ST2352. Stochastic Processes constructed via Conditional Simulation. 09/02/2014 ST2352 Week 4 1
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1 ST35 Sochasic Processes consruced via Condiional Simulaion 09/0/014 ST35 Week 4 1
2 Sochasic Processes consruced via Condiional Simulaion Markov Processes Simulaing Random Tex Google Sugges n grams Random Climaes Fuures ST35 Week 5
3 Sochasic Process Formally, sequence ; Eg ime,order in sequence... running avg ( 1) Z depends on Probdis for depends on pas values of : ha is 1 Disc pmf f y funcion of values running sum Z Char Cs pdf f y Special cases 1 Char f y; y, y, 1 1 of, 1 s f y f y; y (1 order Markov; 0 gram) nd f y; y, y ( order Markov; 1 gram) ST35 Week 5 3
4 Random Walk: Running Sum Excel file n n N approx E E Z Z Z E Z E Z n Var Var Z Z Z Var Z Var Z n SD =Z1-0.5 Sum Z and 0 0, Random Walk Z Z Z ; 0 based on sum U(-0.5,0.5) If Z 1 y 1 1 (some specific value) hen U y, y ST35 Week
5 Climae over 100,000 years Greenland Ice Core Greenland 10,000 year Ice inervals Core Daa Ice Core daa ime series Modern period Oxygen isoope proxy for Greenland emp Temporal srucure for climae (0 yr. resoluion) Frequen small changes, occasional large changes ST35 Week 5 7
6 n-grams Wiki n-gram is a coniguous sequence of n iems from a given sequence of ex or speech n-grams ypically are colleced from a ex or speech corpus Random creaion of senences ha reflec he srucure of he corpus Web Applicaion Mark V Sheney ST35 Week 5 8
7 n grams in Google Sugges I ype iniial par of search: eg 'Mo': Google suggess phrases. Formally: esimaes p =Pr(phrase Firs wochars, JH) uses p i Mo i Mo for each of many phrases ( 3-gram) o rank he phrases. i Esimae: based on corresp rel freqs of searches ha begin wih all wo char combs (eg 'Mo') for all English language users, and users 'like' JH ST35 Week 5 9
8 How does i work? Roughly speaking, his scrip firs finds all 3-word groups wihin he given ex. Then i builds senences by aking he las wo words currenly in he curren senence being buil and finding a valid word o add on o he end. A word is valid if i, along wih he curren las words, complees a 3-word group found in he given ex. Therefore, senences will appear o have local srucure. However, as a whole, hese senences will ypically come ou as complee nonsense. This process is known as a Markov Chain (Mark V. Shaney, ge i?). hp:// ST35 Week 5 11
9 Non-Uniform Random Generaion. Non-Uniform, bu Independen Excel Workshee Indep non-uniform generaes he characers A,B,C and D according o probabiliies ha are no necessarily equal. 1 gram The generaion algorihm does no use informaion abou pas changes. ST35 Week 5 1
10 Random and Independen Draw rand leer from Poss A B C D Wih probabiliy dis Form CumProb, A B C D offse for EXCEL s LOOKUP Inervals EXCEL Draw U = RAND() LOOKUP ( U, CumProb, Poss) Eg U = A U = 0.6 U = ST35 Week 5 13
11 Random bu Dependen Bigram Arbirary choice of firs char Firs specifies disribuion for second Random generaion of second char Specifies disribuion for hird ec Cond Probs for each word A B C D E following A B C D E RAND() RAND() ST35 Week 5 all B C BC D BCD E BCDE 14
