Helper function for PPI++ OLS estimation
Usage
ppi_plusplus_ols(
X_l,
Y_l,
f_l,
X_u,
f_u,
lhat = NULL,
coord = NULL,
w_l = NULL,
w_u = NULL
)Arguments
- X_l
(matrix): n x p matrix of covariates in the labeled data.
- Y_l
(vector): n-vector of labeled outcomes.
- f_l
(vector): n-vector of predictions in the labeled data.
- X_u
(matrix): N x p matrix of covariates in the unlabeled data.
- f_u
(vector): N-vector of predictions in the unlabeled data.
- lhat
(float, optional): Power-tuning parameter (see https://arxiv.org/abs/2311.01453). The default value,
NULL, will estimate the optimal value from the data. Settinglhat = 1recovers PPI with no power tuning, and settinglhat = 0recovers the classical point estimate.- coord
(int, optional): Coordinate for which to optimize
lhat = 1. IfNULL, it optimizes the total variance over all coordinates. Must be in (1, ..., d) where d is the dimension of the estimand.- w_l
(ndarray, optional): Sample weights for the labeled data set. Defaults to a vector of ones.
- w_u
(ndarray, optional): Sample weights for the unlabeled data set. Defaults to a vector of ones.
Value
(list): A list containing the following:
- est
(vector): vector of PPI++ OLS regression coefficient estimates.
- se
(vector): vector of standard errors of the coefficients.
- lambda
(float): estimated power-tuning parameter.
- rectifier_est
(vector): vector of the rectifier OLS regression coefficient estimates.
- var_u
(matrix): covariance matrix for the gradients in the unlabeled data.
- var_l
(matrix): covariance matrix for the gradients in the labeled data.
- grads
(matrix): matrix of gradients for the labeled data.
- grads_hat_unlabeled
(matrix): matrix of predicted gradients for the unlabeled data.
- grads_hat
(matrix): matrix of predicted gradients for the labeled data.
- inv_hessian
(matrix): inverse Hessian matrix.
Details
PPI++: Efficient Prediction Powered Inference (Angelopoulos et al., 2023) https://arxiv.org/abs/2311.01453`
Examples
dat <- simdat(model = "ols")
form <- Y - f ~ X1
X_l <- model.matrix(form, data = dat[dat$set == "labeled",])
Y_l <- dat[dat$set == "labeled", all.vars(form)[1]] |> matrix(ncol = 1)
f_l <- dat[dat$set == "labeled", all.vars(form)[2]] |> matrix(ncol = 1)
X_u <- model.matrix(form, data = dat[dat$set == "unlabeled",])
f_u <- dat[dat$set == "unlabeled", all.vars(form)[2]] |> matrix(ncol = 1)
ppi_plusplus_ols(X_l, Y_l, f_l, X_u, f_u)
#> $est
#> X(Intercept) XX1
#> 0.7776755 0.9294611
#>
#> $se
#> [1] 0.09557173 0.09476422
#>
#> $lambda
#> [1] 0.4730261
#>
#> $rectifier_est
#> X(Intercept) XX1
#> -0.04204282 -0.06020154
#>
#> $var_u
#> [,1] [,2]
#> [1,] 0.2225795 -0.0147331
#> [2,] -0.0147331 0.2180297
#>
#> $var_l
#> [,1] [,2]
#> [1,] 2.51264246 0.01134738
#> [2,] 0.01134738 2.63163397
#>
#> $grads
#> [,1] [,2]
#> [1,] -0.902855646 6.114056e-01
#> [2,] -0.471140491 -5.838350e-01
#> [3,] 1.334473594 -7.148872e-01
#> [4,] 2.867856294 -6.340545e-01
#> [5,] 0.397035186 -7.634147e-01
#> [6,] 1.494598189 -3.225684e-02
#> [7,] -5.585270969 -6.674540e+00
#> [8,] 2.963988952 -2.019371e+00
#> [9,] -0.714560805 -6.728945e-01
#> [10,] 0.240211143 -5.679459e-02
#> [11,] -4.003163670 3.798652e+00
#> [12,] -1.178452470 -1.205931e+00
#> [13,] 0.919850843 -5.725607e-01
#> [14,] 0.900253593 -3.090240e-01
#> [15,] 1.813122247 4.119843e+00
#> [16,] 0.474475535 -5.782018e-01
#> [17,] -1.485333731 -3.145548e+00
#> [18,] 3.048689766 -1.676596e+00
#> [19,] -0.050076737 -1.817154e-02
#> [20,] -0.413663778 -2.961937e-01
#> [21,] 0.164387567 1.206819e-01
#> [22,] 0.132528294 -3.065350e-02
#> [23,] -0.306266900 -3.415679e-01
#> [24,] -0.582958352 1.193308e+00
#> [25,] -0.578757412 -1.350398e+00
#> [26,] -0.970943194 -2.379967e-01
#> [27,] 1.129232265 1.541673e+00
#> [28,] -2.564811083 -8.294792e-02
#> [29,] 0.850334277 -1.450537e-01
#> [30,] -0.249049695 -2.033830e-01
#> [31,] -0.529365184 4.770047e-01
#> [32,] -1.535823380 -1.847004e+00
#> [33,] 0.802826785 4.490469e-01
#> [34,] 0.521152999 4.498351e-01
#> [35,] -1.089756086 -1.273271e+00
#> [36,] 0.120075745 1.113638e-01
#> [37,] -5.091635058 3.929724e+00
#> [38,] -0.938746369 -8.783234e-01
#> [39,] 1.542996366 -1.664438e-01
#> [40,] -0.724955349 4.546202e-01
#> [41,] 3.506225785 -1.337322e+00
#> [42,] -0.165153034 2.068035e-01
#> [43,] 5.448204840 3.186648e+00
#> [44,] -0.807897328 -1.446038e+00
#> [45,] 0.890302915 1.558333e+00
#> [46,] 1.518464450 -1.733331e-01
#> [47,] 2.089064459 1.977090e+00
#> [48,] -3.202229697 7.047847e-01
