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Commit a2ea3575 authored by Erik Strand's avatar Erik Strand
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Write a random number generator and answer 6.2 (c)

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import numpy as np
import matplotlib.pyplot as plt
uniform_samples = np.array([
[0.316139, 1.53805], [0.884079, 1.4898], [0.250138, 2.97285], [0.303945, 1.53857], [0.242934, 4.92948], [0.232131, 4.80524], [0.736897, 3.01305], [0.166709, 0.00494019], [0.493423, 5.12557], [0.805746, 6.00333], [0.0153707, 4.45289], [0.995829, 3.48421], [0.628502, 2.31783], [0.820536, 2.90344], [0.316724, 2.84286], [0.346007, 4.77784], [0.80875, 0.452461], [0.906672, 3.93964], [0.898311, 2.8438], [0.677135, 3.15903], [0.53789, 3.40872], [0.481619, 1.95222], [0.0238657, 3.9772], [0.105797, 4.37167], [0.271287, 1.56883], [0.865473, 3.44895], [0.0315917, 3.28883], [0.08644, 0.883585], [0.932475, 4.51298], [0.438335, 0.860995], [0.146411, 0.723109], [0.917591, 5.07933], [0.532152, 2.81949], [0.0334965, 2.32945], [0.309735, 3.10879], [0.7594, 0.274077], [0.335067, 4.2255], [0.670101, 5.62154], [0.613459, 3.82163], [0.658831, 0.977763], [0.0371085, 1.89177], [0.664055, 2.03241], [0.254855, 4.03807], [0.343052, 3.24564], [0.51877, 4.88984], [0.223581, 2.99578], [0.999282, 3.75056], [0.765606, 1.83726], [0.836824, 2.65238], [0.355774, 1.39154], [0.756062, 4.01192], [0.565774, 3.67607], [0.110647, 0.535916], [0.292856, 5.03828], [0.26325, 5.73852], [0.503795, 1.94685], [0.888686, 0.856414], [0.073202, 1.29747], [0.891196, 2.99395], [0.230949, 5.19199], [0.0676513, 4.81087], [0.117992, 3.87966], [0.498907, 3.71362], [0.615133, 4.89077], [0.131116, 3.64784], [0.624209, 0.902393], [0.316034, 4.16379], [0.940566, 2.70588], [0.311074, 3.61542], [0.378353, 3.20763], [0.0126842, 0.63977], [0.405585, 5.84917], [0.869284, 1.41348], [0.77914, 4.64948], [0.23289, 5.17326], [0.430067, 5.65281], [0.615266, 2.57233], [0.655277, 2.91659], [0.898338, 3.74334], [0.784851, 3.29214], [0.795859, 0.954619], [0.257649, 6.13438], [0.944612, 5.88985], [0.912616, 5.99497], [0.538652, 5.45799], [0.673342, 5.99517], [0.26767, 5.61601], [0.012488, 2.52198], [0.412523, 4.64828], [0.416617, 2.25556], [0.966643, 1.20418], [0.171249, 1.11935], [0.513856, 3.65872], [0.11759, 3.87584], [0.680342, 1.18717], [0.482456, 2.85156], [0.0452426, 4.13019], [0.213038, 4.36991], [0.0740391, 3.89785], [0.340941, 1.10074], [0.38911, 1.26852], [0.235535, 3.34356], [0.445067, 1.37913], [0.334344, 5.18402], [0.724412, 6.01702], [0.74005, 2.09249], [0.170378, 5.65227], [0.158856, 4.85459], [0.0235769, 5.04061], [0.495933, 0.327072], [0.912613, 2.4625], [0.204043, 3.23533], [0.114388, 1.03274], [0.369322, 1.10596], [0.409378, 2.58433], [0.420359, 3.49767], [0.474598, 5.32659], [0.578431, 4.29731], [0.37058, 2.07463], [0.740852, 3.74586], [0.563614, 1.23582], [0.476454, 3.99224], [0.583666, 0.368953], [0.43014, 5.35021], [0.965307, 3.02761], [0.177421, 2.8939], [0.66175, 0.714527], [0.462817, 4.62003], [0.966797, 2.71831], [0.611642, 2.97681], [0.623701, 1.0519], [0.77641, 0.671368], [0.541222, 