The function decomposes a time series into a long-term mean, annual, seasonal and "events" component. The decomposition can be multiplicative or additive, and based on median or mean centering.
Arguments
- x
a monthly time series vector
- event
whether or not an "events" component should be determined
- type
the type of decomposition, either multiplicative ("mult") or additive ("add")
- center
the method of centering, either median or mean
Value
A monthly time series matrix with the following individual time series:
- original
original time series
- annual
annual mean series
- seasonal
repeating seasonal component
- events
optionally, the residual or "events" series
Details
The rationale for this simple approach to decomposing a time series, with examples of its application, is given by Cloern and Jassby (2010). It is motivated by the observation that many important events for estuaries (e.g., persistent dry periods, species invasions) start or stop suddenly. Smoothing to extract the annualized term, which can disguise the timing of these events and make analysis of them unnecessarily difficult, is not used.
A multiplicative decomposition will typically be useful for a biological community- or population-related variable (e.g., chlorophyll-a) that experiences exponential changes in time and is approximately lognormal, whereas an additive decomposition is more suitable for a normal variable. The default centering method is the median, especially appropriate for series that have large, infrequent events.
If event = TRUE, the seasonal component represents a recurring
monthly pattern and the events component a residual series. Otherwise, the
seasonal component becomes the residual series. The latter is appropriate
when seasonal patterns change systematically over time. You can use
plotSeason and seasonTrend to investigate the
way seasonality changes.
References
Cloern, J.E. and Jassby, A.D. (2010) Patterns and scales of phytoplankton variability in estuarine-coastal ecosystems. Estuaries and Coasts 33, 230--241.
See also
Examples
# Apply the function to a single series (Station 27) and plot it:
y <- decompTs(sfbayChla[, 's27'])
y
#> original annual seasonal events
#> Jan 1978 1.1000000 0.7118644 0.7245509 0.7229437
#> Feb 1978 2.8000000 0.7118644 0.9857550 1.3526012
#> Mar 1978 5.5500000 0.7118644 2.7522743 0.9602448
#> Apr 1978 2.7000000 0.7118644 4.0510332 0.3173793
#> May 1978 3.4000000 0.7118644 1.5810082 1.0240602
#> Jun 1978 1.9000000 0.7118644 1.0000000 0.9047619
#> Jul 1978 1.5916667 0.7118644 0.7902392 0.9591228
#> Aug 1978 NA 0.7118644 0.9830918 NA
#> Sep 1978 1.6600000 0.7118644 1.0000000 0.7904762
#> Oct 1978 2.1000000 0.7118644 0.9164258 1.0911958
