example1_lgss: State estimation in a linear Gaussian state space model
In pmhtutorial: Minimal Working Examples for Particle Metropolis-Hastings

Description Usage Details Value Note Author(s) References Examples

Minimal working example of state estimation in a linear Gaussian state space model using Kalman filtering and a fully-adapted particle filter. The code estimates the bias and mean squared error (compared with the Kalman estimate) while varying the number of particles in the particle filter.

1	example1_lgss()

The Kalman filter is a standard implementation without an input. The particle filter is fully adapted (i.e. takes the current observation into account when proposing new particles and computing the weights).

Returns a plot with the generated observations y and the difference in the state estimates obtained by the Kalman filter (the optimal solution) and the particle filter (with 20 particles). Furthermore, the function returns plots of the estimated bias and mean squared error of the state estimate obtained using the particle filter (while varying the number of particles) and the Kalman estimates.

The function returns a list with the elements:

y: The observations generated from the model.
x: The states generated from the model.
kfEstimate: The estimate of the state from the Kalman filter.
pfEstimate: The estimate of the state from the particle filter with 20 particles.

See Section 3.2 in the reference for more details.

Johan Dahlin uni@johandahlin.com

Dahlin, J. & Schon, T. B. "Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models." Journal of Statistical Software, Code Snippets, 88(2): 1–41, 2019.

1	example1_lgss()

