I.Classical probabilities I.4 Markov chains
Markov chains? 2
Andrei A. Markov Russian mathematician 856-922 3
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
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
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
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
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
R R.38 What can we say about the 2 h transition from rain? From 00-06 to 2-8? rain rain.38 dry rain dry rain.38.38 dry rain 0.38 0+0 0.380.56 0.38 0+0 00.44 R R.44..56 9
R R.38 What can we say about the 2 h transition from dry? From 00-06 to 2-8? dry dry dry rain dry rain.38 dry rain 0 0+0 0.380.78 0 0+0 00.22 R R.44.22.56.78 0
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
I.4.2 Some matrice algebra 2
3 39 7 4 3 2 4 3 2 6 5 + 6 2 5 7 6 5 + + 6 4 5 3 6 2 5 A brief repetition of matrix algebra...
A brief repetition of matrix algebra... ( 2) 3 5 4 6 ( 3 + 2 5 ) ( 3 ) ( 2) 3 5 4 6 ( 4 + 2 6) (3 6) 4
5 4 3 2 6 5 4 3 36 29 6 3 + 6 4 4 3 4 3 2 6 5 4 3 3 + 5 2 3 4 3 2 6 5 4 3 6 3 + 6 2 4 A brief repetition of matrix algebra...
Matrix multiplication yields a forecast Last six hours 00-06 h 06-2 h 2.38 Forecast 2-8 h later.44.22.56.78 Observed 2-8 h later Verification Last six hours 00-06 h.47.2.53.79 6
The matrix multiplication continues... Last six hours 00-06 h 4.38 Forecast 00-06 h one day later.33.26.67.74 verification Last six hours 00-06 h Observed 00-06 h one day later.39.24.6.76 7
After repeated multiplications the values converge towards the climate Last six hours 00-06 h 8.38.72.72.28.28 Last six hours 00-06 h.72.72.28.28 8
I.4.3 Vector multiplied into the Markov matrice 9
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
more examples: Obs/analysis Model Forecast (.70.30).38 (.48.52) (.90.0).38 (.58.42).38 ( )..29.7 ( 30.70)..which is almost identical with the input vector 2
I.4.4 Eigen vectors 22
Left eigenvector and -value (.28.72).38 (.28.72) (.28.72) (.28.28).38 (.3.3) eigenvalues 0.47(.28.28) 23
24.38.38 2.56 0.47.2 2.56 0.47 Right eigenvector and -value eigenvalues
The initial transition matrix can be decomposed into a weighted sum of two new matrices.38 n.28.28.72.72 + n 0.47 Eigen value.72.28.72.28 Persistence probabilities Climate probabilities Time folding factor Variability Meteorological interpretation 25
Matrix climate matrix + residual.38 n Persistence.28.28.72.72 + n 0.47.72.28 Variability.72.28 % 00 persistence 80 60 40 28% 20 climate 00 0 2 3 4 5 time 26
I.4.5 Theoretical and observed period lengths 27
.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
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
6 hour transition matrix for Arlanda airport weather Ο Θ 0 2/8 3 5/8 6 8/8 Ο.6 Θ.24.05.03.28.40.23.2.9.06.20.47.3.6.02.07.3.59..03.09.3.3.39 30
0.6 N Ο. 44.6.09.37.4.09.03.02.08.03.23.08.05.9.07.55.20.2. 46.7.06.02.0.05.02 High persistence for clear sky and rain Ο.0.02 N 0.29.06.02.3 Θ Θ.03.04.2.05.27.04.05. 4.06.32.0.0. 04. 02.0.09.2.36.82 Low persistence for showers Ο.7.07 N 0.4.27.26.07 Θ.06.03.0.0.03.25..40.39.0.24.0.4. 37.0.09.04.4.04 Good persistence for overcast and rain Ο Θ.6.33.8.0.0.29.63.34.02.04 N 0.9.40. 2.0.0.0.02.0.00.00.00.0.00.00.00 Very low persistence for variable sky 3
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