#> Nov 1978 2.1500000 0.7118644 0.7131930 1.4355294
#> Dec 1978 1.6666667 0.7118644 0.7591627 1.0454291
#> Jan 1979 1.9000000 0.7288136 0.7245509 1.2196810
#> Feb 1979 1.7666667 0.7288136 0.9857550 0.8335798
#> Mar 1979 2.4333333 0.7288136 2.7522743 0.4112173
#> Apr 1979 3.7666667 0.7288136 4.0510332 0.4324670
#> May 1979 2.2666667 0.7288136 1.5810082 0.6668299
#> Jun 1979 4.8000000 0.7288136 1.0000000 2.2325581
#> Jul 1979 1.5666667 0.7288136 0.7902392 0.9221033
#> Aug 1979 3.9333333 0.7288136 0.9830918 1.8609222
#> Sep 1979 2.1500000 0.7288136 1.0000000 1.0000000
#> Oct 1979 1.2333333 0.7288136 0.9164258 0.6259573
#> Nov 1979 1.1000000 0.7288136 0.7131930 0.7173765
#> Dec 1979 NA 0.7288136 0.7591627 NA
#> Jan 1980 1.3166667 0.5621469 0.7245509 1.0958096
#> Feb 1980 1.9000000 0.5621469 0.9857550 1.1622854
#> Mar 1980 2.0666667 0.5621469 2.7522743 0.4528005
#> Apr 1980 10.2166667 0.5621469 4.0510332 1.5207982
#> May 1980 3.4000000 0.5621469 1.5810082 1.2967999
#> Jun 1980 2.1333333 0.5621469 1.0000000 1.2864322
#> Jul 1980 1.1333333 0.5621469 0.7902392 0.8648230
#> Aug 1980 1.3500000 0.5621469 0.9830918 0.8280716
#> Sep 1980 1.6500000 0.5621469 1.0000000 0.9949749
#> Oct 1980 1.4000000 0.5621469 0.9164258 0.9212105
#> Nov 1980 1.6666667 0.5621469 0.7131930 1.4091909
#> Dec 1980 1.3000000 0.5621469 0.7591627 1.0326107
#> Jan 1981 NA 0.6768362 0.7245509 NA
#> Feb 1981 1.7333333 0.6768362 0.9857550 0.8806585
#> Mar 1981 1.9966667 0.6768362 2.7522743 0.3633359
#> Apr 1981 9.0666667 0.6768362 4.0510332 1.1209243
#> May 1981 NA 0.6768362 1.5810082 NA
#> Jun 1981 NA 0.6768362 1.0000000 NA
#> Jul 1981 NA 0.6768362 0.7902392 NA
#> Aug 1981 NA 0.6768362 0.9830918 NA
#> Sep 1981 NA 0.6768362 1.0000000 NA
#> Oct 1981 NA 0.6768362 0.9164258 NA
#> Nov 1981 NA 0.6768362 0.7131930 NA
#> Dec 1981 NA 0.6768362 0.7591627 NA
#> Jan 1982 2.7888889 0.9453861 0.7245509 1.3801653
#> Feb 1982 4.5375000 0.9453861 0.9857550 1.6505035
#> Mar 1982 6.4944444 0.9453861 2.7522743 0.8460949
#> Apr 1982 9.2983333 0.9453861 4.0510332 0.8230157
#> May 1982 8.2500000 0.9453861 1.5810082 1.8710639
#> Jun 1982 3.4233333 0.9453861 1.0000000 1.2274900
#> Jul 1982 1.3500000 0.9453861 0.7902392 0.6125534
#> Aug 1982 NA 0.9453861 0.9830918 NA
#> Sep 1982 2.1277778 0.9453861 1.0000000 0.7629482
#> Oct 1982 1.8000000 0.9453861 0.9164258 0.7042778
#> Nov 1982 1.7000000 0.9453861 0.7131930 0.8546940
#> Dec 1982 0.9500000 0.9453861 0.7591627 0.4487015
#> Jan 1983 NA 0.5889831 0.7245509 NA
#> Feb 1983 1.4333333 0.5889831 0.9857550 0.8368611
#> Mar 1983 6.9722222 0.5889831 2.7522743 1.4579905
#> Apr 1983 16.3750000 0.5889831 4.0510332 2.3264338