1.44407], [0.0568091, 3.95291], [0.0348342, 0.476022], [0.581254, 5.4285], [0.22117, 5.335], [0.116605, 3.81647], [0.462857, 2.54058], [0.945643, 2.31444], [0.127127, 4.52582], [0.607109, 2.1886], [0.430629, 4.56987], [0.919177, 5.61681], [0.289934, 4.53052], [0.24504, 1.81422], [0.330404, 2.44596], [0.800724, 1.99091], [0.184838, 2.77322], [0.608999, 4.13329], [0.906729, 2.70262], [0.721462, 0.145067], [0.256845, 5.08864], [0.953583, 4.52703], [0.863856, 2.69577], [0.572787, 1.64511], [0.590552, 3.01672], [0.0637656, 1.69892], [0.43782, 3.64487], [0.995935, 5.26482], [0.217211, 2.62284], [0.577637, 4.75985], [0.153151, 5.00148], [0.195604, 0.0414172], [0.640834, 3.31769], [0.121866, 3.18936], [0.730124, 2.91874], [0.335526, 0.488706], [0.504072, 2.80944], [0.24565, 0.302833], [0.440973, 2.31393], [0.776895, 1.80025], [0.118739, 6.09404], [0.623788, 4.2767], [0.550887, 1.84173], [0.232979, 3.70318], [0.23615, 4.37343], [0.524581, 1.22386], [0.246898, 5.53958], [0.202894, 4.98634], [0.286001, 5.88216], [0.425366, 4.17209], [0.839136, 6.16579], [0.152846, 1.15433], [0.127718, 2.43028], [0.643752, 2.71225], [0.711467, 2.8119], [0.00692405, 6.17089], [0.136661, 2.67481], [0.337625, 4.0119], [0.302973, 5.08736], [0.407995, 2.31162], [0.414799, 0.287778], [0.319282, 6.08779], [0.916938, 1.03038], [0.0330362, 3.981], [0.866766, 1.09953], [0.824769, 1.17165], [0.818126, 2.04713], [0.850555, 3.07635], [0.843332, 3.03756], [0.325282, 0.141816], [0.511585, 0.665341], [0.691512, 0.0079404], [0.70871, 0.0484113], [0.990784, 4.1605], [0.32574, 4.79083], [0.518467, 4.42245], [0.499194, 3.79721], [0.883527, 2.92539], [0.746115, 5.29726], [0.961959, 6.25781], [0.7621, 0.0460913], [0.647888, 3.31218], [0.425824, 4.7386], [0.596169, 1.52933], [0.886019, 0.255047], [0.811186, 2.08642], [0.383553, 2.88453], [0.566799, 3.42854], [0.229079, 6.02237], [0.347964, 4.58066], [0.469297, 0.609068], [0.584791, 2.51188], [0.773215, 0.273843], [0.901615, 3.90267], [0.531072, 5.43285], [0.542009, 0.195691], [0.576491, 5.93405], [0.45437, 0.271964], [0.70615, 6.00639], [0.540232, 2.88184], [0.348361, 5.22345], [0.712479, 3.23848], [0.16943, 6.01577], [0.849911, 4.68048], [0.697622, 6.20604], [0.244524, 0.125849], [0.0177815, 1.67685], [0.356754, 3.29358], [0.286912, 2.66177], [0.64719, 3.2218], [0.909099, 5.44031], [0.674461, 2.64036], [0.956485, 5.44809], [0.578809, 1.37124], [0.413589, 3.57804], [0.175083, 3.13049], [0.0722, 5.75772], [0.877597, 3.55408], [0.564702, 0.251538], [0.860967, 1.12883], [0.832252, 5.07107], [0.722957, 0.354606], [0.569914, 0.114306], [0.364545, 0.149493], [0.770904, 4.00695], [0.697573, 4.52299], [0.738008, 0.530964], [0.714397, 3.20427], [0.243293, 1.63335], [0.540487, 1.19527], [0.489669, 0.342826], [0.905504, 5.29871], [0.633282, 2.29499], [0.728603, 3.4997], [0.3684, 1.00774], [0.502245, 5.46267], [0.395379, 3.50997], [0.26305, 1.91402], [0.364664, 2.23289], [0.193228, 3.49421], [0.287871, 3.72551], [0.624874, 5.89791], [0.631869, 4.9136], [0.620421, 0.737388], [0.834527, 2.19681], [0.713555, 2.19335], [0.195524, 0.641834], [0.154655, 5.3152], [0.89331, 