CS 10: Problem solving via Object Oriented Programming Winter 2017
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1 CS 10: Problem solving via Object Oriented Programming Winter 2017 Tim Pierson 260 (255) Sudikoff Day 20 PaGern RecogniIon
2 Agenda 1. PaGern matching vs. recogniion 2. From Finite Automata to Hidden Markov Models 3. Decoding: Viterbi algorithm 4. Training 2
3 Last class we discussed how to use a Finite Automata to match a pagern addresses follow a pagern: mailbox@domain.tld example: tjp@cs.dartmouth.edu com Start a- a- z. edu a- z a- z Finite Automata can represent address pagern Sample addresses can be easily verified if in correct form The address pagern must be followed exactly Any deviaion results in rejecion org... 3
4 SomeImes our input is noisy and does not exactly match a pagern Pa%ern matching vs. recogni2on Matching Recogni2on Is this a duck? 4 Image: duckrace.com
5 SomeImes our input is noisy and does not exactly match a pagern Pa%ern matching vs. recogni2on Looks like a duck Matching Recogni2on Is this a duck? 5 Image: duckrace.com
6 SomeImes our input is noisy and does not exactly match a pagern Pa%ern matching vs. recogni2on Looks like a duck Quacks like a duck Matching Recogni2on Is this a duck? 6 Image: duckrace.com
7 SomeImes our input is noisy and does not exactly match a pagern Pa%ern matching vs. recogni2on Is this a duck? Looks like a duck Quacks like a duck Does not wear cool eyewear Matching Recogni2on Image: duckrace.com 7
8 SomeImes our input is noisy and does not exactly match a pagern Pa%ern matching vs. recogni2on Is this a duck? Looks like a duck Quacks like a duck Does not wear cool eyewear Is it a duck? Matching Recogni2on Image: duckrace.com 8
9 Agenda 1. PaGern matching vs. recogniion 2. From Finite Automata to Hidden Markov Models 3. Decoding: Viterbi algorithm 4. Training 9
10 We can model systems using Finite Automata Weather model: possible states Sunny Cloudy The State of the weather can be: Sunny Cloudy Rainy Rainy 10 Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
11 We can model systems using Finite Automata Weather model: transi2ons 0.8 Sunny Cloudy 0.5 We can observe weather pagerns and determine probability of transi2on between states Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
12 We can model systems using Finite Automata Weather model: Sunny day example 0.8 Sunny Cloudy 0.5 Probability a sunny day is followed by: Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
13 We can model systems using Finite Automata Weather model: Sunny day example 0.8 Sunny Cloudy 0.5 Probability a sunny day is followed by: Another sunny day 80% Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
14 We can model systems using Finite Automata Weather model: Sunny day example 0.8 Sunny Cloudy 0.5 Probability a sunny day is followed by: Another sunny day 80% A cloudy day 15% Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
15 We can model systems using Finite Automata Weather model: Sunny day example 0.8 Sunny Cloudy 0.5 Probability a sunny day is followed by: Another sunny day 80% A cloudy day 15% A rainy day 5% Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
16 Model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
17 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
18 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
19 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
20 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
21 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
22 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Cloudy 0.5 Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy (0.15*0.3) Rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
23 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Rainy Cloudy Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy (0.15*0.3) Could be rainy, then rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
24 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Rainy Cloudy Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy (0.15*0.3) Could be rainy, then rainy Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
25 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance 0.8 Sunny Rainy Cloudy Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy (0.15*0.3) Could be rainy, then rainy (0.05*0.6) Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
26 FA model allows us to answer quesions about the probability of events occurring Weather model: predict two days in advance Sunny 0.2 Cloudy Rainy 0.6 Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf Given today is sunny, what is the probability it will be rainy two days from now? Could be sunny, then rainy (0.8*0.05) Could be cloudy, then rainy (0.15*0.3) Could be rainy, then rainy (0.05*0.6) Total = (0.8*0.05) + (0.15*0.3) + (0.05*0.6) =
