Lspec: logspline estimation of a spectral distribution

Autocorrelations, autocovariances (clspec), spectral densities and line spectrum (dlspec), spectral distributions (plspec) or a random time series(rlspec) from a model fitted with lspec.

clspec(lag, fit, cov = TRUE, mm) 
dlspec(freq, fit) 
plspec(freq, fit, mm) 
rlspec(n, fit, mean = 0, cosmodel = FALSE, mm)

Arguments

lag: vector of integer-valued lags for which the autocorrelations or autocorrelations are to be computed.
fit: lspec object, typically the result of lspec.
cov: compute autocovariances (TRUE) or autocorrelations (FALSE).
mm: number of points used in integration and the fft. Default is the smallest power of two larger than max(fit\$sample, max(lag),1024) for clspec and plspec or the smallest power of two larger than max(fit\$sample, n, max(lag), 1024) for (rlspec).
freq: vector of frequencies. For plspec frequencies should be between $-\pi$ and $\pi$.
n: length of the random time series to be generated.
mean: mean level of the time series to be generated.
cosmodel: indicate that the data should be generated from a model with constant harmonic terms rather than a true Gaussian time series.

Value

Autocovariances or autocorrelations (clspec); values of the spectral distribution at the requested frequencies. (plspec); random time series of length n (rlspec); or a list with three components (dlspec):

d: the spectral density evaluated at the vector of frequencies,
modfreq: modified frequencies of the form $\frac{2\pi j}{T}$ that are close to the frequencies that were requested,
m: mass of the line spectrum at the modified frequencies.

References

Charles Kooperberg, Charles J. Stone, and Young K. Truong (1995). Logspline Estimation of a Possibly Mixed Spectral Distribution. Journal of Time Series Analysis, 16, 359-388.

Charles J. Stone, Mark Hansen, Charles Kooperberg, and Young K. Truong. The use of polynomial splines and their tensor products in extended linear modeling (with discussion) (1997). Annals of Statistics, 25, 1371–1470.

Author

Charles Kooperberg clk@fredhutch.org.