1.4974], [0.599076, 2.17321], [0.629691, 3.34615], [0.528375, 3.87476], [0.403869, 1.45249], [0.390882, 5.95215], [0.586253, 2.74943], [0.836137, 1.37082], [0.869405, 2.03088], [0.933071, 0.259313], [0.941902, 1.41339], [0.272355, 1.8666], [0.327005, 4.4586], [0.597797, 6.24613], [0.759954, 4.25524], [0.471068, 5.95971], [0.695248, 1.12982], [0.262018, 3.62616], [0.899801, 1.22686], [0.399307, 1.22627], [0.428436, 5.17429], [0.363009, 0.981049], [0.319742, 4.5296], [0.0586058, 4.21314], [0.379442, 4.84889], [0.0259881, 0.83453], [0.101946, 5.68623], [0.914525, 3.03097], [0.870034, 0.392111], [0.15175, 5.20442], [0.266402, 4.13861], [0.99309, 5.61726], [0.976213, 5.90458], [0.792115, 5.58688], [0.724482, 3.88723], [0.818681, 5.04634], [0.0703119, 1.54753], [0.0117581, 3.98941], [0.544483, 3.52262], [0.27828, 5.06571], [0.015054, 1.66456], [0.655934, 1.31257], [0.257736, 2.72687], [0.919264, 5.45873], [0.64086, 0.275263], [0.835495, 3.621], [0.00557595, 0.0398099], [0.177317, 5.26501], [0.962819, 5.66191], [0.348624, 4.14932], [0.205989, 5.68698], [0.808027, 4.03449], [0.335294, 0.672217], [0.406757, 0.549728], [0.169154, 5.55067], [0.773526, 3.27322], [0.440182, 5.59558], [0.56557, 6.09021], [0.396451, 4.92944], [0.469633, 4.48506], [0.156041, 4.03379], [0.432878, 6.00957], [0.756545, 0.922819], [0.103337, 4.27633], [0.251104, 5.79919], [0.210859, 0.920667], [0.412373, 6.17291], [0.952659, 3.56628], [0.929708, 4.50027], [0.68241, 3.76135], [0.00446549, 4.02246], [0.976976, 5.90312], [0.686955, 4.09759], [0.245454, 1.12437], [0.505262, 4.8841], [0.6929, 3.79292], [0.75595, 4.77965], [0.567711, 5.4132], [0.228391, 4.94118], [0.869834, 5.18974], [0.81546, 6.26072], [0.852457, 1.30086], [0.963086, 2.47889], [0.0954053, 1.14459], [0.210167, 3.80643], [0.428851, 2.26168], [0.155956, 0.113712], [0.141887, 3.1436], [0.413613, 1.45198], [0.611805, 0.559657], [0.876303, 5.93726], [0.0288781, 3.27641], [0.773662, 4.52049], [0.931162, 0.652057], [0.249621, 0.832998], [0.603456, 2.79613], [0.351061, 1.71474], [0.0939724, 3.82913], [0.581823, 1.73525], [0.530207, 1.04455], [0.508212, 5.08211], [0.827674, 4.06194], [0.946631, 2.43855], [0.281089, 3.23368], [0.24318, 3.56762], [0.678164, 5.04858], [0.119063, 4.75075], [0.309975, 0.740766], [0.0902699, 1.58968], [0.0441766, 2.55825], [0.739316, 2.31625], [0.391236, 4.79012], [0.333367, 5.5571], [0.158938, 5.39712], [0.78125, 3.91203], [0.898978, 0.674403], [0.0361729, 2.89182], [0.734725, 2.25919], [0.0451564, 3.72441], [0.981727, 0.824394], [0.777839, 0.326851], [0.798379, 4.00188], [0.790006, 1.55644], [0.311176, 4.14155], [0.793809, 3.53024], [0.625934, 4.33255], [0.585693, 5.39385], [0.857293, 5.47821], [0.612701, 2.57372], [0.40028, 2.91669], [0.167511, 1.48825], [0.221915, 2.40579], [0.568914, 0.824094], [0.759831, 4.26397], [0.369024, 2.14406], [0.644512, 1.73765], [0.423322, 1.24249], [0.110594, 3.90749], [0.777088, 5.25696], [0.877746, 2.05196], [0.52464, 0.814918], [0.147006, 4.41144], [0.556499, 2.44386], [0.419089, 0.750699], [0.49547, 1.05588], [0.380874, 5.32018], 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[0.413611, 