README.md
In fchamroukhi/RHLP: Regression with Hidden Logistic Process ('RHLP') for Time Series Segmentation
Overview
Flexible and user-friendly probabilistic segmentation of time series
with smooth and/or abrupt regime changes by a mixture
model-based regression approach with a hidden logistic process, fitted
by the EM algorithm.
Installation
You can install the RHLP package from GitHub
with:
# install.packages("devtools")
devtools::install_github("fchamroukhi/RHLP")
To build vignettes for examples of usage, type the command below
instead:
# install.packages("devtools")
devtools::install_github("fchamroukhi/RHLP",
build_opts = c("--no-resave-data", "--no-manual"),
build_vignettes = TRUE)
Use the following command to display vignettes:
browseVignettes("RHLP")
Usage
library(RHLP)
# Application to a toy data set
data("toydataset")
x <- toydataset$x
y <- toydataset$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 - RHLP: Iteration: 1 | log-likelihood: -2119.27308534609
#> EM - RHLP: Iteration: 2 | log-likelihood: -1149.01040321999
#> EM - RHLP: Iteration: 3 | log-likelihood: -1118.20384281234
#> EM - RHLP: Iteration: 4 | log-likelihood: -1096.88260636121
#> EM - RHLP: Iteration: 5 | log-likelihood: -1067.55719357295
#> EM - RHLP: Iteration: 6 | log-likelihood: -1037.26620122646
#> EM - RHLP: Iteration: 7 | log-likelihood: -1022.71743069484
#> EM - RHLP: Iteration: 8 | log-likelihood: -1006.11825447077
#> EM - RHLP: Iteration: 9 | log-likelihood: -1001.18491883952
#> EM - RHLP: Iteration: 10 | log-likelihood: -1000.91250763556
#> EM - RHLP: Iteration: 11 | log-likelihood: -1000.62280600209
#> EM - RHLP: Iteration: 12 | log-likelihood: -1000.3030988811
#> EM - RHLP: Iteration: 13 | log-likelihood: -999.932334880131
#> EM - RHLP: Iteration: 14 | log-likelihood: -999.484219706691
#> EM - RHLP: Iteration: 15 | log-likelihood: -998.928118038989
#> EM - RHLP: Iteration: 16 | log-likelihood: -998.234244664472
#> EM - RHLP: Iteration: 17 | log-likelihood: -997.359536276056
#> EM - RHLP: Iteration: 18 | log-likelihood: -996.152654857298
#> EM - RHLP: Iteration: 19 | log-likelihood: -994.697863447307
#> EM - RHLP: Iteration: 20 | log-likelihood: -993.186583974542
#> EM - RHLP: Iteration: 21 | log-likelihood: -991.81352379631
#> EM - RHLP: Iteration: 22 | log-likelihood: -990.611295217008
#> EM - RHLP: Iteration: 23 | log-likelihood: -989.539226273251
#> EM - RHLP: Iteration: 24 | log-likelihood: -988.55311887915
#> EM - RHLP: Iteration: 25 | log-likelihood: -987.539963690533
#> EM - RHLP: Iteration: 26 | log-likelihood: -986.073920116541
#> EM - RHLP: Iteration: 27 | log-likelihood: -983.263549878169
#> EM - RHLP: Iteration: 28 | log-likelihood: -979.340492188909
#> EM - RHLP: Iteration: 29 | log-likelihood: -977.468559852711
#> EM - RHLP: Iteration: 30 | log-likelihood: -976.653534236095
#> EM - RHLP: Iteration: 31 | log-likelihood: -976.5893387433
#> EM - RHLP: 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("realdataset")
x <- realdataset$x
y <- realdataset$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 - RHLP: Iteration: 1 | log-likelihood: -3321.6485760125
#> EM - RHLP: Iteration: 2 | log-likelihood: -2286.48632282875
#> EM - RHLP: Iteration: 3 | log-likelihood: -2257.60498391374
#> EM - RHLP: Iteration: 4 | log-likelihood: -2243.74506764308
#> EM - RHLP: Iteration: 5 | log-likelihood: -2233.3426635247
#> EM - RHLP: Iteration: 6 | log-likelihood: -2226.89953345319
#> EM - RHLP: Iteration: 7 | log-likelihood: -2221.77999023589
#> EM - RHLP: Iteration: 8 | log-likelihood: -2215.81305295291
#> EM - RHLP: Iteration: 9 | log-likelihood: -2208.25998029539
#> EM - RHLP: Iteration: 10 | log-likelihood: -2196.27872403055
