Modeling Uncertainty in the Earth Sciences Jef Caers Stanford University

Transcription

1 Probability theory and statistical analysis: a review Modeling Uncertainty in the Earth Sciences Jef Caers Stanford University

2 Concepts assumed known Histograms, mean, median, spread, quantiles Probability, conditional probability, Gaussian distribution, Random variable, probability density, cumulative distribution Expectation, variance Scatterplot, correlation coefficient

3 Concepts in statistics Probability, Statistics: a review

4 Graphical analysis But what if we need to compare two data sets and have only very few sample values?

5 Quantile-quantile plot a p-quantile with p [0,1] (a percentile) is defined as that value such that a proportion of 100 p of the data does not exceed this value Dataset Dataset Rank-order Dataset Dataset Percentile 1/6 2/6 3/6 4/6 5/6 6/6

6 Dataset 2 Plotting Dataset 1

7 Concepts in probability theory Probability, Statistics: a review

8 Probability There is a 60% probability/chance of finding iron ore in this region Interpretation 1: The geologist feels that, over the long run, in 60% of similar regions that he studies, one will actually yield iron ore. Interpretation 2: The geologist assesses, based and his/her expertise and prior knowledge that it is more likely that the region will contain iron ore. In fact, 60/100 is a quantitative measure of the geologist s assessment about the hypothesis that the region will contain iron ore, where 0/100 means there is certainly no iron ore and 100/100 means there is certainly iron ore.

9 Conditional probability P event E occurs event F occurs P( E F) E F S P( EF) surface of intersection of circles E and F P( E and F) surface of circle F P( F)

10 Bayes rule P( F E)P( E) P( E F) P( F E)P( E) P( F) Posterior Likelihood Prior Learning from data (and hoping to reduce uncertainty) How much we learn depends on P(E) : how we knew before we had the data P(E F): the (uncertain) relationship between data and the unknown

11 Example 1/10 of all diamond deposits being considered for appraisal are economical Garnet is a mineral that tends to co-occur with diamond: in fact, historically: the probability of garnet exceeding 5ppm for profitable deposits is 4/5 and only 2/5 for non-profitable deposits Your analysis of garnet data for the current deposit reveals that the garnet content equal 6.5 ppm What is the probability that the deposit is profitable?

12 Result E 1 = the deposit is profitable F 1 = the garnet content exceeds 6.5ppm P( E F ) 1 1 P( F E )P( E ) 4 / 5 1 / 10 2 P( F ) 4 / 5 x 1 / 10 2 / 5 x 9 / rule of total probability (removing E by summing over all possibilities) P( F ) P( F E )P( E ) P( F E )P( E ) Note: P( E ) P( E ) 1 1 2

13 Random variable A random variable Z: random variable is a variable whose outcome is unknown but its frequency of outcome is quantified by a probability distribution model Discrete RVs Probability mass function (pmf) Cumulative distribution function (cdf) Continuous RVs Probability density function (pdf) Cumulative distribution function (cdf)

14 Probability mass function Notation p X (a) = P(X=a) For continuous variables p X (a) = P(X=a) = always zero!

15 Probability density function b P( a X b) fx ( x) dx a f X (x) Shaded area represents a probability f X ( x) dx 1 some outcome will occur x 1 a b x 2 x f X ( x) 0 probabilities cannot be negative

16 Likelihood f X (x) Shaded area represents a probability x 1 a b x 2 x f X ( x) has the meaning of a likelihood, not of probability f f X X ( x1) ( x ) 2 this ratio indicate how much more/less likely x will occur vs x 1 2

17 Cumulative distribution function f X (x) 1 F X (x) always between 0 and 1 and never decreases Area A Area A x x F X (x) = P(X x)

18 Examples Poisson p ( i) P( X i) e X i i! is the number of points per unit area f X (x) F X (x) Uniform 1 x a b a b x

19 Examples f X m=0,s=1 m=1,s=1 m=0,s=2 f X 1 1 x m ( x) exp 2ms 2 s 2

20 Empirical distribution function Fˆ ( x) X 1 n=6 data: 10.1 / 15.4 / 8.6 / 9.5 / 20.6 / 3.2 1/ x

21 Modeling from data ˆF 1/(n+1) to allow extrapolation 1 1/ ˆF Linear inter/extrapolation

22 Monte Carlo simulation Aim Mimicking the process of actual sampling Needed: Pseudo random number generator A computer software program that creates (deterministically) a set of uniform random numbers between [0,1], as initiated with a seed (a large odd integer such as 56781) Example: , , , etc Other terminology: drawing, sampling

23 Mechanism 1 F p Use any random number generator that renders a value between [0,1] Any type of cdf x p : value randomly drawn from F x

24 Some first models of uncertainty P( A) P( A B b) f ( X Y y) X Samples: x, x, x,..., x n Samples drawn by Monte Carlo simulation are a valid model of uncertainty

25 Data Transformation Aim To transform the empirical distribution of a dataset into another empirical distribution Why? Certain methods require that the empirical data has a certain shape, such as a standard normal shape To lessen the influence of extreme values for skewed distributions

26 Mechanism cdf data cdf Gaussian 5/6 4/6 3/6 2/6 1/6 x s x y s y

27 Diamond value US $ Correlation or association Diamond size (ct)

28 (linear) correlation coefficient r 1 x x y y n s s n i i 1 i 1 x y n 1 sx x x n 1 i 1 2 i If r larger stronger correlation If r > 0 positive correlation If r < 0 negative correlation If r = 1 perfect linear correlation If r = 0 no linear correlation the range of r restricted to [-1,1]

29 Examples 9 r = r = Y 5 4 Y X X Y r = X