3.68352], [0.41715, 1.41807], [0.608354, 1.52601], [0.747119, 5.42728], [0.802604, 2.23847], [0.0115473, 0.379109], [0.0108027, 2.45516], [0.0134149, 0.628286], [0.986593, 2.25307], [0.105765, 5.76798], [0.0875092, 1.98541], [0.328695, 0.0972343], [0.814501, 3.53776], [0.859199, 2.3819], [0.951873, 4.84189], [0.198208, 3.57169], [0.1317, 4.09346], [0.560435, 0.978005], [0.0406443, 4.80888], [0.980918, 1.58975], [0.177391, 3.19256], [0.847194, 2.02829], [0.931426, 4.90622], [0.8876, 2.43053], [0.548546, 2.66918], [0.969943, 2.08215], [0.661037, 4.51145], [0.782664, 1.68572], [0.831174, 0.588989], [0.669935, 0.0475393], [0.854703, 0.791592], [0.405946, 2.89941], [0.173403, 4.37612], [0.360466, 5.68922], [0.551702, 4.74366], [0.566481, 3.47926], [0.828847, 0.872665], [0.204881, 5.33568], [0.716689, 3.68167], [0.275956, 3.22062], [0.00481157, 4.04429], [0.138507, 1.18719], [0.0807637, 0.629673], [0.953523, 2.98068], [0.193431, 1.71981], [0.260476, 0.396523], [0.707952, 4.98465], 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0.5441], [2.55336, -0.0228017], [1.19706, -0.304907], [-0.2717, -0.155204], [1.3173, 1.16484], [-1.46029, -0.124256], [0.369491, 0.16539], [1.36636, 0.471661], [-0.775922, -0.911821], [0.314819, -1.03536], [0.981746, -1.27827], [0.745869, 0.37079], [-1.78379, -1.04231], [-0.582105, -1.46159], [-0.538042, 1.38547], [-0.506773, 0.680312], [-0.905271, 0.204706], [-0.199785, 0.397766], [1.16465, -0.613275], [0.486524, -0.731617], [-0.288953, -0.0390552], [0.896737, 0.400469], [-0.831548, 0.0576494], [-2.03616, 0.0333684], [0.0677605, -0.602643], [-0.952051, -0.963799], [-0.0556652, -1.36556], [0.0412043, 0.940797], [-0.883829, -0.705589], [1.57109, -0.239523], [1.18736, 1.18196], [-0.500189, 1.59696], [-1.12236, 1.23193], [0.451472, -0.115106], [0.0647658, 1.22229], [0.683932, -0.12665], [0.264998, 0.965813], [2.4754, 0.496097], [-1.04375, 0.334224], [-1.26666, -1.30227], [-0.437887, -0.24741], [-1.34209, 0.0212165], [0.0100202, -1.72318], [0.234954, 0.450138], [-0.998634, 0.981007], [-0.749018, -0.85165], [-2.11172, 0.577559], [0.0815553, -0.0660415], [0.66993, 0.253295], [-0.38844, 0.0315316], [-0.816602, -1.13257], [0.180548, -0.023943], [-0.474977, -0.409587], [-0.721104, 1.19705], [-0.410808, -2.25118], [-0.243645, 0.389952], [-2.48359, -0.0613325], [0.362317, 0.455901], [-0.614325, -0.818379], [-0.221531, 0.387525], [0.76048, 0.716181], [-0.251954, -2.03584], [-0.255745, 0.131604], [-0.276625, 0.196372], [-1.00511, 0.127811], [0.405734, -0.932489], [-0.901784, -1.20458], [-0.304622, -0.238756], [1.53929, -0.712607], [0.656432, -1.02667], [0.639049, 0.888496], [0.538198, 2.39167], [-0.367266, -0.0600696], [-0.768924, 0.324871], [-0.77445, -1.3222], [-1.19619, 1.07086], [0.110543, -1.42439], [-0.373334, -0.0793233], [1.09224, 0.987827], [-2.27172, 1.02365], [0.285715, -0.921374], [1.33082, -0.286332], [1.46311, 0.539082], [-1.61996, 0.669608], [0.438479, 2.60993], [-1.13546, 1.07193], [-1.9663, -1.05068], [0.123316, -0.374968], [0.915063, 0.660079], [-1.98964, 0.287309], [2.16608, 0.24631], [1.8104, -1.23067], [-0.154682, -0.208087], [0.210437, 