Synthetic plots: history and examples

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Synthetic plots history and examples
David R. Brillinger Statistics
Department University of California,
Berkeley www.stat.berkeley.edu/brill
brill_at_stat.berkeley.edu
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SECTIONS

• Model appraisal methods
• II. Synthetic plots
• III. Spatial p.p. galaxies
• IV. Time series river flow
• V. Spatial-temporal p.p. - wildfires
• VI. Trajectories seals, elk
• VII. Summary and discussion
• ?. Explanatories

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• MODEL APPRAISAL
• Science needs appraisal methods
• Cycle
• Model construction lt--gt model appraisal
• Is model compatable with the data?
• Classical chi-squared (df correction)
• The method of synthetics
• Neyman et al

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II. SYNTHETIC PLOTS Simulate realization of
fitted model Put real and synthetic side by
side Assessment Turing test? Compute
same quantity for each?
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III. SPATIAL P. P. - galaxies
Neyman, Scott and Shane (1953) On the spatial
distribution of galaxies Astr J, 117, 92-133
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Figure 1 was constructed assuming , the
Poisson law . it was decided to produce a
synthetic plot When the calculated scheme
of distribution was compared with the actual
distribution of galaxies recorded in Shanes
photographs of the sky see page 192, it became
apparent that the simple mechanism could not
produce a distribution resembling the one we see.
In the real universe there is a much more
pronounced tendency for galaxies to be grouped in
clusters.
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Neyman, Scott Shane (1954) On the index of
clumpiness Astr. J. Suppl. 1, 269-294. In
the third paper , it was shone that the visual
appearance of a synthetic photographic plate,
obtained by means of a large scale sampling
experiment conforming exactly with the
assumptions of the theory, is very similar to the
actual plate. The only difference noticed between
the two is concerned with the small-case
clumpiness of images of galaxies.
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In summary Data (xj,yj), spatial point
process 1.Poisson rejected (visually) 2.
Clustering (Neyman-Scott process) rejected
(visually) 3. More clustering Detail counting
error, variation in limiting magnitude
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Results Scientific American 1956
Turing test?
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Comparison Scott et al (1953)
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IV. TIME SERIES Saugeen River Average monthly
flow 1915 1976 Walkerton
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Data y(t), time series Hippel and
McLeod Periodic autoregression (PAR) Stack years
- 62 by 12 matrix (Buys-Ballot) AR(1)
Xij ?jXi,j-1eij Nonstationary Fit,
generate synthetic series
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Turing test?
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Comparison Spectral ratio
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V. SPATIAL-TEMPORAL P. P. - wildfires
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Risk analysis Pixel model, (xj,yj,tj) logit
PNxyt 1 g1(x,y)g2(lttgt)ht (x,y)
location, lttgt day, t year, g1, g2
smooth Sampled 0s
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Original and Bernoulli simulation
Turing test?
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Comparisons Nearest neighbour distribution
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VI. TRAJECTORIES - Hawaiian monk seal, endangered
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Foraging, resting,
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DEs. Newtonian motion Scalar potential function,
H Planar case, location r (x,y), time
t dr(t) v(t)dt dv(t) - ß v(t)dt ß grad
H(r(t),t)dt v velocity ß coefficient of
friction dr - grad H(r,t)dt µ(r,t)dt, ß gtgt 0
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Examples of H. Point of attraction H(r)
.5s2 log r d r Point of repulsion H(r)
C/r Attraction/repulsion H(r) a(1/r12
1/r6) General parametric H(r) ß10x ß01y
ß20x2 ß11xy ß02y2 Nonparametric spline
expansion
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SDEs.
dr(t) µ(r(t),t)dt s(r(t),t)dB(t) µ
drift, grad H s diffusion B(t) bivariate
Brownian
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Data (x(tj),y(tj)),tj) Solution/approximation.
(r(ti1)-r(ti))/(ti1-ti) µ(r(ti),ti)
s(r(ti),ti) Zi1/v(ti1-ti) Euler
scheme Approximate likelihood Boundary, startup
effects
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Fitted potential general parametric
attraction repulsion
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Synthetic
Turing test?
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Comparison Bagplot cp. Boxplot
centre is bivariate median bag contains
50 with greatest depth fence inflate bag
by 3 Rousseuw, Rutts, Tukey (1999)
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TRAJECTORIES elk/wapiti Rocky Mountain elk
(Cervus elaphus) Banff Starkey Reserve,
Oregon Joint usage possible?
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Data (x(tj),y(tj)), tj) 8 animals, ?t
2hr Foraging, resting, hiding, Model. dr
µ(r)dt sdB(t) µ smooth geography
velocity field
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Boundary (NZ fence) dr µ(r)dt s(r)dB(t)
dA(r) A, support on boundary, keeps particle
constrained Synthetic paths. If generated
point outside, keep pulling back by half til
inside
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Turing test?
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Comparisons Distribution of distances to
centre
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VII. SUMMARY DISCUSSION
Synthetic plots method for appraising complex
data-based models via Monte Carlo Criteria EDA,
formal Four examples time series,
spatial-temporal p.p., trajectories Found
inadequacies in each case
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Corrections like Pearsons for chi-squared Diffic
ulties land mask irregular times large
time differences simulations based on same
data
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Acknowledgements.
Aager, Littman, Preisler, Stewart NSF, FS/USDA
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REFERENCES E. Nelson (1967). Dynamical Theories
of Brownian Motion. Princeton H. S. Niwa (1996).
Newtonian dynamical approach to fish schooling.
J. theor. Biol.
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Part B.
Experiment with explanatory Same 8 animals ATV
days, ?t 5min
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