Temukan distribusi dan ubah ke distribusi normal

Saya memiliki data yang menggambarkan seberapa sering suatu peristiwa berlangsung selama satu jam ("angka per jam", nph) dan berapa lama acara berlangsung ("durasi dalam detik per jam", dph).

Ini adalah data asli:

nph <- c(2.50000000003638, 3.78947368414551, 1.51456310682008, 5.84686774940732, 4.58823529414907, 5.59999999993481, 5.06666666666667, 11.6470588233699, 1.99999999998209, NA, 4.46153846149851, 18, 1.05882352939726, 9.21739130425452, 27.8399999994814, 15.3750000002237, NA, 6.00000000004109, 9.71428571436649, 12.4848484848485, 16.5034965037115, 20.6666666666667, 3.49999999997453, 4.65882352938624, 4.74999999996544, 3.99999999994522, 2.8, 14.2285714286188, 11.0000000000915, NA, 2.66666666666667, 3.76470588230138, 4.70588235287673, 13.2727272728677, 2.0000000000137, 18.4444444444444, 17.5555555555556, 14.2222222222222, 2.00000000001663, 4, 8.46153846146269, 19.2000000001788, 13.9024390245481, 13, 3.00000000004366, NA, 7.36000000006855, 1.61137440758472, 1.50000000000873, 3.36585365857481, 22.3750000003256, 10.8387096775008, 2.92307692305075, 3.48837209304214, 5.17647058827074, 37.6666666666667, 1.17647058824335, 7.45454545462435, 36.2352941171508, 6.82352941167125, 2.22222222222222, 6.13333333333333, 11.4285714286665, 42.7058823523563, 28.1052631584975, 18.3333333333333, 1.24999999999091, 5.1034482758211, 1.82857142855926, 1.30693069306629, 3.22222222222222, 17.2800000001609, 10.5714285715165, 7.81818181826456, 3.14285714288328, 4.05194805197256, 3.6, 23.0909090904203, 0.249999999998181, 10, 27.3043478258106, 2.49999999998181, 2.00000000001663, 9.14285714293317, 4.74999999996544, 29.3999999996577, 16.9999999998021, 15.7777777777778, 1.74999999998727, 3.46666666666667, 2.45161290324422, 2.05231388331614, 2.60000000001513, 15.4054054053569, 4, 12.2222222222222, 2.46153846151642, 8.15384615399219, 2.23529411761644, 15.1111111111111, 0.23529411764867, 10.5454545455661, 17.5714285715747, 2.3030303030303, 1.37931034481651, 8.32000000007749, 5.1578947368105, 24.1999999997183, 15.4782608694085, 21.8749999998408, 2.74999999997999, 9.91304347823578, 3.86206896548623, 1.16959064328441, 2.84210526319272, 12.857142856929, 4, 3.69230769227463, 2, NA, 1.88888888888889, 15.4285714283148, 0.222222222222222, 6.16666666666667, 13.1034482757569, 3.19999999996275, 4.87499999996453, 2.88000000002682, 5.12499999996271, 26.6666666666667, 9.75000000014188, 17.2048192770602, 1.99999999998545, 1.65517241377981, 3.16666666666667, 2.23529411766237, 6.82352941181143, 2.74999999991996, 2.99999999997817, 11.4929577463281, 1.59999999998137, 8.65116279074452, 5.69230769240964, 13.7777777777778, 0.222222222222222, 10.6000000002468, 13.91304347812, 2.75862068963302, NA, 4.26666666666667, 5.64705882356808, 2.74999999997999, 15.047619047619, 16.6666666666667, 1.49999999998909, 4.62499999996635, 5.71428571428571, 1.83206106868927, 2.44444444444444, 2.4, 3.9999999999709, 2.33333333333333, 3.20000000007451, 5.931034482711, 7.14285714273835, 14.7272727274286, 0.352941176465754, 8.40000000019558, 10.1250000001473, 2.66666666666667, NA, 2.66666666666667, 4.7058823529734, 