Observables: (finite or infinte) is the observable part of the model.
Transition matrix: \(P\) [between states]
Initial ditribution: \(\lambda_{i}\) is the probablility that you start in i
Emission matrix: \(B\) rapresent the probability of observing each possible observable given each state of the model. [between states and symbols]
Once I have specified all these ingredients I know the HMM.
library(HMM) # THIS PACKAGE IS USED ONLY IF THE OBSERVABLES ARE CATEGORICAL (DESCRETE),
# CALLED SYMBOLS.Firstly, I have to define a HMM (Hidden Markov Model). So I have to define all the possible states (Unobserved variables: X) and the symbols (Observable variables: Y)
initHMM(States, Symbols, startProbs=NULL, transProbs=NULL, emissionProbs=NULL)hmm1 = initHMM(c("X","Y"), # States: Vector with the names of the states
c("a","b","c") # Symbols: Vector with the names of the symbols.
)
hmm1## $States
## [1] "X" "Y"
##
## $Symbols
## [1] "a" "b" "c"
##
## $startProbs
## X Y
## 0.5 0.5
##
## $transProbs
## to
## from X Y
## X 0.75 0.25
## Y 0.25 0.75
##
## $emissionProbs
## symbols
## states a b c
## X 0.3333333 0.3333333 0.3333333
## Y 0.3333333 0.3333333 0.3333333# All other non specified parameters aree setted as default
# $transProbs (is the transition matrix)
# $emissionProbs (probability of emitting a certain symbol [a,b or c])hmm2 = initHMM(c("X","Y"), # States: Vector with the names of the states
c("a","b"), # Symbols: Vector with the names of the symbols.
c(.3,.7), # startProbs: Vector with the starting probabilities of the states.
matrix(c(.9,.1,.1,.9),2), # transProbs: Stochastic matrix containing the transition probabilities between the states.
matrix(c(.3,.7,.7,.3),2) # emissionProbs: Stochastic matrix containing the emission probabilities of the states.
)
hmm2## $States
## [1] "X" "Y"
##
## $Symbols
## [1] "a" "b"
##
## $startProbs
## X Y
## 0.3 0.7
##
## $transProbs
## to
## from X Y
## X 0.9 0.1
## Y 0.1 0.9
##
## $emissionProbs
## symbols
## states a b
## X 0.3 0.7
## Y 0.7 0.3hmm3 = initHMM(c("A","B"),
c("L","R"),
transProbs = matrix(c(.9,.1,.1,.9),2),
emissionProbs = matrix(c(.5,.51,.5,.49),2)
)
print(hmm3)## $States
## [1] "A" "B"
##
## $Symbols
## [1] "L" "R"
##
## $startProbs
## A B
## 0.5 0.5
##
## $transProbs
## to
## from A B
## A 0.9 0.1
## B 0.1 0.9
##
## $emissionProbs
## symbols
## states L R
## A 0.50 0.50
## B 0.51 0.49Then I have to run the HMM for a certain number of steps.
simHMM(hmm3, 100)## $states
## [1] "A" "A" "A" "A" "A" "A" "B" "B" "B" "A" "A" "A" "A" "A" "A" "B" "A" "A"
## [19] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "B" "A" "B"
## [37] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [55] "B" "B" "B" "B" "B" "B" "A" "A" "A" "A" "B" "B" "B" "B" "B" "B" "B" "B"
## [73] "B" "B" "B" "B" "A" "A" "A" "A" "A" "A" "A" "A" "B" "B" "A" "A" "A" "A"
## [91] "A" "A" "A" "A" "B" "B" "B" "B" "B" "B"
##
## $observation
## [1] "L" "R" "L" "R" "R" "R" "R" "R" "L" "R" "R" "L" "R" "R" "L" "L" "L" "R"
## [19] "L" "R" "L" "L" "L" "R" "L" "L" "L" "R" "R" "L" "L" "L" "L" "L" "R" "R"
## [37] "L" "R" "L" "R" "R" "L" "L" "L" "L" "R" "L" "L" "L" "L" "L" "L" "R" "R"
## [55] "R" "R" "R" "R" "R" "L" "R" "R" "L" "R" "R" "R" "L" "L" "R" "R" "R" "L"
## [73] "L" "L" "R" "L" "L" "R" "R" "L" "R" "R" "L" "R" "L" "L" "R" "L" "L" "R"
## [91] "L" "R" "L" "L" "R" "L" "L" "L" "L" "R"I define a sequence of 800 symbols (Observations), probably they are already given.