#> EM - RHLP: Iteration: 11 | log-likelihood: -2185.40049009242
#> EM - RHLP: Iteration: 12 | log-likelihood: -2180.13934245387
#> EM - RHLP: Iteration: 13 | log-likelihood: -2175.4276274402
#> EM - RHLP: Iteration: 14 | log-likelihood: -2170.86113669353
#> EM - RHLP: Iteration: 15 | log-likelihood: -2165.34927170608
#> EM - RHLP: Iteration: 16 | log-likelihood: -2161.12419211511
#> EM - RHLP: Iteration: 17 | log-likelihood: -2158.63709280617
#> EM - RHLP: Iteration: 18 | log-likelihood: -2156.19846850913
#> EM - RHLP: Iteration: 19 | log-likelihood: -2154.04107470071
#> EM - RHLP: Iteration: 20 | log-likelihood: -2153.24544245686
#> EM - RHLP: Iteration: 21 | log-likelihood: -2151.74944795242
#> EM - RHLP: Iteration: 22 | log-likelihood: -2149.90781423151
#> EM - RHLP: Iteration: 23 | log-likelihood: -2146.40042232588
#> EM - RHLP: Iteration: 24 | log-likelihood: -2142.37530025533
#> EM - RHLP: Iteration: 25 | log-likelihood: -2134.85493291884
#> EM - RHLP: Iteration: 26 | log-likelihood: -2129.67399002071
#> EM - RHLP: Iteration: 27 | log-likelihood: -2126.44739300481
#> EM - RHLP: Iteration: 28 | log-likelihood: -2124.94603052064
#> EM - RHLP: Iteration: 29 | log-likelihood: -2122.51637426267
#> EM - RHLP: Iteration: 30 | log-likelihood: -2121.01493646146
#> EM - RHLP: Iteration: 31 | log-likelihood: -2118.45402063643
#> EM - RHLP: Iteration: 32 | log-likelihood: -2116.9336204919
#> EM - RHLP: Iteration: 33 | log-likelihood: -2114.34424563452
#> EM - RHLP: Iteration: 34 | log-likelihood: -2112.84844186712
#> EM - RHLP: Iteration: 35 | log-likelihood: -2110.34494568025
#> EM - RHLP: Iteration: 36 | log-likelihood: -2108.81734757025
#> EM - RHLP: Iteration: 37 | log-likelihood: -2106.26527191053
#> EM - RHLP: Iteration: 38 | log-likelihood: -2104.96591147986
#> EM - RHLP: Iteration: 39 | log-likelihood: -2102.43927829964
#> EM - RHLP: Iteration: 40 | log-likelihood: -2101.27820194404
#> EM - RHLP: Iteration: 41 | log-likelihood: -2098.81151697567
#> EM - RHLP: Iteration: 42 | log-likelihood: -2097.48008514591
#> EM - RHLP: Iteration: 43 | log-likelihood: -2094.98259556552
#> EM - RHLP: Iteration: 44 | log-likelihood: -2093.66517040802
#> EM - RHLP: Iteration: 45 | log-likelihood: -2091.23625905564
#> EM - RHLP: Iteration: 46 | log-likelihood: -2089.91118603989
#> EM - RHLP: Iteration: 47 | log-likelihood: -2087.67388435026
#> EM - RHLP: Iteration: 48 | log-likelihood: -2086.11373786756
#> EM - RHLP: Iteration: 49 | log-likelihood: -2083.84931461869
#> EM - RHLP: Iteration: 50 | log-likelihood: -2082.16175664198
#> EM - RHLP: Iteration: 51 | log-likelihood: -2080.45137011098
#> EM - RHLP: Iteration: 52 | log-likelihood: -2078.37066132008
#> EM - RHLP: Iteration: 53 | log-likelihood: -2077.06827662071
#> EM - RHLP: Iteration: 54 | log-likelihood: -2074.66718553694
#> EM - RHLP: Iteration: 55 | log-likelihood: -2073.68137124781
#> EM - RHLP: Iteration: 56 | log-likelihood: -2071.20390017789
#> EM - RHLP: Iteration: 57 | log-likelihood: -2069.88260759288
#> EM - RHLP: Iteration: 58 | log-likelihood: -2067.30246728287
#> EM - RHLP: Iteration: 59 | log-likelihood: -2066.08897944236
#> EM - RHLP: Iteration: 60 | log-likelihood: -2064.14482062792
#> EM - RHLP: Iteration: 61 | log-likelihood: -2062.39859624374
#> EM - RHLP: Iteration: 62 | log-likelihood: -2060.73756242314
#> EM - RHLP: Iteration: 63 | log-likelihood: -2058.4448132974
#> EM - RHLP: Iteration: 64 | log-likelihood: -2057.23564743141
#> EM - RHLP: Iteration: 65 | log-likelihood: -2054.73129678764
#> EM - RHLP: Iteration: 66 | log-likelihood: -2053.66525147972
#> EM - RHLP: Iteration: 67 | log-likelihood: -2051.05262427909
