EM Algorithm for Mixtures of Univariate Normals

Return EM algorithm output for mixtures of normal distributions.

Usage

normalmixEM(x, lambda = NULL, mu = NULL, sigma = NULL, k = 2, 
            mean.constr = NULL, sd.constr = NULL,
            epsilon = 1e-08, maxit = 1000, maxrestarts = 20, 
            verb = FALSE, fast = FALSE, ECM = FALSE,
            arbmean = TRUE, arbvar = TRUE)

Details

This is the standard EM algorithm for normal mixtures that maximizes the conditional expected complete-data log-likelihood at each M-step of the algorithm. If desired, the EM algorithm may be replaced by an ECM algorithm (see ECM argument) that alternates between maximizing with respect to the mu and lambda while holding sigma fixed, and maximizing with respect to sigma and lambda while holding mu fixed. In the case where arbmean is FALSE and arbvar is TRUE, there is no closed-form EM algorithm, so the ECM option is forced in this case.

Arguments

x: A vector of length n consisting of the data.
lambda: Initial value of mixing proportions. Automatically repeated as necessary to produce a vector of length k, then normalized to sum to 1. If NULL, then lambda is random from a uniform Dirichlet distribution (i.e., its entries are uniform random and then it is normalized to sum to 1).
mu: Starting value of vector of component means. If non-NULL and a scalar, arbmean is set to FALSE. If non-NULL and a vector, k is set to length(mu). If NULL, then the initial value is randomly generated from a normal distribution with center(s) determined by binning the data.
sigma: Starting value of vector of component standard deviations for algorithm. If non-NULL and a scalar, arbvar is set to FALSE. If non-NULL and a vector, arbvar is set to TRUE and k is set to length(sigma). If NULL, then the initial value is the reciprocal of the square root of a vector of random exponential-distribution values whose means are determined according to a binning method done on the data.
k: Number of components. Initial value ignored unless mu and sigma are both NULL.
mean.constr: Equality constraints on the mean parameters, given as a vector of length k. Each vector entry helps specify the constraints, if any, on the corresponding mean parameter: If NA, the corresponding parameter is unconstrained. If numeric, the corresponding parameter is fixed at that value. If a character string consisting of a single character preceded by a coefficient, such as "0.5a" or "-b", all parameters using the same single character in their constraints will fix these parameters equal to the coefficient times some the same free parameter. For instance, if mean.constr = c(NA, 0, "a", "-a"), then the first mean parameter is unconstrained, the second is fixed at zero, and the third and forth are constrained to be equal and opposite in sign.
sd.constr: Equality constraints on the standard deviation parameters. See mean.constr.
epsilon: The convergence criterion. Convergence is declared when the change in the observed data log-likelihood increases by less than epsilon.
maxit: The maximum number of iterations.
maxrestarts: The maximum number of restarts allowed in case of a problem with the particular starting values chosen due to one of the variance estimates getting too small (each restart uses randomly chosen starting values). It is well-known that when each component of a normal mixture may have its own mean and variance, the likelihood has no maximizer; in such cases, we hope to find a "nice" local maximum with this algorithm instead, but occasionally the algorithm finds a "not nice" solution and one of the variances goes to zero, driving the likelihood to infinity.
verb: If TRUE, then various updates are printed during each iteration of the algorithm.
fast: If TRUE and k==2 and arbmean==TRUE, then use normalmixEM2comp, which is a much faster version of the EM algorithm for this case. This version is less protected against certain kinds of underflow that can cause numerical problems and it does not permit any restarts. If k>2, fast is ignored.
ECM: logical: Should this algorithm be an ECM algorithm in the sense of Meng and Rubin (1993)? If FALSE, the algorithm is a true EM algorithm; if TRUE, then every half-iteration alternately updates the means conditional on the variances or the variances conditional on the means, with an extra E-step in between these updates.
arbmean: If TRUE, then the component densities are allowed to have different mus. If FALSE, then a scale mixture will be fit. Initial value ignored unless mu is NULL.
arbvar: If TRUE, then the component densities are allowed to have different sigmas. If FALSE, then a location mixture will be fit. Initial value ignored unless sigma is NULL.

Value

normalmixEM returns a list of class mixEM with items:

x: The raw data.
lambda: The final mixing proportions.
mu: The final mean parameters.
sigma: The final standard deviations. If arbmean = FALSE, then only the smallest standard deviation is returned. See scale below.
scale: If arbmean = FALSE, then the scale factor for the component standard deviations is returned. Otherwise, this is omitted from the output.
loglik: The final log-likelihood.
posterior: An nxk matrix of posterior probabilities for observations.
all.loglik: A vector of each iteration's log-likelihood. This vector includes both the initial and the final values; thus, the number of iterations is one less than its length.
restarts: The number of times the algorithm restarted due to unacceptable choice of initial values.
ft: A character vector giving the name of the function.

References

McLachlan, G. J. and Peel, D. (2000) Finite Mixture Models, John Wiley and Sons, Inc.
Meng, X.-L. and Rubin, D. B. (1993) Maximum Likelihood Estimation Via the ECM Algorithm: A General Framework, Biometrika 80(2): 267-278.
Benaglia, T., Chauveau, D., Hunter, D. R., and Young, D. mixtools: An R package for analyzing finite mixture models. Journal of Statistical Software, 32(6):1-29, 2009.