#> [49,] -0.087411162 -4.169204e-02
#> [50,] -0.224581731 2.122622e-01
#> [51,] 1.466313001 6.264384e-01
#> [52,] 0.928353332 3.466507e-01
#> [53,] 1.026001293 -6.172357e-01
#> [54,] 1.064456703 1.116203e+00
#> [55,] 0.953967957 1.060336e+00
#> [56,] 0.234843595 -4.548006e-02
#> [57,] 0.355343548 -1.125259e-01
#> [58,] -0.700645041 8.634110e-01
#> [59,] -1.872994161 5.012151e-02
#> [60,] 0.893404968 -1.749060e+00
#> [61,] 2.499459354 -7.381661e-03
#> [62,] -0.815891750 -5.433177e-01
#> [63,] -0.185753600 -9.996290e-03
#> [64,] 1.337114325 1.878338e+00
#> [65,] 2.762923713 3.648273e+00
#> [66,] -3.882633740 -1.188419e+01
#> [67,] 0.139427595 2.726380e-02
#> [68,] -6.582208422 4.652114e+00
#> [69,] -1.584785252 -3.660665e-01
#> [70,] 0.054739244 -4.253634e-02
#> [71,] 0.270391050 1.573632e-01
#> [72,] -0.818055046 3.521197e-01
#> [73,] -4.604878394 -3.722329e-01
#> [74,] 0.772243451 -6.503191e-01
#> [75,] 1.202724381 3.525894e-04
#> [76,] 1.309946365 1.117065e+00
#> [77,] 0.217210745 2.435528e-01
#> [78,] 0.206827409 -1.110492e-01
#> [79,] -0.759858305 -1.143176e-01
#> [80,] 1.099996052 7.017263e-01
#> [81,] -0.959213072 5.810668e-01
#> [82,] -0.513317062 8.467594e-02
#> [83,] -0.411215667 -4.498032e-02
#> [84,] 0.795795814 -5.555848e-01
#> [85,] -0.478430524 1.584804e-01
#> [86,] 0.291300456 6.155891e-01
#> [87,] 1.575749870 2.925830e-01
#> [88,] -1.734828060 -1.240404e+00
#> [89,] -1.046070354 4.097279e+00
#> [90,] 4.065347276 -1.449497e+00
#> [91,] -1.458821169 -1.522046e+00
#> [92,] -2.264234166 -1.909252e+00
#> [93,] -0.830547615 2.993846e-01
#> [94,] -2.164730795 -3.773986e+00
#> [95,] 1.144447003 -1.555206e+00
#> [96,] -1.207963529 2.492817e-01
#> [97,] -0.908931880 9.879605e-01
#> [98,] -1.110321842 -2.544968e-01
#> [99,] 0.597577756 2.037109e-01
#> [100,] -1.751318684 -1.733308e+00
#> [101,] -1.352063825 1.011172e+00
#> [102,] -2.458400790 -1.117725e+00
#> [103,] -0.493274954 -9.764612e-01
#> [104,] 0.766002107 -1.833428e-01
#> [105,] -2.436202420 -2.267918e+00
#> [106,] 1.932812537 3.160919e-01
#> [107,] 1.367367955 -5.135052e-01
#> [108,] 0.849387853 6.851620e-01
#> [109,] 3.102519976 2.121882e-02
#> [110,] -3.587446827 3.499459e+00
#> [111,] 1.501553569 2.406762e-01
#> [112,] 0.586232212 -3.158039e-01
#> [113,] 0.246172499 -1.014369e-02
#> [114,] 1.926025583 5.638783e-01
#> [115,] -0.394500752 3.163908e-02
#> [116,] -0.242960798 2.292559e-01
#> [117,] 0.992873479 1.640902e+00
#> [118,] -1.695143617 -1.088487e-02
#> [119,] 0.722759480 6.401210e-01
#> [120,] 0.412572261 -2.091365e-01
#> [121,] -2.296849300 1.713466e+00
#> [122,] -1.074539270 8.137506e-01
#> [123,] -10.060151997 3.453647e-01
#> [124,] -1.145498232 -2.243211e-02
#> [125,] -1.311899433 -4.479359e-01
#> [126,] 1.214853535 -2.755551e-01
#> [127,] 0.182840346 2.078358e-01
#> [128,] -1.280761934 -2.532955e+00
#> [129,] -0.178419582 1.128394e-01
#> [130,] -1.214949349 1.934967e+00
#> [131,] 0.473836737 5.972697e-02
#> [132,] -0.465493272 6.082372e-01
#> [133,] 2.106392326 1.083241e+00
#> [134,] -1.737478908 -6.645705e-01
#> [135,] 1.261401952 -5.146168e-01
#> [136,] 0.373328492 2.644960e-01
#> [137,] 2.114486038 -4.459456e-01
#> [138,] -1.582021700 1.265509e+00
#> [139,] 0.589229660 5.626898e-01
#> [140,] -1.520724765 1.602371e+00
#> [141,] -0.131139409 8.705297e-02
#> [142,] 0.322532521 -1.304421e-01
#> [143,] 0.106171080 -1.571445e-01
#> [144,] -0.534266416 3.684140e-01
#> [145,] -1.983622454 -1.414859e-01
#> [146,] 0.647067003 1.885951e+00
#> [147,] -1.318926213 1.203926e+00
#> [148,] -0.691388866 -3.223849e-01
#> [149,] -1.318118914 3.510479e-01
#> [150,] -0.510830188 2.408151e-01
#> [151,] 0.709432991 1.271117e+00
#> [152,] 0.026341719 -2.843384e-02
#> [153,] -0.162356913 2.450980e-02
#> [154,] 1.533650938 -1.427742e+00
#> [155,] -1.137724100 1.907210e+00
#> [156,] 1.235019859 4.795421e-01
#> [157,] -0.090294271 -5.280771e-03
#> [158,] 0.271695382 2.119037e-01
#> [159,] -1.388028207 9.292247e-01
#> [160,] -0.037623414 5.905377e-02
#> [161,] -0.228425097 3.154660e-01
#> [162,] -1.260956537 -5.486485e-02
#> [163,] -1.035378380 1.333074e-01
#> [164,] 1.064513386 -1.999618e+00
#> [165,] 1.615139614 -1.084106e+00
#> [166,] -0.845952520 8.216963e-02
#> [167,] 3.161350476 1.989463e+00
#> [168,] -0.591709289 8.010457e-02
#> [169,] 2.024812379 -2.927035e+00
#> [170,] -0.128267168 4.188740e-03
#> [171,] 1.158329383 6.480025e-01
#> [172,] 1.798016849 8.082848e-01
#> [173,] -2.567564708 -9.284915e-02
