pepacf: periodic partial autocorrelation function
In pear: Package for Periodic Autoregression Analysis
Description
Usage
Arguments
Details
Value
Side Effects
References
See Also
Examples
Description
The periodic partial autocorrelation function of a periodic
time series is calculated and plotted if the argument
plot=TRUE. When the period, p=1, this reduces to the usual partial
autocorrelation function as defined in Box and Jenkins (1976)
and is equivalent then to the Splus function acf(type="partial").
As discussed in Box and Jenkins (1976), McLeod (1994) and
Hipel and McLeod (1994) the partial autocorrelation is a
valuable tool in selecting the model order.
Usage
1
Arguments
z
a univariate time series object. Note that the period of
z is given by attr(z, "tsp")[3]. Additional information about
the time series can be provided in a title string by setting
attr(z, "title") to the desired string. This title will then
be displayed on the plot. Abbreviations for the periods may
be provided in attr(z, "abb"). For example, to use the standard
monthly abbreviations: attr(z, "abb")<-month.abb. These abbreviations
will be used to aid one in interpreting the output.
lag.max
maximum lag, if missing default is 0.25*length(z)/p, where
p = attr(z, "tsp")[3]
plot
if plot=TRUE, a plot of the periodic autocorrelations is produced.
acf.out
output from peacf function. If this is provided, execution will
proceed faster, otherwise it is calculated from scratch.
Details
For the detailed derivation of the algorithm see Sakai (1982).
Note that our partial autocorrelation is the negative of that
given in Sakai's paper.
The paper of Noakes et al. (1987) and the book of Hipel and McLeod (1994)
contain examples of this type of plot.
Value
a list containing the following components:
acf.out output list from peacf
pacf matrix of partial autocorrelations
residual.sd matrix of residual standard deviations of
the fitted models of order m, m=1,2,...,lag.max
phi matrix of autoregressive coefficients in the final
model of order lag.max for each period
aic matrix of aic values for each period and lag
bic matrix of bic values for each period and lag
maice vector of length p of the minimum aic models
mbice vector of length p of the minimum bic models
Side Effects
a plot is produced if plot=TRUE
References
Box, G.E.P. and Jenkins, G.M. (1976), "Time Series Analysis: Forecasting
and Control", Holden-Day: San Franciso.
Hipel, K.W. and McLeod, A.I. (1994) "Time Series Modelling of
Water Resources and Environmental Systems" Elsevier, Amsterdam
ISBN 0–444–89270–2. (1013 pages).
McLeod, A.I. (1994), "Diagnostic Checking of Periodic Autoregression"
Journal of Time Series Analysis, Vol. 15, No. 2, pp.221–233.
Noakes, D.J., Hipel, K.W. & McLeod, A.I. (1987).
Forecasting experiments with annual geophysical time series,
The International Journal of Forecasting, V.4, pp.103–115.
Sakai, H. (1982), "Circular lattice filtering using Pagano's Method",
IEEE Transactions, Acoust. Speech, Signal Processing, Vol. 30, pp.279–287.
See Also
peacf, peacf.plot, peplot, acf, acf.plot
Examples
1
2
Example output
$acf.out
$acf.out$means
[1] 6.802323 6.729229 6.701575 7.386260 8.465994 8.842811 8.602368 8.158366
[9] 7.758821 7.533656 7.319756 6.981944
$acf.out$standard.deviations
[1] 0.2644065 0.2569112 0.2744818 0.3637385 0.2291767 0.1758557 0.2064202
[8] 0.1995902 0.2177208 0.2800922 0.3102253 0.3031721
$acf.out$acf
lags
periods lag 1 lag 2 lag 3 lag 4 lag 5
period 1 0.7605628 0.62781334 0.4785081562 0.399444644 0.17661097
period 2 0.7838946 0.56744455 0.5839466885 0.424682988 0.37215526
period 3 0.7607552 0.65286065 0.4225520556 0.454665697 0.29893674
period 4 0.5774642 0.36144104 0.4213526098 0.285934616 0.21419304
period 5 0.3203617 0.23758964 0.1845467328 0.214926812 0.14008272
period 6 0.2411732 -0.25065689 0.0795668857 0.118474978 0.09456793
period 7 0.6095068 -0.03167037 -0.1991385635 0.041008541 0.18859296
period 8 0.7765472 0.49895714 -0.0006171332 -0.152066032 -0.01738546
period 9 0.6880798 0.46788587 0.3688460259 -0.008776051 -0.13996665
period 10 0.6922460 0.34253115 0.3126389048 0.273584893 -0.03691188
period 11 0.6811214 0.44837594 0.2809547895 0.219228567 0.16769948
period 12 0.7625285 0.53863153 0.4387789530 0.179266095 0.12457609
lags
periods lag 6 lag 7 lag 8 lag 9 lag 10 lag 11
period 1 0.13660222 0.16944162 0.06269180 0.05670052 0.12019546 0.13047584
period 2 0.21461752 0.12106743 0.13371361 0.13553223 0.15238960 0.21302776
period 3 0.29526459 0.14633782 0.09488440 0.17967449 0.15599336 0.16506708
period 4 0.13335429 0.18953781 0.09721975 0.07204521 0.14335089 0.04595685
period 5 0.05595161 0.04948945 0.05539409 0.02162948 0.12201415 0.31002916
period 6 0.08392808 0.24240643 0.24311612 0.04806462 0.11143392 0.08472541
period 7 0.16064079 0.13311051 0.21820459 0.08073770 0.03507679 0.18277433
period 8 0.22810334 0.10035739 0.10528346 0.21660781 0.12057986 0.07544637
period 9 0.08347589 0.28375896 0.13397926 0.17252609 0.23556833 0.18463735
period 10 0.01199360 0.16134329 0.25888831 0.20550367 0.16264973 0.21866971
period 11 0.10557517 0.15095225 0.24234106 0.26774014 0.23930645 0.16739872
period 12 0.10023991 0.22334476 0.08978525 0.09381379 0.14871342 0.10675805
lags
periods lag 12 lag 13 lag 14 lag 15 lag 16
period 1 0.049455319 0.089232517 0.04335322 0.077114695 0.140282696
period 2 0.327804252 0.231618700 0.19104061 0.196854567 0.218024105
period 3 0.295577067 0.280404763 0.17519688 0.088432269 0.086179599
period 4 0.004271505 0.198384054 0.11759177 0.131683658 0.114607274
period 5 0.118450203 -0.234620975 -0.02490185 -0.021567902 -0.070902345
period 6 0.118363563 -0.006027259 -0.14595339 -0.029299173 0.157225203
period 7 0.153868629 0.028870972 -0.05790376 -0.177364733 -0.085811158
period 8 0.232977189 0.171541098 0.03875821 0.002776710 -0.139645970
