Calculates the spectral density for a known ARMA process
arma_spec.Rdarma_spec() gives the spectral density across a given set of \(\omega\)s for a known ARMA process.
This is an extra function to help users compare spectral density estimates obtained from
HBEST() or HBEST_fast() against the "truth" when the data generation process is known.
Examples
omega <- seq(0, pi, length.out = 1000)
phi <- 0.5
arma_spec(omega = omega, phi = phi)
#> [1] 4.0000000 3.9999209 3.9996836 3.9992881 3.9987346 3.9980231 3.9971540
#> [8] 3.9961273 3.9949433 3.9936024 3.9921048 3.9904509 3.9886411 3.9866760
#> [15] 3.9845558 3.9822813 3.9798528 3.9772711 3.9745367 3.9716503 3.9686127
#> [22] 3.9654244 3.9620864 3.9585994 3.9549642 3.9511818 3.9472529 3.9431786
#> [29] 3.9389598 3.9345975 3.9300928 3.9254466 3.9206601 3.9157344 3.9106707
#> [36] 3.9054700 3.9001336 3.8946627 3.8890586 3.8833225 3.8774558 3.8714596
#> [43] 3.8653355 3.8590847 3.8527086 3.8462087 3.8395863 3.8328429 3.8259800
#> [50] 3.8189990 3.8119014 3.8046888 3.7973626 3.7899245 3.7823759 3.7747184
#> [57] 3.7669536 3.7590831 3.7511085 3.7430314 3.7348535 3.7265762 3.7182014
#> [64] 3.7097306 3.7011655 3.6925077 3.6837590 3.6749209 3.6659951 3.6569833
#> [71] 3.6478872 3.6387085 3.6294488 3.6201098 3.6106932 3.6012006 3.5916338
#> [78] 3.5819945 3.5722841 3.5625046 3.5526574 3.5427443 3.5327670 3.5227270
#> [85] 3.5126260 3.5024656 3.4922476 3.4819734 3.4716448 3.4612633 3.4508305
#> [92] 3.4403480 3.4298174 3.4192402 3.4086181 3.3979525 3.3872450 3.3764971
#> [99] 3.3657104 3.3548863 3.3440263 3.3331319 3.3222046 3.3112458 3.3002569
#> [106] 3.2892394 3.2781947 3.2671242 3.2560292 3.2449111 3.2337713 3.2226111
#> [113] 3.2114318 3.2002347 3.1890212 3.1777924 3.1665497 3.1552943 3.1440274
#> [120] 3.1327502 3.1214640 3.1101698 3.0988689 3.0875624 3.0762514 3.0649371
#> [127] 3.0536205 3.0423028 3.0309849 3.0196679 3.0083529 2.9970409 2.9857328
#> [134] 2.9744297 2.9631324 2.9518421 2.9405595 2.9292857 2.9180214 2.9067676
#> [141] 2.8955252 2.8842950 2.8730778 2.8618744 2.8506858 2.8395125 2.8283555
#> [148] 2.8172155 2.8060931 2.7949892 2.7839045 2.7728396 2.7617952 2.7507720
#> [155] 2.7397706 2.7287917 2.7178359 2.7069038 2.6959960 2.6851131 2.6742556
#> [162] 2.6634242 2.6526192 2.6418413 2.6310910 2.6203687 2.6096750 2.5990103
#> [169] 2.5883750 2.5777698 2.5671948 2.5566507 2.5461378 2.5356564 2.5252070
#> [176] 2.5147900 2.5044058 2.4940545 2.4837368 2.4734527 2.4632027 2.4529871
#> [183] 2.4428062 2.4326602 2.4225495 2.4124743 2.4024348 2.3924314 2.3824642
#> [190] 2.3725336 2.3626396 2.3527825 2.3429626 2.3331799 2.3234348 2.3137273
#> [197] 2.3040577 2.2944261 2.2848327 2.2752775 2.2657608 2.2562827 2.2468432
#> [204] 2.2374426 2.2280808 2.2187581 2.2094745 2.2002300 2.1910248 2.1818590
#> [211] 2.1727326 2.1636456 2.1545982 2.1455903 2.1366221 2.1276935 2.1188046
#> [218] 2.1099554 2.1011459 2.0923761 2.0836461 2.0749559 2.0663054 2.0576947