#> May 1983 16.6402778 0.5889831 1.5810082 6.0576146
#> Jun 1983 5.3750000 0.5889831 1.0000000 3.0935252
#> Jul 1983 1.4291667 0.5889831 0.7902392 1.0408772
#> Aug 1983 1.7375000 0.5889831 0.9830918 1.0171990
#> Sep 1983 2.0375000 0.5889831 1.0000000 1.1726619
#> Oct 1983 1.4666667 0.5889831 0.9164258 0.9211053
#> Nov 1983 1.5000000 0.5889831 0.7131930 1.2104849
#> Dec 1983 1.4500000 0.5889831 0.7591627 1.0992800
#> Jan 1984 1.9000000 0.8545198 0.7245509 1.0402568
#> Feb 1984 2.7666667 0.8545198 0.9857550 1.1133808
#> Mar 1984 2.9500000 0.8545198 2.7522743 0.4251931
#> Apr 1984 9.8333333 0.8545198 4.0510332 0.9629214
#> May 1984 3.5033333 0.8545198 1.5810082 0.8790290
#> Jun 1984 1.2500000 0.8545198 1.0000000 0.4958678
#> Jul 1984 1.6750000 0.8545198 0.7902392 0.8408375
#> Aug 1984 2.2750000 0.8545198 0.9830918 0.9180011
#> Sep 1984 2.9000000 0.8545198 1.0000000 1.1504132
#> Oct 1984 1.3500000 0.8545198 0.9164258 0.5843759
#> Nov 1984 NA 0.8545198 0.7131930 NA
#> Dec 1984 NA 0.8545198 0.7591627 NA
#> Jan 1985 NaN 0.5197740 0.7245509 NaN
#> Feb 1985 NaN 0.5197740 0.9857550 NaN
#> Mar 1985 12.2916667 0.5197740 2.7522743 2.9126109
#> Apr 1985 8.1333333 0.5197740 4.0510332 1.3093815
#> May 1985 1.5666667 0.5197740 1.5810082 0.6462580
#> Jun 1985 1.4416667 0.5197740 1.0000000 0.9402174
#> Jul 1985 0.7916667 0.5197740 0.7902392 0.6533520
#> Aug 1985 NA 0.5197740 0.9830918 NA
#> Sep 1985 1.9750000 0.5197740 1.0000000 1.2880435
#> Oct 1985 1.5000000 0.5197740 0.9164258 1.0674742
#> Nov 1985 1.1000000 0.5197740 0.7131930 1.0058866
#> Dec 1985 NA 0.5197740 0.7591627 NA
#> Jan 1986 1.2333333 0.4731638 0.7245509 1.2194893
#> Feb 1986 1.2000000 0.4731638 0.9857550 0.8721249
#> Mar 1986 3.9944444 0.4731638 2.7522743 1.0397552
#> Apr 1986 25.4633333 0.4731638 4.0510332 4.5031445
#> May 1986 4.0083333 0.4731638 1.5810082 1.8163358
#> Jun 1986 1.5500000 0.4731638 1.0000000 1.1104478
#> Jul 1986 1.4666667 0.4731638 0.7902392 1.3296559
#> Aug 1986 NA 0.4731638 0.9830918 NA
#> Sep 1986 NA 0.4731638 1.0000000 NA
#> Oct 1986 1.3250000 0.4731638 0.9164258 1.0358217
#> Nov 1986 1.2000000 0.4731638 0.7131930 1.2054261
#> Dec 1986 1.1000000 0.4731638 0.7591627 1.0380643
#> Jan 1987 1.4000000 0.6610169 0.7245509 0.9908879
#> Feb 1987 1.4000000 0.6610169 0.9857550 0.7283237
#> Mar 1987 5.1000000 0.6610169 2.7522743 0.9502631
#> Apr 1987 5.9000000 0.6610169 4.0510332 0.7468813
#> May 1987 5.0500000 0.6610169 1.5810082 1.6380330
#> Jun 1987 2.9000000 0.6610169 1.0000000 1.4871795
#> Jul 1987 1.7000000 0.6610169 0.7902392 1.1032037
#> Aug 1987 2.0000000 0.6610169 0.9830918 1.0432810
#> Sep 1987 1.9500000 0.6610169 1.0000000 1.0000000
#> Oct 1987 1.9500000 0.6610169 0.9164258 1.0911958
