Package 'samurais' reference manual

Title:	Statistical Models for the Unsupervised Segmentation of Time-Series ('SaMUraiS')
Description:	Provides a variety of original and flexible user-friendly statistical latent variable models and unsupervised learning algorithms to segment and represent time-series data (univariate or multivariate), and more generally, longitudinal data, which include regime changes. 'samurais' is built upon the following packages, each of them is an autonomous time-series segmentation approach: Regression with Hidden Logistic Process ('RHLP'), Hidden Markov Model Regression ('HMMR'), Multivariate 'RHLP' ('MRHLP'), Multivariate 'HMMR' ('MHMMR'), Piece-Wise regression ('PWR'). For the advantages/differences of each of them, the user is referred to our mentioned paper references.
Authors:	Faicel Chamroukhi [aut] , Marius Bartcus [aut], Florian Lecocq [aut, cre]
Maintainer:	Florian Lecocq <[email protected]>
License:	GPL (>= 3)
Version:	0.1.0
Built:	2024-11-01 11:46:10 UTC
Source:	CRAN

SaMUraiS: StAtistical Models for the UnsupeRvised segmentAtIon of time-Series

Description

samurais is a toolbox including many original and flexible user-friendly statistical latent variable models and efficient unsupervised algorithms to segment and represent time-series data (univariate or multivariate), and more generally, longitudinal data, which include regime changes.

samurais contains the following time series segmentation models:

RHLP;
HMM/HMMR;
PWR;
MRHLP;
MHMMR;

For the advantages/differences of each of them, the user is referred to our mentioned paper references.

To learn more about samurais, start with the vignettes: browseVignettes(package = "samurais")

Author(s)

Maintainer: Florian Lecocq [email protected]

Authors:

Faicel Chamroukhi [email protected] (0000-0002-5894-3103)
Marius Bartcus [email protected]

References

Chamroukhi, F., and Hien D. Nguyen. 2019. Model-Based Clustering and Classification of Functional Data. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery. https://doi.org/10.1002/widm.1298.

Chamroukhi, F. 2015. Statistical Learning of Latent Data Models for Complex Data Analysis. Habilitation Thesis (HDR), Universite de Toulon. https://chamroukhi.com/Dossier/FChamroukhi-Habilitation.pdf.

Trabelsi, D., S. Mohammed, F. Chamroukhi, L. Oukhellou, and Y. Amirat. 2013. An Unsupervised Approach for Automatic Activity Recognition Based on Hidden Markov Model Regression. IEEE Transactions on Automation Science and Engineering 3 (10): 829–335. https://chamroukhi.com/papers/Chamroukhi-MHMMR-IeeeTase.pdf.

Chamroukhi, F., D. Trabelsi, S. Mohammed, L. Oukhellou, and Y. Amirat. 2013. Joint Segmentation of Multivariate Time Series with Hidden Process Regression for Human Activity Recognition. Neurocomputing 120: 633–44. https://chamroukhi.com/papers/chamroukhi_et_al_neucomp2013b.pdf.

Chamroukhi, F., A. Same, G. Govaert, and P. Aknin. 2010. A Hidden Process Regression Model for Functional Data Description. Application to Curve Discrimination. Neurocomputing 73 (7-9): 1210–21. https://chamroukhi.com/papers/chamroukhi_neucomp_2010.pdf.

Chamroukhi, F. 2010. Hidden Process Regression for Curve Modeling, Classification and Tracking. Ph.D. Thesis, Universite de Technologie de Compiegne. https://chamroukhi.com/papers/FChamroukhi-Thesis.pdf.

Chamroukhi, F., A. Same, G. Govaert, and P. Aknin. 2009. Time Series Modeling by a Regression Approach Based on a Latent Process. Neural Networks 22 (5-6): 593–602. https://chamroukhi.com/papers/Chamroukhi_Neural_Networks_2009.pdf.

emHMMR implemens the EM (Baum-Welch) algorithm to fit a HMMR model.

Description

emHMMR implements the maximum-likelihood parameter estimation of the HMMR model by the Expectation-Maximization (EM) algorithm, known as Baum-Welch algorithm in the context of HMMs.

Usage

emHMMR(X, Y, K, p = 3, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE)
emHMMR(X, Y, K, p = 3, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
`K`	The number of regimes/segments (HMMR components).
`p`	Optional. The order of the polynomial regression. By default, `p` is set at 3.
`variance_type`	Optional character indicating if the model is "homoskedastic" or "heteroskedastic" (i.e same variance or different variances for each of the K regmies). By default the model is "heteroskedastic".
`n_tries`	Optional. Number of runs of the EM algorithm. The solution providing the highest log-likelihood will be returned. If `n_tries` > 1, then for the first run, parameters are initialized by uniformly segmenting the data into K segments, and for the next runs, parameters are initialized by randomly segmenting the data into K contiguous segments.
`max_iter`	Optional. The maximum number of iterations for the EM algorithm.
`threshold`	Optional. A numeric value specifying the threshold for the relative difference of log-likelihood between two steps of the EM as stopping criteria.
`verbose`	Optional. A logical value indicating whether or not values of the log-likelihood should be printed during EM iterations.

Details

emHMMR function implements the EM algorithm for the HMMR model. This function starts with an initialization of the parameters done by the method initParam of the class ParamHMMR, then it alternates between the E-Step (method of the class StatHMMR) and the M-Step (method of the class ParamHMMR) until convergence (until the relative variation of log-likelihood between two steps of the EM algorithm is less than the threshold parameter).

Value

EM returns an object of class ModelHMMR.

Examples

data(univtoydataset)
hmmr <- emHMMR(univtoydataset$x, univtoydataset$y, K = 5, p = 1, verbose = TRUE)

hmmr$summary()

hmmr$plot()

data(univtoydataset)
hmmr <- emHMMR(univtoydataset$x, univtoydataset$y, K = 5, p = 1, verbose = TRUE)

hmmr$summary()

hmmr$plot()

emMHMMR implemens the EM (Baum-Welch) algorithm to fit a MHMMR model.

Description

emMHMMR implements the maximum-likelihood parameter estimation of the MHMMR model by the Expectation-Maximization (EM) algorithm, known as Baum-Welch algorithm in the context of HMMs.

Usage

emMHMMR(X, Y, K, p = 3, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE)
emMHMMR(X, Y, K, p = 3, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Matrix of size $(m, d)$ representing a $d$ dimension time series observed at points $1,\dots,m$ .
`K`	The number of regimes (MHMMR components).
`p`	Optional. The order of the polynomial regression. By default, `p` is set at 3.
`variance_type`	Optional character indicating if the model is "homoskedastic" or "heteroskedastic". By default the model is "heteroskedastic".
`n_tries`	Optional. Number of runs of the EM algorithm. The solution providing the highest log-likelihood will be returned. If `n_tries` > 1, then for the first run, parameters are initialized by uniformly segmenting the data into K segments, and for the next runs, parameters are initialized by randomly segmenting the data into K contiguous segments.
`max_iter`	Optional. The maximum number of iterations for the EM algorithm.
`threshold`	Optional. A numeric value specifying the threshold for the relative difference of log-likelihood between two steps of the EM as stopping criteria.
`verbose`	Optional. A logical value indicating whether or not values of the log-likelihood should be printed during EM iterations.

Details

emMHMMR function implements the EM algorithm. This function starts with an initialization of the parameters done by the method initParam of the class ParamMHMMR, then it alternates between the E-Step (method of the class StatMHMMR) and the M-Step (method of the class ParamMHMMR) until convergence (until the relative variation of log-likelihood between two steps of the EM algorithm is less than the threshold parameter).

Value

EM returns an object of class ModelMHMMR.

Examples

data(multivtoydataset)

mhmmr <- emMHMMR(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

mhmmr$summary()

mhmmr$plot()
data(multivtoydataset)

mhmmr <- emMHMMR(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

mhmmr$summary()

mhmmr$plot()

emMRHLP implemens the EM algorithm to fit a MRHLP model.

