samurais is an open source toolbox (available in R and in Matlab) including many original and flexible user-friendly statistical latent variable models and unsupervised algorithms to segment and represent, time-series data (univariate or multivariate), and more generally, longitudinal data which include regime changes.
Our samurais use mainly the following efficient "sword" packages to segment data: Regression with Hidden Logistic Process (RHLP), Hidden Markov Model Regression (HMMR), Piece-Wise regression (PWR), Multivariate 'RHLP' (MRHLP), and Multivariate 'HMMR' (MHMMR).
The models and algorithms are developed and written in Matlab by Faicel Chamroukhi, and translated and designed into R packages by Florian Lecocq, Marius Bartcus and Faicel Chamroukhi.
You can install the samurais package from GitHub with:
# install.packages("devtools")
devtools::install_github("fchamroukhi/SaMUraiS")To build vignettes for examples of usage, type the command below instead:
# install.packages("devtools")
devtools::install_github("fchamroukhi/SaMUraiS",
build_opts = c("--no-resave-data", "--no-manual"),
build_vignettes = TRUE)Use the following command to display vignettes:
browseVignettes("samurais")library(samurais)RHLP
# Application to a toy data set
data("univtoydataset")
x <- univtoydataset$x
y <- univtoydataset$y
K <- 5 # Number of regimes (mixture components)
p <- 3 # Dimension of beta (order of the polynomial regressors)
q <- 1 # Dimension of w (order of the logistic regression: to be set to 1 for segmentation)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model
n_tries <- 1
max_iter = 1500
threshold <- 1e-6
verbose <- TRUE
verbose_IRLS <- FALSE
rhlp <- emRHLP(X = x, Y = y, K, p, q, variance_type, n_tries,
max_iter, threshold, verbose, verbose_IRLS)
#> EM: Iteration : 1 || log-likelihood : -2119.27308534609
#> EM: Iteration : 2 || log-likelihood : -1149.01040321999
#> EM: Iteration : 3 || log-likelihood : -1118.20384281234
#> EM: Iteration : 4 || log-likelihood : -1096.88260636121
#> EM: Iteration : 5 || log-likelihood : -1067.55719357295
#> EM: Iteration : 6 || log-likelihood : -1037.26620122646
#> EM: Iteration : 7 || log-likelihood : -1022.71743069484
#> EM: Iteration : 8 || log-likelihood : -1006.11825447077
#> EM: Iteration : 9 || log-likelihood : -1001.18491883952
#> EM: Iteration : 10 || log-likelihood : -1000.91250763556
#> EM: Iteration : 11 || log-likelihood : -1000.62280600209
#> EM: Iteration : 12 || log-likelihood : -1000.3030988811
#> EM: Iteration : 13 || log-likelihood : -999.932334880131
#> EM: Iteration : 14 || log-likelihood : -999.484219706691
#> EM: Iteration : 15 || log-likelihood : -998.928118038989
#> EM: Iteration : 16 || log-likelihood : -998.234244664472
#> EM: Iteration : 17 || log-likelihood : -997.359536276056
#> EM: Iteration : 18 || log-likelihood : -996.152654857298
#> EM: Iteration : 19 || log-likelihood : -994.697863447307
#> EM: Iteration : 20 || log-likelihood : -993.186583974542
#> EM: Iteration : 21 || log-likelihood : -991.81352379631
#> EM: Iteration : 22 || log-likelihood : -990.611295217008
#> EM: Iteration : 23 || log-likelihood : -989.539226273251
#> EM: Iteration : 24 || log-likelihood : -988.55311887915
#> EM: Iteration : 25 || log-likelihood : -987.539963690533
#> EM: Iteration : 26 || log-likelihood : -986.073920116541
#> EM: Iteration : 27 || log-likelihood : -983.263549878169
#> EM: Iteration : 28 || log-likelihood : -979.340492188909
#> EM: Iteration : 29 || log-likelihood : -977.468559852711
#> EM: Iteration : 30 || log-likelihood : -976.653534236095