#> [174,] -0.808760930 -1.367525e+00
#> [175,] 0.432116439 5.387745e-01
#> [176,] 1.737886828 -7.857079e-02
#> [177,] 1.239439204 -5.932220e-01
#> [178,] 1.062766903 -7.795114e-01
#> [179,] 1.384349177 2.626654e+00
#> [180,] 0.658511365 7.290400e-01
#> [181,] -0.079112044 8.177798e-02
#> [182,] 7.882839136 2.496855e+00
#> [183,] 1.418716693 -5.207099e-01
#> [184,] -0.795622618 4.132705e-01
#> [185,] 1.648978442 5.385378e-01
#> [186,] -2.858341080 2.647462e+00
#> [187,] -0.005285383 -7.423771e-03
#> [188,] -0.035396302 2.028736e-03
#> [189,] 1.431731254 1.538579e-01
#> [190,] 1.182456181 -1.430551e-01
#> [191,] -2.814355308 1.686411e+00
#> [192,] 0.956925798 8.746875e-03
#> [193,] -0.742854339 8.716333e-01
#> [194,] -0.419370871 -4.583058e-01
#> [195,] -0.220933655 3.367091e-01
#> [196,] 0.509965964 1.920087e-01
#> [197,] -0.648229235 4.712979e-01
#> [198,] 0.597307804 -1.059883e+00
#> [199,] -0.400331075 2.422264e-01
#> [200,] 0.357128552 2.472662e-01
#> [201,] -0.141909054 5.955277e-02
#> [202,] -1.496274215 6.620693e-01
#> [203,] -0.346584023 -1.716995e-01
#> [204,] -0.469908950 -8.926937e-02
#> [205,] 1.614568576 -4.888635e-01
#> [206,] 0.815853510 7.471921e-01
#> [207,] -0.890523585 -6.704737e-01
#> [208,] 1.088811535 -5.624024e-01
#> [209,] -0.588470110 -1.275980e-01
#> [210,] 1.796278025 -1.474258e+00
#> [211,] -0.563197540 4.513137e-01
#> [212,] 0.961549386 6.678526e-01
#> [213,] 0.483561106 -5.637917e-02
#> [214,] 2.331417839 2.604374e+00
#> [215,] 0.496715361 2.629518e-02
#> [216,] -2.538555701 4.671533e+00
#> [217,] 0.046069261 6.106023e-04
#> [218,] -0.208244272 -6.501683e-01
#> [219,] 0.858410905 -9.640494e-01
#> [220,] -0.068538123 4.100061e-03
#> [221,] -0.555096883 -2.389209e-01
#> [222,] -1.588888023 5.122022e+00
#> [223,] 1.064968616 2.255056e-02
#> [224,] -1.792738521 5.349271e-01
#> [225,] -0.528683202 4.664409e-01
#> [226,] 0.590887706 -9.844975e-01
#> [227,] -0.502610606 3.714781e-01
#> [228,] 0.401183518 -1.560076e-01
#> [229,] -0.407319312 1.567209e-01
#> [230,] 1.004534333 2.012632e+00
#> [231,] -1.239798147 -2.550253e+00
#> [232,] 2.470972072 1.811256e+00
#> [233,] -0.886598344 -5.012420e-01
#> [234,] 0.433799281 -3.848927e-02
#> [235,] -0.687731293 -1.690562e+00
#> [236,] -1.681429354 -7.755083e-01
#> [237,] -0.613024317 5.501610e-01
#> [238,] 0.278102732 -1.136934e-01
#> [239,] -0.331595870 6.240896e-02
#> [240,] 0.070089152 -4.074931e-02
#> [241,] 0.918273859 7.390863e-01
#> [242,] 0.293620655 -1.392897e-01
#> [243,] -1.300015176 8.534762e-01
#> [244,] -0.734091256 -7.360349e-01
#> [245,] 0.774623620 8.580835e-02
#> [246,] 0.386210350 -9.145060e-01
#> [247,] 2.440660562 1.549569e+00
#> [248,] -1.555079642 1.297047e+00
#> [249,] 0.591716994 2.761750e-01
#> [250,] 2.197957992 1.442009e+00
#> [251,] 0.298507243 6.765830e-01
#> [252,] 1.779672528 -6.307539e-01
#> [253,] -1.534926300 1.514100e+00
#> [254,] 0.357124847 9.138021e-02
#> [255,] 0.949216834 4.916260e-02
#> [256,] -0.941107282 3.777974e-01
#> [257,] 0.658349743 7.610657e-01
#> [258,] 0.726983338 -1.594745e+00
#> [259,] -0.197180528 6.788761e-02
#> [260,] 7.843581715 -1.653019e+01
#> [261,] -0.698899695 -2.587408e-01
#> [262,] -0.452460392 7.101409e-02
#> [263,] 1.571443591 -6.944026e-01
#> [264,] -0.034673430 -3.270216e-02
#> [265,] 0.340020656 4.565734e-01
#> [266,] 0.409194876 4.765148e-01
#> [267,] 2.519812409 -1.658724e+00
#> [268,] 0.898442863 -2.264291e-01
#> [269,] -0.851508191 -1.209455e-01
#> [270,] -0.194081612 1.524446e-01
#> [271,] 0.513098691 -1.160589e+00
#> [272,] 1.350169047 -4.959419e-01
#> [273,] -6.108175411 -3.305738e+00
#> [274,] 0.292333382 -1.056131e-01
#> [275,] 1.733291931 5.987292e-01
#> [276,] 0.453375876 4.188619e-01
#> [277,] 1.943581738 -9.860094e-01
#> [278,] 0.325253134 6.649674e-02
#> [279,] 0.588185841 -1.005581e-01
#> [280,] -2.018163152 -2.466100e+00
#> [281,] -0.501816669 -1.167445e+00
#> [282,] -3.325995513 -7.287768e+00
#> [283,] 1.521982886 1.164055e+00
#> [284,] -1.297804989 1.188212e+00
#> [285,] -1.242609674 -1.771420e+00
#> [286,] 0.442731029 -9.521087e-01
#> [287,] -3.353409256 7.100244e-02
#> [288,] 1.356627538 2.028155e+00
#> [289,] 1.321425494 -1.176106e+00
#> [290,] 0.335507283 1.624104e-01
#> [291,] -3.684518760 2.071452e+00
#> [292,] 1.083138166 -4.494018e-01
#> [293,] 0.521641327 5.855491e-01
#> [294,] -2.090614849 2.061038e-01
#> [295,] 1.549294891 -1.995358e+00
#> [296,] -0.181185623 -2.338699e-01
#> [297,] 2.691334743 -3.288095e-01