Examples

##Analyzing the Old Faithful geyser data with a 2-component mixture of normals.

data(faithful)
attach(faithful)
set.seed(100)
system.time(out<-normalmixEM(waiting, arbvar = FALSE, epsilon = 1e-03))
#> number of iterations= 8 
#>    user  system elapsed 
#>   0.002   0.000   0.003 
out
#> $x
#>   [1] 79 54 74 62 85 55 88 85 51 85 54 84 78 47 83 52 62 84 52 79 51 47 78 69 74
#>  [26] 83 55 76 78 79 73 77 66 80 74 52 48 80 59 90 80 58 84 58 73 83 64 53 82 59
#>  [51] 75 90 54 80 54 83 71 64 77 81 59 84 48 82 60 92 78 78 65 73 82 56 79 71 62
#>  [76] 76 60 78 76 83 75 82 70 65 73 88 76 80 48 86 60 90 50 78 63 72 84 75 51 82
#> [101] 62 88 49 83 81 47 84 52 86 81 75 59 89 79 59 81 50 85 59 87 53 69 77 56 88
#> [126] 81 45 82 55 90 45 83 56 89 46 82 51 86 53 79 81 60 82 77 76 59 80 49 96 53
#> [151] 77 77 65 81 71 70 81 93 53 89 45 86 58 78 66 76 63 88 52 93 49 57 77 68 81
#> [176] 81 73 50 85 74 55 77 83 83 51 78 84 46 83 55 81 57 76 84 77 81 87 77 51 78
#> [201] 60 82 91 53 78 46 77 84 49 83 71 80 49 75 64 76 53 94 55 76 50 82 54 75 78
#> [226] 79 78 78 70 79 70 54 86 50 90 54 54 77 79 64 75 47 86 63 85 82 57 82 67 74
#> [251] 54 83 73 73 88 80 71 83 56 79 78 84 58 83 43 60 75 81 46 90 46 74
#> 
#> $lambda
#> [1] 0.3606935 0.6393065
#> 
#> $mu
#> [1] 54.60884 80.08679
#> 
#> $sigma
#> [1] 5.870228 5.870228
#> 
#> $loglik
#> [1] -1034.002
#> 
#> $posterior
#>              comp.1       comp.2
#>   [1,] 1.023014e-04 9.998977e-01
#>   [2,] 9.999083e-01 9.174080e-05
#>   [3,] 4.108021e-03 9.958920e-01
#>   [4,] 9.671243e-01 3.287574e-02
#>   [5,] 1.211525e-06 9.999988e-01
#>   [6,] 9.998079e-01 1.921400e-04
#>   [7,] 1.318367e-07 9.999999e-01
#>   [8,] 1.211525e-06 9.999988e-01
#>   [9,] 9.999900e-01 9.983929e-06
#>  [10,] 1.211525e-06 9.999988e-01
#>  [11,] 9.999083e-01 9.174080e-05
#>  [12,] 2.537644e-06 9.999975e-01
#>  [13,] 2.142555e-04 9.997857e-01
#>  [14,] 9.999995e-01 5.186927e-07
#>  [15,] 5.315308e-06 9.999947e-01
#>  [16,] 9.999791e-01 2.091200e-05
#>  [17,] 9.671243e-01 3.287574e-02
#>  [18,] 2.537644e-06 9.999975e-01
#>  [19,] 9.999791e-01 2.091200e-05
#>  [20,] 1.023014e-04 9.998977e-01
#>  [21,] 9.999900e-01 9.983929e-06
#>  [22,] 9.999995e-01 5.186927e-07
#>  [23,] 2.142555e-04 9.997857e-01
#>  [24,] 1.425940e-01 8.574060e-01
#>  [25,] 4.108021e-03 9.958920e-01
#>  [26,] 5.315308e-06 9.999947e-01
#>  [27,] 9.998079e-01 1.921400e-04
#>  [28,] 9.393222e-04 9.990607e-01
#>  [29,] 2.142555e-04 9.997857e-01
#>  [30,] 1.023014e-04 9.998977e-01
#>  [31,] 8.566098e-03 9.914339e-01
#>  [32,] 4.486720e-04 9.995513e-01
#>  [33,] 6.044788e-01 3.955212e-01
#>  [34,] 4.884342e-05 9.999512e-01
#>  [35,] 4.108021e-03 9.958920e-01
#>  [36,] 9.999791e-01 2.091200e-05
#>  [37,] 9.999989e-01 1.086448e-06
#>  [38,] 4.884342e-05 9.999512e-01
#>  [39,] 9.963145e-01 3.685472e-03
#>  [40,] 3.004960e-08 1.000000e+00
#>  [41,] 4.884342e-05 9.999512e-01
#>  [42,] 9.982371e-01 1.762916e-03
#>  [43,] 2.537644e-06 9.999975e-01
#>  [44,] 9.982371e-01 1.762916e-03
#>  [45,] 8.566098e-03 9.914339e-01
#>  [46,] 5.315308e-06 9.999947e-01
#>  [47,] 8.702168e-01 1.297832e-01
#>  [48,] 9.999562e-01 4.380105e-05
#>  [49,] 1.113332e-05 9.999889e-01
#>  [50,] 9.963145e-01 3.685472e-03
#>  [51,] 1.965473e-03 9.980345e-01