0.35854], [-1.42009, -1.62322], [-0.0909902, -1.03053], [0.944582, 0.49327], [-0.762145, 1.98068], [-0.999201, -2.12108], [0.412362, -0.248292], [0.455183, 0.702731], [0.236988, -0.517877], [-0.0374885, -0.17247], [0.96271, -2.17589], [0.0113102, 0.126853], [-1.00728, 0.0198978], [-0.773542, 1.40667], [-0.566324, -1.63676], [-0.767773, 0.438585], [0.337854, 0.534918], [0.0287283, 0.317772], [1.3919, 0.877594], [-0.798186, 1.30224], [-2.07755, -0.312064], [-1.01399, 0.940973], [-1.25153, -1.5387], [-0.524975, 0.080868], [0.0797614, 0.239796], [0.0956565, -0.698444], [0.147941, 1.84792], [0.8016, 0.238835], [0.23246, 0.0151746], [-2.12457, 0.881792], [-0.711904, -0.104244], [-0.245624, 0.512708], [1.12235, 1.65715], [0.00496017, -0.701278], [-0.516653, 0.276595], [-0.949625, -0.31033], [0.439616, -1.74055], [-1.07232, -0.638487], [0.126046, 1.74187], [0.500682, 0.687388], [-1.29569, 0.424968], [-1.24563, 1.3822], [0.584518, 1.0448], [1.21201, 0.951916], [0.832242, -0.333032], [0.746457, 2.17353], [-0.49986, -0.771412], [-0.929274, 0.99857], [1.89281, 1.26686], [-1.01927, 1.17337], [1.035, -1.12982], [-0.764936, 0.897869], [0.0295534, -0.640278], [-0.345498, 1.0377], [-0.0434688, 0.833539], [1.1402, -0.675048], [-1.25897, 0.689338], [-0.12245, -0.21025], [-0.0848153, 0.201959], [0.226738, 1.12701], [-0.00638499, 0.618052], [-1.03348, 1.18184], [-0.892638, 0.0380599], [0.377239, 0.504302], [1.81161, -1.70934], [-1.00696, -0.683504], [0.621547, -0.925813], [0.687788, 1.56845], [0.272612, 1.94963], [-0.198476, 0.0661005], [-1.6875, -1.30918], [0.949896, 1.09731], [-1.11979, 0.78039], [-0.461724, -1.34513], [0.5761, -0.716619], [0.859432, -0.0241924], [-1.02742, -0.538449], [1.35795, 1.0619], [0.472289, 0.603105], [1.81824, 0.334394], [1.36778, 0.632521], [-0.82999, -1.26714], [1.53366, 0.128943], [-0.388881, 0.285737], [0.853306, -0.882685], [0.25278, 1.84658], [1.12651, 0.495177], [1.94484, -0.882168], [1.30812, -1.56248], [-1.41455, 0.49967], [-0.843078, -0.146927], [-1.27123, -0.663297], [2.22449, 1.38502], [-0.001271, -1.26174], [1.22906, 0.286519], [-1.00135, -0.0820875], [-1.89613, 1.17284], [-0.56098, 3.04576], [0.140465, 0.238299], [-1.34982, 2.31735], [0.536465, -0.382739], [1.26624, -1.44119], [0.745741, -0.702564], [-0.517982, -0.831608], [-0.627377, 0.388396], [-0.140789, 0.176391], [0.315242, -0.185723], [0.794804, 1.00042], [-0.647678, -0.235116], [1.78143, 1.04622], [-0.262964, -1.78253], [-0.486572, -1.77751], [0.904881, -0.89635], [-0.0508336, -1.35195], [0.640106, -0.456426], [0.252893, 0.0155731], [0.217156, 1.40347], [-0.394186, 1.19028], [1.25799, -0.227223], [0.858356, 0.378674], [1.26293, 0.189622], [-0.712565, -0.504085], [2.46637, 1.56416], [-0.567978, 1.30861], [-1.10977, -0.345549], [0.471204, 0.00806457], [1.80547, -1.12261], [-0.0560927, 2.48337], [0.08416, 1.6267], [2.28704, 0.669478], [0.543775, -1.11671], [1.02993, -1.33416], [0.310876, 1.73437], [0.354757, -0.492108], [0.733383, 1.67913], [-0.597031, 1.25056], [-1.03228, -0.0507122], [0.439803, 1.25489], [1.51004, 0.766595], [-1.83451, -0.787465], [1.02143, 1.28207], [-0.855308, -0.512009], [1.34727, 0.190189], [-1.30282, 1.31732], [-0.280476, -0.250465], [0.353894, -1.27819], [-0.629537, -1.06361], [1.93548, -0.119851], [-1.03482, -1.07915]