#> EM: Iteration : 31 || log-likelihood : -976.5893387433
#> EM: Iteration : 32 || log-likelihood : -976.589338067237
rhlp$summary()
#> ---------------------
#> Fitted RHLP model
#> ---------------------
#>
#> RHLP model with K = 5 components:
#>
#> log-likelihood nu AIC BIC ICL
#> -976.5893 33 -1009.589 -1083.959 -1083.176
#>
#> Clustering table (Number of observations in each regimes):
#>
#> 1 2 3 4 5
#> 100 120 200 100 150
#>
#> Regression coefficients:
#>
#> Beta(K = 1) Beta(K = 2) Beta(K = 3) Beta(K = 4) Beta(K = 5)
#> 1 6.031875e-02 -5.434903 -2.770416 120.7699 4.027542
#> X^1 -7.424718e+00 158.705091 43.879453 -474.5888 13.194261
#> X^2 2.931652e+02 -650.592347 -94.194780 597.7948 -33.760603
#> X^3 -1.823560e+03 865.329795 67.197059 -244.2386 20.402153
#>
#> Variances:
#>
#> Sigma2(K = 1) Sigma2(K = 2) Sigma2(K = 3) Sigma2(K = 4) Sigma2(K = 5)
#> 1.220624 1.110243 1.079394 0.9779734 1.028332
rhlp$plot()
# Application to a real data set
data("univrealdataset")
x <- univrealdataset$x
y <- univrealdataset$y2
K <- 5 # Number of regimes (mixture components)
p <- 3 # Dimension of beta (order of the polynomial regressors)
q <- 1 # Dimension of w (order of the logistic regression: to be set to 1 for segmentation)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model
n_tries <- 1
max_iter = 1500
threshold <- 1e-6
verbose <- TRUE
verbose_IRLS <- FALSE
rhlp <- emRHLP(X = x, Y = y, K, p, q, variance_type, n_tries,
max_iter, threshold, verbose, verbose_IRLS)
#> EM: Iteration : 1 || log-likelihood : -3321.6485760125
#> EM: Iteration : 2 || log-likelihood : -2286.48632282875
#> EM: Iteration : 3 || log-likelihood : -2257.60498391374
#> EM: Iteration : 4 || log-likelihood : -2243.74506764308
#> EM: Iteration : 5 || log-likelihood : -2233.3426635247
#> EM: Iteration : 6 || log-likelihood : -2226.89953345319
#> EM: Iteration : 7 || log-likelihood : -2221.77999023589
#> EM: Iteration : 8 || log-likelihood : -2215.81305295291
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#> EM: Iteration : 10 || log-likelihood : -2196.27872403055
#> EM: Iteration : 11 || log-likelihood : -2185.40049009242
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#> EM: Iteration : 172 || log-likelihood : -1946.21315057723
rhlp$summary()
#> ---------------------
#> Fitted RHLP model
#> ---------------------
#>
#> RHLP model with K = 5 components:
#>
#> log-likelihood nu AIC BIC ICL
#> -1946.213 33 -1979.213 -2050.683 -2050.449
#>
#> Clustering table (Number of observations in each regimes):
#>
#> 1 2 3 4 5
#> 16 129 180 111 126
#>
#> Regression coefficients:
#>
#> Beta(K = 1) Beta(K = 2) Beta(K = 3) Beta(K = 4) Beta(K = 5)
#> 1 2187.539 330.05723 1508.2809 -13446.7332 6417.62830
#> X^1 -15032.659 -107.79782 -1648.9562 11321.4509 -3571.94090
#> X^2 -56433.432 14.40154 786.5723 -3062.2825 699.55894
#> X^3 494014.670 56.88016 -118.0693 272.7844 -45.42922
#>
#> Variances:
#>
#> Sigma2(K = 1) Sigma2(K = 2) Sigma2(K = 3) Sigma2(K = 4) Sigma2(K = 5)
#> 8924.363 49.22616 78.2758 105.6606 15.66317
rhlp$plot()
HMMR
# Application to a toy data set
data("univtoydataset")
x <- univtoydataset$x
y <- univtoydataset$y
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
hmmr <- emHMMR(X = x, Y = y, K, p, variance_type,
n_tries, max_iter, threshold, verbose)
#> EM: Iteration : 1 || log-likelihood : -1556.39696825601
#> EM: Iteration : 2 || log-likelihood : -1022.47935723687
#> EM: Iteration : 3 || log-likelihood : -1019.51830707432
#> EM: Iteration : 4 || log-likelihood : -1019.51780361388
hmmr$summary()
#> ---------------------