#> EM - RHLP: Iteration: 68 | log-likelihood: -2049.89030367995
#> EM - RHLP: Iteration: 69 | log-likelihood: -2047.68843285481
#> EM - RHLP: Iteration: 70 | log-likelihood: -2046.16052536146
#> EM - RHLP: Iteration: 71 | log-likelihood: -2044.92677581091
#> EM - RHLP: Iteration: 72 | log-likelihood: -2042.67687818721
#> EM - RHLP: Iteration: 73 | log-likelihood: -2041.77608506749
#> EM - RHLP: Iteration: 74 | log-likelihood: -2039.40345316134
#> EM - RHLP: Iteration: 75 | log-likelihood: -2038.20062153928
#> EM - RHLP: Iteration: 76 | log-likelihood: -2036.05846372404
#> EM - RHLP: Iteration: 77 | log-likelihood: -2034.52492449426
#> EM - RHLP: Iteration: 78 | log-likelihood: -2033.44774900177
#> EM - RHLP: Iteration: 79 | log-likelihood: -2031.15837908019
#> EM - RHLP: Iteration: 80 | log-likelihood: -2030.29908045026
#> EM - RHLP: Iteration: 81 | log-likelihood: -2028.08193331457
#> EM - RHLP: Iteration: 82 | log-likelihood: -2026.82779637097
#> EM - RHLP: Iteration: 83 | log-likelihood: -2025.51219569808
#> EM - RHLP: Iteration: 84 | log-likelihood: -2023.47136697978
#> EM - RHLP: Iteration: 85 | log-likelihood: -2022.86702240332
#> EM - RHLP: Iteration: 86 | log-likelihood: -2021.05803372565
#> EM - RHLP: Iteration: 87 | log-likelihood: -2019.68013062929
#> EM - RHLP: Iteration: 88 | log-likelihood: -2018.57796815284
#> EM - RHLP: Iteration: 89 | log-likelihood: -2016.51065270015
#> EM - RHLP: Iteration: 90 | log-likelihood: -2015.84957111014
#> EM - RHLP: Iteration: 91 | log-likelihood: -2014.25626618564
#> EM - RHLP: Iteration: 92 | log-likelihood: -2012.83069679254
#> EM - RHLP: Iteration: 93 | log-likelihood: -2012.36700738444
#> EM - RHLP: Iteration: 94 | log-likelihood: -2010.80319327333
#> EM - RHLP: Iteration: 95 | log-likelihood: -2009.62231094925
#> EM - RHLP: Iteration: 96 | log-likelihood: -2009.18020396728
#> EM - RHLP: Iteration: 97 | log-likelihood: -2007.70135886708
#> EM - RHLP: Iteration: 98 | log-likelihood: -2006.56703696874
#> EM - RHLP: Iteration: 99 | log-likelihood: -2006.01673291469
#> EM - RHLP: Iteration: 100 | log-likelihood: -2004.41194242792
#> EM - RHLP: Iteration: 101 | log-likelihood: -2003.4625414477
#> EM - RHLP: Iteration: 102 | log-likelihood: -2002.88040058763
#> EM - RHLP: Iteration: 103 | log-likelihood: -2001.35926477816
#> EM - RHLP: Iteration: 104 | log-likelihood: -2000.57003100128
#> EM - RHLP: Iteration: 105 | log-likelihood: -2000.13742634303
#> EM - RHLP: Iteration: 106 | log-likelihood: -1998.8742667185
#> EM - RHLP: Iteration: 107 | log-likelihood: -1997.9672441114
#> EM - RHLP: Iteration: 108 | log-likelihood: -1997.53617878001
#> EM - RHLP: Iteration: 109 | log-likelihood: -1996.26856906479
#> EM - RHLP: Iteration: 110 | log-likelihood: -1995.29073069489
#> EM - RHLP: Iteration: 111 | log-likelihood: -1994.96901833912
#> EM - RHLP: Iteration: 112 | log-likelihood: -1994.04338389315
#> EM - RHLP: Iteration: 113 | log-likelihood: -1992.93228304533
#> EM - RHLP: Iteration: 114 | log-likelihood: -1992.58825334521
#> EM - RHLP: Iteration: 115 | log-likelihood: -1992.08820485443
#> EM - RHLP: Iteration: 116 | log-likelihood: -1990.99459284997
#> EM - RHLP: Iteration: 117 | log-likelihood: -1990.39820233453
#> EM - RHLP: Iteration: 118 | log-likelihood: -1990.25156085256
#> EM - RHLP: Iteration: 119 | log-likelihood: -1990.02689844513
#> EM - RHLP: Iteration: 120 | log-likelihood: -1989.4524459209
#> EM - RHLP: Iteration: 121 | log-likelihood: -1988.77939887023
#> EM - RHLP: Iteration: 122 | log-likelihood: -1988.43670301286
#> EM - RHLP: Iteration: 123 | log-likelihood: -1988.05097380424
#> EM - RHLP: Iteration: 124 | log-likelihood: -1987.13583867675
