I.Classical probabilities

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Christopher Black
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1 I.Classical probabilities I.4 Markov chains

2 Markov chains? 2

3 Andrei A. Markov Russian mathematician

4 Markov chains as pedagogic, analytical and prognostic tool. Helps us understand probabilities and their additions and co-variations (also new Bayesian Markov Chain Monte Carlo MCMC techniques) 2. Helps us analyse data in a new and interesting way 3. Provides a good complement to traditional statistical post-processing 4

5 I.4. A simple contingency table? Warning No warning Event occurs Hit (H) Missed event (M) Even does not occur False alarm (F) Correct negative (N) Hit rate (HR) H/(H+M) False alarm rate (FAR) F/(F+N) not F/(F+H) 5

6 A simple contingency table? Following 6 hours rain dry First 6 hours rain 7 28 cases dry 6 72 cases 28 cases 72 cases We note that the climatological rain probability is 28% 6

7 A transition table for rain fall at Stockholm- Arlanda airport (00 typical cases) 62% (7/28) probability of continued rain Following 6 hours rain dry 38% (/28) probability of rain to dry First 6 hours rain 7 28 cases 5% (/72) probability of dry to rain dry 6 28 cases 72 cases 72 cases 85% (6/72) probability of staying dry 7

8 Transition probabilities 62% probability of continued rain Next 6 hours R 38% (/28) probability of rain to dry 5% (/72) probability of dry to rain Last 6 hours R R.38 rain dry 85% (6/72) probability of staying dry 8

9 R R.38 What can we say about the 2 h transition from rain? From to 2-8? rain rain.38 dry rain dry rain dry rain R R

10 R R.38 What can we say about the 2 h transition from dry? From to 2-8? dry dry dry rain dry rain.38 dry rain R R

11 The matrice can also be regarded as an algebraic transition matrix Next 6 hours R Last 6 hours R.38 Then the full force of 3000 years of algebra can be applied, in particular matrix algebra

12 I.4.2 Some matrice algebra 2

13 A brief repetition of matrix algebra...

14 A brief repetition of matrix algebra... ( 2) ( ) ( 3 ) ( 2) ( ) (3 6) 4

15 A brief repetition of matrix algebra...

16 Matrix multiplication yields a forecast Last six hours h 06-2 h 2.38 Forecast 2-8 h later Observed 2-8 h later Verification Last six hours h

17 The matrix multiplication continues... Last six hours h 4.38 Forecast h one day later verification Last six hours h Observed h one day later

18 After repeated multiplications the values converge towards the climate Last six hours h Last six hours h

19 I.4.3 Vector multiplied into the Markov matrice 9

20 Similarities with forecast models: Obs/analysis Model Forecast ( 0) Rain.38 (.38) ( 0 ) Dry.38 ( ) (.70.30) 70% rain 30% dry.38 (.48.52) 20

21 more examples: Obs/analysis Model Forecast (.70.30).38 (.48.52) (.90.0).38 (.58.42).38 ( ) ( 30.70)..which is almost identical with the input vector 2

22 I.4.4 Eigen vectors 22

23 Left eigenvector and -value (.28.72).38 (.28.72) (.28.72) (.28.28).38 (.3.3) eigenvalues 0.47(.28.28) 23

24 Right eigenvector and -value eigenvalues

25 The initial transition matrix can be decomposed into a weighted sum of two new matrices.38 n n 0.47 Eigen value Persistence probabilities Climate probabilities Time folding factor Variability Meteorological interpretation 25

26 Matrix climate matrix + residual.38 n Persistence n Variability % 00 persistence % 20 climate time 26

27 I.4.5 Theoretical and observed period lengths 27

28 .38 Mean period length for rain ( - 0) - 6 hours 5.8 hours Generally (τ) with p ii as the persistence probability of the class: τ t ( pii ) 28

29 Calculated and observed duration of rain and showers Theoretical period length 5.8 h Observed period length 5.9 h Theoretical period length 40 h Observed period length 57 h 29

30 6 hour transition matrix for Arlanda airport weather Ο Θ 0 2/8 3 5/8 6 8/8 Ο.6 Θ

31 0.6 N Ο High persistence for clear sky and rain Ο.0.02 N Θ Θ Low persistence for showers Ο.7.07 N Θ Good persistence for overcast and rain Ο Θ N Very low persistence for variable sky 3

32 END 32

Understanding MCMC. Marcel Lüthi, University of Basel. Slides based on presentation by Sandro Schönborn

Understanding MCMC. Marcel Lüthi, University of Basel. Slides based on presentation by Sandro Schönborn Understanding MCMC Marcel Lüthi, University of Basel Slides based on presentation by Sandro Schönborn 1 The big picture which satisfies detailed balance condition for p(x) an aperiodic and irreducable