])
def make_uniform_plot():
fig = plt.figure(figsize=[6, 1.5])
left, bottom, width, height = 0.1, 0.15, 0.8, 0.9
ax = fig.add_axes([left, bottom, width, height])
ax.scatter(uniform_samples[:, 1], uniform_samples[:, 0], color='k', marker='.', s=4)
ax.set_aspect(1)
ax.set_xlabel('$x_2$')
ax.set_ylabel("$x_1$")
fig.savefig("../assets/img/03_uniform_samples.png", transparent=True)
def make_gaussian_plot():
fig = plt.figure(figsize=[6, 6])
left, bottom, width, height = 0.1, 0.1, 0.8, 0.8
ax = fig.add_axes([left, bottom, width, height])
ax.scatter(transformed_samples[:, 0], transformed_samples[:, 1], color='k', marker='.', s=4)
ax.set_aspect(1)
ax.set_xlabel('$y_1$')
ax.set_ylabel("$y_2$")
ax.set_xlim(-3.5, 3.5)
ax.set_ylim(-3.5, 3.5)
fig.savefig("../assets/img/03_gaussian_samples.png", transparent=True)
if __name__ == "__main__":
make_uniform_plot()
make_gaussian_plot()
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*.swp
*.swo
.DS_Store
build
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cmake_minimum_required(VERSION 3.13) # 3.13 is required for target_link_options
project(Rand CXX)
if(NOT CMAKE_BUILD_TYPE)
set(CMAKE_BUILD_TYPE "Release")
endif()
message(STATUS "Build type: ${CMAKE_BUILD_TYPE}")
find_package (Eigen3 3.3 REQUIRED NO_MODULE)
include(cmake/shared_settings.cmake)
add_subdirectory(rand)
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# This file defines an interface library used to add common compile flags to all libraries and
# executables.
#---------------------------------------------------------------------------------------------------
add_library(shared_settings INTERFACE)
# Speed flags
target_compile_options(shared_settings INTERFACE -march=native -ffast-math)
# Warning flags
target_compile_options(shared_settings INTERFACE
-Wall
-Wcast-align
-Wcast-qual
-Wextra
-Wundef
-Wzero-as-null-pointer-constant
-pedantic
)
target_compile_features(shared_settings INTERFACE cxx_std_17)
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add_executable(rand
main.cpp
)
target_link_libraries(rand shared_settings Eigen3::Eigen)
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#include <iostream>
#include <Eigen/Dense>
#include <cmath>
#include <vector>
#include "xorshift.h"
//--------------------------------------------------------------------------------------------------
int main() {
XorShift64 rand;
uint32_t constexpr n_warmup = 1000;
uint32_t constexpr n_samples = 100000000;
uint32_t constexpr n_printed = 1000;
for (uint32_t i = 0; i < n_warmup; ++i) {
rand.advance();
}
Eigen::ArrayX2d samples;
samples.resize(n_samples, 2);
// Draw uniform samples.
for (uint32_t i = 0; i < n_samples; ++i) {
samples(i, 0) = rand.draw_double();
samples(i, 1) = rand.draw_double();
}
samples.col(1) *= 2 * 3.1415926;
// Print uniform samples.
std::cout << "Uniform samples\n";
for (uint32_t i = 0; i < n_printed; ++i) {
std::cout << '[' << samples(i, 0) << ", " << samples(i, 1) << "], ";
}
std::cout << "\n\n";
// Transform samples.