#> EM - RHLP: Iteration: 125 | log-likelihood: -1986.24508709354
#> EM - RHLP: Iteration: 126 | log-likelihood: -1985.66862327892
#> EM - RHLP: Iteration: 127 | log-likelihood: -1984.91555844651
#> EM - RHLP: Iteration: 128 | log-likelihood: -1984.02840365821
#> EM - RHLP: Iteration: 129 | log-likelihood: -1983.69130067161
#> EM - RHLP: Iteration: 130 | log-likelihood: -1983.59891631866
#> EM - RHLP: Iteration: 131 | log-likelihood: -1983.46950685882
#> EM - RHLP: Iteration: 132 | log-likelihood: -1983.16677154063
#> EM - RHLP: Iteration: 133 | log-likelihood: -1982.7130488681
#> EM - RHLP: Iteration: 134 | log-likelihood: -1982.36482921383
#> EM - RHLP: Iteration: 135 | log-likelihood: -1982.09501016661
#> EM - RHLP: Iteration: 136 | log-likelihood: -1981.45901315766
#> EM - RHLP: Iteration: 137 | log-likelihood: -1980.56116931257
#> EM - RHLP: Iteration: 138 | log-likelihood: -1979.78682525118
#> EM - RHLP: Iteration: 139 | log-likelihood: -1978.57039689029
#> EM - RHLP: Iteration: 140 | log-likelihood: -1977.62583903156
#> EM - RHLP: Iteration: 141 | log-likelihood: -1976.44993964017
#> EM - RHLP: Iteration: 142 | log-likelihood: -1975.34352117182
#> EM - RHLP: Iteration: 143 | log-likelihood: -1973.94511304916
#> EM - RHLP: Iteration: 144 | log-likelihood: -1972.69707782729
#> EM - RHLP: Iteration: 145 | log-likelihood: -1971.24412635765
#> EM - RHLP: Iteration: 146 | log-likelihood: -1970.06230181165
#> EM - RHLP: Iteration: 147 | log-likelihood: -1968.63106242841
#> EM - RHLP: Iteration: 148 | log-likelihood: -1967.54773416029
#> EM - RHLP: Iteration: 149 | log-likelihood: -1966.19481640747
#> EM - RHLP: Iteration: 150 | log-likelihood: -1965.07487280506
#> EM - RHLP: Iteration: 151 | log-likelihood: -1963.69466194804
#> EM - RHLP: Iteration: 152 | log-likelihood: -1962.43103040224
#> EM - RHLP: Iteration: 153 | log-likelihood: -1961.13942311651
#> EM - RHLP: Iteration: 154 | log-likelihood: -1959.76348415393
#> EM - RHLP: Iteration: 155 | log-likelihood: -1958.66111557445
#> EM - RHLP: Iteration: 156 | log-likelihood: -1957.08412155615
#> EM - RHLP: Iteration: 157 | log-likelihood: -1956.38405033098
#> EM - RHLP: Iteration: 158 | log-likelihood: -1955.13976323662
#> EM - RHLP: Iteration: 159 | log-likelihood: -1954.0307602366
#> EM - RHLP: Iteration: 160 | log-likelihood: -1953.28771131999
#> EM - RHLP: Iteration: 161 | log-likelihood: -1951.68947232015
#> EM - RHLP: Iteration: 162 | log-likelihood: -1950.97779043109
#> EM - RHLP: Iteration: 163 | log-likelihood: -1950.82786273359
#> EM - RHLP: Iteration: 164 | log-likelihood: -1950.39568293481
#> EM - RHLP: Iteration: 165 | log-likelihood: -1949.51404624208
#> EM - RHLP: Iteration: 166 | log-likelihood: -1948.906374824
#> EM - RHLP: Iteration: 167 | log-likelihood: -1948.43487893552
#> EM - RHLP: Iteration: 168 | log-likelihood: -1947.2118394595
#> EM - RHLP: Iteration: 169 | log-likelihood: -1946.34871715855
#> EM - RHLP: Iteration: 170 | log-likelihood: -1946.22041468711
#> EM - RHLP: Iteration: 171 | log-likelihood: -1946.2132265072
#> EM - RHLP: 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()
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 RHLP model for the following time series Y:
data("toydataset")
x <- toydataset$x
y <- toydataset$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$plot(what = "estimatedsignal")
fchamroukhi/RHLP documentation built on Sept. 19, 2019, 8:04 a.m.
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Overview
Flexible and user-friendly probabilistic segmentation of time series with smooth and/or abrupt regime changes by a mixture model-based regression approach with a hidden logistic process, fitted by the EM algorithm.