27 Markov property suggests it doesn t really mager how we got into the current State Given current State, can predict likelihood of future states Given that we can observe the state we are in, it doesn t really mager how 0.2 we got there: Sunny Cloudy Probability of weather at Ime n, given the weather at Ime n- 1, and at n- 2, Is approximately equal to the probability of Rainy weather at Ime n given only the weather at n P(w n w n- 1,w n- 2,w n- 3 ) P(w n w n- 1 ) Adapted from: hgps://pdfs.semanicscholar.org/b328/2eb b80760fea5845e158168daee62.pdf
28 Model works well if we can directly observe the state, what if we cannot? Some2mes we cannot directly observe the state You re being held prisoner and want to know the weather outside. You can t see outside, but you can observe if the guard brings an umbrella. You observe photos of your friends. You don t know what city they were in, but do know something about the ciies. Can you guess what ciies they visited? You want to ask for a raise, but only if the boss is in a good mood. How can you tell if the boss is in a good mood if you can t tell by looking? 28
29 Want to ask the boss for raise when the boss s state is a Good mood Gather stats about likelihood of states Hidden States Good Start Bad Can t know boss s mood for sure simply by looking (state is hidden) Want to know current state (Good or Bad) Could ask everyday and record staisics about it Assume boss answers truthfully Ask 100 Imes 60 Imes good 40 Imes bad Boss slightly more likely to be in good mood 29 (60% chance)
30 In addiion to states, find likelihood of transi.oning from one state to another Gather stats about state transi2ons Hidden States Start Good Bad Watch boss on day aper asking about mood, ask again next day Calculate probability of staying in same mood or transiioning to another mood (hidden state) Similar to how weather transiioned states 30
31 Once have states and transiions, might find something we can directly observe Might be able to observe music playing Hidden States Start Good Bad Might observe what music the boss plays Blues, Jazz or Rock Record stats about music choice when in either mood (hidden states) Blues Jazz Rock 31
32 Once have states and transiions, might find something we can directly observe Might be able to observe music playing Hidden States Start Good Bad Might observe what music the boss plays Blues, Jazz or Rock Record stats about music choice when in either mood (hidden states) Blues Jazz Rock 32
33 This is a Hidden Markov Model (HMM) Hidden Markov Model Hidden States Start Good Bad States (boss s mood) are hidden, can t be directly observed But, we can observe something (music) that can help us calculate the most likely hidden state Blues Jazz Rock 33
34 So is today a good day to ask for a raise? So far we have no music observa2on Hidden States Start Good Bad Blues Jazz Rock Given no other informaion, it s a pregy good bet the boss in Good mood Good mood = 0.6 Bad mood = 0.4 Yes, on any given day boss is slightly more likely to be in a good mood 34
35 By observing music, we might be able to get a beger sense of the boss s mood! Observe Rock music Hidden States Start Good Bad Blues Jazz Rock Say today we observe the boss is playing Rock music Should we ask for a raise? Good mood = 0.6*0.5 = 0.3 Bad mood = 0.4*0.1 = 0.04 Most likely a good day 35 to ask!
36 Bayes theorem can give us the actual probabiliies of each hidden state Observe Rock music Hidden States Start Good Bad Blues Jazz Rock Given the boss is playing Rock music, use Bayes Theorem P(A B) = P(B A)*P(A) P(B) P(G R) = P(R G)*P(G) P(R) P(R G) = 0.5, P(G) = 0.6 P(R) = 0.34 G=Good, B=Bad, R=Rock P(G R) = 0.5*0.6/0.34 =
37 Agenda 1. PaGern matching vs. recogniion 2. From Finite Automata to Hidden Markov Models 3. Decoding: Viterbi algorithm 4. Training 37
38 We can esimate the most likely hidden state based on observaions Start Good Bad Given no observaions, can make a guess at true state Guess state with highest score 38
39 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock 0.6*0.5 Start Good 0.3 If we make an observaion, we might be able to increase our accuracy MulIply previous score by likelihood of observaion 0.4*0.1 Bad 0.04 Most likely in a Good mood (~8X more likely) Should ask for a raise 39
40 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock 0.6*0.5 Start 0.4*0.1 Good 0.3 Bad 0.04 If we make an observaion, we might be able to increase our accuracy MulIply previous score by likelihood of observaion Most likely in a Good mood (~8X more likely) Should ask for a raise 40
41 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock Day 2: Observe Jazz 0.6*0.5 Start 0.4*0.1 Good 0.3 Bad What if you chicken out and don t ask today? Tomorrow you observe Jazz, should you ask now?