#> Nov 1987 0.6600000 0.6610169 0.7131930 0.4745721
#> Dec 1987 1.3000000 0.6610169 0.7591627 0.8781604
#> Jan 1988 NA 0.9355932 0.7245509 NA
#> Feb 1988 4.1200000 0.9355932 0.9857550 1.5143252
#> Mar 1988 5.2400000 0.9355932 2.7522743 0.6898116
#> Apr 1988 6.8200000 0.9355932 4.0510332 0.6099714
#> May 1988 2.2400000 0.9355932 1.5810082 0.5133397
#> Jun 1988 2.9800000 0.9355932 1.0000000 1.0797101
#> Jul 1988 2.7600000 0.9355932 0.7902392 1.2654396
#> Aug 1988 2.7133333 0.9355932 0.9830918 1.0000000
#> Sep 1988 1.7000000 0.9355932 1.0000000 0.6159420
#> Oct 1988 3.2000000 0.9355932 0.9164258 1.2651546
#> Nov 1988 1.6400000 0.9355932 0.7131930 0.8331586
#> Dec 1988 1.6000000 0.9355932 0.7591627 0.7636178
#> Jan 1989 NA 0.8491525 0.7245509 NA
#> Feb 1989 3.6500000 0.8491525 0.9857550 1.4781420
#> Mar 1989 10.3000000 0.8491525 2.7522743 1.4939559
#> Apr 1989 10.9700000 0.8491525 4.0510332 1.0810184
#> May 1989 4.1066667 0.8491525 1.5810082 1.0369256
#> Jun 1989 2.1400000 0.8491525 1.0000000 0.8542914
#> Jul 1989 2.2300000 0.8491525 0.7902392 1.1265191
#> Aug 1989 2.0000000 0.8491525 0.9830918 0.8121349
#> Sep 1989 2.1800000 0.8491525 1.0000000 0.8702595
#> Oct 1989 2.7800000 0.8491525 0.9164258 1.2109878
#> Nov 1989 NA 0.8491525 0.7131930 NA
#> Dec 1989 1.2000000 0.8491525 0.7591627 0.6310135
#> Jan 1990 2.1400000 0.8338983 0.7245509 1.2006316
#> Feb 1990 NaN 0.8338983 0.9857550 NaN
#> Mar 1990 5.3850000 0.8338983 2.7522743 0.7953511
#> Apr 1990 14.9933333 0.8338983 4.0510332 1.5045176
#> May 1990 3.1000000 0.8338983 1.5810082 0.7970627
#> Jun 1990 2.0000000 0.8338983 1.0000000 0.8130081
#> Jul 1990 NA 0.8338983 0.7902392 NA
#> Aug 1990 2.4600000 0.8338983 0.9830918 1.0171990
#> Sep 1990 NA 0.8338983 1.0000000 NA
#> Oct 1990 NA 0.8338983 0.9164258 NA
#> Nov 1990 NA 0.8338983 0.7131930 NA
#> Dec 1990 1.9000000 0.8338983 0.7591627 1.0173810
#> Jan 1991 1.7000000 0.6949153 0.7245509 1.1445273
#> Feb 1991 1.9000000 0.6949153 0.9857550 0.9402228
#> Mar 1991 3.6600000 0.6949153 2.7522743 0.6486875
#> Apr 1991 8.1628571 0.6949153 4.0510332 0.9829299
#> May 1991 6.3400000 0.6949153 1.5810082 1.9561461
#> Jun 1991 4.9600000 0.6949153 1.0000000 2.4195122
#> Jul 1991 NaN 0.6949153 0.7902392 NaN
#> Aug 1991 1.9000000 0.6949153 0.9830918 0.9427698
#> Sep 1991 2.2000000 0.6949153 1.0000000 1.0731707
#> Oct 1991 NA 0.6949153 0.9164258 NA
#> Nov 1991 0.7000000 0.6949153 0.7131930 0.4787812
#> Dec 1991 1.1800000 0.6949153 0.7591627 0.7582166
#> Jan 1992 1.6000000 0.7932203 0.7245509 0.9437028
#> Feb 1992 2.3066667 0.7932203 0.9857550 1.0000000
#> Mar 1992 4.8466667 0.7932203 2.7522743 0.7525504
#> Apr 1992 11.6050000 0.7932203 4.0510332 1.2242313
