is the data viewed as temporary or worth

archiving? Is more data generated than can be stored? Proprietary? Subject to privacy issues?

Backup policy? Organization that holds the data

Tech savvy? Have money? Well-organized in general?

Redundant sites? Banks do this, but they have $ and are protecting $

The Importance of Prediction

Scientists do a lot of things: Prediction underlies much

Scientific assertions Numbers: Equations Concepts: Rules, decision trees, production sys Probabilities: Graphical models

Next time!

Equations

“Strongest” form of knowledgeDefines relationships between quantities

F = ma Force and acceleration acting in same direction Mass is scalar

For N variables: N-1 knowns -> 1 unknown

Scalar or Vector F = ma

Equations, cont'd

Are they always generalizations?The Drake Equation:

N = R* * fp * n

e * f

l * f

i * f

c * L

Where:N = Number of civilizations emitting radio transmissionsR* = rate of Sun-like star formationfp = fraction of stars with planets

ne = average number of inhabitable planets

fl,f

i,f

c = probability of life, intelligence and civilization

L = average duration of a civilizationSo far N = 0, is this equation useful?Do we know any of these terms well?What if N was non-zero, would the eqn be useful?

Computing with symbols

Rules If dog(X) and wet(X) then smelly(X)

Decision trees

Production systems Like rules, but special memory to hold what is

true

Simulations

Dimensions Deterministic or stochastic? Discrete or continuous? Steady-state or time-varying? Linear or non-linear? Batch or real-time?

Phases of development1. Real system to logical abstraction2. Appropriate datastructs/algorithms/math technique3. Implement algorithm as program4. Validate implementation

Real System to Logical Abstraction

1. Define the problem What are the objectives? What question(s) would you like to answer? What do you anticipate answering in the future?

2. When in doubt, leave detail out You may not need it

“A theory should be as simple as it needs to be, and no simpler” A. Einstein

You can always add it in future3. Spend the time turning vague statement of

goals into better defined one(s) Revisit as necessary (That's just good software engineering!)

Appropriate Datastructure/Algorithm/Math Technique

Setting up the model Start with what's known Go simple -> complex Iterate Develop large models modularly Model only what is necessary Make (and state) assumptions and hypotheses State constraints Alternate between top-down & bottom up

Best data structure/algorithm? Data Structures and algorithms class

Best math technique? Numerical methods class

Validation

Define some “test cases” before hand Define (input,output) pairings Should range in complexity Should be checked:

By scientists, or By well-known (e.g. “textbook”) cases, or By hand (and then double-checked by someone else)

Just good software engineering!

Implementation

Procedural languagesObject oriented languagesArtificial Intelligence LanguagesProduction systemsSimulation languages

Procedural Languages

Examples Fortran, C

Fast Compilers map language to machine code easily Decades worth of optimizing compilers

Large body of libraries for scientific Among first programming languages Decades worth of optimizing libraries

Have to think in machine terms, not domain terms

Object Oriented Languages

Examples C++, Java, C#, (Eiffel, Smalltalk)

Object Oriented Nature Classes can represent scientific objects Methods can represent scientific transitions C++ can link with C libraries

Still may be a hassle to program objects

Artificial Intelligence Languages

Examples Prolog, Lisp, Scheme

Prolog: Predicate Logic (Horn Clause) basedFacts:

sentiveNose(sniffer).dog(fido). wet(fido).

Rules:smelly(X) :- dog(X), wet(X).offendedBy(Y,X) :- smelly(X), sensitiveNose(Y).

Possible queries:Is sniffer offended by fido? offendedBy(sniffer,fido).Does fido offend anyone? offendedBy(A,fido).Is sniffer offended by anyone? offendedBy(sniffer,B).Is anyone offended by anyone else? offendedBy(A,B).