#>  [52,] 3.004960e-08 1.000000e+00
#>  [53,] 9.999083e-01 9.174080e-05
#>  [54,] 4.884342e-05 9.999512e-01
#>  [55,] 9.999083e-01 9.174080e-05
#>  [56,] 5.315308e-06 9.999947e-01
#>  [57,] 3.652235e-02 9.634777e-01
#>  [58,] 8.702168e-01 1.297832e-01
#>  [59,] 4.486720e-04 9.995513e-01
#>  [60,] 2.331945e-05 9.999767e-01
#>  [61,] 9.963145e-01 3.685472e-03
#>  [62,] 2.537644e-06 9.999975e-01
#>  [63,] 9.999989e-01 1.086448e-06
#>  [64,] 1.113332e-05 9.999889e-01
#>  [65,] 9.923115e-01 7.688534e-03
#>  [66,] 6.849218e-09 1.000000e+00
#>  [67,] 2.142555e-04 9.997857e-01
#>  [68,] 2.142555e-04 9.997857e-01
#>  [69,] 7.619717e-01 2.380283e-01
#>  [70,] 8.566098e-03 9.914339e-01
#>  [71,] 1.113332e-05 9.999889e-01
#>  [72,] 9.995976e-01 4.023697e-04
#>  [73,] 1.023014e-04 9.998977e-01
#>  [74,] 3.652235e-02 9.634777e-01
#>  [75,] 9.671243e-01 3.287574e-02
#>  [76,] 9.393222e-04 9.990607e-01
#>  [77,] 9.923115e-01 7.688534e-03
#>  [78,] 2.142555e-04 9.997857e-01
#>  [79,] 9.393222e-04 9.990607e-01
#>  [80,] 5.315308e-06 9.999947e-01
#>  [81,] 1.965473e-03 9.980345e-01
#>  [82,] 1.113332e-05 9.999889e-01
#>  [83,] 7.355866e-02 9.264413e-01
#>  [84,] 7.619717e-01 2.380283e-01
#>  [85,] 8.566098e-03 9.914339e-01
#>  [86,] 1.318367e-07 9.999999e-01
#>  [87,] 9.393222e-04 9.990607e-01
#>  [88,] 4.884342e-05 9.999512e-01
#>  [89,] 9.999989e-01 1.086448e-06
#>  [90,] 5.784074e-07 9.999994e-01
#>  [91,] 9.923115e-01 7.688534e-03
#>  [92,] 3.004960e-08 1.000000e+00
#>  [93,] 9.999952e-01 4.766558e-06
#>  [94,] 2.142555e-04 9.997857e-01
#>  [95,] 9.335308e-01 6.646924e-02
#>  [96,] 1.777579e-02 9.822242e-01
#>  [97,] 2.537644e-06 9.999975e-01
#>  [98,] 1.965473e-03 9.980345e-01
#>  [99,] 9.999900e-01 9.983929e-06
#> [100,] 1.113332e-05 9.999889e-01
#> [101,] 9.671243e-01 3.287574e-02
#> [102,] 1.318367e-07 9.999999e-01
#> [103,] 9.999977e-01 2.275659e-06
#> [104,] 5.315308e-06 9.999947e-01
#> [105,] 2.331945e-05 9.999767e-01
#> [106,] 9.999995e-01 5.186927e-07
#> [107,] 2.537644e-06 9.999975e-01
#> [108,] 9.999791e-01 2.091200e-05
#> [109,] 5.784074e-07 9.999994e-01
#> [110,] 2.331945e-05 9.999767e-01
#> [111,] 1.965473e-03 9.980345e-01
#> [112,] 9.963145e-01 3.685472e-03
#> [113,] 6.294156e-08 9.999999e-01
#> [114,] 1.023014e-04 9.998977e-01
#> [115,] 9.963145e-01 3.685472e-03
#> [116,] 2.331945e-05 9.999767e-01
#> [117,] 9.999952e-01 4.766558e-06
#> [118,] 1.211525e-06 9.999988e-01
#> [119,] 9.963145e-01 3.685472e-03
#> [120,] 2.761437e-07 9.999997e-01
#> [121,] 9.999562e-01 4.380105e-05
#> [122,] 1.425940e-01 8.574060e-01
#> [123,] 4.486720e-04 9.995513e-01
#> [124,] 9.995976e-01 4.023697e-04
#> [125,] 1.318367e-07 9.999999e-01
#> [126,] 2.331945e-05 9.999767e-01
#> [127,] 9.999999e-01 1.182259e-07
#> [128,] 1.113332e-05 9.999889e-01
#> [129,] 9.998079e-01 1.921400e-04
#> [130,] 3.004960e-08 1.000000e+00
#> [131,] 9.999999e-01 1.182259e-07
#> [132,] 5.315308e-06 9.999947e-01
#> [133,] 9.995976e-01 4.023697e-04
#> [134,] 6.294156e-08 9.999999e-01
#> [135,] 9.999998e-01 2.476346e-07
#> [136,] 1.113332e-05 9.999889e-01
#> [137,] 9.999900e-01 9.983929e-06
#> [138,] 5.784074e-07 9.999994e-01
#> [139,] 9.999562e-01 4.380105e-05
#> [140,] 1.023014e-04 9.998977e-01
#> [141,] 2.331945e-05 9.999767e-01