#> May 1992 3.8100000 0.7932203 1.5810082 1.0298524
#> Jun 1992 2.9600000 0.7932203 1.0000000 1.2649573
#> Jul 1992 2.5200000 0.7932203 0.7902392 1.3627811
#> Aug 1992 2.3400000 0.7932203 0.9830918 1.0171990
#> Sep 1992 2.1000000 0.7932203 1.0000000 0.8974359
#> Oct 1992 NA 0.7932203 0.9164258 NA
#> Nov 1992 1.7000000 0.7932203 0.7131930 1.0186523
#> Dec 1992 1.3000000 0.7932203 0.7591627 0.7318004
#> Jan 1993 1.3400000 0.5864407 0.7245509 1.0690298
#> Feb 1993 1.7600000 0.5864407 0.9857550 1.0320425
#> Mar 1993 22.1333333 0.5864407 2.7522743 4.6484589
#> Apr 1993 9.8000000 0.5864407 4.0510332 1.3983445
#> May 1993 2.8900000 0.5864407 1.5810082 1.0566171
#> Jun 1993 1.7000000 0.5864407 1.0000000 0.9826590
#> Jul 1993 NA 0.5864407 0.7902392 NA
#> Aug 1993 NA 0.5864407 0.9830918 NA
#> Sep 1993 NaN 0.5864407 1.0000000 NaN
#> Oct 1993 1.2400000 0.5864407 0.9164258 0.7821288
#> Nov 1993 1.1600000 0.5864407 0.7131930 0.9401666
#> Dec 1993 NaN 0.5864407 0.7591627 NaN
#> Jan 1994 1.3000000 0.8677966 0.7245509 0.7008652
#> Feb 1994 2.7666667 0.8677966 0.9857550 1.0963466
#> Mar 1994 12.5600000 0.8677966 2.7522743 1.7826166
#> Apr 1994 4.5280000 0.8677966 4.0510332 0.4366170
#> May 1994 3.9500000 0.8677966 1.5810082 0.9759398
#> Jun 1994 5.7200000 0.8677966 1.0000000 2.2343750
#> Jul 1994 2.5600000 0.8677966 0.7902392 1.2654396
#> Aug 1994 1.9000000 0.8677966 0.9830918 0.7549524
#> Sep 1994 1.9600000 0.8677966 1.0000000 0.7656250
#> Oct 1994 2.0800000 0.8677966 0.9164258 0.8865966
#> Nov 1994 1.7800000 0.8677966 0.7131930 0.9749289
#> Dec 1994 NA 0.8677966 0.7591627 NA
#> Jan 1995 1.5000000 1.2259887 0.7245509 0.5724188
#> Feb 1995 7.6100000 1.2259887 0.9857550 2.1345542
#> Mar 1995 22.9283333 1.2259887 2.7522743 2.3034155
#> Apr 1995 12.8600000 1.2259887 4.0510332 0.8777416
#> May 1995 5.3733333 1.2259887 1.5810082 0.9397259
#> Jun 1995 1.6800000 1.2259887 1.0000000 0.4645161
#> Jul 1995 1.2200000 1.2259887 0.7902392 0.4268672
#> Aug 1995 1.2000000 1.2259887 0.9830918 0.3375038
#> Sep 1995 1.8600000 1.2259887 1.0000000 0.5142857
#> Oct 1995 6.4200000 1.2259887 0.9164258 1.9369983
#> Nov 1995 NA 1.2259887 0.7131930 NA
#> Dec 1995 NA 1.2259887 0.7591627 NA
#> Jan 1996 0.9800000 0.5084746 0.7245509 0.9017080
#> Feb 1996 1.1900000 0.5084746 0.9857550 0.8047977
#> Mar 1996 6.5950000 0.5084746 2.7522743 1.5974667
#> Apr 1996 11.4733333 0.5084746 4.0510332 1.8881328
#> May 1996 3.8300000 0.5084746 1.5810082 1.6150032
#> Jun 1996 0.4400000 0.5084746 1.0000000 0.2933333
#> Jul 1996 1.1000000 0.5084746 0.7902392 0.9279890
#> Aug 1996 1.1400000 0.5084746 0.9830918 0.7730713
#> Sep 1996 1.6000000 0.5084746 1.0000000 1.0666667
#> Oct 1996 2.2000000 0.5084746 0.9164258 1.6004205