42 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock Day 2: Observe Jazz 0.6*0.5 Start 0.4*0.1 Good 0.3 Bad *0.7* *0.3*0.3 Good Update rule: Current* Transi2on* Observa2on Most likely state has highest value Bad Now only about 3X more likely to be in Good mood 42
43 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock 0.6*0.5 Start 0.4*0.1 Good *0.6*0.3 Bad 0.04 Day 2: Observe Jazz 0.3*0.7* *0.3* *0.6*0.3 Good Update rule: Current* Transi2on* Observa2on Most likely state has highest value Bad Now only about 3X more likely to be in Good mood 43
44 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock 0.6*0.5 Start 0.4*0.1 Good *0.4*0.4 Bad Day 2: Observe Jazz 0.3*0.7* *0.3* *0.6*0.3 Good *0.4*0.1 Bad Day 3: Observe Blues 0.084*0.7* *0.3* *0.6*0.6 Good Bad
45 We can esimate the most likely hidden state based on observaions Day 1: Observe Rock 0.6*0.5 Start 0.4*0.1 Good *0.4*0.4 Bad Day 2: Observe Jazz 0.3*0.7* *0.3* *0.6*0.3 Good *0.4*0.1 Bad Day 3: Observe Blues 0.084*0.7* *0.3* *0.6*0.6 Good Bad Given observaions of {Rock, Jazz, Blues} The boss s mood mostly likely was {Good, Good, Bad}
46 Viterbi algorithm back tracks to find most likely state sequence given observaions Day 1: Observe Rock Start 0.6*.5 0.4*0.1 Good *0.4*0.4 Bad Day 2: Observe Jazz 0.3*0.7* *0.3* *0.6*0.3 Good *0.4*0.1 Bad Day 3: Observe Blues 0.084*0.7* *0.3* *0.6*0.6 Good Bad Given observaions of {Rock, Jazz, Blues} The boss s mood mostly likely was {Good, Good, Bad}
47 HMMs and Viterbi algorithm used in a number of fields such a monitoring health Start Day 1: Sensor input Good Healthy Day 2: Sensor input Healthy Day 3: Sensor input Healthy Prof. Campbell s BeWell app uses smart phone sensor data and HMM to esimate health behavior of users over Ime Unhealthy Unhealthy Unhealthy Given sequence of sensor data, what was the subject s most likely health state on each day Lane N, Mohammod M, Lin M, Yang X, Lu H, Ali S, et al. BeWell: A smartphone applicaion to monitor, model and promote wellbeing. Interna.onal Conference on Pervasive Compu.ng Technologies for Healthcare;
48 HMMs allow us to determine the most likely sequence of state transiions Key points We can t directly the hidden state and know the true state with certainty If there is something we can observe, we might be able to infer the true state with greater accuracy than guessing Given a sequence of observaions we can determine the most likely state transiions over Ime 48
49 Agenda 1. PaGern matching vs. recogniion 2. From Finite Automata to Hidden Markov Models 3. Decoding: Viterbi algorithm 4. Training 49
50 To build an HMM we start with previous observaions called training data Annotated training data gives transi2on probabili2es Situa2on: Have a diary with of number of ice cream cones eaten each day (observaions) Want to reconstruct the weather (hidden state) when the cones were eaten 50
51 To build an HMM we start with previous observaions called training data Annotated training data gives transi2on probabili2es Diary entries: 1. Hot day today! I chowed down three whole cones. 2. Hot again. But I only ate two cones; need to run to the store and get more ice cream. 3. Cold today. SIll, the ice cream was calling me, and I ate one cone. 4. Cold again. Kind of depressed, so ate a couple cones despite the weather. 5. SIll cold. Only in the mood for one cone. 6. Nice hot day. Yay! Was able to eat a cone each for breakfast, lunch, and dinner. 7. Hot but was out all day and only had enough cash on me for one ice cream. 8. Brrrr, the weather turned cold really quickly. Only one cone today. 9. Even colder. SIll ate one cone. 10. Defying the coninued coldness by eaing three cones. 51