Installation
You can install the RHLP package from GitHub with:
# install.packages("devtools")
devtools::install_github("fchamroukhi/RHLP")
To build vignettes for examples of usage, type the command below instead:
# install.packages("devtools")
devtools::install_github("fchamroukhi/RHLP",
build_opts = c("--no-resave-data", "--no-manual"),
build_vignettes = TRUE)
Use the following command to display vignettes:
browseVignettes("RHLP")
Usage
library(RHLP)
# Application to a toy data set
data("toydataset")
x <- toydataset$x
y <- toydataset$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 - RHLP: Iteration: 1 | log-likelihood: -2119.27308534609
#> EM - RHLP: Iteration: 2 | log-likelihood: -1149.01040321999
#> EM - RHLP: Iteration: 3 | log-likelihood: -1118.20384281234
#> EM - RHLP: Iteration: 4 | log-likelihood: -1096.88260636121
#> EM - RHLP: Iteration: 5 | log-likelihood: -1067.55719357295
#> EM - RHLP: Iteration: 6 | log-likelihood: -1037.26620122646
#> EM - RHLP: Iteration: 7 | log-likelihood: -1022.71743069484
#> EM - RHLP: Iteration: 8 | log-likelihood: -1006.11825447077
#> EM - RHLP: Iteration: 9 | log-likelihood: -1001.18491883952
#> EM - RHLP: Iteration: 10 | log-likelihood: -1000.91250763556
#> EM - RHLP: Iteration: 11 | log-likelihood: -1000.62280600209
#> EM - RHLP: Iteration: 12 | log-likelihood: -1000.3030988811
#> EM - RHLP: Iteration: 13 | log-likelihood: -999.932334880131
#> EM - RHLP: Iteration: 14 | log-likelihood: -999.484219706691
#> EM - RHLP: Iteration: 15 | log-likelihood: -998.928118038989
#> EM - RHLP: Iteration: 16 | log-likelihood: -998.234244664472
#> EM - RHLP: Iteration: 17 | log-likelihood: -997.359536276056
#> EM - RHLP: Iteration: 18 | log-likelihood: -996.152654857298
#> EM - RHLP: Iteration: 19 | log-likelihood: -994.697863447307
#> EM - RHLP: Iteration: 20 | log-likelihood: -993.186583974542
#> EM - RHLP: Iteration: 21 | log-likelihood: -991.81352379631
#> EM - RHLP: Iteration: 22 | log-likelihood: -990.611295217008
#> EM - RHLP: Iteration: 23 | log-likelihood: -989.539226273251
#> EM - RHLP: Iteration: 24 | log-likelihood: -988.55311887915
#> EM - RHLP: Iteration: 25 | log-likelihood: -987.539963690533
#> EM - RHLP: Iteration: 26 | log-likelihood: -986.073920116541
#> EM - RHLP: Iteration: 27 | log-likelihood: -983.263549878169
#> EM - RHLP: Iteration: 28 | log-likelihood: -979.340492188909
#> EM - RHLP: Iteration: 29 | log-likelihood: -977.468559852711
#> EM - RHLP: Iteration: 30 | log-likelihood: -976.653534236095
#> EM - RHLP: Iteration: 31 | log-likelihood: -976.5893387433
#> EM - RHLP: 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$plot()
# Application to a real data set
data("realdataset")
x <- realdataset$x
y <- realdataset$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 - RHLP: Iteration: 1 | log-likelihood: -3321.6485760125
#> EM - RHLP: Iteration: 2 | log-likelihood: -2286.48632282875
#> EM - RHLP: Iteration: 3 | log-likelihood: -2257.60498391374
#> EM - RHLP: Iteration: 4 | log-likelihood: -2243.74506764308
#> EM - RHLP: Iteration: 5 | log-likelihood: -2233.3426635247
#> EM - RHLP: Iteration: 6 | log-likelihood: -2226.89953345319
#> EM - RHLP: Iteration: 7 | log-likelihood: -2221.77999023589
#> EM - RHLP: Iteration: 8 | log-likelihood: -2215.81305295291
#> EM - RHLP: Iteration: 9 | log-likelihood: -2208.25998029539
#> EM - RHLP: Iteration: 10 | log-likelihood: -2196.27872403055
#> EM - RHLP: Iteration: 11 | log-likelihood: -2185.40049009242
#> EM - RHLP: Iteration: 12 | log-likelihood: -2180.13934245387
#> EM - RHLP: Iteration: 13 | log-likelihood: -2175.4276274402
#> EM - RHLP: Iteration: 14 | log-likelihood: -2170.86113669353
