R/generate_data.R
generate_piecewise_fractional_brownian.RdThis function generates a list of realizations of a piecewise fractional Brownian motion with random noise. A piecewise fractional Brownian motion is defined by a non constant Hurst parameter along the sampling points. We observe the process at regularly spaced time \(t_i = \frac{i}{M_n}\), where \(i = 0, \dots, M_n\). We define a segmentation \(\tau = (\tau_k)_{k=0, \dots, K+1}\), with \(0 = \tau_0 < \tau_1 < \dots < \tau_{K} < \tau_{K+1} = 1\). So, on the interval \([\tau_k, \tau_{k+1}]\), for \(k = 0, \dots, K\), the process is a fractional Brownian motion with Hurst parameter \(H_k\).
generate_piecewise_fractional_brownian( N = 100, M = 100, H = c(0.2, 0.5, 0.8), sigma = 0.05, pdf = NULL )
| N | An integer, number of curves to simulate. |
|---|---|
| M | An integer, expected number of points in the trajectory. The number of points follows a Poisson distribution with mean \(M\). |
| H | A vector of numeric, Hurst coefficients. \(0 < H_k < 1\) |
| sigma | A vector of numeric, standard deviation of the noise to add to the piecewise fractional Brownian motion. Should have the length of H. It adds heteroscedastic noise to the data. |
A function for the generation of the sampling points. |
A tibble containing the following elements:
...1: The sampling points
...2 The true trajectory
...3 The trajectory contaminated by noise with standard deviation \(\sigma\)
generate_piecewise_fractional_brownian(100, 50, c(0.2, 0.5, 0.8), 0.1) #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> New names: #> * `` -> ...1 #> * `` -> ...2 #> * `` -> ...3 #> [[1]] #> [[1]]$t #> [1] 0.00000000 0.01449275 0.02898551 0.04347826 0.05797101 0.07246377 #> [7] 0.08695652 0.10144928 0.11594203 0.13043478 0.14492754 0.15942029 #> [13] 0.17391304 0.18840580 0.20289855 0.21739130 0.23188406 0.24637681 #> [19] 0.26086957 0.27536232 0.28985507 0.30434783 0.31884058 0.33333333 #> [25] 0.33333333 0.34782609 0.36231884 0.37681159 0.39130435 0.40579710 #> [31] 0.42028986 0.43478261 0.44927536 0.46376812 0.47826087 0.49275362 #> [37] 0.50724638 0.52173913 0.53623188 0.55072464 0.56521739 0.57971014 #> [43] 0.59420290 0.60869565 0.62318841 0.63768116 0.65217391 0.66666667 #> [49] 0.66666667 0.68181818 0.69696970 0.71212121 0.72727273 0.74242424 #> [55] 0.75757576 0.77272727 0.78787879 0.80303030 0.81818182 0.83333333 #> [61] 0.84848485 0.86363636 0.87878788 0.89393939 0.90909091 0.92424242 #> [67] 0.93939394 0.95454545 0.96969697 0.98484848 1.00000000 #> #> [[1]]$x #> [1] -0.29090743 -0.12539168 -0.81913322 -0.81516034 0.03749846 -0.71168776 #> [7] -0.92763642 -1.01433273 -0.68557876 -0.85871183 -0.94947242 -0.92401524 #> [13] -1.46820905 -1.34667677 -0.63472253 -1.59106478 -0.40584185 -1.02059703 #> [19] -0.42385541 -1.30321804 -0.33058527 -0.84093023 -0.72307816 -0.29974726 #> [25] -0.54929349 -0.29691438 -0.56731959 -0.88984280 -0.85845083 -1.04213487 #> [31] -0.72174920 -0.65854129 -0.61081602 -0.76054203 -0.42626259 -0.51164210 #> [37] -0.66697951 -0.55011361 -0.50394748 -0.62078710 -0.73118349 -0.56070469 #> [43] -0.19449320 0.06576275 -0.10638393 0.05261436 0.06842330 0.42718594 #> [49] 0.13982296 0.07959831 0.08893443 -0.01415321 0.19962216 -0.10375487 #> [55] 0.03839462 -0.29123067 -0.49217435 -0.33904680 -0.59047055 -0.67367346 #> [61] -0.65769831 -0.83930412 -0.76501096 -0.84896385 -0.99794913 -1.04393657 #> [67] -1.21531728 -1.06482929 -1.06349202 -0.97907121 -1.06847632 #> #> [[1]]$x_true #> [1] -0.34816278 -0.26737324 -0.92242128 -0.85111078 -0.09852262 -0.64376990 #> [7] -0.95476228 -1.00051864 -0.61040837 -0.74437964 -1.05149912 -0.78827493 #> [13] -1.28513887 -1.35555891 -0.72937249 -1.49440881 -0.56151306 -0.79868219 #> [19] -0.54427997 -1.33570632 -0.30087125 -0.83279860 -0.62796217 -0.42171222 #> [25] -0.59928009 -0.41671169 -0.57419908 -0.84098918 -0.88774831 -1.05223410 #> [31] -0.58057689 -0.64527665 -0.65362955 -0.58522762 -0.56264195 -0.50448569 #> [37] -0.70758780 -0.49811464 -0.57730521 -0.73673045 -0.77727690 -0.59865742 #> [43] -0.13914447 0.03797381 -0.05749440 -0.01166704 0.08828335 0.21348144 #> [49] 0.11255793 0.14625907 0.11064242 0.01860783 0.02395225 -0.09469767 #> [55] -0.18456103 -0.41275109 -0.46146497 -0.42463120 -0.47814969 -0.60438809 #> [61] -0.72165549 -0.80635214 -0.69909775 -0.82098511 -0.95216646 -0.87920294 #> [67] -1.05142087 -1.06524652 -1.08118589 -1.04074080 -1.06467261 #> #> #> [[2]] #> [[2]]$t #> [1] 0.00000000 0.01587302 0.03174603 0.04761905 0.06349206 0.07936508 #> [7] 0.09523810 0.11111111 0.12698413 0.14285714 0.15873016 0.17460317 #> [13] 0.19047619 0.20634921 0.22222222 0.23809524 0.25396825 0.26984127 #> [19] 0.28571429 0.30158730 0.31746032 0.33333333 0.33333333 0.34920635 #> [25] 0.36507937 0.38095238 0.39682540 0.41269841 0.42857143 0.44444444 #> [31] 0.46031746 0.47619048 0.49206349 0.50793651 0.52380952 0.53968254 #> [37] 0.55555556 0.57142857 0.58730159 0.60317460 0.61904762 0.63492063 #> [43] 0.65079365 0.66666667 0.66666667 0.68253968 0.69841270 0.71428571 #> [49] 0.73015873 0.74603175 0.76190476 0.77777778 0.79365079 0.80952381 #> [55] 0.82539683 0.84126984 0.85714286 0.87301587 0.88888889 0.90476190 #> [61] 0.92063492 0.93650794 0.95238095 0.96825397 0.98412698 1.00000000 #> #> [[2]]$x #> [1] 0.5852844 0.6003414 0.6441141 1.4329007 1.0217839 0.5207435 1.2673833 #> [8] 1.8231714 1.5436712 1.1839542 0.7434542 0.6794726 0.7397757 1.6956218 #> [15] 1.5101861 0.6685422 0.9954779 1.2428142 0.1333141 1.3701529 0.4019598 #> [22] 1.1803166 1.4108870 1.6324002 1.3392744 1.1187579 0.9046282 1.1270606 #> [29] 1.1949271 0.7806829 1.4155396 0.9982684 0.9756488 0.9081231 0.7824243 #> [36] 0.6988739 0.7886909 0.8552240 0.6815784 0.5577192 1.0906981 1.0587784 #> [43] 1.2750252 1.4667752 1.5416496 1.6259446 1.4015042 1.3027382 1.2065802 #> [50] 1.2651532 1.3473052 1.2001301 1.0573824 1.1393478 1.1134194 1.1923936 #> [57] 1.0567722 0.8943055 0.8808548 0.8741925 0.9191489 0.7222151 0.5953175 #> [64] 0.5431312 0.8251980 0.6046360 #> #> [[2]]$x_true #> [1] 0.5442340 0.4794926 0.6553109 1.3775168 1.0425605 0.5772356 1.3475137 #> [8] 1.9331758 1.5637751 1.3164235 0.8327749 0.6016183 0.8118376 1.6308902 #> [15] 1.3193018 0.7637223 0.8957766 1.2936204 0.2139223 1.3339861 0.4517980 #> [22] 1.2831638 1.3021316 1.5134562 1.3824862 1.1147718 0.8558737 1.0939580 #> [29] 1.2303399 0.9051596 1.4311637 0.9840759 0.9952173 0.9698317 0.7800919 #> [36] 0.9273992 0.9702430 1.0094862 0.7808328 0.5991675 1.0067342 1.0561337 #> [43] 1.2509507 1.4752418 1.5522682 1.5627310 1.5219539 1.3906394 1.2250319 #> [50] 1.2006579 1.2940016 1.2632776 1.1178347 1.1929499 1.2155379 1.1542522 #> [57] 1.1442471 1.0873874 0.9288680 0.7501088 0.7120999 0.6539814 0.6326594 #> [64] 0.6557943 0.6928934 0.6802347 #> #> #> [[3]] #> [[3]]$t #> [1] 0.00000000 0.01960784 0.03921569 0.05882353 0.07843137 0.09803922 #> [7] 0.11764706 0.13725490 0.15686275 0.17647059 0.19607843 0.21568627 #> [13] 0.23529412 0.25490196 0.27450980 0.29411765 0.31372549 