Artificial Intelligence Langs (2)

Lisp It's all functions operating on lists:

(defun computeProperties (object newObj) (if (and (getProperties object dog) (getProperties object wet) ) (addProperty object smelly newObj) ))

Artificial Intelligence Langs (3)

Advantages Encourage thinking in domain space Prolog: modular rules Lisp: functions operating on lists good for

symbolic processing

Disadvantages: Prolog: not good for floating pt numbers

Exact matching not compatible with float pt rounding Lisp:

Handling loops and variables somewhat contrived

Production Systems

Examples SOAR, OPS5, Mycin?

Models human thought

Production sys = Rules + Working Memory Rules = if-then statements

if dog(X) and wet(X) then smelly(X)if smelly(X) and sensitiveNose(Y) then offendedBy(Y,X)

Working memory = models human memorydog(fido), wet(fido), sensitiveNose(sniffer)

Computation1st round: compute smelly(fido)2nd round: compute offendedBy(sniffer,fido)

Production Systems, cont'dAdvantages

Productions are inherently modular Can have constraints on working memory elements

(“wme's”) Better models human cognitive limitations

RETE algorithm matches wmes and rules efficiently Rules = laws, WMEs = state

When rules conflict, what happens? Architecture has to do something OPS5: More specific rule wins SOAR: Create problem space to resolve issue MYCIN: Rules w/certainty factors (weighted vote) Is what the architecture does what you want?

Simulations Languages

Examples QSIM, SPICE (analog circuits), Scilab

Qualitative simulations Very precise, not very general:

d2x/dt2(t) = -9.8 m/sec2, x(t0) = 2m, v(t

0) = 0 m/sec

Less precise, more generald2x/dt2(t) = -g, x(t

0) = x

0, v(t

0) = v

0

Even less precise, even more generaldx/dt = M- (monotonically dec. fnc),x(t

0) = 0 .. x

0 .. infinity

QSIM (1)Values

Either landmark values0, MAX_CAPACITY, t

0, inf

Or ranges between landmark valuesNot full but not empty: (0,MAX_CAPACITY),Time before t

1 but after t

0: (t

0,t

1)

Variables:Have both a value and derivative

<0,inc> <(0,MAX_CAPACITY),inc><MAX_CAPACITY,std>

Functions:Either monotonically increasing or decreasing

M+, M-

Derivatives: either increasing, decreasing or steady:inc, dec, std.

QSIM (2): Classic Example

U-Tube: Two tubes connected by pipe Both have

Some initial fluid levelamtA: 0..AMAX..inf amtB: 0..BMAX..inf

Pressure dependent upon levelpressureA = M+(amtA) pressureB = M+(amtB)

QSIM (3)Other constraints:

Total fluid only in A & B, and is constant:amtA + amtB = total constant(total)

Flow depends on pressure difference:pAB = pressureA – pressureBflowAB = M+(pAB)d(amtB)/dt = flowAB d(amtA)/dt = -flowAB

Knowledge of quantities:total: 0..infamtA: 0..AMAX..inf amtB: 0..BMAX..infpressureA: 0..inf pressureB: 0..infpAB: -inf..0..infflowAB: -inf..0..inf

Correspondence between valuespressureA & amtA: (0,0), (inf,inf)pressureB & amtB: (0,0), (inf,inf)flowAB & pAB: (-inf, -inf), (0,0), (inf,inf)

QSIM (4)

Note correspondence w/ordinary diffy eqnd(amtB)/dt = f(g(amtA) – h(amtB))f,g,h M+

Qualitative state is dynamicImagine filling tank A (ignore tank B)Need to represent amtA and its derivative:amtA(t):

t0: <0,inc>

(t0,t

1): <(0,AMAX),inc>

t1: <AMAX,std>

QSIM (5): Predicting behavior

1. Give initial (at least some) conditions:

“Tank A full” t = t

0: amtA = <AMAX,?>

“Tank B empty”t = t

0: amtB = <0,?>

QSIM (6):

2. Correspondence propagation on init stateamtB = 0 therefore pressureB = 0amtA = AMAX therefore pressureA =

(0,inf)amtA=AMAX && amtB = 0 therefore total = (0,inf)pressureA = (0,inf) and pressureB = 0 therefore pAB = (0,inf)constant(total) therefore d(total)/dt = std pAB = (0,inf) therefore flowAB = (0,inf)flowAB = (0,inf) therefore d(amtA)/dt = dec d(amtB)/dt = incd(amtA)/dt = dec therefore d(pressureA)/dt = dec.d(amtB)/dt = dec therefore d(pressureB)/dt = dec.