4.83333333333333, 9.31034482751146, 24.5882352937809, 2.13333333333333, 10.1739130434525, 5.56521739124801, 2.12658227848728, 1.88888888888889, 5.80000000013504, 7.14285714291654, 1.71428571429997, 1.99999999994179, NA, 5.00000000007276, NA, 0.129032258062578, 8.22222222222222, 7.16666666666667, 4.13793103444954, 2.82352941178404, 3.07692307697818, 4.00000000004902, 4.74999999986176, 9.75000000014188, 20.1333333333333, 2.66666666666667, 6.78947368416893, 1.46666666666667, 1.73195876289076, 4.76923076931619, 2.88888888888889, 7.4285714286332, 5.2, 3.384615384676, 4.7727272727399, 6.59999999992317, 11.4545454546667, 1.41176470586302, 11.1999999998696, 6.08000000005662, 4, 4.71428571432492, 5.00000000004158, 6.8, 6.83870967747072, 14.2500000002074, 5.49999999983993, 2.4, 4.71910112354612, 4, 1.72185430463842, 2.44444444444444, 4.30769230776946, 6.30769230780528, 3.53846153852491, 4.35294117641097, NA, 5.99999999990022, NA, NA, 7.42857142857143, 10.1333333333333, 6.79999999992084, 5.54838709681587, 1.83333333333333, 7.06666666666667, 2.9090909091217, 10.8000000001006, NA, 2.13333333333333, NA, 5.09090909090909, 4.21052631570563, 4.00000000003326, 4.28571428571429, 4.28571428574992, 2.49999999998181, 2.76923076928037, 4.99999999985448, 3.87500000005639, NA, NA, 12.2105263159391, 5.44444444444444, 2.6249999999809, 3.74193548389907, 3.28571428574161, 4.88888888888889, 9.33333333333333, 4.21621621620295, NA, 0.8, 4.5306122448549, 4.14285714289159, 3.1137724550985, 0.266666666666667, 5.27272727261567, 1.84615384613731, 8.36363636372488, 2.42857142853104, NA, 2.42857142853104, 8.28571428578318, 1.64705882350685, 8.2, 6.88888888888889, 1.74999999998727, 7.6, 3.33333333333333, 6.24999999995453, 9.56521739120752, 4.93333333333333, 16.4, 2.53333333333333, 7.2, 1.33333333333333, 3.3962264151018, 2, 9.38461538453135, 1.57142857144164, 3.45454545458201, 5.37499999996089, 7.74193548375467, 3.38461538458508, 7, NA, 4.54545454545455, 14.5, 1.93939393939394, 4.33333333333333, 4, 6.58823529402741, 2.90909090902933, 3.32530120480995, 25.6666666666667, 2, 6.54545454545455, 4.4, 3.54378818739119, 1.62499999998818, 4.22222222222222, 2.53333333333333, 14.6666666666667, 2.96296296296296, NA, 3.00000000004366, 16.1999999998114, 1.55555555555556, 3.11111111111111, NA, 4.8, 3.99999999997339, 4, 6.37499999995362, 2.7999999999674, NA, 32.8, 2.49999999998181, 11.0561797754255, NA, 2.75229357793903, 1.7142857142572, 7.66666666666667, 7.28571428577487, 2.36363636358633, 2.14285714287496, 6.27272727274387, 3.62499999997362, 19.6666666666667, 1.71428571427431, 6.60869565210701, 5.57894736838687, 5.84615384610149, 3.03030303030303, 1.33333333333333, 4.87499999996453, 4.71428571432492, 4.74418604653732, 13.0588235292329, 3.12500000004547, NA, 3.37500000004911, 2.41525423729648, 2.37499999998272, 4.54545454550265, 6.28571428576655, 2.55555555555556, 3.17647058819179, 5.59999999993481, 5.85714285719156, 7.42857142844789, NA, 4.83333333333333, 5.33333333333333, 4.48484848484848, 2.93333333333333, 3.83333333333333, 5.52941176474375, 9.33333333333333, 5.16666666666667, 18, 2.82352941178404, 5.54838709681587, 3.55555555555556, 