#> Fitted HMMR model
#> ---------------------
#>
#> HMMR model with K = 5 components:
#>
#> log-likelihood nu AIC BIC
#> -1019.518 49 -1068.518 -1178.946
#>
#> Clustering table (Number of observations in each regimes):
#>
#> 1 2 3 4 5
#> 100 120 200 100 150
#>
#> Regression coefficients:
#>
#> Beta(K = 1) Beta(K = 2) Beta(K = 3) Beta(K = 4) Beta(K = 5)
#> 1 6.031872e-02 -5.326689 -2.65064 120.8612 3.858683
#> X^1 -7.424715e+00 157.189455 43.13601 -474.9870 13.757279
#> X^2 2.931651e+02 -643.706204 -92.68115 598.3726 -34.384734
#> X^3 -1.823559e+03 855.171715 66.18499 -244.5175 20.632196
#>
#> Variances:
#>
#> Sigma2(K = 1) Sigma2(K = 2) Sigma2(K = 3) Sigma2(K = 4) Sigma2(K = 5)
#> 1.220624 1.111487 1.080043 0.9779724 1.028399
hmmr$plot(what = c("smoothed", "regressors", "loglikelihood"))
# Application to a real data set
data("univrealdataset")
x <- univrealdataset$x
y <- univrealdataset$y2
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
hmmr <- emHMMR(X = x, Y = y, K, p, variance_type,
n_tries, max_iter, threshold, verbose)
#> EM: Iteration : 1 || log-likelihood : -2733.41028643114
#> EM: Iteration : 2 || log-likelihood : -2303.24018378559
#> EM: Iteration : 3 || log-likelihood : -2295.0470677529
#> EM: Iteration : 4 || log-likelihood : -2288.57866215726
#> EM: Iteration : 5 || log-likelihood : -2281.36756202518
#> EM: Iteration : 6 || log-likelihood : -2273.50303676091
#> EM: Iteration : 7 || log-likelihood : -2261.70334656117
#> EM: Iteration : 8 || log-likelihood : -2243.43509121433
#> EM: Iteration : 9 || log-likelihood : -2116.4610801575
#> EM: Iteration : 10 || log-likelihood : -2046.73194777839
#> EM: Iteration : 11 || log-likelihood : -2046.68328282973
#> EM: Iteration : 12 || log-likelihood : -2046.67329222076
#> EM: Iteration : 13 || log-likelihood : -2046.66915144265
#> EM: Iteration : 14 || log-likelihood : -2046.66694236131
#> EM: Iteration : 15 || log-likelihood : -2046.66563379017
hmmr$summary()
#> ---------------------
#> Fitted HMMR model
#> ---------------------
#>
#> HMMR model with K = 5 components:
#>
#> log-likelihood nu AIC BIC
#> -2046.666 49 -2095.666 -2201.787
#>
#> Clustering table (Number of observations in each regimes):
#>
#> 1 2 3 4 5
#> 14 214 99 109 126
#>
#> Regression coefficients:
#>
#> Beta(K = 1) Beta(K = 2) Beta(K = 3) Beta(K = 4) Beta(K = 5)
#> 1 2152.64 379.75158 5211.1759 -14306.4654 6417.62823
#> X^1 -12358.67 -373.37266 -5744.7879 11987.6666 -3571.94086
#> X^2 -103908.33 394.49359 2288.9418 -3233.8021 699.55894
#> X^3 722173.26 -98.60485 -300.7686 287.4567 -45.42922
#>
#> Variances:
#>
#> Sigma2(K = 1) Sigma2(K = 2) Sigma2(K = 3) Sigma2(K = 4) Sigma2(K = 5)
#> 9828.793 125.3346 58.71053 105.8328 15.66317
hmmr$plot(what = c("smoothed", "regressors", "loglikelihood"))
PWR
# Application to a toy data set
data("univtoydataset")
x <- univtoydataset$x
y <- univtoydataset$y
K <- 5 # Number of segments
p <- 3 # Polynomial degree
pwr <- fitPWRFisher(X = x, Y = y, K, p)
pwr$summary()
#> --------------------
#> Fitted PWR model
#> --------------------
#>
#> PWR model with K = 5 components:
#>
#> Clustering table (Number of observations in each regimes):
#>
#> 1 2 3 4 5
#> 100 120 200 100 150
#>
#> Regression coefficients:
#>
#> Beta(K = 1) Beta(K = 2) Beta(K = 3) Beta(K = 4) Beta(K = 5)
#> 1 6.106872e-02 -5.450955 -2.776275 122.7045 4.020809
#> X^1 -7.486945e+00 158.922010 43.915969 -482.8929 13.217587
#> X^2 2.942201e+02 -651.540876 -94.269414 609.6493 -33.787416
#> X^3 -1.828308e+03 866.675017 67.247141 -249.8667 20.412380
#>
#> Variances:
#>
#> Sigma2(K = 1) Sigma2(K = 2) Sigma2(K = 3) Sigma2(K = 4) Sigma2(K = 5)
#> 1.220624 1.110193 1.079366 0.9779733 1.028329
pwr$plot()
# Application to a real data set
data("univrealdataset")
x <- univrealdataset$x
y <- univrealdataset$y2
K <- 5 # Number of segments
p <- 3 # Polynomial degree
pwr <- fitPWRFisher(X = x, Y = y, K, p)
pwr$summary()
#> --------------------
#> Fitted PWR model
#> --------------------
#>
#> PWR model with K = 5 components:
#>
#> Clustering table (Number of observations in each regimes):
#>
#> 1 2 3 4 5
#> 15 130 178 113 126
#>
#> Regression coefficients:
#>
#> Beta(K = 1) Beta(K = 2) Beta(K = 3) Beta(K = 4) Beta(K = 5)
#> 1 2163.323 334.23747 1458.6530 -11445.9003 6418.36449
#> X^1 -13244.753 -125.04633 -1578.1793 9765.9713 -3572.38535
#> X^2 -86993.374 35.33532 753.8468 -2660.5976 699.64809
#> X^3 635558.069 49.12683 -113.1589 238.3246 -45.43516
#>
#> Variances:
#>
#> Sigma2(K = 1) Sigma2(K = 2) Sigma2(K = 3) Sigma2(K = 4) Sigma2(K = 5)
#> 9326.335 50.71573 75.23989 110.6818 15.66317
pwr$plot()
MRHLP
# Application to a toy data set
data("multivtoydataset")
x <- multivtoydataset$x
y <- multivtoydataset[,c("y1", "y2", "y3")]
K <- 5 # Number of regimes (mixture components)
p <- 1 # Dimension of beta (order of the polynomial regressors)
q <- 1 # Dimension of w (order of the logistic regression: to be set to 1 for segmentation)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model
n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE
verbose_IRLS <- FALSE
mrhlp <- emMRHLP(X = x, Y = y, K, p, q, variance_type, n_tries,
max_iter, threshold, verbose, verbose_IRLS)
#> EM: Iteration : 1 || log-likelihood : -4807.6644322901
#> EM: Iteration : 2 || log-likelihood : -3314.25165556383
#> EM: Iteration : 3 || log-likelihood : -3216.8871750704
#> EM: Iteration : 4 || log-likelihood : -3126.33556053822
#> EM: Iteration : 5 || log-likelihood : -2959.59933830667
#> EM: Iteration : 6 || log-likelihood : -2895.65953485704
#> EM: Iteration : 7 || log-likelihood : -2892.93263500326
#> EM: Iteration : 8 || log-likelihood : -2889.34084959654
#> EM: Iteration : 9 || log-likelihood : -2884.56422084139
#> EM: Iteration : 10 || log-likelihood : -2878.29772085061
#> EM: Iteration : 11 || log-likelihood : -2870.61242183846
#> EM: Iteration : 12 || log-likelihood : -2862.86238149363
#> EM: Iteration : 13 || log-likelihood : -2856.85351443338
#> EM: Iteration : 14 || log-likelihood : -2851.74642203885
#> EM: Iteration : 15 || log-likelihood : -2850.00381259526
#> EM: Iteration : 16 || log-likelihood : -2849.86516522686
#> EM: Iteration : 17 || log-likelihood : -2849.7354103643
#> EM: Iteration : 18 || log-likelihood : -2849.56953544124
#> EM: Iteration : 19 || log-likelihood : -2849.40322468732
#> EM: Iteration : 20 || log-likelihood : -2849.40321381274
mrhlp$summary()
#> ----------------------
#> Fitted MRHLP model
#> ----------------------
#>
#> MRHLP model with K = 5 regimes
#>
#> log-likelihood nu AIC BIC ICL
#> -2849.403 68 -2917.403 -3070.651 -3069.896
#>
#> 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:
#>
#> 1.19063336 0.12765794 0.05537134
#> 0.12765794 0.87144062 -0.05213162
#> 0.05537134 -0.05213162 0.87885166
#> ------------------
#> Regime 2 (K = 2):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 6.924025 4.9368460 10.288339
#> X^1 1.118034 0.4726707 -1.409218
#>
#> Covariance matrix:
#>
#> 1.0690431 -0.18293369 0.12602459
#> -0.1829337 1.05280632 0.01390041
#> 0.1260246 0.01390041 0.75995058
#> ------------------