## [1] "R" "L" "R" "R" "L" "R" "R" "R" "R" "R" "R" "R" "R" "R" "L" "L" "R" "L"
## [19] "R" "R" "L" "L" "R" "R" "R" "L" "R" "R" "R" "R" "L" "R" "R" "L" "R" "R"
## [37] "R" "R" "R" "R" "L" "R" "L" "R" "R" "L" "R" "R" "R" "R" "R" "R" "R" "L"
## [55] "R" "R" "R" "R" "R" "R" "R" "R" "R" "L" "L" "R" "R" "R" "R" "R" "L" "R"
## [73] "L" "R" "R" "R" "R" "R" "R" "L" "R" "R" "R" "R" "R" "L" "R" "R" "R" "R"
## [91] "R" "L" "R" "R" "R" "L" "R" "R" "R" "R" "R" "L" "R" "R" "R" "R" "L" "R"
## [109] "R" "L" "L" "L" "R" "L" "L" "R" "L" "R" "L" "R" "R" "R" "R" "R" "R" "R"
## [127] "R" "R" "R" "R" "R" "R" "L" "R" "R" "R" "L" "L" "R" "R" "L" "R" "R" "R"
## [145] "R" "R" "R" "L" "L" "R" "R" "R" "L" "R" "L" "R" "L" "R" "R" "R" "R" "R"
## [163] "R" "R" "L" "R" "R" "R" "R" "R" "L" "L" "R" "R" "R" "R" "R" "R" "R" "L"
## [181] "R" "R" "R" "R" "R" "R" "L" "R" "R" "R" "R" "L" "R" "R" "R" "L" "R" "R"
## [199] "R" "R" "R" "R" "L" "R" "R" "R" "R" "R" "L" "R" "R" "R" "R" "R" "R" "R"
## [217] "R" "R" "R" "R" "R" "L" "R" "L" "R" "L" "R" "R" "R" "R" "R" "R" "R" "L"
## [235] "L" "L" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R"
## [253] "L" "R" "R" "R" "R" "L" "R" "R" "R" "R" "R" "L" "L" "R" "R" "R" "L" "R"
## [271] "R" "R" "R" "R" "L" "R" "L" "R" "R" "R" "R" "L" "R" "L" "R" "R" "R" "R"
## [289] "L" "R" "R" "R" "R" "R" "R" "L" "R" "R" "R" "R" "L" "R" "L" "R" "L" "L"
## [307] "R" "R" "R" "R" "R" "R" "L" "L" "L" "R" "L" "L" "R" "L" "R" "R" "R" "R"
## [325] "L" "R" "R" "L" "R" "L" "R" "R" "R" "R" "R" "R" "R" "R" "R" "R" "L" "R"
## [343] "R" "R" "R" "L" "R" "R" "L" "R" "R" "L" "R" "R" "L" "R" "R" "R" "R" "L"
## [361] "R" "L" "L" "R" "L" "R" "L" "R" "R" "R" "L" "L" "R" "R" "L" "R" "R" "R"
## [379] "L" "R" "L" "R" "L" "R" "L" "R" "R" "R" "R" "R" "R" "R" "L" "R" "R" "R"
## [397] "L" "L" "R" "L" "L" "L" "R" "L" "L" "L" "R" "R" "L" "L" "L" "L" "R" "L"
## [415] "L" "L" "L" "L" "L" "L" "L" "L" "R" "L" "L" "R" "R" "L" "L" "L" "R" "L"
## [433] "L" "L" "L" "R" "R" "L" "L" "L" "L" "L" "L" "R" "L" "R" "L" "L" "L" "L"
## [451] "R" "L" "L" "L" "R" "L" "R" "R" "L" "L" "L" "L" "L" "L" "R" "L" "L" "L"
## [469] "L" "L" "R" "L" "L" "L" "L" "L" "L" "L" "L" "R" "L" "L" "R" "L" "L" "L"
## [487] "L" "L" "L" "R" "R" "L" "L" "L" "L" "L" "R" "L" "R" "L" "R" "L" "R" "R"
## [505] "L" "L" "L" "R" "R" "L" "L" "L" "L" "L" "L" "L" "L" "L" "L" "L" "L" "L"
## [523] "L" "L" "R" "R" "R" "L" "L" "R" "R" "L" "R" "L" "L" "L" "L" "L" "L" "L"
## [541] "R" "L" "L" "L" "L" "L" "L" "L" "L" "L" "L" "R" "L" "L" "L" "L" "L" "R"
## [559] "L" "L" "L" "R" "L" "L" "L" "R" "L" "R" "L" "L" "L" "L" "L" "L" "L" "R"
## [577] "R" "R" "L" "L" "L" "L" "R" "L" "L" "L" "L" "R" "L" "L" "L" "R" "L" "R"
## [595] "R" "L" "L" "L" "L" "L" "L" "R" "L" "L" "R" "R" "L" "L" "L" "R" "L" "L"
## [613] "L" "L" "L" "L" "R" "L" "R" "R" "L" "L" "L" "L" "R" "L" "R" "R" "R" "L"
## [631] "R" "L" "R" "L" "L" "L" "L" "L" "R" "L" "L" "R" "R" "L" "L" "L" "L" "L"
## [649] "L" "L" "L" "L" "L" "L" "L" "L" "L" "L" "R" "L" "L" "L" "L" "L" "L" "L"
## [667] "L" "L" "L" "R" "R" "L" "L" "L" "L" "L" "L" "R" "L" "L" "R" "R" "L" "L"
## [685] "L" "R" "L" "R" "L" "L" "L" "R" "L" "L" "R" "L" "L" "L" "R" "L" "L" "L"
## [703] "L" "L" "L" "R" "L" "R" "R" "R" "L" "L" "L" "L" "L" "L" "L" "L" "R" "L"
## [721] "L" "L" "R" "L" "L" "R" "R" "L" "L" "L" "L" "R" "R" "R" "R" "L" "L" "L"
## [739] "L" "L" "L" "L" "L" "R" "L" "L" "L" "L" "L" "R" "L" "L" "R" "L" "L" "L"
## [757] "L" "R" "L" "L" "L" "L" "R" "L" "L" "L" "R" "L" "L" "L" "L" "L" "R" "R"
## [775] "L" "L" "L" "R" "L" "L" "L" "L" "L" "L" "L" "L" "L" "L" "R" "L" "R" "L"
## [793] "L" "L" "L" "L" "R" "L" "L" "L"We have to give a HMM model and the symbols we observed, and it reconstructs the most probable sequence of states (It reconstruts the hidden model).
It can answer to the question: “at a certain time what is the most probable state ?”
viterbi(hmm3, observation)## [1] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [19] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [37] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [55] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [73] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [91] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [109] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [127] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [145] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [163] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [181] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [199] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [217] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [235] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [253] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [271] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [289] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [307] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [325] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [343] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [361] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [379] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "A"
## [397] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [415] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [433] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [451] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [469] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [487] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [505] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [523] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [541] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [559] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [577] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [595] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [613] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [631] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [649] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [667] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [685] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [703] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [721] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [739] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [757] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [775] "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B" "B"
## [793] "B" "B" "B" "B" "B" "B" "B" "B"Given a model what is the probability that I observe this signal (sequence of observations) ?