Description

emMRHLP implements the maximum-likelihood parameter estimation of the MRHLP model by the Expectation-Maximization (EM) algorithm.

Usage

emMRHLP(X, Y, K, p = 3, q = 1, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE, verbose_IRLS = FALSE)
emMRHLP(X, Y, K, p = 3, q = 1, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE, verbose_IRLS = FALSE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Matrix of size $(m, d)$ representing a $d$ dimension function of `X` observed at points $1,\dots,m$ . `Y` is the observed response/output.
`K`	The number of regimes (MRHLP components).
`p`	Optional. The order of the polynomial regression. By default, `p` is set at 3.
`q`	Optional. The dimension of the logistic regression. For the purpose of segmentation, it must be set to 1 (which is the default value).
`variance_type`	Optional character indicating if the model is "homoskedastic" or "heteroskedastic". By default the model is "heteroskedastic".
`n_tries`	Optional. Number of runs of the EM algorithm. The solution providing the highest log-likelihood will be returned. If `n_tries` > 1, then for the first run, parameters are initialized by uniformly segmenting the data into K segments, and for the next runs, parameters are initialized by randomly segmenting the data into K contiguous segments.
`max_iter`	Optional. The maximum number of iterations for the EM algorithm.
`threshold`	Optional. A numeric value specifying the threshold for the relative difference of log-likelihood between two steps of the EM as stopping criteria.
`verbose`	Optional. A logical value indicating whether or not values of the log-likelihood should be printed during EM iterations.
`verbose_IRLS`	Optional. A logical value indicating whether or not values of the criterion optimized by IRLS should be printed at each step of the EM algorithm.

Details

emMRHLP function implements the EM algorithm of the MRHLP model. This function starts with an initialization of the parameters done by the method initParam of the class ParamMRHLP, then it alternates between the E-Step (method of the class StatMRHLP) and the M-Step (method of the class ParamMRHLP) until convergence (until the relative variation of log-likelihood between two steps of the EM algorithm is less than the threshold parameter).

Value

EM returns an object of class ModelMRHLP.

Examples

data(multivtoydataset)

mrhlp <- emMRHLP(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

mrhlp$summary()

mrhlp$plot()
data(multivtoydataset)

mrhlp <- emMRHLP(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

mrhlp$summary()

mrhlp$plot()

emRHLP implements the EM algorithm to fit a RHLP model.

Description

emRHLP implements the maximum-likelihood parameter estimation of the RHLP model by the Expectation-Maximization (EM) algorithm.

Usage

emRHLP(X, Y, K, p = 3, q = 1, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE, verbose_IRLS = FALSE)
emRHLP(X, Y, K, p = 3, q = 1, variance_type = c("heteroskedastic",
  "homoskedastic"), n_tries = 1, max_iter = 1500, threshold = 1e-06,
  verbose = FALSE, verbose_IRLS = FALSE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
`K`	The number of regimes (RHLP components).
`p`	Optional. The order of the polynomial regression. By default, `p` is set at 3.
`q`	Optional. The dimension of the logistic regression. For the purpose of segmentation, it must be set to 1 (which is the default value).
`variance_type`	Optional character indicating if the model is "homoskedastic" or "heteroskedastic". By default the model is "heteroskedastic".
`n_tries`	Optional. Number of runs of the EM algorithm. The solution providing the highest log-likelihood will be returned. If `n_tries` > 1, then for the first run, parameters are initialized by uniformly segmenting the data into K segments, and for the next runs, parameters are initialized by randomly segmenting the data into K contiguous segments.
`max_iter`	Optional. The maximum number of iterations for the EM algorithm.
`threshold`	Optional. A numeric value specifying the threshold for the relative difference of log-likelihood between two steps of the EM as stopping criteria.
`verbose`	Optional. A logical value indicating whether or not values of the log-likelihood should be printed during EM iterations.
`verbose_IRLS`	Optional. A logical value indicating whether or not values of the criterion optimized by IRLS should be printed at each step of the EM algorithm.

Details

emRHLP function implements the EM algorithm for the RHLP model. This function starts with an initialization of the parameters done by the method initParam of the class ParamRHLP, then it alternates between the E-Step (method of the class StatRHLP) and the M-Step (method of the class ParamRHLP) until convergence (until the relative variation of log-likelihood between two steps of the EM algorithm is less than the threshold parameter).

Value

EM returns an object of class ModelRHLP.

Examples

data(univtoydataset)

rhlp <- emRHLP(univtoydataset$x, univtoydataset$y, K = 3, p = 1, verbose = TRUE)

rhlp$summary()

rhlp$plot()
data(univtoydataset)

rhlp <- emRHLP(univtoydataset$x, univtoydataset$y, K = 3, p = 1, verbose = TRUE)

rhlp$summary()

rhlp$plot()

fitPWRFisher implements an optimized dynamic programming algorithm to fit a PWR model.

Description

fitPWRFisher is used to fit a Piecewise Regression (PWR) model by maximum-likelihood via an optimized dynamic programming algorithm. The estimation performed by the dynamic programming algorithm provides an optimal segmentation of the time series.

Usage

fitPWRFisher(X, Y, K, p = 3)
fitPWRFisher(X, Y, K, p = 3)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
`K`	The number of regimes/segments (PWR components).
`p`	Optional. The order of the polynomial regression. By default, `p` is set at 3.

Details

fitPWRFisher function implements an optimized dynamic programming algorithm of the PWR model. This function starts with the calculation of the "cost matrix" then it estimates the transition points given K the number of regimes thanks to the method computeDynamicProgram (method of the class ParamPWR).

Value

fitPWRFisher returns an object of class ModelPWR.

Examples

data(univtoydataset)

pwr <- fitPWRFisher(univtoydataset$x, univtoydataset$y, K = 5, p = 1)

pwr$summary()

pwr$plot()
data(univtoydataset)

pwr <- fitPWRFisher(univtoydataset$x, univtoydataset$y, K = 5, p = 1)

pwr$summary()

pwr$plot()

hmmProcess calculates the probability distribution of a random process following a Markov chain

Description

hmmProcess calculates the probability distribution of a random process following a Markov chain

Usage

hmmProcess(prior, trans_mat, n)
hmmProcess(prior, trans_mat, n)

Arguments

`prior`	Numeric vector or a one row matrix of length K representing the prior probabilities of the Markov chain.
`trans_mat`	Matrix of size $(K, K)$ representing the transition matrix of the Markov chain.
`n`	Numeric. Number of variables of the Markov chain.

Details

hmmProcess calculates the distribution $P(Z_{1},\dots,Z_{n};\pi,A)$ of a Markov chain $(Z_{1},\dots,Z_{n})$ with prior probability $\pi$ and transition matrix $A$ .

The calculation is based on the following formula:

$P(Z_{i} = k) = \sum_{l} P(Z_{i} = k, Z_{i-1} = l) = \sum_{l} P(Z_{i} = k | Z_{i-1} = l) \times P(Z_{i-1} = l) = \sum_{l} A_{lk} \times P(Z_{i-1})$

Value

Matrix of size $(n, K)$ giving the distribution of process given the K-state Markov chain parameters.

A Reference Class which represents multivariate data.

Description

MData is a reference class which represents multivariate objects. The data can be ordered by time (multivariate time series). In the last case, the field X represents the time.

Fields

X: Numeric vector of length m.
Y: Matrix of size $(m, d)$ representing a $d$ dimension function of X observed at points $1,\dots,m$ .

mkStochastic ensures that it is a stochastic vector, matrix or array.

Description

mkStochastic ensures that it is a stochastic vector, matrix or array.

Usage

mkStochastic(M)
mkStochastic(M)

Arguments

`M`	A vector, matrix or array to transform.

