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Bell Curve Stuff

How Noise and Uncertainty might look like

In many applications, we model noise in the system as a bell curve type distribution. This structure lends itself to simple models for filtering and prediction, and also allows itself to be applied in multiple dimensions.
Published

July 1, 2026

Modified

November 7, 2026

How did the Bell Curve originate?

  • Random Variables; Probability. An Intro paragraph
  • Gauss and his work
  • Where does this show up?

Idea of Dimensions

What Does Gaussian Noise look like?

sigma <- matrix(c(4, 1, 1, 1, 4, 1, 1, 1, 4), ncol = 3, nrow = 3)
mean <- c(0, 0, 0)
# sigma
# mean
rmvnorm(n = 500, mean = mean, sigma = sigma, method = "chol")
               [,1]        [,2]          [,3]
  [1,]  6.108725092  3.68531399  3.3434129551
  [2,] -2.067249115 -1.51208570  0.1436642334
  [3,] -1.526964355 -1.47922780 -0.4854985608
  [4,] -1.097837796  0.33557521  1.8502756190
  [5,]  1.381668438  0.78423306 -1.8244626254
  [6,] -4.581846002  0.37733365 -0.7719952411
  [7,]  1.817340208 -2.62046227  1.6719046501
  [8,] -0.357094329  3.57215738  3.8784263141
  [9,]  2.067847627  2.26916480  0.2339682150
 [10,]  3.315700323  2.03315793  1.2225323153
 [11,] -2.552478958 -1.39265458 -0.2565954660
 [12,] -0.797952805 -2.45122397  0.9874823567
 [13,]  1.600011987 -0.78999745 -0.1539783203
 [14,] -2.111534864 -0.16083604  0.1051217882
 [15,]  1.330825160  2.64518775  1.3156107340
 [16,] -0.230113234  1.79195137 -2.6471130424
 [17,]  2.618395576 -1.81505242  1.5948043802
 [18,]  5.160844845  3.90000910 -0.2053557526
 [19,]  0.911843509 -1.24704526 -1.1733185165
 [20,]  2.093072783 -0.12598531  2.1670706052
 [21,] -1.065114177  0.38407430  0.4172529487
 [22,] -0.500633013  2.52868658 -0.5300790014
 [23,] -0.048377262 -0.20542347  2.0429457909
 [24,] -0.705782319  2.08722020  3.2307794658
 [25,]  2.026020157 -0.50951837  0.7276442611
 [26,]  1.433528595  2.34032703  0.9581269218
 [27,] -4.854987325 -0.85270975  1.2841294596
 [28,]  0.282556881  0.40257089  0.2289067240
 [29,] -0.350436740  3.28239137  0.5946569226
 [30,]  1.068833512  0.74338889  1.1397952678
 [31,]  1.256325016  0.80483902  1.3556166745
 [32,] -1.787516028 -1.70143412 -1.8737242703
 [33,] -0.764469389  3.36036299  1.8589567757
 [34,]  1.807042909 -0.40534265  1.1773272385
 [35,]  0.495553916 -0.11809954 -3.1131067901
 [36,]  2.857574733  3.04166080  3.1254193316
 [37,] -4.886764069  0.68720352  1.8599749581
 [38,]  1.673871576  1.45630802  2.7258068011
 [39,]  1.034430783  2.51606865 -0.3471859534
 [40,]  2.043398472 -0.97511586 -2.5248500822
 [41,] -4.560078683  1.04069412  1.2338648833
 [42,] -1.683538542  0.09815340  3.8086243206
 [43,] -1.111242593  1.25120368 -3.2853693399
 [44,]  1.984816870  0.22114104 -2.6602498579
 [45,] -3.593903058  1.40663581 -1.3310258885
 [46,] -0.889945164 -2.23613541 -1.5989159531
 [47,] -1.018491993  0.42331003 -2.7110375511
 [48,]  2.589850957  1.95417733  0.8390221671
 [49,]  2.954466085 -1.07048745  3.3530677206
 [50,]  0.832045437  0.04142833  1.9058566784
 [51,] -0.949720148 -0.52917537 -0.4633729969
 [52,]  1.339400442  1.45157759  1.5320866027
 [53,]  2.208250950  0.66452641  3.8019626196
 [54,]  2.492079229  4.26018856  0.1833297345
 [55,]  0.902256226  0.27777185 -0.0069603125
 [56,] -0.618827406  3.35645919  0.8432452461
 [57,]  3.581222573  3.43561496  5.8771342136
 [58,] -0.764577942  0.46814297  0.2473092494