I have to compute the posterior probabilities of being in the state X at time k given that you have observed a certain sequence of observations up to time k.
posterior(hmm3, observation)## index
## states 1 2 3 4 5 6 7
## A 0.5118627 0.512556 0.5161046 0.5181861 0.5189053 0.5227965 0.5255561
## B 0.4881373 0.487444 0.4838954 0.4818139 0.4810947 0.4772035 0.4744439
## index
## states 8 9 10 11 12 13 14
## A 0.5273231 0.5281867 0.5281904 0.5273343 0.5255754 0.5228249 0.5189442
## B 0.4726769 0.4718133 0.4718096 0.4726657 0.4744246 0.4771751 0.4810558
## index
## states 15 16 17 18 19 20 21
## A 0.5137376 0.5114462 0.5119544 0.5107881 0.5123885 0.5123361 0.5106283
## B 0.4862624 0.4885538 0.4880456 0.4892119 0.4876115 0.4876639 0.4893717
## index
## states 22 23 24 25 26 27 28
## A 0.5116793 0.515542 0.5179092 0.5189002 0.518565 0.5213845 0.523002
## B 0.4883207 0.484458 0.4820908 0.4810998 0.481435 0.4786155 0.476998
## index
## states 29 30 31 32 33 34 35
## A 0.523499 0.5229007 0.5211768 0.5227376 0.5231647 0.5224796 0.5251442
## B 0.476501 0.4770993 0.4788232 0.4772624 0.4768353 0.4775204 0.4748558
## index
## states 36 37 38 39 40 41 42
## A 0.5267956 0.5275171 0.527345 0.5262707 0.52424 0.5211506 0.5213447
## B 0.4732044 0.4724829 0.472655 0.4737293 0.47576 0.4788494 0.4786553
## index
## states 43 44 45 46 47 48 49
## A 0.5203352 0.5225686 0.5236595 0.5236632 0.5270756 0.5295715 0.5312766
## B 0.4796648 0.4774314 0.4763405 0.4763368 0.4729244 0.4704285 0.4687234
## index
## states 50 51 52 53 54 55 56
## A 0.5322771 0.5326233 0.5323326 0.5313905 0.5297493 0.5318195 0.5332122
## B 0.4677229 0.4673767 0.4676674 0.4686095 0.4702507 0.4681805 0.4667878
## index
## states 57 58 59 60 61 62 63
## A 0.5339976 0.5342153 0.5338761 0.5329632 0.5314304 0.5292004 0.5261608
## B 0.4660024 0.4657847 0.4661239 0.4670368 0.4685696 0.4707996 0.4738392
## index
## states 64 65 66 67 68 69 70
## A 0.5221585 0.5214902 0.5241222 0.5256898 0.526272 0.525898 0.524549
## B 0.4778415 0.4785098 0.4758778 0.4743102 0.473728 0.474102 0.475451
## index
## states 71 72 73 74 75 76 77
## A 0.5221571 0.5230988 0.5229249 0.526123 0.5283569 0.5297391 0.5303394
## B 0.4778429 0.4769012 0.4770751 0.473877 0.4716431 0.4702609 0.4696606
## index
## states 78 79 80 81 82 83 84
## A 0.5301879 0.5292772 0.5275612 0.5294477 0.5305374 0.5308855 0.5305092
## B 0.4698121 0.4707228 0.4724388 0.4705523 0.4694626 0.4691145 0.4694908
## index
## states 85 86 87 88 89 90 91
## A 0.5293898 0.5274708 0.5291497 0.5300171 0.5301165 0.529453 0.5279931
## B 0.4706102 0.4725292 0.4708503 0.4699829 0.4698835 0.470547 0.4720069
## index
## states 92 93 94 95 96 97 98
## A 0.5256634 0.5268417 0.5270925 0.5264284 0.5248159 0.5266693 0.5275864
## B 0.4743366 0.4731583 0.4729075 0.4735716 0.4751841 0.4733307 0.4724136
## index
## states 99 100 101 102 103 104 105
## A 0.5276134 0.5267517 0.5249579 0.5221414 0.5226578 0.5220366 0.5202464
## B 0.4723866 0.4732483 0.4750421 0.4778586 0.4773422 0.4779634 0.4797536
## index
## states 106 107 108 109 110 111 112
## A 0.5171971 0.5127348 0.5111372 0.5078236 0.5026271 0.4997911 0.4991727
## B 0.4828029 0.4872652 0.4888628 0.4921764 0.4973729 0.5002089 0.5008273
## index
## states 113 114 115 116 117 118 119
## A 0.5007408 0.5000731 0.501637 0.5055111 0.5073873 0.5118614 0.5146552
## B 0.4992592 0.4999269 0.498363 0.4944889 0.4926127 0.4881386 0.4853448
## index
## states 120 121 122 123 124 125 126
## A 0.5204102 0.5249124 0.5283888 0.5310146 0.5329222 0.5342078 0.5349362
## B 0.4795898 0.4750876 0.4716112 0.4689854 0.4670778 0.4657922 0.4650638
## index
## states 127 128 129 130 131 132 133
## A 0.5351441 0.5348419 0.5340145 0.5326201 0.5305885 0.5278172 0.5241665
## B 0.4648559 0.4651581 0.4659855 0.4673799 0.4694115 0.4721828 0.4758335
## index
## states 134 135 136 137 138 139 140
## A 0.5239497 0.5226601 0.5202327 0.5165451 0.5159118 0.5183008 0.5193333
## B 0.4760503 0.4773399 0.4797673 0.4834549 0.4840882 0.4816992 0.4806667
## index
## states 141 142 143 144 145 146 147
## A 0.5190613 0.5219687 0.5237035 0.524353 0.52395 0.5224742 0.5198511
## B 0.4809387 0.4780313 0.4762965 0.475647 0.47605 0.4775258 0.4801489
## index
## states 148 149 150 151 152 153 154
## A 0.5159486 