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Overview

Multiple Hidden Markov Model Regression (HMMR) for the segmentation of multivariate time series with regime changes.

The model assumes that the time series is governed by a sequence of hidden discrete regimes/states, where each regime/state has multivariate Gaussian regressors emission densities. The model parameters are estimated by MLE via the EM algorithm.

Installation

You can install the development version of MHMMR from GitHub with:

# install.packages("devtools")
devtools::install_github("fchamroukhi/MHMMR")

To build vignettes for examples of usage, type the command below instead:

# install.packages("devtools")
devtools::install_github("fchamroukhi/MHMMR", 
                         build_opts = c("--no-resave-data", "--no-manual"), 
                         build_vignettes = TRUE)

Use the following command to display vignettes:

browseVignettes("MHMMR")

Usage

library(MHMMR)
# Application to a simulated data set
data("toydataset")
x <- toydataset$x
y <- toydataset[, c("y1", "y2", "y3")]

K <- 5 # Number of regimes (states)
p <- 1 # Dimension of beta (order of the polynomial regressors)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model

n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE

mhmmr <- emMHMMR(X = x, Y = y, K, p, variance_type, n_tries, 
                 max_iter, threshold, verbose)
#> EM - MHMMR: Iteration: 1 | log-likelihood: -4539.37845473736
#> EM - MHMMR: Iteration: 2 | log-likelihood: -3075.7862970485
#> EM - MHMMR: Iteration: 3 | log-likelihood: -2904.71126233611
#> EM - MHMMR: Iteration: 4 | log-likelihood: -2883.23456594806
#> EM - MHMMR: Iteration: 5 | log-likelihood: -2883.12446634454
#> EM - MHMMR: Iteration: 6 | log-likelihood: -2883.12436399888

mhmmr$summary()
#> ----------------------
#> Fitted MHMMR model
#> ----------------------
#> 
#> MHMMR model with K = 5 regimes
#> 
#>  log-likelihood nu       AIC      BIC
#>       -2883.124 84 -2967.124 -3156.43
#> 
#> Clustering table:
#>   1   2   3   4   5 
#> 100 120 200 100 150 
#> 
#> 
#> ------------------
#> Regime 1 (k = 1):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1    0.11943184   0.6087582   -2.038486
#> X^1 -0.08556857   4.1038126    2.540536
#> 
#> Covariance matrix:
#>                                    
#>  1.19064336  0.12765794  0.05537134
#>  0.12765794  0.87145062 -0.05213162
#>  0.05537134 -0.05213162  0.87886166
#> ------------------
#> Regime 2 (k = 2):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1      6.921139   4.9377164   10.290536
#> X^1    1.131946   0.4684922   -1.419758
#> 
#> Covariance matrix:
#>                                   
#>   1.0688949 -0.18240787 0.12675972
#>  -0.1824079  1.05317924 0.01419686
#>   0.1267597  0.01419686 0.76030310
#> ------------------
#> Regime 3 (k = 3):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1     3.6576562   6.3642526    8.493765
#> X^1   0.6155173  -0.8844373   -1.137027
#> 
#> Covariance matrix:
#>                                     
#>   1.02647251 -0.05491451 -0.01930098
#>  -0.05491451  1.18921808  0.01510035
#>  -0.01930098  0.01510035  1.00352482
#> ------------------
#> Regime 4 (k = 4):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1     -1.439637   -4.463014    2.952470
#> X^1    0.703211    3.649717   -4.187703
#> 
#> Covariance matrix:
#>                                     
#>   0.88001190 -0.03249118 -0.03411075
#>  -0.03249118  1.12088583 -0.07881351
#>  -0.03411075 -0.07881351  0.86061127
#> ------------------
#> Regime 5 (k = 5):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1     3.4982408   2.5357751    7.652113
#> X^1   0.0574791  -0.7286824   -3.005802
#> 
#> Covariance matrix:
#>                                  
#>  1.13331209 0.25869951 0.03163467
#>  0.25869951 1.21231741 0.04746018
#>  0.03163467 0.04746018 0.80242715

mhmmr$plot(what = c("smoothed", "regressors", "loglikelihood"))

# Application to a real data set (human activity recognition data)
data("realdataset")
x <- realdataset$x
y <- realdataset[, c("y1", "y2", "y3")]

K <- 5 # Number of regimes (states)
p <- 3 # Dimension of beta (order of the polynomial regressors)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model

n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE

mhmmr <- emMHMMR(X = x, Y = y, K, p, variance_type, n_tries, 
                 max_iter, threshold, verbose)
#> EM - MHMMR: Iteration: 1 | log-likelihood: 817.206309249687
#> EM - MHMMR: Iteration: 2 | log-likelihood: 1793.49320726452
#> EM - MHMMR: Iteration: 3 | log-likelihood: 1908.47251424374
#> EM - MHMMR: Iteration: 4 | log-likelihood: 2006.7976746047
#> EM - MHMMR: Iteration: 5 | log-likelihood: 3724.91911814713
#> EM - MHMMR: Iteration: 6 | log-likelihood: 3846.02584774854
#> EM - MHMMR: Iteration: 7 | log-likelihood: 3957.04953794437
#> EM - MHMMR: Iteration: 8 | log-likelihood: 4008.60804596975
#> EM - MHMMR: Iteration: 9 | log-likelihood: 4011.09964067314
#> EM - MHMMR: Iteration: 10 | log-likelihood: 4014.35810165377
#> EM - MHMMR: Iteration: 11 | log-likelihood: 4026.38632031497
#> EM - MHMMR: Iteration: 12 | log-likelihood: 4027.13758668835
#> EM - MHMMR: Iteration: 13 | log-likelihood: 4027.13639613206