#> Nov 1996 2.2600000 0.5084746 0.7131930 2.1125651
#> Dec 1996 1.4000000 0.5084746 0.7591627 1.2294246
#> Jan 1997 4.0000000 0.9152542 0.7245509 2.0446893
#> Feb 1997 12.5600000 0.9152542 0.9857550 4.7190751
#> Mar 1997 6.8550000 0.9152542 2.7522743 0.9224694
#> Apr 1997 11.3650000 0.9152542 4.0510332 1.0390582
#> May 1997 1.9400000 0.9152542 1.5810082 0.4544686
#> Jun 1997 1.3000000 0.9152542 1.0000000 0.4814815
#> Jul 1997 2.7000000 0.9152542 0.7902392 1.2654396
#> Aug 1997 1.7400000 0.9152542 0.9830918 0.6555283
#> Sep 1997 2.9200000 0.9152542 1.0000000 1.0814815
#> Oct 1997 2.6200000 0.9152542 0.9164258 1.0588641
#> Nov 1997 2.6200000 0.9152542 0.7131930 1.3606000
#> Dec 1997 NA 0.9152542 0.7591627 NA
#> Jan 1998 2.7600000 1.1389831 0.7245509 1.1337072
#> Feb 1998 1.6866667 1.1389831 0.9857550 0.5092382
#> Mar 1998 20.1250000 1.1389831 2.7522743 2.1762305
#> Apr 1998 18.6840000 1.1389831 4.0510332 1.3726657
#> May 1998 6.3200000 1.1389831 1.5810082 1.1897170
#> Jun 1998 4.0900000 1.1389831 1.0000000 1.2172619
#> Jul 1998 1.9800000 1.1389831 0.7902392 0.7457055
#> Aug 1998 1.8000000 1.1389831 0.9830918 0.5449280
#> Sep 1998 3.9600000 1.1389831 1.0000000 1.1785714
#> Oct 1998 NA 1.1389831 0.9164258 NA
#> Nov 1998 1.8400000 1.1389831 0.7131930 0.7678413
#> Dec 1998 NA 1.1389831 0.7591627 NA
#> Jan 1999 2.4200000 1.1322034 0.7245509 1.0000000
#> Feb 1999 1.8600000 1.1322034 0.9857550 0.5649337
#> Mar 1999 13.6500000 1.1322034 2.7522743 1.4848906
#> Apr 1999 11.5200000 1.1322034 4.0510332 0.8514129
#> May 1999 4.8000000 1.1322034 1.5810082 0.9089932
#> Jun 1999 3.3400000 1.1322034 1.0000000 1.0000000
#> Jul 1999 NA 1.1322034 0.7902392 NA
#> Aug 1999 3.3800000 1.1322034 0.9830918 1.0293810
#> Sep 1999 2.2600000 1.1322034 1.0000000 0.6766467
#> Oct 1999 16.1800000 1.1322034 0.9164258 5.2860923
#> Nov 1999 2.3600000 1.1322034 0.7131930 0.9907372
#> Dec 1999 2.2000000 1.1322034 0.7591627 0.8676435
#> Jan 2000 3.2800000 1.1000000 0.7245509 1.3950515
#> Feb 2000 3.2300000 1.1000000 0.9857550 1.0097616
#> Mar 2000 9.3000000 1.1000000 2.7522743 1.0413016
#> Apr 2000 13.3700000 1.1000000 4.0510332 1.0170701
#> May 2000 3.2600000 1.1000000 1.5810082 0.6354316
#> Jun 2000 1.7200000 1.1000000 1.0000000 0.5300462
#> Jul 2000 1.7800000 1.1000000 0.7902392 0.6941394
#> Aug 2000 2.6000000 1.1000000 0.9830918 0.8150131
#> Sep 2000 3.0000000 1.1000000 1.0000000 0.9244992
#> Oct 2000 2.6200000 1.1000000 0.9164258 0.8810271
#> Nov 2000 39.4400000 1.1000000 0.7131930 17.0417869
#> Dec 2000 25.6200000 1.1000000 0.7591627 10.3999091
#> Jan 2001 NA 1.2813559 0.7245509 NA
#> Feb 2001 2.6650000 1.2813559 0.9857550 0.7152147
#> Mar 2001 20.2600000 1.2813559 2.7522743 1.9474034