Examples

data(co2)
co2.detrend <- lm(co2~c(1:length(co2)))$residuals
fit <- lspec(co2.detrend)
clspec(0:12,fit)
#>  [1]  6.7850276  6.0562328  4.2967554  2.1677376  0.3352617 -0.8238498
#>  [7] -1.2194218 -0.8578529  0.2669647  2.0641153  4.1544602  5.8644938
#> [13]  6.5146536
plspec((0:314)/100, fit)
#>   [1] 3.392514 3.411057 4.477251 4.494846 4.511564 4.527199 4.541621 4.554737
#>   [9] 4.566526 4.576996 4.586204 4.594222 4.601148 4.607085 4.612143 4.616428
#>  [17] 4.620043 4.623084 4.625633 4.627769 4.629557 4.631056 4.632313 4.633372
#>  [25] 4.634266 4.635025 4.635672 4.636229 4.636711 4.637132 4.637503 4.637834
#>  [33] 4.638133 4.638406 4.638659 4.638896 4.639123 4.639344 4.639562 4.639782
#>  [41] 4.640008 4.640245 4.640499 4.640778 4.641092 4.641454 4.641881 4.642401
#>  [49] 4.643050 4.643886 4.644975 4.646360 4.647996 6.605395 6.606959 6.608227
#>  [57] 6.609180 6.609866 6.610360 6.610729 6.611026 6.611286 6.611529 6.611766
#>  [65] 6.612000 6.612231 6.612460 6.612686 6.612910 6.613132 6.613351 6.613568
#>  [73] 6.613782 6.613995 6.614205 6.614413 6.614619 6.614823 6.615025 6.615224
#>  [81] 6.615422 6.615618 6.615812 6.616003 6.616193 6.616381 6.616568 6.616752
#>  [89] 6.616935 6.617115 6.617294 6.617472 6.617647 6.617821 6.617994 6.618164
#>  [97] 6.618333 6.618501 6.618667 6.618831 6.618994 6.619156 6.619315 6.619474
#> [105] 6.619631 6.767021 6.767175 6.767328 6.767479 6.767629 6.767778 6.767925
#> [113] 6.768072 6.768217 6.768360 6.768503 6.768644 6.768784 6.768923 6.769060
#> [121] 6.769197 6.769332 6.769467 6.769600 6.769732 6.769863 6.769993 6.770122
#> [129] 6.770249 6.770376 6.770502 6.770627 6.770751 6.770873 6.770995 6.771116
#> [137] 6.771236 6.771355 6.771474 6.771591 6.771707 6.771823 6.771937 6.772051
#> [145] 6.772164 6.772276 6.772388 6.772498 6.772608 6.772717 6.772825 6.772933
#> [153] 6.773039 6.773145 6.773251 6.773355 6.773459 6.773562 6.773664 6.773766
#> [161] 6.773867 6.773967 6.774067 6.774166 6.774264 6.774362 6.774459 6.774556
#> [169] 6.774652 6.774747 6.774842 6.774936 6.775029 6.775122 6.775214 6.775306
#> [177] 6.775398 6.775488 6.775579 6.775668 6.775757 6.775846 6.775934 6.776022
#> [185] 6.776109 6.776196 6.776282 6.776367 6.776453 6.776537 6.776622 6.776706
#> [193] 6.776789 6.776872 6.776954 6.777037 6.777118 6.777199 6.777280 6.777361
#> [201] 6.777441 6.777521 6.777600 6.777679 6.777757 6.777835 6.777913 6.777991
#> [209] 6.778068 6.778144 6.778221 6.778297 6.778372 6.778448 6.778523 6.778598
#> [217] 6.778672 6.778746 6.778820 6.778893 6.778966 6.779039 6.779112 6.779184
#> [225] 6.779256 6.779328 6.779399 6.779470 6.779541 6.779612 6.779682 6.779752
#> [233] 6.779822 6.779892 6.779961 6.780030 6.780099 6.780168 6.780236 6.780305
#> [241] 6.780373 6.780440 6.780508 6.780575 6.780643 6.780709 6.780776 6.780843
#> [249] 6.780909 6.780975 6.781041 6.781107 6.781173 6.781238 6.781303 6.781369
#> [257] 6.781433 6.781498 6.781563 6.781627 6.781692 6.781756 6.781820 6.781884
#> [265] 6.781947 6.782011 6.782074 6.782138 6.782201 6.782264 6.782327 6.782390
#> [273] 6.782453 6.782515 6.782578 6.782640 6.782702 6.782764 6.782827 6.782889
#> [281] 6.782950 6.783012 6.783074 6.783136 6.783197 6.783259 6.783320 6.783381
#> [289] 6.783442 6.783504 6.783565 6.783626 6.783687 6.783748 6.783808 6.783869
#> [297] 6.783930 6.783991 6.784051 6.784112 6.784173 6.784233 6.784294 6.784354
#> [305] 6.784415 6.784475 6.784535 6.784596 6.784656 6.784716 6.784777 6.784837