period 9 0.141201719 0.117100327 0.11240910 -0.005549963 0.074465684
period 10 0.218647259 0.180272071 0.15227946 0.224600757 -0.002263883
period 11 0.199158185 0.175461323 0.18999489 0.129609617 0.166094149
period 12 0.149219929 0.091770314 0.11724368 0.179356115 0.021027990
lags
periods lag 17 lag 18 lag 19 lag 20
period 1 0.09244696 0.11764486 0.104274822 0.252082540
period 2 0.17055676 0.08520229 0.157702069 0.176325018
period 3 0.14455326 0.13606276 -0.011466155 -0.017398013
period 4 0.13841436 0.07917652 0.135022580 -0.026378876
period 5 -0.10751993 -0.08619534 -0.205238460 -0.165789384
period 6 0.01085247 -0.04211419 0.019922399 0.028277957
period 7 0.09862085 0.01710679 -0.002822277 -0.005141135
period 8 -0.08697952 0.09437301 0.057698893 -0.035828107
period 9 -0.01157336 0.06295157 0.251669518 0.228956761
period 10 0.05798679 0.21507431 0.289823146 0.377964423
period 11 0.16270087 0.20784731 0.179953724 0.250192058
period 12 0.02730439 0.10004517 0.244679423 0.055016152
$acf.out$benchmark.sd
[1] 0.1125088
$acf.out$sub.lengths
[1] 78 78 79 79 79 79 79 79 79 79 79 79
$acf.out$period
[1] 12
$acf.out$title
[1] "Fraser River at Hope, mean monthly flow (cms), 1912.3-1991.12"
$acf.out$periodicity.test
$acf.out$periodicity.test$Q1
[1] 413.8426
$acf.out$periodicity.test$Q1.sl
[1] 0
$acf.out$portmanteau.test
$acf.out$portmanteau.test$QM
[,1] [,2] [,3] [,4]
[1,] 110.01158 115.46054 118.26778 127.59270
[2,] 125.50857 135.00505 157.56146 168.75634
[3,] 117.36989 131.30939 150.03778 153.86909
[4,] 60.90238 68.01549 73.82885 78.48846
[5,] 20.47729 22.39789 35.73090 43.31427
[6,] 11.87399 23.04682 26.54660 28.77465
[7,] 35.50314 43.38915 50.85643 52.25002
[8,] 69.15743 79.86792 87.25158 90.50027
[9,] 66.99880 82.15009 88.62415 98.65782
[10,] 60.86842 73.68416 89.92403 112.04591
[11,] 64.78689 82.29441 94.48628 109.86784
[12,] 87.82874 95.64255 102.71727 108.64554
$acf.out$portmanteau.test$QM.df
[,1] [,2] [,3] [,4]
[1,] 5 10 15 20
[2,] 5 10 15 20
[3,] 5 10 15 20
[4,] 5 10 15 20
[5,] 5 10 15 20
[6,] 5 10 15 20
[7,] 5 10 15 20
[8,] 5 10 15 20
[9,] 5 10 15 20
[10,] 5 10 15 20
[11,] 5 10 15 20
[12,] 5 10 15 20
attr(,"type")
[1] "acf"
$pacf
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.7605628 0.11394630 0.08036702 0.056158942 -0.02436818 0.032424679
[2,] 0.7838946 -0.07133404 0.27948636 -0.021247269 0.10958794 0.004781437
[3,] 0.7607552 0.14022568 -0.13315920 0.061227364 -0.08065277 0.080544563
[4,] 0.5774642 -0.14695213 0.19627865 0.011156909 -0.10627583 -0.017377189
[5,] 0.3203617 0.06800249 0.04152799 0.052265652 -0.01991474 -0.101059549
[6,] 0.2411732 -0.35669286 0.27823338 0.026359184 0.06820623 0.036784104
[7,] 0.6095068 -0.23222233 0.03792279 0.028399006 0.24517730 0.070738314
[8,] 0.7765472 0.05134407 0.02247208 0.004211267 -0.11427340 0.317866302
[9,] 0.6880798 -0.14532284 0.11028352 -0.058607095 -0.03123170 0.219152565
[10,] 0.6922460 -0.25548745 0.23970725 0.037725352 -0.04590016 0.223689177
[11,] 0.6811214 -0.04377081 0.13187985 -0.101777713 -0.03467196 0.201880032
[12,] 0.7625285 0.04065401 0.17805634 -0.254824594 0.04354441 -0.025707433
[,7] [,8] [,9] [,10] [,11]
[1,] 0.11972231 -0.20537472 0.107710870 0.04223548 -0.052539230
[2,] -0.11173972 -0.03394285 0.226643980 0.07005224 0.062536466
[3,] -0.09684996 0.06220516 0.133932906 0.10059050 0.111023159
[4,] 0.06496209 0.02680676 0.003610312 0.02872669 -0.077278963
[5,] 0.01177495 -0.03872308 0.028208530 0.18181171 0.289691862
[6,] 0.29923771 0.12088234 -0.123178754 0.18073730 -0.175606095
[7,] -0.02052832 -0.04947714 -0.244471962 0.11090044 0.146012703
[8,] -0.22761489 0.03450800 0.068703724 0.03286754 0.040683283
[9,] 0.10502882 -0.01500722 0.123917973 -0.05036435 0.044343965
[10,] -0.02996325 0.06979801 0.062421181 -0.11674434 0.044980035
[11,] 0.10759542 0.12454993 -0.055427348 0.03567654 0.001596226
[12,] 0.25559544 -0.12990716 -0.211864375 0.01988046 -0.092620196
[,12] [,13] [,14] [,15] [,16]
[1,] -0.114226548 0.009312263 -0.158978664 -0.0215874578 0.04388896
[2,] 0.345705871 0.039090202 -0.027503968 0.1418962828 0.11777371
[3,] 0.146638772 -0.103495975 -0.077284801 -0.1198539901 -0.11064221
[4,] -0.039060421 0.161314356 -0.099284334 0.2210078904 0.04801038
[5,] -0.006741508 -0.231475022 0.006925223 -0.0002148399 -0.10682016
[6,] -0.029372176 -0.047942536 -0.206453710 0.1217146863 0.28230509
[7,] 0.131770909 -0.065107952 -0.067382186 -0.1865451435 0.04356740
[8,] 0.109459085 -0.029429078 -0.020510811 0.0911316558 -0.04983819
[9,] 0.025423008 -0.171328202 0.158483439 -0.0638270944 0.06234286
[10,] 0.079601612 0.046189957 0.139738812 0.2091884742 -0.18512371
[11,] 0.009642685 0.004636237 0.053615427 -0.1009629393 -0.02471343
[12,] 0.156338815 -0.095694310 0.077986486 0.0818659785 -0.18640910
[,17] [,18] [,19] [,20]
[1,] 0.168735845 0.17485129 -0.02349203 0.10680345
[2,] -0.152383067 -0.10341523 0.08498022 0.09287694
[3,] 0.065566342 0.02116472 -0.23018673 -0.18536488
[4,] 0.166060773 -0.11276839 0.10732484 -0.12534145
[5,] -0.114350039 -0.10049617 -0.30990940 0.02663347
[6,] -0.086077257 0.04415289 0.07955377 0.04475262
[7,] -0.006033092 0.06277297 -0.04330494 -0.07967384
[8,] -0.048357739 -0.09464232 0.04474486 -0.21045467
[9,] 0.035403540 0.09572668 0.18265452 0.10921502
[10,] 0.175661951 0.37141764 0.11106115 -0.02393132
[11,] 0.246353164 0.15188595 0.04319799 -0.06561984
[12,] -0.132112869 0.03813748 0.09329108 -0.08430969
$residual.sd
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.2644065 0.1716697 0.1705516 0.1699999 0.1697317 0.1696813 0.1695920
[2,] 0.2569112 0.1595146 0.1591082 0.1527677 0.1527332 0.1518133 0.1518116
[3,] 0.2744818 0.1781494 0.1763892 0.1748184 0.1744904 0.1739219 0.1733569
[4,] 0.3637385 0.2969620 0.2937380 0.2880243 0.2880064 0.2863753 0.2863321