0.5150708 0.5171733 0.5178629 0.5171741 0.5150724 0.5159512
## B 0.4840514 0.4849292 0.4828267 0.4821371 0.4828259 0.4849276 0.4840488
## index
## states 155 156 157 158 159 160 161
## A 0.5153559 0.5177552 0.5187706 0.5229514 0.5260081 0.5280948 0.5293168
## B 0.4846441 0.4822448 0.4812294 0.4770486 0.4739919 0.4719052 0.4706832
## index
## states 162 163 164 165 166 167 168
## A 0.5297357 0.5293725 0.528209 0.5261865 0.527698 0.5283251 0.5280992
## B 0.4702643 0.4706275 0.471791 0.4738135 0.472302 0.4716749 0.4719008
## index
## states 169 170 171 172 173 174 175
## A 0.5270089 0.5249994 0.5219692 0.5222634 0.5258966 0.5285541 0.5303696
## B 0.4729911 0.4750006 0.4780308 0.4777366 0.4741034 0.4714459 0.4696304
## index
## states 176 177 178 179 180 181 182
## A 0.5314348 0.5318034 0.5314939 0.5304907 0.5287432 0.5306571 0.531835
## B 0.4685652 0.4681966 0.4685061 0.4695093 0.4712568 0.4693429 0.468165
## index
## states 183 184 185 186 187 188 189
## A 0.5323364 0.5321865 0.5313778 0.5298695 0.5275856 0.5289054 0.5294013
## B 0.4676636 0.4678135 0.4686222 0.4701305 0.4724144 0.4710946 0.4705987
## index
## states 190 191 192 193 194 195 196
## A 0.5290985 0.5279815 0.525994 0.5275311 0.5281753 0.527959 0.5268713
## B 0.4709015 0.4720185 0.474006 0.4724689 0.4718247 0.472041 0.4731287
## index
## states 197 198 199 200 201 202 203
## A 0.5293518 0.5310306 0.5319924 0.5322855 0.5319248 0.5308921 0.5291353
## B 0.4706482 0.4689694 0.4680076 0.4677145 0.4680752 0.4691079 0.4708647
## index
## states 204 205 206 207 208 209 210
## A 0.5310593 0.5322677 0.5328212 0.5327478 0.5320439 0.5306739 0.5330612
## B 0.4689407 0.4677323 0.4671788 0.4672522 0.4679561 0.4693261 0.4669388
## index
## states 211 212 213 214 215 216 217
## A 0.5348333 0.5360796 0.5368628 0.5372225 0.5371768 0.5367234 0.5358394
## B 0.4651667 0.4639204 0.4631372 0.4627775 0.4628232 0.4632766 0.4641606
## index
## states 218 219 220 221 222 223 224
## A 0.5344802 0.5325774 0.530035 0.5267249 0.5224801 0.5215855 0.5194992
## B 0.4655198 0.4674226 0.469965 0.4732751 0.4775199 0.4784145 0.4805008
## index
## states 225 226 227 228 229 230 231
## A 0.5206141 0.5204887 0.523614 0.525649 0.5266964 0.526809 0.5259924
## B 0.4793859 0.4795113 0.476386 0.474351 0.4733036 0.473191 0.4740076
## index
## states 232 233 234 235 236 237 238
## A 0.5242056 0.5213584 0.5173072 0.5163485 0.5184338 0.5236684 0.5278141
## B 0.4757944 0.4786416 0.4826928 0.4836515 0.4815662 0.4763316 0.4721859
## index
## states 239 240 241 242 243 244 245
## A 0.5310799 0.5336306 0.5355946 0.537071 0.5381342 0.5388378 0.5392173
## B 0.4689201 0.4663694 0.4644054 0.462929 0.4618658 0.4611622 0.4607827
## index
## states 246 247 248 249 250 251 252
## A 0.5392918 0.5390651 0.5385258 0.5376466 0.5363832 0.534672 0.5324267
## B 0.4607082 0.4609349 0.4614742 0.4623534 0.4636168 0.465328 0.4675733
## index
## states 253 254 255 256 257 258 259
## A 0.529534 0.5303421 0.5303986 0.5297064 0.5282306 0.5258968 0.5270826
## B 0.470466 0.4696579 0.4696014 0.4702936 0.4717694 0.4741032 0.4729174
## index
## states 260 261 262 263 264 265 266
## A 0.5273531 0.5267217 0.5251568 0.5225793 0.5188594 0.5183091 0.5209007
## B 0.4726469 0.4732783 0.4748432 0.4774207 0.4811406 0.4816909 0.4790993
## index
## states 267 268 269 270 271 272 273
## A 0.5222662 0.5224742 0.5215355 0.5238993 0.5251876 0.5254652 0.5247461
## B 0.4777338 0.4775258 0.4784645 0.4761007 0.4748124 0.4745348 0.4752539
## index
## states 274 275 276 277 278 279 280
## A 0.5229942 0.5201211 0.5204802 0.5195924 0.5219103 0.5230528 0.5230775
## B 0.4770058 0.4798789 0.4795198 0.4804076 0.4780897 0.4769472 0.4769225
## index
## states 281 282 283 284 285 286 287
## A 0.5219857 0.5197222 0.520671 0.5203824 0.5233391 0.5251919 0.5260341
## B 0.4780143 0.4802778 0.479329 0.4796176 0.4766609 0.4748081 0.4739659
## index
## states 288 289 290 291 292 293 294
## A 0.5259083 0.524808 0.5271733 0.5286274 0.5292437 0.5290532 0.5280464
## B 0.4740917 0.475192 0.4728267 0.4713726 0.4707563 0.4709468 0.4719536
## index
## states 295 296 297 298 299 300 301
## A 0.5261724 0.5233368 0.5238936 0.5233747 0.5217542 0.5189503 0.5148215
## B 0.4738276 0.4766632 0.4761064 0.4766253 0.4782458 0.4810497 0.4851785
