This function assigns a physical activity range to each observation of a time-series
(such as a sequence of impulse counts recorded by an accelerometer) using
hidden Markov models (HMM). The activity ranges are defined by thresholds called
cut-off points. Basically, this function combines HMM_training,
HMM_decoding and cut_off_point_method.
See Details for further information.
HMM_based_method(
x,
cut_points,
distribution_class,
min_m = 2,
max_m = 6,
n = 100,
max_scaled_x = NA,
names_activity_ranges = NA,
discr_logL = FALSE,
discr_logL_eps = 0.5,
dynamical_selection = TRUE,
training_method = "EM",
Mstep_numerical = FALSE,
BW_max_iter = 50,
BW_limit_accuracy = 0.001,
BW_print = TRUE,
DNM_max_iter = 50,
DNM_limit_accuracy = 0.001,
DNM_print = 2,
decoding_method = "global",
bout_lengths = NULL,
plotting = 0
)HMM_based_method returns a list containing the output of the trained
hidden Markov model, including the selected number of states m (i.e., number of
physical activities) and plots key figures.
a list object containing the trained hidden Markov
model including the selected number of states m (see HMM_training
for further details).
a list object containing the output of the decoding
(see HMM_decoding for further details)
a list object containing the output of the
cut-off point method. The counts x are classified into the activity ranges
by the corresponding sequence of hidden PA-levels, which were decoded by the HMM
(see cut_off_point_method for further details).
a vector object of length T containing non-negative observations of a
time-series, such as a sequence of accelerometer impulse counts, which are assumed to
be realizations of the (hidden Markov state dependent) observation process of a HMM.
a vector object containing cut-off points to separate activity ranges.
For instance, the vector c(7,15,23) separates the four activity ranges
[0,7), [7,15), [15,23) and [23,Inf).
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: Poisson (pois); generalized Poisson
(genpois); normal (norm)).
miminum number of hidden states in the hidden Markov chain.
Default value is 2.
maximum number of hidden states in the hidden Markov chain.
Default value is 6.
a single numerical value specifying the number of samples.
Default value is 100.
an optional numerical value, to be used to scale the observations
of the time-series x before the hidden Markov model is trained and decoded
(see Details). Default value is NA.
an optional character string vector to name the activity
ranges induced by the cut-points. This vector must contain one element more than the
vector cut_points.
a logical object indicating whether the discrete log-likelihood
should be used (for "norm") for estimating the model specific parameters instead
of the general log-likelihood. See MacDonald & Zucchini (2009, Paragraph 1.2.3)
for further details. Default is FALSE.
a single numerical value to approximate the discrete
log-likelihood for a hidden Markov model based on nomal distributions
(for distribution_class="norm"). The default value is 0.5.
a logical value indicating whether the method of dynamical
initial parameter selection should be applied (see HMM_training
for details). Default is TRUE.
a logical value indicating whether the Baum-Welch algorithm
("EM") or the method of direct numerical maximization ("numerical")
should be applied for estimating the model specific parameters of the HMM.
See Baum_Welch_algorithm and
direct_numerical_maximization for further details.
Default is training_method = "EM".
a logical object indicating whether the Maximization Step of the Baum-Welch algorithm shall be performed by numerical maximization. Default is FALSE.
a single numerical value representing the maximum number of
iterations in the Baum-Welch algorithm. Default value is 50.
a single numerical value representing the convergence
criterion of the Baum-Welch algorithm. Default value is 0.001.
a logical object indicating whether the log-likelihood at each
iteration-step shall be printed. Default is TRUE.
a single numerical value representing the maximum number of iterations
of the numerical maximization using the nlm-function (used to perform the M-step of the
Baum-Welch-algorithm). Default value is 50.
a single numerical value representing the convergence
criterion of the numerical maximization algorithm using the nlm
function (used to perform the M-step of the Baum-Welch-algorithm).
Default value is 0.001.
a single numerical value to determine the level of printing of the
nlm-function. See nlm-function for further informations.
The value 0 suppresses, that no printing will be outputted.
Default value is 2 for full printing.
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".
a vector object (with even number of elemets) to define the range
of the bout intervals (see Details for the definition of bouts). For instance,
bout_lengths = c(1,1,2,2,3,10,11,20,1,20) defines the five bout intervals
[1,1] (1 count); [2,2] (2 counts); [3,10] (3-10 counts); [11,20] (11-20 counts);
[1,20] (1-20 counts - overlapping with other bout intervalls is possible).
Default value is bout_lengths=NULL.
a numeric value between 0 and 5 (generates different outputs).
NA suppresses graphical output. Default value is 0.
0: output 1-5
1: summary of all results
2: time series of activity counts, classified into activity ranges
3: time series of bouts (and, if available, the sequence of the estimated
hidden physical activity levels, extracted by decoding a trained HMM,
in green colour)
4: barplots of absolute and relative frequencies of time spent in different
activity ranges
5: barplots of relative frequencies of the lenghts of bout intervals
(overall and by activity ranges )
Vitali Witowski (2013).
The function combines HMM_training, HMM_decoding and
cut_off_point_method as follows:
Step 1: HMM_training trains the most likely HMM for a given
time-series of accelerometer counts.
Step 2: HMM_decoding decodes the trained HMM (Step 1) into the
most likely sequence of hidden states corresponding to the given time-series of
observations (respectively the most likely sequence of physical activity levels
corresponding to the time-series of accelerometer counts).
Step 3. cut_off_point_method assigns an activity range to each
accelerometer count by its hidden physical activity level (extracted in Step 2).
Brachmann, B. (2011). Hidden-Markov-Modelle fuer Akzelerometerdaten. Diploma Thesis, University Bremen - Bremen Institute for Prevention Research and Social Medicine (BIPS).
MacDonald, I. L., Zucchini, W. (2009) Hidden Markov Models for Time Series: An Introduction Using R, Boca Raton: Chapman & Hall.
Witowski, V., Foraita, R., Pitsiladis, Y., Pigeot, I., Wirsik, N. (2014) Using hidden Markov models to improve quantifying physical activity in accelerometer data - A simulation study. PLOS ONE. 9(12), e114089. tools:::Rd_expr_doi("10.1371/journal.pone.0114089")
initial_parameter_training, Baum_Welch_algorithm,
direct_numerical_maximization, AIC_HMM,
BIC_HMM, HMM_training, Viterbi_algorithm,
local_decoding_algorithm, cut_off_point_method
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)
# Assumptions (number of states, probability vector,
# transition matrix, and distribution parameters)
m <- 4
delta <- c(0.25, 0.25, 0.25, 0.25)
gamma <- 0.7 * diag(m) + rep(0.3 / m)
distribution_class <- "pois"
distribution_theta <- list(lambda = c(4, 9, 17, 25))
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