#> Regime 3 (K = 3):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 3.6535241 6.3654379 8.488318
#> X^1 0.6233579 -0.8866887 -1.126692
#>
#> Covariance matrix:
#>
#> 1.02591553 -0.05445227 -0.02019896
#> -0.05445227 1.18941700 0.01565240
#> -0.02019896 0.01565240 1.00257195
#> ------------------
#> 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:
#>
#> 0.88000190 -0.03249118 -0.03411075
#> -0.03249118 1.12087583 -0.07881351
#> -0.03411075 -0.07881351 0.86060127
#> ------------------
#> 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:
#>
#> 1.13330209 0.25869951 0.03163467
#> 0.25869951 1.21230741 0.04746018
#> 0.03163467 0.04746018 0.80241715
mrhlp$plot()
# Application to a real data set (human activity recogntion data)
data("multivrealdataset")
x <- multivrealdataset$x
y <- multivrealdataset[,c("y1", "y2", "y3")]
K <- 5 # Number of regimes (mixture components)
p <- 3 # Dimension of beta (order of the polynomial regressors)
q <- 1 # Dimension of w (order of the logistic regression: to be set to 1 for segmentation)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model
n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE
verbose_IRLS <- FALSE
mrhlp <- emMRHLP(X = x, Y = y, K, p, q, variance_type, n_tries,
max_iter, threshold, verbose, verbose_IRLS)
#> EM: Iteration : 1 || log-likelihood : -792.888668727036
#> EM: Iteration : 2 || log-likelihood : 6016.45835957306
#> EM: Iteration : 3 || log-likelihood : 6362.81791662824
#> EM: Iteration : 4 || log-likelihood : 6615.72233403002
#> EM: Iteration : 5 || log-likelihood : 6768.32107943849
#> EM: Iteration : 6 || log-likelihood : 6840.97339565987
#> EM: Iteration : 7 || log-likelihood : 6860.97262839295
#> EM: Iteration : 8 || log-likelihood : 6912.25605673784
#> EM: Iteration : 9 || log-likelihood : 6945.96718258737
#> EM: Iteration : 10 || log-likelihood : 6951.28584396645
#> EM: Iteration : 11 || log-likelihood : 6952.37644678517
#> EM: Iteration : 12 || log-likelihood : 6954.80510338749
#> EM: Iteration : 13 || log-likelihood : 6958.99033092484
#> EM: Iteration : 14 || log-likelihood : 6964.81099837456
#> EM: Iteration : 15 || log-likelihood : 6999.90358068156
#> EM: Iteration : 16 || log-likelihood : 7065.39327246318
#> EM: Iteration : 17 || log-likelihood : 7166.23398344994
#> EM: Iteration : 18 || log-likelihood : 7442.73330846285
#> EM: Iteration : 19 || log-likelihood : 7522.65416438396
#> EM: Iteration : 20 || log-likelihood : 7524.41524338024
#> EM: Iteration : 21 || log-likelihood : 7524.57590110924
#> EM: Iteration : 22 || log-likelihood : 7524.73808801417
#> EM: Iteration : 23 || log-likelihood : 7524.88684996651
#> EM: Iteration : 24 || log-likelihood : 7524.9753964817
#> EM: Iteration : 25 || log-likelihood : 7524.97701548847
mrhlp$summary()
#> ----------------------
#> Fitted MRHLP model
#> ----------------------
#>
#> MRHLP model with K = 5 regimes
#>
#> log-likelihood nu AIC BIC ICL
#> 7524.977 98 7426.977 7146.696 7147.535
#>
#> Clustering table:
#> 1 2 3 4 5
#> 413 344 588 423 485
#>
#>
#> ------------------
#> Regime 1 (K = 1):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 1.64847721 2.33823068 9.40173242
#> X^1 -0.31396583 0.38235782 -0.10031616
#> X^2 0.23954454 -0.30105177 0.07812145
#> X^3 -0.04725267 0.06166899 -0.01586579
#>
#> Covariance matrix:
#>
#> 0.0200740364 -0.004238036 0.0004011388
#> -0.0042380363 0.006082904 -0.0012973026
#> 0.0004011388 -0.001297303 0.0013201963
#> ------------------
#> Regime 2 (K = 2):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 -106.0250571 -31.4671946 -107.9697464
#> X^1 45.2035210 21.2126134 72.0220177