## index
## states 302 303 304 305 306 307 308
## A 0.5136614 0.5109119 0.5109355 0.5092336 0.5102207 0.5139468 0.5160974
## B 0.4863386 0.4890881 0.4890645 0.4907664 0.4897793 0.4860532 0.4839026
## index
## states 309 310 311 312 313 314 315
## A 0.5167811 0.5160323 0.5138132 0.510012 0.5044371 0.5013134 0.5004835
## B 0.4832189 0.4839677 0.4861868 0.489988 0.4955629 0.4986866 0.4995165
## index
## states 316 317 318 319 320 321 322
## A 0.5019056 0.5011502 0.5026802 0.5065726 0.5085202 0.5131223 0.5161071
## B 0.4980944 0.4988498 0.4973198 0.4934274 0.4914798 0.4868777 0.4838929
## index
## states 323 324 325 326 327 328 329
## A 0.5176252 0.5177531 0.5164971 0.5182927 0.5187314 0.5178355 0.5200578
## B 0.4823748 0.4822469 0.4835029 0.4817073 0.4812686 0.4821645 0.4799422
## index
## states 330 331 332 333 334 335 336
## A 0.5210117 0.5252427 0.5284647 0.53084 0.5324884 0.533493 0.5339045
## B 0.4789883 0.4747573 0.4715353 0.46916 0.4675116 0.466507 0.4660955
## index
## states 337 338 339 340 341 342 343
## A 0.5337437 0.5330023 0.5316431 0.5295974 0.5267623 0.5274899 0.5273227
## B 0.4662563 0.4669977 0.4683569 0.4704026 0.4732377 0.4725101 0.4726773
## index
## states 344 345 346 347 348 349 350
## A 0.5262521 0.5242243 0.5211368 0.5213323 0.5203235 0.5180597 0.5189254
## B 0.4737479 0.4757757 0.4788632 0.4786677 0.4796765 0.4819403 0.4810746
## index
## states 351 352 353 354 355 356 357
## A 0.518466 0.5166586 0.5179107 0.5177869 0.516281 0.5178159 0.51797
## B 0.481534 0.4833414 0.4820893 0.4822131 0.483719 0.4821841 0.48203
## index
## states 358 359 360 361 362 363 364
## A 0.5167512 0.5140981 0.5098769 0.5083777 0.5050243 0.5041503 0.5057115
## B 0.4832488 0.4859019 0.4901231 0.4916223 0.4949757 0.4958497 0.4942885
## index
## states 365 366 367 368 369 370 371
## A 0.5052855 0.5073516 0.5075126 0.510277 0.5112825 0.5105799 0.5081338
## B 0.4947145 0.4926484 0.4924874 0.489723 0.4887175 0.4894201 0.4918662
## index
## states 372 373 374 375 376 377 378
## A 0.5083219 0.5111539 0.512271 0.5117295 0.5140017 0.5147018 0.5138652
## B 0.4916781 0.4888461 0.487729 0.4882705 0.4859983 0.4852982 0.4861348
## index
## states 379 380 381 382 383 384 385
## A 0.5114496 0.5118339 0.5105376 0.5119955 0.511781 0.5143828 0.5154316
## B 0.4885504 0.4881661 0.4894624 0.4880045 0.488219 0.4856172 0.4845684
## index
## states 386 387 388 389 390 391 392
## A 0.5194794 0.5222289 0.5238189 0.5243295 0.5237863 0.5221622 0.519375
## B 0.4805206 0.4777711 0.4761811 0.4756705 0.4762137 0.4778378 0.480625
## index
## states 393 394 395 396 397 398 399
## A 0.5152845 0.5141855 0.5115236 0.5071645 0.5008885 0.496887 0.4949585
## B 0.4847155 0.4858145 0.4884764 0.4928355 0.4991115 0.503113 0.5050415
## index
## states 400 401 402 403 404 405 406
## A 0.4905042 0.4878037 0.4867209 0.4872012 0.4847694 0.483803 0.4842535
## B 0.5094958 0.5121963 0.5132791 0.5127988 0.5152306 0.516197 0.5157465
## index
## states 407 408 409 410 411 412 413
## A 0.4861436 0.4850688 0.480975 0.4781573 0.4764735 0.475839 0.4762216
## B 0.5138564 0.5149312 0.519025 0.5218427 0.5235265 0.524161 0.5237784
## index
## states 414 415 416 417 418 419 420
## A 0.4731448 0.4709505 0.4695281 0.4688059 0.4687475 0.4693499 0.4706436
## B 0.5268552 0.5290495 0.5304719 0.5311941 0.5312525 0.5306501 0.5293564
## index
## states 421 422 423 424 425 426 427
## A 0.4726937 0.4756035 0.4795197 0.4801405 0.4819945 0.4851752 0.4853426
## B 0.5273063 0.5243965 0.5204803 0.5198595 0.5180055 0.5148248 0.5146574
## index
## states 428 429 430 431 432 433 434
## A 0.4825051 0.4810196 0.4808115 0.4818702 0.479751 0.4788455 0.4791081
## B 0.5174949 0.5189804 0.5191885 0.5181298 0.520249 0.5211545 0.5208919
## index
## states 435 436 437 438 439 440 441
## A 0.4805522 0.4832503 0.4828393 0.4792984 0.4769491 0.475673 0.4754058
## B 0.5194478 0.5167497 0.5171607 0.5207016 0.5230509 0.524327 0.5245942
## index
## states 442 443 444 445 446 447 448
## A 0.476134 0.4778943 0.4807755 0.4804242 0.4813202 0.4790109 0.4778783
## B 0.523866 0.5221057 0.5192245 0.5195758 0.5186798 0.5209891 0.5221217
## index
## states 449 450 451 452 453 454 455
## A 0.4778653 0.4789712 0.4812518 0.4803236 0.4806378 0.4822101 0.4851198
## B 0.5221347 