HMM_decoding: Algorithm for Decoding Hidden Markov Models (local or global)
In HMMpa: Analysing Accelerometer Data Using Hidden Markov Models
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
The function decodes a hidden Markov model into a most likely sequence of hidden states. Furthermore this function provides estimated observation values along the most likely sequence of hidden states. See Details for more information.
Usage
1
2
3
HMM_decoding(x, m, delta, gamma, distribution_class,
distribution_theta, decoding_method = "global",
discr_logL = FALSE, discr_logL_eps = 0.5)
Arguments
x
a vector object containing the time-series of observations that are assumed to be realizations of the (hidden Markov state dependent) observation process of the model.
m
a (finite) number of states in the hidden Markov chain.
delta
a vector object containing values for the marginal probability distribution of the m states of the Markov chain at the time point t=1.
gamma
a matrix (ncol=nrow=m) containing values for the transition matrix of the hidden Markov chain.
distribution_class
a single character string object with the abbreviated name of the m observation distributions of the Markov dependent observation process. The following distributions are supported by this algorithm: Poisson (pois); generalized Poisson (genpois); normal (norm); geometric (geom).
distribution_theta
a list object containing the parameter values for the m observation distributions that are dependent on the hidden Markov state.
decoding_method
a string object to choose the applied decoding-method to decode the HMM given the time-series of observations x. Possible values are "global" (for the use of the Viterbi_algorithm) and "local" (for the use of the
local_decoding_algorithm). Default value is "global".
discr_logL
a logical object. It is TRUE if the discrete log-likelihood shall be calculated (for distribution_class="norm" instead of the general log-likelihood). Default is FALSE.
discr_logL_eps
a single numerical value to approximately determine the discrete log-likelihood for a hidden Markov model based on nomal distributions (for "norm"). The default value is 0.5.
Details
More precisely, the function works as follows:
Step 1:
In a first step, the algorithm decodes a HMM into the most likely sequence of hidden states, given a time-series of observations. The user can choose between a global and a local approch.
If decoding_method="global" is applied, the function calls Viterbi_algorithm to determine the sequence of most likely hidden states for all time points simultaneously.
If decoding_method="local" is applied, the function calls local_decoding_algorithm to determine the most likely hidden state for each time point seperately.
Step 2:
In a second step, this function links each observation to the mean of the distribution, that corresponds to the decoded state at this point in time.
Value
HMM_decoding returns a list containing the following two components:
decoding_method
a string object indicating the applied decoding method.
decoding
a numerical vector containing the most likely sequence of hidden states as decoded by the Viterbi_algorithm (if "global" was applied) or by the
local_decoding_algorithm (if "local" was applied).
decoding_distr_means
a numerical vector of estimated oberservation values along the most likely seuquence of hidden states (see decoding and Step 2).
Author(s)
Vitali Witowski (2013).
References
MacDonald, I. L., Zucchini, W. (2009) Hidden Markov Models for Time Series: An Introduction Using R, Boca Raton: Chapman & Hall.
See Also
local_decoding_algorithm,
Viterbi_algorithm
Examples
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################################################################
### i) HMM-training ###########################################
################################################################
################################################################
### Fictitious observations ####################################
################################################################
x <- c(1,16,19,34,22,6,3,5,6,3,4,1,4,3,5,7,9,8,11,11,
14,16,13,11,11,10,12,19,23,25,24,23,20,21,22,22,18,7,
5,3,4,3,2,3,4,5,4,2,1,3,4,5,4,5,3,5,6,4,3,6,4,8,9,12,
9,14,17,15,25,23,25,35,29,36,34,36,29,41,42,39,40,43,
37,36,20,20,21,22,23,26,27,28,25,28,24,21,25,21,20,21,
11,18,19,20,21,13,19,18,20,7,18,8,15,17,16,13,10,4,9,
7,8,10,9,11,9,11,10,12,12,5,13,4,6,6,13,8,9,10,13,13,
11,10,5,3,3,4,9,6,8,3,5,3,2,2,1,3,5,11,2,3,5,6,9,8,5,
2,5,3,4,6,4,8,15,12,16,20,18,23,18,19,24,23,24,21,26,
36,38,37,39,45,42,41,37,38,38,35,37,35,31,32,30,20,39,
40,33,32,35,34,36,34,32,33,27,28,25,22,17,18,16,10,9,
5,12,7,8,8,9,19,21,24,20,23,19,17,18,17,22,11,12,3,9,
10,4,5,13,3,5,6,3,5,4,2,5,1,2,4,4,3,2,1)
### Train hidden Markov model for different number of
### states m=2,...,6 and select the optimal model
m_trained_HMM <-
HMM_training(x = x,
min_m = 2,
max_m = 6,
distribution_class="pois")$trained_HMM_with_selected_m
################################################################
################################################################
### ii) Global decoding ########################################
################################################################
################################################################
### Decode the trained HMM using the Viterbi algorithm to get
### the estimated sequence of hidden physical activity levels
global_decoding <-