#> [142,] 9.923115e-01 7.688534e-03
#> [143,] 1.113332e-05 9.999889e-01
#> [144,] 4.486720e-04 9.995513e-01
#> [145,] 9.393222e-04 9.990607e-01
#> [146,] 9.963145e-01 3.685472e-03
#> [147,] 4.884342e-05 9.999512e-01
#> [148,] 9.999977e-01 2.275659e-06
#> [149,] 3.558324e-10 1.000000e+00
#> [150,] 9.999562e-01 4.380105e-05
#> [151,] 4.486720e-04 9.995513e-01
#> [152,] 4.486720e-04 9.995513e-01
#> [153,] 7.619717e-01 2.380283e-01
#> [154,] 2.331945e-05 9.999767e-01
#> [155,] 3.652235e-02 9.634777e-01
#> [156,] 7.355866e-02 9.264413e-01
#> [157,] 2.331945e-05 9.999767e-01
#> [158,] 3.269958e-09 1.000000e+00
#> [159,] 9.999562e-01 4.380105e-05
#> [160,] 6.294156e-08 9.999999e-01
#> [161,] 9.999999e-01 1.182259e-07
#> [162,] 5.784074e-07 9.999994e-01
#> [163,] 9.982371e-01 1.762916e-03
#> [164,] 2.142555e-04 9.997857e-01
#> [165,] 6.044788e-01 3.955212e-01
#> [166,] 9.393222e-04 9.990607e-01
#> [167,] 9.335308e-01 6.646924e-02
#> [168,] 1.318367e-07 9.999999e-01
#> [169,] 9.999791e-01 2.091200e-05
#> [170,] 3.269958e-09 1.000000e+00
#> [171,] 9.999977e-01 2.275659e-06
#> [172,] 9.991576e-01 8.424283e-04
#> [173,] 4.486720e-04 9.995513e-01
#> [174,] 2.583518e-01 7.416482e-01
#> [175,] 2.331945e-05 9.999767e-01
#> [176,] 2.331945e-05 9.999767e-01
#> [177,] 8.566098e-03 9.914339e-01
#> [178,] 9.999952e-01 4.766558e-06
#> [179,] 1.211525e-06 9.999988e-01
#> [180,] 4.108021e-03 9.958920e-01
#> [181,] 9.998079e-01 1.921400e-04
#> [182,] 4.486720e-04 9.995513e-01
#> [183,] 5.315308e-06 9.999947e-01
#> [184,] 5.315308e-06 9.999947e-01
#> [185,] 9.999900e-01 9.983929e-06
#> [186,] 2.142555e-04 9.997857e-01
#> [187,] 2.537644e-06 9.999975e-01
#> [188,] 9.999998e-01 2.476346e-07
#> [189,] 5.315308e-06 9.999947e-01
#> [190,] 9.998079e-01 1.921400e-04
#> [191,] 2.331945e-05 9.999767e-01
#> [192,] 9.991576e-01 8.424283e-04
#> [193,] 9.393222e-04 9.990607e-01
#> [194,] 2.537644e-06 9.999975e-01
#> [195,] 4.486720e-04 9.995513e-01
#> [196,] 2.331945e-05 9.999767e-01
#> [197,] 2.761437e-07 9.999997e-01
#> [198,] 4.486720e-04 9.995513e-01
#> [199,] 9.999900e-01 9.983929e-06
#> [200,] 2.142555e-04 9.997857e-01
#> [201,] 9.923115e-01 7.688534e-03
#> [202,] 1.113332e-05 9.999889e-01
#> [203,] 1.434630e-08 1.000000e+00
#> [204,] 9.999562e-01 4.380105e-05
#> [205,] 2.142555e-04 9.997857e-01
#> [206,] 9.999998e-01 2.476346e-07
#> [207,] 4.486720e-04 9.995513e-01
#> [208,] 2.537644e-06 9.999975e-01
#> [209,] 9.999977e-01 2.275659e-06
#> [210,] 5.315308e-06 9.999947e-01
#> [211,] 3.652235e-02 9.634777e-01
#> [212,] 4.884342e-05 9.999512e-01
#> [213,] 9.999977e-01 2.275659e-06
#> [214,] 1.965473e-03 9.980345e-01
#> [215,] 8.702168e-01 1.297832e-01
#> [216,] 9.393222e-04 9.990607e-01
#> [217,] 9.999562e-01 4.380105e-05
#> [218,] 1.561145e-09 1.000000e+00
#> [219,] 9.998079e-01 1.921400e-04
#> [220,] 9.393222e-04 9.990607e-01
#> [221,] 9.999952e-01 4.766558e-06
#> [222,] 1.113332e-05 9.999889e-01
#> [223,] 9.999083e-01 9.174080e-05
#> [224,] 1.965473e-03 9.980345e-01
#> [225,] 2.142555e-04 9.997857e-01
#> [226,] 1.023014e-04 9.998977e-01
#> [227,] 2.142555e-04 9.997857e-01
#> [228,] 2.142555e-04 9.997857e-01
#> [229,] 7.355866e-02 9.264413e-01
#> [230,] 1.023014e-04 9.998977e-01
#> [231,] 7.355866e-02 9.264413e-01