0.5210288 0.5187482 0.5196764 0.5193622 0.5177899 0.5148802
## index
## states 456 457 458 459 460 461 462
## A 0.4850135 0.4863847 0.4848031 0.4801891 0.476812 0.4745016 0.4731415
## B 0.5149865 0.5136153 0.5151969 0.5198109 0.523188 0.5254984 0.5268585
## index
## states 463 464 465 466 467 468 469
## A 0.4726632 0.4730426 0.4742987 0.472 0.4705258 0.4698021 0.4697922
## B 0.5273368 0.5269574 0.5257013 0.528 0.5294742 0.5301979 0.5302078
## index
## states 470 471 472 473 474 475 476
## A 0.4704956 0.471948 0.4697284 0.4682193 0.4673446 0.4670603 0.467352
## B 0.5295044 0.528052 0.5302716 0.5317807 0.5326554 0.5329397 0.532648
## index
## states 477 478 479 480 481 482 483
## A 0.4682344 0.4697519 0.4719812 0.4750344 0.4745692 0.4750573 0.4765235
## B 0.5317656 0.5302481 0.5280188 0.5249656 0.5254308 0.5249427 0.5234765
## index
## states 484 485 486 487 488 489 490
## A 0.4745456 0.47352 0.4733952 0.4741649 0.4758677 0.4785897 0.4824678
## B 0.5254544 0.52648 0.5266048 0.5258351 0.5241323 0.5214103 0.5175322
## index
## states 491 492 493 494 495 496 497
## A 0.4831974 0.4808153 0.4797004 0.4797965 0.4811086 0.4837026 0.4877093
## B 0.5168026 0.5191847 0.5202996 0.5202035 0.5188914 0.5162974 0.5122907
## index
## states 498 499 500 501 502 503 504
## A 0.4888288 0.4916173 0.491714 0.493624 0.4929426 0.4941358 0.4927632
## B 0.5111712 0.5083827 0.508286 0.506376 0.5070574 0.5058642 0.5072368
## index
## states 505 506 507 508 509 510 511
## A 0.4887554 0.4864136 0.4856197 0.4863337 0.4840923 0.4787826 0.4746398
## B 0.5112446 0.5135864 0.5143803 0.5136663 0.5159077 0.5212174 0.5253602
## index
## states 512 513 514 515 516 517 518
## A 0.4714552 0.4690681 0.4673585 0.46624 0.4656563 0.465578 0.4660011
## B 0.5285448 0.5309319 0.5326415 0.53376 0.5343437 0.534422 0.5339989
## index
## states 519 520 521 522 523 524 525
## A 0.4669471 0.4684634 0.4706267 0.4735458 0.4773679 0.4822857 0.4885469
## B 0.5330529 0.5315366 0.5293733 0.5264542 0.5226321 0.5177143 0.5114531
## index
## states 526 527 528 529 530 531 532
## A 0.4919611 0.4927004 0.4908022 0.4906716 0.492302 0.491275 0.4875387
## B 0.5080389 0.5072996 0.5091978 0.5093284 0.507698 0.508725 0.5124613
## index
## states 533 534 535 536 537 538 539
## A 0.4854072 0.4802731 0.4763805 0.4735333 0.4715879 0.4704464 0.4700512
## B 0.5145928 0.5197269 0.5236195 0.5264667 0.5284121 0.5295536 0.5299488
## index
## states 540 541 542 543 544 545 546
## A 0.4703823 0.4714566 0.4688345 0.4668781 0.4654888 0.4645965 0.4641563
## B 0.5296177 0.5285434 0.5311655 0.5331219 0.5345112 0.5354035 0.5358437
## index
## states 547 548 549 550 551 552 553
## A 0.4641461 0.4645653 0.465435 0.4667991 0.4687263 0.4713137 0.4701976
## B 0.5358539 0.5354347 0.534565 0.5332009 0.5312737 0.5286863 0.5298024
## index
## states 554 555 556 557 558 559 560
## A 0.4698154 0.4701477 0.4712112 0.4730597 0.4757862 0.4750317 0.4752539
## B 0.5301846 0.5298523 0.5287888 0.5269403 0.5242138 0.5249683 0.5247461
## index
## states 561 562 563 564 565 566 567
## A 0.4764639 0.4787228 0.477647 0.4776791 0.4788209 0.4811298 0.4802239
## B 0.5235361 0.5212772 0.522353 0.5223209 0.5211791 0.5188702 0.5197761
## index
## states 568 569 570 571 572 573 574
## A 0.4805553 0.4776432 0.4758394 0.4750529 0.4752441 0.4764226 0.4786479
## B 0.5194447 0.5223568 0.5241606 0.5249471 0.5247559 0.5235774 0.5213521
## index
## states 575 576 577 578 579 580 581
## A 0.4820321 0.4867457 0.4885236 0.4874555 0.4834874 0.4809212 0.4796276
## B 0.5179679 0.5132543 0.5114764 0.5125445 0.5165126 0.5190788 0.5203724
## index
## states 582 583 584 585 586 587 588
## A 0.4795415 0.4806584 0.4785371 0.4775686 0.4777041 0.4789505 0.4813706
## B 0.5204585 0.5193416 0.5214629 0.5224314 0.5222959 0.5210495 0.5186294
## index
## states 589 590 591 592 593 594 595
## A 0.4805879 0.4810607 0.4828129 0.4859328 0.4860772 0.4877524 0.4865431
## B 0.5194121 0.5189393 0.5171871 0.5140672 0.5139228 0.5122476 0.5134569
## index
## states 596 597 598 599 600 601 602
## A 0.4823883 0.4795804 0.477978 0.4775002 0.478123 0.4798778 0.482853
## B 0.5176117 0.5204196 0.522022 0.5224998 0.521877 0.5201222 0.517147
## index
## states 603 604 605 606 607 608 609
## A 0.4826991 0.4839067 0.4865367 