0.33333333 #> [19] 0.33333333 0.35294118 0.37254902 0.39215686 0.41176471 0.43137255 #> [25] 0.45098039 0.47058824 0.49019608 0.50980392 0.52941176 0.54901961 #> [31] 0.56862745 0.58823529 0.60784314 0.62745098 0.64705882 0.66666667 #> [37] 0.66666667 0.68518519 0.70370370 0.72222222 0.74074074 0.75925926 #> [43] 0.77777778 0.79629630 0.81481481 0.83333333 0.85185185 0.87037037 #> [49] 0.88888889 0.90740741 0.92592593 0.94444444 0.96296296 0.98148148 #> [55] 1.00000000 #> #> [[3]]$x #> [1] -1.4641091 -1.2984498 -1.2972029 -1.1146778 -0.4401957 -1.1916949 #> [7] -0.2040197 -0.7552950 -1.4350292 -0.7219001 -0.2128568 -0.3987813 #> [13] -1.0976877 -0.8196462 -0.5905943 -0.9049199 -2.1795442 -1.5813781 #> [19] -1.5819979 -1.9323797 -1.8189838 -1.4961944 -2.0022326 -1.6510248 #> [25] -1.5935910 -2.1736316 -1.6912648 -1.5281184 -1.6549340 -1.7077839 #> [31] -1.9893255 -2.3426470 -2.2918876 -2.8995259 -2.7639886 -2.7212818 #> [37] -2.6557387 -2.6524267 -2.7294313 -2.1912597 -2.2408256 -2.3524263 #> [43] -2.6294901 -2.5827640 -2.9106266 -2.9099700 -3.0827065 -3.1019130 #> [49] -3.2076812 -3.3659197 -3.4292826 -3.4477986 -3.2835909 -3.6840172 #> [55] -3.7197968 #> #> [[3]]$x_true #> [1] -1.3398060 -1.4207634 -1.1020024 -1.1344722 -0.4995397 -1.2589408 #> [7] -0.1547979 -0.9404475 -1.3296112 -0.8313222 -0.1159716 -0.5336233 #> [13] -1.1265002 -0.7611819 -0.6303375 -1.0380104 -2.3777257 -1.5471253 #> [19] -1.6112547 -1.9216044 -1.7809051 -1.7057309 -1.9852480 -1.5960666 #> [25] -1.5707078 -1.8831068 -1.6848228 -1.6487019 -1.7407033 -1.7119376 #> [31] -1.9631464 -2.1996174 -2.2477395 -2.7014276 -2.7250563 -2.5898609 #> [37] -2.6492820 -2.6316136 -2.5254380 -2.4450548 -2.3775787 -2.5408390 #> [43] -2.6757890 -2.7562651 -2.7783021 -2.9082000 -3.0184321 -3.1123811 #> [49] -3.1808115 -3.3081188 -3.4968465 -3.4889785 -3.5363303 -3.6703549 #> [55] -3.7456519 #> #> #> [[4]] #> [[4]]$t #> [1] 0.00000000 0.02083333 0.04166667 0.06250000 0.08333333 0.10416667 #> [7] 0.12500000 0.14583333 0.16666667 0.18750000 0.20833333 0.22916667 #> [13] 0.25000000 0.27083333 0.29166667 0.31250000 0.33333333 0.33333333 #> [19] 0.35416667 0.37500000 0.39583333 0.41666667 0.43750000 0.45833333 #> [25] 0.47916667 0.50000000 0.52083333 0.54166667 0.56250000 0.58333333 #> [31] 0.60416667 0.62500000 0.64583333 0.66666667 0.66666667 0.68888889 #> [37] 0.71111111 0.73333333 0.75555556 0.77777778 0.80000000 0.82222222 #> [43] 0.84444444 0.86666667 0.88888889 0.91111111 0.93333333 0.95555556 #> [49] 0.97777778 1.00000000 #> #> [[4]]$x #> [1] -0.27053084 -0.90860868 -1.69096205 -1.03335797 -1.22586960 -0.36267806 #> [7] 0.10685003 -0.86429146 -0.27796554 -1.02898763 -1.22061040 -0.81067163 #> [13] -0.02224438 -0.20612667 -0.52041660 -0.42482545 -0.68345510 -1.05034182 #> [19] -1.04325779 -0.73102231 -1.05634521 -1.01812870 -1.11715906 -1.24533103 #> [25] -1.26759762 -1.75556300 -1.58934788 -1.67420365 -1.47986755 -1.88720373 #> [31] -2.26180005 -2.69003281 -3.11902817 -2.76566719 -2.78961781 -2.81154137 #> [37] -2.62039336 -2.66716280 -2.89061398 -2.80588651 -2.60264086 -2.44045446 #> [43] -2.46301223 -2.25890267 -2.36619066 -2.13313141 -2.02579155 -2.00957010 #> [49] -2.09852228 -2.15877545 #> #> [[4]]$x_true #> [1] -0.38447012 -0.82423150 -1.73545478 -1.00127894 -1.05467913 -0.43436207 #> [7] 0.12555976 -0.78472922 -0.15580118 -1.01350401 -1.25030640 -0.75558865 #> [13] 0.06966404 -0.35002477 -0.58688850 -0.50750591 -0.89377754 -1.14492092 #> [19] -0.96675570 -0.69248880 -0.98689284 -0.87038213 -0.95456299 -1.41508921 #> [25] -1.31627291 -1.66292013 -1.59060939 -1.49283712 -1.66582152 -1.90804600 #> [31] -2.08819215 -2.63648760 -3.02161583 -2.62865493 -2.73494816 -2.78488945 #> [37] -2.79847478 -2.79086608 -2.72872105 -2.67145608 -2.64560136 -2.52022709 #> [43] -2.39051418 -2.37793250 -2.27088632 -2.11197071 -2.01801184 -2.02812742 #> [49] -2.02201873 -2.19099058 #> #> #> [[5]] #> [[5]]$t #> [1] 0.00000000 0.02777778 0.05555556 0.08333333 0.11111111 0.13888889 #> [7] 0.16666667 0.19444444 0.22222222 0.25000000 0.27777778 0.30555556 #> [13] 0.33333333 0.33333333 0.36111111 0.38888889 0.41666667 0.44444444 #> [19] 0.47222222 0.50000000 0.52777778 0.55555556 0.58333333 0.61111111 #> [25] 0.63888889 0.66666667 0.66666667 0.69696970 0.72727273 0.75757576 #> [31] 0.78787879 0.81818182 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0.09259259 #> [7] 0.11111111 0.12962963 0.14814815 0.16666667 0.18518519 0.20370370 #> [13] 0.22222222 0.24074074 0.25925926 0.27777778 0.29629630 0.31481481 #> [19] 0.33333333 0.33333333 0.35185185 0.37037037 0.38888889 0.40740741 #> [25] 0.42592593 0.44444444 0.46296296 0.48148148 0.50000000 0.51851852 #> [31] 0.53703704 0.55555556 0.57407407 0.59259259 0.61111111 0.62962963 #> [37] 0.64814815 0.66666667 0.66666667 0.68518519 0.70370370 0.72222222 #> [43] 0.74074074 0.75925926 0.77777778 0.79629630 0.81481481 0.83333333 #> [49] 0.85185185 0.87037037 0.88888889 0.90740741 0.92592593 0.94444444 #> [55] 0.96296296 0.98148148 1.00000000 #> #> [[8]]$x #> [1] 0.85473990 0.48432381 0.85516281 0.16067073 0.91888553 0.74708530 #> [7] 1.30618272 0.93013088 -0.06331076 0.12778696 0.85029352 0.35225172 #> [13] 0.38358681 -0.85388386 0.56621746 0.31749851 0.45372282 -0.10307654 #> [19] 0.04002159 0.24737026 0.32756063 0.52199123 0.58097625 0.35689775 #> [25] 0.46554198 0.57089219 1.10169065 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0.81809207 0.82590864 0.86852763 0.82262584 0.83666195 #> [55] 0.94878037 0.99907143 1.05331298 #> #> #> [[9]] #> [[9]]$t #> [1] 0.00000000 0.01851852 0.03703704 0.05555556 0.07407407 0.09259259 #> [7] 0.11111111 0.12962963 0.14814815 0.16666667 0.18518519 0.20370370 #> [13] 0.22222222 0.24074074 0.25925926 0.27777778 0.29629630 0.31481481 #> [19] 0.33333333 0.33333333 0.35185185 0.37037037 0.38888889 0.40740741 #> [25] 0.42592593 0.44444444 0.46296296 0.48148148 0.50000000 0.51851852 #> [31] 0.53703704 0.55555556 0.57407407 0.59259259 0.61111111 0.62962963 #> [37] 0.64814815 0.66666667 0.66666667 0.68421053 0.70175439 0.71929825 #> [43] 0.73684211 0.75438596 0.77192982 0.78947368 0.80701754 0.82456140 #> [49] 0.84210526 0.85964912 0.87719298 0.89473684 0.91228070 0.92982456 #> [55] 0.94736842 0.96491228 0.98245614 1.00000000 #> #> [[9]]$x #> [1] -0.06943062 -0.36603787 -0.94459043 -0.48367836 -1.35204929 -1.44250851 #> [7] -1.31599832 -1.21075325 -1.75612677 -0.57951275 -1.51875271 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0.435961811 0.113751225 0.391332945 #> [31] 0.553331340 0.679022324 0.871550007 0.339961837 0.810320361 #> [36] 0.727988821 0.184548591 0.003559792 0.045383302 0.101240167 #> [41] 0.072168623 0.049421069 0.086521090 0.118753048 0.159129575 #> [46] 0.281074321 0.280914771 0.316104875 0.348068804 0.427456544 #> [51] 0.388972016 0.462831983 0.523844574 0.386140570 0.436794521 #> [56] 0.409010779 0.415606528 0.341846794 #> #> #> [[10]] #> [[10]]$t #> [1] 0.00000000 0.01960784 0.03921569 0.05882353 0.07843137 0.09803922 #> [7] 