 [59,]  2.293302076 -0.51506388  0.5444383101
 [60,]  0.348388306  0.14931754 -0.5615714192
 [61,] -2.295472157  1.81543078  0.8711807681
 [62,]  1.642070826 -1.68507025 -0.6489471251
 [63,] -0.506024864  0.75998442  1.6457721691
 [64,]  1.948587289 -0.14573461  0.5317793121
 [65,]  1.891807914 -1.97140379  0.1228302837
 [66,]  0.103770242 -0.60261899 -2.5884593066
 [67,] -1.152681501  1.06872365  1.2309644712
 [68,] -3.595921845 -3.26168740  1.0348216059
 [69,] -1.487702627 -0.01978442 -1.6982885178
 [70,] -1.139085047  0.09955348 -0.8744712147
 [71,] -0.488985774  0.29242215  2.5217535097
 [72,]  3.510505142 -0.43526613  1.1644455665
 [73,]  2.998820330 -0.21265394 -0.2288809878
 [74,] -1.452967766  1.85774014 -2.3113599889
 [75,] -2.133488234  0.62306390  0.2392035797
 [76,] -2.287601086  2.94858937 -5.6155008421
 [77,] -0.164188407 -4.23125403 -3.2172045439
 [78,]  1.091421774  0.99221576 -0.2044836276
 [79,]  0.588355091  0.51876986 -2.6740462177
 [80,]  2.044441787 -1.10480102 -2.5731061622
 [81,] -0.106947048 -0.66882579  0.4626078246
 [82,] -0.466253558  0.95240358 -2.9268444206
 [83,] -1.102556788  0.90606317  0.6820424619
 [84,] -3.381492478  0.37404510 -0.7345173593
 [85,] -2.836832747 -0.88012006 -1.5242269398
 [86,] -1.025908970  1.95860709 -0.0003333742
 [87,] -0.316031461 -1.08054441  1.3061416314
 [88,]  0.673641948  0.22296255 -1.6224653235
 [89,]  3.805338944  2.06673468  1.5434580511
 [90,]  1.594624885  1.04258376  0.9083011128
 [91,] -2.157414050 -2.88929851 -1.6082687714
 [92,]  1.327001618 -2.45906929 -0.6679286820
 [93,] -0.298495561  0.70078974 -2.1749779216
 [94,]  1.367023753 -1.67317169 -0.1120779151
 [95,]  0.949115457  4.23611915  3.2523212850
 [96,]  2.978396586  0.15753702  0.0407310209
 [97,]  1.584524463 -2.53549189 -1.9845702713
 [98,] -1.825558391  0.90845122 -0.2305464723
 [99,]  0.412035264 -0.87232573 -2.6264472725
[100,] -0.353159579  2.66211468 -1.3573654883
[101,] -2.016819588 -3.06549090  1.3459598328
[102,]  0.188754616 -0.68042243  1.8940364357
[103,]  1.499330354 -1.78148820 -0.8064998948
[104,]  1.712810174  0.09573920 -0.4051206184
[105,] -1.765091035  1.37570975  0.9892061504
[106,]  0.586652926 -0.15044614 -4.3110295641
[107,] -0.466338090 -0.71035094 -0.9485390261
[108,]  0.486658694 -0.41396138 -0.3014861203
[109,]  1.303509215  1.60294964  1.3714090757
[110,]  0.408802283 -0.20529791  1.0603793858
[111,]  3.789620520  2.07617545 -0.9876190875
[112,]  2.635529309 -2.92654957 -1.3750905306
[113,]  0.429096602  0.03990322  2.0916904508
[114,]  3.904726674  4.26664790 -0.6292651060
[115,] -2.653284356 -2.87832849 -2.9800280407
[116,]  0.300827206  1.03422723 -0.4988818512
[117,] -1.186851076 -0.99585822  0.2430106919
[118,]  4.569810671 -1.43177519  0.5605902152
[119,]  3.972624340  6.28229737  1.1316911780
[120,] -1.133717470 -1.37996214 -1.7456532122
[121,]  1.039971257 -0.92378545  1.0071290665
[122,] -1.214894658  1.93437950  1.0038531049
[123,]  0.290088982 -0.71841050 -1.1569601431
[124,] -1.595482402 -3.93514066 -2.5574688933
[125,]  0.570003324  3.07368208  0.4575180574
[126,]  0.751323766 -2.04733734 -0.3593810464
[127,] -0.207651213 -2.54934776  0.4650097021
[128,] -0.341166056  0.45635582  0.9276548314
[129,] -0.401392848 -0.57871497  1.8284635762
[130,] -0.910523587  0.54015235 -3.9904628991
[131,]  4.240078771  0.49176664  1.9487175351
[132,]  1.397731329 -2.31443929 -2.6392497169
[133,]  2.460261674  3.98313475 -1.2767614745