$y
                [,1]
  [1,]            NA
  [2,]  0.0003209167
  [3,] -1.4171876933
  [4,] -0.6844286346
  [5,] -1.7869998257
  [6,] -2.9652946891
  [7,] -1.0273774834
  [8,] -0.9667058113
  [9,] -0.0487628562
 [10,] -1.0177320818
 [11,]  0.2251563321
 [12,] -0.6821954608
 [13,] -1.2345221398
 [14,] -2.0695261925
 [15,] -2.2988913408
 [16,] -1.7858956528
 [17,] -3.1819232610
 [18,] -1.3935375542
 [19,] -2.5825176273
 [20,] -1.6430460900
 [21,] -1.4900533694
 [22,] -0.0583758915
 [23,] -0.7318061973
 [24,] -1.5819737110
 [25,] -0.9819148487
 [26,] -1.3259373049
 [27,] -1.4776882549
 [28,]  0.4985562147
 [29,]  0.7980381312
 [30,] -0.3093693253
 [31,] -0.8887897731
 [32,] -1.9716284879
 [33,] -2.2408192356
 [34,] -0.5181341205
 [35,]  0.2077597677
 [36,]  0.6300232637
 [37,]  0.8038709135
 [38,] -0.3601106308
 [39,] -1.6263608154
 [40,] -2.8305893859
 [41,] -2.9900429608
 [42,] -1.7861642927
 [43,] -0.5383692468
 [44,] -0.0013970711
 [45,]  0.8688560047
 [46,] -0.7062376287
 [47,] -0.9596344432
 [48,] -0.5202850434
 [49,] -0.0238720703
 [50,] -1.4288153250
 [51,] -0.8677101420
 [52,] -1.4626765344
 [53,] -2.0972465242
 [54,] -2.2031984892
 [55,] -1.1994553666
 [56,]  0.3657256278
 [57,] -0.9775700809
 [58,] -1.3135532259
 [59,] -0.0294752844
 [60,]  1.0862199082
 [61,]  2.1096602205
 [62,]  1.0498488134
 [63,]  0.6208727029
 [64,]  1.0875238382
 [65,]  1.0267489433
 [66,]  0.3650766844
 [67,]  1.8568884978
 [68,] -0.7198401534
 [69,] -1.0471051008
 [70,] -0.1436851731
 [71,]  0.6005771832
 [72,] -1.4170268139
 [73,] -2.7885584642
 [74,] -1.7226649995
 [75,] -0.8310914867
 [76,]  0.7155385692
 [77,]  0.1140203634
 [78,]  1.1728873558
 [79,] -0.1542599856
 [80,]  0.7822120618
 [81,] -0.8571891544
 [82,]  0.0738335359
 [83,]  0.5476950070
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 [85,]  0.8305617396
 [86,]  0.7732834788
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 [88,] -0.0872142165
 [89,] -0.5881382523
 [90,] -0.3394842783
 [91,]  0.7832591392
 [92,]  1.0361457735
 [93,] -0.6513099266
 [94,]  0.3525291504
 [95,] -0.1678790506
 [96,]  0.5489507717
 [97,]  0.6207985200
 [98,] -0.4179080866
 [99,] -1.7176969279
[100,] -3.1034730992
[101,] -3.4383642039
[102,] -1.2477060021
[103,]  0.5168845798
[104,] -0.7089749525
[105,]  0.3562447274
[106,]  0.0396071992
[107,]  0.3948884624
[108,] -0.4122338691
[109,]  0.0452268570
[110,]  0.8016532251
[111,]  0.0818959290
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[114,]  0.2343218019
[115,]  0.3811284883
[116,]  0.8570966614
[117,]  0.1376084887
[118,] -0.0491383203
[119,]  0.4184707116
[120,] -0.8070722003
[121,] -1.7020807559
[122,]  0.5035940957
[123,]  1.2234615797