Details

mkStochastic ensures that the giving argument is a stochastic vector, matrix or array, i.e., that the sum over the last dimension is 1.

Value

A vector, matrix or array for which the sum over the last dimension is 1.

A Reference Class which represents a fitted HMMR model.

Description

ModelHMMR represents an estimated HMMR model.

Fields

param: An object of class ParamHMMR. It contains the estimated values of the parameters.
stat: An object of class StatHMMR. It contains all the statistics associated to the HMMR model.

Methods

plot(what = c("predicted", "filtered", "smoothed", "regressors", "loglikelihood"), ...)

Plot method.

what

The type of graph requested:

"predicted" = Predicted time series and predicted regime probabilities (fields predicted and predict_prob of class StatHMMR).
"filtered" = Filtered time series and filtering regime probabilities (fields filtered and filter_prob of class StatHMMR).
"smoothed" = Smoothed time series, and segmentation (fields smoothed and klas of the class StatHMMR).
"regressors" = Polynomial regression components (fields regressors and tau_tk of class StatHMMR).
"loglikelihood" = Value of the log-likelihood for each iteration (field stored_loglik of class StatHMMR).

...

Other graphics parameters.

By default, all the graphs mentioned above are produced.

summary(digits = getOption("digits"))

Summary method.

digits: The number of significant digits to use when printing.

Examples

data(univtoydataset)

hmmr <- emHMMR(univtoydataset$x, univtoydataset$y, K = 5, p = 1, verbose = TRUE)

# hmmr is a ModelHMMR object. It contains some methods such as 'summary' and 'plot'
hmmr$summary()
hmmr$plot()

# hmmr has also two fields, stat and param which are reference classes as well

# Log-likelihood:
hmmr$stat$loglik

# Parameters of the polynomial regressions:
hmmr$param$beta
data(univtoydataset)

hmmr <- emHMMR(univtoydataset$x, univtoydataset$y, K = 5, p = 1, verbose = TRUE)

# hmmr is a ModelHMMR object. It contains some methods such as 'summary' and 'plot'
hmmr$summary()
hmmr$plot()

# hmmr has also two fields, stat and param which are reference classes as well

# Log-likelihood:
hmmr$stat$loglik

# Parameters of the polynomial regressions:
hmmr$param$beta

A Reference Class which represents a fitted MHMMR model.

Description

ModelMHMMR represents an estimated MHMMR model.

Fields

param: A ParamMHMMR object. It contains the estimated values of the parameters.
stat: A StatMHMMR object. It contains all the statistics associated to the MHMMR model.

Methods

plot(what = c("predicted", "filtered", "smoothed", "regressors", "loglikelihood"), ...)

Plot method.

what

The type of graph requested:

"predicted" = Predicted time series and predicted regime probabilities (fields predicted and predict_prob of class StatMHMMR).
"filtered" = Filtered time series and filtering regime probabilities (fields filtered and filter_prob of class StatMHMMR).
"smoothed" = Smoothed time series, and segmentation (fields smoothed and klas of class StatMHMMR).
"regressors" = Polynomial regression components (fields regressors and tau_tk of class StatMHMMR).
"loglikelihood" = Value of the log-likelihood for each iteration (field stored_loglik of class StatMHMMR).

...

Other graphics parameters.

By default, all the above graphs are produced.

summary(digits = getOption("digits"))

Summary method.

digits: The number of significant digits to use when printing.

Examples

data(multivtoydataset)

mhmmr <- emMHMMR(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

# mhmmr is a ModelMHMMR object. It contains some methods such as 'summary' and 'plot'
mhmmr$summary()
mhmmr$plot()

# mhmmr has also two fields, stat and param which are reference classes as well

# Log-likelihood:
mhmmr$stat$loglik

# Parameters of the polynomial regressions:
mhmmr$param$beta
data(multivtoydataset)

mhmmr <- emMHMMR(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

# mhmmr is a ModelMHMMR object. It contains some methods such as 'summary' and 'plot'
mhmmr$summary()
mhmmr$plot()

# mhmmr has also two fields, stat and param which are reference classes as well

# Log-likelihood:
mhmmr$stat$loglik

# Parameters of the polynomial regressions:
mhmmr$param$beta

A Reference Class which represents a fitted MRHLP model.

Description

ModelMRHLP represents an estimated MRHLP model.

Fields

param: A ParamMRHLP object. It contains the estimated values of the parameters.
stat: A StatMRHLP object. It contains all the statistics associated to the MRHLP model.

Methods

plot(what = c("regressors", "estimatedsignal", "loglikelihood"), ...)

Plot method.

what

The type of graph requested:

"regressors" = Polynomial regression components (fields polynomials and pi_ik of class StatMRHLP).
"estimatedsignal" = Estimated signal (fields Ex and klas of class StatMRHLP).
"loglikelihood" = Value of the log-likelihood for each iteration (field stored_loglik of class StatMRHLP).

...

Other graphics parameters.

By default, all the graphs mentioned above are produced.

summary(digits = getOption("digits"))

Summary method.

digits: The number of significant digits to use when printing.

Examples

data(multivtoydataset)

mrhlp <- emMRHLP(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

# mrhlp is a ModelMRHLP object. It contains some methods such as 'summary' and 'plot'
mrhlp$summary()
mrhlp$plot()

# mrhlp has also two fields, stat and param which are reference classes as well

# Log-likelihood:
mrhlp$stat$loglik

# Parameters of the polynomial regressions:
mrhlp$param$beta
data(multivtoydataset)

mrhlp <- emMRHLP(multivtoydataset$x, multivtoydataset[,c("y1", "y2", "y3")],
                 K = 5, p = 1, verbose = TRUE)

# mrhlp is a ModelMRHLP object. It contains some methods such as 'summary' and 'plot'
mrhlp$summary()
mrhlp$plot()

# mrhlp has also two fields, stat and param which are reference classes as well

# Log-likelihood:
mrhlp$stat$loglik

# Parameters of the polynomial regressions:
mrhlp$param$beta

A Reference Class which represents a fitted PWR model.

Description

ModelPWR represents an estimated PWR model.

Fields

param: A ParamPWR object. It contains the estimated values of the parameters.
stat: A StatPWR object. It contains all the statistics associated to the PWR model.

Methods

plot(what = c("regressors", "segmentation"), ...)

Plot method.

what

The type of graph requested:

"regressors" = Polynomial regression components (field regressors of class StatPWR).
"segmentation" = Estimated signal (field mean_function of class StatPWR).

...

Other graphics parameters.

By default, all the graphs mentioned above are produced.

summary(digits = getOption("digits"))

Summary method.

digits: The number of significant digits to use when printing.

Examples

data(univtoydataset)

pwr <- fitPWRFisher(univtoydataset$x, univtoydataset$y, K = 5, p = 1)

# pwr is a ModelPWR object. It contains some methods such as 'summary' and 'plot'
pwr$summary()
pwr$plot()

# pwr has also two fields, stat and param which are reference classes as well

# Value of the objective function:
pwr$stat$objective

# Parameters of the polynomial regressions:
pwr$param$beta
data(univtoydataset)

pwr <- fitPWRFisher(univtoydataset$x, univtoydataset$y, K = 5, p = 1)

# pwr is a ModelPWR object. It contains some methods such as 'summary' and 'plot'
pwr$summary()
pwr$plot()

# pwr has also two fields, stat and param which are reference classes as well

# Value of the objective function:
pwr$stat$objective

# Parameters of the polynomial regressions:
pwr$param$beta

A Reference Class which represents a fitted RHLP model.

Description

ModelRHLP represents an estimated RHLP model.

Fields

param: A ParamRHLP object. It contains the estimated values of the parameters.
stat: A StatRHLP object. It contains all the statistics associated to the RHLP model.

Methods

plot(what = c("regressors", "estimatedsignal", "loglikelihood"), ...)