0.11764706 0.13725490 0.15686275 0.17647059 0.19607843 0.21568627 #> [13] 0.23529412 0.25490196 0.27450980 0.29411765 0.31372549 0.33333333 #> [19] 0.33333333 0.35294118 0.37254902 0.39215686 0.41176471 0.43137255 #> [25] 0.45098039 0.47058824 0.49019608 0.50980392 0.52941176 0.54901961 #> [31] 0.56862745 0.58823529 0.60784314 0.62745098 0.64705882 0.66666667 #> [37] 0.66666667 0.68518519 0.70370370 0.72222222 0.74074074 0.75925926 #> [43] 0.77777778 0.79629630 0.81481481 0.83333333 0.85185185 0.87037037 #> [49] 0.88888889 0.90740741 0.92592593 0.94444444 0.96296296 0.98148148 #> [55] 1.00000000 #> #> [[10]]$x #> [1] 1.4749716 1.9931595 1.5230100 2.0398376 1.3636937 0.9661566 1.5646828 #> [8] 1.6817598 1.1457214 1.1629889 0.7925075 1.1479838 1.4292760 0.8185629 #> [15] 0.7836142 1.4921214 1.4144518 1.7074171 1.3007835 1.4446546 1.5600985 #> [22] 2.0220583 1.5188466 1.5498106 2.1848180 1.8966133 2.3056470 1.9988625 #> [29] 2.1330128 1.9940512 1.7792919 2.1133296 2.0002426 1.6670335 1.4927820 #> [36] 1.0938158 0.9803841 1.0175732 1.1068929 0.7382010 0.7018778 0.8189283 #> [43] 0.8712070 0.8813994 0.8518682 0.7113365 0.7566006 0.5447579 0.6386584 #> [50] 0.5346615 0.5671563 0.2071402 0.2545852 0.0809026 0.1005300 #> #> [[10]]$x_true #> [1] 1.2410368 1.9150454 1.6675816 2.1645832 1.2836634 0.9469755 1.4821093 #> [8] 1.5841450 1.0521327 1.1371904 0.7301258 0.9734458 1.4667774 0.6761917 #> [15] 0.7224025 1.5531357 1.3879978 1.6409907 1.3829938 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1.19954280 0.73449738 0.35572277 0.09043183 #> [19] 0.05316431 0.31715508 0.40265712 0.19613620 0.16347780 0.70853578 #> [25] 0.47623737 0.74928504 0.98581314 0.72586550 0.05737453 -0.36220379 #> [31] -0.43268937 -0.06864179 -0.09641254 -0.15475921 -0.28603996 -0.24815381 #> [37] -0.04183380 0.01367505 -0.16282801 -0.25088814 -0.06678551 0.15934197 #> [43] 0.18278176 0.35737091 0.39650488 0.49170469 0.54462623 #> #> #> [[27]] #> [[27]]$t #> [1] 0.00000000 0.02083333 0.04166667 0.06250000 0.08333333 0.10416667 #> [7] 0.12500000 0.14583333 0.16666667 0.18750000 0.20833333 0.22916667 #> [13] 0.25000000 0.27083333 0.29166667 0.31250000 0.33333333 0.33333333 #> [19] 0.35416667 0.37500000 0.39583333 0.41666667 0.43750000 0.45833333 #> [25] 0.47916667 0.50000000 0.52083333 0.54166667 0.56250000 0.58333333 #> [31] 0.60416667 0.62500000 0.64583333 0.66666667 0.66666667 0.68750000 #> [37] 0.70833333 0.72916667 0.75000000 0.77083333 0.79166667 0.81250000 #> [43] 0.83333333 0.85416667 0.87500000 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1.00000000 #> #> [[29]]$x #> [1] -0.15573712 0.23163722 0.42780654 -0.16505977 -0.34006234 -0.31927409 #> [7] -1.65840569 -1.39199915 -1.36646785 -1.48434656 -1.59440065 -0.95278883 #> [13] -1.63525148 -1.78060089 -1.99288634 -1.50100512 -0.35757586 -0.62722304 #> [19] -0.37402051 -0.65833253 -0.74805211 -0.60527700 -0.90714968 -0.63760003 #> [25] -0.31935236 -0.29749062 0.18114155 0.12453715 -0.15915948 -0.27074460 #> [31] 0.47418874 0.39798877 0.31922426 0.61122983 0.36940590 0.51140825 #> [37] 0.37106199 0.37526772 0.42030984 0.54664530 0.39283985 0.62074712 #> [43] 0.47819651 0.38762070 0.11191826 0.37783875 0.16686718 0.13312326 #> [49] 0.37888392 0.27994695 -0.03946583 0.02005064 0.27762734 0.25673171 #> #> [[29]]$x_true #> [1] -0.302110219 0.256345531 0.309997797 -0.087545803 -0.336594254 #> [6] -0.484894257 -1.680049935 -1.384268016 -1.301839099 -1.546288557 #> [11] -1.453258864 -1.019229766 -1.694608276 -1.860325398 -1.943839728 #> [16] -1.398959786 -0.364570317 -0.746614544 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0.87179487 0.89743590 0.92307692 0.94871795 #> [43] 0.97435897 1.00000000 #> #> [[30]]$x #> [1] -1.0396004 -0.9151069 -1.6324680 -1.1531412 -0.3243006 -1.0044265 #> [7] -0.1201862 -0.7639290 0.1758765 -0.4425674 -0.6955005 -0.0213823 #> [13] -1.3631164 -0.8556212 -0.4953226 -0.4093906 -0.3220277 -0.5031118 #> [19] -0.2936568 -0.3251453 -0.1529938 -0.2097405 -0.1411436 -0.4613810 #> [25] -0.4586287 -0.6064285 -0.0527892 -0.2323190 0.3091592 0.3641157 #> [31] 0.0939796 0.3378192 0.1013459 -0.1194118 -0.2196683 -0.3553638 #> [37] -0.3917578 -0.3648352 -0.6450451 -0.6504715 -0.6988407 -0.6141089 #> [43] -0.9674525 -0.8226070 #> #> [[30]]$x_true #> [1] -1.10454196 -0.99323742 -1.61227965 -1.21890098 -0.43137668 -1.06989999 #> [7] -0.09919985 -0.64924444 0.19786817 -0.44368069 -0.61769026 -0.07580160 #> [13] -1.30798581 -0.83721594 -0.45302388 -0.32915910 -0.21210093 -0.29900322 #> [19] -0.43568316 -0.35411020 -0.22014612 -0.08328271 -0.18420481 -0.53082602 #> [25] -0.39222336 -0.43861019 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0.15549119 #> #> #> [[32]] #> [[32]]$t #> [1] 0.00000000 0.02222222 0.04444444 0.06666667 0.08888889 0.11111111 #> [7] 0.13333333 0.15555556 0.17777778 0.20000000 0.22222222 0.24444444 #> [13] 0.26666667 0.28888889 0.31111111 0.33333333 0.33333333 0.35555556 #> [19] 0.37777778 0.40000000 0.42222222 0.44444444 0.46666667 0.48888889 #> [25] 0.51111111 0.53333333 0.55555556 0.57777778 0.60000000 0.62222222 #> [31] 0.64444444 0.66666667 0.66666667 0.69047619 0.71428571 0.73809524 #> [37] 0.76190476 0.78571429 0.80952381 0.83333333 0.85714286 0.88095238 #> [43] 0.90476190 0.92857143 0.95238095 0.97619048 1.00000000 #> #> [[32]]$x #> [1] -0.45464718 -1.35487066 -0.64835127 -1.17774691 -0.29657919 -1.42619559 #> [7] -0.44050434 -0.83483768 -0.76766879 -0.07687915 -0.74414782 0.24556893 #> [13] 0.67671860 0.42448204 0.27029710 -0.30042380 -0.48831021 -0.27362282 #> [19] -0.34221478 -0.23057851 -0.00907746 -0.08614154 0.11409063 0.05277485 #> [25] 0.25765163 0.27477153 0.81318492 0.79523825 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0.358771959 #> [41] 0.442906080 0.413400289 0.290554461 0.154627063 0.168261984 #> [46] 0.201054953 #> #> #> [[50]] #> [[50]]$t #> [1] 0.00000000 0.02222222 0.04444444 0.06666667 0.08888889 0.11111111 #> [7] 0.13333333 0.15555556 0.17777778 0.20000000 0.22222222 0.24444444 #> [13] 0.26666667 0.28888889 0.31111111 0.33333333 0.33333333 0.35555556 #> [19] 0.37777778 0.40000000 0.42222222 0.44444444 0.46666667 0.48888889 #> [25] 0.51111111 0.53333333 0.55555556 0.57777778 0.60000000 0.62222222 #> [31] 0.64444444 0.66666667 0.66666667 0.68750000 0.70833333 0.72916667 #> [37] 0.75000000 0.77083333 0.79166667 0.81250000 0.83333333 0.85416667 #> [43] 0.87500000 0.89583333 0.91666667 0.93750000 0.95833333 0.97916667 #> [49] 1.00000000 #> #> [[50]]$x #> [1] -0.20372976 -0.14213240 0.56638024 0.71523199 0.40802834 -0.18351577 #> [7] -0.33064464 -0.42626404 0.53486561 0.63843795 -0.70078943 -0.58386833 #> [13] -0.81795197 -0.54157840 0.16093747 0.04546873 0.23635679 0.14572758 #> [19] -0.03086760 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[[68]]$x_true #> [1] -0.608124981 0.135600651 -0.158245691 0.969838389 0.076379377 #> [6] 0.316959294 0.790727018 0.936646481 1.020626679 1.067551660 #> [11] 1.076569588 0.847637780 0.634122800 0.528024890 0.240567120 #> [16] 0.508920166 