0.4862213 0.4829448 0.4810426 0.4804189
## B 0.5173009 0.5160933 0.5134633 0.5137787 0.5170552 0.5189574 0.5195811
## index
## states 610 611 612 613 614 615 616
## A 0.4810423 0.4784465 0.4769991 0.4766273 0.4773122 0.4790884 0.4820454
## B 0.5189577 0.5215535 0.5230009 0.5233727 0.5226878 0.5209116 0.5179546
## index
## states 617 618 619 620 621 622 623
## A 0.4863322 0.4876629 0.4906043 0.4908033 0.4882699 0.4873774 0.4880809
## B 0.5136678 0.5123371 0.5093957 0.5091967 0.5117301 0.5126226 0.5119191
## index
## states 624 625 626 627 628 629 630
## A 0.4904159 0.4944999 0.4960356 0.4996014 0.5008739 0.4999172 0.4966832
## B 0.5095841 0.5055001 0.5039644 0.5003986 0.4991261 0.5000828 0.5033168
## index
## states 631 632 633 634 635 636 637
## A 0.4955115 0.4918422 0.4899931 0.4853704 0.4822444 0.4804575 0.4799198
## B 0.5044885 0.5081578 0.5100069 0.5146296 0.5177556 0.5195425 0.5200802
## index
## states 638 639 640 641 642 643 644
## A 0.4806041 0.4825449 0.4813415 0.4814315 0.4828194 0.4810768 0.4761158
## B 0.5193959 0.5174551 0.5186585 0.5185685 0.5171806 0.5189232 0.5238842
## index
## states 645 646 647 648 649 650 651
## A 0.4721876 0.4690943 0.4666799 0.4648227 0.4634293 0.4624293 0.4617724
## B 0.5278124 0.5309057 0.5333201 0.5351773 0.5365707 0.5375707 0.5382276
## index
## states 652 653 654 655 656 657 658
## A 0.4614255 0.461371 0.4616062 0.462143 0.4630084 0.4642461 0.4659184
## B 0.5385745 0.538629 0.5383938 0.537857 0.5369916 0.5357539 0.5340816
## index
## states 659 660 661 662 663 664 665
## A 0.4681097 0.4664381 0.4653115 0.4646729 0.4644903 0.4647545 0.4654786
## B 0.5318903 0.5335619 0.5346885 0.5353271 0.5355097 0.5352455 0.5345214
## index
## states 666 667 668 669 670 671 672
## A 0.4666993 0.468478 0.4709044 0.4741008 0.4782283 0.4789959 0.4764424
## B 0.5333007 0.531522 0.5290956 0.5258992 0.5217717 0.5210041 0.5235576
## index
## states 673 674 675 676 677 678 679
## A 0.4749368 0.4744031 0.4748145 0.4761917 0.4786042 0.4821734 0.4825795
## B 0.5250632 0.5255969 0.5251855 0.5238083 0.5213958 0.5178266 0.5174205
## index
## states 680 681 682 683 684 685 686
## A 0.4843411 0.4875471 0.4878581 0.4852897 0.4842129 0.4845735 0.4863896
## B 0.5156589 0.5124529 0.5121419 0.5147103 0.5157871 0.5154265 0.5136104
## index
## states 687 688 689 690 691 692 693
## A 0.4852532 0.4856063 0.4829678 0.4817045 0.4817527 0.4831151 0.4813616
## B 0.5147468 0.5143937 0.5170322 0.5182955 0.5182473 0.5168849 0.5186384
## index
## states 694 695 696 697 698 699 700
## A 0.4809026 0.4817149 0.4793416 0.4781616 0.4781154 0.4792007 0.4769751
## B 0.5190974 0.5182851 0.5206584 0.5218384 0.5218846 0.5207993 0.5230249
## index
## states 701 702 703 704 705 706 707
## A 0.475824 0.4756894 0.4765645 0.4784933 0.4815731 0.4859591 0.4873704
## B 0.524176 0.5243106 0.5234355 0.5215067 0.5184269 0.5140409 0.5126296
## index
## states 708 709 710 711 712 713 714
## A 0.4903777 0.4906313 0.4881439 0.4827901 0.4788042 0.4759854 0.4741916
## B 0.5096223 0.5093687 0.5118561 0.5172099 0.5211958 0.5240146 0.5258084
## index
## states 715 716 717 718 719 720 721
## A 0.4733323 0.4733644 0.4742894 0.4761539 0.4790519 0.4786315 0.4793684
## B 0.5266677 0.5266356 0.5257106 0.5238461 0.5209481 0.5213685 0.5206316
## index
## states 722 723 724 725 726 727 728
## A 0.4812999 0.4845233 0.4847007 0.4863401 0.489524 0.4899115 0.4875221
## B 0.5187001 0.5154767 0.5152993 0.5136599 0.510476 0.5100885 0.5124779
## index
## states 729 730 731 732 733 734 735
## A 0.4867361 0.487514 0.489895 0.493999 0.4955296 0.4945639 0.4910532
## B 0.5132639 0.512486 0.510105 0.506001 0.5044704 0.5054361 0.5089468
## index
## states 736 737 738 739 740 741 742
## A 0.4848206 0.4800585 0.4765268 0.4740477 0.4724962 0.471794 0.4719058
## B 0.5151794 0.5199415 0.5234732 0.5259523 0.5275038 0.528206 0.5280942
## index
## states 743 744 745 746 747 748 749
## A 0.4728373 0.4746353 0.4728952 0.4720245 0.4719794 0.4727577 0.4743984
## B 0.5271627 0.5253647 0.5271048 0.5279755 0.5280206 0.5272423 0.5256016
## index
## states 750 751 752 753 754 755 756
## A 0.4769844 0.4761491 0.4763466 0.4775868 0.4754357 0.4742819 0.474067
## B 0.5230156 0.5238509 0.5236534 0.5224132 0.5245643 0.5257181 0.525933
## index