#> [232,] 9.999083e-01 9.174080e-05
#> [233,] 5.784074e-07 9.999994e-01
#> [234,] 9.999952e-01 4.766558e-06
#> [235,] 3.004960e-08 1.000000e+00
#> [236,] 9.999083e-01 9.174080e-05
#> [237,] 9.999083e-01 9.174080e-05
#> [238,] 4.486720e-04 9.995513e-01
#> [239,] 1.023014e-04 9.998977e-01
#> [240,] 8.702168e-01 1.297832e-01
#> [241,] 1.965473e-03 9.980345e-01
#> [242,] 9.999995e-01 5.186927e-07
#> [243,] 5.784074e-07 9.999994e-01
#> [244,] 9.335308e-01 6.646924e-02
#> [245,] 1.211525e-06 9.999988e-01
#> [246,] 1.113332e-05 9.999889e-01
#> [247,] 9.991576e-01 8.424283e-04
#> [248,] 1.113332e-05 9.999889e-01
#> [249,] 4.218472e-01 5.781528e-01
#> [250,] 4.108021e-03 9.958920e-01
#> [251,] 9.999083e-01 9.174080e-05
#> [252,] 5.315308e-06 9.999947e-01
#> [253,] 8.566098e-03 9.914339e-01
#> [254,] 8.566098e-03 9.914339e-01
#> [255,] 1.318367e-07 9.999999e-01
#> [256,] 4.884342e-05 9.999512e-01
#> [257,] 3.652235e-02 9.634777e-01
#> [258,] 5.315308e-06 9.999947e-01
#> [259,] 9.995976e-01 4.023697e-04
#> [260,] 1.023014e-04 9.998977e-01
#> [261,] 2.142555e-04 9.997857e-01
#> [262,] 2.537644e-06 9.999975e-01
#> [263,] 9.982371e-01 1.762916e-03
#> [264,] 5.315308e-06 9.999947e-01
#> [265,] 1.000000e+00 2.694728e-08
#> [266,] 9.923115e-01 7.688534e-03
#> [267,] 1.965473e-03 9.980345e-01
#> [268,] 2.331945e-05 9.999767e-01
#> [269,] 9.999998e-01 2.476346e-07
#> [270,] 3.004960e-08 1.000000e+00
#> [271,] 9.999998e-01 2.476346e-07
#> [272,] 4.108021e-03 9.958920e-01
#> 
#> $all.loglik
#> [1] -1168.602 -1077.998 -1059.123 -1042.454 -1035.211 -1034.109 -1034.011
#> [8] -1034.002 -1034.002
#> 
#> $restarts
#> [1] 0
#> 
#> $ft
#> [1] "normalmixEM"
#> 
#> attr(,"class")
#> [1] "mixEM"
system.time(out2<-normalmixEM(waiting, arbvar = FALSE, epsilon = 1e-03, fast=TRUE))
#> number of iterations= 6 
#>    user  system elapsed 
#>   0.001   0.000   0.001 
out2 # same thing but much faster
#> $x
#>   [1] 79 54 74 62 85 55 88 85 51 85 54 84 78 47 83 52 62 84 52 79 51 47 78 69 74
#>  [26] 83 55 76 78 79 73 77 66 80 74 52 48 80 59 90 80 58 84 58 73 83 64 53 82 59
#>  [51] 75 90 54 80 54 83 71 64 77 81 59 84 48 82 60 92 78 78 65 73 82 56 79 71 62
#>  [76] 76 60 78 76 83 75 82 70 65 73 88 76 80 48 86 60 90 50 78 63 72 84 75 51 82
#> [101] 62 88 49 83 81 47 84 52 86 81 75 59 89 79 59 81 50 85 59 87 53 69 77 56 88
#> [126] 81 45 82 55 90 45 83 56 89 46 82 51 86 53 79 81 60 82 77 76 59 80 49 96 53
#> [151] 77 77 65 81 71 70 81 93 53 89 45 86 58 78 66 76 63 88 52 93 49 57 77 68 81
#> [176] 81 73 50 85 74 55 77 83 83 51 78 84 46 83 55 81 57 76 84 77 81 87 77 51 78
#> [201] 60 82 91 53 78 46 77 84 49 83 71 80 49 75 64 76 53 94 55 76 50 82 54 75 78
#> [226] 79 78 78 70 79 70 54 86 50 90 54 54 77 79 64 75 47 86 63 85 82 57 82 67 74
#> [251] 54 83 73 73 88 80 71 83 56 79 78 84 58 83 43 60 75 81 46 90 46 74
#> 
#> $lambda
#> [1] 0.3609393 0.6390607
#> 
#> $mu
#> [1] 54.61664 80.09219
#> 
#> $sigma
#> [1] 5.868839
#> 
#> $loglik
#> [1] -1034.002
#> 
#> $posterior
#>              comp.1       comp.2
#>   [1,] 1.034690e-04 9.998965e-01
#>   [2,] 9.999094e-01 9.062883e-05
#>   [3,] 4.155418e-03 9.958446e-01
#>   [4,] 9.675013e-01 3.249868e-02
#>   [5,] 1.225105e-06 9.999988e-01
#>   [6,] 9.998102e-01 1.898178e-04
#>   [7,] 1.333010e-07 9.999999e-01