#> [298,] -1.541383322 1.271061e+00
#> [299,] 0.149813406 -9.712114e-02
#> [300,] 1.644469131 -5.002835e-01
#>
#> $grads_hat_unlabeled
#> [,1] [,2]
#> [1,] -0.8675342631 -0.9686230172
#> [2,] -1.2957332435 0.7818620340
#> [3,] -1.0304206594 0.4936779847
#> [4,] -1.9868688839 1.4965345955
#> [5,] -1.0171596979 -0.2241327525
#> [6,] 1.1404579897 0.5192694486
#> [7,] -1.0197621678 0.7693006133
#> [8,] 0.3302323415 -0.1498204394
#> [9,] -1.0890407726 0.0105939339
#> [10,] -1.6912975042 -2.6056799239
#> [11,] -0.4395947274 0.2360411457
#> [12,] 0.5023405650 0.5044115745
#> [13,] -0.1547736872 0.4046551658
#> [14,] -1.0205959804 -0.3413033443
#> [15,] 0.8428515923 -0.1001647552
#> [16,] -0.1290160769 -0.1243457197
#> [17,] 0.5060008539 -0.9133988345
#> [18,] -0.0308389439 0.0592089051
#> [19,] 0.2495967742 -0.0247230112
#> [20,] 0.5150288137 0.8802526506
#> [21,] -0.9948711301 0.9261749560
#> [22,] -0.3911628155 -0.0547370877
#> [23,] -0.8779790040 0.7146835330
#> [24,] 1.4066692528 -1.3759124531
#> [25,] 0.6598975615 -1.1975854378
#> [26,] -0.2922826142 0.5521419552
#> [27,] -0.4171175194 -0.1555246716
#> [28,] 2.1148629014 -0.3708061950
#> [29,] -0.7798811853 -0.2556563525
#> [30,] 0.3474902518 0.4276980385
#> [31,] 2.3864085878 0.0467669161
#> [32,] 0.3013450635 -0.0819212039
#> [33,] 0.5925285749 -0.2211674504
#> [34,] 0.1232730291 -0.0113759106
#> [35,] 0.8987552310 0.6163306959
#> [36,] -0.5272937253 -1.0459806161
#> [37,] -0.2745193682 0.0210678545
#> [38,] -2.2730201500 2.3019068802
#> [39,] 0.6964339204 1.4441252771
#> [40,] -1.2942539320 3.3752883215
#> [41,] -0.5883911949 -0.3848943411
#> [42,] 0.5914978522 0.6081355285
#> [43,] 0.6173619298 -0.5468516032
#> [44,] 1.1737445295 -1.6569798979
#> [45,] -0.0012962263 -0.0006466135
#> [46,] -0.9213146683 0.4387590925
#> [47,] 1.3916163588 2.6318027356
#> [48,] 0.6415686263 1.1482772795
#> [49,] -0.5162909939 -0.2643708024
#> [50,] 0.7901610091 -1.6315275805
#> [51,] 1.9084487392 0.1680172099
#> [52,] 0.5880142093 0.1866458830
#> [53,] 0.0331540458 0.0110824855
#> [54,] 0.7749500217 0.4281007616
#> [55,] 0.4019433666 -0.1013582359
#> [56,] -1.8942008326 2.9644628421
#> [57,] 1.4721384850 -1.2341184411
#> [58,] -0.3360462575 -0.2727253562
#> [59,] -0.0914363086 -0.0692510802
#> [60,] -0.6411116470 0.1134234439
#> [61,] 1.0199693469 0.6868291822
#> [62,] 0.5302811101 -0.7522253699
#> [63,] -0.5511422835 0.2221860976
#> [64,] -0.0686995450 -0.0253595204
#> [65,] -0.2230915752 0.1571853929
#> [66,] -0.1760126713 0.1338573903
#> [67,] -1.8172042922 -1.0359557051
#> [68,] -1.2191127821 1.4743395971
#> [69,] -1.1509330792 0.7765056348
#> [70,] 0.1705749123 -0.0516211074
#> [71,] 1.0054220331 0.8719811092
#> [72,] 0.3849921270 0.3561605608
#> [73,] -0.1660412110 -0.0049316583
#> [74,] -0.1429286769 -0.0292546723
#> [75,] 1.1724965407 0.1070567709
#> [76,] 0.7382213801 0.1273707515
#> [77,] 1.1668556304 0.8850479313
#> [78,] -0.1443547173 0.1539264437
#> [79,] -0.7792626292 -0.1545770734
#> [80,] -2.2981026298 -0.0532718122
#> [81,] 1.3315059073 1.4515543567
#> [82,] 0.2202515579 0.1072703212
#> [83,] -0.0689663876 -0.0562356255
#> [84,] 0.6495382202 -0.6860667598
#> [85,] 0.4884122461 -0.7377501206
#> [86,] -0.0550358962 -0.0056382460
#> [87,] -0.0572679246 0.0013516510
#> [88,] 0.5030590248 0.1243926928
#> [89,] 0.7900721055 -0.7403218161
#> [90,] 0.8887268861 -0.8584115970
#> [91,] 1.9157336754 -2.8598951406
#> [92,] 0.4756976713 -0.3098289695
#> [93,] 0.3667851206 -0.2495069506
#> [94,] 0.7940098254 0.7364083192
#> [95,] -0.3305076750 -0.4395349194
#> [96,] 0.4253550317 0.8512109024
#> [97,] 0.7243577783 -0.7754312326
#> [98,] -0.6165974490 0.3778988819
#> [99,] -0.6147477690 0.8461177563
#> [100,] -0.4672910887 0.0183979717
#> [101,] 0.0277245373 0.0127532500
#> [102,] 1.0193787990 0.9047886769
#> [103,] 0.2382798673 -0.1617229808
#> [104,] 2.1402251877 -0.3127023468
#> [105,] -0.2030367732 0.2873773998
#> [106,] -0.5919115084 -0.0178671006
#> [107,] -0.6820095384 1.2130275847
#> [108,] -0.2391175100 0.0210517867
#> [109,] 0.3760791759 0.3296716428
#> [110,] -0.6098199526 -0.1570861591
#> [111,] 1.0851475911 0.6685958408
#> [112,] 0.0006791731 -0.0005416342
#> [113,] -1.4176454052 0.4851723168
#> [114,] 0.4364186625 -0.5000442081
#> [115,] -1.7394765035 1.8839426421
#> [116,] 1.4686830261 -0.7749904809
#> [117,] -1.0113399602 -0.5766349550
#> [118,] 1.2287495702 -0.8884747540
#> [119,] -1.9471397945 -0.5448994702
#> [120,] 0.9327058769 -0.3792313147