[134,]  2.075289588  0.81767646  2.8933040804
[135,] -2.967542726 -1.80620805 -0.8769459871
[136,]  1.034075690 -0.93507677  1.8010621607
[137,]  3.512958022  2.22607020  1.6578523373
[138,]  4.192310539 -0.56097017  2.3707409928
[139,] -0.621001306  0.65431450 -0.9086744172
[140,]  0.439954600  0.47619086  1.5709481588
[141,]  2.492863168  0.83143808  0.7646381333
[142,]  0.864200905  1.49781757 -3.7576278155
[143,]  0.100161682 -1.92179632 -0.8889920080
[144,]  3.156708978  1.09727826  0.4377470793
[145,]  0.233347246  0.51237485  1.3499904722
[146,] -0.995228997 -2.49502206  0.5242132205
[147,] -3.817388954 -1.73741039 -1.6453275128
[148,] -1.216298402  2.57499390 -2.5883216852
[149,] -4.370856827 -0.28390173  1.1135101278
[150,] -2.649847003 -4.03134446  1.6013619071
[151,]  3.059800599  2.39145473  4.0529170729
[152,] -2.398653948 -2.62078038 -0.3540023955
[153,] -1.738157078  4.14168462  4.2783598743
[154,]  2.001378807  2.42846787  1.2683292969
[155,]  1.578408134  0.18295809 -2.5768592646
[156,]  1.616233871 -0.41694982  1.1038821657
[157,] -1.082766453 -3.08745355 -2.7133028331
[158,]  4.246683536  2.50763250  0.8370541444
[159,]  1.282842777  0.52468552 -2.8760303649
[160,]  0.251863894 -0.59474545  0.2852624834
[161,]  2.425178077  0.87956842 -1.8714300298
[162,] -1.569806312 -1.29392175  0.5465540899
[163,]  0.992773660 -0.77021884  2.5225086952
[164,]  1.362672917 -0.12927199  0.8467696291
[165,]  0.325061024 -1.08517359 -1.6129779137
[166,] -1.694460056  3.14808801  1.9613842832
[167,] -0.899682929  1.92413046  1.1109556210
[168,]  4.762986197 -1.51613094  4.7728368475
[169,]  1.625489448 -0.44632313  0.8111200520
[170,] -0.825279224  0.92318724  0.0307620579
[171,]  4.459214752 -2.91571555 -3.0188122173
[172,] -4.753151052 -1.17912605 -1.8627193131
[173,]  0.542773865  1.39100175 -0.2852248951
[174,] -5.226716952 -0.70327564 -1.1461725991
[175,]  0.612204412  1.43607750  2.8345470408
[176,] -0.929867267 -0.17072280  3.0279050128
[177,] -1.183667500 -2.21892075 -2.3966738485
[178,]  0.242593273  2.96826051  2.9710046152
[179,]  0.942084049  1.25179944 -0.8029044644
[180,] -3.285391585  1.41024676 -1.2446745622
[181,] -2.836318841  2.61344667  2.5039006587
[182,] -0.087786655  2.36846856  3.0509663089
[183,]  1.699551524 -3.11798951 -1.2020236295
[184,] -1.506958263  5.49575381 -2.4319643893
[185,]  0.840889293  2.40116482 -2.5618262168
[186,] -2.778664162 -1.46473636 -0.2432913117
[187,]  0.487940127 -1.70556391  1.0359909596
[188,] -2.136734160 -0.87283001 -0.3415039976
[189,] -0.119770234  0.31029858  2.5696264284
[190,] -3.012077642  3.94062817 -1.5333028640
[191,]  2.070366369  6.55959044  2.7968776122
[192,]  1.873613663  4.00146318  0.8707067323
[193,] -0.957909389 -1.78582858 -0.9464055514
[194,] -1.616365069 -0.06088238  0.5699854949
[195,]  0.363931251  1.19956977  0.1840583041
[196,] -0.047177999 -5.38955420 -1.5807181417
[197,]  0.508035953 -0.82603678 -2.6794195312
[198,] -0.816947584  2.44841983  1.4985935461
[199,]  0.141135950  0.46817926  0.3351302743
[200,] -0.657618153 -0.07571685  1.4712947599
[201,] -0.485446258 -2.04903306 -0.6549611430
[202,]  1.793383603 -3.23802189  2.0241156090
[203,] -2.335109160 -0.40901283  0.6650706183
[204,]  1.196205209 -0.37401105 -3.7161176289
[205,] -0.347853621  0.99466111  0.7403238807
[206,] -1.709647858  3.20669351  0.3358224717
[207,] -1.757475011 -1.53296223  0.3989537679
[208,] -2.076429895  0.52005534 -1.2516077665