1.25237191650965, 2, 2.16666666666667, 7.16666666666667, 3.00000000002495, 2.83333333333333, 2.48275862068966, 4.42857142860825, 11.1428571426718, NA, 5.52380952380952, 34.3448275859312, 4.75000000006912, 3.26315789471685, 10.2857142857998, 10.5555555555556, 5.00000000004158, 19.0843373493441, 20.6153846152, 2.24999999998363, 8.59259259259259, 4.25806451616101, 2.85714285716014, 5.1578947368105, 8.66666666666667, 3.14285714280487, 6.30769230763582, 6.79999999992084, 8.07692307663376, 5.73333333333333, 8.46153846146269, 2.34482758618807, 4.31999999991953, 4.57142857135254, 2.87500000004184, 2.28571428567627, 0.857142857149985, 10.2352941175069, 3.26086956520914, NA, 13.3333333333333, 2.75000000004002, 6.45161290312889, 3.61290322575218, 1.48854961831995, 3.37499999997544, 4.0540540540413, 5.73333333333333, 3.85714285707871, 3, 6.31578947364551, 1.55555555555556, 7.84615384608358, 0.4, 7.66666666666667, NA, 7.85185185185185, 2.59090909091595, 7.28571428577487, 5.74999999995816, 3.28571428574161, 16.043478260829, 15.8000000003679, 2.50000000003638, NA, 2.06451612904776, 1.82163187855948, 0.874999999993634, 13.2000000001229, 6.92307692301493, 3.7142857143166, 3.00000000001343, 5.83333333333333, 3.86666666666667, 9.39999999989057, 2.49999999998181, 6.24000000005811, 4.58823529414907, 3.72413793109428, 3.21428571427235, 6.85714285719988, 8.42857142864151, 5.23076923086291, 10.5454545455661, 14.1428571429747, 4.00000000005821, 4.08791208795393, 8.47058823517811, 3.94422310755509, 3.62500000005275, 6.0000000001397, 1.33333333333333, 3.73333333333333, 6.31578947352942, NA, 4.53333333333333, 8.46153846169001, 0.470588235287673, 2.28571428571429, 22.7142857144746, 8.00000000012846, 2.8108108108285, 4.57142857146658, 5.87500000008549, 6.42857142862488, 19.2258064513241, 13.4666666666667, 3.46666666666667, 4.90322580648844, 3.51515151515152, 1.56862745098755, 1.53846153844776, 3.63636363636364, 4.71428571432492, 3.06666666666667, 4.61538461546728, NA, 2.83333333333333, 5.53846153841194, 1.80645161287609, 9.14285714285714, 2.42857142853104, 3.2, 5.00000000007276, 4.42857142860825, 6.12500000008913, 3.24999999990541, 4.16326530608288, 14.6666666666667, 5.37499999996089, 7.43478260867684, 9.93548387104236, 3.73205741626378, 2.24999999998363, 13.7777777777778, 4.74074074074074, 7.4285714286332, 3.61904761904762, 7.13513513511269, 5.28571428575824, 5, 2.5882352940822, 11.5000000001673, 27.1249999998026, 2.875, 2.81081081077544, 9.42857142864983, 7.05882352931509, 3.83333333333333, 16.8695652172205, 16.7692307690806, 10.1333333333333, 5.45454545455989, 7.8750000001146, 1.6883116883219, 2.66666666666667, 11.7857142856653, 3.33333333333333, 6.33333333333333, 7.39999999991385, 12.5882352942039, 4.00000000003326, 6.72727272734392, 3.03030303030303, 6, 30.6666666666667, 3.74999999997272, 3.00000000003011, 8.00000000006652, 8.00000000006009, 2.57142857144995, 10.695652173886, 14.2666666666667, 7.75000000011278, 2.51162790697674, 6.33333333333333, 3.28125000004775, 1.88888888888889, 10.4000000002421, 4.87499999996453, 13.7142857143998, 8.5, NA, 4.87499999996453, 8.181818181645, 1.24999999999091, 4.38095238095238, 27.1764705878631, 2.37499999998272, 2.94117647060838, 11.7142857143831, 5.99999999996324, 2.37499999998272, 14.7637795275455, 14.313253012008)