[124,]  1.3848231823
[125,]  1.1046053019
[126,]  0.5984968178
[127,]  1.5428005232
[128,] -0.4641846233
[129,] -0.8542140270
[130,] -1.5261739517
[131,] -2.2555141395
[132,] -1.1576164632
[133,] -3.2322968791
[134,] -1.3644534696
[135,] -1.2078768936
[136,] -0.7257136935
[137,] -0.8184656758
[138,] -1.0460415738
[139,]  1.7783037901
[140,]  0.4575918909
[141,]  1.2711641027
[142,]  1.1147562996
[143,] -0.1902424561
[144,] -0.3111834866
[145,] -0.0949047805
[146,]  0.3536996027
[147,]  0.4135066749
[148,] -0.5250109393
[149,]  0.0667708767
[150,]  1.0236034963
[151,]  0.7181858299
[152,]  2.0046986382
[153,]  0.9077927709
[154,]  0.7790784302
[155,]  0.1912033838
[156,] -1.4352184570
[157,] -0.3449373388
[158,]  2.0286756074
[159,]  1.5224477900
[160,]  2.4124872903
[161,]  4.6216801527
[162,]  4.5606237681
[163,]  4.1655734596
[164,]  2.7620022767
[165,]  1.7726492591
[166,]  1.3898503380
[167,]  1.5833358570
[168,] -0.9299775398
[169,] -0.8107543597
[170,] -0.6250473180
[171,] -0.1541083817
[172,]  1.1067684197
[173,]  0.4122461856
[174,] -0.2996294337
[175,] -0.9243977825
[176,]  1.2895256600
[177,]  0.2846134319
[178,] -1.3747339792
[179,]  1.0860423150
[180,]  2.0867609423
[181,]  2.3427763781
[182,]  2.7734469278
[183,]  0.4971678414
[184,] -0.6726235350
[185,]  0.3815858278
[186,]  2.4558619670
[187,]  1.9080271275
[188,]  0.9924010304
[189,]  1.2431613121
[190,]  1.0237790659
[191,] -0.6340481190
[192,] -1.0697942282
[193,] -0.1011008540
[194,] -1.5890390071
[195,] -1.8570122599
[196,]  0.5294503752
[197,] -0.2725477537
[198,]  0.0558945911
[199,]  0.1967847290
[200,]  1.2792998757
[201,]  0.3239701982
[202,]  0.1883623021
[203,]  2.2794019161
[204,]  0.7450805424
[205,] -1.3385985737
[206,]  0.2019887355
[207,]  0.0637177586
[208,] -0.2632557150
[209,]  1.5612644364
[210,] -0.4124138300
[211,] -0.0308214428
[212,]  0.6676389301
[213,] -0.1096525246
[214,]  0.4505206940
[215,] -0.2268479223
[216,]  1.3906749026
[217,]  0.9118228299
[218,]  0.7164064950
[219,]  0.1645681673
[220,]  0.9974894100
[221,] -0.1550790685
[222,]  1.3562476702
[223,]  2.1083521456
[224,]  1.3899603764
[225,]  1.7222307202
[226,]  1.0930239577
[227,]  1.9283408906
[228,] -1.0285784639
[229,] -1.0493328695
[230,] -1.1013613879
[231,]  0.0278984230
[232,] -0.5058597614
[233,] -0.4861780311
[234,] -0.6161346046
[235,]  2.2761757408
[236,]  2.8248956950
[237,]  1.3885538550
[238,]  2.1657465553
[239,]  0.5205357838
[240,]  0.1957492498
[241,] -1.9508949068
[242,] -2.0135022249
[243,] -0.0799742008
[244,] -1.6495083566
[245,] -1.2692081515
[246,] -2.4072600401
[247,] -0.9344244249
[248,]  0.6169820505
[249,] -1.7977821823
[250,] -0.2526471842
[251,] -1.1675263008

