This function estimates the log power spectral density against the log frequency, and calculates a slope \(\alpha\).
calc_PSD(chain, plot = FALSE)Arguments
Value
Returns a list with log frequencies, log PSDs, and slope and intercept estimates.
Details
A number of studies have reported that cognitive activities contain a long-range slowly decaying autocorrelation. In the frequency domain, this is expressed as \(S(f)\) ~ \(1/f^{-\alpha}\), with \(f\) being frequency, \(S(f)\) being spectral power, and \(\alpha\) \(\epsilon\) \([0.5,1.5]\) is considered \(1/f\) scaling. See See Zhu et al. (2018) for a comparison of Levy Flight and PSD measures for different samplers in multimodal representations.
References
Zhu J, Sanborn AN, Chater N (2018). “Mental Sampling in Multimodal Representations.” Advances in Neural Information Processing Systems, 31, 5748–5759.
Examples
set.seed(1)
chain1 <- sampler_mh(1, "norm", c(0,1), diag(1))
calc_PSD(chain1[[1]], plot= TRUE)
#> $log_freq
#> [1] -3.0103000 -2.7092700 -2.5331787 -2.4082400 -2.3113300 -2.2321487
#> [7] -2.1652019 -2.1072100 -2.0560574 -2.0103000 -1.9689073 -1.9311187
#> [13] -1.8963566 -1.8641719 -1.8342087 -1.8061800 -1.7798510 -1.7550275
#> [19] -1.7315464 -1.7092700 -1.6880807 -1.6678773 -1.6485721 -1.6300887
#> [25] -1.6123599 -1.5953266 -1.5789362 -1.5631419 -1.5479020 -1.5331787
#> [31] -1.5189383 -1.5051500 -1.4917860 -1.4788210 -1.4662319 -1.4539975
#> [37] -1.4420982 -1.4305164 -1.4192353 -1.4082400 -1.3975161 -1.3870507
#> [43] -1.3768315 -1.3668473 -1.3570874 -1.3475421 -1.3382021 -1.3290587
#> [49] -1.3201039 -1.3113300 -1.3027298 -1.2942966 -1.2860241 -1.2779062
#> [55] -1.2699373 -1.2621119 -1.2544251 -1.2468720 -1.2394479 -1.2321487
#> [61] -1.2249701 -1.2179083 -1.2109594 -1.2041200 -1.1973866 -1.1907560
#> [67] -1.1842252 -1.1777910 -1.1714509 -1.1652019 -1.1590416 -1.1529675
#> [73] -1.1469771 -1.1410682 -1.1352387 -1.1294864 -1.1238092 -1.1182054
#> [79] -1.1126729 -1.1072100 -1.1018149 -1.0964861 -1.0912219 -1.0860207
#> [85] -1.0808810 -1.0758015 -1.0707807 -1.0658173 -1.0609099 -1.0560574
#> [91] -1.0512586 -1.0465121 -1.0418170 -1.0371721 -1.0325764 -1.0280287
#> [97] -1.0235282 -1.0190739 -1.0146648 -1.0103000 -1.0059786 -1.0016998
#> [103] -0.9974627 -0.9932666 -0.9891107 -0.9849941 -0.9809162 -0.9768762
#> [109] -0.9728735 -0.9689073 -0.9649770 -0.9610819 -0.9572215 -0.9533951
#> [115] -0.9496021 -0.9458420 -0.9421141 -0.9384179 -0.9347530 -0.9311187
#> [121] -0.9275146 -0.9239401 -0.9203948 -0.9168783 -0.9133899 -0.9099294
#> [127] -0.9064962 -0.9030900 -0.8997102 -0.8963566 -0.8930287 -0.8897260
#> [133] -0.8864483 -0.8831952 -0.8799662 -0.8767610 -0.8735794 -0.8704209
#> [139] -0.8672852 -0.8641719 -0.8610808 -0.8580116 -0.8549639 -0.8519375
#> [145] -0.8489320 -0.8459471 -0.8429826 -0.8400382 -0.8371137 -0.8342087
#> [151] -0.8313230 -0.8284564 -0.8256085 -0.8227792 -0.8199683 -0.8171754
#> [157] -0.8144003 -0.8116429 -0.8089028 -0.8061800 -0.8034741 -0.8007849