#> [121,] -1.1786849396 -0.7741483421
#> [122,] -0.6149742348 0.3162140930
#> [123,] 0.7562573526 -0.8059829698
#> [124,] -0.7908112249 0.6757954372
#> [125,] -1.2229367920 0.6884049455
#> [126,] -0.0934273926 -0.0424022151
#> [127,] -0.4042595530 0.0805428260
#> [128,] -0.5528717049 -0.3661707411
#> [129,] 0.3550591199 -0.5548568501
#> [130,] -1.4694859111 1.6944833146
#> [131,] 0.4314376705 0.2601025482
#> [132,] -1.1386027821 -0.6664524235
#> [133,] 0.7624736130 0.3779251765
#> [134,] -0.4823692111 1.0282426782
#> [135,] 0.3907375919 -0.3169464439
#> [136,] 1.5160774373 1.1787008497
#> [137,] -0.9311969142 -0.7543877173
#> [138,] -0.5822413877 -0.4628925178
#> [139,] -0.4671334925 0.0631902263
#> [140,] -0.4455079016 -0.2504169860
#> [141,] -0.6395210537 -0.5188661928
#> [142,] 0.7647449095 0.8202432289
#> [143,] 0.3599114222 0.0230298086
#> [144,] -0.4625211209 0.8703168687
#> [145,] 0.8426839593 -0.4474971058
#> [146,] 0.0589944784 -0.0824857096
#> [147,] -0.5771116435 0.0139074365
#> [148,] 0.3676640233 -0.3490778311
#> [149,] -1.2092638486 -0.6675817674
#> [150,] -1.5045145844 -0.7829160745
#> [151,] -1.3297441789 1.0544509259
#> [152,] 0.3752144365 -0.0473575591
#> [153,] -0.3813120840 -0.0184272637
#> [154,] 0.5398834132 0.1762945780
#> [155,] -1.2373791596 1.9103758287
#> [156,] 1.1266339265 -0.8962629564
#> [157,] 0.2296688032 -0.2725145990
#> [158,] 1.5754620460 -0.3123760361
#> [159,] -0.1213234474 -0.2312492021
#> [160,] 1.4235390976 -0.2948560401
#> [161,] -0.5518196108 0.7118245536
#> [162,] -0.1296523972 0.0593901374
#> [163,] -0.0987362827 0.0615276115
#> [164,] 2.0770946112 -4.8567494242
#> [165,] 0.9867326132 -0.1560091118
#> [166,] 1.6896438738 -2.2122034547
#> [167,] 1.2640771840 1.0357432859
#> [168,] 0.4421008216 -0.3228228594
#> [169,] -0.3669735687 0.2552670360
#> [170,] -0.1495310413 0.1514943195
#> [171,] -0.5339256247 -1.5157825474
#> [172,] -2.0001471724 -1.8758430479
#> [173,] 0.9748624959 -1.2867044736
#> [174,] 0.7851278186 -0.9245353464
#> [175,] 2.4917270175 -0.0826547509
#> [176,] -1.5748334366 1.2146706793
#> [177,] 0.2146664319 0.1517691248
#> [178,] 0.4171004638 0.7426898706
#> [179,] 0.0903058441 -0.0567387467
#> [180,] 2.2098494960 -1.4421654264
#> [181,] -1.3972653377 0.3949333176
#> [182,] 0.3659288014 0.1504689617
#> [183,] -1.0921788563 -1.0231927992
#> [184,] 1.6469495300 0.0859799333
#> [185,] -1.3098085971 -2.0367417462
#> [186,] 1.5847532442 1.3545542785
#> [187,] -0.7415401676 0.4656585622
#> [188,] -1.6240947337 -2.6423593442
#> [189,] 0.0070095670 0.0066656993
#> [190,] -1.4210025399 -0.7344665870
#> [191,] -0.8326939771 -0.2140479621
#> [192,] 0.8348540021 1.1135753670
#> [193,] -0.1122077495 0.0721504469
#> [194,] -1.5683885677 -2.0160742443
#> [195,] 0.6565565051 -0.4684988714
#> [196,] -1.0681758175 0.2324488493
#> [197,] 0.6068840390 -0.6209524080
#> [198,] 0.0750447955 -0.0171648761
#> [199,] -0.2801150869 -0.5521190629
#> [200,] 0.5405768681 0.7635137446
#> [201,] -0.5467012960 -0.4408550026
#> [202,] -0.8475315601 -1.1501314895
#> [203,] 0.3943677948 -0.0448688851
#> [204,] 0.4631888879 0.0018583084
#> [205,] -0.8446527005 1.9856260230
#> [206,] 0.2863805070 0.1270153112
#> [207,] -1.8295719822 1.4536425122
#> [208,] -2.1177922602 1.4032025693
#> [209,] -1.4713935950 0.0969689241
#> [210,] -0.7357805987 1.1038831779
#> [211,] 0.3772950768 -0.1704168102
#> [212,] -0.7391828056 -1.5219812437
#> [213,] -1.0375942582 -0.2332250915
#> [214,] -0.0484508703 -0.0320042507
#> [215,] 0.7209617663 1.2291720618
#> [216,] -3.0963084752 -4.3833427577
#> [217,] 0.5670717020 1.0722932106
#> [218,] 1.0774672487 0.7756675291
#> [219,] -1.9065544592 0.9760957501
#> [220,] 0.4926765975 -0.3376240290
#> [221,] 1.6431804204 1.3614942618
#> [222,] -0.6522615376 -0.3666569011
#> [223,] 0.5990687788 -0.7159199492
#> [224,] 1.1504102983 -2.1291874368
#> [225,] -0.1134315639 0.0497249362
#> [226,] -0.2265217800 0.0390819791
#> [227,] -0.6600287506 -0.2573924874
#> [228,] -0.2985356287 0.2432588341
#> [229,] 0.1299461109 -0.0176272207
#> [230,] -0.8374841422 -0.3906145632
#> [231,] 0.3193586935 -0.5057024898
#> [232,] 0.6770847256 0.7179541987
#> [233,] -1.0872787345 -0.2801846306
#> [234,] -0.0355280103 0.0268802671
#> [235,] 0.6486705151 0.6384211446
#> [236,] -1.3935051273 1.0956841482
#> [237,] -0.9499862587 -0.3267061741
#> [238,] -0.6628232472 0.2609415831
#> [239,] 0.0075626078 0.0049522254
#> [240,] -1.8492289620 -2.8578587519