[209,] -1.812065010 -3.53513665 -0.3376260891
[210,]  1.774374552  0.17714903  0.1215383443
[211,] -0.010773934 -0.29183984 -4.5203485486
[212,]  1.657680669  3.61764132  3.6063367653
[213,] -0.226651547 -0.67561090  2.1072539445
[214,]  0.184801637  1.34033791  3.1491771389
[215,]  3.010248289 -1.29417409 -1.3129726935
[216,]  0.524226137  1.42265596 -1.9853044388
[217,]  2.230117491 -1.90595679  0.8831544831
[218,] -2.028353307 -1.50002481 -0.3751046798
[219,]  0.753207714 -0.65918806 -2.2781340704
[220,]  0.187675447 -1.03358958 -0.8540686632
[221,]  0.443090450  2.25940713 -1.0864965601
[222,] -2.675588104 -1.98598747 -0.4007525419
[223,]  1.328631047 -1.20407369  0.0819834048
[224,]  0.687393218 -0.03806806 -0.9870353962
[225,] -4.915722379 -2.62826072 -2.5454521008
[226,] -0.703681430  3.50823511 -2.2150446542
[227,] -7.162891355 -0.96223875 -2.5743919768
[228,] -3.381697784 -0.14748242  0.4452456338
[229,] -1.201404679 -1.95441552 -3.5300774435
[230,] -3.069824597 -0.07130158  5.3333682795
[231,] -1.203572780  1.53308758 -0.0513617404
[232,] -2.302627999 -0.67687179 -2.4053126182
[233,]  1.333121350 -0.76126803  2.1507326984
[234,] -0.738106222  1.26582955 -2.9146893840
[235,] -0.425098011  1.76257460 -1.8603613448
[236,]  0.587386583 -1.89712345 -0.3146100936
[237,]  0.730170305 -0.41875399  1.1156902730
[238,] -1.979437735  0.18285585  0.7017906877
[239,] -1.843212269 -0.14377625 -2.2856767469
[240,] -1.271208285 -0.66037704  0.1665091700
[241,]  2.811195221  1.41090078  2.9520214182
[242,] -0.402100536  1.83723577  1.7620892325
[243,] -1.988349794  2.40536849  1.6643155565
[244,]  0.914694398  1.39188236  0.2269715753
[245,]  1.777096907  0.43081537 -1.0748283727
[246,]  1.191711286  1.61223980 -0.8684405695
[247,]  1.259516147 -1.16948995  0.5836615189
[248,] -0.379911002 -1.44030710 -1.3298285010
[249,] -0.215379312 -0.20150386 -0.7953068181
[250,] -0.447432637  1.38767943 -0.3157546342
[251,]  1.125883334 -1.99095431  1.8251343315
[252,] -1.395429678 -1.43592447 -0.4583970911
[253,]  0.257682350  1.82200344  1.2680453053
[254,]  0.346478218 -0.85947481 -0.5396686128
[255,] -0.105477887 -3.20922324 -3.1875246409
[256,]  2.309098482  4.16779586  1.4780244792
[257,] -1.034375704 -2.05937475 -0.1648468498
[258,] -2.329818810 -0.25495077 -0.5885482053
[259,] -0.617889508 -2.72537678 -1.2708274368
[260,]  0.040963231 -1.18603356 -0.2235405408
[261,] -2.198584277  4.10894080 -2.0712432508
[262,] -0.032081364 -2.91016137  1.4186637414
[263,]  0.498732592 -0.41155315  1.6438950963
[264,]  0.915335562 -1.56803976 -2.8448838559
[265,]  3.342201960  2.74168482  1.4331019911
[266,] -0.867813956  2.50211918  1.1616609087
[267,] -2.483840795 -2.68409935 -2.2085190416
[268,]  1.142104995  3.44010993  1.8774044546
[269,]  1.571692566  1.88518580 -0.1939821628
[270,] -2.005902347 -2.54097009 -3.6308632541
[271,]  0.487344267  2.55134936  4.2928157120
[272,]  0.401268812  2.69384847  0.2361107267
[273,]  0.439893132  0.25063250 -0.0352600440
[274,]  1.268490571 -1.25512018  2.1830394320
[275,]  2.170812909 -0.53422020  0.7720285894
[276,] -2.558347633  1.47352863  0.3702659252
[277,] -0.096993411 -1.61466456 -1.7668037216
[278,] -0.983958536 -4.49279674  1.7371156640
[279,]  1.640246924  0.36675646  1.0027984188
[280,] -1.128211833  1.41212407  2.7880399390
[281,] -3.998022053 -0.46647432 -4.1002650437
[282,]  3.086289723  0.84416252  2.8126282029
[283,] -0.982075820 -2.01914823 -2.0656280448