[5,] 0.2291767 0.2170981 0.2165955 0.2164087 0.2161129 0.2160700 0.2149638
[6,] 0.1758557 0.1706648 0.1594388 0.1531431 0.1530899 0.1527334 0.1526300
[7,] 0.2064202 0.1636460 0.1591723 0.1590578 0.1589937 0.1541409 0.1537548
[8,] 0.1995902 0.1257536 0.1255877 0.1255560 0.1255549 0.1247324 0.1182632
[9,] 0.2177208 0.1579859 0.1563087 0.1553553 0.1550882 0.1550126 0.1512443
[10,] 0.2800922 0.2021318 0.1954235 0.1897260 0.1895910 0.1893911 0.1845921
[11,] 0.3102253 0.2271378 0.2269201 0.2249382 0.2237701 0.2236355 0.2190309
[12,] 0.3031721 0.1961386 0.1959764 0.1928448 0.1864785 0.1863016 0.1862400
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 0.1683722 0.1647831 0.1638244 0.1636783 0.1634522 0.1623824 0.1623753
[2,] 0.1508609 0.1507740 0.1468505 0.1464897 0.1462030 0.1371885 0.1370837
[3,] 0.1725419 0.1722078 0.1706562 0.1697907 0.1687410 0.1669169 0.1660205
[4,] 0.2857272 0.2856246 0.2856227 0.2855048 0.2846510 0.2844338 0.2807086
[5,] 0.2149489 0.2147877 0.2147022 0.2111239 0.2020709 0.2020663 0.1965783
[6,] 0.1456363 0.1445683 0.1434674 0.1411047 0.1389120 0.1388520 0.1386924
[7,] 0.1537224 0.1535341 0.1488753 0.1479570 0.1463713 0.1450950 0.1447871
[8,] 0.1151590 0.1150904 0.1148185 0.1147564 0.1146614 0.1139724 0.1139231
[9,] 0.1504078 0.1503909 0.1492317 0.1490423 0.1488957 0.1488476 0.1466467
[10,] 0.1845092 0.1840592 0.1837003 0.1824441 0.1822595 0.1816811 0.1814872
[11,] 0.2177594 0.2160638 0.2157317 0.2155943 0.2155940 0.2155840 0.2155817
[12,] 0.1800538 0.1785281 0.1744753 0.1744409 0.1736910 0.1715552 0.1707679
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] 0.1603102 0.1602729 0.1601184 0.1578226 0.1553913 0.1553484 0.1544598
[2,] 0.1370318 0.1356453 0.1347012 0.1331281 0.1324143 0.1319353 0.1313651
[3,] 0.1655240 0.1643308 0.1633219 0.1629704 0.1629339 0.1585586 0.1558107
[4,] 0.2793216 0.2724146 0.2721004 0.2683225 0.2666109 0.2650710 0.2629805
[5,] 0.1965736 0.1965736 0.1954489 0.1941668 0.1931839 0.1836727 0.1836075
[6,] 0.1357044 0.1346955 0.1292167 0.1287371 0.1286116 0.1282040 0.1280755
[7,] 0.1444580 0.1419223 0.1417875 0.1417849 0.1415053 0.1413726 0.1409231
[8,] 0.1138991 0.1134252 0.1132842 0.1131517 0.1126438 0.1125310 0.1100107
[9,] 0.1447934 0.1444981 0.1442171 0.1441266 0.1434648 0.1410513 0.1402075
[10,] 0.1797065 0.1757306 0.1726931 0.1700078 0.1578465 0.1568700 0.1568250
[11,] 0.2152716 0.2141716 0.2141062 0.2075075 0.2051000 0.2049085 0.2044669
[12,] 0.1702478 0.1696764 0.1667023 0.1652411 0.1651209 0.1644008 0.1638155
$phi
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.6589492 0.041082637 -0.08977826 0.12565403 -0.10277954 -0.102763468
[2,] 0.8700298 -0.301093399 0.20157789 -0.03079655 0.03393160 0.065703678
[3,] 0.6618710 0.296971024 -0.25808640 0.11528032 -0.07865125 0.136771321
[4,] 0.8973904 -0.400732861 0.34100591 0.18605764 -0.26247150 0.004142777
[5,] 0.1549853 0.001942657 0.20813377 -0.09963635 0.13985599 -0.105093148
[6,] 0.2770527 -0.246038698 0.22451675 -0.23322979 0.15748177 -0.128203864
[7,] 0.7864776 -0.237065015 0.01235239 -0.14879102 0.22981196 0.073481194
[8,] 0.6670789 0.119126606 -0.01232244 0.07829449 -0.26767349 0.375857046
[9,] 0.9123711 -0.287911056 0.10652915 -0.03023542 -0.08993360 0.100222406
[10,] 1.1128403 -0.677791408 0.33951405 0.27197651 -0.06824997 0.158925955
[11,] 0.8789312 -0.322224260 0.47434117 -0.06176384 -0.16674364 0.114327571
[12,] 0.7161806 -0.022203567 0.38491587 -0.40752050 0.23592099 -0.183254331
[,7] [,8] [,9] [,10] [,11]
[1,] 0.41223045 -0.281475832 0.075295918 0.052775980 0.093885847
[2,] -0.02403290 -0.150209166 0.126514333 0.040870751 -0.221951330
[3,] -0.06389537 0.059240977 0.058567200 0.111787889 -0.004527702
[4,] -0.11991548 0.447798726 -0.272157740 -0.034793875 -0.043184470
[5,] 0.02438340 -0.083512145 -0.248288098 0.001657224 0.531696759
[6,] 0.15505353 0.254789381 -0.429838277 0.345595587 -0.180467909
[7,] 0.04386750 0.048949462 -0.208703034 -0.037122445 0.078404957
[8,] -0.16581268 -0.006679296 0.004783692 0.022239096 -0.004865177
[9,] 0.03283126 -0.016228703 0.057910805 -0.011939482 -0.073043914
[10,] -0.18136690 -0.003376425 0.231934113 -0.132002763 -0.078057042
[11,] 0.02943037 0.114889880 -0.120501648 0.081932483 -0.105471235
[12,] 0.28094506 0.014338909 -0.164159584 0.089705473 -0.230064377
[,12] [,13] [,14] [,15] [,16]
[1,] -0.096202219 0.081918305 -0.147399718 -0.03276490 -0.04080620
[2,] 0.299968480 0.035270350 -0.087899574 0.06970029 0.10257729
[3,] 0.208768945 -0.074517462 -0.012265190 -0.07296191 -0.07833666
[4,] -0.217336111 0.331934747 -0.557544774 0.40650388 -0.17355963
[5,] 0.022668978 -0.124821036 -0.032526777 0.12974870 0.01190982
[6,] -0.102042285 0.045385447 -0.099158723 -0.13778805 0.31528167
[7,] 0.136939410 -0.083801071 -0.008961802 -0.11133849 0.04609912
[8,] 0.113656691 0.002241603 -0.052129274 0.05759674 -0.01127975
[9,] 0.225526303 -0.352039303 0.180044656 -0.09116181 0.02571675
[10,] -0.002541963 0.136873641 -0.144552766 0.45285473 -0.21635699
[11,] 0.070471051 -0.089367450 0.268642667 -0.22507271 -0.14274563
[12,] 0.112930513 -0.053384602 0.014021250 0.22097278 -0.05678251
[,17] [,18] [,19] [,20]
[1,] -0.028979821 0.28967523 -0.08276141 0.08954410
[2,] -0.008722016 -0.21444746 0.05345975 0.09956896
[3,] 0.026720990 0.20297024 -0.06105521 -0.26524599
[4,] 0.410928012 -0.34009349 0.43449414 -0.28297603
[5,] -0.042674433 0.05999727 -0.33315523 0.03340474
[6,] -0.096708046 -0.01134837 0.04418287 0.03154346
[7,] -0.040293802 0.07779931 0.01369336 -0.06643308
[8,] 0.021053308 -0.11102518 0.15793778 -0.12743820
[9,] -0.017134201 -0.06903196 0.11162207 0.10550274
[10,] 0.014879420 0.17715996 0.11215068 -0.02874923
[11,] 0.342763103 0.15047913 0.06496907 -0.06990429
[12,] -0.202188163 -0.02555086 0.11074691 -0.05261904
$aic
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -207.5218 -272.9005 -271.9199 -270.4253 -268.6717 -266.7180 -264.8001