0.139156342 0.552328228 0.188142215 -0.317117471 #> [21] -0.567687797 -0.244766788 -0.063842634 -0.043116335 0.153878709 #> [26] 0.231666726 0.131730313 0.007760145 0.020551776 0.296288779 #> [31] 0.206412419 0.276781393 0.354149193 0.253611990 0.281492374 #> [36] 0.172369133 0.087723018 0.041255647 -0.111592925 -0.082877581 #> [41] -0.140312274 -0.094354885 -0.120071805 -0.192686597 -0.208324233 #> [46] -0.173221213 #> #> #> [[69]] #> [[69]]$t #> [1] 0.00000000 0.02380952 0.04761905 0.07142857 0.09523810 0.11904762 #> [7] 0.14285714 0.16666667 0.19047619 0.21428571 0.23809524 0.26190476 #> [13] 0.28571429 0.30952381 0.33333333 0.33333333 0.35714286 0.38095238 #> [19] 0.40476190 0.42857143 0.45238095 0.47619048 0.50000000 0.52380952 #> [25] 0.54761905 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[7] -0.11248573 -0.12046178 0.93014864 0.78913133 0.99498091 0.61447215 #> [13] 0.77357693 0.11582893 0.71167051 1.03638477 0.58266539 0.64204258 #> [19] 0.97364236 0.48874873 -0.01827167 0.03327990 0.51263160 0.67911821 #> [25] 0.39055691 0.45745014 0.95620383 1.14889915 0.52562189 0.55802915 #> [31] 0.59069401 0.86206490 1.05154872 1.13095628 1.29004673 1.22814628 #> [37] 1.17647789 1.36449748 1.58427065 1.73345374 1.62662618 1.45017144 #> [43] 1.76435831 1.68277954 1.59606168 1.56664829 #> #> [[70]]$x_true #> [1] -0.08486562 0.02183112 1.12416469 1.53822928 0.08133632 -0.07782168 #> [7] -0.26046006 -0.10249159 0.99049338 0.93252338 0.90436837 0.74261850 #> [13] 0.88868316 0.03465959 0.90041431 1.08500975 0.63887166 0.62539560 #> [19] 1.05306727 0.59088732 0.18882418 0.13032070 0.43409915 0.68476419 #> [25] 0.32925578 0.52234685 0.90387282 1.12164063 0.53132235 0.58402152 #> [31] 0.65634416 0.96013995 1.06583041 1.13593790 1.15981716 1.27183551 #> [37] 1.41435552 1.49403820 1.55163041 1.59110634 1.64525668 1.56330144 #> [43] 1.49721424 1.52545546 1.58848041 1.77169683 #> #> #> [[71]] #> [[71]]$t #> [1] 0.00000000 0.01851852 0.03703704 0.05555556 0.07407407 0.09259259 #> [7] 0.11111111 0.12962963 0.14814815 0.16666667 0.18518519 0.20370370 #> [13] 0.22222222 0.24074074 0.25925926 0.27777778 0.29629630 0.31481481 #> [19] 0.33333333 0.33333333 0.35185185 0.37037037 0.38888889 0.40740741 #> [25] 0.42592593 0.44444444 0.46296296 0.48148148 0.50000000 0.51851852 #> [31] 0.53703704 0.55555556 0.57407407 0.59259259 0.61111111 0.62962963 #> [37] 0.64814815 0.66666667 0.66666667 0.68518519 0.70370370 0.72222222 #> [43] 0.74074074 0.75925926 0.77777778 0.79629630 0.81481481 0.83333333 #> [49] 0.85185185 0.87037037 0.88888889 0.90740741 0.92592593 0.94444444 #> [55] 0.96296296 0.98148148 1.00000000 #> #> [[71]]$x #> [1] -0.08798294 -0.68972107 0.06465659 0.86730636 0.45145077 0.42123429 #> [7] 0.06251663 -0.74298467 0.04194398 0.18351449 0.40396696 -0.02370735 #> [13] 0.02481706 0.04969997 0.33117472 0.48804892 0.27386242 1.05655696 #> [19] 1.00716110 1.19625188 1.24848646 1.89257880 2.49578145 2.42843200 #> [25] 2.02834614 1.72864913 1.38511591 1.72373972 1.47617917 1.48768440 #> [31] 1.39266738 1.25332518 1.67376500 1.49551586 1.67346033 1.53401949 #> [37] 0.98408473 1.09001368 0.90312676 0.62355657 0.85181234 0.67651767 #> [43] 0.41674944 0.53207605 0.41827726 0.64223034 0.64644663 0.46936200 #> [49] 0.55628545 0.82753511 0.92086381 0.97900588 1.05751848 1.04101513 #> [55] 0.85191001 0.74088788 0.64349511 #> #> [[71]]$x_true #> [1] 0.04748072 -0.54824261 0.20112255 0.91504751 0.44424494 0.48070820 #> [7] 0.14223742 -0.76463174 -0.05782438 