Plot method.

what

The type of graph requested:

"regressors" = Polynomial regression components (fields polynomials and pi_ik of class StatRHLP).
"estimatedsignal" = Estimated signal (fields Ex and klas of class StatRHLP).
"loglikelihood" = Value of the log-likelihood for each iteration (field stored_loglik of class StatRHLP).

...

Other graphics parameters.

By default, all the graphs mentioned above are produced.

summary(digits = getOption("digits"))

Summary method.

digits: The number of significant digits to use when printing.

Examples

data(univtoydataset)

rhlp <- emRHLP(univtoydataset$x, univtoydataset$y, K = 3, p = 1, verbose = TRUE)

# rhlp is a ModelMHMMR object. It contains some methods such as 'summary' and 'plot'
rhlp$summary()
rhlp$plot()

# rhlp has also two fields, stat and param which are reference classes as well

# Log-likelihood:
rhlp$stat$loglik

# Parameters of the polynomial regressions:
rhlp$param$beta
data(univtoydataset)

rhlp <- emRHLP(univtoydataset$x, univtoydataset$y, K = 3, p = 1, verbose = TRUE)

# rhlp is a ModelMHMMR object. It contains some methods such as 'summary' and 'plot'
rhlp$summary()
rhlp$plot()

# rhlp has also two fields, stat and param which are reference classes as well

# Log-likelihood:
rhlp$stat$loglik

# Parameters of the polynomial regressions:
rhlp$param$beta

Time series representing the three acceleration components recorded over time with body mounted accelerometers during the activity of a given person.

Description

This dataset is provided for illustration only and represents the three acceleration components recorded over time with body mounted accelerometers during the activity of a given person. These data consist therefore of multidimensional time series with several regime changes over time, each regime is associated with an activity.

Usage

multivrealdataset
multivrealdataset

Format

A data frame with 2253 rows and 4 columns:

x: The covariate variable (the sampling time).
y1: X axis component of the acceleration.
y2: Y axis component of the acceleration.
y3: Z axis component of the acceleration.

A simulated non-stationary multidimensional time series with regime changes.

Description

A simulated non-stationary multidimensional time series with five regimes (segments). This time series is used for illustration.

Usage

multivtoydataset
multivtoydataset

Format

A data frame with 670 rows and 4 columns:

x

The covariate variable (the sampling time for time series).

y1

The first dimension of the time series. The latter has been generated as follows:

First regime: 100 values of standard Normally distributed random numbers.
Second regime: 120 values of Normally distributed random numbers with mean 7 and unit variance.
Third regime: 200 values of Normally distributed random numbers with mean 4 and unit variance.
Fourth regime: 100 values of Normally distributed random numbers with mean -1 and unit variance.
Fifth regime: 150 values of Normally distributed random numbers with mean 3.5 and unit variance.

y2

The second dimension of the time series. The latter has been generated as follows:

First regime: 100 values of Normally distributed random numbers with mean 1 and unit variance.
Second regime: 120 values of Normally distributed random numbers with mean 5 and unit variance.
Third regime: 200 values of Normally distributed random numbers with mean 6 and unit variance.
Fourth regime: 100 values of Normally distributed random numbers with mean -2 and unit variance.
Fifth regime: 150 values of Normally distributed random numbers with mean 2 and unit variance.

y3

The third dimension of the time series. The latter has been generated as follows:

First regime: 100 values of Normally distributed random numbers with mean -2 and unit variance.
Second regime: 120 values of Normally distributed random numbers with mean 10 and unit variance.
Third regime: 200 values of Normally distributed random numbers with mean 8 and unit variance.
Fourth regime: 100 values of Normally distributed random numbers and unit variance.
Fifth regime: 150 values of Normally distributed random numbers with mean 5 and unit variance.

A Reference Class which contains parameters of a HMMR model.

Description

ParamHMMR contains all the parameters of a HMMR model. The parameters are calculated by the initialization Method and then updated by the Method implementing the M-Step of the EM algorithm.

Fields

X: Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
Y: Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
m: Numeric. Length of the response/output vector Y.
K: The number of regimes (HMMR components).
p: The order of the polynomial regression.
variance_type: Character indicating if the model is homoskedastic (variance_type = "homoskedastic") or heteroskedastic (variance_type = "heteroskedastic"). By default the model is heteroskedastic.
prior: The prior probabilities of the Markov chain. prior is a row matrix of dimension $(1, K)$ .
trans_mat: The transition matrix of the Markov chain. trans_mat is a matrix of dimension $(K, K)$ .
mask: Mask applied to the transition matrices trans_mat. By default, a mask of order one is applied.
beta: Parameters of the polynomial regressions. $\boldsymbol{\beta} = (\boldsymbol{\beta}_{1},\dots,\boldsymbol{\beta}_{K})$ is a matrix of dimension $(p + 1, K)$ , with p the order of the polynomial regression. p is fixed to 3 by default.
sigma2: The variances for the K regimes. If HMMR model is heteroskedastic (variance_type = "heteroskedastic") then sigma2 is a matrix of size $(K, 1)$ (otherwise HMMR model is homoskedastic (variance_type = "homoskedastic") and sigma2 is a matrix of size $(1, 1)$ ).
nu: The degree of freedom of the HMMR model representing the complexity of the model.
phi: A list giving the regression design matrices for the polynomial and the logistic regressions.

Methods

initParam(try_algo = 1)

Method to initialize parameters mask, prior, trans_mat, beta and sigma2.

If try_algo = 1 then beta and sigma2 are initialized by segmenting the time series Y uniformly into K contiguous segments. Otherwise, beta and sigma2 are initialized by segmenting randomly the time series Y into K segments.

MStep(statHMMR)

Method which implements the M-step of the EM algorithm to learn the parameters of the HMMR model based on statistics provided by the object statHMMR of class StatHMMR (which contains the E-step).

A Reference Class which contains parameters of a MHMMR model.

Description

ParamMHMMR contains all the parameters of a MHMMR model. The parameters are calculated by the initialization Method and then updated by the Method implementing the M-Step of the EM algorithm.

Fields

mData: MData object representing the sample (covariates/inputs X and observed multivariate responses/outputs Y).
K: The number of regimes (MHMMR components).
p: The order of the polynomial regression.
variance_type: Character indicating if the model is homoskedastic (variance_type = "homoskedastic") or heteroskedastic (variance_type = "heteroskedastic"). By default the model is heteroskedastic.
prior: The prior probabilities of the Markov chain. prior is a row matrix of dimension $(1, K)$ .
trans_mat: The transition matrix of the Markov chain. trans_mat is a matrix of dimension $(K, K)$ .
mask: Mask applied to the transition matrices trans_mat. By default, a mask of order one is applied.
beta: Parameters of the polynomial regressions. $\boldsymbol{\beta} = (\boldsymbol{\beta}_{1},\dots,\boldsymbol{\beta}_{K})$ is an array of dimension $(p + 1, d, K)$ , with p the order of the polynomial regression. p is fixed to 3 by default.
sigma2: The variances for the K regimes. If MRHLP model is heteroskedastic (variance_type = "heteroskedastic") then sigma2 is an array of size $(d, d, K)$ (otherwise MRHLP model is homoskedastic (variance_type = "homoskedastic") and sigma2 is a matrix of size $(d, d)$ ).
nu: The degree of freedom of the MHMMR model representing the complexity of the model.
phi: A list giving the regression design matrices for the polynomial and the logistic regressions.

Methods

initParam(try_algo = 1)

Method to initialize parameters prior, trans_mat, beta and sigma2.

MStep(statMHMMR)

Method which implements the M-step of the EM algorithm to learn the parameters of the MHMMR model based on statistics provided by the object statMHMMR of class StatMHMMR (which contains the E-step).

A Reference Class which contains the parameters of a MRHLP model.