#> [241,] 0.2261759373 -0.0967110924
#> [242,] 1.1966750980 0.5769139703
#> [243,] 1.1582664195 -0.4681848035
#> [244,] -1.0002482404 0.6833589528
#> [245,] -0.8551857375 0.1427084082
#> [246,] -0.8271689920 0.4603922573
#> [247,] -0.5624282740 0.5764987638
#> [248,] -1.0348725003 -1.8652888487
#> [249,] 1.2444190716 0.9179392611
#> [250,] 0.9707473107 -0.8929448533
#> [251,] 0.5675493431 -0.3645164671
#> [252,] -1.0496990067 -1.0534659875
#> [253,] -0.2905085677 -0.1198758057
#> [254,] 1.4362418477 -2.2023052969
#> [255,] -0.8249718053 1.3081838097
#> [256,] -1.7823415027 1.1057302444
#> [257,] 1.2641098337 -0.3298916079
#> [258,] 0.8982870177 -1.2068332382
#> [259,] 1.1232180324 -1.2559449995
#> [260,] -0.6524129007 -0.6855133451
#> [261,] 1.0748328735 -1.0226328512
#> [262,] -0.3882311110 -0.4693730042
#> [263,] -0.6163929196 0.7883836929
#> [264,] -0.7701061366 -0.2192475229
#> [265,] -0.1087859106 -0.0630019099
#> [266,] 1.4340788671 -1.4055702415
#> [267,] 0.3909407315 -0.1320247380
#> [268,] -0.1013132450 -0.0188188221
#> [269,] -1.6302032593 1.1218597259
#> [270,] -0.1756609878 0.3456772090
#> [271,] 0.5680824062 -1.0238324317
#> [272,] -2.7394311084 -0.0629648767
#> [273,] -0.1458463995 0.0310500042
#> [274,] 1.0042242731 -0.2788581010
#> [275,] -0.8931992161 0.2437078228
#> [276,] 2.3230454665 3.7385996882
#> [277,] -0.9053023551 0.0233845944
#> [278,] 0.5261836527 -0.8788280465
#> [279,] 0.0267943682 0.0271839002
#> [280,] -1.8680882668 -2.0869227138
#> [281,] -0.6125281040 0.7776852493
#> [282,] 1.5484303659 0.3796657526
#> [283,] -1.2838321979 -0.9110610686
#> [284,] 0.2955735566 -0.2289537564
#> [285,] -0.2378363212 -0.2491129612
#> [286,] 0.2288771638 0.0710035669
#> [287,] 0.8805354511 0.4934623342
#> [288,] -0.2503908325 0.3228304955
#> [289,] 0.8074080115 0.0650495015
#> [290,] -0.4243312848 0.1362060062
#> [291,] 1.6984032253 -2.4960125107
#> [292,] 0.3749012602 -0.6290526787
#> [293,] 0.0129321873 0.0115376284
#> [294,] -0.1797962613 -0.0510970867
#> [295,] -0.6124159360 0.4004025666
#> [296,] 0.2088770280 0.2238162087
#> [297,] -0.4645496967 0.0323739696
#> [298,] 1.0473562201 1.3250554830
#> [299,] 0.1369001671 0.1797925444
#> [300,] 1.2240353813 1.0617996289
#>
#> $grads_hat
#> [,1] [,2]
#> [1,] 0.5519993207 -0.3738089220
#> [2,] -0.0633674197 -0.0785245950
#> [3,] 0.7286730495 -0.3903554720
#> [4,] -0.1939572162 0.0428820137
#> [5,] 1.5217286359 -2.9259624739
#> [6,] -0.0755429150 0.0016303883
#> [7,] -2.5103837457 -2.9999721572
#> [8,] -0.2121276928 0.1445229713
#> [9,] 0.6219995703 0.5857305857
#> [10,] 0.7227507512 -0.1708843681
#> [11,] -2.4399618077 2.3153101695
#> [12,] 1.2711524405 1.3007922002
#> [13,] -0.7253895916 0.4515183968
#> [14,] 1.0943046563 -0.3756345460
#> [15,] -0.0336895963 -0.0765507451
#> [16,] 0.7937916532 -0.9673243254
#> [17,] 0.8890136134 1.8826982379
#> [18,] 0.8575496646 -0.4716006806
#> [19,] -0.6711273447 -0.2435346436
#> [20,] -1.0992502274 -0.7870908492
#> [21,] -1.9250916744 -1.4132683492
#> [22,] -0.0068973386 0.0015953393
#> [23,] -1.1901165799 -1.3272921607
#> [24,] -0.4937823188 1.0107656486
#> [25,] -0.4786953779 -1.1169265044
#> [26,] 0.5975722116 0.1464763741
#> [27,] -0.2704358396 -0.3692097071
#> [28,] -1.8738349863 -0.0606012336
#> [29,] 1.6153536486 -0.2755540368
#> [30,] -1.1568127508 -0.9446952725
#> [31,] 0.3149395908 -0.2837883322
#> [32,] -1.7034236663 -2.0485624082
#> [33,] 1.3661322685 0.7641217572
#> [34,] -0.1819292012 -0.1570328467
#> [35,] 0.4804524661 0.5613606903
#> [36,] 0.4362879866 0.4046337552
#> [37,] -1.9385875552 1.4962018920
#> [38,] 0.6545731014 0.6124411422
#> [39,] -0.5415486130 0.0584171166
#> [40,] -0.4685902687 0.2938534353
#> [41,] 2.0229225302 -0.7715702930
#> [42,] 0.6226033322 -0.7796196807
#> [43,] 1.9684359764 1.1513358264
#> [44,] 0.1957863253 0.3504337344
#> [45,] 0.6639520832 1.1621419670
#> [46,] 0.3648498323 -0.0416476962
#> [47,] -1.0498120311 -0.9935415818
#> [48,] 0.5583753083 -0.1228938696
#> [49,] -0.8192256459 -0.3907416933
#> [50,] -1.1274077200 1.0655631580
#> [51,] -0.2992184225 -0.1278321305
#> [52,] 0.5852582902 0.2185377231
#> [53,] -1.3906497872 0.8366058021
#> [54,] 0.1046711074 0.1097594487
#> [55,] -0.8047446058 -0.8944741589
#> [56,] 0.6783700559 -0.1313738743
#> [57,] -0.5938301948 0.1880469862
#> [58,] -0.0811560252 0.1000092767
#> [59,] -1.1595428245 0.0310294834
#> [60,] -0.2614206480 0.5117951681
#> [61,] 0.2079760552 -0.0006142163