dph <- c(3.12500000004547, 6.69473684199041, 4.3106796117187, 11.6937354988146, 103.882352941888, 10.9999999998719, 7.33333333333333, 20.3529411761918, 5.23076923072239, NA, 4.61538461534328, 47.5555555555556, 2.94117647054795, 18.9565217389385, 44.3199999991745, 28.5000000004147, NA, 10.4705882353658, 19.000000000158, 25.8181818181818, 43.2167832173461, 51.5555555555556, 8.37499999993906, 6.91764705878563, 9.37499999993179, 5.64705882345207, 4.53333333333333, 27.4285714286627, 14.4285714286914, NA, 1.6, 5.76470588227399, 4.70588235287673, 55.2727272733122, 2.11764705883803, 30.8888888888889, 41.2222222222222, 23.4444444444444, 2.42857142859162, 6.2, 17.0769230767702, 21.2800000001982, 40.8292682931466, 14.5, 6.25000000009095, NA, 15.0400000001401, 5.68720379147547, 2.40000000001397, NA, 26.3750000003838, 18.0645161291679, 3.99999999996418, 6.13953488375417, 8.47058823535212, 128.666666666667, 2.23529411766237, 34.1818181821799, 115.999999998411, 5.99999999991782, 5.77777777777778, 10.6666666666667, 15.4285714286997, 54.8235294110138, 81.315789475428, 42.3333333333333, 1.74999999998727, 7.99999999993577, 4.34285714282825, 1.90099009900552, 5.22222222222222, 39.840000000371, 25.1428571430662, 7.81818181826456, 8.57142857149985, 15.2727272728196, 6.4, 93.0909090889387, 0.374999999997272, 23.1666666666667, 29.3913043475286, 0.874999999993634, 1.71428571429997, 13.5714285715414, 5.49999999995998, 134.799999998431, 77.7999999990943, 18, 2.24999999998363, 5.73333333333333, 3.09677419357165, 2.29376257547098, 5.70000000003318, 23.1891891891162, 14, 13.5555555555556, 1.69230769229254, 9.23076923093455, 4.35294117641097, 48.6666666666667, 0.352941176473005, 16.0000000001693, 56.7142857147573, 1.81818181818182, 1.37931034481651, 19.6800000001833, 6.63157894732779, 134.999999998428, 41.0434782604541, 26.8749999998045, 3.62499999997362, 16.5652173912624, 10.3448275861238, 1.28654970761285, 2.94736842108875, 13.4285714283481, 7.6, 3.2307692307403, 2, NA, 3.44444444444444, 93.1428571413081, 0.111111111111111, 13.6666666666667, 28.1379310342568, 2.39999999997206, 7.8749999999427, 4.00000000003725, 6.99999999994907, 60, 26.8750000003911, 30.5060240963, 3.12499999997726, 3.17241379307798, 4.83333333333333, 9.29411764712247, 12.7058823530282, 4.24999999987631, 6.99999999994907, 9.97183098578469, 2.39999999997206, 8.93023255818789, 15.3846153848909, 94, 0.111111111111111, 21.4000000004983, 29.9130434779581, 1.24137931033486, NA, 15.8666666666667, 7.17647058828444, 1.49999999998909, 37.9047619047619, 27.6666666666667, 1.74999999998727, 9.37499999993179, 17.3333333333333, 11.603053435032, 5.33333333333333, 2.8, 7.99999999994179, 3.5, 1.60000000003725, 7.31034482752751, 6.42857142846452, 56.7272727278731, 0, 21.6000000005029, 28.8750000004202, 1.6, NA, 4.5, 5.64705882356808, 7.16666666666667, 36.2068965514334, 40.235294117096, 4.8, 22.3043478260305, 8.86956521730152, 3.94936708861923, 3.33333333333333, 12.6000000002934, 20.0000000001663, 1.28571428572498, 