[2,] -212.0079 -284.3567 -282.7546 -287.0985 -285.1337 -284.0762 -282.0779
[3,] -204.2735 -270.5710 -270.1399 -269.5532 -267.8499 -266.3655 -264.8797
[4,] -159.7886 -189.8359 -189.5606 -190.6643 -188.6741 -187.5714 -185.5953
[5,] -232.7754 -239.3302 -237.6963 -235.8327 -234.0488 -232.0801 -230.8911
[6,] -274.6184 -277.3525 -286.1030 -290.4684 -288.5233 -286.8917 -284.9987
[7,] -249.2990 -283.9879 -286.3673 -284.4810 -282.5448 -285.4423 -283.8386
[8,] -254.6153 -325.6021 -323.8106 -321.8505 -319.8519 -318.8903 -325.3051
[9,] -240.8776 -289.5495 -289.2357 -288.2024 -286.4743 -284.5513 -286.4397
[10,] -201.0765 -250.6160 -253.9486 -256.6235 -254.7361 -252.9027 -254.9579
[11,] -184.9321 -232.1873 -230.3388 -229.7249 -228.5475 -226.6425 -227.9297
[12,] -188.5658 -255.3715 -253.5022 -254.0474 -257.3514 -255.5014 -253.5536
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] -263.9262 -265.2875 -264.1977 -262.3370 -260.5526 -259.5770 -257.5838
[2,] -281.0580 -279.1479 -281.2611 -279.6448 -277.9505 -285.8783 -283.9976
[3,] -263.6242 -261.9305 -261.3604 -260.1639 -259.1437 -258.8609 -257.7117
[4,] -183.9294 -181.9862 -179.9872 -178.0524 -176.5256 -174.6463 -174.7292
[5,] -228.9021 -227.0206 -225.0835 -225.7390 -230.6636 -228.6672 -231.0177
[6,] -290.4096 -289.5725 -288.7803 -289.4040 -289.8785 -287.9467 -286.1285
[7,] -281.8719 -280.0656 -282.9341 -281.9117 -281.6142 -280.9980 -279.3336
[8,] -327.5078 -325.6019 -323.9757 -322.0611 -320.1919 -319.1442 -317.2126
[9,] -285.3160 -283.3338 -282.5563 -280.7569 -278.9124 -276.9635 -277.3171
[10,] -253.0289 -251.4147 -249.7231 -248.8072 -246.9672 -245.4694 -243.6381
[11,] -226.8496 -226.0847 -224.3278 -222.4284 -220.4286 -218.4359 -216.4376
[12,] -256.8909 -256.2355 -257.8636 -255.8948 -254.5754 -254.5303 -253.2571
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] -257.5805 -255.6169 -253.7673 -254.0203 -254.4422 -252.4853 -251.3801
[2,] -282.0566 -281.6431 -280.7326 -280.5652 -279.4038 -277.9692 -276.6449
[3,] -256.1850 -255.3280 -254.3011 -252.6415 -250.6769 -252.9777 -253.7399
[4,] -173.5118 -175.4680 -173.6503 -173.8594 -172.8705 -171.7857 -171.0367
[5,] -229.0215 -227.0215 -225.9281 -224.9679 -223.7698 -229.7468 -227.8029
[6,] -287.5696 -286.7487 -291.3098 -289.8973 -288.0514 -286.5530 -284.7114
[7,] -277.6931 -278.4912 -276.6413 -274.6441 -272.9561 -271.1043 -269.6074
[8,] -315.2459 -313.9047 -312.1012 -310.2861 -308.9969 -307.1553 -308.7341
[9,] -277.3267 -275.6492 -273.9569 -272.0559 -270.7832 -271.4638 -270.4118
[10,] -243.1960 -244.7309 -245.4858 -245.9619 -255.6889 -254.6694 -252.7147
[11,] -214.6650 -213.4745 -211.5227 -214.4689 -214.3127 -212.4603 -210.8012
[12,] -251.7390 -250.2702 -251.0642 -250.4552 -248.5702 -247.2608 -245.8243
$bic
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -207.5218 -270.5438 -267.2065 -263.3552 -259.2449 -254.9345 -250.6598
[2,] -212.0079 -282.0000 -278.0412 -280.0284 -275.7069 -272.2926 -267.9377
[3,] -204.2735 -268.2016 -265.4010 -262.4449 -258.3721 -254.5183 -250.6630
[4,] -159.7886 -187.4664 -184.8217 -183.5559 -179.1963 -175.7242 -171.3786
[5,] -232.7754 -236.9607 -232.9574 -228.7244 -224.5710 -220.2329 -216.6744
[6,] -274.6184 -274.9830 -281.3641 -283.3601 -279.0455 -275.0445 -270.7820
[7,] -249.2990 -281.6185 -281.6284 -277.3727 -273.0670 -273.5951 -269.6219
[8,] -254.6153 -323.2326 -319.0717 -314.7422 -310.3741 -307.0431 -311.0884
[9,] -240.8776 -287.1800 -284.4968 -281.0941 -276.9965 -272.7041 -272.2230
[10,] -201.0765 -248.2465 -249.2097 -249.5152 -245.2583 -241.0554 -240.7412
[11,] -184.9321 -229.8179 -225.5999 -222.6166 -219.0697 -214.7953 -213.7130
[12,] -188.5658 -253.0021 -248.7633 -246.9390 -247.8736 -243.6541 -239.3369
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] -247.4292 -246.4339 -242.9874 -238.7699 -234.6288 -231.2965 -226.9466
[2,] -264.5610 -260.2942 -260.0507 -256.0777 -252.0267 -257.5978 -253.3603
[3,] -247.0380 -242.9749 -240.0354 -236.4694 -233.0798 -230.4276 -226.9089
[4,] -167.3432 -163.0306 -158.6622 -154.3579 -150.4617 -146.2129 -143.9264
[5,] -212.3159 -208.0650 -203.7585 -202.0445 -204.5997 -200.2338 -200.2149
[6,] -273.8234 -270.6169 -267.4553 -265.7095 -263.8146 -259.5133 -255.3257
[7,] -265.2858 -261.1100 -261.6091 -258.2172 -255.5503 -252.5646 -248.5307
[8,] -310.9216 -306.6463 -302.6507 -298.3666 -294.1280 -290.7108 -286.4098
[9,] -268.7298 -264.3782 -261.2313 -257.0624 -252.8485 -248.5301 -246.5143
[10,] -236.4427 -232.4591 -228.3981 -225.1127 -220.9033 -217.0360 -212.8353
[11,] -210.2634 -207.1291 -203.0027 -198.7339 -194.3647 -190.0025 -185.6348
[12,] -240.3048 -237.2799 -236.5385 -232.2003 -228.5115 -226.0969 -222.4542
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] -224.5866 -220.2662 -216.0599 -213.9562 -212.0214 -207.7078 -204.2459
[2,] -249.0627 -246.2925 -243.0252 -240.5011 -236.9831 -233.1917 -229.5107
[3,] -223.0127 -219.7863 -216.3899 -212.3608 -208.0268 -207.9582 -206.3509
[4,] -140.3396 -139.9263 -135.7391 -133.5788 -130.2204 -126.7662 -123.6477
[5,] -195.8492 -191.4798 -188.0169 -184.6873 -181.1198 -184.7273 -180.4139
[6,] -254.3973 -251.2070 -253.3986 -249.6167 -245.4014 -241.5335 -237.3224
[7,] -244.5208 -242.9495 -238.7301 -234.3635 -230.3060 -226.0848 -222.2185
[8,] -282.0736 -278.3630 -274.1900 -270.0055 -266.3469 -262.1357 -261.3452
[9,] -244.1544 -240.1075 -236.0457 -231.7753 -228.1331 -226.4443 -223.0228
[10,] -210.0237 -209.1892 -207.5746 -205.6813 -213.0389 -209.6499 -205.3257
[11,] -181.4928 -177.9327 -173.6116 -174.1883 -171.6627 -167.4408 -163.4122
[12,] -218.5667 -214.7285 -213.1530 -210.1746 -205.9201 -202.2413 -198.4354
$maice
[1] 1 3 1 3 1 16 2 7 1 3 1 9
$mbice
[1] 1 1 1 1 1 3 2 1 1 3 1 1
attr(,"type")
[1] "pacf"
pear documentation built on May 2, 2019, 9:16 a.m.