#> EM - RHLP: Iteration: 15 | log-likelihood: -2165.34927170608
#> EM - RHLP: Iteration: 16 | log-likelihood: -2161.12419211511
#> EM - RHLP: Iteration: 17 | log-likelihood: -2158.63709280617
#> EM - RHLP: Iteration: 18 | log-likelihood: -2156.19846850913
#> EM - RHLP: Iteration: 19 | log-likelihood: -2154.04107470071
#> EM - RHLP: Iteration: 20 | log-likelihood: -2153.24544245686
#> EM - RHLP: Iteration: 21 | log-likelihood: -2151.74944795242
#> EM - RHLP: Iteration: 22 | log-likelihood: -2149.90781423151
#> EM - RHLP: Iteration: 23 | log-likelihood: -2146.40042232588
#> EM - RHLP: Iteration: 24 | log-likelihood: -2142.37530025533
#> EM - RHLP: Iteration: 25 | log-likelihood: -2134.85493291884
#> EM - RHLP: Iteration: 26 | log-likelihood: -2129.67399002071
#> EM - RHLP: Iteration: 27 | log-likelihood: -2126.44739300481
#> EM - RHLP: Iteration: 28 | log-likelihood: -2124.94603052064
#> EM - RHLP: Iteration: 29 | log-likelihood: -2122.51637426267
#> EM - RHLP: Iteration: 30 | log-likelihood: -2121.01493646146
#> EM - RHLP: Iteration: 31 | log-likelihood: -2118.45402063643
#> EM - RHLP: Iteration: 32 | log-likelihood: -2116.9336204919
#> EM - RHLP: Iteration: 33 | log-likelihood: -2114.34424563452
#> EM - RHLP: Iteration: 34 | log-likelihood: -2112.84844186712
#> EM - RHLP: Iteration: 35 | log-likelihood: -2110.34494568025
#> EM - RHLP: Iteration: 36 | log-likelihood: -2108.81734757025
#> EM - RHLP: Iteration: 37 | log-likelihood: -2106.26527191053
#> EM - RHLP: Iteration: 38 | log-likelihood: -2104.96591147986
#> EM - RHLP: Iteration: 39 | log-likelihood: -2102.43927829964
#> EM - RHLP: Iteration: 40 | log-likelihood: -2101.27820194404
#> EM - RHLP: Iteration: 41 | log-likelihood: -2098.81151697567
#> EM - RHLP: Iteration: 42 | log-likelihood: -2097.48008514591
#> EM - RHLP: Iteration: 43 | log-likelihood: -2094.98259556552
#> EM - RHLP: Iteration: 44 | log-likelihood: -2093.66517040802
#> EM - RHLP: Iteration: 45 | log-likelihood: -2091.23625905564
#> EM - RHLP: Iteration: 46 | log-likelihood: -2089.91118603989
#> EM - RHLP: Iteration: 47 | log-likelihood: -2087.67388435026
#> EM - RHLP: Iteration: 48 | log-likelihood: -2086.11373786756
#> EM - RHLP: Iteration: 49 | log-likelihood: -2083.84931461869
#> EM - RHLP: Iteration: 50 | log-likelihood: -2082.16175664198
#> EM - RHLP: Iteration: 51 | log-likelihood: -2080.45137011098
#> EM - RHLP: Iteration: 52 | log-likelihood: -2078.37066132008
#> EM - RHLP: Iteration: 53 | log-likelihood: -2077.06827662071
#> EM - RHLP: Iteration: 54 | log-likelihood: -2074.66718553694
#> EM - RHLP: Iteration: 55 | log-likelihood: -2073.68137124781
#> EM - RHLP: Iteration: 56 | log-likelihood: -2071.20390017789
#> EM - RHLP: Iteration: 57 | log-likelihood: -2069.88260759288
#> EM - RHLP: Iteration: 58 | log-likelihood: -2067.30246728287
#> EM - RHLP: Iteration: 59 | log-likelihood: -2066.08897944236
#> EM - RHLP: Iteration: 60 | log-likelihood: -2064.14482062792
#> EM - RHLP: Iteration: 61 | log-likelihood: -2062.39859624374
#> EM - RHLP: Iteration: 62 | log-likelihood: -2060.73756242314
#> EM - RHLP: Iteration: 63 | log-likelihood: -2058.4448132974
#> EM - RHLP: Iteration: 64 | log-likelihood: -2057.23564743141
#> EM - RHLP: Iteration: 65 | log-likelihood: -2054.73129678764
#> EM - RHLP: Iteration: 66 | log-likelihood: -2053.66525147972
#> EM - RHLP: Iteration: 67 | log-likelihood: -2051.05262427909
#> EM - RHLP: Iteration: 68 | log-likelihood: -2049.89030367995
#> EM - RHLP: Iteration: 69 | log-likelihood: -2047.68843285481
#> EM - RHLP: Iteration: 70 | log-likelihood: -2046.16052536146
#> EM - RHLP: Iteration: 71 | log-likelihood: -2044.92677581091