#> [62,] -0.8184110693 -0.5449953310
#> [63,] -0.3319045644 -0.0178613730
#> [64,] 1.3800407574 1.9386399491
#> [65,] 0.1108469243 0.1463666285
#> [66,] 0.2757549941 0.8440467157
#> [67,] -0.9138612548 -0.1786972685
#> [68,] -2.5200910231 1.7811272583
#> [69,] -0.0571795113 -0.0132077862
#> [70,] -0.1296708742 0.1007636106
#> [71,] 0.2270589202 0.1321445920
#> [72,] 0.3288734189 -0.1415587197
#> [73,] -2.2619918374 -0.1828469070
#> [74,] 0.5925363750 -0.4989847475
#> [75,] 1.4573787006 0.0004272436
#> [76,] 1.2235928101 1.0434266697
#> [77,] -0.6870168298 -0.7703343644
#> [78,] 0.5496727514 -0.2951288769
#> [79,] -0.3835663345 -0.0577059965
#> [80,] 0.0643343417 0.0410411448
#> [81,] -0.6401052975 0.3877594540
#> [82,] 0.3443223480 -0.0567988480
#> [83,] 0.0718487692 0.0078590900
#> [84,] -1.0215939378 0.7132257352
#> [85,] -1.3940749064 0.4617881797
#> [86,] -0.5525545553 -1.1676829592
#> [87,] -0.3150129814 -0.0584911701
#> [88,] -0.0178670023 -0.0127749304
#> [89,] -0.7465092012 2.9239490467
#> [90,] 2.3409019937 -0.8346470144
#> [91,] 0.4605648680 0.4805257178
#> [92,] -1.5143362372 -1.2769214247
#> [93,] 0.8115057335 -0.2925206324
#> [94,] -1.7939744306 -3.1276103302
#> [95,] 0.3302204679 -0.4487413590
#> [96,] -0.2452012039 0.0506010161
#> [97,] -1.5605388328 1.6962226069
#> [98,] -0.4247636768 -0.0973600548
#> [99,] 0.0357915000 0.0122011204
#> [100,] -2.1164917086 -2.0947256163
#> [101,] 0.4026848599 -0.3011571837
#> [102,] -1.4714922522 -0.6690220730
#> [103,] -1.4815151563 -2.9327295129
#> [104,] -0.5885808399 0.1408769917
#> [105,] 0.1150658195 0.1071174655
#> [106,] 1.6492600116 0.2697197578
#> [107,] 1.2317557530 -0.4625770222
#> [108,] 0.8436040322 0.6804964433
#> [109,] 1.1572912544 0.0079149724
#> [110,] -2.0000108484 1.9509575884
#> [111,] -0.2401369281 -0.0384902915
#> [112,] 0.2762000610 -0.1487892653
#> [113,] -1.1546918186 0.0475797846
#> [114,] 0.5545552702 0.1623559218
#> [115,] -0.0898088576 0.0072026985
#> [116,] 0.6334987466 -0.5977643624
#> [117,] -0.8347114654 -1.3795106329
#> [118,] -1.1928305011 -0.0076594110
#> [119,] 1.0488808347 0.9289544398
#> [120,] 0.4328386136 -0.2194096551
#> [121,] -1.6530125325 1.2331593456
#> [122,] 1.1034124570 -0.8356163682
#> [123,] -2.8699727516 0.0985260843
#> [124,] -0.2664784107 -0.0052184052
#> [125,] 0.2047381384 0.0699059406
#> [126,] -0.5873460093 0.1332228206
#> [127,] -0.1602317251 -0.1821364190
#> [128,] -0.5019129415 -0.9926303002
#> [129,] 0.3727796456 -0.2357601896
#> [130,] 0.6179750989 -0.9842066742
#> [131,] 0.4353456015 0.0548751727
#> [132,] -0.0596795843 0.0779803815
#> [133,] -0.3088160853 -0.1588128332
#> [134,] -0.4208083756 -0.1609555174
#> [135,] -0.0148432246 0.0060556220
#> [136,] -0.1562168395 -0.1106765910
#> [137,] 1.4531707913 -0.3064740739
#> [138,] 0.6755382404 -0.5403842547
#> [139,] 0.2483806409 0.2371931556
#> [140,] 0.6032909202 -0.6356812047
#> [141,] -0.5004686829 0.3322211328
#> [142,] 1.4906966365 -0.6028837966
#> [143,] 0.9357818307 -1.3850570565
#> [144,] 0.4545160389 -0.3134205260
#> [145,] -1.2643063234 -0.0901792350
#> [146,] -0.2975296057 -0.8671843791
#> [147,] 1.0860796941 -0.9913821257
#> [148,] -0.5157345667 -0.2404797929
#> [149,] -0.1104686239 0.0294205453
#> [150,] -0.3406197299 0.1605746418
#> [151,] 0.9490580243 1.7004624844
#> [152,] -0.0710304114 0.0766718143
#> [153,] 0.4537591090 -0.0685005826
#> [154,] 0.3183186554 -0.2963365537
#> [155,] -0.6425247766 1.0770886803
#> [156,] -0.6681540345 -0.2594355159
#> [157,] 0.7018590598 0.0410475318
#> [158,] 0.5660155336 0.4414531659
#> [159,] -0.8561294790 0.5731415528
#> [160,] 0.1923586148 -0.3019263643
#> [161,] -1.1588864719 1.6004778004
#> [162,] 0.0640222404 0.0027856397
#> [163,] 0.0298481377 -0.0038430191
#> [164,] -0.2910737183 0.5467628308
#> [165,] 0.8573978819 -0.5754983109
#> [166,] -0.7129925642 0.0692548775
#> [167,] -0.2809337337 -0.1767938369
#> [168,] -0.2226733713 0.0301451318
#> [169,] -0.0835361127 0.1207584285
#> [170,] 0.4514561620 -0.0147429175
#> [171,] 0.5010751475 0.2803157389
#> [172,] -0.2497552803 -0.1122755905
#> [173,] -0.7329551687 -0.0265053752
#> [174,] -0.9930393482 -1.6791189990
#> [175,] 0.8963572033 1.1176024491
#> [176,] -0.2743495899 0.0124034910
#> [177,] 0.1849479164 -0.0885200144
#> [178,] -0.7800607284 0.5721538929
#> [179,] 0.1274888416 0.2418963879
#> [180,] 0.6777453418 0.7503339971
#> [181,] 1.3349968979 -1.3799839864
#> [182,] 2.7555987078 0.8728240115