## states 757 758 759 760 761 762 763
## A 0.4747803 0.4764578 0.4746878 0.4738774 0.4739856 0.4750179 0.4770264
## B 0.5252197 0.5235422 0.5253122 0.5261226 0.5260144 0.5249821 0.5229736
## index
## states 764 765 766 767 768 769 770
## A 0.4756158 0.4752112 0.4757922 0.4773882 0.4755831 0.4747824 0.4749459
## B 0.5243842 0.5247888 0.5242078 0.5226118 0.5244169 0.5252176 0.5250541
## index
## states 771 772 773 774 775 776 777
## A 0.4760817 0.4782472 0.4815513 0.4816615 0.4785833 0.4766605 0.4757964
## B 0.5239183 0.5217528 0.5184487 0.5183385 0.5214167 0.5233395 0.5242036
## index
## states 778 779 780 781 782 783 784
## A 0.4759473 0.4726252 0.4701595 0.468426 0.4673374 0.4668389 0.4669051
## B 0.5240527 0.5273748 0.5298405 0.531574 0.5326626 0.5331611 0.5330949
## index
## states 785 786 787 788 789 790 791
## A 0.4675396 0.4687743 0.4706714 0.4733266 0.4768735 0.4769935 0.4781886
## B 0.5324604 0.5312257 0.5293286 0.5266734 0.5231265 0.5230065 0.5218114
## index
## states 792 793 794 795 796 797 798
## A 0.4760224 0.4748829 0.4747126 0.4755029 0.4772936 0.4801751 0.4797941
## B 0.5239776 0.5251171 0.5252874 0.5244971 0.5227064 0.5198249 0.5202059
## index
## states 799 800
## A 0.4806288 0.4827213
## B 0.5193712 0.5172787The HMM is unknown, and I want to estimate the parameters of the model.
The inputs are the obsrevations and the output are the model’s parameters estimation through Maximum Likelihood.
Even though I give as input a model with some parameters (ex: transition matrix, emission matrix,…), it will estimate other parameters [different from the ones I gave as input] only based on the sequences of observations and hidden states.
bw = baumWelch(hmm3, observation, maxIterations = 100)
print(bw$hmm) # optimal parameters to the HMM## $States
## [1] "A" "B"
##
## $Symbols
## [1] "L" "R"
##
## $startProbs
## A B
## 0.5 0.5
##
## $transProbs
## to
## from A B
## A 9.974780e-01 0.002521982
## B 4.076426e-12 1.000000000
##
## $emissionProbs
## symbols
## states L R
## A 0.2465230 0.7534770
## B 0.7490964 0.2509036# So these are not the inizialized paraemters but the estimated ones by the
# observations. So the transitin matrix and the emission matrix are estimated
# from the data (observations/symbols)Note that the given transition matrix and the emission matrix given as input are different relatively to the estiamated ones.
print(hmm3)## $States
## [1] "A" "B"
##
## $Symbols
## [1] "L" "R"
##
## $startProbs
## A B
## 0.5 0.5
##
## $transProbs
## to
## from A B
## A 0.9 0.1
## B 0.1 0.9
##
## $emissionProbs
## symbols
## states L R
## A 0.50 0.50
## B 0.51 0.49print(bw$hmm)## $States
## [1] "A" "B"
##
## $Symbols
## [1] "L" "R"
##
## $startProbs
## A B
## 0.5 0.5
##
## $transProbs
## to
## from A B
## A 9.974780e-01 0.002521982
## B 4.076426e-12 1.000000000
##
## $emissionProbs
## symbols
## states L R
## A 0.2465230 0.7534770
## B 0.7490964 0.2509036A dishonest Casino uses a fair dice most of the time, but switches to the loaded dice once in a while. The probabilities of the fair die are (1/6, …, 1/6) for throwing (“1”,…,“6”). The probabilities of the loaded die are (1/10, …, 1/10, 1/2) for throwing (“1”,…,“5”,“6”). The observer doesn’t know which die is actually taken (the state is hidden), but the sequence of throws (observations) can be used to infer which die (state) was used.
Can we detect which die is in use at any given time, just by observing the sequence of rolls?
# P(Fair --> Fair) = 0.95
# P(Fair --> Loaded) = 0.05
# P(Loaded --> Fair) = 0.1
# P(Loaded --> Loaded) = 0.9
States= c('Fair','Loaded')
Symbols = c(1,2,3,4,5,6)
transProbs = matrix(c(0.95, 0.05, # between states
0.1, 0.9),2) # '2' stands for 2 cols
emissProbs = matrix(c(1/6, 1/10, # from states to symbols
1/6, 1/10,
1/6, 1/10,
1/6, 1/10,
1/6, 1/10,
1/6, 1/2) ,2) # '2' stands for 2 cols
startProbs = c(0.5,0.5) # I start randomly with one of the two.
hmm = initHMM(States, Symbols,startProbs, transProbs, emissProbs)Suppose I have this sequence of observations.
observations = c(3, 2, 2, 1, 2, 3, 6, 6, 6, 6)We can identify which die has been used ?
This is a smoothing problem, so:
viterbi(hmm, observations)## [1] "Fair" "Fair" "Fair" "Fair" "Fair" "Fair" "Loaded" "Loaded"
## [9] "Loaded" "Loaded"