HMM_decoding(x = x,
m = m_trained_HMM$m,
delta = m_trained_HMM$delta,
gamma = m_trained_HMM$gamma,
distribution_class = m_trained_HMM$distribution_class,
distribution_theta = m_trained_HMM$distribution_theta,
decoding_method = "global")
### Globally most likely sequence of hidden states,
### i.e. in this case sequence of activity levels
global_decoding$decoding
par(mfrow = c(1,1))
plot(global_decoding$decoding)
### Plot the observed impulse counts and the most likely
### sequence (green) according to the Viterbi algorithm that
### generated these observations
plot(x)
lines(global_decoding$decoding_distr_means, col = "green")
################################################################
################################################################
### iii) Local decoding ########################################
################################################################
################################################################
### Decode the trained HMM using the local decoding algorithm
### to get the estimated sequence of hidden physical activity
### levels
local_decoding <-
HMM_decoding(x = x,
m = m_trained_HMM$m,
delta = m_trained_HMM$delta,
gamma = m_trained_HMM$gamma,
distribution_class = m_trained_HMM$distribution_class,
distribution_theta = m_trained_HMM$distribution_theta,
decoding_method = "local")
### Locally most likely sequence of hidden states,
### i.e. in this case sequence of activity levels
local_decoding$decoding
par(mfrow=c(1,1))
plot(local_decoding$decoding)
### Plot the observed impulse counts and the most likely
### sequence (green) according to the local decoding algorithm
### that generated these observations
plot(x)
lines(local_decoding$decoding_distr_means, col = "red")
################################################################
################################################################
### iv) Comparison of global and local decoding ################
################################################################
################################################################
### Comparison of global decoding (green), local decoding (red)
### and the connection to the closest mean (blue)
print(global_decoding$decoding)
print(local_decoding$decoding)
### Plot comparison
par(mfrow = c(2,2))
plot(global_decoding$decoding[seq(230,260)], col = "green",
ylab = "global decoding", main = "(zooming)")
plot(x[seq(230,260)], ylab = "global decoding",
main = "(zooming x[seq(230,260)])")
lines(global_decoding$decoding_distr_means[seq(230,260)],
col = "green")
plot(local_decoding$decoding[seq(230,260)], col = "red",
ylab = "local decoding", main = "(zooming)")
plot(x[seq(230,260)], ylab = "local decoding",
main = "(zooming x[seq(230,260)])")
lines(local_decoding$decoding_distr_means[seq(230,260)],
col = "red")
par(mfrow = c(1,1))
HMMpa documentation built on May 2, 2019, 7:58 a.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
The function decodes a hidden Markov model into a most likely sequence of hidden states. Furthermore this function provides estimated observation values along the most likely sequence of hidden states. See Details for more information.
Usage
1 2 3 | HMM_decoding(x, m, delta, gamma, distribution_class,
distribution_theta, decoding_method = "global",
discr_logL = FALSE, discr_logL_eps = 0.5)
|
Arguments
x |
a vector object containing the time-series of observations that are assumed to be realizations of the (hidden Markov state dependent) observation process of the model. |
m |
a (finite) number of states in the hidden Markov chain. |
delta |
a vector object containing values for the marginal probability distribution of the |
gamma |
a matrix ( |
distribution_class |
a single character string object with the abbreviated name of the |
distribution_theta |
a list object containing the parameter values for the |
decoding_method |
a string object to choose the applied decoding-method to decode the HMM given the time-series of observations
|
discr_logL |
a logical object. It is |
discr_logL_eps |
a single numerical value to approximately determine the discrete log-likelihood for a hidden Markov model based on nomal distributions (for |
Details
More precisely, the function works as follows:
Step 1:
In a first step, the algorithm decodes a HMM into the most likely sequence of hidden states, given a time-series of observations. The user can choose between a global and a local approch.
If decoding_method="global" is applied, the function calls Viterbi_algorithm to determine the sequence of most likely hidden states for all time points simultaneously.
If decoding_method="local" is applied, the function calls local_decoding_algorithm to determine the most likely hidden state for each time point seperately.
Step 2: In a second step, this function links each observation to the mean of the distribution, that corresponds to the decoded state at this point in time.
Value
HMM_decoding returns a list containing the following two components:
decoding_method |
a string object indicating the applied decoding method. |
decoding |
a numerical vector containing the most likely sequence of hidden states as decoded by the
|
decoding_distr_means |
a numerical vector of estimated oberservation values along the most likely seuquence of hidden states (see |
Author(s)
Vitali Witowski (2013).