#> [163] -0.7981124 -0.7954561 -0.7928160 -0.7901919 -0.7875835 -0.7849907
#> [169] -0.7824133 -0.7798510 -0.7773038 -0.7747715 -0.7722539 -0.7697507
#> [175] -0.7672619 -0.7647873 -0.7623267 -0.7598800 -0.7574469 -0.7550275
#> [181] -0.7526214 -0.7502286 -0.7478489 -0.7454821 -0.7431282 -0.7407870
#> [187] -0.7384584 -0.7361421 -0.7338382 -0.7315464 -0.7292666 -0.7269987
#> [193] -0.7247426 -0.7224982 -0.7202653 -0.7180439 -0.7158337 -0.7136348
#> [199] -0.7114469 -0.7092700 -0.7071039 -0.7049486 -0.7028039 -0.7006698
#> [205] -0.6985461 -0.6964327 -0.6943296 -0.6922366 -0.6901537 -0.6880807
#> [211] -0.6860175 -0.6839641 -0.6819204 -0.6798862 -0.6778615 -0.6758462
#> [217] -0.6738402 -0.6718435 -0.6698558 -0.6678773 -0.6659077 -0.6639470
#> [223] -0.6619951 -0.6600519 -0.6581174 -0.6561915 -0.6542741 -0.6523651
#> [229] -0.6504645 -0.6485721 -0.6466880 -0.6448120 -0.6429440 -0.6410841
#> [235] -0.6392321 -0.6373880 -0.6355516 -0.6337230 -0.6319021 -0.6300887
#> [241] -0.6282829 -0.6264846 -0.6246937 -0.6229101 -0.6211339 -0.6193648
#> [247] -0.6176030 -0.6158483 -0.6141006 -0.6123599 -0.6106262 -0.6088994
#> [253] -0.6071794 -0.6054662 -0.6037598 -0.6020600 -0.6003668 -0.5986803
#> [259] -0.5970002 -0.5953266 -0.5936594 -0.5919987 -0.5903442 -0.5886960
#> [265] -0.5870541 -0.5854183 -0.5837887 -0.5821652 -0.5805477 -0.5789362
#> [271] -0.5773307 -0.5757311 -0.5741373 -0.5725494 -0.5709673 -0.5693909
#> [277] -0.5678202 -0.5662552 -0.5646958 -0.5631419 -0.5615936 -0.5600508
#> [283] -0.5585135 -0.5569816 -0.5554551 -0.5539339 -0.5524181 -0.5509075
#> [289] -0.5494021 -0.5479020 -0.5464070 -0.5449171 -0.5434323 -0.5419526
#> [295] -0.5404779 -0.5390082 -0.5375435 -0.5360837 -0.5346288 -0.5331787
#> [301] -0.5317335 -0.5302930 -0.5288573 -0.5274264 -0.5260001 -0.5245785
#> [307] -0.5231616 -0.5217492 -0.5203415 -0.5189383 -0.5175396 -0.5161454
#> [313] -0.5147556 -0.5133703 -0.5119894 -0.5106129 -0.5092407 -0.5078728
#> [319] -0.5065093 -0.5051500 -0.5037949 -0.5024441 -0.5010974 -0.4997549
#> [325] -0.4984166 -0.4970824 -0.4957522 -0.4944261 -0.4931041 -0.4917860
#> [331] -0.4904720 -0.4891619 -0.4878557 -0.4865535 -0.4852551 -0.4839607
#> [337] -0.4826701 -0.4813833 -0.4801003 -0.4788210 -0.4775456 -0.4762739
#> [343] -0.4750058 -0.4737415 -0.4724809 -0.4712239 -0.4699705 -0.4687207
#> [349] -0.4674745 -0.4662319 -0.4649928 -0.4637573 -0.4625253 -0.4612967
#> [355] -0.4600716 -0.4588500 -0.4576317 -0.4564169 -0.4552055 -0.4539975
#> [361] -0.4527928 -0.4515914 -0.4503933 -0.4491986 -0.4480071 -0.4468189
#> [367] -0.4456339 -0.4444521 -0.4432736 -0.4420982 -0.4409260 -0.4397570
#> [373] -0.4385911 -0.4374284 -0.4362687 -0.4351121 -0.4339586 -0.4328082
#> [379] -0.4316607 -0.4305164 -0.4293750 -0.4282366 -0.4271012 -0.4259687
#> [385] -0.4248392 -0.4237127 -0.4225890 -0.4214682 -0.4203504 -0.4192353
#> [391] -0.4181232 -0.4170139 -0.4159074 -0.4148037 -0.4137029 -0.4126048
#> [397] -0.4115094 -0.4104169 -0.4093271 -0.4082400 -0.4071556 -0.4060739