0.749999999978172, NA, 6.25000000009095, NA, 0.258064516125156, 18.6666666666667, 17, 5.51724137926605, 2.58823529413537, 11.0769230771215, 5.26315789480134, 11.4999999996653, 34.1250000004966, 42.4, 6.53333333333333, 33.1578947366389, 4.4, 4.9484536082593, 11.2307692309704, 5.11111111111111, 23.8571428573412, 0.4, 2.30769230773364, 6.81818181819986, 8.19999999990454, 26.7272727275556, 0.352941176465754, 24.1999999997183, 7.04000000006557, 2.5, 7.14285714291654, 11.4285714286665, 12.1333333333333, 2.83870967744068, 42.7500000006221, 4.99999999985448, 3.33333333333333, 10.112359550456, 16.8, 4.23841059603303, 2.22222222222222, 14.4615384617975, 15.6923076925887, 3.23076923082709, 1.05882352939726, NA, 7.42857142844789, NA, NA, 16.952380952381, 12.4, 6.29999999992666, 85.4193548393512, 4.33333333333333, 11.8666666666667, 6.0000000000635, 19.6800000001833, NA, 3.46666666666667, NA, 13.0909090909091, 12.6315789471169, 5.14285714289991, 9.14285714285714, 12.1428571429581, 2.87499999997908, 1.692307692338, 10.2499999997017, 5.00000000007276, NA, NA, 19.578947368661, 10.4444444444444, 1.74999999998727, 4.77419354842295, 8.57142857149985, 9.66666666666667, 13.5238095238095, 7.29729729727434, NA, 1.6, 9.18367346930048, 6.85714285719988, 4.5508982036055, 0.666666666666667, 10.90909090886, 2.61538461536119, 6.1818181818836, 1.57142857140244, NA, 1.99999999996674, 24.4285714287746, 0.941176470575345, 16.6, 17.6666666666667, 0.999999999992724, 10.2666666666667, 7.5, 11.2499999999181, 11.9999999998785, 12.8, 29.7333333333333, 5.33333333333333, 13.6, 1.84615384615385, 12.7924528302168, 2.4, 23.6923076920955, 2.42857142859162, 4.90909090914286, 3.62499999997362, 11.4193548385381, 4.92307692303284, 17, NA, 16.9090909090909, 20.8333333333333, 0.96969696969697, 8, 11.8333333333333, 10.2352941175069, 5.81818181805867, 6.07228915660947, 39.3333333333333, 4.13333333333333, 9.6969696969697, 11.2, 7.94297352346302, 2.12499999998454, 4.66666666666667, 2.66666666666667, 11.3333333333333, 3.7037037037037, NA, 2.87500000004184, 24.3999999997159, 1.88888888888889, 10.4444444444444, NA, 3.73333333333333, 7.08571428566715, 15.8333333333333, 11.2499999999181, 2.59999999996973, NA, 43.6, 3.24999999997635, 22.9213483149066, NA, 5.22935779808415, 1.85714285711197, 14.3333333333333, 15.4285714286997, 4.363636363544, 1.8571428571583, 7.36363636365585, 6.37499999995362, 51.3333333333333, 3.42857142854862, 1.043478260859, 4.94736842102232, 2.76923076920597, 5.09090909090909, 2.5, 7.49999999994543, 9.71428571436649, 7.25581395352766, 29.8823529407672, 6.62500000009641, NA, 6.12500000008913, 5.59322033900236, 5.12499999996271, 5.45454545460318, 7.00000000005821, 2.44444444444444, 3.05882352936987, 16.9999999998021, 7.71428571434986, 16.8571428568625, NA, 8.83333333333333, 6.77777777777778, 2.78787878787879, 5.06666666666667, 8.83333333333333, 9.17647058829813, 14.1666666666667, 5.5, 36.6666666666667, 4.23529411767606, 7.48387096779814, 5.33333333333333, 2.73244781783923, 2.13333333333333, 2.5, 11.5, 6.42857142862488, 3, 1.79310344827586, 8.00000000006652, 24.8571428567295, NA, 6.09523809523809, 68.5517241373807, 21.2500000003092, 