[284,]  1.950759383  1.18642594  0.0461521368
[285,]  1.679901033  0.73003249  0.8311402870
[286,] -0.133338001  0.05880074 -0.7283483904
[287,]  4.251734651  1.70855757  0.0316597063
[288,]  1.782439658 -2.72867644  2.7714469018
[289,] -1.309974496 -1.96868699 -2.5755051553
[290,] -2.146665844 -3.21168699  1.6614581727
[291,] -2.166466130 -2.18573721 -3.5905665547
[292,]  0.916700141 -0.31598568 -3.2124812831
[293,]  3.052066649  3.30550364 -1.0703300053
[294,]  1.554669800 -0.91018086  1.2845875773
[295,]  0.014266014 -1.13874181 -0.2455478112
[296,]  1.075606997  2.61256062  1.7020641914
[297,]  0.110618170 -1.47264463  1.0705655873
[298,]  1.347677767 -0.26650524 -1.3982063499
[299,] -1.048916618  0.23089344 -0.8260718225
[300,] -1.509437655  0.90279389  1.2672030931
[301,]  0.508570773  0.84005449 -4.0123527406
[302,] -0.243834825 -0.68553449  0.8554083089
[303,]  0.837848098  1.18995176  0.6000486778
[304,] -0.043426771  1.08476073 -0.9371854701
[305,] -1.434735891 -0.80631926 -0.6601869113
[306,] -1.900885482  0.17447981 -1.4816876073
[307,]  1.062240260  0.67660391 -0.9517609515
[308,] -1.371708066 -1.39604126  2.7519894114
[309,]  2.737073252  1.66848287  0.2716911541
[310,] -0.357007898 -1.74545726 -0.0270405232
[311,] -1.483709043  0.18895216 -0.7134711614
[312,]  2.392749704  1.45067941  2.6358892571
[313,] -4.863441194 -1.61101036  0.8948181004
[314,] -2.135251542  1.11141102 -2.7447959952
[315,] -1.145726435 -0.04536365 -2.2342888685
[316,] -1.278766729 -1.39628774  0.5511643259
[317,]  0.214486933 -3.12362636 -1.1837351974
[318,]  0.198960326 -1.65820210 -0.1971743479
[319,]  1.203292508  0.77844045  0.2361732142
[320,] -3.689548468  1.37864574 -3.6965943548
[321,]  0.640605194 -2.35940595 -1.4645135930
[322,]  3.781304470  0.94333777  0.9559362583
[323,]  1.190038718  2.22296510 -0.8568225211
[324,] -1.798566499  0.67778796  1.5088342681
[325,]  0.444761288 -1.02623530  3.0799053913
[326,]  2.074186931  1.85584129 -1.8481088095
[327,] -0.929860593 -1.26618116 -0.9235378282
[328,] -2.770388830 -2.59944812  0.4725059679
[329,] -0.881004645  2.51248128  1.6726035518
[330,]  2.501390625  4.72332433 -0.2025133961
[331,] -0.108433557  1.14358770  0.1014550547
[332,] -0.016826318  0.14848553  2.1238274611
[333,] -2.791001557 -3.08741382  0.3620107340
[334,] -1.154222977 -1.71163002 -0.3662302406
[335,]  0.682836745  1.10857026 -1.4313681997
[336,]  1.007778315  0.57289727 -1.5093248392
[337,]  2.115464248  2.26960399 -0.2350319223
[338,]  1.312910661  1.36157834  0.7356854344
[339,] -4.258555089 -0.14323635 -1.1515763535
[340,] -2.434799522 -0.43358380 -0.8303109973
[341,] -1.490290474 -2.15537809  3.4554416558
[342,]  0.666860018  1.93814040  0.4492045559
[343,] -0.282566103 -0.42778048 -2.3644821386
[344,] -1.887584062  0.51735990  1.0846580667
[345,]  0.293717456 -1.90495359 -2.8538599248
[346,] -3.497568659 -3.45988825 -1.6755875595
[347,] -1.372122086  4.14241549 -0.8433802094
[348,]  0.982802208  2.72511418  2.8625718748
[349,]  1.833433785 -2.44946896 -1.6723662083
[350,] -0.759894201  1.25944937  1.5081744547
[351,] -3.559901021 -2.44848603 -2.0525216453
[352,] -0.847320309 -0.81420598 -0.0608537432
[353,]  1.278087170  0.82080651  2.5738451725
[354,] -0.092099599 -1.80909162 -1.9554005482
[355,] -1.920626109  0.44634810 -0.3985009875
[356,] -0.059203760  1.16893878  0.5758808952
[357,]  1.643379023 -0.97927018 -0.3832295656
[358,] -1.063317384 -2.71714422 -0.8886436424