Description

ParamMRHLP contains all the parameters of a MRHLP model. The parameters are calculated by the initialization Method and then updated by the Method implementing the M-Step of the EM algorithm.

Fields

mData: MData object representing the sample (covariates/inputs X and observed responses/outputs Y).
K: The number of regimes (MRHLP components).
p: The order of the polynomial regression.
q: The dimension of the logistic regression. For the purpose of segmentation, it must be set to 1.
variance_type: Character indicating if the model is homoskedastic (variance_type = "homoskedastic") or heteroskedastic (variance_type = "heteroskedastic"). By default the model is heteroskedastic.
W: Parameters of the logistic process. $\boldsymbol{W} = (\boldsymbol{w}_{1},\dots,\boldsymbol{w}_{K-1})$ is a matrix of dimension $(q + 1, K - 1)$ , with q the order of the logistic regression. q is fixed to 1 by default.
beta: Parameters of the polynomial regressions. $\boldsymbol{\beta} = (\boldsymbol{\beta}_{1},\dots,\boldsymbol{\beta}_{K})$ is an array of dimension $(p + 1, d, K)$ , with p the order of the polynomial regression. p is fixed to 3 by default.
sigma2: The variances for the K regimes. If MRHLP model is heteroskedastic (variance_type = "heteroskedastic") then sigma2 is an array of size $(d, d, K)$ (otherwise MRHLP model is homoskedastic (variance_type = "homoskedastic") and sigma2 is a matrix of size $(d, d)$ ).
nu: The degree of freedom of the MRHLP model representing the complexity of the model.
phi: A list giving the regression design matrices for the polynomial and the logistic regressions.

Methods

initParam(try_algo = 1)

Method to initialize parameters W, beta and sigma2.

If try_algo = 1 then beta and sigma2 are initialized by segmenting the time series Y uniformly into K contiguous segments. Otherwise, W, beta and sigma2 are initialized by segmenting randomly the time series Y into K segments.

MStep(statMRHLP, verbose_IRLS)

Method which implements the M-step of the EM algorithm to learn the parameters of the MRHLP model based on statistics provided by the object statMRHLP of class StatMRHLP (which contains the E-step).

A Reference Class which contains the parameters of a PWR model.

Description

ParamPWR contains all the parameters of a PWR model. The parameters are calculated by the initialization Method and then updated by the Method dynamic programming (here dynamic programming)

Fields

X: Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
Y: Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
m: Numeric. Length of the response/output vector Y.
K: The number of regimes (PWR components).
p: The order of the polynomial regression. p is fixed to 3 by default.
gamma: Set of transition points. gamma is a column matrix of size $(K + 1, 1)$ .
beta: Parameters of the polynomial regressions. beta is a matrix of dimension $(p + 1, K)$ , with p the order of the polynomial regression. p is fixed to 3 by default.
sigma2: The variances for the K regimes. sigma2 is a matrix of size $(K, 1)$ .
phi: A list giving the regression design matrices for the polynomial and the logistic regressions.

Methods

computeDynamicProgram(C1, K): Method which implements the dynamic programming based on the cost matrix C1 and the number of regimes/segments K.
computeParam(): Method which estimates the parameters beta and sigma2 knowing the transition points gamma.

A Reference Class which contains parameters of a RHLP model.

Description

ParamRHLP contains all the parameters of a RHLP model. The parameters are calculated by the initialization Method and then updated by the Method implementing the M-Step of the EM algorithm.

Fields

X: Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
Y: Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
m: Numeric. Length of the response/output vector Y.
K: The number of regimes (RHLP components).
p: The order of the polynomial regression.
q: The dimension of the logistic regression. For the purpose of segmentation, it must be set to 1.
variance_type: Character indicating if the model is homoskedastic (variance_type = "homoskedastic") or heteroskedastic (variance_type = "heteroskedastic"). By default the model is heteroskedastic.
W: Parameters of the logistic process. $\boldsymbol{W} = (\boldsymbol{w}_{1},\dots,\boldsymbol{w}_{K-1})$ is a matrix of dimension $(q + 1, K - 1)$ , with q the order of the logistic regression. q is fixed to 1 by default.
beta: Parameters of the polynomial regressions. $\boldsymbol{\beta} = (\boldsymbol{\beta}_{1},\dots,\boldsymbol{\beta}_{K})$ is a matrix of dimension $(p + 1, K)$ , with p the order of the polynomial regression. p is fixed to 3 by default.
sigma2: The variances for the K regimes. If RHLP model is heteroskedastic (variance_type = "heteroskedastic") then sigma2 is a matrix of size $(K, 1)$ (otherwise RHLP model is homoskedastic (variance_type = "homoskedastic") and sigma2 is a matrix of size $(1, 1)$ ).
nu: The degree of freedom of the RHLP model representing the complexity of the model.
phi: A list giving the regression design matrices for the polynomial and the logistic regressions.

Methods

initParam(try_algo = 1)

Method to initialize parameters W, beta and sigma2.

MStep(statRHLP, verbose_IRLS)

Method which implements the M-step of the EM algorithm to learn the parameters of the RHLP model based on statistics provided by the object statRHLP of class StatRHLP (which contains the E-step).

selectHMMR implements a model selection procedure to select an optimal HMMR model with unknown structure.

Description

selectHMMR implements a model selection procedure to select an optimal HMMR model with unknown structure.

Usage

selectHMMR(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)
selectHMMR(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
`Kmin`	The minimum number of regimes (HMMR components).
`Kmax`	The maximum number of regimes (HMMR components).
`pmin`	The minimum order of the polynomial regression.
`pmax`	The maximum order of the polynomial regression.
`criterion`	The criterion used to select the HMMR model ("BIC", "AIC").
`verbose`	Optional. A logical value indicating whether or not a summary of the selected model should be displayed.

Details

selectHMMR selects the optimal HMMR model among a set of model candidates by optimizing a model selection criteria, including the Bayesian Information Criterion (BIC). This function first fits the different HMMR model candidates by varying the number of regimes K from Kmin to Kmax and the order of the polynomial regression p from pmin to pmax. The model having the highest value of the chosen selection criterion is then selected.

Value

selectHMMR returns an object of class ModelHMMR representing the selected HMMR model according to the chosen criterion.

Examples

data(univtoydataset)

selectedhmmr <- selectHMMR(X = univtoydataset$x, Y = univtoydataset$y,
                           Kmin = 2, Kmax = 6, pmin = 0, pmax = 2)

selectedhmmr$plot()
data(univtoydataset)

selectedhmmr <- selectHMMR(X = univtoydataset$x, Y = univtoydataset$y,
                           Kmin = 2, Kmax = 6, pmin = 0, pmax = 2)

selectedhmmr$plot()

selectMHMMR implements a model selection procedure to select an optimal MHMMR model with unknown structure.

Description

selectMHMMR implements a model selection procedure to select an optimal MHMMR model with unknown structure.

Usage

selectMHMMR(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)
selectMHMMR(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Matrix of size $(m, d)$ representing a $d$ dimension time series observed at points $1,\dots,m$ .
`Kmin`	The minimum number of regimes (c components).
`Kmax`	The maximum number of regimes (MHMMR components).
`pmin`	The minimum order of the polynomial regression.
`pmax`	The maximum order of the polynomial regression.
`criterion`	The criterion used to select the MHMMR model ("BIC", "AIC").
`verbose`	Optional. A logical value indicating whether or not a summary of the selected model should be displayed.

Details

selectMHMMR selects the optimal MHMMR model among a set of model candidates by optimizing a model selection criteria, including the Bayesian Information Criterion (BIC). This function first fits the different MHMMR model candidates by varying the number of regimes K from Kmin to Kmax and the order of the polynomial regression p from pmin to pmax. The model having the highest value of the chosen selection criterion is then selected.

Value

selectMHMMR returns an object of class ModelMHMMR representing the selected MHMMR model according to the chosen criterion.