$x
              [,1]
  [1,]  0.00000000
  [2,]  0.01874617
  [3,] -1.35727092
  [4,] -0.72340806
  [5,] -1.75063222
  [6,] -2.93964685
  [7,] -1.10295563
  [8,] -1.06545028
  [9,] -0.05769758
 [10,] -0.99821704
 [11,]  0.17685848
 [12,] -0.46366678
 [13,] -1.02261602
 [14,] -2.03216004
 [15,] -2.21167546
 [16,] -1.76051760
 [17,] -3.17412865
 [18,] -1.41203015
 [19,] -2.43896619
 [20,] -1.46713741
 [21,] -1.42489707
 [22,]  0.01787860
 [23,] -0.81525359
 [24,] -1.57909218
 [25,] -0.95179398
 [26,] -1.39146007
 [27,] -1.44423260
 [28,]  0.28477950
 [29,]  0.71940389
 [30,] -0.36265902
 [31,] -0.91788852
 [32,] -1.92601086
 [33,] -2.27483080
 [34,] -0.63974670
 [35,]  0.25588063
 [36,]  0.75465523
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 [40,] -2.78755060
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 [44,] -0.01728013
 [45,]  0.64680404
 [46,] -0.69884205
 [47,] -0.94048621
 [48,] -0.63581988
 [49,]  0.11809244
 [50,] -1.51810791
 [51,] -0.99041298
 [52,] -1.50461407
 [53,] -2.16840392
 [54,] -2.25951596
 [55,] -1.03365028
 [56,]  0.25293027
 [57,] -1.09045690
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 [59,] -0.03718976
 [60,]  1.01203129
 [61,]  2.01456832
 [62,]  1.02956064
 [63,]  0.74043073
 [64,]  1.17900429
 [65,]  1.13301122
 [66,]  0.48577617
 [67,]  1.79354491
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[100,] -2.96842163
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[107,]  0.26871696
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[110,]  0.84274232
[111,]  0.18601156
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[113,]  0.29014209
[114,]  0.16097026
[115,]  0.21803931
[116,]  0.72414018
[117,]  0.26428164
[118,] -0.05093716
[119,]  0.33886986
[120,] -0.58652503
[121,] -1.56794976
[122,]  0.42915172
[123,]  1.18394600
[124,]  1.39707819
[125,]  1.14039329
[126,]  0.49563975
[127,]  1.44962240
[128,] -0.37357707
[129,] -0.96053063
[130,] -1.41281113
[131,] -2.15422378
[132,] -1.20539051
[133,] -3.23392243
[134,] -1.44508830
[135,] -0.96421533
[136,] -0.69213902
[137,] -0.78288043
[138,] -1.06831512
[139,]  1.62867398
[140,]  0.50429728
[141,]  1.04114641
[142,]  1.10836995
[143,] -0.30828348
[144,] -0.18982413
[145,] -0.06917232
[146,]  0.21492717
[147,]  0.35427501
[148,] -0.56426827
[149,] -0.03833359
[150,]  1.12704729
[151,]  0.59081740
[152,]  1.94565769
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[154,]  0.74680293
[155,]  0.11451899
[156,] -1.31761378
[157,] -0.47661387
[158,]  2.03545255
[159,]  1.32280434
[160,]  2.53532105
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[162,]  4.46482928
[163,]  4.18250102
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[165,]  1.81745708
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[168,] -0.86091199
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[170,] -0.63465971
[171,] -0.20958773
[172,]  1.07825508
[173,]  0.34989838
[174,] -0.46157130
[175,] -0.96285189
[176,]  1.39238240
[177,]  0.15749883
[178,] -1.48696132
[179,]  1.04321765
[180,]  1.98359193
[181,]  2.14156820
[182,]  2.70080599
[183,]  0.62594736
[184,] -0.49895496
[185,]  0.45748114
[186,]  2.49818786
[187,]  1.86799095
[188,]  1.03531541
[189,]  1.02118369
[190,]  1.08015757
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[192,] -1.03474717
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[194,] -1.48383936
[195,] -1.79571814
[196,]  0.54399143
[197,] -0.23143015
[198,]  0.16921927
[199,]  0.19298475
[200,]  1.06580331
[201,]  0.19952555
[202,]  0.07018715
[203,]  2.23878442
[204,]  0.94072921
[205,] -1.24449879
[206,]  0.26441267
[207,]  0.06543985
[208,] -0.41054939
[209,]  1.86140757
[210,] -0.37592991
[211,]  0.09258635
[212,]  0.54433318
[213,] -0.18487877
[214,]  0.47338643
[215,] -0.34979953
[216,]  1.25616548
[217,]  0.81843258
[218,]  0.63332128
[219,]  0.17049630
[220,]  1.13526892
[221,] -0.15275104
[222,]  1.43218400
[223,]  2.08093065
[224,]  1.48031563
[225,]  1.58338182
[226,]  1.22228246
[227,]  2.08067935
[228,] -0.95782554
[229,] -1.00580204
[230,] -1.09743030
[231,]  0.06389326
[232,] -0.68103432
[233,] -0.59681433
[234,] -0.46678859
[235,]  2.35054518
[236,]  2.82810387
[237,]  1.58111821
[238,]  2.19367004
[239,]  0.39568667
[240,]  0.10455163
[241,] -2.07985596
[242,] -1.98800194
[243,] -0.16384321
[244,] -1.58788309
[245,] -1.08480648
[246,] -2.43002771
[247,] -0.90130408
[248,]  0.61125597
[249,] -1.74549115
[250,] -0.22833305
[251,] -1.08069635