#> EM - RHLP: Iteration: 72 | log-likelihood: -2042.67687818721
#> EM - RHLP: Iteration: 73 | log-likelihood: -2041.77608506749
#> EM - RHLP: Iteration: 74 | log-likelihood: -2039.40345316134
#> EM - RHLP: Iteration: 75 | log-likelihood: -2038.20062153928
#> EM - RHLP: Iteration: 76 | log-likelihood: -2036.05846372404
#> EM - RHLP: Iteration: 77 | log-likelihood: -2034.52492449426
#> EM - RHLP: Iteration: 78 | log-likelihood: -2033.44774900177
#> EM - RHLP: Iteration: 79 | log-likelihood: -2031.15837908019
#> EM - RHLP: Iteration: 80 | log-likelihood: -2030.29908045026
#> EM - RHLP: Iteration: 81 | log-likelihood: -2028.08193331457
#> EM - RHLP: Iteration: 82 | log-likelihood: -2026.82779637097
#> EM - RHLP: Iteration: 83 | log-likelihood: -2025.51219569808
#> EM - RHLP: Iteration: 84 | log-likelihood: -2023.47136697978
#> EM - RHLP: Iteration: 85 | log-likelihood: -2022.86702240332
#> EM - RHLP: Iteration: 86 | log-likelihood: -2021.05803372565
#> EM - RHLP: Iteration: 87 | log-likelihood: -2019.68013062929
#> EM - RHLP: Iteration: 88 | log-likelihood: -2018.57796815284
#> EM - RHLP: Iteration: 89 | log-likelihood: -2016.51065270015
#> EM - RHLP: Iteration: 90 | log-likelihood: -2015.84957111014
#> EM - RHLP: Iteration: 91 | log-likelihood: -2014.25626618564
#> EM - RHLP: Iteration: 92 | log-likelihood: -2012.83069679254
#> EM - RHLP: Iteration: 93 | log-likelihood: -2012.36700738444
#> EM - RHLP: Iteration: 94 | log-likelihood: -2010.80319327333
#> EM - RHLP: Iteration: 95 | log-likelihood: -2009.62231094925
#> EM - RHLP: Iteration: 96 | log-likelihood: -2009.18020396728
#> EM - RHLP: Iteration: 97 | log-likelihood: -2007.70135886708
#> EM - RHLP: Iteration: 98 | log-likelihood: -2006.56703696874
#> EM - RHLP: Iteration: 99 | log-likelihood: -2006.01673291469
#> EM - RHLP: Iteration: 100 | log-likelihood: -2004.41194242792
#> EM - RHLP: Iteration: 101 | log-likelihood: -2003.4625414477
#> EM - RHLP: Iteration: 102 | log-likelihood: -2002.88040058763
#> EM - RHLP: Iteration: 103 | log-likelihood: -2001.35926477816
#> EM - RHLP: Iteration: 104 | log-likelihood: -2000.57003100128
#> EM - RHLP: Iteration: 105 | log-likelihood: -2000.13742634303
#> EM - RHLP: Iteration: 106 | log-likelihood: -1998.8742667185
#> EM - RHLP: Iteration: 107 | log-likelihood: -1997.9672441114
#> EM - RHLP: Iteration: 108 | log-likelihood: -1997.53617878001
#> EM - RHLP: Iteration: 109 | log-likelihood: -1996.26856906479
#> EM - RHLP: Iteration: 110 | log-likelihood: -1995.29073069489
#> EM - RHLP: Iteration: 111 | log-likelihood: -1994.96901833912
#> EM - RHLP: Iteration: 112 | log-likelihood: -1994.04338389315
#> EM - RHLP: Iteration: 113 | log-likelihood: -1992.93228304533
#> EM - RHLP: Iteration: 114 | log-likelihood: -1992.58825334521
#> EM - RHLP: Iteration: 115 | log-likelihood: -1992.08820485443
#> EM - RHLP: Iteration: 116 | log-likelihood: -1990.99459284997
#> EM - RHLP: Iteration: 117 | log-likelihood: -1990.39820233453
#> EM - RHLP: Iteration: 118 | log-likelihood: -1990.25156085256
#> EM - RHLP: Iteration: 119 | log-likelihood: -1990.02689844513
#> EM - RHLP: Iteration: 120 | log-likelihood: -1989.4524459209
#> EM - RHLP: Iteration: 121 | log-likelihood: -1988.77939887023
#> EM - RHLP: Iteration: 122 | log-likelihood: -1988.43670301286
#> EM - RHLP: Iteration: 123 | log-likelihood: -1988.05097380424
#> EM - RHLP: Iteration: 124 | log-likelihood: -1987.13583867675
#> EM - RHLP: Iteration: 125 | log-likelihood: -1986.24508709354
#> EM - RHLP: Iteration: 126 | log-likelihood: -1985.66862327892
#> EM - RHLP: Iteration: 127 | log-likelihood: -1984.91555844651
#> EM - RHLP: Iteration: 128 | log-likelihood: -1984.02840365821