#> [225] 2.0491237 2.0405925 2.0321009 2.0236491 2.0152369 2.0068643 1.9985313
#> [232] 1.9902379 1.9819839 1.9737694 1.9655944 1.9574587 1.9493622 1.9413050
#> [239] 1.9332870 1.9253081 1.9173682 1.9094673 1.9016052 1.8937819 1.8859974
#> [246] 1.8782514 1.8705440 1.8628750 1.8552444 1.8476521 1.8400979 1.8325817
#> [253] 1.8251035 1.8176632 1.8102606 1.8028956 1.7955681 1.7882780 1.7810252
#> [260] 1.7738096 1.7666310 1.7594894 1.7523846 1.7453164 1.7382848 1.7312897
#> [267] 1.7243308 1.7174081 1.7105215 1.7036707 1.6968558 1.6900765 1.6833326
#> [274] 1.6766242 1.6699510 1.6633129 1.6567097 1.6501414 1.6436077 1.6371086
#> [281] 1.6306438 1.6242133 1.6178169 1.6114544 1.6051258 1.5988308 1.5925693
#> [288] 1.5863412 1.5801464 1.5739846 1.5678557 1.5617596 1.5556961 1.5496652
#> [295] 1.5436665 1.5377000 1.5317656 1.5258630 1.5199922 1.5141529 1.5083451
#> [302] 1.5025686 1.4968231 1.4911087 1.4854251 1.4797721 1.4741497 1.4685576
#> [309] 1.4629958 1.4574640 1.4519622 1.4464901 1.4410476 1.4356346 1.4302509
#> [316] 1.4248964 1.4195709 1.4142743 1.4090064 1.4037671 1.3985561 1.3933735
#> [323] 1.3882190 1.3830924 1.3779937 1.3729227 1.3678792 1.3628631 1.3578742
#> [330] 1.3529124 1.3479776 1.3430695 1.3381882 1.3333333 1.3285049 1.3237026
#> [337] 1.3189264 1.3141762 1.3094518 1.3047530 1.3000798 1.2954319 1.2908093
#> [344] 1.2862118 1.2816392 1.2770914 1.2725684 1.2680698 1.2635957 1.2591459
#> [351] 1.2547201 1.2503184 1.2459406 1.2415865 1.2372559 1.2329489 1.2286651
#> [358] 1.2244046 1.2201671 1.2159526 1.2117609 1.2075918 1.2034453 1.1993212
#> [365] 1.1952194 1.1911398 1.1870822 1.1830465 1.1790325 1.1750403 1.1710696
#> [372] 1.1671203 1.1631922 1.1592854 1.1553996 1.1515347 1.1476906 1.1438671
#> [379] 1.1400643 1.1362819 1.1325198 1.1287779 1.1250561 1.1213542 1.1176723
#> [386] 1.1140100 1.1103674 1.1067443 1.1031406 1.0995562 1.0959910 1.0924448
#> [393] 1.0889176 1.0854092 1.0819195 1.0784485 1.0749959 1.0715618 1.0681460
#> [400] 1.0647483 1.0613688 1.0580072 1.0546635 1.0513376 1.0480293 1.0447386
#> [407] 1.0414654 1.0382095 1.0349708 1.0317494 1.0285449 1.0253575 1.0221868
#> [414] 1.0190330 1.0158958 1.0127752 1.0096710 1.0065832 1.0035117 1.0004563
#> [421] 0.9974171 0.9943938 0.9913864 0.9883949 0.9854190 0.9824588 0.9795141
#> [428] 0.9765848 0.9736709 0.9707722 0.9678887 0.9650203 0.9621669 0.9593284
#> [435] 0.9565047 0.9536958 0.9509014 0.9481217 0.9453564 0.9426056 0.9398690
#> [442] 0.9371467 0.9344385 0.9317444 0.9290643 0.9263980 0.9237456 0.9211069
#> [449] 0.9184819 0.9158705 0.9132725 0.9106880 0.9081169 0.9055590 0.9030143
#> [456] 0.9004827 0.8979642 0.8954586 0.8929659 0.8904861 0.8880190 0.8855645
#> [463] 0.8831227 0.8806934 0.8782765 0.8758720 0.8734799 0.8711000 0.8687322
#> [470] 0.8663766 0.8640330 0.8617013 0.8593816 0.8570736 0.8547775 0.8524930
#> [477] 0.8502202 0.8479589 0.8457091 0.8434708 0.8412438 0.8390281 0.8368237