6.21052631575142, 19.2857142858747, 15.1111111111111, 5.5714285714749, 42.6506024095189, 42.615384615003, 4.87499999996453, 13.3333333333333, 11.8709677420246, 8.83116883122224, 6.31578947364551, 9.83333333333333, 1.99999999996674, 7.69230769223881, 4.39999999994878, 17.3076923070723, 8.13333333333333, 16.461538461391, 1.65517241377981, 7.03999999986887, 10.2857142855432, 2.12500000003092, 1.14285714283814, 1.14285714286665, 13.1764705880548, 3.7826086956426, NA, 28.1333333333333, 3.75000000005457, 8.38709677406756, 6.83870967731663, 3.20610687022758, 6.49999999995271, 6.32432432430443, 13.8666666666667, 8.42857142843125, 2.83333333333333, 13.4210526314967, 3.33333333333333, 14.1538461537194, 0.933333333333333, 15.8333333333333, NA, 8.2962962962963, 5.31818181819589, 13.5714285715414, 10.1249999999263, 6.28571428576655, 39.260869565118, 26.6000000006193, 4.00000000005821, NA, 3.74193548389907, 5.35104364326849, 0.749999999994543, 12.0000000001118, 4.30769230765373, 6.57142857148322, 6.00000000002686, 13.3333333333333, 5.33333333333333, 16.1999999998114, 1.87499999998636, 13.1200000001222, 11.0588235294875, 2.0689655172746, 5.57142857140541, 17.1428571429997, 12.8571428572498, 10.4615384617258, 27.2727272730159, 25.5714285716412, 9.25000000013461, 12.3956043957313, 20.8235294114795, 4.54183266930586, 6.25000000009095, 14.000000000326, 1.33333333333333, 8.13333333333333, 7.15789473666668, NA, 62.6666666666667, 18.0000000003224, 0.117647058821918, 6.66666666666667, 43.8571428575075, 8.55172413806835, 5.40540540543942, 7.71428571434986, 11.0000000001601, 18.2857142858663, 52.6451612895318, 26.4, 5.6, 13.1612903226795, 5.93939393939394, 2.48366013073029, 1.53846153844776, 2.36363636363636, 4.14285714289159, 1.33333333333333, 9.23076923093455, NA, 2.83333333333333, 10.9230769229791, 2.19354838706382, 18.6666666666667, 3.57142857136918, 1.6, 8.50000000012369, 9.85714285722482, 11.2500000001637, 1.74999999994907, 6.367346938715, 33, 10.8749999999209, 23.9999999999393, 23.4838709679183, 3.73205741626378, 2.74999999997999, 20.6666666666667, 4.14814814814815, 13.2857142858248, 4.57142857142857, 15.2432432431953, 5.85714285719156, 10, 2.5882352940822, 20.5000000002983, 58.3749999995753, 1.875, 5.08108108101713, 13.5714285715414, 10.8235294116165, 2.66666666666667, 27.4782608692871, 30.9230769228, 17.6, 7.77272727274784, 15.7500000002292, 2.46753246754739, 2.77777777777778, 12.6428571428046, 3.6, 11.2222222222222, 6.79999999992084, 20.705882353083, 2.85714285716662, 14.1818181819683, 3.51515151515152, 11.7777777777778, 57.8888888888889, 3.9999999999709, 5.58620689660779, 15.4285714286997, 11.3548387097627, 1.00000000000832, 23.9999999999393, 25.3333333333333, 20.1250000002929, 4.88372093023256, 13.1111111111111, 2.57812500003752, 2.66666666666667, 12.0000000002794, 7.74999999994361, 23.2857142859079, 10.3333333333333, NA, 4.74999999996544, 12.545454545189, 1.74999999998727, 8, 55.999999999233, 2.12499999998454, 5.05882352944641, 24.5714285716329, 8.21052631573917, 1.99999999998545, 29.17322834643, 30.5060240963)
par(mfrow = c(2, 2))
hist(nph)
hist(dph)
qqnorm(nph)
qqline(nph)
qqnorm(dph)
qqline(dph)