mhmmr$summary()
#> ----------------------
#> Fitted MHMMR model
#> ----------------------
#> 
#> MHMMR model with K = 5 regimes
#> 
#>  log-likelihood  nu      AIC      BIC
#>        4027.136 114 3913.136 3587.095
#> 
#> Clustering table:
#>   1   2   3   4   5 
#> 461 297 587 423 485 
#> 
#> 
#> ------------------
#> Regime 1 (k = 1):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2)  Beta(d = 3)
#> 1    1.41265303  2.42222746  9.381994682
#> X^1  0.47242692  0.09217574 -0.023282898
#> X^2 -0.28135064 -0.10169173  0.018998710
#> X^3  0.04197568  0.02620151 -0.004217078
#> 
#> Covariance matrix:
#>                                       
#>   0.12667921 -0.019381009 -0.018810846
#>  -0.01938101  0.109202105 -0.001402791
#>  -0.01881085 -0.001402791  0.026461790
#> ------------------
#> Regime 2 (k = 2):
#> 
#> Regression coefficients:
#> 
#>     Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1    -3.6868321   2.4724043    7.794639
#> X^1  -6.8471097   4.6786664   14.749215
#> X^2   2.9742521  -1.4716819   -4.646020
#> X^3  -0.2449644   0.1076065    0.335142
#> 
#> Covariance matrix:
#>                                      
#>   0.22604244 -0.032716477 0.013626769
#>  -0.03271648  0.032475350 0.008585402
#>   0.01362677  0.008585402 0.041960228
#> ------------------
#> Regime 3 (k = 3):
#> 
#> Regression coefficients:
#> 
#>      Beta(d = 1)  Beta(d = 2)   Beta(d = 3)
#> 1    0.776245522  0.014437427 -0.1144683124
#> X^1  2.627158141  0.048519275 -0.3883099866
#> X^2 -0.255314738 -0.008318957  0.0283047828
#> X^3  0.008129981  0.000356239 -0.0007003718
#> 
#> Covariance matrix:
#>                                           
#>   0.0012000978 -0.0002523608 -0.0001992900
#>  -0.0002523608  0.0006584694  0.0002391577
#>  -0.0001992900  0.0002391577  0.0014228769
#> ------------------
#> Regime 4 (k = 4):
#> 
#> Regression coefficients:
#> 
#>      Beta(d = 1)   Beta(d = 2)  Beta(d = 3)
#> 1    0.002894474 -0.0002900823 -0.001513232
#> X^1  0.029936273 -0.0029993910 -0.015647636
#> X^2  0.232798943 -0.0233058753 -0.121611904
#> X^3 -0.013209774  0.0019141508  0.009151938
#> 
#> Covariance matrix:
#>                                     
#>   0.21455830 -0.07328139 -0.08824736
#>  -0.07328139  0.17055704  0.45218611
#>  -0.08824736  0.45218611  1.76616982
#> ------------------
#> Regime 5 (k = 5):
#> 
#> Regression coefficients:
#> 
#>       Beta(d = 1)   Beta(d = 2)   Beta(d = 3)
#> 1    9.416685e-05  0.0001347198  0.0005119141
#> X^1  1.259159e-03  0.0018014389  0.0068451694
#> X^2  1.265758e-02  0.0181095390  0.0688126905
#> X^3 -4.344666e-04 -0.0005920827 -0.0022723501
#> 
#> Covariance matrix:
#>                                       
#>   0.009259719 -0.000696446 0.006008102
#>  -0.000696446  0.003732296 0.001056145
#>   0.006008102  0.001056145 0.016144263

mhmmr$plot(what = c("smoothed", "regressors", "loglikelihood"))

Model selection

In this package, it is possible to select models based on information criteria such as BIC, AIC and ICL.

The selection can be done for the two following parameters:

  • K: The number of regimes;
  • p: The order of the polynomial regression.

Let’s select a MHMMR model for the following multivariate time series Y:

data("toydataset")
x <- toydataset$x
y <- toydataset[, c("y1", "y2", "y3")]
matplot(x, y, type = "l", xlab = "x", ylab = "Y", lty = 1)

selectedmhmmr <- selectMHMMR(X = x, Y = y, Kmin = 2, Kmax = 6, pmin = 0, pmax = 3)
#> The MHMMR model selected via the "BIC" has K = 5 regimes 
#>  and the order of the polynomial regression is p = 0.
#> BIC = -3118.9815385353
#> AIC = -2963.48045745801

selectedmhmmr$plot(what = "smoothed")

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Joint segmentation of multivariate time series with a Multiple Hidden Markov Model Regression (MHMMR)

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