#> [484] 0.8346304 0.8324483 0.8302771 0.8281170 0.8259678 0.8238295 0.8217020
#> [491] 0.8195852 0.8174791 0.8153836 0.8132986 0.8112242 0.8091602 0.8071066
#> [498] 0.8050634 0.8030304 0.8010076 0.7989949 0.7969924 0.7949999 0.7930174
#> [505] 0.7910448 0.7890821 0.7871293 0.7851861 0.7832527 0.7813290 0.7794149
#> [512] 0.7775103 0.7756152 0.7737296 0.7718534 0.7699865 0.7681289 0.7662805
#> [519] 0.7644414 0.7626113 0.7607904 0.7589785 0.7571756 0.7553817 0.7535966
#> [526] 0.7518204 0.7500530 0.7482943 0.7465444 0.7448031 0.7430704 0.7413463
#> [533] 0.7396307 0.7379235 0.7362248 0.7345345 0.7328525 0.7311788 0.7295134
#> [540] 0.7278561 0.7262070 0.7245660 0.7229331 0.7213083 0.7196914 0.7180824
#> [547] 0.7164814 0.7148882 0.7133028 0.7117252 0.7101554 0.7085932 0.7070387
#> [554] 0.7054918 0.7039525 0.7024207 0.7008965 0.6993796 0.6978702 0.6963682
#> [561] 0.6948735 0.6933861 0.6919060 0.6904331 0.6889674 0.6875088 0.6860574
#> [568] 0.6846130 0.6831757 0.6817453 0.6803220 0.6789056 0.6774960 0.6760934
#> [575] 0.6746976 0.6733085 0.6719262 0.6705507 0.6691818 0.6678196 0.6664640
#> [582] 0.6651150 0.6637726 0.6624366 0.6611072 0.6597842 0.6584676 0.6571575
#> [589] 0.6558537 0.6545562 0.6532650 0.6519801 0.6507014 0.6494289 0.6481626
#> [596] 0.6469024 0.6456483 0.6444004 0.6431584 0.6419225 0.6406926 0.6394686
#> [603] 0.6382506 0.6370385 0.6358322 0.6346318 0.6334372 0.6322484 0.6310654
#> [610] 0.6298880 0.6287164 0.6275505 0.6263902 0.6252356 0.6240865 0.6229430
#> [617] 0.6218050 0.6206726 0.6195456 0.6184241 0.6173081 0.6161974 0.6150922
#> [624] 0.6139923 0.6128977 0.6118084 0.6107245 0.6096457 0.6085723 0.6075040
#> [631] 0.6064409 0.6053829 0.6043301 0.6032825 0.6022399 0.6012023 0.6001698
#> [638] 0.5991424 0.5981199 0.5971024 0.5960898 0.5950822 0.5940795 0.5930816
#> [645] 0.5920886 0.5911005 0.5901171 0.5891386 0.5881648 0.5871958 0.5862315
#> [652] 0.5852719 0.5843170 0.5833667 0.5824211 0.5814801 0.5805438 0.5796120
#> [659] 0.5786847 0.5777620 0.5768438 0.5759302 0.5750210 0.5741162 0.5732159
#> [666] 0.5723200 0.5714286 0.5705415 0.5696587 0.5687804 0.5679063 0.5670366
#> [673] 0.5661711 0.5653099 0.5644529 0.5636002 0.5627517 0.5619074 0.5610673
#> [680] 0.5602314 0.5593995 0.5585718 0.5577483 0.5569288 0.5561133 0.5553020
#> [687] 0.5544946 0.5536913 0.5528920 0.5520967 0.5513054 0.5505180 0.5497346
#> [694] 0.5489550 0.5481794 0.5474077 0.5466398 0.5458758 0.5451157 0.5443594
#> [701] 0.5436069 0.5428581 0.5421132 0.5413720 0.5406346 0.5399009 0.5391710
#> [708] 0.5384447 0.5377221 0.5370032 0.5362880 0.5355764 0.5348684 0.5341641
#> [715] 0.5334634 0.5327662 0.5320727 0.5313826 0.5306962 0.5300132 0.5293338
#> [722] 0.5286579 0.5279855 0.5273166 0.5266511 0.5259891 0.5253305 0.5246754
#> [729] 0.5240237 0.5233753 0.5227304 0.5220888 0.5214506 0.5208157 0.5201842
#> [736] 0.5195560 0.5189311 0.5183095 0.5176912 0.5170762 0.5164645 0.5158560