#>   [8,] 1.225105e-06 9.999988e-01
#>   [9,] 9.999901e-01 9.861904e-06
#>  [10,] 1.225105e-06 9.999988e-01
#>  [11,] 9.999094e-01 9.062883e-05
#>  [12,] 2.566176e-06 9.999974e-01
#>  [13,] 2.167079e-04 9.997833e-01
#>  [14,] 9.999995e-01 5.122838e-07
#>  [15,] 5.375251e-06 9.999946e-01
#>  [16,] 9.999793e-01 2.065711e-05
#>  [17,] 9.675013e-01 3.249868e-02
#>  [18,] 2.566176e-06 9.999974e-01
#>  [19,] 9.999793e-01 2.065711e-05
#>  [20,] 1.034690e-04 9.998965e-01
#>  [21,] 9.999901e-01 9.861904e-06
#>  [22,] 9.999995e-01 5.122838e-07
#>  [23,] 2.167079e-04 9.997833e-01
#>  [24,] 1.440290e-01 8.559710e-01
#>  [25,] 4.155418e-03 9.958446e-01
#>  [26,] 5.375251e-06 9.999946e-01
#>  [27,] 9.998102e-01 1.898178e-04
#>  [28,] 9.501303e-04 9.990499e-01
#>  [29,] 2.167079e-04 9.997833e-01
#>  [30,] 1.034690e-04 9.998965e-01
#>  [31,] 8.664773e-03 9.913352e-01
#>  [32,] 4.538218e-04 9.995462e-01
#>  [33,] 6.072941e-01 3.927059e-01
#>  [34,] 4.939925e-05 9.999506e-01
#>  [35,] 4.155418e-03 9.958446e-01
#>  [36,] 9.999793e-01 2.065711e-05
#>  [37,] 9.999989e-01 1.073060e-06
#>  [38,] 4.939925e-05 9.999506e-01
#>  [39,] 9.963584e-01 3.641574e-03
#>  [40,] 3.038129e-08 1.000000e+00
#>  [41,] 4.939925e-05 9.999506e-01
#>  [42,] 9.982582e-01 1.741818e-03
#>  [43,] 2.566176e-06 9.999974e-01
#>  [44,] 9.982582e-01 1.741818e-03
#>  [45,] 8.664773e-03 9.913352e-01
#>  [46,] 5.375251e-06 9.999946e-01
#>  [47,] 8.715501e-01 1.284499e-01
#>  [48,] 9.999567e-01 4.326865e-05
#>  [49,] 1.125926e-05 9.999887e-01
#>  [50,] 9.963584e-01 3.641574e-03
#>  [51,] 1.988132e-03 9.980119e-01
#>  [52,] 3.038129e-08 1.000000e+00
#>  [53,] 9.999094e-01 9.062883e-05
#>  [54,] 4.939925e-05 9.999506e-01
#>  [55,] 9.999094e-01 9.062883e-05
#>  [56,] 5.375251e-06 9.999946e-01
#>  [57,] 3.693347e-02 9.630665e-01
#>  [58,] 8.715501e-01 1.284499e-01
#>  [59,] 4.538218e-04 9.995462e-01
#>  [60,] 2.358403e-05 9.999764e-01
#>  [61,] 9.963584e-01 3.641574e-03
#>  [62,] 2.566176e-06 9.999974e-01
#>  [63,] 9.999989e-01 1.073060e-06
#>  [64,] 1.125926e-05 9.999887e-01
#>  [65,] 9.924024e-01 7.597573e-03
#>  [66,] 6.924351e-09 1.000000e+00
#>  [67,] 2.167079e-04 9.997833e-01
#>  [68,] 2.167079e-04 9.997833e-01
#>  [69,] 7.641095e-01 2.358905e-01
#>  [70,] 8.664773e-03 9.913352e-01
#>  [71,] 1.125926e-05 9.999887e-01
#>  [72,] 9.996025e-01 3.975211e-04
#>  [73,] 1.034690e-04 9.998965e-01
#>  [74,] 3.693347e-02 9.630665e-01
#>  [75,] 9.675013e-01 3.249868e-02
#>  [76,] 9.501303e-04 9.990499e-01
#>  [77,] 9.924024e-01 7.597573e-03
#>  [78,] 2.167079e-04 9.997833e-01
#>  [79,] 9.501303e-04 9.990499e-01
#>  [80,] 5.375251e-06 9.999946e-01
#>  [81,] 1.988132e-03 9.980119e-01
#>  [82,] 1.125926e-05 9.999887e-01
#>  [83,] 7.435684e-02 9.256432e-01
#>  [84,] 7.641095e-01 2.358905e-01
#>  [85,] 8.664773e-03 9.913352e-01
#>  [86,] 1.333010e-07 9.999999e-01
#>  [87,] 9.501303e-04 9.990499e-01
#>  [88,] 4.939925e-05 9.999506e-01
#>  [89,] 9.999989e-01 1.073060e-06
#>  [90,] 5.848711e-07 9.999994e-01
#>  [91,] 9.924024e-01 7.597573e-03
#>  [92,] 3.038129e-08 1.000000e+00
#>  [93,] 9.999953e-01 4.708141e-06
#>  [94,] 2.167079e-04 9.997833e-01
#>  [95,] 9.342648e-01 6.573517e-02
#>  [96,] 1.797922e-02 9.820208e-01
#>  [97,] 2.566176e-06 9.999974e-01