[359,]  1.594027699 -1.58833889 -0.7611776359
[360,] -3.869033058 -1.55165896 -3.4049479184
[361,]  3.509107729 -1.98023796  2.7226796974
[362,]  3.089831989 -1.95728494  3.2680718505
[363,]  0.856027740 -1.80689431  3.8907800972
[364,] -1.992238212 -2.76704602  1.3029965371
[365,]  1.740783131  0.49627397  0.8095546090
[366,]  1.155841869 -0.44387070 -0.7231259935
[367,] -0.966429115 -0.78396878 -1.3257240281
[368,]  0.062832540  0.94489710 -2.3048883015
[369,]  0.478434866 -3.80204875  2.5686393154
[370,]  0.488311005  1.19522734 -0.2808768222
[371,]  1.911856988  1.45700162 -2.5561477946
[372,] -1.468060877 -0.79537133  2.2057277973
[373,]  1.379327020 -0.54414263  0.2380297180
[374,]  1.259076260  3.45958419  3.3736219380
[375,] -2.584472225  0.95144528  0.1663277561
[376,] -1.900860753 -3.18086051 -0.5311493368
[377,]  2.901956888  1.36959975  3.3560078981
[378,] -2.950300000  4.27409995  3.6049705560
[379,] -6.795082288 -1.52664418 -0.4524427273
[380,]  0.196967320 -2.09244163 -2.3660202714
[381,] -0.790804270  3.54307156  0.5922541673
[382,] -2.779962514  0.35257799 -0.2131173884
[383,] -4.177858084 -3.50582556  0.9216293788
[384,] -2.541843391 -0.56159958 -1.6684765482
[385,] -0.216315613 -3.48208717  2.1193924818
[386,] -2.148521259 -1.91183088 -3.8673412698
[387,]  3.992937662  0.66409274  0.4203566724
[388,] -3.491801108 -0.35313027 -3.6559905451
[389,] -0.823068581  1.76796736  3.2966611857
[390,]  1.776636512 -1.56165538 -0.4608490914
[391,] -3.613767319 -2.90091519 -4.3965096887
[392,]  0.726485648  1.86759101  2.7387711823
[393,]  0.040588009 -1.04284644  1.3113751868
[394,]  0.734818160 -0.38052494 -1.8450844660
[395,]  0.352206105 -0.57285438 -0.5044549584
[396,] -0.835343760  0.88348441  0.0876725848
[397,]  0.992623152 -1.19112383 -3.9508305520
[398,] -0.587825613 -2.94879664 -2.7766916527
[399,] -0.421094167  1.98617399 -0.7111319039
[400,]  0.728837122  0.86339185  0.2592838929
[401,]  1.995305872  2.58080408  1.4202938021
[402,]  0.546641996 -2.11741944  1.1837023463
[403,] -2.565953818 -2.97404332 -0.6643382372
[404,] -0.909150626  0.69209134 -1.5825260348
[405,] -0.218811040 -2.15076504 -0.2032815200
[406,]  0.533655375  2.73941519 -1.6755985287
[407,] -2.125889305 -0.42891796 -4.7103989575
[408,]  0.561572071  0.26877729  1.6766487711
[409,] -1.012137615 -1.17261766  1.3585307316
[410,] -2.122950731 -0.18382195  5.4964797854
[411,] -4.212210110  1.98746778  0.7645510733
[412,]  0.647359643  0.69340174  0.5014315937
[413,] -2.624125470  0.59841746 -2.6874899341
[414,]  0.305100869  0.83101684  0.2872098367
[415,] -0.369761352  0.07322986 -3.7224014842
[416,] -1.634409551 -1.23263502 -0.5967398363
[417,] -1.543717227  0.37127909  0.8684884666
[418,]  3.139558327  0.92581474 -3.5687889758
[419,]  1.320294357 -0.74919365 -4.2016220182
[420,]  0.600549106  0.96750496  0.1718831031
[421,] -1.701467608  1.87099969  2.5694279843
[422,] -1.369599361  1.87161501  2.8342415065
[423,] -1.949899033  0.98516752  0.3519145528
[424,] -0.592501380 -1.61897678  0.3347920725
[425,]  0.467640280  0.66522604  0.2132899570
[426,]  2.701828017 -1.03493787  1.9730057460
[427,]  0.855080447  0.75720637 -3.2971792077
[428,] -3.368642878  1.00615594 -1.2985996254
[429,] -4.169989587  0.55377499  1.5260203214
[430,] -1.231436438 -1.47649358 -0.8938460426
[431,]  2.775549617  0.41452349  3.1104606604
[432,]  0.426992346 -1.44566144  3.1537940345
[433,] -4.080439555 -0.90278400 -1.2204484233