12 Non-Uniform, non-independen Random Generaion. Non-Uniform and non-independen. Excel Workshee Order 1 Bigram generaes random words, where he probabiliies for he nex word depend on he previous word. Excel Workshee Order Trigram generaes random words, where he probabiliies for he nex word depend on he previous wo words. ST35 Week 5 15
13 Random bu Dependen Trigram Arbirary choice of firs wo chars; hen Firs specify disribuion for 3 rd Random generaion of 3 rd char Mos recen specify disribuion for 4 h Ec ST35 Week 5 16
14 Properies of Sochasic Sysems Given iniial values ini and generaing mechanism can generae random srings,..,.... ini s...and hence sudy properies of random given of subses of subse 1 ini given Subse Subse ini Subse given subse 1 Subse ST35 Week 5 17
15 Probabiliy Theory: Firs Order, Bigram Given 1 = B Prob dis for One Sep A B C D E B Challenge Prob dis for 3 Two Sep A B C D E B Cond Probs for each word A B C D E following A B C D E Simulaion approach: Repea many imes Compue rel freq of ( 1 ) pairs wih 1 =B ST35 Week 5 18
16 Probabiliy Theory: Firs Order, Bigram Given 1 = B Prob dis for One Sep A B C D E B Challenge Prob dis for 3 Two Sep A B C D E B Even Ideniy A B 3 1 Hence A, some value B 3 1 A, A OR OR E B 3 1 A B 3 1 A some value B 3 1 Pr A, y B 3 1 Pr Pr, Poss y ST35 Week 5 19
17 Probabiliy Theory: Firs Order, Bigram Given 1 = B Prob dis for One Sep A B C D E B Challenge Prob dis for 3 Two Sep A B C D E B Pr A B 3 1 Challenge Pr 3 1 A A B 3 1 Pr A, E B 3 1 A A 3 A B 1 Pr A EPr E B Pr, Pr Pr 0.01 C B? 3 1 ST35 Week 5 0
18 Probabiliy Theory Decomposiion More general 1s Order Markov saemem pmf f y y f y y f y y dy k k k k k Poss y 1, 1 y y y y y dy k k Poss y Poss y k k k y y y y dy k k 1 k ST35 Week 5 1
19 Probabiliy Theory: Firs Order, Bigram Given 1 = B Prob dis for n +1 Cond Probs for each word A B C D E following A B C D E sep probabiliies P P 1 -sep probabiliies P P 1 n-sep probabiliies P P n. In imporan cases P n n row-vecor Equilibrium dis Eigen heory ST35 Week 5
20 Sochasic Sysem 1 s Order Auo-Regression Excel Workshee AR1 1 1 i y 1 y N 1 y,1 y ~ N y,1 1 ~ N(0,1) Properies of, given s 1 ST35 Week 5 3
21 1 s Order Auo-Regression: -seps ;, ~ N(0,1) indep y N y, a 1 ST35 Week 5 4
22 1 s Order Auo-Regression: -seps Var y for 1 for 1 ST35 Week 5 5
23 Probabiliy Theory Decomposiion More general 1s Order Markov saemem pdf f y y f y y f y y dy k k k k k Poss y 1, 1 y y y y y dy k k Poss y Poss y k k k y y y y dy k k 1 k ST35 Week 5 6
24 Revision: Normal Dis Convoluion Sum of wo indep Normals X ;, ~ N 0, indep 1 1 y y 1 1 f y e Recall e dy i y 1 y1 1 y j pdf, e e , exp 1 j pdf X y x y y x y x exp 1 e exp X x 1 pdf X f x y x y dy y X ~ N 0, y 1 e 1 4 x ST35 Week 5 7 y 1 dy 1
25 Bivariae Normal ST35 Week 5 8
26 Join and Marginal Disribuion AR1 ~ N(0,1) 1 1 i Suppose know Prob Dis and eg ~ N,, E Corr, ~ N, 1 1 ~ BVN, 1 Cov E E E, ST35 Week 5 9
27 Random Generaion from Bivariae Normal Dis General algorihm Acc-Rej Faser: Exploi properies of Normal To generae BVN, Generae Z N 0,1 ; Z N 0,1 Form 1 Z 1 a b 1 b a Z for suiable ab, 0, BVN Choose a ba ab a 1 ; 1 b 1 1 ST35 Week 5 30 a
28 Random Generaion from Bivariae Normal Dis To generae BVN, 1 X Generae BVN, X 0 1 Form 1 X X ST35 Week 5 31
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