52 To build an HMM we start with previous observaions called training data Annotated training data gives transi2on probabili2es Diary entries: 1. Hot day today! I chowed down three whole cones. 2. Hot again. But I only ate two cones; need to run to the store and get more ice cream. 3. Cold today. SIll, the ice cream was calling me, and I ate one cone. 4. Cold again. Kind of depressed, so ate a couple cones despite the weather. 5. SIll cold. Only in the mood for one cone. 6. Nice hot day. Yay! Was able to eat a cone each for breakfast, lunch, and dinner. 7. Hot but was out all day and only had enough cash on me for one ice cream. 8. Brrrr, the weather turned cold really quickly. Only one cone today. 9. Even colder. SIll ate one cone. 10. Defying the coninued coldness by eaing three cones. Hidden states: Hot (4 days) or Cold (6 days) Observa2ons: 1, 2, or 3 ice cream cones eaten 52
53 To build an HMM we start with previous observaions called training data Count of observa2ons: 4 hot days, 6 cold days Start 4 6 Hidden States Hot Cold 53
54 To build an HMM we start with previous observaions called training data Count of observa2ons: transi2ons between hidden states Hidden States Start Hot 2 1 Cold 54
55 To build an HMM we start with previous observaions called training data Count of observa2ons: cones eaten when hot Hidden States Start Hot Cold 1 cone 2 cones 3 cones 55
56 To build an HMM we start with previous observaions called training data Count of observa2ons: cones eaten when cold Hidden States Start Hot 2 1 Cold cone 2 cones cones 56
57 Convert observaions into probabiliies by dividing count by total count Probabili2es based on observa2ons Hidden States Start Hot Cold cone 2 cones 3 cones 57
58 Convert observaions into probabiliies by dividing by total count, then use logs Probabili2es based on observa2ons Hidden States Start Hot Cold Repeatedly muliplying probabiliies quickly leads to very small numbers This can cause numerical precision issues 1 cone 2 cones 3 cones log(mn) = log(m) + log(n) To calculate Viterbi, add logs of each factor instead of muliplying 58
59 Convert observaions into probabiliies by dividing by total count, then use logs Log probabili2es based on observa2ons Hidden States Hot Start Cold Repeatedly muliplying probabiliies quickly leads to very small numbers This can cause numerical precision issues 1 cone 2 cones 3 cones log(mn) = log(m) + log(n) To calculate Viterbi, add logs of each factor instead of muliplying 59
60 Example # ObservaIon nextstate currrentstate currscore+transscore +observaion Start n/a Start n/a 0 0 nextscore Most likely {Hot,Hot,Hot} 60
61 Example # ObservaIon nextstate currrentstate currscore+transscore +observaion Start n/a Start n/a 0 0 nextscore 0 Two cones Cold Start Hot Start Most likely {Cold} 61
62 Example # ObservaIon nextstate currrentstate currscore+transscore +observaion Start n/a Start n/a 0 0 nextscore 0 Two cones Cold Start Hot Start Three cones Cold Cold Cold Hot Hot Cold Hot Hot Most likely {Hot,Hot} 62
63 Example # ObservaIon nextstate currrentstate currscore+transscore +observaion Start n/a Start n/a 0 0 nextscore 0 Two cones Cold Start Hot Start Three cones Cold Cold Cold Hot Hot Cold Hot Hot Two cones Cold Cold Cold Hot Hot Cold Hot Hot Most likely {Hot,Hot,Hot} 63
64 Example # ObservaIon nextstate currrentstate currscore+transscore +observaion Start n/a Start n/a 0 0 nextscore 0 Two cones Cold Start Hot Start Three cones Cold Cold Cold Hot Hot Cold Hot Hot Two cones Cold Cold Cold Hot Hot Cold Hot Hot Most likely {Hot,Hot,Hot} 64
65 65
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