Examples

data(multivtoydataset)
x <- multivtoydataset$x
y <- multivtoydataset[, c("y1", "y2", "y3")]

selectedmhmmr <- selectMHMMR(X = x, Y = y, Kmin = 2, Kmax = 6,
                             pmin = 0, pmax = 2)

selectedmhmmr$summary()
data(multivtoydataset)
x <- multivtoydataset$x
y <- multivtoydataset[, c("y1", "y2", "y3")]

selectedmhmmr <- selectMHMMR(X = x, Y = y, Kmin = 2, Kmax = 6,
                             pmin = 0, pmax = 2)

selectedmhmmr$summary()

selecMRHLP implements a model selection procedure to select an optimal MRHLP model with unknown structure.

Description

selecMRHLP implements a model selection procedure to select an optimal MRHLP model with unknown structure.

Usage

selectMRHLP(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)
selectMRHLP(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Matrix of size $(m, d)$ representing a $d$ dimension function of `X` observed at points $1,\dots,m$ . `Y` is the observed response/output.
`Kmin`	The minimum number of regimes (MRHLP components).
`Kmax`	The maximum number of regimes (MRHLP components).
`pmin`	The minimum order of the polynomial regression.
`pmax`	The maximum order of the polynomial regression.
`criterion`	The criterion used to select the MRHLP model ("BIC", "AIC").
`verbose`	Optional. A logical value indicating whether or not a summary of the selected model should be displayed.

Details

selectMRHLP selects the optimal MRHLP model among a set of model candidates by optimizing a model selection criteria, including the Bayesian Information Criterion (BIC). This function first fits the different MRHLP model candidates by varying the number of regimes K from Kmin to Kmax and the order of the polynomial regression p from pmin to pmax. The model having the highest value of the chosen selection criterion is then selected.

Value

selectMRHLP returns an object of class ModelMRHLP representing the selected MRHLP model according to the chosen criterion.

Examples

data(multivtoydataset)

# Let's select a MRHLP model on a multivariate time series with 3 regimes:
data <- multivtoydataset[1:320, ]
x <- data$x
y <- data[, c("y1", "y2", "y3")]

selectedmrhlp <- selectMRHLP(X = x, Y = y, Kmin = 2, Kmax = 4,
                             pmin = 0, pmax = 1)

selectedmrhlp$summary()
data(multivtoydataset)

# Let's select a MRHLP model on a multivariate time series with 3 regimes:
data <- multivtoydataset[1:320, ]
x <- data$x
y <- data[, c("y1", "y2", "y3")]

selectedmrhlp <- selectMRHLP(X = x, Y = y, Kmin = 2, Kmax = 4,
                             pmin = 0, pmax = 1)

selectedmrhlp$summary()

selecRHLP implements a model selection procedure to select an optimal RHLP model with unknown structure.

Description

selecRHLP implements a model selection procedure to select an optimal RHLP model with unknown structure.

Usage

selectRHLP(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)
selectRHLP(X, Y, Kmin = 1, Kmax = 10, pmin = 0, pmax = 4,
  criterion = c("BIC", "AIC"), verbose = TRUE)

Arguments

`X`	Numeric vector of length m representing the covariates/inputs $x_{1},\dots,x_{m}$ .
`Y`	Numeric vector of length m representing the observed response/output $y_{1},\dots,y_{m}$ .
`Kmin`	The minimum number of regimes (RHLP components).
`Kmax`	The maximum number of regimes (RHLP components).
`pmin`	The minimum order of the polynomial regression.
`pmax`	The maximum order of the polynomial regression.
`criterion`	The criterion used to select the RHLP model ("BIC", "AIC").
`verbose`	Optional. A logical value indicating whether or not a summary of the selected model should be displayed.

Details

selectRHLP selects the optimal MRHLP model among a set of model candidates by optimizing a model selection criteria, including the Bayesian Information Criterion (BIC). This function first fits the different RHLP model candidates by varying the number of regimes K from Kmin to Kmax and the order of the polynomial regression p from pmin to pmax. The model having the highest value of the chosen selection criterion is then selected.

Value

selectRHLP returns an object of class ModelRHLP representing the selected RHLP model according to the chosen criterion.

Examples

data(univtoydataset)

# Let's select a RHLP model on a time series with 3 regimes:
data <- univtoydataset[1:320,]

selectedrhlp <- selectRHLP(X = data$x, Y = data$y,
                           Kmin = 2, Kmax = 4, pmin = 0, pmax = 1)

selectedrhlp$summary()
data(univtoydataset)

# Let's select a RHLP model on a time series with 3 regimes:
data <- univtoydataset[1:320,]

selectedrhlp <- selectRHLP(X = data$x, Y = data$y,
                           Kmin = 2, Kmax = 4, pmin = 0, pmax = 1)

selectedrhlp$summary()

A Reference Class which contains statistics of a HMMR model.

Description

StatHMMR contains all the statistics associated to a HMMR model. It mainly includes the E-Step of the EM algorithm calculating the posterior distribution of the hidden variables (ie the smoothing probabilities), as well as the calculation of the prediction and filtering probabilities, the log-likelhood at each step of the algorithm and the obtained values of model selection criteria..

Fields

tau_tk: Matrix of size $(m, K)$ giving the posterior probability that the observation $Y_{i}$ originates from the $k$ -th regression model.
alpha_tk: Matrix of size $(m, K)$ giving the forwards probabilities: $P(Y_{1},\dots,Y_{t}, z_{t} = k)$ .
beta_tk: Matrix of size $(m, K)$ , giving the backwards probabilities: $P(Y_{t+1},\dots,Y_{m} | z_{t} = k)$ .
xi_tkl: Array of size $(m - 1, K, K)$ giving the joint post probabilities: $xi_tk[t, k, l] = P(z_{t} = k, z_{t-1} = l | \boldsymbol{Y})$ for $t = 2,\dots,m$ .
f_tk: Matrix of size $(m, K)$ giving the cumulative distribution function $f(y_{t} | z{_t} = k)$ .
log_f_tk: Matrix of size $(m, K)$ giving the logarithm of the cumulative distribution f_tk.
loglik: Numeric. Log-likelihood of the HMMR model.
stored_loglik: Numeric vector. Stored values of the log-likelihood at each iteration of the EM algorithm.
klas: Column matrix of the labels issued from z_ik. Its elements are $klas(i) = k$ , $k = 1,\dots,K$ .
z_ik: Hard segmentation logical matrix of dimension $(m, K)$ obtained by the Maximum a posteriori (MAP) rule: $z\_ik = 1 \ \textrm{if} \ z\_ik = \textrm{arg} \ \textrm{max}_{s} \ P(z_{i} = s | \boldsymbol{Y}) = tau\_tk;\ 0 \ \textrm{otherwise}$ , $k = 1,\dots,K$ .
state_probs: Matrix of size $(m, K)$ giving the distribution of the Markov chain. $P(z_{1},\dots,z_{m};\pi,\boldsymbol{A})$ with $\pi$ the prior probabilities (field prior of the class ParamHMMR) and $\boldsymbol{A}$ the transition matrix (field trans_mat of the class ParamHMMR) of the Markov chain.
BIC: Numeric. Value of BIC (Bayesian Information Criterion).
AIC: Numeric. Value of AIC (Akaike Information Criterion).
regressors: Matrix of size $(m, K)$ giving the values of the estimated polynomial regression components.
predict_prob: Matrix of size $(m, K)$ giving the prediction probabilities: $P(z_{t} = k | y_{1},\dots,y_{t-1})$ .
predicted: Row matrix of size $(m, 1)$ giving the sum of the polynomial components weighted by the prediction probabilities predict_prob.
filter_prob: Matrix of size $(m, K)$ giving the filtering probabilities $Pr(z_{t} = k | y_{1},\dots,y_{t})$ .
filtered: Row matrix of size $(m, 1)$ giving the sum of the polynomial components weighted by the filtering probabilities.
smoothed_regressors: Matrix of size $(m, K)$ giving the polynomial components weighted by the posterior probability tau_tk.
smoothed: Row matrix of size $(m, 1)$ giving the sum of the polynomial components weighted by the posterior probability tau_tk.