R Package Documentation
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Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Description Usage Arguments Details Value Side Effects References See Also Examples
Description
The periodic partial autocorrelation function of a periodic time series is calculated and plotted if the argument plot=TRUE. When the period, p=1, this reduces to the usual partial autocorrelation function as defined in Box and Jenkins (1976) and is equivalent then to the Splus function acf(type="partial"). As discussed in Box and Jenkins (1976), McLeod (1994) and Hipel and McLeod (1994) the partial autocorrelation is a valuable tool in selecting the model order.
Usage
1 |
Arguments
z |
a univariate time series object. Note that the period of z is given by attr(z, "tsp")[3]. Additional information about the time series can be provided in a title string by setting attr(z, "title") to the desired string. This title will then be displayed on the plot. Abbreviations for the periods may be provided in attr(z, "abb"). For example, to use the standard monthly abbreviations: attr(z, "abb")<-month.abb. These abbreviations will be used to aid one in interpreting the output. |
lag.max |
maximum lag, if missing default is 0.25*length(z)/p, where p = attr(z, "tsp")[3] |
plot |
if plot=TRUE, a plot of the periodic autocorrelations is produced. |
acf.out |
output from peacf function. If this is provided, execution will proceed faster, otherwise it is calculated from scratch. |
Details
For the detailed derivation of the algorithm see Sakai (1982). Note that our partial autocorrelation is the negative of that given in Sakai's paper.
The paper of Noakes et al. (1987) and the book of Hipel and McLeod (1994) contain examples of this type of plot.
Value
a list containing the following components: acf.out output list from peacf pacf matrix of partial autocorrelations residual.sd matrix of residual standard deviations of the fitted models of order m, m=1,2,...,lag.max phi matrix of autoregressive coefficients in the final model of order lag.max for each period aic matrix of aic values for each period and lag bic matrix of bic values for each period and lag maice vector of length p of the minimum aic models mbice vector of length p of the minimum bic models
Side Effects
a plot is produced if plot=TRUE
References
Box, G.E.P. and Jenkins, G.M. (1976), "Time Series Analysis: Forecasting and Control", Holden-Day: San Franciso.
Hipel, K.W. and McLeod, A.I. (1994) "Time Series Modelling of Water Resources and Environmental Systems" Elsevier, Amsterdam ISBN 0–444–89270–2. (1013 pages).
McLeod, A.I. (1994), "Diagnostic Checking of Periodic Autoregression" Journal of Time Series Analysis, Vol. 15, No. 2, pp.221–233.
Noakes, D.J., Hipel, K.W. & McLeod, A.I. (1987). Forecasting experiments with annual geophysical time series, The International Journal of Forecasting, V.4, pp.103–115.
Sakai, H. (1982), "Circular lattice filtering using Pagano's Method", IEEE Transactions, Acoust. Speech, Signal Processing, Vol. 30, pp.279–287.
See Also
peacf, peacf.plot, peplot, acf, acf.plot
Examples
1 2 |
Example output
$acf.out
$acf.out$means
[1] 6.802323 6.729229 6.701575 7.386260 8.465994 8.842811 8.602368 8.158366
[9] 7.758821 7.533656 7.319756 6.981944
$acf.out$standard.deviations
[1] 0.2644065 0.2569112 0.2744818 0.3637385 0.2291767 0.1758557 0.2064202
[8] 0.1995902 0.2177208 0.2800922 0.3102253 0.3031721
$acf.out$acf
lags
periods lag 1 lag 2 lag 3 lag 4 lag 5
period 1 0.7605628 0.62781334 0.4785081562 0.399444644 0.17661097
period 2 0.7838946 0.56744455 0.5839466885 0.424682988 0.37215526
period 3 0.7607552 0.65286065 0.4225520556 0.454665697 0.29893674
period 4 0.5774642 0.36144104 0.4213526098 0.285934616 0.21419304
period 5 0.3203617 0.23758964 0.1845467328 0.214926812 0.14008272
period 6 0.2411732 -0.25065689 0.0795668857 0.118474978 0.09456793
period 7 0.6095068 -0.03167037 -0.1991385635 0.041008541 0.18859296
period 8 0.7765472 0.49895714 -0.0006171332 -0.152066032 -0.01738546
period 9 0.6880798 0.46788587 0.3688460259 -0.008776051 -0.13996665
period 10 0.6922460 0.34253115 0.3126389048 0.273584893 -0.03691188
period 11 0.6811214 0.44837594 0.2809547895 0.219228567 0.16769948
period 12 0.7625285 0.53863153 0.4387789530 0.179266095 0.12457609
lags
periods lag 6 lag 7 lag 8 lag 9 lag 10 lag 11
period 1 0.13660222 0.16944162 0.06269180 0.05670052 0.12019546 0.13047584
period 2 0.21461752 0.12106743 0.13371361 0.13553223 0.15238960 0.21302776
period 3 0.29526459 0.14633782 0.09488440 0.17967449 0.15599336 0.16506708
period 4 0.13335429 0.18953781 0.09721975 0.07204521 0.14335089 0.04595685
period 5 0.05595161 0.04948945 0.05539409 0.02162948 0.12201415 0.31002916
period 6 0.08392808 0.24240643 0.24311612 0.04806462 0.11143392 0.08472541
period 7 0.16064079 0.13311051 0.21820459 0.08073770 0.03507679 0.18277433
period 8 0.22810334 0.10035739 0.10528346 0.21660781 0.12057986 0.07544637
period 9 0.08347589 0.28375896 0.13397926 0.17252609 0.23556833 0.18463735
period 10 0.01199360 0.16134329 0.25888831 0.20550367 0.16264973 0.21866971
period 11 0.10557517 0.15095225 0.24234106 0.26774014 0.23930645 0.16739872
period 12 0.10023991 0.22334476 0.08978525 0.09381379 0.14871342 0.10675805
lags
periods lag 12 lag 13 lag 14 lag 15 lag 16
period 1 0.049455319 0.089232517 0.04335322 0.077114695 0.140282696
period 2 0.327804252 0.231618700 0.19104061 0.196854567 0.218024105
period 3 0.295577067 0.280404763 0.17519688 0.088432269 0.086179599
period 4 0.004271505 0.198384054 0.11759177 0.131683658 0.114607274
period 5 0.118450203 -0.234620975 -0.02490185 -0.021567902 -0.070902345
period 6 0.118363563 -0.006027259 -0.14595339 -0.029299173 0.157225203
period 7 0.153868629 0.028870972 -0.05790376 -0.177364733 -0.085811158
