peacf: Periodic Autocorrelation Function
In pear: Package for Periodic Autoregression Analysis

Description Usage Arguments Details Value Side Effects References See Also Examples

The periodic autocorrelation for a time series with period p may be defined as for period (m=1,...,p) and lag, l=1,2,... by r(m, l) = c(m, l)/sqrt(c(m, 0)*c(m-l, 0)) where c(m, l) is the periodic autocovariance defined by c(m, l) = sum(z[t] * z[t-l]) where the sum is over all data in period m. When p=1, peacf produces output which is equivalent to that produced by Splus function acf.

1	peacf(z, lag.max, plot=TRUE)

`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.

The use of the periodic autocorrelation and its plot are discussed in McLeod (1994) and a portmanteau model adequacy test is developed therein. The periodicity test is discussed in McLeod (1993). For more details, see the references below.

a list is returned with the following components: means periodic means standard.deviations periodic standard deviations acf matrix of periodic autocorrelations benchmark.sd 1/sqrt(nyrs), nyrs=ceiling(length(z)/p) sub.lengths number of observations in each period period p = attr(z, "tsp")[3] title title = attr(z, "title") periodicity.test list: test for periodic correlation at lag 1 The components of this list are: Q1 = scalar value, the test statistic Q1.sl = signifiance level (upper tail) portmanteau.test list: portmanteau test at various lags The components of this list are: QM = matrix of portmanteau statistics for each period and lag QM.df = corresponding df of QM

a plot may be produced if plot=TRUE

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. (1993), "Parsimony, Model Adequacy and Periodic Correlation in Time Series Forecasting", International Statistical Review, Vol. 61, No. 3, pp.387–393. McLeod, A.I. (1994), "Diagnostic Checking of Periodic Autoregression" Journal of Time Series Analysis, Vol. 15, No. 2, pp.221–233.

peacf.plot, pepacf, peplot, peboxplot, pear, acf, acf.plot

1 2	data(Fraser) peacf(log(Fraser))

$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

$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
           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

$benchmark.sd
[1] 0.1125088

$sub.lengths
 [1] 78 78 79 79 79 79 79 79 79 79 79 79

$period
[1] 12

$title
[1] "Fraser River at Hope, mean monthly flow (cms),  1912.3-1991.12"

$periodicity.test
$periodicity.test$Q1
[1] 413.8426

$periodicity.test$Q1.sl
[1] 0


$portmanteau.test
$portmanteau.test$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

$portmanteau.test$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"