#> [403] -0.4049949 -0.4039186 -0.4028449 -0.4017739 -0.4007055 -0.3996398
#> [409] -0.3985766 -0.3975161 -0.3964581 -0.3954027 -0.3943499 -0.3932996
#> [415] -0.3922519 -0.3912066 -0.3901639 -0.3891237 -0.3880859 -0.3870507
#> [421] -0.3860179 -0.3849875 -0.3839596 -0.3829341 -0.3819110 -0.3808904
#> [427] -0.3798721 -0.3788562 -0.3778427 -0.3768315 -0.3758227 -0.3748162
#> [433] -0.3738121 -0.3728102 -0.3718107 -0.3708135 -0.3698185 -0.3688258
#> [439] -0.3678354 -0.3668473 -0.3658614 -0.3648777 -0.3638962 -0.3629170
#> [445] -0.3619399 -0.3609651 -0.3599924 -0.3590219 -0.3580536 -0.3570874
#> [451] -0.3561234 -0.3551615 -0.3542018 -0.3532441 -0.3522886 -0.3513351
#> [457] -0.3503838 -0.3494345 -0.3484873 -0.3475421 -0.3465990 -0.3456580
#> [463] -0.3447190 -0.3437820 -0.3428470 -0.3419140 -0.3409831 -0.3400541
#> [469] -0.3391271 -0.3382021 -0.3372790 -0.3363580 -0.3354388 -0.3345216
#> [475] -0.3336063 -0.3326930 -0.3317816 -0.3308721 -0.3299644 -0.3290587
#> [481] -0.3281549 -0.3272529 -0.3263528 -0.3254546 -0.3245582 -0.3236637
#> [487] -0.3227710 -0.3218801 -0.3209911 -0.3201039 -0.3192185 -0.3183349
#> [493] -0.3174530 -0.3165730 -0.3156948 -0.3148183 -0.3139436 -0.3130706
#> [499] -0.3121994 -0.3113300 -0.3104622 -0.3095962 -0.3087320 -0.3078694
#> [505] -0.3070086 -0.3061494 -0.3052920 -0.3044362 -0.3035822 -0.3027298
#> [511] -0.3018791 -0.3010300
#>
#> $log_psd
#> [1] 1.054538094 -0.005930573 0.896786005 1.601153185 1.309203752
#> [6] 0.741801273 1.050227678 1.106918969 0.476371236 0.718913806
#> [11] 0.980239063 -0.947765261 0.802918292 0.412632319 -0.228496611
#> [16] 0.547708384 0.840352208 1.251342067 0.058799573 0.521948989
#> [21] -0.038402312 -0.655631219 0.911660674 -0.244662100 -0.333394109
#> [26] -0.545201065 0.158920085 0.781421260 0.986483649 1.024663776
#> [31] 0.396074099 1.040048502 0.665157519 0.611152390 1.091004665
#> [36] 0.239480726 0.585810241 0.632275406 0.586461558 0.400097363
#> [41] 0.435268345 -0.811692044 0.484834706 -0.135284380 -0.786454705
#> [46] 0.530231443 0.310909536 -1.378052324 0.224985374 -0.085111407
#> [51] 0.521496950 0.336236148 0.479501668 0.439830966 0.306298043
#> [56] 0.499145188 0.597450019 0.641347778 1.019261082 0.495493648
#> [61] -0.588163269 -1.068117959 -0.652041316 -0.533067666 0.223713776
#> [66] 0.619575998 0.313565577 -0.212928453 0.705966068 0.311988095
#> [71] 0.541442412 -0.566372980 0.297204838 0.371315806 0.357277088
#> [76] -0.942967528 -0.006309224 -0.161103006 -0.069185213 0.256348277
#> [81] 0.179048043 0.014246940 0.293014073 0.164090080 0.519949731
#> [86] 0.720432771 -0.393030620 0.209787414 0.325830141 0.197958377
#> [91] -0.435851122 0.362577011 0.595811630 -0.864417793 -0.288802475
#> [96] -0.192087883 -0.465824519 -0.343159860 -0.763506884 0.032398155
#> [101] -0.831714982 -0.310152145 -0.598359679 -0.391072602 -0.224747442
#> [106] -0.153355967 -0.233026051 -0.192211816 -0.119864436 0.109808409