#> [313] 6.784897 6.784958 6.785018
dlspec((0:314)/100, fit)
#> $d
#>   [1] 1.859826215 1.843522777 1.795550614 1.719450123 1.620536361 1.504674826
#>   [7] 1.377776009 1.245386288 1.112388399 0.982818091 0.859788365 0.745502539
#>  [13] 0.641332489 0.547937872 0.465404705 0.393386093 0.331232873 0.278106708
#>  [19] 0.233072221 0.195167720 0.163456164 0.137059095 0.115176819 0.097098018
#>  [25] 0.082201763 0.069954360 0.059903024 0.051667853 0.044933185 0.039439057
#>  [31] 0.034973234 0.031364070 0.028474348 0.026196143 0.024446714 0.023165445
#>  [37] 0.022311830 0.021864601 0.021822130 0.022204417 0.023057152 0.024458702
#>  [43] 0.026531391 0.029459333 0.033516564 0.039111744 0.046860119 0.057701288
#>  [49] 0.073095556 0.095129931 0.123543945 0.152762556 0.171549173 0.167514335
#>  [55] 0.142897118 0.110609235 0.080748360 0.057786769 0.042136017 0.032537701
#>  [61] 0.027425943 0.024944003 0.023886598 0.023500545 0.023247762 0.022998786
#>  [67] 0.022753457 0.022511716 0.022273505 0.022038764 0.021807438 0.021579470
#>  [73] 0.021354806 0.021133393 0.020915178 0.020700109 0.020488134 0.020279205
#>  [79] 0.020073273 0.019870288 0.019670205 0.019472976 0.019278555 0.019086900
#>  [85] 0.018897964 0.018711706 0.018528083 0.018347053 0.018168575 0.017992609
#>  [91] 0.017819116 0.017648058 0.017479395 0.017313090 0.017149108 0.016987411
#>  [97] 0.016827965 0.016670735 0.016515685 0.016362784 0.016211997 0.016063292
#> [103] 0.015916638 0.015772003 0.015629357 0.015488668 0.015349908 0.015213047
#> [109] 0.015078057 0.014944909 0.014813576 0.014684030 0.014556245 0.014430194
#> [115] 0.014305852 0.014183192 0.014062191 0.013942824 0.013825066 0.013708894
#> [121] 0.013594284 0.013481214 0.013369661 0.013259603 0.013151018 0.013043885
#> [127] 0.012938183 0.012833890 0.012730988 0.012629455 0.012529272 0.012430420
#> [133] 0.012332880 0.012236633 0.012141661 0.012047945 0.011955469 0.011864214
#> [139] 0.011774163 0.011685300 0.011597607 0.011511069 0.011425670 0.011341393
#> [145] 0.011258223 0.011176146 0.011095145 0.011015206 0.010936315 0.010858458
#> [151] 0.010781619 0.010705786 0.010630945 0.010557082 0.010484185 0.010412240
#> [157] 0.010341235 0.010271157 0.010201994 0.010133734 0.010066364 0.009999874
#> [163] 0.009934251 0.009869485 0.009805563 0.009742476 0.009680213 0.009618762
#> [169] 0.009558113 0.009498256 0.009439182 0.009380879 0.009323338 0.009266551
#> [175] 0.009210506 0.009155195 0.009100608 0.009046738 0.008993574 0.008941108
#> [181] 0.008889331 0.008838236 0.008787813 0.008738055 0.008688953 0.008640500
#> [187] 0.008592687 0.008545508 0.008498954 0.008453019 0.008407694 0.008362973
#> [193] 0.008318849 0.008275314 0.008232362 0.008189987 0.008148181 0.008106938
#> [199] 0.008066251 0.008026115 0.007986523 0.007947469 0.007908948 0.007870952
#> [205] 0.007833477 0.007796516 0.007760065 0.007724117 0.007688667 0.007653710
#> [211] 0.007619240 0.007585253 0.007551743 0.007518706 0.007486136 0.007454028
#> [217] 0.007422378 0.007391182 0.007360433 0.007330129 0.007300265 0.007270836
#> [223] 0.007241838 0.007213266 0.007185117 0.007157387 0.007130071 0.007103166
#> [229] 0.007076667 0.007050572 0.007024876 0.006999575 0.006974666 0.006950145
#> [235] 0.006926010 0.006902256 0.006878880 0.006855878 0.006833249 0.006810988
#> [241] 0.006789092 0.006767559 0.006746384 0.006725566 0.006705102 0.006684988
#> [247] 0.006665222 0.006645801 0.006626722 0.006607983 0.006589581 0.006571514