#> X^2 -5.7330338 -4.1285514 -13.9857795
#> X^3 0.2343552 0.2485377 0.8374817
#>
#> Covariance matrix:
#>
#> 0.11899225 -0.03866052 -0.06693441
#> -0.03866052 0.17730401 0.04036629
#> -0.06693441 0.04036629 0.11983979
#> ------------------
#> Regime 3 (K = 3):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 9.0042249443 -1.247752962 -2.492119515
#> X^1 0.2191555621 0.418071041 0.310449523
#> X^2 -0.0242080660 -0.043802827 -0.039012607
#> X^3 0.0008494208 0.001474635 0.001427627
#>
#> Covariance matrix:
#>
#> 4.103351e-04 -0.0001330363 5.289199e-05
#> -1.330363e-04 0.0006297205 2.027763e-04
#> 5.289199e-05 0.0002027763 1.374405e-03
#> ------------------
#> Regime 4 (K = 4):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 -1029.9071752 334.4975068 466.0981076
#> X^1 199.9531885 -68.7252041 -105.6436899
#> X^2 -12.6550086 4.6489685 7.6555642
#> X^3 0.2626998 -0.1032161 -0.1777453
#>
#> Covariance matrix:
#>
#> 0.058674116 -0.017661572 0.002139975
#> -0.017661572 0.047588713 0.007867532
#> 0.002139975 0.007867532 0.067150809
#> ------------------
#> Regime 5 (K = 5):
#>
#> Regression coefficients:
#>
#> Beta(d = 1) Beta(d = 2) Beta(d = 3)
#> 1 27.247199195 -14.393798357 19.741283724
#> X^1 -3.530625667 2.282492947 -1.511225702
#> X^2 0.161234880 -0.101613670 0.073003292
#> X^3 -0.002446104 0.001490288 -0.001171127
#>
#> Covariance matrix:
#>
#> 6.900384e-03 -0.001176838 2.966199e-05
#> -1.176838e-03 0.003596238 -2.395420e-04
#> 2.966199e-05 -0.000239542 5.573451e-04
mrhlp$plot()
MHMMR
# Application to a simulated data set
data("multivtoydataset")
x <- multivtoydataset$x
y <- multivtoydataset[,c("y1", "y2", "y3")]
K <- 5 # Number of regimes (states)
p <- 1 # Dimension of beta (order of the polynomial regressors)
variance_type <- "heteroskedastic" # "heteroskedastic" or "homoskedastic" model
n_tries <- 1
max_iter <- 1500
threshold <- 1e-6
verbose <- TRUE
mhmmr <- emMHMMR(X = x, Y = y, K, p, variance_type, n_tries,
max_iter, threshold, verbose)
#> EM: Iteration : 1 || log-likelihood : -4539.37845473736
#> EM: Iteration : 2 || log-likelihood : -3075.7862970485
#> EM: Iteration : 3 || log-likelihood : -2904.71126233611
#> EM: Iteration : 4 || log-likelihood : -2883.23456594806
#> EM: Iteration : 5 || log-likelihood : -2883.12446634454
#> EM: Iteration : 6 || log-likelihood : -2883.12436399888
mhmmr$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$plot(what = c("smoothed", "regressors", "loglikelihood"))
# Application to a real data set (human activity recognition data)
data("multivrealdataset")
x <- multivrealdataset$x
y <- multivrealdataset[,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: Iteration : 1 || log-likelihood : 817.206309249687
#> EM: Iteration : 2 || log-likelihood : 1793.49320726452
#> EM: Iteration : 3 || log-likelihood : 1908.47251424374
#> EM: Iteration : 4 || log-likelihood : 2006.7976746047
#> EM: Iteration : 5 || log-likelihood : 3724.91911814713
#> EM: Iteration : 6 || log-likelihood : 3846.02584774854
#> EM: Iteration : 7 || log-likelihood : 3957.04953794437
#> EM: Iteration : 8 || log-likelihood : 4008.60804596975
#> EM: Iteration : 9 || log-likelihood : 4011.09964067314
#> EM: Iteration : 10 || log-likelihood : 4014.35810165377
#> EM: Iteration : 11 || log-likelihood : 4026.38632031497
#> EM: Iteration : 12 || log-likelihood : 4027.13758668835
#> EM: 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"))
samurais also implements model selection procedures to select an optimal model based on information criteria including BIC, AIC and ICL.
