Nessun titolo diapositiva

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Statistics and Probability in Geosciences
from Least Squares to Random Fields
Fernando Sansó
DIIAR - Politecnico di Milano Polo Regionale di
Como
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Statistics of Geosciences
Laplace Legendre Gauss Poisson Markov
From the availability of new electronic HW, new
impulses to model theory and field theory
Jeffreys Baarda Moritz Krarup Tarantola-Vallette G
eman-Geman
Statistics drifts away from its origins, so much
entangled with geo-sciences (and astronomy)
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recognizing that Statistics is the art and
science of ambiguous knowledge,
I claim that the whole probabilistic, Bayesian,
point of view can be taken as a unifying
foundation for all spatial information sciences.
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Scientific concepts are born of an abstraction
process, namely when we observe natural
phenomena, after eliminatinga multitude of tiny
details, we can grasp the common and regular
elements on which axioms rules and laws can be
built. (From Platos ideas (????) to Eucledes
(????????) elements).
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• Examples
• a straight line who has ever seen one?
• a Eucledean triangle measuring its angles at the
  astronomical level has become one of the means to
  decide about the curvature of the universe
• the Galilean equivalence principle, which emerges
  from physical experiments only by abstracting
  from friction, by assuming a constant gravity
  etc.

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Psychologically one can understand why at the
beginning of modern science the incertitude was
considered as the enemy, classified as
measurement error. This is how modern
statistics was born from the very beginning as
error theory, based on a probabilistic
interpretation via the central theorem and used
in an inferential approach, to produce best
estimates of the parameters of interest,
according to a proto-maximum likelihood
criterion.
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• Historical examples
• The astronomical measurement of Jupiter diameters
  to test the hypothesis that its figure was
  ellipsoidal and rotating around the minor axis.
• The geodetic measurements of arcs of meridians,
  performed by the French Academy in France, in
  Lapland and on the Andes, to measure the
  eccentricity of the Earth.

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• One fundamental step in the development of the
  understanding of statistics has been the clear
  establishment of the so called Gauss-Markov
  linear standard model, with all its developments
  in least squares theory
•  
• this is understood by explaining what are
• the deterministic model,
• the stochastic model.

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• The deterministic model in Gauss-Markov theory
• (discrete and finite-dimensional)
• every experiment can be described by nm
  variables organized in two vectors
• measurable quantities
• parameters
•   these variables are deterministic, i.e. in
  principle they can be given a fixed numerical
  value in the experiment analysed, and they are
  related by geometric and physical laws.

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General mathematical form of the physics of the
experiment   From the observations themselves or
from prior knowledge we have approximate values
and we put   after linearization we
have   and we assume to be able to solve for ?
. In the end we have a linear model of
observation equations
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Once linearized, the deterministic model has the
meaning that y cannot wander over the whole ?m,
but it is constrained to
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The stochastic Model (reduced to pure 2nd order
information) We assume now we observe the vector
y , so that we draw a vector Y0 from an
m-dimensional variate
L.S. problem find on V, somehow related to Y0
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L.S. Principle Let be given
by Justification (Markov theorem) Among all
linear unbiased estimators
putting
we have
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• By L.S. theory, complemented by suitable
  numerical techniques, several very large geodetic
  problems have been solved
• adjustment of large geodetic networks
• (N.A. datum 40.000 parameters 1980)
• satellite orbit control (from 1970)
• analytic photogrammetry
• discrete finite models of the gravity field
  (e.g. by buried masses or by truncated
  spherical harmonics expansion

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• From L.S. theory new problems have evolved
• testing theory as applied to

• Correctness of the model
• (?2 test on )
• Values of the parameters (significance of
  input factors in linear regression analysis)
• Outliers identification and rejection
  (Baardas data snooping) and the natural
  evolution towards
• robust estimators (L1-estimators etc.)

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• Mixed models
• with two types of parameters x (continuous), b
  (integers) like with GPS observations where b are
  initial phase ambiguities.
• Note
• the numerical complexity if we adopt a simple
  trial and error strategy for b if we have a base
  line with 10 visible satellites and for each
  double difference we want to try 3 values, we
  have to perform
• 39 20.000
• adjustments.

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Variance elements estimation or random effects
model When we dont know C?? but we have a
model this corresponds to the following
stochastic model when ?i are basically non
observable (or hidden) random parameters.
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Examples (ITRF 2005) We estimate 3N
coordinated of Earth stations x1, x2, , xN by
different spatial techniques (e.g. GPS, LR,
etc.).   Each technique has a vector of adjusted
coordinates in its own reference frame
where, with respect to a unified reference
system, Note due to imperfect
modelling, one can assume that the estimate of
is unrealistic.
and
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If we called all the equations we get we
get and in the next ITRF, the IERS is going to
estimate together with
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• The Bayesian revolution
• Probability is an axiomatic index measuring the
  subjective state of knowledge of a certain
  system,
• every system is thus described by a number of
  random variables through their joint
  distribution,
• every observation modifies the state of knowledge
  of the system, namely the distribution of the
  relevant variables, through the operation of
  probabilistic conditioning.

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According to this vision, the physical laws are
only verified in the mean, when we average on a
population of effects that cannot be controlled,
and which can be described only by a probability
distribution, expressing our prior knowledge of
the phenomenon. (De Finetti)
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• Linear Bayesian Models
• We start from observation equations
• where all variables are random X, N are the
  primitive variables of the observation process
  and we assume them to be independent and
  described by some prior distribution
• Y, the variable we sample by observations, is
  a derived variable with joint prior

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According to Bayes theorem the observation Y0
enters to condition the X distribution,
namely (Posterior) Example (random
networks) we measure two distances of P from
known P1 and P2
D1
P
P1
Prior of P
D2
P2
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We measure D1
Effect of D1
D1
D1
P1
P1
P2
P2
We measure D2
Effect of D1 and D2
D1
Posteriorof P
P1
P1
D2
D2
P2
P2
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A general Bayesian network is a network where
points are random and measurements change their
distribution in a sense (apart from Heisenbergs
principle) there is a striking similarity with
quantum theory.
Let us now restrict the Bayes concept to the
linear regression case.
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The solution is then written as
Where we see that it is a combination of the
observations with the prior knowledge
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Note now we dont have any more a rank
deficiency because gt 0 for sure, so that we
can have n gt m and and even
n ? ,
i.e. X is in reality a random field
functionals of u
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• Examples
• Cartography u(P) is DEM
• Image analysis u(P) is density of flux through
  a sensor element
• Physical Geodesy u(P) is the anomalous Earth
  potential

(point height)
(gravity anomaly)
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Important remark it is easy to prove that
controls the prior regularity of the field.
elevations profile
image profile
gravity profile
can be considered as a hyperparameter and
estimated through an infinite dimensional
calculus (Malliavin Calculus).   Here statistics
is fused with functional analysis to properly
define the space of estimators.
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Conclusions If modern statistics, which was
born at the beginning together with geodesy and
astronomy to treat measurement errors, has slowly
drifted away to become mate of mechanics, then of
radio-signals analysis and finally of economic
sciences, nowadays we are entitled to say that
Earth sciences with their need of estimating
spatial fields, are giving to statistics a
serious scientific contribution, pushing it along
the road of modern probability theory and
functional analysis.