Ini adalah distribusinya:

Karena data jelas tidak terdistribusi normal, banyak uji statistik tidak dapat diterapkan pada data ini. Tapi mungkin saya bisa mengubah data menjadi distribusi normal?

Bagaimana saya bisa mengetahui distribusi mana ini?
Dan bagaimana saya bisa mentransfer data ke distribusi normal?

Tujuannya adalah untuk melakukan analisis varians (MANOVA) atau semacamnya (data yang disajikan di sini adalah dua variabel dependen).

Data terlihat seperti memiliki distribusi eksponensial . Untuk transformasi, log sederhana tampaknya berfungsi dengan baik.

hist(log(dph), freq=FALSE, ylim=c(0, .4))
lines(seq(-6, 6, by=0.01), dnorm(seq(-6, 6, by=0.01), 2, 1), col="red")
qqnorm(log(dph), ylim=c(0, 5))
qqline(log(dph), col="red")

Setiap distribusi kontinu dapat diubah menjadi distribusi normal melalui proses yang disebut Gaussianization (Chen & Gopinath, 2001) . Untuk distribusi univariat, Gaussianization sederhana. Jika variabel acak$Y$ memiliki fungsi distribusi kumulatif (CDF) $F_Y$ dan $\Phi$ adalah CDF dari standar normal, lalu

akan memiliki distribusi normal standar. Ini mudah dilihat, karena CDF dari$X$ adalah

Jika $Y$ didistribusikan secara eksponensial dengan kurs $\lambda$, maka data dapat ditransformasikan melalui

yang terlihat mirip dengan logaritma:

Saya tidak menggunakan R , tapi saya yakin Anda dapat menemukan implementasi CDF terbalik (juga dikenal sebagai fungsi kuantil ) dari normal,$\Phi^{-1}$.

1. Bagaimana saya bisa mengetahui distribusi mana ini? Di sini Anda dapat menggunakan beberapa uji statistik dari paket R fitdistrplus. Dari paket Anda akan menemukan crateria pas yaitu AIC, BIC dll. Pemasangan distribusi 'gamma atau distribusi mor seperti "normal". Inilah metodenya.

   • ESTIMASI LIKELIHOOD MAKSIMUM
   • ESTIMASI PENCARIAN MOMEN
   • ESTIMASI PENCARIAN QUANTILE
   • ESTIMASI KEBAIKAN MAKSIMUM MAKSIMUM (Statistik Kebaikan dan Kriteria Kebaikan)

Kemudian akhirnya Anda akan menemukan di antara beberapa model teoritis yang terbaik yang menyerupai data yang Anda amati.

2. Dan bagaimana saya bisa mentransfer data ke distribusi normal? Di sini Anda dapat menggunakan Box Cox Transfom

   Box_Cox_tran=function(x, lambda, jacobian.adjusted = FALSE) 
   {
     bc1 <- function(x, lambda) 
     {
       if (any(x[!is.na(x)] <= 0)) 
         stop("First argument must be strictly positive.")
       z <- if (abs(lambda) <= 1e-06) 
         log(x)
       else ((x^lambda) - 1)/lambda
       if (jacobian.adjusted == TRUE) {
         z * (exp(mean(log(x), na.rm = TRUE)))^(1 - lambda)
       }
       else z
     }
     out <- x
     out <- if (is.matrix(out) | is.data.frame(out)) {
       if (is.null(colnames(out))) 
         colnames(out) <- paste("Z", 1:dim(out)[2], sep = "")
       for (j in 1:ncol(out)) {
         out[, j] <- bc1(out[, j], lambda[j])
       }
       colnames(out) <- paste(colnames(out), round(lambda, 2), 
                              sep = "^")
       out
     }
     else bc1(out, lambda)
     out
   }
   

Inilah exmaple kerja saya:

# ---------------------------------------------------------------------------------------------------------------------------
# Objective three starts Here
# (3)= Bivariate modelling of annual maxima using traditional approach 
# a)    First transform onbserved seasonal maxima into normal distribution using Box-Cox Transformations(x to z)
# b)    Finaly, Estimate Pearson coefficient using traditional bivariate normal distribution
# ---------------------------------------------------------------------------------------------------------------------------
rm(list=ls())
Sys.setenv(LANGUAGE="en")  # to set languege from Polish to English
setwd("C:/Users/sdebele/Desktop/From_oldcomp/Old_Computer/Seasonal_APP/Data/Data_Winter&Summer")
# Loading the required package here
library(MASS)
library(geoR)
require(scales)
require(plyr)
require(car)
library(ggplot2)
require(alr3)
library(ggplot2)
library(reshape2)
library(nortest)
require(AID)
require(distr)
require(fBasics)
# -----------------------------------------------------------------------------------------------------------------------------
# Here the Box-Cox Transformations equations
# x(lambda)=x^lamda-1/lambda, if lambda is not zero
# else log(x) if lambda=0
#--------------------------------------------------------------------------------------------------------------------------------
# Here is the data for six guaging stations of dependant ( 51.12% to 89.85%)
filenames=c("ZAPALOW.txt","GORLICZYNA.txt","SARZYNA.txt","OSUCHY.txt","HARASIUKI.txt","RUDJASTKOWSKA.txt")
# ---------------------------------------------------------------------------------------------------------------------------
# (1)= For ZAPALOW hydrological guaging stations starts here
# --------------------------------------------------------------------------------------------------------------------------------
ZAPALOW=read.table(file=filenames[1],head=T,sep="\t")
newZAPALOW <- na.omit(ZAPALOW) # to eliminte the missing value from the data sets 
Years=newZAPALOW$Year
    Winter=newZAPALOW$Winter
Summer=newZAPALOW$Sumer
    source("Box_Cox_Transfom.R") # R_script containing the tranformation equations 
    # estimation of lambda using AID R package 
    # boxcoxnc(Sumer, method="ac", lam=seq(-2,2,0.01), plotit=TRUE, rep=30, p.method="BY")
    # boxcoxnc(Winter, method="ac", lam=seq(-2,2,0.01), plotit=TRUE, rep=30, p.method="BY")
    Trans_Win=boxcoxnc(Winter)
    Trans_Sum=boxcoxnc(Summer)
    Winter_trans=Box_Cox_tran(Winter,Trans_Win$result[1,1],jacobian.adjusted=T)
Summer_trans=Box_Cox_tran(Summer,Trans_Sum$result[1,1],jacobian.adjusted=T)
    newZAPALOW[,4]=Winter_trans
    newZAPALOW[,5]=Summer_trans
    colnames(newZAPALOW)= c("Year","Winter " ,"Summer","Winter_Trans","Summer_Trans")
    par(mfrow=c(2,2))
    par("lwd"=2)
    ## Plot histogram with overlayed normal distribution.
    hist(newZAPALOW[,4],main="",xlab="Discharge",freq=FALSE,col="lightblue")
    curve(dnorm(x,mean=mean(newZAPALOW[,4]),sd=sd(newZAPALOW[,4])), add=TRUE, col="darkred",lwd=2)
    qq.plot(newZAPALOW[,4], dist= "norm", col=palette()[1], ylab="Sample Quantiles",
            main="Normal Probability Plot", pch=19)
    #b <- mydata[,c(2,3)] # select interesting columns
    result <- shapiro.test(newZAPALOW[,4]) # checking for normality test 
    result$p.value
ad.test(newZAPALOW[,4]) # checking for normality test 
## Plot histogram with overlayed normal distribution.
hist(newZAPALOW[,5],main="",xlab="Discharge",freq=FALSE,col="lightblue")
curve(dnorm(x,mean=mean(newZAPALOW[,5]),sd=sd(newZAPALOW[,5])), add=TRUE, col="darkred",lwd=2)
qq.plot(newZAPALOW[,5], dist= "norm", col=palette()[1], ylab="Sample Quantiles",
        main="Normal Probability Plot", pch=19)
result <- shapiro.test(newZAPALOW[,5]) # checking for normality test 
result$p.value
ad.test(newZAPALOW[,5]) # checking for normality test 
write.table(newZAPALOW, "newZAPALOW_trans.txt", sep="\t")
For sure this will be helpfull for you.