References
MacDonald, I. L., Zucchini, W. (2009) Hidden Markov Models for Time Series: An Introduction Using R, Boca Raton: Chapman & Hall.
See Also
local_decoding_algorithm,
Viterbi_algorithm
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | ################################################################
### i) HMM-training ###########################################
################################################################
################################################################
### Fictitious observations ####################################
################################################################
x <- c(1,16,19,34,22,6,3,5,6,3,4,1,4,3,5,7,9,8,11,11,
14,16,13,11,11,10,12,19,23,25,24,23,20,21,22,22,18,7,
5,3,4,3,2,3,4,5,4,2,1,3,4,5,4,5,3,5,6,4,3,6,4,8,9,12,
9,14,17,15,25,23,25,35,29,36,34,36,29,41,42,39,40,43,
37,36,20,20,21,22,23,26,27,28,25,28,24,21,25,21,20,21,
11,18,19,20,21,13,19,18,20,7,18,8,15,17,16,13,10,4,9,
7,8,10,9,11,9,11,10,12,12,5,13,4,6,6,13,8,9,10,13,13,
11,10,5,3,3,4,9,6,8,3,5,3,2,2,1,3,5,11,2,3,5,6,9,8,5,
2,5,3,4,6,4,8,15,12,16,20,18,23,18,19,24,23,24,21,26,
36,38,37,39,45,42,41,37,38,38,35,37,35,31,32,30,20,39,
40,33,32,35,34,36,34,32,33,27,28,25,22,17,18,16,10,9,
5,12,7,8,8,9,19,21,24,20,23,19,17,18,17,22,11,12,3,9,
10,4,5,13,3,5,6,3,5,4,2,5,1,2,4,4,3,2,1)
### Train hidden Markov model for different number of
### states m=2,...,6 and select the optimal model
m_trained_HMM <-
HMM_training(x = x,
min_m = 2,
max_m = 6,
distribution_class="pois")$trained_HMM_with_selected_m
################################################################
################################################################
### ii) Global decoding ########################################
################################################################
################################################################
### Decode the trained HMM using the Viterbi algorithm to get
### the estimated sequence of hidden physical activity levels
global_decoding <-
HMM_decoding(x = x,
m = m_trained_HMM$m,
delta = m_trained_HMM$delta,
gamma = m_trained_HMM$gamma,
distribution_class = m_trained_HMM$distribution_class,
distribution_theta = m_trained_HMM$distribution_theta,
decoding_method = "global")
### Globally most likely sequence of hidden states,
### i.e. in this case sequence of activity levels
global_decoding$decoding
par(mfrow = c(1,1))
plot(global_decoding$decoding)
### Plot the observed impulse counts and the most likely
### sequence (green) according to the Viterbi algorithm that
### generated these observations
plot(x)
lines(global_decoding$decoding_distr_means, col = "green")
################################################################
################################################################
### iii) Local decoding ########################################
################################################################
################################################################
### Decode the trained HMM using the local decoding algorithm
### to get the estimated sequence of hidden physical activity
### levels
local_decoding <-
HMM_decoding(x = x,
m = m_trained_HMM$m,
delta = m_trained_HMM$delta,
gamma = m_trained_HMM$gamma,
distribution_class = m_trained_HMM$distribution_class,
distribution_theta = m_trained_HMM$distribution_theta,
decoding_method = "local")
### Locally most likely sequence of hidden states,
### i.e. in this case sequence of activity levels
local_decoding$decoding
par(mfrow=c(1,1))
plot(local_decoding$decoding)
### Plot the observed impulse counts and the most likely
### sequence (green) according to the local decoding algorithm
### that generated these observations
plot(x)
lines(local_decoding$decoding_distr_means, col = "red")
################################################################
################################################################
### iv) Comparison of global and local decoding ################
################################################################
################################################################
### Comparison of global decoding (green), local decoding (red)
### and the connection to the closest mean (blue)
print(global_decoding$decoding)
print(local_decoding$decoding)
### Plot comparison
par(mfrow = c(2,2))
plot(global_decoding$decoding[seq(230,260)], col = "green",
ylab = "global decoding", main = "(zooming)")
plot(x[seq(230,260)], ylab = "global decoding",
main = "(zooming x[seq(230,260)])")
lines(global_decoding$decoding_distr_means[seq(230,260)],
col = "green")
plot(local_decoding$decoding[seq(230,260)], col = "red",
ylab = "local decoding", main = "(zooming)")
plot(x[seq(230,260)], ylab = "local decoding",
main = "(zooming x[seq(230,260)])")
lines(local_decoding$decoding_distr_means[seq(230,260)],
col = "red")
par(mfrow = c(1,1))
|
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