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Dynamic Indeterminism in Science David R. Brillinger Statistics Department University of California, Berkeley www.stat.berkeley.edu/~brill brill@stat.berkeley.edu. 1. INTRODUCTION. I. Neyman II. Stochastics III. Population dynamics

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Dynamic Indeterminism in Science

David R. Brillinger

Statistics Department

University of California, Berkeley

www.stat.berkeley.edu/~brill

brill@stat.berkeley.edu

slide2

1. INTRODUCTION

I. Neyman

II. Stochastics

III. Population dynamics

IV. Moving particles

V. Discussion

A succession of examples, some JN’s, some DRB + collaborators’

slide3

I. NEYMAN

1894 Born, Bendery, Monrovia

1916 Candidate in Mathematics, U. of Kharkov

  1917-1921 Lecturer, Institute of Technology, Kharkov

  1921-1923 Statistician, Agricultural Research Inst, Bydgoszcz, Poland

  1923 Ph.D. (Mathematics), University of Warsaw

  1923-1934 Lecturer, University of Warsaw

Head, Biometric Laboratory, NenckiInst.

1934-1938 Lecturer, then Reader, University College

slide4

1938 Professor of Mathematics, UC Berkeley

  • Statistics Department, UCB
  • 1961 Professor Emeritus, UCB
  • 1981 Died, Oakland, California
slide6

2. THE MAN.

Polish ancestry and very Polish.

“His devotion to Poland and its culture and traditions was very marked, and when his influence on statistics and statisticians had become world wide it was fashionable ... to say that `we have all learned to speak statistics with a Polish accent' …”

D.G. Kendall (1982)

Twinkle in the eye - coat

Own money for visitors and students

Drinks at Faculty Club

“To the ladies present, and …”

Soccer

“I was one of the forwards, not on the center, …, but on the left. … I could run fast.”

slide7

Many, many visitors to Berkeley

“He seemed to know personally all the statisticians of the world.”

T. L. Page (1982)

Strong social conscience

“this is in connection with the current developments in the South, including the arrests of large numbers of youngsters, their suspension or dismissal from schools, the tricks used to prevent Negroes from voting, …”

Neyman and others (1963)

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3. NEYMAN’S WORK.

“… the delight I experience in trying to fathom the chance mechanisms of phenomena in the empirical world.”

Neyman(1970)

215 research papers

From 1948, 55 out of 140 with E.L.Scott

slide9

Special influences.

K. Pearson (The Grammar of Science), R. A. Fisher (Statistical Methods for Research Workers)

“… there is not the slightest doubt that his (RAF’s) many remarkable achievements had a profound influence on my own thinking and work.”

Neyman (1967)

Applied at the start (agriculture) and at the end (Using Our Discipline to Enhance Human Welfare)

slide10

Applications.

Agriculture, astronomy, cancer, entomology, oceanography, public health, weather modification, …

Theory.

CIs, testing, sampling, optimality, C(α), BAN, …

slide11

How were models validated?

Observed and expected

Formal tests with broad alternatives

Chi-squared

“appears reasonable”, “satisfactory fit”, …

“… the method of synthetic photographic plates”

Neyman, Scott, Shane (1952)

One simulates realizations of a fitted model

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Synthetic

Photographic plate

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Discovered variability beyond elementary clustering

“When the calculated scheme of distribution was compared with the actual …, it became apparent that the simple mechanism could not produce a distribution resembling the one we see.”

Neyman and Scott (1956)

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II. STOCHASTICS

“The essence of dynamic indeterminism in science consists in an effort to invent a hypothetical chance mechanism, called a 'stochastic model', operating on various clearly defined hypothetical entities, such that the resulting frequencies of various possible outcomes correspond approximately to those actually observed.”

Neyman(1960)

“… stochastic is used as a synonym of indeterministic.”

Neyman and Scott (1959)

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4. RANDOM PROCESSES.

Time series.

Chapter in Neyman (1938)

Markov.

“Markov is when the probability of going - let's say - between today and tomorrow, whatever, depends only on where you are today. That's Markovian. If it depends on something that happened yesterday, or before yesterday, that is a generalization of Markovian.”

Neyman in Reid (1998)

States of health, Fix and Neyman (1951)

slide16

State space model.

Vector contains basic information concerning evolution

Can incorporate background knowledge

Can make situation Markov

Evolution/dynamic equation

Measurement equation

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III. POPULATION DYNAMICS

6. SARDINES.

In 1940s Neyman called upon to study the declining sardine catches along the West Coast.

slide19

“Certain publications dealing with the survival rates of the sardines begin with the assumption that both the natural death rate and the fishing mortality are independent of the age of the sardines, …”

Neyman(1948)

Na,t: fish aged a available year t

N(t) = [Na,t]: state vector

na,t: expected number caught

qa: natural mortality age a

Qt: fishing mortality year t

Model: Na+1,t+1 = Na,t(1-qa)(1-Qt)

H0: qb = qb+1 = … = qa , a > b

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“… the death rate has a component which increases with the increase in age of the sardines. It may be presumed that this component is due to natural causes.”

Neyman(1948)

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Tables of fitted and observed.

“While in certain instances the differences between Tables IV and VII are considerable, it will be recognized that the general character of variation in the figures of both tables is essentially similar.”

(ibid)

How to study further? HA?

