Hidden Markov Model Parameters
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1 .PPT 5/04/00 Lecture 6 HMM Traiig Traiig Hidde Markov Model Iitial model etimate Viterbi traiig Baum-Welch traiig 8.7.PPT 5/04/ Hidde Markov Model Parameter c c c 3 a a a 3 t t t 3 c a t A Hidde Markov Model for a word ecifie the followig arameter for each tate : The mea ad variace for each of the F elemet of the arameter vector: µ ad σ. Thee allow u to calculate d (x: the outut robability deity of iut frame x i tate. The traitio robabilitie a,j to every oible ucceor tate. a,j i ofte zero for all j excet j ad j+ it i the called a left-to-right, o ki model. For a Hidde Markov Model with tate we therefore have aroud (F+ arameter. A tyical word might have 5 ad F39 givig 00 arameter i all. Determiig the correct value for all thee arameter i called traiig the model. We trai the model by gettig lot of eole to ay the word lot of time. 00
2 .PPT 5/04/ Traiig whole-word model Record >0 examle of each word i the vocabulary. Decide how may tate to have for each word model. Chooe a iitial aligmet for each traiig word. Reeat the followig te util aligmet coverge: Etimate the model arameter from the aliged traiig examle Alig all the traiig word with the ew model uig Viterbi aligmet. 5 aliged examle of cat : Mea ad Variace: There are 3 frame aliged to tate. Take the mea ad variace of the 3 feature vector to get the outut ditributio for tate. Traitio Probabilitie: ice 5 of the 3 frame are followed by a frame, the traitio robability from tate to tate 3 i 5/3. Legal traitio that do t occur i the traiig data get a fixed low robability of, ay, 0.00.PPT 5/04/ Iitial Aligmet cat Calculate the Euclidea ditace betwee the feature vector from coecutive frame Plot the cumulative um of thee ditace ad divide the rage u ito N equal egmet Cumulative um grah will be teeet where the ectrum i chagig mot raidly tate will be hort whe ectrum i chagig raidly ad log where it i cotat. Iitial aligmet doe t matter all that much. ome recogier jut divide the eech igal ito equal legth chuk. 00
3 .PPT 5/04/ Probabilitic Aligmet Viterbi aligmet fid the bet aligmet of a eech igal with a model. There may be everal other aligmet that are almot a good a the bet oe. Our traiig rocedure i called Viterbi reetimatio of the model. Viterbi reetimatio oly coider the bet aligmet of each traiig examle ad igore all other oible aligmet Baum-Welch reetimatio coider all oible aligmet ad weight them with their roductio robabilitie. We defie two quatitie: P(,t i the total robability deity that the model geerate x, x,, x t ummed over all aligmet havig frame t i tate. Q(,t i the total robability deity that the model geerate x t+, x t+,, x T ummed over all aligmet havig frame t i tate. It follow that P(,t Q(,t i the robability deity that the model geerate the oberved eech igal ad that frame t i i tate..ppt 5/04/ tate Aigmet Probabilitie For ay aligmet, frame x t mut be i oe of the tate,,,. Hece P (, t Qt (, P where P i the total robability of geeratig the word ummed over all oible aligmet. With Viterbi reetimatio, we ued the bet aligmet to aig x t to oe articular tate. With Baum-Welch reetimatio we aig art of x t to may differet tate. The fractio of x t that it i aiged to tate i give by Pt (, Qt (, At (, where At (, P The total umber of frame aiged to tate i give by T A (, t 00 t
4 .PPT 5/04/ Baum-Welch Reetimatio We etimate the mea feature vector i each tate by takig a weighted average of all frame from all N traiig examle of the word. m N T t N T t A (, t t A (, t x imilarly, we etimate the covariace matrix by takig a weighted average of the quared deviatio from the mea. C N T t N T A (, t ( x m ( x m t t t A (, t T If the elemet of the feature vector are ideedet, the C will be a diagoal matrix..ppt 5/04/ Baum-Welch Traitio Probabilitie We calculate the traitio robability a k by fidig out what fractio of frame that were aiged to tate had the followig frame aiged to tate k. The total robability deity of all aligmet with x t i tate ad x t+ i tate k i Pt (, a d ( x Qkt (, + k k t + the fractio of aligmet with thi roerty i therefore P (, t a d ( x Q (, k t+ / P k k, t+ We calculate a k by addig u the total umber of time that coecutive frame are aiged to tate ad k reectively ad dividig by the total umber of frame that are aiged to tate a N T t P (, t a d ( x Q (, k t+ / P k k, t+ k N T t A (, t 00
5 .PPT 5/04/ Forward Probability Calculatio x 0 Frame: tart 3 P(,9 P(,9 P(3,9 P(3,0 P(,t i the total robability deity that the model geerate x, x,, x t ummed over all aligmet havig frame t i tate. Ay aligmet goig through (3,0 mut go through either (,9, (,9 or (3,9. Hece: 3 P(3,0 P(,9 a 3 d3( x 0 I geeral, we ca calculate P(*,t from P(*,t : Pkt (, Pt (, a d ( x k k t.ppt 5/04/ Forward Probability Recurio The calculatio of P(,t i very imilar to the calculatio of B(,t excet that we add together the robabilitie at each tage itead of jut takig the bigget. P(, d (x ; P(, 0 for for t:t for k: ed ed Pkt (, Pt (, a d ( x k k t ice all aligmet have to have the lat frame i tate, the total robability deity over all aligmet i give by: P P(,T a a i the exit robability from the fial tate 00
6 .PPT 5/04/ Backward Probability Recurio Q(k,t i the total robability deity that the model geerate x t+, x t+,, x T ummed over all aligmet havig frame t i tate k. All aligmet havig frame t i tate k mut have frame t+ i ome other tate. We ca therefore calculate Q(k,t i term of Q(*,t+. We ue a recurio goig backward i time: Q(,T a ; Q(,T 0 for for tt : : for k: Qkt (, ak d ( x t + Qt (, + ed ed ice all aligmet mut have frame i tate, we have P d ( x Q (,.PPT Log Probabilitie Probability roduct get very mall o we ue log robabilitie to avoid uderflow roblem. The Viterbi calculatio oly require robability roduct but the forward/backward calculatio alo require robability um. Log of roduct are eay but log of um are difficult: log( log( + log( log( + log( + log + We ca alway chooe o 0 /. We ca the aroximate log(+ / over thi rage a a fuctio of log( / : log( + e x log( x x x log( log( 00
7 .PPT 5/04/ ummary of HMM Traiig Procedure The followig te mut be erformed for each differet word that you eed to recogie: Make a crude model: we have ee oe of everal oible method. Do Viterbi traiig util the model coverge: Alig each of your traiig examle with your curret model. Calculate ew value for m ad C by averagig over all frame that alig with tate. Calculate ew value for a k by takig the fractio of the above frame for which the followig frame aliged with tate k. Do a few (u to 0 iteratio of Baum-Welch traiig: Calculate P(,t ad Q(,t for all t ad. Thee are called the forward ad backward robabilitie reectively. Calculate ew value for m ad C by formig a weighted average over all the traiig frame. Calculate ew value for a k by takig the ratio of the df of ath through both ad k to that of all ath through. 00
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