#> [253] 0.006553779 0.006536374 0.006519296 0.006502543 0.006486114 0.006470004
#> [259] 0.006454213 0.006438738 0.006423578 0.006408729 0.006394191 0.006379960
#> [265] 0.006366036 0.006352416 0.006339099 0.006326083 0.006313365 0.006300945
#> [271] 0.006288820 0.006276990 0.006265452 0.006254205 0.006243247 0.006232578
#> [277] 0.006222195 0.006212097 0.006202283 0.006192751 0.006183501 0.006174531
#> [283] 0.006165840 0.006157427 0.006149291 0.006141430 0.006133843 0.006126531
#> [289] 0.006119491 0.006112722 0.006106225 0.006099997 0.006094039 0.006088349
#> [295] 0.006082926 0.006077771 0.006072882 0.006068258 0.006063900 0.006059806
#> [301] 0.006055976 0.006052410 0.006049106 0.006046065 0.006043286 0.006040769
#> [307] 0.006038513 0.006036519 0.006034786 0.006033313 0.006032101 0.006031149
#> [313] 0.006030457 0.006030025 0.006029854
#> 
#> $modfreq
#>   [1] 0.00000000 0.01342561 0.01342561 0.02685122 0.04027683 0.05370244
#>   [7] 0.05370244 0.06712805 0.08055366 0.09397927 0.09397927 0.10740488
#>  [13] 0.12083049 0.13425610 0.13425610 0.14768171 0.16110732 0.17453293
#>  [19] 0.17453293 0.18795853 0.20138414 0.21480975 0.21480975 0.22823536
#>  [25] 0.24166097 0.25508658 0.25508658 0.26851219 0.28193780 0.29536341
#>  [31] 0.29536341 0.30878902 0.32221463 0.33564024 0.33564024 0.34906585
#>  [37] 0.36249146 0.37591707 0.37591707 0.38934268 0.40276829 0.41619390
#>  [43] 0.41619390 0.42961951 0.44304512 0.45647073 0.45647073 0.46989634
#>  [49] 0.48332195 0.48332195 0.49674756 0.51017317 0.52359878 0.52359878
#>  [55] 0.53702439 0.55044999 0.56387560 0.56387560 0.57730121 0.59072682
#>  [61] 0.60415243 0.60415243 0.61757804 0.63100365 0.64442926 0.64442926
#>  [67] 0.65785487 0.67128048 0.68470609 0.68470609 0.69813170 0.71155731
#>  [73] 0.72498292 0.72498292 0.73840853 0.75183414 0.76525975 0.76525975
#>  [79] 0.77868536 0.79211097 0.80553658 0.80553658 0.81896219 0.83238780
#>  [85] 0.84581341 0.84581341 0.85923902 0.87266463 0.88609024 0.88609024
#>  [91] 0.89951585 0.91294145 0.92636706 0.92636706 0.93979267 0.95321828
#>  [97] 0.96664389 0.96664389 0.98006950 0.99349511 0.99349511 1.00692072
#> [103] 1.02034633 1.03377194 1.03377194 1.04719755 1.06062316 1.07404877
#> [109] 1.07404877 1.08747438 1.10089999 1.11432560 1.11432560 1.12775121
#> [115] 1.14117682 1.15460243 1.15460243 1.16802804 1.18145365 1.19487926
#> [121] 1.19487926 1.20830487 1.22173048 1.23515609 1.23515609 1.24858170
#> [127] 1.26200731 1.27543291 1.27543291 1.28885852 1.30228413 1.31570974
#> [133] 1.31570974 1.32913535 1.34256096 1.35598657 1.35598657 1.36941218
#> [139] 1.38283779 1.39626340 1.39626340 1.40968901 1.42311462 1.43654023
#> [145] 1.43654023 1.44996584 1.46339145 1.46339145 1.47681706 1.49024267
#> [151] 1.50366828 1.50366828 1.51709389 1.53051950 1.54394511 1.54394511
#> [157] 1.55737072 1.57079633 1.58422194 1.58422194 1.59764755 1.61107316
#> [163] 1.62449877 1.62449877 1.63792437 1.65134998 1.66477559 1.66477559
#> [169] 1.67820120 1.69162681 1.70505242 1.70505242 1.71847803 1.73190364
#> [175] 1.74532925 1.74532925 1.75875486 1.77218047 1.78560608 1.78560608
#> [181] 1.79903169 1.81245730 1.82588291 1.82588291 1.83930852 1.85273413
#> [187] 1.86615974 1.86615974 1.87958535 1.89301096 1.90643657 1.90643657
#> [193] 1.91986218 1.93328779 1.93328779 1.94671340 1.96013901 1.97356462
#> [199] 1.97356462 1.98699023 2.00041583 2.01384144 2.01384144 2.02726705