#> Apr 2001 21.6866667 1.2813559 4.0510332 1.4162346
#> May 2001 4.5800000 1.2813559 1.5810082 0.7663719
#> Jun 2001 6.3600000 1.2813559 1.0000000 1.6825397
#> Jul 2001 3.9800000 1.2813559 0.7902392 1.3323940
#> Aug 2001 NA 1.2813559 0.9830918 NA
#> Sep 2001 2.9400000 1.2813559 1.0000000 0.7777778
#> Oct 2001 3.3400000 1.2813559 0.9164258 0.9641783
#> Nov 2001 2.6800000 1.2813559 0.7131930 0.9941134
#> Dec 2001 3.5800000 1.2813559 0.7591627 1.2475454
#> Jan 2002 NA 1.4237288 0.7245509 NA
#> Feb 2002 NA 1.4237288 0.9857550 NA
#> Mar 2002 15.9500000 1.4237288 2.7522743 1.3798112
#> Apr 2002 8.3000000 1.4237288 4.0510332 0.4878238
#> May 2002 10.1200000 1.4237288 1.5810082 1.5240426
#> Jun 2002 NA 1.4237288 1.0000000 NA
#> Jul 2002 2.6000000 1.4237288 0.7902392 0.7833674
#> Aug 2002 4.1800000 1.4237288 0.9830918 1.0123552
#> Sep 2002 4.2000000 1.4237288 1.0000000 1.0000000
#> Oct 2002 4.2000000 1.4237288 0.9164258 1.0911958
#> Nov 2002 3.9600000 1.4237288 0.7131930 1.3220224
#> Dec 2002 3.6800000 1.4237288 0.7591627 1.1541537
#> Jan 2003 4.4600000 1.5288136 0.7245509 1.3648641
#> Feb 2003 30.3200000 1.5288136 0.9857550 6.8199890
#> Mar 2003 26.6475000 1.5288136 2.7522743 2.1467833
#> Apr 2003 9.0500000 1.5288136 4.0510332 0.4953432
#> May 2003 4.5600000 1.5288136 1.5810082 0.6395201
#> Jun 2003 4.2200000 1.5288136 1.0000000 0.9356984
#> Jul 2003 4.3200000 1.5288136 0.7902392 1.2121284
#> Aug 2003 5.1700000 1.5288136 0.9830918 1.1660574
#> Sep 2003 5.5400000 1.5288136 1.0000000 1.2283814
#> Oct 2003 3.5200000 1.5288136 0.9164258 0.8516650
#> Nov 2003 2.6000000 1.5288136 0.7131930 0.8083319
#> Dec 2003 2.0800000 1.5288136 0.7591627 0.6075079
#> Jan 2004 2.3000000 1.6508475 0.7245509 0.6518234
#> Feb 2004 5.3200000 1.6508475 0.9857550 1.1081886
#> Mar 2004 17.0866667 1.6508475 2.7522743 1.2747842
#> Apr 2004 6.0533333 1.6508475 4.0510332 0.3068314
#> May 2004 4.9200000 1.6508475 1.5810082 0.6390017
#> Jun 2004 2.8600000 1.6508475 1.0000000 0.5872690
#> Jul 2004 3.1400000 1.6508475 0.7902392 0.8159097
#> Aug 2004 4.6800000 1.6508475 0.9830918 0.9775136
#> Sep 2004 7.8400000 1.6508475 1.0000000 1.6098563
#> Oct 2004 5.0400000 1.6508475 0.9164258 1.1292868
#> Nov 2004 3.9800000 1.6508475 0.7131930 1.1459008
#> Dec 2004 4.8200000 1.6508475 0.7591627 1.3037166
#> Jan 2005 2.5400000 1.6813559 0.7245509 0.7067782
#> Feb 2005 3.9300000 1.6813559 0.9857550 0.8037887
#> Mar 2005 4.9600000 1.6813559 2.7522743 0.3633359
#> Apr 2005 8.7300000 1.6813559 4.0510332 0.4344770
#> May 2005 6.8300000 1.6813559 1.5810082 0.8709734
#> Jun 2005 4.6800000 1.6813559 1.0000000 0.9435484
#> Jul 2005 NA 1.6813559 0.7902392 NA