$xhatKF
  [1]  0.0000000000  0.0001604584 -1.4031939083 -0.6880518905 -1.7744850666
  [6] -2.9492009534 -1.0390411174 -0.9648602955 -0.0554082142 -1.0081199793
 [11]  0.2154943030 -0.6738866637 -1.2273428472 -2.0582121571 -2.2914548026
 [16] -1.7852329257 -3.1637758239 -1.4031803626 -2.5674508870 -1.6458281945
 [21] -1.4875357456 -0.0687865606 -0.7251083179 -1.5717514430 -0.9838536496
 [26] -1.3201469872 -1.4728872250  0.4827697870  0.7937453601 -0.3004612374
 [31] -0.8822570467 -1.9587299560 -2.2332198376 -0.5295245830  0.2018034728
 [36]  0.6253099417  0.8005733704 -0.3506524839 -1.6129361183 -2.8146290098
 [41] -2.9813870185 -1.7905940823 -0.5462916610 -0.0054176884  0.8602606385
 [46] -0.6929304847 -0.9553025237 -0.5222168839 -0.0274935906 -1.4149492629
 [51] -0.8696154945 -1.4546961369 -2.0873385434 -2.1969193078 -1.2038689893
 [56]  0.3532338463 -0.9653356122 -1.3077480943 -0.0388427938  1.0752373818
 [61]  2.0968276961  1.0549963873  0.6225503288  1.0814128726  1.0246251175
 [66]  0.3690487622  1.8413297226 -0.6991538566 -1.0419578445 -0.1499652272
 [71]  0.5935559894 -1.3986903672 -2.7714297449 -1.7261695087 -0.8356557789
 [76]  0.7023215490  0.1180842975  1.1622103513 -0.1441580907  0.7734452514
 [81] -0.8430367706  0.0668806781  0.5427959386 -0.0389787096  0.8220955965
 [86]  0.7717403864 -0.4807291239 -0.0899056384 -0.5830109915 -0.3404470284
 [91]  0.7730324211  1.0316520249 -0.6372779102  0.3443515959 -0.1636829547
 [96]  0.5423366216  0.6186908746 -0.4092240315 -1.7038054242 -3.0854967930
[101] -3.4272941443 -1.2607308562  0.5024844569 -0.6982830385  0.3475800628
[106]  0.0417840844  0.3913086941 -0.4052849078  0.0417884857  0.7940682023
[111]  0.0869537332 -0.2152094933  0.2552380066  0.2338994449  0.3791029855
[116]  0.8514567780  0.1425415242 -0.0476017987  0.4139986197 -0.7960678226
[121] -1.6911998682  0.4861458187  1.2150047274  1.3801600885  1.1039211023
[126]  0.6007560820  1.5320456380 -0.4482997560 -0.8491135474 -1.5174169075
[131] -2.2445109405 -1.1627935257 -3.2090567023 -1.3747170600 -1.2061356136
[136] -0.7274751485 -0.8157789145 -1.0417660679  1.7530999212  0.4660327997
[141]  1.2620890034  1.1131001891 -0.1801489323 -0.3094497622 -0.0962555746
[146]  0.3495059829  0.4120161105 -0.5167985724  0.0622968377  1.0139844524
[151]  0.7186023697  1.9902658820  0.9135521658  0.7781536850  0.1950673422
[156] -1.4196457288 -0.3520249569  2.0061001552  1.5222718033  2.3999742786
[161]  4.5938957507  4.5496427192  4.1581555395  2.7655137482  1.7756178991
[166]  1.3892778818  1.5780050807 -0.9091667424 -0.8094853250 -0.6248707341
[171] -0.1572056059  1.0947094470  0.4162713740 -0.2936049037 -0.9174638009
[176]  1.2700526127  0.2911902930 -1.3590469531  1.0653118057  2.0740805995
[181]  2.3350249062  2.7633818546  0.5126800085 -0.6622142679  0.3729380057
[186]  2.4344340074  1.9072177135  0.9967140016  1.2382810380  1.0228429579
[191] -0.6202511294 -1.0638408610 -0.1079618292 -1.5741895313 -1.8503522525
[196]  0.5105721561 -0.2660934756  0.0533791092  0.1952412563  1.2681448656
[201]  0.3301454366  0.1889456868  2.2583527175  0.7544219404 -1.3198463876
[206]  0.1902527254  0.0644953694 -0.2601872179  1.5439696609 -0.3969506746
[211] -0.0334494415  0.6608178712 -0.1036926581  0.4453187838 -0.2213255337
[216]  1.3753468637  0.9130013539  0.7160947933  0.1682360885  0.9889098668
[221] -0.1462489355  1.3418130676  2.0975011676  1.3917639504  1.7155506487
[226]  1.0949306614  1.9174391968 -1.0042900466 -1.0464171134 -1.0982444322
[231]  0.0195131608 -0.5007346117 -0.4850887206 -0.6136501067  2.2492311248
[236]  2.8136904331  1.3956603494  2.1547280955  0.5313229349  0.1977455969
[241] -1.9302246930 -2.0079306353 -0.0940153237 -1.6339604627 -1.2687774789
[246] -2.3929264394 -0.9428952416  0.6039435206 -1.7756198431 -0.2632724303

$xhatPF
               [,1]
  [1,]  0.000000000
  [2,]  0.004062193
  [3,] -1.442807161
  [4,] -0.689801096
  [5,] -1.799507513
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  [9,] -0.129250053
 [10,] -0.986301533
 [11,]  0.200857470
 [12,] -0.670172226
 [13,] -1.236706176
 [14,] -2.041638642
 [15,] -2.273095402
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 [24,] -1.580222379
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 [27,] -1.459093790
 [28,]  0.486922459
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[109,]  0.073423271
[110,]  0.820745544
[111,]  0.101244143
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[139,]  1.723352533
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