#> [205] 2.04069266 2.05411827 2.05411827 2.06754388 2.08096949 2.09439510
#> [211] 2.09439510 2.10782071 2.12124632 2.13467193 2.13467193 2.14809754
#> [217] 2.16152315 2.17494876 2.17494876 2.18837437 2.20179998 2.21522559
#> [223] 2.21522559 2.22865120 2.24207681 2.25550242 2.25550242 2.26892803
#> [229] 2.28235364 2.29577925 2.29577925 2.30920486 2.32263047 2.33605608
#> [235] 2.33605608 2.34948169 2.36290730 2.37633290 2.37633290 2.38975851
#> [241] 2.40318412 2.41660973 2.41660973 2.43003534 2.44346095 2.44346095
#> [247] 2.45688656 2.47031217 2.48373778 2.48373778 2.49716339 2.51058900
#> [253] 2.52401461 2.52401461 2.53744022 2.55086583 2.56429144 2.56429144
#> [259] 2.57771705 2.59114266 2.60456827 2.60456827 2.61799388 2.63141949
#> [265] 2.64484510 2.64484510 2.65827071 2.67169632 2.68512193 2.68512193
#> [271] 2.69854754 2.71197315 2.72539876 2.72539876 2.73882436 2.75224997
#> [277] 2.76567558 2.76567558 2.77910119 2.79252680 2.80595241 2.80595241
#> [283] 2.81937802 2.83280363 2.84622924 2.84622924 2.85965485 2.87308046
#> [289] 2.88650607 2.88650607 2.89993168 2.91335729 2.91335729 2.92678290
#> [295] 2.94020851 2.95363412 2.95363412 2.96705973 2.98048534 2.99391095
#> [301] 2.99391095 3.00733656 3.02076217 3.03418778 3.03418778 3.04761339
#> [307] 3.06103900 3.07446461 3.07446461 3.08789022 3.10131582 3.11474143
#> [313] 3.11474143 3.12816704 3.14159265
#> 
#> $m
#>   [1]        NA 1.0479728 1.0479728 0.0000000 0.0000000 0.0000000 0.0000000
#>   [8] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [15] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [22] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [29] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [36] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [43] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [50] 0.0000000 0.0000000 0.0000000 1.9556825 1.9556825 0.0000000 0.0000000
#>  [57] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [64] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [71] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [78] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [85] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [92] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#>  [99] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [106] 0.1472341 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [113] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [120] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [127] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [134] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [141] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [148] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [155] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [162] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [169] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [176] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [183] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [190] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [197] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [204] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [211] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [218] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [225] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [232] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [239] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [246] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [253] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [260] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [267] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [274] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [281] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [288] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [295] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [302] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> [309] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
#> 
rlspec(length(co2),fit)
#>   [1] -1.35361180 -1.42647725 -1.78413085 -2.13873961 -1.40967992 -0.95718169
#>   [7] -0.91970665 -1.29698779 -2.01878640 -2.80046120 -2.55756333 -2.04544261
#>  [13] -1.76917873 -1.95358552 -2.43056939 -1.53944201 -0.93055900 -1.05367233
#>  [19] -1.18267917 -1.57208337 -1.45951876 -1.62246257 -1.31541680 -1.17339413
#>  [25] -1.27330169 -1.13789696 -1.37110552 -1.37696243 -1.07244906 -0.77970547
#>  [31] -0.69727390 -0.99744669 -0.97977686 -1.26434506 -0.85344182 -1.15003125
#>  [37] -1.28647154 -1.69539187 -1.97228910 -1.87783541 -1.59690012 -0.72394371
#>  [43] -0.46218310 -0.96940549 -1.49207087 -1.55494506 -1.24757804 -1.49636651
#>  [49] -1.80552479 -1.26157490 -1.62860097 -1.39092413 -0.27389324 -0.48004550
#>  [55] -0.62755042 -1.43368363 -1.14674805 -1.53774787 -1.62898243 -1.65841362
#>  [61] -1.60824665 -2.00509320 -2.08905285 -1.61311010 -0.94785112 -0.38347615
#>  [67] -0.97677614 -0.96192211 -1.23403452 -1.43725459 -1.71978478 -1.92177948
#>  [73] -1.66665491 -1.54740797 -1.94877690 -1.15790485 -0.53592599 -0.47425599
#>  [79] -1.16819792 -1.70521931 -1.96245165 -2.00615035 -1.67902127 -1.74325928
#>  [85] -1.63051812 -1.57255186 -1.51340661 -1.15618571 -0.15159722 -0.15156262
#>  [91] -0.49950031 -1.46483733 -1.49640367 -0.84575980 -1.00990057 -0.91263594
#>  [97] -0.37854194 -1.19675079 -1.05033403 -0.40468167 -0.20228257  0.66561069
#> [103]  0.44550475  0.36692982 -0.36743181 -0.50795572 -0.09803698  0.05087616
#> [109] -0.15097375 -0.11273637 -0.36937523  0.21361087  0.13161534  0.19424687
#> [115]  0.42414248 -0.46953270 -1.04387855 -1.59037841 -0.97525561 -0.54045430
#> [121] -0.66115238 -1.51393440 -0.97888579 -0.27697874 -0.21001599  0.41899940
#> [127]  0.08667141 -0.16607020 -0.40741414 -0.25445915 -0.01848729 -0.43468643
#> [133]  0.01758540  0.29547032  0.55262860  0.75100924  1.60184405  1.12460375
#> [139]  0.98763576 -0.30093543  0.16036097 -0.15250069  0.56465810  0.84119585
#> [145]  0.84225710  0.69656118  0.40031478  1.14411293  1.36948664  1.27091320
#> [151]  1.31934378  0.56740793  0.16047124  0.49015053  0.40405431  0.21357149
#> [157]  1.14179605  0.85362504  0.40093307  0.18735516  0.65990131  0.88197324
#> [163]  0.87498987 -0.26580281 -0.78194972 -0.64282378 -1.02795780 -0.34242241
#> [169] -0.71962276 -1.07053032 -0.43801352 -0.30078387 -0.07401013  0.06416799
#> [175] -0.11176455 -0.47457892 -0.63467786 -0.49019873 -0.13698046  0.04556003
#> [181]  0.38344092 -0.22743471  0.13425815  0.43586969  1.20790433  1.81397146
#> [187]  1.65400271  1.47056533  0.83408413  0.95516682  0.94648802  1.03969982
#> [193]  1.32041147  1.59362313  1.65677611  1.72584876  2.82858716  3.44746654
#> [199]  3.23317154  2.44873059  2.18424456  2.17461175  1.72705022  2.17214676
#> [205]  1.52016852  1.12667602  0.97206827  1.83398788  1.88800107  2.39321543