#>  [98,] 1.988132e-03 9.980119e-01
#>  [99,] 9.999901e-01 9.861904e-06
#> [100,] 1.125926e-05 9.999887e-01
#> [101,] 9.675013e-01 3.249868e-02
#> [102,] 1.333010e-07 9.999999e-01
#> [103,] 9.999978e-01 2.247693e-06
#> [104,] 5.375251e-06 9.999946e-01
#> [105,] 2.358403e-05 9.999764e-01
#> [106,] 9.999995e-01 5.122838e-07
#> [107,] 2.566176e-06 9.999974e-01
#> [108,] 9.999793e-01 2.065711e-05
#> [109,] 5.848711e-07 9.999994e-01
#> [110,] 2.358403e-05 9.999764e-01
#> [111,] 1.988132e-03 9.980119e-01
#> [112,] 9.963584e-01 3.641574e-03
#> [113,] 6.363847e-08 9.999999e-01
#> [114,] 1.034690e-04 9.998965e-01
#> [115,] 9.963584e-01 3.641574e-03
#> [116,] 2.358403e-05 9.999764e-01
#> [117,] 9.999953e-01 4.708141e-06
#> [118,] 1.225105e-06 9.999988e-01
#> [119,] 9.963584e-01 3.641574e-03
#> [120,] 2.792201e-07 9.999997e-01
#> [121,] 9.999567e-01 4.326865e-05
#> [122,] 1.440290e-01 8.559710e-01
#> [123,] 4.538218e-04 9.995462e-01
#> [124,] 9.996025e-01 3.975211e-04
#> [125,] 1.333010e-07 9.999999e-01
#> [126,] 2.358403e-05 9.999764e-01
#> [127,] 9.999999e-01 1.167572e-07
#> [128,] 1.125926e-05 9.999887e-01
#> [129,] 9.998102e-01 1.898178e-04
#> [130,] 3.038129e-08 1.000000e+00
#> [131,] 9.999999e-01 1.167572e-07
#> [132,] 5.375251e-06 9.999946e-01
#> [133,] 9.996025e-01 3.975211e-04
#> [134,] 6.363847e-08 9.999999e-01
#> [135,] 9.999998e-01 2.445666e-07
#> [136,] 1.125926e-05 9.999887e-01
#> [137,] 9.999901e-01 9.861904e-06
#> [138,] 5.848711e-07 9.999994e-01
#> [139,] 9.999567e-01 4.326865e-05
#> [140,] 1.034690e-04 9.998965e-01
#> [141,] 2.358403e-05 9.999764e-01
#> [142,] 9.924024e-01 7.597573e-03
#> [143,] 1.125926e-05 9.999887e-01
#> [144,] 4.538218e-04 9.995462e-01
#> [145,] 9.501303e-04 9.990499e-01
#> [146,] 9.963584e-01 3.641574e-03
#> [147,] 4.939925e-05 9.999506e-01
#> [148,] 9.999978e-01 2.247693e-06
#> [149,] 3.596871e-10 1.000000e+00
#> [150,] 9.999567e-01 4.326865e-05
#> [151,] 4.538218e-04 9.995462e-01
#> [152,] 4.538218e-04 9.995462e-01
#> [153,] 7.641095e-01 2.358905e-01
#> [154,] 2.358403e-05 9.999764e-01
#> [155,] 3.693347e-02 9.630665e-01
#> [156,] 7.435684e-02 9.256432e-01
#> [157,] 2.358403e-05 9.999764e-01
#> [158,] 3.305716e-09 1.000000e+00
#> [159,] 9.999567e-01 4.326865e-05
#> [160,] 6.363847e-08 9.999999e-01
#> [161,] 9.999999e-01 1.167572e-07
#> [162,] 5.848711e-07 9.999994e-01
#> [163,] 9.982582e-01 1.741818e-03
#> [164,] 2.167079e-04 9.997833e-01
#> [165,] 6.072941e-01 3.927059e-01
#> [166,] 9.501303e-04 9.990499e-01
#> [167,] 9.342648e-01 6.573517e-02
#> [168,] 1.333010e-07 9.999999e-01
#> [169,] 9.999793e-01 2.065711e-05
#> [170,] 3.305716e-09 1.000000e+00
#> [171,] 9.999978e-01 2.247693e-06
#> [172,] 9.991677e-01 8.323094e-04
#> [173,] 4.538218e-04 9.995462e-01
#> [174,] 2.606042e-01 7.393958e-01
#> [175,] 2.358403e-05 9.999764e-01
#> [176,] 2.358403e-05 9.999764e-01
#> [177,] 8.664773e-03 9.913352e-01
#> [178,] 9.999953e-01 4.708141e-06
#> [179,] 1.225105e-06 9.999988e-01
#> [180,] 4.155418e-03 9.958446e-01
#> [181,] 9.998102e-01 1.898178e-04
#> [182,] 4.538218e-04 9.995462e-01
#> [183,] 5.375251e-06 9.999946e-01
#> [184,] 5.375251e-06 9.999946e-01
#> [185,] 9.999901e-01 9.861904e-06
#> [186,] 2.167079e-04 9.997833e-01
#> [187,] 2.566176e-06 9.999974e-01