period 8 0.232977189 0.171541098 0.03875821 0.002776710 -0.139645970
period 9 0.141201719 0.117100327 0.11240910 -0.005549963 0.074465684
period 10 0.218647259 0.180272071 0.15227946 0.224600757 -0.002263883
period 11 0.199158185 0.175461323 0.18999489 0.129609617 0.166094149
period 12 0.149219929 0.091770314 0.11724368 0.179356115 0.021027990
lags
periods lag 17 lag 18 lag 19 lag 20
period 1 0.09244696 0.11764486 0.104274822 0.252082540
period 2 0.17055676 0.08520229 0.157702069 0.176325018
period 3 0.14455326 0.13606276 -0.011466155 -0.017398013
period 4 0.13841436 0.07917652 0.135022580 -0.026378876
period 5 -0.10751993 -0.08619534 -0.205238460 -0.165789384
period 6 0.01085247 -0.04211419 0.019922399 0.028277957
period 7 0.09862085 0.01710679 -0.002822277 -0.005141135
period 8 -0.08697952 0.09437301 0.057698893 -0.035828107
period 9 -0.01157336 0.06295157 0.251669518 0.228956761
period 10 0.05798679 0.21507431 0.289823146 0.377964423
period 11 0.16270087 0.20784731 0.179953724 0.250192058
period 12 0.02730439 0.10004517 0.244679423 0.055016152
$acf.out$benchmark.sd
[1] 0.1125088
$acf.out$sub.lengths
[1] 78 78 79 79 79 79 79 79 79 79 79 79
$acf.out$period
[1] 12
$acf.out$title
[1] "Fraser River at Hope, mean monthly flow (cms), 1912.3-1991.12"
$acf.out$periodicity.test
$acf.out$periodicity.test$Q1
[1] 413.8426
$acf.out$periodicity.test$Q1.sl
[1] 0
$acf.out$portmanteau.test
$acf.out$portmanteau.test$QM
[,1] [,2] [,3] [,4]
[1,] 110.01158 115.46054 118.26778 127.59270
[2,] 125.50857 135.00505 157.56146 168.75634
[3,] 117.36989 131.30939 150.03778 153.86909
[4,] 60.90238 68.01549 73.82885 78.48846
[5,] 20.47729 22.39789 35.73090 43.31427
[6,] 11.87399 23.04682 26.54660 28.77465
[7,] 35.50314 43.38915 50.85643 52.25002
[8,] 69.15743 79.86792 87.25158 90.50027
[9,] 66.99880 82.15009 88.62415 98.65782
[10,] 60.86842 73.68416 89.92403 112.04591
[11,] 64.78689 82.29441 94.48628 109.86784
[12,] 87.82874 95.64255 102.71727 108.64554
$acf.out$portmanteau.test$QM.df
[,1] [,2] [,3] [,4]
[1,] 5 10 15 20
[2,] 5 10 15 20
[3,] 5 10 15 20
[4,] 5 10 15 20
[5,] 5 10 15 20
[6,] 5 10 15 20
[7,] 5 10 15 20
[8,] 5 10 15 20
[9,] 5 10 15 20
[10,] 5 10 15 20
[11,] 5 10 15 20
[12,] 5 10 15 20
attr(,"type")
[1] "acf"
$pacf
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.7605628 0.11394630 0.08036702 0.056158942 -0.02436818 0.032424679
[2,] 0.7838946 -0.07133404 0.27948636 -0.021247269 0.10958794 0.004781437
[3,] 0.7607552 0.14022568 -0.13315920 0.061227364 -0.08065277 0.080544563
[4,] 0.5774642 -0.14695213 0.19627865 0.011156909 -0.10627583 -0.017377189
[5,] 0.3203617 0.06800249 0.04152799 0.052265652 -0.01991474 -0.101059549
[6,] 0.2411732 -0.35669286 0.27823338 0.026359184 0.06820623 0.036784104
[7,] 0.6095068 -0.23222233 0.03792279 0.028399006 0.24517730 0.070738314
[8,] 0.7765472 0.05134407 0.02247208 0.004211267 -0.11427340 0.317866302
[9,] 0.6880798 -0.14532284 0.11028352 -0.058607095 -0.03123170 0.219152565
[10,] 0.6922460 -0.25548745 0.23970725 0.037725352 -0.04590016 0.223689177
[11,] 0.6811214 -0.04377081 0.13187985 -0.101777713 -0.03467196 0.201880032
[12,] 0.7625285 0.04065401 0.17805634 -0.254824594 0.04354441 -0.025707433
[,7] [,8] [,9] [,10] [,11]
[1,] 0.11972231 -0.20537472 0.107710870 0.04223548 -0.052539230
[2,] -0.11173972 -0.03394285 0.226643980 0.07005224 0.062536466
[3,] -0.09684996 0.06220516 0.133932906 0.10059050 0.111023159
[4,] 0.06496209 0.02680676 0.003610312 0.02872669 -0.077278963
[5,] 0.01177495 -0.03872308 0.028208530 0.18181171 0.289691862
[6,] 0.29923771 0.12088234 -0.123178754 0.18073730 -0.175606095
[7,] -0.02052832 -0.04947714 -0.244471962 0.11090044 0.146012703
[8,] -0.22761489 0.03450800 0.068703724 0.03286754 0.040683283
[9,] 0.10502882 -0.01500722 0.123917973 -0.05036435 0.044343965
[10,] -0.02996325 0.06979801 0.062421181 -0.11674434 0.044980035
[11,] 0.10759542 0.12454993 -0.055427348 0.03567654 0.001596226
[12,] 0.25559544 -0.12990716 -0.211864375 0.01988046 -0.092620196
[,12] [,13] [,14] [,15] [,16]
[1,] -0.114226548 0.009312263 -0.158978664 -0.0215874578 0.04388896
[2,] 0.345705871 0.039090202 -0.027503968 0.1418962828 0.11777371
[3,] 0.146638772 -0.103495975 -0.077284801 -0.1198539901 -0.11064221
[4,] -0.039060421 0.161314356 -0.099284334 0.2210078904 0.04801038
[5,] -0.006741508 -0.231475022 0.006925223 -0.0002148399 -0.10682016
[6,] -0.029372176 -0.047942536 -0.206453710 0.1217146863 0.28230509
[7,] 0.131770909 -0.065107952 -0.067382186 -0.1865451435 0.04356740
[8,] 0.109459085 -0.029429078 -0.020510811 0.0911316558 -0.04983819
[9,] 0.025423008 -0.171328202 0.158483439 -0.0638270944 0.06234286
[10,] 0.079601612 0.046189957 0.139738812 0.2091884742 -0.18512371
[11,] 0.009642685 0.004636237 0.053615427 -0.1009629393 -0.02471343
[12,] 0.156338815 -0.095694310 0.077986486 0.0818659785 -0.18640910
[,17] [,18] [,19] [,20]
[1,] 0.168735845 0.17485129 -0.02349203 0.10680345
[2,] -0.152383067 -0.10341523 0.08498022 0.09287694
[3,] 0.065566342 0.02116472 -0.23018673 -0.18536488
[4,] 0.166060773 -0.11276839 0.10732484 -0.12534145
[5,] -0.114350039 -0.10049617 -0.30990940 0.02663347
[6,] -0.086077257 0.04415289 0.07955377 0.04475262
[7,] -0.006033092 0.06277297 -0.04330494 -0.07967384
[8,] -0.048357739 -0.09464232 0.04474486 -0.21045467
[9,] 0.035403540 0.09572668 0.18265452 0.10921502
[10,] 0.175661951 0.37141764 0.11106115 -0.02393132
[11,] 0.246353164 0.15188595 0.04319799 -0.06561984
[12,] -0.132112869 0.03813748 0.09329108 -0.08430969
$residual.sd
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0.2644065 0.1716697 0.1705516 0.1699999 0.1697317 0.1696813 0.1695920
[2,] 0.2569112 0.1595146 0.1591082 0.1527677 0.1527332 0.1518133 0.1518116
[3,] 0.2744818 0.1781494 0.1763892 0.1748184 0.1744904 0.1739219 0.1733569
[4,] 0.3637385 0.2969620 0.2937380 0.2880243 0.2880064 0.2863753 0.2863321