#> [743] 0.5152507 0.5146487 0.5140498 0.5134542 0.5128618 0.5122725 0.5116865
#> [750] 0.5111036 0.5105238 0.5099472 0.5093737 0.5088033 0.5082360 0.5076718
#> [757] 0.5071107 0.5065526 0.5059977 0.5054457 0.5048968 0.5043510 0.5038081
#> [764] 0.5032683 0.5027315 0.5021976 0.5016667 0.5011388 0.5006139 0.5000919
#> [771] 0.4995728 0.4990567 0.4985435 0.4980332 0.4975257 0.4970212 0.4965196
#> [778] 0.4960208 0.4955249 0.4950318 0.4945416 0.4940543 0.4935697 0.4930880
#> [785] 0.4926090 0.4921329 0.4916595 0.4911890 0.4907212 0.4902561 0.4897939
#> [792] 0.4893343 0.4888775 0.4884235 0.4879721 0.4875235 0.4870776 0.4866343
#> [799] 0.4861938 0.4857559 0.4853207 0.4848882 0.4844583 0.4840311 0.4836065
#> [806] 0.4831845 0.4827652 0.4823485 0.4819344 0.4815229 0.4811140 0.4807076
#> [813] 0.4803039 0.4799027 0.4795041 0.4791081 0.4787146 0.4783236 0.4779352
#> [820] 0.4775493 0.4771660 0.4767851 0.4764068 0.4760310 0.4756576 0.4752868
#> [827] 0.4749184 0.4745525 0.4741891 0.4738282 0.4734697 0.4731136 0.4727600
#> [834] 0.4724088 0.4720601 0.4717138 0.4713699 0.4710284 0.4706894 0.4703527
#> [841] 0.4700184 0.4696865 0.4693570 0.4690299 0.4687052 0.4683828 0.4680627
#> [848] 0.4677451 0.4674298 0.4671168 0.4668061 0.4664978 0.4661919 0.4658882
#> [855] 0.4655869 0.4652878 0.4649911 0.4646967 0.4644046 0.4641148 0.4638272
#> [862] 0.4635420 0.4632590 0.4629783 0.4626998 0.4624236 0.4621497 0.4618780
#> [869] 0.4616086 0.4613414 0.4610764 0.4608137 0.4605532 0.4602950 0.4600389
#> [876] 0.4597851 0.4595335 0.4592841 0.4590369 0.4587919 0.4585491 0.4583085
#> [883] 0.4580700 0.4578338 0.4575997 0.4573678 0.4571381 0.4569105 0.4566851
#> [890] 0.4564619 0.4562408 0.4560218 0.4558050 0.4555904 0.4553779 0.4551675
#> [897] 0.4549593 0.4547532 0.4545492 0.4543473 0.4541476 0.4539500 0.4537545
#> [904] 0.4535611 0.4533698 0.4531806 0.4529935 0.4528085 0.4526256 0.4524447
#> [911] 0.4522660 0.4520894 0.4519148 0.4517423 0.4515719 0.4514035 0.4512373
#> [918] 0.4510731 0.4509109 0.4507508 0.4505928 0.4504368 0.4502829 0.4501311
#> [925] 0.4499812 0.4498335 0.4496877 0.4495441 0.4494024 0.4492628 0.4491252
#> [932] 0.4489897 0.4488561 0.4487246 0.4485952 0.4484677 0.4483423 0.4482189
#> [939] 0.4480975 0.4479781 0.4478608 0.4477454 0.4476320 0.4475207 0.4474114
#> [946] 0.4473040 0.4471987 0.4470954 0.4469940 0.4468947 0.4467973 0.4467020
#> [953] 0.4466086 0.4465172 0.4464278 0.4463404 0.4462550 0.4461716 0.4460901
#> [960] 0.4460106 0.4459332 0.4458576 0.4457841 0.4457125 0.4456430 0.4455753
#> [967] 0.4455097 0.4454460 0.4453843 0.4453246 0.4452668 0.4452110 0.4451572
#> [974] 0.4451053 0.4450554 0.4450075 0.4449615 0.4449175 0.4448754 0.4448354
#> [981] 0.4447972 0.4447610 0.4447268 0.4446946 0.4446643 0.4446359 0.4446095
#> [988] 0.4445851 0.4445626 0.4445421 0.4445236 0.4445070 0.4444923 0.4444796
#> [995] 0.4444689 0.4444601 0.4444532 0.4444484 0.4444454 0.4444444
## output is a vector of length 1000