QSIM (7)

(Init state correspondence propagate, cont'd)Additional constraint

d(pressureA)/dt = dec && d(pressureB)/dt = inc therefore d(pAB)/dt = dec

Last propagationd(pAB)/dt = dec therefore d(flowAB)/dt = dec

So we have at t = t0:

amtA = <AMAX,dec> pressureA = <(0,inf),dec>amtB = <0,inc> pressureB = <0,inc>pAB = <(0,inf), dec> flowAB = <(0,inf), dec>Total = <(0,inf), std>

QSIM (8)

3. Predicting the next state:What events can make the next state?

1. A variable can reach a limit or landmark2. A variable may move off a landmark3. A variable may start or stop moving

6 variables moving to limits/landmarks(theoretically): 46 - 1 = 4095 possibilities (actual): constraints & correspondences reduce choices

t = (t

0,t

1):

amtA = <(0,AMAX),dec> pressureA = <(0,inf),dec>amtB = <(0,BMAX),inc> pressureB = <(0,inf),inc>pAB = <(0,inf), dec> flowAB = <(0,inf), dec>Total = <(0,inf), std>

QSIM (9)What happens after?

t = (t0,t

1):

amtA = <(0,AMAX),dec> pressureA = <(0,inf),dec>amtB = <(0,BMAX),inc> pressureB = <(0,inf),inc>pAB = <(0,inf), dec> flowAB = <(0,inf),

dec>total = <(0,inf), std>

Well:amtA is decreasing toward 0 (won't get there)amtB is increasing toward BMAX.flowAB is decreasing toward 0.So either:

1. flowAB gets to 0 AND amtB gets to BMAX, or2. flowAB gets to 0 BEFORE amtB gets to BMAX, or3. amtB gets to BMAX BEFORE flowAB gets to 0

(Tank B overflows)

QSIM (10)1. flowAB gets to 0 AND amtB gets to BMAX:

t = t1(a)

:amtA = <(0,AMAX),std> pressureA = <(0,inf),std>amtB = <BMAX,std> pressureB = <(0,inf),std>pAB = <0,std> flowAB = <0,std>Total = <(0,inf), std>

3. amtB gets to BMAX BEFORE flowAB gets to 0(Tank B overflows)t = t

1(c):

amtA = <(0,AMAX),dec> pressureA = <(0,inf),dec>amtB = <BMAX,inc> pressureB = <(0,inf),inc>pAB = <(0,inf),dec> flowAB = <(0,inf),dec>Total = <(0,inf),std>

QSIM (11)

2. flowAB gets to 0 BEFORE amtB gets to BMAXt = t

1(b):

amtA = <(0,AMAX),std> pressureA = <(0,inf),std>amtB = <(0,BMAX),std> pressureB = <(0,inf),std>pAB = <0,std> flowAB = <0,std>Total = <(0,inf), std>

amtB's level becomes a new landmark so:amtA: 0..a

0..AMAX..inf

pressureA: 0..p0..inf

amtB: 0..a1..BMAX..inf

pressureB: 0..p1..inf

pAB: -inf..0..infflowAB: -inf..0..inftotal: 0..to

0..inf

QSIM (12)

Output: graph of state transitions:

QSIM (13)QSIM algorithm:

1. Init queue w/QState(t0), complete its description

2. If queue empty or exceed resource limit then stop, otherwise pop state S from queue

3. For each var vi in S get successors from table

4. Determine all successor states consistent w/restrictions If S is time-interval, delete copies of itself

5. For each successor Si, assert:

successor(S,Si) predecessor(S

i,S)

6. Apply global filters on successors7. Add successors to queue

Don't add inconsistent, quiescent, cycles, transition or t = inf states

8. Go to 2

Next Time

Probabilistic reasoning Graphical Models

Machine Learning and CSD