Eigen::ArrayXd radii = Eigen::sqrt(-2 * Eigen::log(samples.col(0)));
samples.col(0) = radii * Eigen::sin(samples.col(1));
samples.col(1) = radii * Eigen::cos(samples.col(1));
// Print transformed samples.
std::cout << "Transformed samples\n";
for (uint32_t i = 0; i < n_printed; ++i) {
std::cout << '[' << samples(i, 0) << ", " << samples(i, 1) << "], ";
}
std::cout << "\n\n";
// Calculate expectations of y_1, y_1^2, and y_1^3.
double e_1 = samples.col(1).mean();
double e_2 = samples.col(1).square().mean();
double e_3 = samples.col(1).cube().mean();
std::cout << "C1 = " << e_1 << '\n';
std::cout << "C2 = " << e_2 - e_1 * e_1 << '\n';
std::cout << "C3 = " << e_3 - 3 * e_1 * e_2 + 2 * e_1 * e_1 * e_1 << '\n';
}
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#ifndef XORSHIFT_H
#define XORSHIFT_H
#include <stdint.h>
//--------------------------------------------------------------------------------------------------
// from Numerical Recipes in C, 3rd Edition
class XorShift64 {
public:
XorShift64 (): x_(0xdefceed) {}
XorShift64 (uint64_t seed): x_(seed) {}
inline XorShift64& advance();
inline uint64_t draw_uint64() { advance(); return x_; }
inline uint32_t draw_uint32() { advance(); return static_cast<uint32_t>(x_); }
inline double draw_double() { advance(); return 5.42101086242752217e-20 * x_; }
private:
uint64_t x_;
static constexpr uint32_t a1_ = 21;
static constexpr uint32_t a2_ = 35;
static constexpr uint32_t a3_ = 4;
};
//..................................................................................................
XorShift64& XorShift64::advance() {
x_ = x_ ^ (x_ >> a1_);
x_ = x_ ^ (x_ << a2_);
x_ = x_ ^ (x_ >> a3_);
return *this;
}
#endif
_psets/03.md
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......@@ -322,3 +322,63 @@ $$
This is the two dimensional normal normal distribution with mean $$(0, 0)$$ and identity covariance
matrix.
### (c)
{:.question}
Write a uniform random number generator, and transform it by the equations above. Numerically
evaluate the first three cumulants of its output.
I implemented a 64 bit [xorshift](https://en.wikipedia.org/wiki/Xorshift) random number generator
(code lives [here](https://gitlab.cba.mit.edu/erik/nmm_2020_site/_code/pset_03_rand)). These
generators are mathematically equivalent to LFSRs, but they are more efficient to implement in
software. I got the special constants from Numerical Recipes in C, third edition.
First, let's make sure it works. Here are 1,000 samples drawn from a uniform distribution on a
rectangle of area $$2 \pi$$.
![uniform samples](../assets/img/03_uniform_samples.png)
And here are the transformed samples.
![gaussian samples](../assets/img/03_gaussian_samples.png)
Looks good to me.
Cumulants can be defined for multivariate distributions, but since I already worked out the
univariate cases above I'm just going to calculate them for $$y_1$$ alone. I computed $$\langle y_1
\rangle$$, $$\langle y_1^2 \rangle$$, and $$\langle y_1^3 \rangle$$ experimentally by taking the
average over a large sample of $$y_1$$, $$y_1^2$$, and $$y_1^3$$, respectively. Then I just plugged
the resulting values into the formulas I found in the previous problem.
With 100,000 samples, I get these values.
$$
\begin{aligned}
C_1 &= 0.00106 \\
C_2 &= 1.00525 \\
C_3 &= -0.00516
\end{aligned}
$$
With 1,000,000 samples, I get these.
$$
\begin{aligned}
C_1 &= 7.524 \times 10^{-5} \\
C_2 &= 0.9992 \\
C_3 &= -0.003815
\end{aligned}
$$
And with one hundred million samples, I get these.
$$
\begin{aligned}
C_1 &= 8.541 \times 10^{-5} \\
C_2 &= 0.9999 \\
C_3 &= 4.013 \times 10^{-4}
\end{aligned}
$$
These are consistent with the predicted values.
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assets/img/03_gaussian_samples.png

16.8 KiB

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assets/img/03_uniform_samples.png

9.5 KiB

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