The selection can be done for the two following parameters:
- K: The number of regimes (segments);
- p: The order of the polynomial regression.
Instructions below can be used to illustrate the model on provided simulated and real data sets.
RHLP
Let's select a RHLP model for the following time series:
data("univtoydataset")
x = univtoydataset$x
y = univtoydataset$y
plot(x, y, type = "l", xlab = "x", ylab = "Y")
selectedrhlp <- selectRHLP(X = x, Y = y, Kmin = 2, Kmax = 6, pmin = 0, pmax = 3)
#> The RHLP model selected via the "BIC" has K = 5 regimes
#> and the order of the polynomial regression is p = 0.
#> BIC = -1041.40789532438
#> AIC = -1000.84239591291
selectedrhlp$plot(what = "estimatedsignal")
HMMR
Let's select a HMMR model for the following time series:
data("univtoydataset")
x = univtoydataset$x
y = univtoydataset$y
plot(x, y, type = "l", xlab = "x", ylab = "Y")
selectedhmmr <- selectHMMR(X = x, Y = y, Kmin = 2, Kmax = 6, pmin = 0, pmax = 3)
#> The HMMR model selected via the "BIC" has K = 5 regimes
#> and the order of the polynomial regression is p = 0.
#> BIC = -1136.39152222095
#> AIC = -1059.76780111041
selectedhmmr$plot(what = "smoothed")
MRHLP
Let's select a MRHLP model for the following multivariate time series:
data("multivtoydataset")
x <- multivtoydataset$x
y <- multivtoydataset[, c("y1", "y2", "y3")]
matplot(x, y, type = "l", xlab = "x", ylab = "Y", lty = 1)
selectedmrhlp <- selectMRHLP(X = x, Y = y, Kmin = 2, Kmax = 6, pmin = 0, pmax = 3)
#> Warning in emMRHLP(X = X1, Y = Y1, K, p): EM log-likelihood is decreasing
#> from -3105.78591044952to -3105.78627830471 !
#> The MRHLP model selected via the "BIC" has K = 5 regimes
#> and the order of the polynomial regression is p = 0.
#> BIC = -3033.20042397111
#> AIC = -2913.75756459291
selectedmrhlp$plot(what = "estimatedsignal")
MHMMR
Let's select a MHMMR model for the following multivariate time series:
data("multivtoydataset")
x <- multivtoydataset$x
y <- multivtoydataset[, c("y1", "y2", "y3")]
matplot(x, y, type = "l", xlab = "x", ylab = "Y", lty = 1)
selectedmhmmr <- selectMHMMR(X = x, Y = y, Kmin = 2, Kmax = 6, pmin = 0, pmax = 3)
#> The MHMMR model selected via the "BIC" has K = 5 regimes
#> and the order of the polynomial regression is p = 0.
#> BIC = -3118.9815385353
#> AIC = -2963.48045745801
selectedmhmmr$plot(what = "smoothed")
Chamroukhi, Faicel, and Hien D. Nguyen. 2019. “Model-Based Clustering and Classification of Functional Data.” Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, jan. https://chamroukhi.com/papers/MBCC-FDA.pdf.
Chamroukhi, F. 2015. “Statistical Learning of Latent Data Models for Complex Data Analysis.” Habilitation Thesis (HDR), Université de Toulon. https://chamroukhi.com/FChamroukhi-HDR.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 (november): 633–44. https://chamroukhi.com/papers/chamroukhi_et_al_neucomp2013b.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. 2010. “Hidden Process Regression for Curve Modeling, Classification and Tracking.” Ph.D. Thesis, Université de Technologie de Compiègne. https://chamroukhi.com/FChamroukhi-PhD.pdf.
Chamroukhi, F., A. Samé, 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., A. Samé, 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.
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