Neyman et al (1952), astronomy

EDA: plot |X-Y| versus (X+Y)/2

slide24

7. Lucilia cuprina.

Guckenheimer, Gutttorp, Oster & DRB in late 70s studied A. J. Nicholson’s blowfly data.

slide25

Obtaining the data

Population maintained with limited food for 2 years

Started with pulse

Counts of eggs, emerging, deaths every other day

Life stages

egg: .5 – 1.0 day

larva: 5-10 days

pupa: 6-8 days

adult: 1-35 days

slide27

Question: Dynamical system leading to chaos?

State space setup.

Na,t: number aged a on occasion t

Et: number emerging = N0,t

Nt: state vector = [Na,t]

Nt: number of adults = 1’Nt

Dt: number dying = Nt-1 + Et – Nt

qa,t: Prob{individual aged a dies aged a | history}

Dt | history fluctuates about Σaqa,tNa,t

slide28

Age and density dependent model,

qa,t = 1 – (1-αa)(1-βNt)(1-γNt-1)

αa: dies | age a

βNt: dies | Nt adults

γNt-1: dies |Nt-1, preceding time

NLS, weights Nt2

slide31

Blowfly conclusions.

Death rate age/density dependent

Nonlinear dynamic system, chaos possible

“Nicholson was using the flies as a computer.”

P.A.P. Moran (late 70s)

slide32

IV MOVING PARTICLES

8. CLOUD SEEDING.

JN started work in early 50s

California, Arizona, Switzerland

Emphasized importance of randomization

Hail suppression experiment

Grossversuch III, Ticino

Suitable days (thunderstorm forecast)

Silver iodide seeding from ground generators

slide35

DRB (1995)

Particles born at Ticino at times σj

Point process, {σj}, has rate pM(t)

t, time of day

Travel times independent, density f(.)

Particles arrive at Zurich at rate pN(t)

pN(t) = ∫ pM(t-u)f(u)du

slide36

μR: mean rain per particle

X: cumulative process of rain

pX(t): rate of rainfall

pX(t) = μR∫ pM(t-u)f(u) du

E{X(t)} = ∫0t pX(v)dv

α: rate of unrelated rainfall

slide37

Running mean [X(t+1)-X(t-2)]/3

pM(t) = C, A < t < B

Regression function.

α + C0μR [∫ab F(u)du- ∫cd F(u)du]

a = t-2-A, b = t+1-A, c = t-2-B, d = t+1-B

Travel velocities, gamma

OLS, weights 53 and 38

slide38

Approximate 95% CI for travel time.

5.50 ± 1.96(.76)

Seeding started at 7.5 hr

CI for T, arrival time of effect

13.0 ± 1.5 hr

slide40

12. Equations of motion.

DEs. Newtonian motion

Described by potential function, H

Planar case, location r = (x,y)’, time t

dr(t) = v(t)dt

dv(t) = - βv(t)dt – β H(r(t),t)dt

v: velocity β: coefficient of friction

dr = - H(r,t)dt = μ(r,t)dt, β >> 0

Advantage of H - modelling

slide41

SDEs.

dr(t) = μ(r(t),t)dt + σ(r(t),dt)dB(t)

μ: drift (2-)vector

σ: diffusion (2 by 2-)matrix

{B(t)}: bivariate Brownian

(Continuous Gaussian random walk)

SDE benefits,

conceptualization, extension

slide42

Solution/approximation.

(r(ti+1)-r(ti))/(ti+1-ti) =

μ(r(ti),ti) + σZi+1/√(ti+1-ti)

Euler scheme

Approximate likelihood

slide43

14. ELK. DRB et al(2001 - 2004)

Starkey Reserve, Oregon

Can elk, deer, cows, humans coexist?

NE pasture

slide44

Rocky Mountain elk (Cervus elaphus)

8 animals, control days, Δt = 2hr

Part A.

Model.

dr = μ(r)dt + σdB(t)

μ smooth - geography

Nonparametric fit

Estimate of μ(r):velocity field

slide46

Synthetic path.

Boundary (NZ fence)

dr(t)= μ(r(t),t)dt + σ(r(t),dt)dB(t) +dA(r(t),t)

A, support on boundary, keeps particle in

What is the behavior at the fence?

slide48

Part B.

Experiment with explanatory

Same 8 animals

ATV days, Δt = 5min

slide51

Model.

dr(t)= μ(r(t))dt + υ(|r(t)-x(t-τ)|)dt + σdB(t)

x(t): location of ATV at time t

τ: time lag

slide53

V. DISCUSSION

Examples of dynamic indeterminism

JN’s EDA.

Residuals.

“... one can observe a substantial number of consecutive differences that are all negative while all the others are positive. ... the `goodness of fit' is subject to a rather strong doubt, irrespective of the actual computed value of χ2, even if it happens to be small.”

Neyman (1980)

(X-Y) vs. (X+Y)/2 plot

slide54

Lunch time conversations, Neyman Seminars, drinks at Faculty Club, hooplas, …

JN: the gentleman of statistics

Role models – JN, JWT, …. I was lucky.

slide55

Acknowledgements.

Aager, Guckenheimer, Guttorp, Kie, Oster, Preisler, Stewart, Wisdom

Cattaneo, Guha, Lasiecki

Lovett, Spector

NSF, FS/USDA