Methods

computeLikelihood(paramHMMR)

Method to compute the log-likelihood based on some parameters given by the object paramHMMR of class ParamHMMR.

computeStats(paramHMMR)

Method used in the EM algorithm to compute statistics based on parameters provided by the object paramHMMR of class ParamHMMR.

EStep(paramHMMR)

Method used in the EM algorithm to update statistics based on parameters provided by the object paramHMMR of class ParamHMMR (prior and posterior probabilities).

MAP()

MAP calculates values of the fields z_ik and klas by applying the Maximum A Posteriori Bayes allocation rule.

$z\_ik = 1 \ \textrm{if} \ z\_ik = \textrm{arg} \ \textrm{max}_{s} \ P(z_{i} = s | \boldsymbol{Y}) = tau\_tk;\ 0 \ \textrm{otherwise}$

A Reference Class which contains statistics of a MHMMR model.

Description

StatMHMMR contains all the statistics associated to a MHMMR model. It mainly includes the E-Step of the EM algorithm calculating the posterior distribution of the hidden variables (ie the smoothing probabilities), as well as the calculation of the prediction and filtering probabilities, the log-likelhood at each step of the algorithm and the obtained values of model selection criteria..

Fields

tau_tk: Matrix of size $(m, K)$ giving the posterior probability that the observation $Y_{i}$ originates from the $k$ -th regression model.
alpha_tk: Matrix of size $(m, K)$ giving the forwards probabilities: $P(Y_{1},\dots,Y_{t}, z_{t} = k)$ .
beta_tk: Matrix of size $(m, K)$ , giving the backwards probabilities: $P(Y_{t+1},\dots,Y_{m} | z_{t} = k)$ .
xi_tkl: Array of size $(m - 1, K, K)$ giving the joint post probabilities: $xi_tk[t, k, l] = P(z_{t} = k, z_{t-1} = l | \boldsymbol{Y})$ for $t = 2,\dots,m$ .
f_tk: Matrix of size $(m, K)$ giving the cumulative distribution function $f(y_{t} | z{_t} = k)$ .
log_f_tk: Matrix of size $(m, K)$ giving the logarithm of the cumulative distribution f_tk.
loglik: Numeric. Log-likelihood of the MHMMR model.
stored_loglik: Numeric vector. Stored values of the log-likelihood at each iteration of the EM algorithm.
klas: Column matrix of the labels issued from z_ik. Its elements are $klas(i) = k$ , $k = 1,\dots,K$ .
z_ik: Hard segmentation logical matrix of dimension $(m, K)$ obtained by the Maximum a posteriori (MAP) rule: $z\_ik = 1 \ \textrm{if} \ z\_ik = \textrm{arg} \ \textrm{max}_{s} \ P(z_{i} = s | \boldsymbol{Y}) = tau\_tk;\ 0 \ \textrm{otherwise}$ , $k = 1,\dots,K$ .
state_probs: Matrix of size $(m, K)$ giving the distribution of the Markov chain. $P(z_{1},\dots,z_{m};\pi,\boldsymbol{A})$ with $\pi$ the prior probabilities (field prior of the class ParamMHMMR) and $\boldsymbol{A}$ the transition matrix (field trans_mat of the class ParamMHMMR) of the Markov chain.
BIC: Numeric. Value of BIC (Bayesian Information Criterion).
AIC: Numeric. Value of AIC (Akaike Information Criterion).
regressors: Matrix of size $(m, K)$ giving the values of the estimated polynomial regression components.
predict_prob: Matrix of size $(m, K)$ giving the prediction probabilities: $P(z_{t} = k | y_{1},\dots,y_{t-1})$ .
predicted: Row matrix of size $(m, 1)$ giving the sum of the polynomial components weighted by the prediction probabilities predict_prob.
filter_prob: Matrix of size $(m, K)$ giving the filtering probabilities $Pr(z_{t} = k | y_{1},\dots,y_{t})$ .
filtered: Row matrix of size $(m, 1)$ giving the sum of the polynomial components weighted by the filtering probabilities.
smoothed_regressors: Matrix of size $(m, K)$ giving the polynomial components weighted by the posterior probability tau_tk.
smoothed: Row matrix of size $(m, 1)$ giving the sum of the polynomial components weighted by the posterior probability tau_tk.

Methods

computeLikelihood(paramMHMMR)

Method to compute the log-likelihood based on some parameters given by the object paramMHMMR of class ParamMHMMR.

computeStats(paramMHMMR)

Method used in the EM algorithm to compute statistics based on parameters provided by the object paramMHMMR of class ParamMHMMR.

EStep(paramMHMMR)

Method used in the EM algorithm to update statistics based on parameters provided by the object paramMHMMR of class ParamMHMMR (prior and posterior probabilities).

MAP()

MAP calculates values of the fields z_ik and klas by applying the Maximum A Posteriori Bayes allocation rule.

$z\_ik = 1 \ \textrm{if} \ z\_ik = \textrm{arg} \ \textrm{max}_{s} \ P(z_{i} = s | \boldsymbol{Y}) = tau\_tk;\ 0 \ \textrm{otherwise}$

A Reference Class which contains statistics of a MRHLP model.

Description

StatMRHLP contains all the statistics associated to a MRHLP model. It mainly includes the E-Step of the EM algorithm calculating the posterior distribution of the hidden variables, as well as the calculation of the log-likelhood at each step of the algorithm and the obtained values of model selection criteria..

Fields

pi_ik: Matrix of size $(m, K)$ representing the prior/logistic probabilities $\pi_{k}(x_{i}; \boldsymbol{\Psi}) = P(z_{i} = k | \boldsymbol{x}; \Psi)$ of the latent variable $z_{i}, i = 1,\dots,m$ .
z_ik: Hard segmentation logical matrix of dimension $(m, K)$ obtained by the Maximum a posteriori (MAP) rule: $z\_ik = 1 \ \textrm{if} \ z\_ik = \textrm{arg} \ \textrm{max}_{s} \ \pi_{s}(x_{i}; \boldsymbol{\Psi});\ 0 \ \textrm{otherwise}$ , $k = 1,\dots,K$ .
klas: Column matrix of the labels issued from z_ik. Its elements are $klas(i) = k$ , $k = 1,\dots,K$ .
tau_ik: Matrix of size $(m, K)$ giving the posterior probability that the observation $Y_{i}$ originates from the $k$ -th regression model.
polynomials: Array of size $(m, d, K)$ giving the values of the estimated polynomial regression components.
weighted_polynomials: Array of size $(m, d, K)$ giving the values of the estimated polynomial regression components weighted by the prior probabilities pi_ik.
Ex: Matrix of size (m, d). Ex is the curve expectation (estimated signal): sum of the polynomial components weighted by the logistic probabilities pi_ik.
loglik: Numeric. Observed-data log-likelihood of the MRHLP model.
com_loglik: Numeric. Complete-data log-likelihood of the MRHLP model.
stored_loglik: Numeric vector. Stored values of the log-likelihood at each EM iteration.
stored_com_loglik: Numeric vector. Stored values of the Complete log-likelihood at each EM iteration.
BIC: Numeric. Value of BIC (Bayesian Information Criterion).
ICL: Numeric. Value of ICL (Integrated Completed Likelihood).
AIC: Numeric. Value of AIC (Akaike Information Criterion).
log_piik_fik: Matrix of size $(m, K)$ giving the values of the logarithm of the joint probability $P(y_{i}, \ z_{i} = k | \boldsymbol{x}, \boldsymbol{\Psi})$ , $i = 1,\dots,m$ .
log_sum_piik_fik: Column matrix of size m giving the values of $\textrm{log} \sum_{k = 1}^{K} P(y_{i}, \ z_{i} = k | \boldsymbol{x}, \boldsymbol{\Psi})$ , $i = 1,\dots,m$ .