#> [111] -0.532277499 -1.335012067 -0.446878562 0.387819763 -0.087412091
#> [116] -0.024134873 -1.294408178 0.193234652 0.314985026 0.162694463
#> [121] -0.539619911 -0.742814703 0.112558848 -1.947710897 -0.044353531
#> [126] 0.498327762 -0.471629180 0.336832098 -1.767766795 0.334021768
#> [131] -1.608558089 0.497933889 -0.570644047 -0.208727936 0.417026916
#> [136] -1.191198291 -1.709041837 -0.662302881 -1.038396752 -0.077509275
#> [141] -0.706162183 -0.659848470 -0.438814465 -0.500946905 -0.860534362
#> [146] -0.184104927 -0.379431803 0.007007244 -0.267054774 -0.386885222
#> [151] -0.558951827 0.014568020 -0.858400419 -1.253367694 -1.287959490
#> [156] -0.563332788 -1.455253757 0.255484980 -0.176658499 -0.554037587
#> [161] -0.976738141 0.006566158 -1.470941667 -0.560464342 -1.292305988
#> [166] -0.999639529 -0.893609511 0.007900654 -0.152395199 -0.076965409
#> [171] -0.407967854 -0.555087748 -1.021345541 0.177982044 0.050146590
#> [176] -0.404040705 -0.048403806 -0.885126266 -0.111177426 -2.079027236
#> [181] -0.680386656 -0.146519980 -0.545073762 -0.258600403 -1.080893828
#> [186] -0.628182969 -0.129147070 -0.837270091 -0.617147494 -1.063842777
#> [191] -1.088082532 -0.276180149 -0.820986957 -0.179011903 -0.256519367
#> [196] -2.043987185 -0.571579877 -0.776621810 -0.639094554 -1.937187119
#> [201] 0.022428963 -1.111089553 -0.946893226 -0.138354554 -0.503735766
#> [206] 0.251512960 0.231893011 -0.112281154 -0.251267848 -1.345284419
#> [211] -0.701411065 -0.221702484 -0.621962743 -0.605566437 -0.495989787
#> [216] -1.733415982 -1.148515974 -0.736742337 -0.514894634 -1.481815478
#> [221] -0.715906608 -0.433662530 -0.404045424 -0.591452763 -2.208591122
#> [226] -1.083207717 -1.040727885 -0.848945691 -1.353516441 -1.077935842
#> [231] -0.554788715 -1.277367843 -0.455405152 -0.623707314 -0.086366855
#> [236] -0.595637049 -0.882144465 -0.848267051 -0.985600771 -0.582683540
#> [241] -0.493928965 -0.505227070 -0.958882781 -0.742852340 -1.332795646
#> [246] -1.141692752 -1.452059555 -0.958998025 -1.215110726 -0.857156194
#> [251] -0.348144150 -0.041252959 -0.848471613 -1.135994569 -0.527410417
#> [256] -1.426159949 -1.493608452 -0.856821037 -1.602342454 -1.024834513
#> [261] -0.984268423 -1.090296272 -1.500832151 -1.827388684 -0.787067200
#> [266] -0.088587544 -0.202123799 -0.747879234 -0.574507984 -1.229443130
#> [271] -0.581914955 0.031384948 -0.751373157 -1.101605096 -0.738614017
#> [276] -1.150183853 -1.004119107 -0.066138822 -0.994497941 -1.084560472
#> [281] -1.570321484 -0.313505300 -2.574538194 -1.966883380 -1.664486526
#> [286] -1.173864700 -1.883940407 -0.673299766 0.005458407 -0.723097158
#> [291] -0.246154152 -0.081586887 -0.393382204 -1.530208660 -0.982162147
#> [296] -0.347917630 -0.527442242 -1.462908794 -1.679760845 -0.992691603
#> [301] -0.543335392 -0.985396925 -1.789440594 -0.688201595 -0.549226335
#> [306] -0.589164378 -0.667119541 -1.162588548 -0.975605067 -1.032407428
#> [311] -0.897512669 -1.071753264 -1.767297816 -0.527452389 -0.418921927