#> EM - RHLP: Iteration: 129 | log-likelihood: -1983.69130067161
#> EM - RHLP: Iteration: 130 | log-likelihood: -1983.59891631866
#> EM - RHLP: Iteration: 131 | log-likelihood: -1983.46950685882
#> EM - RHLP: Iteration: 132 | log-likelihood: -1983.16677154063
#> EM - RHLP: Iteration: 133 | log-likelihood: -1982.7130488681
#> EM - RHLP: Iteration: 134 | log-likelihood: -1982.36482921383
#> EM - RHLP: Iteration: 135 | log-likelihood: -1982.09501016661
#> EM - RHLP: Iteration: 136 | log-likelihood: -1981.45901315766
#> EM - RHLP: Iteration: 137 | log-likelihood: -1980.56116931257
#> EM - RHLP: Iteration: 138 | log-likelihood: -1979.78682525118
#> EM - RHLP: Iteration: 139 | log-likelihood: -1978.57039689029
#> EM - RHLP: Iteration: 140 | log-likelihood: -1977.62583903156
#> EM - RHLP: Iteration: 141 | log-likelihood: -1976.44993964017
#> EM - RHLP: Iteration: 142 | log-likelihood: -1975.34352117182
#> EM - RHLP: Iteration: 143 | log-likelihood: -1973.94511304916
#> EM - RHLP: Iteration: 144 | log-likelihood: -1972.69707782729
#> EM - RHLP: Iteration: 145 | log-likelihood: -1971.24412635765
#> EM - RHLP: Iteration: 146 | log-likelihood: -1970.06230181165
#> EM - RHLP: Iteration: 147 | log-likelihood: -1968.63106242841
#> EM - RHLP: Iteration: 148 | log-likelihood: -1967.54773416029
#> EM - RHLP: Iteration: 149 | log-likelihood: -1966.19481640747
#> EM - RHLP: Iteration: 150 | log-likelihood: -1965.07487280506
#> EM - RHLP: Iteration: 151 | log-likelihood: -1963.69466194804
#> EM - RHLP: Iteration: 152 | log-likelihood: -1962.43103040224
#> EM - RHLP: Iteration: 153 | log-likelihood: -1961.13942311651
#> EM - RHLP: Iteration: 154 | log-likelihood: -1959.76348415393
#> EM - RHLP: Iteration: 155 | log-likelihood: -1958.66111557445
#> EM - RHLP: Iteration: 156 | log-likelihood: -1957.08412155615
#> EM - RHLP: Iteration: 157 | log-likelihood: -1956.38405033098
#> EM - RHLP: Iteration: 158 | log-likelihood: -1955.13976323662
#> EM - RHLP: Iteration: 159 | log-likelihood: -1954.0307602366
#> EM - RHLP: Iteration: 160 | log-likelihood: -1953.28771131999
#> EM - RHLP: Iteration: 161 | log-likelihood: -1951.68947232015
#> EM - RHLP: Iteration: 162 | log-likelihood: -1950.97779043109
#> EM - RHLP: Iteration: 163 | log-likelihood: -1950.82786273359
#> EM - RHLP: Iteration: 164 | log-likelihood: -1950.39568293481
#> EM - RHLP: Iteration: 165 | log-likelihood: -1949.51404624208
#> EM - RHLP: Iteration: 166 | log-likelihood: -1948.906374824
#> EM - RHLP: Iteration: 167 | log-likelihood: -1948.43487893552
#> EM - RHLP: Iteration: 168 | log-likelihood: -1947.2118394595
#> EM - RHLP: Iteration: 169 | log-likelihood: -1946.34871715855
#> EM - RHLP: Iteration: 170 | log-likelihood: -1946.22041468711
#> EM - RHLP: Iteration: 171 | log-likelihood: -1946.2132265072
#> EM - RHLP: 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$plot()
Model selection
In this package, it is possible to select models based on information criteria such as BIC, AIC and ICL.
The selection can be done for the two following parameters:
- K: The number of regimes;
- p: The order of the polynomial regression.
Let’s select a RHLP model for the following time series Y:
data("toydataset")
x <- toydataset$x
y <- toydataset$y
plot(x, y, type = "l", xlab = "x", ylab = "Y")
selectedrhlp <- selectRHLP(X = x, Y = y, Kmin = 2, Kmax = 6, pmin = 0, pmax = 3)
#> The RHLP model selected via the "BIC" has K = 5 regimes
#> and the order of the polynomial regression is p = 0.
#> BIC = -1041.40789532438
#> AIC = -1000.84239591291
selectedrhlp$plot(what = "estimatedsignal")
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