0.21295354 0.57176791 0.07476940 #> [13] 0.03509919 0.12277279 0.45963967 0.41134390 0.28382178 1.14419588 #> [19] 0.98563209 1.27186588 1.39748041 1.81108828 2.37578149 2.43191418 #> [25] 1.97954871 1.75376771 1.52756101 1.57724910 1.49973938 1.44183278 #> [31] 1.41240037 1.39432973 1.70121773 1.63878418 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0.68770597 #> [7] 0.27303380 0.46950424 0.27959261 -0.14789285 -0.57118579 0.28756127 #> [13] 0.07447020 0.69232325 0.26258077 0.41634537 0.61788738 0.55285616 #> [19] 0.49185842 -0.05133161 -0.22212838 -0.31458246 -0.23936632 -0.55412233 #> [25] -0.30184449 -0.32732899 -0.93017624 -0.33707058 -0.77183667 -0.37922091 #> [31] 0.08710335 0.11137023 0.06698931 -0.13443568 -0.02996032 -0.05690801 #> [37] -0.20804598 -0.01762739 0.04782213 0.15817509 0.11457467 0.26991891 #> [43] -0.02692924 0.24427368 0.06583031 -0.02412551 0.29670299 0.31876526 #> [49] 0.26321791 #> #> [[72]]$x_true #> [1] 0.293386862 0.636764561 0.427236666 0.127991090 0.637593528 #> [6] 0.702688020 0.403604173 0.443919261 0.426925039 -0.211590112 #> [11] -0.563539807 0.266065950 -0.039922973 0.608667498 0.281323860 #> [16] 0.513190330 0.493918964 0.497566558 0.532389798 -0.096616336 #> [21] -0.315268795 -0.334556339 -0.225865389 -0.561162885 -0.333120001 #> [26] -0.318759707 -0.853176674 -0.403844273 -0.598544978 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#> #> [[73]]$x #> [1] -0.607412901 -0.172873609 -0.926184951 -0.552086703 -0.011820618 #> [6] -0.442021017 -0.383776022 -0.681957863 0.244712499 -1.321435656 #> [11] -0.214527083 -0.398184912 -0.886607701 -0.820846886 -0.297824108 #> [16] -0.851296517 -0.741496382 -0.013262056 -0.674708516 -0.886444771 #> [21] -1.275448031 -1.244566317 -0.846060903 -0.744660368 -0.623716264 #> [26] -0.578947740 -0.469184927 -0.074390843 0.329693667 0.052273217 #> [31] 0.389975853 0.262327404 0.310591787 0.266982854 -0.050369690 #> [36] -0.048398287 0.150173878 -0.248210394 -0.049483286 -0.055108801 #> [41] 0.045047846 0.277072797 0.120300551 -0.039053871 0.120449566 #> [46] -0.005745625 0.150572068 0.377803879 0.396709343 0.471790422 #> [51] 0.385812654 0.184974330 0.015576058 -0.027667193 0.141160454 #> [56] 0.074630550 #> #> [[73]]$x_true #> [1] -0.572932222 -0.178503733 -0.866510695 -0.476682389 -0.005301984 #> [6] -0.368465451 -0.303281121 -0.552809833 0.293003064 -1.305153332 #> [11] -0.247507242 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[[91]]$x_true #> [1] 1.0619126 1.0252764 0.1897154 1.2262869 0.6105722 0.8519819 #> [7] 0.9904757 0.7052381 1.0121423 1.1119809 0.9443466 1.2269005 #> [13] 1.4547950 1.2715123 0.9071958 0.9291924 0.1360110 0.3363822 #> [19] 1.1300350 0.6802054 -0.1511470 -0.2834438 0.5117512 0.6514755 #> [25] 0.8968204 0.9172338 0.8682881 0.8928901 0.7096388 0.4784026 #> [31] 0.4900145 0.6740502 0.3265656 0.5862440 0.6545301 0.6712962 #> [37] 0.5516771 0.6162020 0.5385955 0.6610210 0.6051056 0.3667692 #> [43] 0.5365479 0.7943000 0.5417817 0.6121226 0.4825284 0.3691824 #> [49] 0.3301078 0.2930346 0.3332296 0.4494639 0.5018802 0.6678913 #> [55] 0.7334467 0.9291443 0.9330148 0.8628823 0.8273685 0.7649417 #> [61] 0.7165505 0.6501248 0.6886933 0.7854633 0.9614697 1.0427246 #> [67] 1.0001592 0.9594214 0.8839468 0.7535149 #> #> #> [[92]] #> [[92]]$t #> [1] 0.00000000 0.02222222 0.04444444 0.06666667 0.08888889 0.11111111 #> [7] 0.13333333 0.15555556 0.17777778 0.20000000 0.22222222 0.24444444 #> [13] 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