[5,] 0.2291767 0.2170981 0.2165955 0.2164087 0.2161129 0.2160700 0.2149638
[6,] 0.1758557 0.1706648 0.1594388 0.1531431 0.1530899 0.1527334 0.1526300
[7,] 0.2064202 0.1636460 0.1591723 0.1590578 0.1589937 0.1541409 0.1537548
[8,] 0.1995902 0.1257536 0.1255877 0.1255560 0.1255549 0.1247324 0.1182632
[9,] 0.2177208 0.1579859 0.1563087 0.1553553 0.1550882 0.1550126 0.1512443
[10,] 0.2800922 0.2021318 0.1954235 0.1897260 0.1895910 0.1893911 0.1845921
[11,] 0.3102253 0.2271378 0.2269201 0.2249382 0.2237701 0.2236355 0.2190309
[12,] 0.3031721 0.1961386 0.1959764 0.1928448 0.1864785 0.1863016 0.1862400
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] 0.1683722 0.1647831 0.1638244 0.1636783 0.1634522 0.1623824 0.1623753
[2,] 0.1508609 0.1507740 0.1468505 0.1464897 0.1462030 0.1371885 0.1370837
[3,] 0.1725419 0.1722078 0.1706562 0.1697907 0.1687410 0.1669169 0.1660205
[4,] 0.2857272 0.2856246 0.2856227 0.2855048 0.2846510 0.2844338 0.2807086
[5,] 0.2149489 0.2147877 0.2147022 0.2111239 0.2020709 0.2020663 0.1965783
[6,] 0.1456363 0.1445683 0.1434674 0.1411047 0.1389120 0.1388520 0.1386924
[7,] 0.1537224 0.1535341 0.1488753 0.1479570 0.1463713 0.1450950 0.1447871
[8,] 0.1151590 0.1150904 0.1148185 0.1147564 0.1146614 0.1139724 0.1139231
[9,] 0.1504078 0.1503909 0.1492317 0.1490423 0.1488957 0.1488476 0.1466467
[10,] 0.1845092 0.1840592 0.1837003 0.1824441 0.1822595 0.1816811 0.1814872
[11,] 0.2177594 0.2160638 0.2157317 0.2155943 0.2155940 0.2155840 0.2155817
[12,] 0.1800538 0.1785281 0.1744753 0.1744409 0.1736910 0.1715552 0.1707679
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] 0.1603102 0.1602729 0.1601184 0.1578226 0.1553913 0.1553484 0.1544598
[2,] 0.1370318 0.1356453 0.1347012 0.1331281 0.1324143 0.1319353 0.1313651
[3,] 0.1655240 0.1643308 0.1633219 0.1629704 0.1629339 0.1585586 0.1558107
[4,] 0.2793216 0.2724146 0.2721004 0.2683225 0.2666109 0.2650710 0.2629805
[5,] 0.1965736 0.1965736 0.1954489 0.1941668 0.1931839 0.1836727 0.1836075
[6,] 0.1357044 0.1346955 0.1292167 0.1287371 0.1286116 0.1282040 0.1280755
[7,] 0.1444580 0.1419223 0.1417875 0.1417849 0.1415053 0.1413726 0.1409231
[8,] 0.1138991 0.1134252 0.1132842 0.1131517 0.1126438 0.1125310 0.1100107
[9,] 0.1447934 0.1444981 0.1442171 0.1441266 0.1434648 0.1410513 0.1402075
[10,] 0.1797065 0.1757306 0.1726931 0.1700078 0.1578465 0.1568700 0.1568250
[11,] 0.2152716 0.2141716 0.2141062 0.2075075 0.2051000 0.2049085 0.2044669
[12,] 0.1702478 0.1696764 0.1667023 0.1652411 0.1651209 0.1644008 0.1638155
$phi
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.6589492 0.041082637 -0.08977826 0.12565403 -0.10277954 -0.102763468
[2,] 0.8700298 -0.301093399 0.20157789 -0.03079655 0.03393160 0.065703678
[3,] 0.6618710 0.296971024 -0.25808640 0.11528032 -0.07865125 0.136771321
[4,] 0.8973904 -0.400732861 0.34100591 0.18605764 -0.26247150 0.004142777
[5,] 0.1549853 0.001942657 0.20813377 -0.09963635 0.13985599 -0.105093148
[6,] 0.2770527 -0.246038698 0.22451675 -0.23322979 0.15748177 -0.128203864
[7,] 0.7864776 -0.237065015 0.01235239 -0.14879102 0.22981196 0.073481194
[8,] 0.6670789 0.119126606 -0.01232244 0.07829449 -0.26767349 0.375857046
[9,] 0.9123711 -0.287911056 0.10652915 -0.03023542 -0.08993360 0.100222406
[10,] 1.1128403 -0.677791408 0.33951405 0.27197651 -0.06824997 0.158925955
[11,] 0.8789312 -0.322224260 0.47434117 -0.06176384 -0.16674364 0.114327571
[12,] 0.7161806 -0.022203567 0.38491587 -0.40752050 0.23592099 -0.183254331
[,7] [,8] [,9] [,10] [,11]
[1,] 0.41223045 -0.281475832 0.075295918 0.052775980 0.093885847
[2,] -0.02403290 -0.150209166 0.126514333 0.040870751 -0.221951330
[3,] -0.06389537 0.059240977 0.058567200 0.111787889 -0.004527702
[4,] -0.11991548 0.447798726 -0.272157740 -0.034793875 -0.043184470
[5,] 0.02438340 -0.083512145 -0.248288098 0.001657224 0.531696759
[6,] 0.15505353 0.254789381 -0.429838277 0.345595587 -0.180467909
[7,] 0.04386750 0.048949462 -0.208703034 -0.037122445 0.078404957
[8,] -0.16581268 -0.006679296 0.004783692 0.022239096 -0.004865177
[9,] 0.03283126 -0.016228703 0.057910805 -0.011939482 -0.073043914
[10,] -0.18136690 -0.003376425 0.231934113 -0.132002763 -0.078057042
[11,] 0.02943037 0.114889880 -0.120501648 0.081932483 -0.105471235
[12,] 0.28094506 0.014338909 -0.164159584 0.089705473 -0.230064377
[,12] [,13] [,14] [,15] [,16]
[1,] -0.096202219 0.081918305 -0.147399718 -0.03276490 -0.04080620
[2,] 0.299968480 0.035270350 -0.087899574 0.06970029 0.10257729
[3,] 0.208768945 -0.074517462 -0.012265190 -0.07296191 -0.07833666
[4,] -0.217336111 0.331934747 -0.557544774 0.40650388 -0.17355963
[5,] 0.022668978 -0.124821036 -0.032526777 0.12974870 0.01190982
[6,] -0.102042285 0.045385447 -0.099158723 -0.13778805 0.31528167
[7,] 0.136939410 -0.083801071 -0.008961802 -0.11133849 0.04609912
[8,] 0.113656691 0.002241603 -0.052129274 0.05759674 -0.01127975
[9,] 0.225526303 -0.352039303 0.180044656 -0.09116181 0.02571675
[10,] -0.002541963 0.136873641 -0.144552766 0.45285473 -0.21635699
[11,] 0.070471051 -0.089367450 0.268642667 -0.22507271 -0.14274563
[12,] 0.112930513 -0.053384602 0.014021250 0.22097278 -0.05678251
[,17] [,18] [,19] [,20]
[1,] -0.028979821 0.28967523 -0.08276141 0.08954410
[2,] -0.008722016 -0.21444746 0.05345975 0.09956896
[3,] 0.026720990 0.20297024 -0.06105521 -0.26524599
[4,] 0.410928012 -0.34009349 0.43449414 -0.28297603
[5,] -0.042674433 0.05999727 -0.33315523 0.03340474
[6,] -0.096708046 -0.01134837 0.04418287 0.03154346
[7,] -0.040293802 0.07779931 0.01369336 -0.06643308
[8,] 0.021053308 -0.11102518 0.15793778 -0.12743820
[9,] -0.017134201 -0.06903196 0.11162207 0.10550274
[10,] 0.014879420 0.17715996 0.11215068 -0.02874923
[11,] 0.342763103 0.15047913 0.06496907 -0.06990429
[12,] -0.202188163 -0.02555086 0.11074691 -0.05261904
$aic