#> [183,] 1.8551150164 -0.6808806722
#> [184,] 0.0615295468 -0.0319603133
#> [185,] 0.0003968346 0.0001296017
#> [186,] -1.7197313881 1.5928554879
#> [187,] 0.2302137558 0.3233548177
#> [188,] -0.1000678510 0.0057353793
#> [189,] -0.8176964115 -0.0878719871
#> [190,] -0.7512654440 0.0908890767
#> [191,] -2.0038900865 1.2007660012
#> [192,] 1.4302104714 0.0130729797
#> [193,] -1.4317162886 1.6799141885
#> [194,] -0.8269082290 -0.9036795147
#> [195,] 0.5636085191 -0.8589552278
#> [196,] 1.0947013702 0.4121690106
#> [197,] -0.4291255694 0.3119975917
#> [198,] 0.9777056509 -1.7348736233
#> [199,] -0.5175246460 0.3131361156
#> [200,] -0.2300444387 -0.1592765775
#> [201,] 0.4557093615 -0.1912404706
#> [202,] -0.5151791976 0.2279557767
#> [203,] 0.7054862938 0.3495016052
#> [204,] -0.3020766059 -0.0573859836
#> [205,] 1.2266535048 -0.3714095263
#> [206,] 0.5146627508 0.4713492561
#> [207,] 0.9305779484 0.7006305162
#> [208,] 0.7983100600 -0.4123500707
#> [209,] -0.4998367077 -0.1083796245
#> [210,] -0.7551298746 0.6197572992
#> [211,] -0.6890215022 0.5521416718
#> [212,] -0.4165376031 -0.2893098534
#> [213,] 0.9204813985 -0.1073204169
#> [214,] 1.1017390443 1.2307277390
#> [215,] 0.9840570094 0.0520941315
#> [216,] -0.3084832526 0.5676809676
#> [217,] -0.1830765284 -0.0024264977
#> [218,] 0.5247811892 1.6384416643
#> [219,] 0.0838777759 -0.0942000167
#> [220,] 0.5738762066 -0.0343301982
#> [221,] -0.4618042888 -0.1987665920
#> [222,] 0.2614313864 -0.8427638783
#> [223,] -0.5197121317 -0.0110048288
#> [224,] 1.2420997977 -0.3706245422
#> [225,] -0.0246049042 0.0217081466
#> [226,] -0.3369390185 0.5613852208
#> [227,] -0.2428129543 0.1794623907
#> [228,] 1.1289079747 -0.4389965936
#> [229,] -0.3563336919 0.1371036123
#> [230,] -0.2426517256 -0.4861642413
#> [231,] -0.3352657971 -0.6896386209
#> [232,] -0.3786477330 -0.2775539554
#> [233,] -1.1191703822 -0.6327275886
#> [234,] 0.7575485460 -0.0672142410
#> [235,] -0.5459713043 -1.3420918264
#> [236,] 0.9662868639 0.4456705001
#> [237,] -1.0062424725 0.9030560922
#> [238,] 0.2674009504 -0.1093182798
#> [239,] -0.1415620770 0.0266431020
#> [240,] -0.9962490714 0.5792118096
#> [241,] 0.0784666140 0.0631550190
#> [242,] -0.1786415575 0.0847451409
#> [243,] -0.5590441598 0.3670194591
#> [244,] -0.2522958257 -0.2529638137
#> [245,] 0.7721034609 0.0855291837
#> [246,] -0.0596807062 0.1413177145
#> [247,] 2.4535617074 1.5577596220
#> [248,] -1.7542564089 1.4631741756
#> [249,] -1.1476951147 -0.5356693678
#> [250,] 0.3694343877 0.2423739730
#> [251,] -0.7857850270 -1.7810246751
#> [252,] -1.2507973747 0.4433092403
#> [253,] -0.7452429009 0.7351311619
#> [254,] -0.1236928852 -0.0316502254
#> [255,] -0.2609392161 -0.0135147724
#> [256,] -1.7066951389 0.6851344752
#> [257,] -0.3029294846 -0.3501926256
#> [258,] 1.5976073605 -3.5045866630
#> [259,] 1.0253975652 -0.3530358266
#> [260,] 3.0116266746 -6.3469442848
#> [261,] -0.7425386338 -0.2748964174
#> [262,] 0.4829533019 -0.0757999829
#> [263,] 0.2273505895 -0.1004635716
#> [264,] -0.1664656685 -0.1570017023
#> [265,] 0.1003161283 0.1347026325
#> [266,] -0.6342788469 -0.7386291405
#> [267,] 2.1101141279 -1.3890305000
#> [268,] 0.5459106988 -0.1375825408
#> [269,] -0.9030594136 -0.1282676523
#> [270,] 0.8625666968 -0.6775172626
#> [271,] 0.0427925440 -0.0967934183
#> [272,] -0.2045079275 0.0751195257
#> [273,] -2.0296264040 -1.0984317353
#> [274,] 0.4416658927 -0.1595634296
#> [275,] 0.7936924073 0.2741643080
#> [276,] -0.5616441075 -0.5188880257
#> [277,] 1.5434330998 -0.7830077429
#> [278,] 0.3080575441 0.0629811696
#> [279,] -1.3293239165 0.2272654644
#> [280,] -0.1367728714 -0.1671300008
#> [281,] -0.5940067554 -1.3819200271
#> [282,] -0.7927054937 -1.7369396237
#> [283,] 1.2822615461 0.9807094235
#> [284,] -1.3295636952 1.2172892592
#> [285,] -0.6904257238 -0.9842460457
#> [286,] 0.3981428140 -0.8562201299
#> [287,] 1.0743840681 -0.0227481584
#> [288,] -1.0063176812 -1.5044426714
#> [289,] 0.3748118356 -0.3335932461
#> [290,] -0.4594095091 -0.2223882684
#> [291,] -0.6689850499 0.3761061841
#> [292,] 0.4829046221 -0.2003606020
#> [293,] -0.6744704796 -0.7571017915
#> [294,] -0.2140479992 0.0211019754
#> [295,] 1.3169391199 -1.6961040180
#> [296,] -1.1687251812 -1.5085617193
#> [297,] 0.5756035686 -0.0703234319
#> [298,] -0.8278149712 0.6826355450
#> [299,] 1.1596362094 -0.7517697657
#> [300,] -1.1153759781 0.3393217811
#>
#> $inv_hessian
#> [,1] [,2]
#> [1,] 1.00062255 0.02459982
#> [2,] 0.02459982 0.97204820
#>