Methods

computeLikelihood(reg_irls)

Method to compute the log-likelihood. reg_irls is the value of the regularization part in the IRLS algorithm.

computeStats(paramMRHLP)

Method used in the EM algorithm to compute statistics based on parameters provided by the object paramMRHLP of class ParamMRHLP.

EStep(paramMRHLP)

Method used in the EM algorithm to update statistics based on parameters provided by the object paramMRHLP of class ParamMRHLP (prior and posterior probabilities).

MAP()

MAP calculates values of the fields z_ik and klas by applying the Maximum A Posteriori Bayes allocation rule.

$z_{ik} = 1 \ \textrm{if} \ k = \textrm{arg} \ \textrm{max}_{s} \ \pi_{s}(x_{i}; \boldsymbol{\Psi});\ 0 \ \textrm{otherwise}$

A Reference Class which contains statistics of a PWR model.

Description

StatPWR contains all the statistics associated to a PWR model.

Fields

z_ik: Logical matrix of dimension $(m, K)$ giving the class vector.
klas: Column matrix of the labels issued from z_ik. Its elements are $klas(i) = k$ , $k = 1,\dots,K$ .
mean_function: Approximation of the time series given the estimated parameters. mean_function is a matrix of size $(m, 1)$ .
regressors: Matrix of size $(m, K)$ giving the values of the estimated polynomial regression components.
objective: Numeric. Value of the objective function.

Methods

computeStats(paramPWR): Method used at the end of the dynamic programming algorithm to compute statistics based on parameters provided by paramPWR.

A Reference Class which contains statistics of a RHLP model.

Description

StatRHLP contains all the statistics associated to a RHLP model. It mainly includes the E-Step of the EM algorithm calculating the posterior distribution of the hidden variables, as well as the calculation of the log-likelhood at each step of the algorithm and the obtained values of model selection criteria..

Fields

pi_ik: Matrix of size $(m, K)$ representing the prior/logistic probabilities $\pi_{k}(x_{i}; \boldsymbol{\Psi}) = P(z_{i} = k | \boldsymbol{x}; \Psi)$ of the latent variable $z_{i}, i = 1,\dots,m$ .
z_ik: Hard segmentation logical matrix of dimension $(m, K)$ obtained by the Maximum a posteriori (MAP) rule: $z\_ik = 1 \ \textrm{if} \ z\_ik = \textrm{arg} \ \textrm{max}_{s} \ \pi_{s}(x_{i}; \boldsymbol{\Psi});\ 0 \ \textrm{otherwise}$ , $k = 1,\dots,K$ .
klas: Column matrix of the labels issued from z_ik. Its elements are $klas(i) = k$ , $k = 1,\dots,K$ .
tau_ik: Matrix of size $(m, K)$ giving the posterior probability that the observation $Y_{i}$ originates from the $k$ -th regression model.
polynomials: Matrix of size $(m, K)$ giving the values of the estimated polynomial regression components.
Ex: Column matrix of dimension m. Ex is the curve expectation (estimated signal): sum of the polynomial components weighted by the logistic probabilities pi_ik.
loglik: Numeric. Observed-data log-likelihood of the RHLP model.
com_loglik: Numeric. Complete-data log-likelihood of the RHLP model.
stored_loglik: Numeric vector. Stored values of the log-likelihood at each EM iteration.
stored_com_loglik: Numeric vector. Stored values of the Complete log-likelihood at each EM iteration.
BIC: Numeric. Value of BIC (Bayesian Information Criterion).
ICL: Numeric. Value of ICL (Integrated Completed Likelihood).
AIC: Numeric. Value of AIC (Akaike Information Criterion).
log_piik_fik: Matrix of size $(m, K)$ giving the values of the logarithm of the joint probability $P(y_{i}, \ z_{i} = k | \boldsymbol{x}, \boldsymbol{\Psi})$ , $i = 1,\dots,m$ .
log_sum_piik_fik: Column matrix of size m giving the values of $\textrm{log} \sum_{k = 1}^{K} P(y_{i}, \ z_{i} = k | \boldsymbol{x}, \boldsymbol{\Psi})$ , $i = 1,\dots,m$ .

Methods

computeLikelihood(reg_irls)

Method to compute the log-likelihood. reg_irls is the value of the regularization part in the IRLS algorithm.

computeStats(paramRHLP)

Method used in the EM algorithm to compute statistics based on parameters provided by the object paramRHLP of class ParamRHLP.

EStep(paramRHLP)

Method used in the EM algorithm to update statistics based on parameters provided by the object paramRHLP of class ParamRHLP (prior and posterior probabilities).

MAP()

MAP calculates values of the fields z_ik and klas by applying the Maximum A Posteriori Bayes allocation rule.

$z_{ik} = 1 \ \textrm{if} \ k = \textrm{arg} \ \textrm{max}_{s} \ \pi_{s}(x_{i}; \boldsymbol{\Psi});\ 0 \ \textrm{otherwise}$

Time series representing the electrical power consumption during a railway switch operation

Description

This dataset is provided for illustration only; It is issued from the switch railway monitoring domain. The switch mechanism enables trains to be guided from one track to another at a railway junction. During each switch operation, a set of measurements are recorded. Each measurement represents the consumed electrical power. The resulting time series present regime changes.

Usage

univrealdataset
univrealdataset

Format

A data frame with 562 rows and 3 columns:

x: The covariate variables which are the sampling time in this time-series case.
y1: Measurements of the electrical power consumed during time for a first example of switch operations.
y2: Measurements of the electrical power consumed during during time for another example of switch operations.

A simulated non-stationary time series with regime changes.

Description

A simulated non-stationary time series with regime changes. This time series is used for illustration.

Usage

univtoydataset
univtoydataset

Format

A data frame with 670 rows and 2 columns:

x

The covariate variable which is the time in this time-series case.

y

The time series. The latter has been generated as follows:

First regime: 100 values of standard Normally distributed random numbers.
Second regime: 120 values of Normally distributed random numbers with mean 7 and unit variance.
Third regime: 200 values of Normally distributed random numbers with mean 4 and unit variance.
Fourth regime: 100 values of Normally distributed random numbers with mean -2 and unit variance.
Fifth regime: 150 values of Normally distributed random numbers with mean 3.5 and unit variance.

Package 'samurais'

Help Index

SaMUraiS: StAtistical Models for the UnsupeRvised segmentAtIon of time-Series

Description

Author(s)

References

See Also

emHMMR implemens the EM (Baum-Welch) algorithm to fit a HMMR model.

Description

Usage

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Details

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Examples

emMHMMR implemens the EM (Baum-Welch) algorithm to fit a MHMMR model.

Description

Usage

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Details

Value

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Examples

emMRHLP implemens the EM algorithm to fit a MRHLP model.

Description

Usage

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Details

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Examples

emRHLP implements the EM algorithm to fit a RHLP model.

Description

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Value

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Examples

fitPWRFisher implements an optimized dynamic programming algorithm to fit a PWR model.

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Details

Value

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Examples

hmmProcess calculates the probability distribution of a random process following a Markov chain

Description

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Details

Value

A Reference Class which represents multivariate data.

Description

Fields

mkStochastic ensures that it is a stochastic vector, matrix or array.

Description

Usage

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Details

Value

A Reference Class which represents a fitted HMMR model.

Description

Fields

Methods

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Examples

A Reference Class which represents a fitted MHMMR model.

Description

Fields

Methods

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Examples

A Reference Class which represents a fitted MRHLP model.

Description

Fields

Methods

See Also

Examples