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Dynamic Indeterminism in Science
David R. Brillinger Statistics Department University of California, Berkeley
www.stat.berkeley.edu/~brill [email protected]
I. Neyman
II. Stochastics
III. Population dynamics
IV. Moving particles
V. Discussion
A succession of examples, some JN’s, some DRB + collaborators’
1. INTRODUCTION
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 WarsawHead, Biometric Laboratory, Nencki Inst.
1934-1938 Lecturer, then Reader, University College
1955 Statistics Department, UCB
1961 Professor Emeritus, UCB
1981 Died, Oakland, California
1938 Professor of Mathematics, UC Berkeley
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 - coatOwn 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.”
2. THE MAN.
“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)
Many, many visitors to Berkeley
“… the delight I experience in trying to fathom the chance mechanisms of phenomena in the empirical world.”
Neyman(1970)
215 research papersFrom 1948, 55 out of 140 with E.L.Scott
3. NEYMAN’S WORK.
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)
Special influences.
Agriculture, astronomy, cancer, entomology, oceanography, public health, weather modification, …
Theory.
CIs, testing, sampling, optimality, C(α), BAN, …
Applications.
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
How were models validated?
Photographic plate Synthetic
“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)
Discovered variability beyond elementary clustering
“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)
II. STOCHASTICS
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)
4. RANDOM PROCESSES.
Vector contains basic information concerning evolution
Can incorporate background knowledge
Can make situation Markov
Evolution/dynamic equation
Measurement equation
State space model.
6. SARDINES. In 1940s Neyman called upon to study the declining sardine catches along the West Coast.
III. POPULATION DYNAMICS
Season 41-2 42-3 43-4 44-5 45-6
Age=1 926.0 718.0 1030.0 951.0 493.0
2 6206.0 2512.0 1308.0 2481.0 1634.0
3 3207.0 4496.0 2245.0 1457.0 1529.0
4 868.0 1792.0 2688.0 1298.0 799.0
5 361.0 478.0 929.0 1368.0 407.0
6 95.1 169.4 327.0 498.5 299.2
7 47.2 36.0 98.4 148.0 111.2
Sardines (arbitrary units) landed on West Coast
Na,t: fish aged a available year t
N(t) = [Na,ta,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
“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)
“… steady increase in fishing effort … 1943-8”
“… 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)
“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
Tables of fitted and observed.
Guckenheimer, Gutttorp, Oster & DRB in late 70s studied A. J. Nicholson’s blowfly data.
7. Lucilia cuprina.
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
Obtaining the data
State space setup.
Na,t: number aged a on occasion tEt: 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 Σa qa,tNa,t
Question: Dynamical system leading to chaos?
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
Age and density dependent model,
Death rate age/density dependent
Nonlinear dynamic system, chaos possible
“Nicholson was using the flies as a computer.”P.A.P. Moran (late 70s)
Blowfly conclusions.
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
IV MOVING PARTICLES
Data: 3 hr rainfall at Zurich, 120km
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
DRB (1995)
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
μR: mean rain per particle
pM(t) = C, A < t < B
Regression function.
α + C0 μR [∫ab F(u)du- ∫c
d 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
Running mean [X(t+1)-X(t-2)]/3
5.50 ± 1.96(.76)
Seeding started at 7.5 hr
CI for T, arrival time of effect
13.0 ± 1.5 hr
Approximate 95% CI for travel time.
DEs. Newtonian motion
Described by potential function, H
Planar case, location r = (x,y)’, time t
dr(t) = v(t)dt
dv(t) = - β v(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
12. Equations of motion.
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
SDEs.
(r(ti+1)-r(ti))/(ti+1-ti) =
μ(r(ti),ti) + σ Zi+1/√(ti+1-ti)
Euler scheme
Approximate likelihood
Solution/approximation.
Starkey Reserve, OregonCan elk, deer, cows, humans coexist?NE pasture
14. ELK. DRB et al(2001 - 2004)
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
Rocky Mountain elk (Cervus elaphus)
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?
Synthetic path.
Experiment with explanatory
Same 8 animals
ATV days, Δt = 5min
Part B.
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
Model.
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
V. DISCUSSION
JN: the gentleman of statistics
Role models – JN, JWT, …. I was lucky.
Lunch time conversations, Neyman Seminars, drinks at Faculty Club, hooplas, …
Aager, Guckenheimer, Guttorp, Kie, Oster, Preisler, Stewart, Wisdom
Cattaneo, Guha, Lasiecki
Lovett, Spector
NSF, FS/USDA
Acknowledgements.
