Statistics and Probability in Geosciences:

1. Statistics and Probability in Geosciences: from Least Squares to Random Fields Fernando Sansó DIIAR - Politecnico di Milano – Polo Regionale di Como Athens - 11 May 2004

2. 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 Geman-Geman Statistics drifts away from its origins, so much entangled with geo-sciences (and astronomy) Athens - 11 May 2004

3. 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. Athens - 11 May 2004

4. 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 Plato’s ideas () to Euclede’s () elements). Athens - 11 May 2004

5. 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. Athens - 11 May 2004

6. 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. Athens - 11 May 2004

7. 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. Athens - 11 May 2004

8. 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. Athens - 11 May 2004

9. The deterministic model in Gauss-Markov theory • (discrete and finite-dimensional): • every experiment can be described by n+m 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. Athens - 11 May 2004

10. 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 Athens - 11 May 2004

11. a • V • Range [A] • Once linearized, the deterministic model has the meaning that y cannot wander over the whole m, but it is constrained to Athens - 11 May 2004

12. The stochastic Model • (reduced to pure 2nd order information) • We assume now we observe the vector y , so that we draw a vector Y0from an m-dimensional variate: • L.S. problem find on V, somehow related to Y0 Athens - 11 May 2004

13. L.S. Principle • Let be given by: • Justification (Markov theorem) • Among all linear unbiased estimators • putting • we have Athens - 11 May 2004

14. 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: Athens - 11 May 2004

15. From L.S. theory new problems have evolved: • testing theory as applied to • Correctness of the model (2test on ) • Values of the parameters(significance of input factors in linear regression analysis) • Outliers identification and rejection(Baarda’s data snooping) and the natural evolution towards robust estimators (L1-estimators etc.) Athens - 11 May 2004

16. 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. Athens - 11 May 2004

17. Variance elements estimation or random effects model • When we don’t know C but we have a model • this corresponds to the following stochastic model • when i are basically non observable (or hidden) random parameters. Athens - 11 May 2004

18. 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 Athens - 11 May 2004

19. If we called all the equations we get • we get • and in the next ITRF, the IERS is going to estimate • together with Athens - 11 May 2004

20. 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. Athens - 11 May 2004

21. 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) Athens - 11 May 2004

22. 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 Athens - 11 May 2004

23. According to Bayes theorem the observation Y0enters 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 Athens - 11 May 2004

24. 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 Athens - 11 May 2004

25. A general Bayesian network is a network where points are random and measurements change their distribution; in a sense (apart from Heisenberg’s principle) there is a striking similarity with quantum theory. • Let us now restrict the Bayes concept to the linear regression case. Athens - 11 May 2004

26. The solution is then written as • Where we see that it is a combination of the observations with the prior knowledge Athens - 11 May 2004

27. Note: now we don’t have any more a rank deficiency because > 0 for sure, so that we can have n > m andand even • n = + , • i.e. X is in reality a random field: • functionals of u Athens - 11 May 2004

28. 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) Athens - 11 May 2004

29. 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. Athens - 11 May 2004

30. 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. Athens - 11 May 2004