#> [316] -0.531363976 -0.872359160 -0.373650312 -0.511534369 -1.396032280
#> [321] -0.510388733 -0.930865691 -1.354269434 -0.708128129 -0.455274155
#> [326] -0.881680462 -0.367335760 -0.479165545 -0.564813510 -0.661648410
#> [331] -0.598596528 -0.369491867 -1.303838552 -1.515003145 -2.396819147
#> [336] -0.992833186 -0.838494332 -0.475711285 -0.800530776 -1.519869857
#> [341] -1.467204492 -0.957496671 -1.210811790 -0.719092611 -1.064969055
#> [346] -1.905875589 -0.900746011 -0.897181008 -1.052658456 -0.478566712
#> [351] -1.035188663 -0.671720270 -0.891828139 -1.467153024 -1.436529546
#> [356] -0.567359237 -1.423769024 -0.790405330 -0.226271868 -1.202107902
#> [361] -0.420966011 -0.711835773 -1.550476669 -2.204942292 -1.035644894
#> [366] -2.453910108 -1.089036968 -0.808353513 -1.497442120 -0.777611798
#> [371] -1.537722679 -0.748009770 -0.776734598 -1.496370488 -1.793892855
#> [376] -0.696761893 -0.803249387 -0.242958840 -0.766042691 -0.823672993
#> [381] -1.319640375 -0.625085327 -0.909283742 -0.996162562 -0.512014030
#> [386] -0.927366146 -2.092619650 -1.729046815 -1.978460270 -0.365199587
#> [391] -1.634755052 -0.471163236 -1.567404280 -0.594859999 -1.418290827
#> [396] -0.646662464 -0.944619197 -1.071528342 -0.955411932 -1.817570201
#> [401] -0.918877982 -0.931661230 -0.624721532 -0.712273738 -1.153020275
#> [406] -1.613620998 -0.917322703 -0.793416141 -0.894551836 -2.607707172
#> [411] -1.240151826 -0.722646227 -0.702873748 -1.034136256 -1.246630468
#> [416] -1.050565910 -0.622353182 -0.788004848 -1.574125510 -1.800918180
#> [421] -1.737923727 -0.663960578 -0.520521964 -0.991953747 -2.122109729
#> [426] -0.213542767 -0.566605754 -1.289690980 -1.517274146 -0.651500738
#> [431] -0.548713701 -0.579778587 -0.860295170 -1.052880232 -1.826162236
#> [436] -0.889157466 -1.470119081 -0.251311715 -0.522117915 -1.615672866
#> [441] -1.061542492 -1.089459141 -0.988343229 -0.659730255 -1.059064706
#> [446] -1.260482425 -0.889107064 -1.731250346 -0.851466795 -3.175954921
#> [451] -0.830156082 -1.325711132 -1.344164970 -0.830244362 -1.440286684
#> [456] -0.867341809 -0.829160258 -1.327695269 -0.772884586 -0.628726849
#> [461] -2.377824290 -1.474767516 -1.138639094 -1.986039614 -2.832403070
#> [466] -1.201307896 -1.092649498 -0.736087930 -0.660648698 -1.290598140
#> [471] -0.469362091 -1.640822284 -1.068981000 -0.927477664 -0.600844152
#> [476] -0.633548522 -0.694711656 -2.044497659 -1.692033080 -1.624225941
#> [481] -0.544588005 -1.910827057 -0.574557797 -0.727806611 -1.561624453
#> [486] -1.566586217 -1.091925673 -0.769763375 -1.720842617 -1.134723803
#> [491] -1.378731371 -2.036199983 -0.586310544 -0.364816995 -0.769486125
#> [496] -2.435281780 -1.089717542 -0.989942018 -0.832263632 -0.989879911
#> [501] -1.342964026 -0.411006576 -1.890059277 -2.199023759 -1.020232177
#> [506] -0.964199158 -0.579841860 -0.586818550 -0.857262338 -0.678559717
#> [511] -1.062687163 -1.442719846
#>
#> $polyfit
#> [1] -0.7943447 -1.1243849
#>