#> Aug 2005 5.8700000 1.6813559 0.9830918 1.2038222
#> Sep 2005 5.1700000 1.6813559 1.0000000 1.0423387
#> Oct 2005 4.1500000 1.6813559 0.9164258 0.9129965
#> Nov 2005 6.8200000 1.6813559 0.7131930 1.9279494
#> Dec 2005 3.7000000 1.6813559 0.7591627 0.9826190
#> Jan 2006 2.6600000 1.6271186 0.7245509 0.7648416
#> Feb 2006 16.7100000 1.6271186 0.9857550 3.5315571
#> Mar 2006 9.9066667 1.6271186 2.7522743 0.7498849
#> Apr 2006 27.7733333 1.6271186 4.0510332 1.4283050
#> May 2006 17.6800000 1.6271186 1.5810082 2.3297370
#> Jun 2006 NA 1.6271186 1.0000000 NA
#> Jul 2006 2.5800000 1.6271186 0.7902392 0.6801738
#> Aug 2006 3.5066667 1.6271186 0.9830918 0.7431204
#> Sep 2006 4.6400000 1.6271186 1.0000000 0.9666667
#> Oct 2006 4.8000000 1.6271186 0.9164258 1.0911958
#> Nov 2006 8.9400000 1.6271186 0.7131930 2.6114950
#> Dec 2006 3.0600000 1.6271186 0.7591627 0.8397409
#> Jan 2007 3.8800000 2.3728814 0.7245509 0.7650059
#> Feb 2007 5.2400000 2.3728814 0.9857550 0.7593889
#> Mar 2007 8.1100000 2.3728814 2.7522743 0.4209506
#> Apr 2007 17.0600000 2.3728814 4.0510332 0.6016102
#> May 2007 NA 2.3728814 1.5810082 NA
#> Jun 2007 7.0000000 2.3728814 1.0000000 1.0000000
#> Jul 2007 8.0800000 2.3728814 0.7902392 1.4606788
#> Aug 2007 17.0400000 2.3728814 0.9830918 2.4761530
#> Sep 2007 7.9700000 2.3728814 1.0000000 1.1385714
#> Oct 2007 5.3400000 2.3728814 0.9164258 0.8324265
#> Nov 2007 4.6400000 2.3728814 0.7131930 0.9294218
#> Dec 2007 5.6000000 2.3728814 0.7591627 1.0537925
#> Jan 2008 3.4800000 1.9898305 0.7245509 0.8182241
#> Feb 2008 3.2066667 1.9898305 0.9857550 0.5541747
#> Mar 2008 17.3100000 1.9898305 2.7522743 1.0714385
#> Apr 2008 27.2900000 1.9898305 4.0510332 1.1476240
#> May 2008 13.7500000 1.9898305 1.5810082 1.4815983
#> Jun 2008 6.6400000 1.9898305 1.0000000 1.1311755
#> Jul 2008 5.8600000 1.9898305 0.7902392 1.2632838
#> Aug 2008 6.6100000 1.9898305 0.9830918 1.1454319
#> Sep 2008 5.8800000 1.9898305 1.0000000 1.0017036
#> Oct 2008 5.0800000 1.9898305 0.9164258 0.9443398
#> Nov 2008 2.9500000 1.9898305 0.7131930 0.7046555
#> Dec 2008 2.9000000 1.9898305 0.7591627 0.6507662
#> Jan 2009 2.3400000 1.5711864 0.7245509 0.6967825
#> Feb 2009 2.5666667 1.5711864 0.9857550 0.5617599
#> Mar 2009 2.6266667 1.5711864 2.7522743 0.2059034
#> Apr 2009 8.7100000 1.5711864 4.0510332 0.4638768
#> May 2009 5.1600000 1.5711864 1.5810082 0.7041511
#> Jun 2009 4.6300000 1.5711864 1.0000000 0.9989213
#> Jul 2009 4.7100000 1.5711864 0.7902392 1.2859159
#> Aug 2009 4.6400000 1.5711864 0.9830918 1.0182963
#> Sep 2009 NA 1.5711864 1.0000000 NA
#> Oct 2009 NA 1.5711864 0.9164258 NA
#> Nov 2009 NA 1.5711864 0.7131930 NA
#> Dec 2009 NA 1.5711864 0.7591627 NA
plot(y, nc=1, main="")