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -207.5218 -272.9005 -271.9199 -270.4253 -268.6717 -266.7180 -264.8001
[2,] -212.0079 -284.3567 -282.7546 -287.0985 -285.1337 -284.0762 -282.0779
[3,] -204.2735 -270.5710 -270.1399 -269.5532 -267.8499 -266.3655 -264.8797
[4,] -159.7886 -189.8359 -189.5606 -190.6643 -188.6741 -187.5714 -185.5953
[5,] -232.7754 -239.3302 -237.6963 -235.8327 -234.0488 -232.0801 -230.8911
[6,] -274.6184 -277.3525 -286.1030 -290.4684 -288.5233 -286.8917 -284.9987
[7,] -249.2990 -283.9879 -286.3673 -284.4810 -282.5448 -285.4423 -283.8386
[8,] -254.6153 -325.6021 -323.8106 -321.8505 -319.8519 -318.8903 -325.3051
[9,] -240.8776 -289.5495 -289.2357 -288.2024 -286.4743 -284.5513 -286.4397
[10,] -201.0765 -250.6160 -253.9486 -256.6235 -254.7361 -252.9027 -254.9579
[11,] -184.9321 -232.1873 -230.3388 -229.7249 -228.5475 -226.6425 -227.9297
[12,] -188.5658 -255.3715 -253.5022 -254.0474 -257.3514 -255.5014 -253.5536
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] -263.9262 -265.2875 -264.1977 -262.3370 -260.5526 -259.5770 -257.5838
[2,] -281.0580 -279.1479 -281.2611 -279.6448 -277.9505 -285.8783 -283.9976
[3,] -263.6242 -261.9305 -261.3604 -260.1639 -259.1437 -258.8609 -257.7117
[4,] -183.9294 -181.9862 -179.9872 -178.0524 -176.5256 -174.6463 -174.7292
[5,] -228.9021 -227.0206 -225.0835 -225.7390 -230.6636 -228.6672 -231.0177
[6,] -290.4096 -289.5725 -288.7803 -289.4040 -289.8785 -287.9467 -286.1285
[7,] -281.8719 -280.0656 -282.9341 -281.9117 -281.6142 -280.9980 -279.3336
[8,] -327.5078 -325.6019 -323.9757 -322.0611 -320.1919 -319.1442 -317.2126
[9,] -285.3160 -283.3338 -282.5563 -280.7569 -278.9124 -276.9635 -277.3171
[10,] -253.0289 -251.4147 -249.7231 -248.8072 -246.9672 -245.4694 -243.6381
[11,] -226.8496 -226.0847 -224.3278 -222.4284 -220.4286 -218.4359 -216.4376
[12,] -256.8909 -256.2355 -257.8636 -255.8948 -254.5754 -254.5303 -253.2571
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] -257.5805 -255.6169 -253.7673 -254.0203 -254.4422 -252.4853 -251.3801
[2,] -282.0566 -281.6431 -280.7326 -280.5652 -279.4038 -277.9692 -276.6449
[3,] -256.1850 -255.3280 -254.3011 -252.6415 -250.6769 -252.9777 -253.7399
[4,] -173.5118 -175.4680 -173.6503 -173.8594 -172.8705 -171.7857 -171.0367
[5,] -229.0215 -227.0215 -225.9281 -224.9679 -223.7698 -229.7468 -227.8029
[6,] -287.5696 -286.7487 -291.3098 -289.8973 -288.0514 -286.5530 -284.7114
[7,] -277.6931 -278.4912 -276.6413 -274.6441 -272.9561 -271.1043 -269.6074
[8,] -315.2459 -313.9047 -312.1012 -310.2861 -308.9969 -307.1553 -308.7341
[9,] -277.3267 -275.6492 -273.9569 -272.0559 -270.7832 -271.4638 -270.4118
[10,] -243.1960 -244.7309 -245.4858 -245.9619 -255.6889 -254.6694 -252.7147
[11,] -214.6650 -213.4745 -211.5227 -214.4689 -214.3127 -212.4603 -210.8012
[12,] -251.7390 -250.2702 -251.0642 -250.4552 -248.5702 -247.2608 -245.8243
$bic
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -207.5218 -270.5438 -267.2065 -263.3552 -259.2449 -254.9345 -250.6598
[2,] -212.0079 -282.0000 -278.0412 -280.0284 -275.7069 -272.2926 -267.9377
[3,] -204.2735 -268.2016 -265.4010 -262.4449 -258.3721 -254.5183 -250.6630
[4,] -159.7886 -187.4664 -184.8217 -183.5559 -179.1963 -175.7242 -171.3786
[5,] -232.7754 -236.9607 -232.9574 -228.7244 -224.5710 -220.2329 -216.6744
[6,] -274.6184 -274.9830 -281.3641 -283.3601 -279.0455 -275.0445 -270.7820
[7,] -249.2990 -281.6185 -281.6284 -277.3727 -273.0670 -273.5951 -269.6219
[8,] -254.6153 -323.2326 -319.0717 -314.7422 -310.3741 -307.0431 -311.0884
[9,] -240.8776 -287.1800 -284.4968 -281.0941 -276.9965 -272.7041 -272.2230
[10,] -201.0765 -248.2465 -249.2097 -249.5152 -245.2583 -241.0554 -240.7412
[11,] -184.9321 -229.8179 -225.5999 -222.6166 -219.0697 -214.7953 -213.7130
[12,] -188.5658 -253.0021 -248.7633 -246.9390 -247.8736 -243.6541 -239.3369
[,8] [,9] [,10] [,11] [,12] [,13] [,14]
[1,] -247.4292 -246.4339 -242.9874 -238.7699 -234.6288 -231.2965 -226.9466
[2,] -264.5610 -260.2942 -260.0507 -256.0777 -252.0267 -257.5978 -253.3603
[3,] -247.0380 -242.9749 -240.0354 -236.4694 -233.0798 -230.4276 -226.9089
[4,] -167.3432 -163.0306 -158.6622 -154.3579 -150.4617 -146.2129 -143.9264
[5,] -212.3159 -208.0650 -203.7585 -202.0445 -204.5997 -200.2338 -200.2149
[6,] -273.8234 -270.6169 -267.4553 -265.7095 -263.8146 -259.5133 -255.3257
[7,] -265.2858 -261.1100 -261.6091 -258.2172 -255.5503 -252.5646 -248.5307
[8,] -310.9216 -306.6463 -302.6507 -298.3666 -294.1280 -290.7108 -286.4098
[9,] -268.7298 -264.3782 -261.2313 -257.0624 -252.8485 -248.5301 -246.5143
[10,] -236.4427 -232.4591 -228.3981 -225.1127 -220.9033 -217.0360 -212.8353
[11,] -210.2634 -207.1291 -203.0027 -198.7339 -194.3647 -190.0025 -185.6348
[12,] -240.3048 -237.2799 -236.5385 -232.2003 -228.5115 -226.0969 -222.4542
[,15] [,16] [,17] [,18] [,19] [,20] [,21]
[1,] -224.5866 -220.2662 -216.0599 -213.9562 -212.0214 -207.7078 -204.2459
[2,] -249.0627 -246.2925 -243.0252 -240.5011 -236.9831 -233.1917 -229.5107
[3,] -223.0127 -219.7863 -216.3899 -212.3608 -208.0268 -207.9582 -206.3509
[4,] -140.3396 -139.9263 -135.7391 -133.5788 -130.2204 -126.7662 -123.6477
[5,] -195.8492 -191.4798 -188.0169 -184.6873 -181.1198 -184.7273 -180.4139
[6,] -254.3973 -251.2070 -253.3986 -249.6167 -245.4014 -241.5335 -237.3224
[7,] -244.5208 -242.9495 -238.7301 -234.3635 -230.3060 -226.0848 -222.2185
[8,] -282.0736 -278.3630 -274.1900 -270.0055 -266.3469 -262.1357 -261.3452
[9,] -244.1544 -240.1075 -236.0457 -231.7753 -228.1331 -226.4443 -223.0228
[10,] -210.0237 -209.1892 -207.5746 -205.6813 -213.0389 -209.6499 -205.3257
[11,] -181.4928 -177.9327 -173.6116 -174.1883 -171.6627 -167.4408 -163.4122
[12,] -218.5667 -214.7285 -213.1530 -210.1746 -205.9201 -202.2413 -198.4354
$maice
[1] 1 3 1 3 1 16 2 7 1 3 1 9
$mbice
[1] 1 1 1 1 1 3 2 1 1 3 1 1
attr(,"type")
[1] "pacf"
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