[434,] -1.414792856  1.10295778  1.0327745175
[435,] -2.714700414  2.28502032  3.6162993486
[436,] -1.805582923  1.42293225  0.4798182250
[437,]  0.223834886  0.74600212  1.8478767633
[438,] -1.108252106  1.43168148  0.7200370702
[439,]  0.685728784 -0.34024041  0.6128831961
[440,]  0.504594628  1.14551473 -3.1110421660
[441,]  0.533022117 -2.68209842 -1.8824578768
[442,]  0.212055211  0.09264403 -2.0299243153
[443,]  2.514810888 -3.22337964  0.9687506859
[444,]  1.539577575  0.21984963  0.5177311888
[445,] -0.128509758 -2.90373885  1.4051269816
[446,] -0.249453732 -1.51127807 -0.3492236959
[447,]  0.024603504  2.98267118 -1.7709404368
[448,]  3.389588324 -2.34359476  0.3347068364
[449,]  0.404334982  1.39040672 -4.2253178143
[450,]  1.299761828 -2.12256179 -3.0328228705
[451,]  1.634835323  1.04678674 -1.7681977935
[452,] -0.004586708 -0.93500335 -0.1457514247
[453,]  3.426048927  5.66315052  2.2774185239
[454,] -0.961423879  2.79499632  2.8092048614
[455,] -0.416494315  2.82343926  0.1886266282
[456,] -1.004432993 -0.08792559 -1.2749057053
[457,]  0.472769208 -2.09326692 -4.3436680299
[458,]  1.653930011  0.94943037  2.9420515045
[459,]  3.672559473  0.42238902  0.8761868429
[460,]  3.071623929  2.35508471  2.3709109546
[461,]  2.625491371  2.48462903  0.4329329674
[462,]  1.668497179  2.41403148 -0.1960614319
[463,]  1.242357310  3.48922646  1.6889826300
[464,] -1.294547176 -0.28218556  1.6153189502
[465,] -1.031999886 -3.04953784  0.5299679700
[466,]  1.097083256 -0.24338758 -0.2605282097
[467,] -0.035850282  0.94866531  0.2480152323
[468,]  1.804799263 -1.20299638  1.7792976369
[469,]  3.270920120  2.30467027  1.8614083408
[470,] -1.549957825 -0.30259225  0.3204201925
[471,]  1.070192550 -2.44234793  2.7960933559
[472,] -0.470259126  0.48787613  3.0922200747
[473,] -2.690021558 -2.88040716 -2.0038179653
[474,]  2.759666907  1.14964858  0.8158001488
[475,]  0.415127337  1.57740202 -1.0780838285
[476,]  0.981286811  1.04780358  0.8316162361
[477,]  0.879358320  1.89485986  4.5616167859
[478,] -1.417493197  1.10790628  3.6350260570
[479,] -4.369787495  0.18140044 -2.6941663740
[480,]  0.353804939 -3.08715357  1.7269216572
[481,]  0.755867573  2.30812300 -0.9726053086
[482,] -2.212242962 -0.12458130  0.2177193999
[483,]  0.325073466  2.40168613 -2.2316642191
[484,]  3.623392401  2.11201325  2.1857560442
[485,] -0.774159358 -2.49356822 -1.1201852753
[486,] -1.596736855 -1.30021491 -4.9033207131
[487,] -4.356631571 -3.24774711 -0.2836213771
[488,] -1.312465671  0.82632574 -0.0091351355
[489,]  4.684567794 -0.89826142  0.6962677077
[490,]  1.455176067  0.02370090  2.2977290081
[491,] -0.540390965 -1.35609655  1.2267494856
[492,] -2.630545494  1.87536318  0.8408973130
[493,] -0.500295137 -4.57079628 -1.4705053275
[494,] -1.192759997 -1.21215493 -2.2732814692
[495,] -1.588379306  0.65611120  0.9917375273
[496,]  0.191043187 -1.02730183  0.4844300194
[497,]  1.733424576  2.59626950  0.9358692600
[498,]  0.963938751 -1.80416190 -3.8073854394
[499,]  0.561046046  1.04507796 -0.8509966023
[500,]  1.067022026 -2.77011197  0.8063214877
data_3d <- mvtnorm::rmvnorm(mean = mean, n = 1000, sigma = sigma) %>%
  as_tibble() |>
  janitor::clean_names()
Warning: The `x` argument of `as_tibble.matrix()` must have unique column names if
`.name_repair` is omitted as of tibble 2.0.0.
ℹ Using compatibility `.name_repair`.
# 2D density on x-y
kd <- kde2d(data_3d$v1, data_3d$v2, n = 100)