#> [188,] 9.999998e-01 2.445666e-07
#> [189,] 5.375251e-06 9.999946e-01
#> [190,] 9.998102e-01 1.898178e-04
#> [191,] 2.358403e-05 9.999764e-01
#> [192,] 9.991677e-01 8.323094e-04
#> [193,] 9.501303e-04 9.990499e-01
#> [194,] 2.566176e-06 9.999974e-01
#> [195,] 4.538218e-04 9.995462e-01
#> [196,] 2.358403e-05 9.999764e-01
#> [197,] 2.792201e-07 9.999997e-01
#> [198,] 4.538218e-04 9.995462e-01
#> [199,] 9.999901e-01 9.861904e-06
#> [200,] 2.167079e-04 9.997833e-01
#> [201,] 9.924024e-01 7.597573e-03
#> [202,] 1.125926e-05 9.999887e-01
#> [203,] 1.450416e-08 1.000000e+00
#> [204,] 9.999567e-01 4.326865e-05
#> [205,] 2.167079e-04 9.997833e-01
#> [206,] 9.999998e-01 2.445666e-07
#> [207,] 4.538218e-04 9.995462e-01
#> [208,] 2.566176e-06 9.999974e-01
#> [209,] 9.999978e-01 2.247693e-06
#> [210,] 5.375251e-06 9.999946e-01
#> [211,] 3.693347e-02 9.630665e-01
#> [212,] 4.939925e-05 9.999506e-01
#> [213,] 9.999978e-01 2.247693e-06
#> [214,] 1.988132e-03 9.980119e-01
#> [215,] 8.715501e-01 1.284499e-01
#> [216,] 9.501303e-04 9.990499e-01
#> [217,] 9.999567e-01 4.326865e-05
#> [218,] 1.578163e-09 1.000000e+00
#> [219,] 9.998102e-01 1.898178e-04
#> [220,] 9.501303e-04 9.990499e-01
#> [221,] 9.999953e-01 4.708141e-06
#> [222,] 1.125926e-05 9.999887e-01
#> [223,] 9.999094e-01 9.062883e-05
#> [224,] 1.988132e-03 9.980119e-01
#> [225,] 2.167079e-04 9.997833e-01
#> [226,] 1.034690e-04 9.998965e-01
#> [227,] 2.167079e-04 9.997833e-01
#> [228,] 2.167079e-04 9.997833e-01
#> [229,] 7.435684e-02 9.256432e-01
#> [230,] 1.034690e-04 9.998965e-01
#> [231,] 7.435684e-02 9.256432e-01
#> [232,] 9.999094e-01 9.062883e-05
#> [233,] 5.848711e-07 9.999994e-01
#> [234,] 9.999953e-01 4.708141e-06
#> [235,] 3.038129e-08 1.000000e+00
#> [236,] 9.999094e-01 9.062883e-05
#> [237,] 9.999094e-01 9.062883e-05
#> [238,] 4.538218e-04 9.995462e-01
#> [239,] 1.034690e-04 9.998965e-01
#> [240,] 8.715501e-01 1.284499e-01
#> [241,] 1.988132e-03 9.980119e-01
#> [242,] 9.999995e-01 5.122838e-07
#> [243,] 5.848711e-07 9.999994e-01
#> [244,] 9.342648e-01 6.573517e-02
#> [245,] 1.225105e-06 9.999988e-01
#> [246,] 1.125926e-05 9.999887e-01
#> [247,] 9.991677e-01 8.323094e-04
#> [248,] 1.125926e-05 9.999887e-01
#> [249,] 4.247170e-01 5.752830e-01
#> [250,] 4.155418e-03 9.958446e-01
#> [251,] 9.999094e-01 9.062883e-05
#> [252,] 5.375251e-06 9.999946e-01
#> [253,] 8.664773e-03 9.913352e-01
#> [254,] 8.664773e-03 9.913352e-01
#> [255,] 1.333010e-07 9.999999e-01
#> [256,] 4.939925e-05 9.999506e-01
#> [257,] 3.693347e-02 9.630665e-01
#> [258,] 5.375251e-06 9.999946e-01
#> [259,] 9.996025e-01 3.975211e-04
#> [260,] 1.034690e-04 9.998965e-01
#> [261,] 2.167079e-04 9.997833e-01
#> [262,] 2.566176e-06 9.999974e-01
#> [263,] 9.982582e-01 1.741818e-03
#> [264,] 5.375251e-06 9.999946e-01
#> [265,] 1.000000e+00 2.661072e-08
#> [266,] 9.924024e-01 7.597573e-03
#> [267,] 1.988132e-03 9.980119e-01
#> [268,] 2.358403e-05 9.999764e-01
#> [269,] 9.999998e-01 2.445666e-07
#> [270,] 3.038129e-08 1.000000e+00
#> [271,] 9.999998e-01 2.445666e-07
#> [272,] 4.155418e-03 9.958446e-01
#> 
#> $all.loglik
#> [1] -1095.264 -1047.825 -1036.111 -1034.115 -1034.006 -1034.002 -1034.002
#> 
#> $restarts
#> [1] 0
#> 
#> $ft
#> [1] "normalmixEM"
#> 
#> attr(,"class")
#> [1] "mixEM"