#> [211]  2.64920670  1.35397647  0.29546038  0.25330395 -0.10095840  0.59509996
#> [217]  0.31323579 -0.31534346  0.20750988  0.15365502  1.01131745  0.91992454
#> [223]  1.33495659  0.69943571  0.63374982  0.26774708 -0.23067668  0.25716882
#> [229]  0.65382547  0.23576084  0.26924326  0.77880706  1.98517482  2.14583288
#> [235]  1.42899393  0.86244365  0.10970332 -0.07348761  0.45743046  0.26904638
#> [241]  0.25514810 -0.16988037 -0.08733330  0.56549418  0.85621910  1.49262364
#> [247]  1.60175870  1.20803383  0.32439052  0.70689919  0.98288473  1.00701141
#> [253]  0.80193553  0.95253606  0.95483040  1.98806815  2.33774408  2.37979741
#> [259]  2.34467210  1.17324373  1.25548460  0.87190961  1.19755472  0.89910893
#> [265]  0.89175045  0.63050645  0.34126886  1.35283379  1.63053909  1.93362387
#> [271]  1.17506690  0.50466268 -0.34150593  0.05169589  0.43854824  0.80251842
#> [277]  0.99055217  0.43654745  0.68318101  1.14398219  1.15644697  0.94868324
#> [283]  1.30142193  0.72022939  0.23880142 -0.08776532 -0.02140494  0.67921466
#> [289]  0.82981891  0.55149192  0.32233802  0.41526327  1.12487000  1.43180883
#> [295]  1.34259066  0.22433036 -0.42976092 -0.59255815 -0.62183880 -0.59476129
#> [301] -0.82864659 -0.39335029 -0.79661390 -0.66808798 -0.02323936  0.01723087
#> [307] -0.25771871 -0.59567770 -0.59306975 -1.16192465 -1.17212515 -1.22753538
#> [313] -1.32712271 -1.06130728 -1.19770773 -0.83201226  0.01458677  0.45583114
#> [319]  0.30151742 -0.25041239 -0.97481735 -0.78255510 -0.90370029 -0.39080716
#> [325] -0.11499869 -0.77645536 -0.92589864 -0.30977747  0.48593851  0.58932239
#> [331]  0.85488546  0.17495527  0.06273051  0.02920057  0.02491464 -0.08510931
#> [337] -0.66874381 -0.84786734 -0.31589962 -0.40189437  0.37998478  0.44035286
#> [343]  0.35077607 -0.01583461 -0.61983154 -0.20143080 -0.09542037  0.26977713
#> [349]  0.20164251 -0.52968538 -0.01144806 -0.17029816  0.24448537  0.83194930
#> [355]  0.57784477 -0.62183348 -0.38283718 -0.58621892 -0.47076360 -0.62673604
#> [361] -0.77680929 -0.90545762 -0.87482324 -0.59521560 -0.06184945  0.64490561
#> [367]  0.21500502 -0.58147098 -0.65863463 -0.97778440 -1.09779427 -0.48652867
#> [373] -1.18415687 -0.73308981 -0.70215338 -0.02697964  0.64175971  0.80405958
#> [379]  0.66585491  0.52978344  0.41732618 -0.14152585  0.18101026 -0.35005432
#> [385] -0.58873640 -0.86095991 -0.86938634 -0.86714565 -0.15820574  0.63483431
#> [391]  0.17881833 -0.45618679 -1.11920844 -1.06057173 -1.27380957 -1.58200382
#> [397] -1.78829904 -2.09214075 -2.32818143 -1.96555775 -1.25551555 -0.56130309
#> [403] -0.67202251 -0.72041114 -2.01156153 -2.01000002 -1.87696673 -1.92241665
#> [409] -1.96687383 -1.93193624 -2.13733680 -1.76084039 -0.63292400 -0.08478270
#> [415] -0.22743235 -0.72699603 -1.27782807 -1.41099352 -1.55359160 -1.65745311
#> [421] -1.87319170 -2.10943867 -1.89110838 -1.71900024 -1.06579646 -0.65623269
#> [427] -0.86055635 -1.11988019 -1.59639846 -1.87389769 -1.61468134 -1.53538847
#> [433] -2.11169148 -2.00490888 -2.05002438 -1.10607188 -0.52692460 -0.02892174
#> [439] -0.37800033 -0.78138675 -1.52793085 -1.69907667 -1.33880187 -1.03779062
#> [445] -2.01629915 -2.00055278 -1.85734144 -1.73022602 -1.09605242 -0.47843058
#> [451]  0.09767245 -0.71164938 -1.84110562 -2.05546959 -2.18200600 -2.34308650
#> [457] -2.72437346 -2.51929239 -2.67843906 -1.90499001 -1.00382250 -0.43154191
#> [463] -0.32908284 -0.86763174 -1.64142368 -1.94330324 -2.39850592 -2.28405068