# put the density on the xy plane at the minimum z
z0 <- min(data_3d$v3)

data_3d |>
  plot_ly(x = ~v1, y = ~v2, z = ~v3) |>
  add_trace(type = "scatter3d", mode = "markers", size = 3) |>
  add_surface(
    x = kd$x,
    y = kd$y,
    z = matrix(z0, nrow = length(kd$y), ncol = length(kd$x)),
    surfacecolor = t(kd$z),
    colorscale = "Viridis",
    opacity = 0.5,
    showscale = FALSE,
    contours = list(
      z = list(show = "truehist")
    )
  )
p2 <- data_3d |> ggplot(aes(x = v1, y = v2, z = v3)) +
  ggcube::geom_point_3d() +
  coord_3d() +
  # place 2D density plot on zmin face
  stat_density_2d(
    position = position_on_face(faces = "zmin", axes = c("x", "y")),
    geom = "polygon", alpha = .1, linewidth = .25
  ) +
  stat_density_2d(
    position = position_on_face(faces = "ymin", axes = c("x", "z")),
    geom = "polygon", alpha = .1, linewidth = .25
  )

p2
Warning: The following aesthetics were dropped during statistical transformation: z.
ℹ This can happen when ggplot fails to infer the correct grouping structure in
  the data.
ℹ Did you forget to specify a `group` aesthetic or to convert a numerical
  variable into a factor?
The following aesthetics were dropped during statistical transformation: z.
ℹ This can happen when ggplot fails to infer the correct grouping structure in
  the data.
ℹ Did you forget to specify a `group` aesthetic or to convert a numerical
  variable into a factor?
Warning: Unknown or uninitialised column: `subgroup`.
Unknown or uninitialised column: `subgroup`.

# animate_3d(p2)

References

  1. bzarg blog. How a Kalman Filter Works, in Pictures. https://www.bzarg.com/p/how-a-kalman-filter-works-in-pictures/

  2. Ramsey Faragher. Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Derivation.

    Download PDF File

    Unable to display PDF file. Download instead.

  3. Tim Babb.The Kalman Filter in Pictures.https://www.bzarg.com/p/how-a-kalman-filter-works-in-pictures/

  4. The Kalman Filter Website for Students. https://thekalmanfilter.com/

  5. Chapter 6. The Kalman Filter. https://bookdown.org/probability/bookdown-demo/kalman-filters.html in Stochastic Processes! An interactive introduction.

  6. Kalman Filter in Julia. https://github.com/wkearn/Kalman.jl

  7. Chuong B. Do. (October 10, 2008). The Multivariate Gaussian Distribution. https://cs229.stanford.edu/section/gaussians.pdf

  8. https://ccrma.stanford.edu/~jos/filters/

Additional Reading

  1. Ramsey Faragher https://sites.google.com/view/ramseyfaragher/papers
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