Band-pass filter in R: weird behaviour at the end of time series

Question

I have this time series

t

            Jan         Feb         Mar         Apr         May         Jun         Jul         Aug         Sep         Oct         Nov         Dec
1922 -0.25108773 -0.27732553 -0.29703807 -0.30274000 -0.30323653 -0.28441682 -0.24106527 -0.18705071 -0.17440826 -0.17291725 -0.19116734 -0.21678948
1923 -0.24487998 -0.26658925 -0.28613991 -0.29674346 -0.29335742 -0.28325761 -0.23326680 -0.18697904 -0.18443807 -0.18144226 -0.18190910 -0.21574376
1924 -0.24465806 -0.27349425 -0.29925888 -0.30386766 -0.30250722 -0.27464960 -0.23390958 -0.19300616 -0.17910621 -0.17869576 -0.19611839 -0.20447324
1925 -0.25326812 -0.27344637 -0.29352971 -0.30947682 -0.30872025 -0.27604449 -0.24065208 -0.19676031 -0.17172229 -0.18484153 -0.19542607 -0.21841577
1926 -0.25214568 -0.27450911 -0.29438956 -0.30392114 -0.30619846 -0.29089168 -0.24829621 -0.20204202 -0.18621514 -0.18808172 -0.19708748 -0.22629595
1927 -0.25107357 -0.27204514 -0.29494695 -0.30751442 -0.30800040 -0.28569694 -0.24655626 -0.19547608 -0.19018517 -0.18866641 -0.20132372 -0.22084811
1928 -0.24733214 -0.27490388 -0.28780308 -0.30407576 -0.30857301 -0.28629658 -0.23872777 -0.19590465 -0.18437917 -0.18274289 -0.19936931 -0.22368973
1929 -0.25531870 -0.27264628 -0.29418746 -0.30385231 -0.31022219 -0.27931003 -0.23404912 -0.19538227 -0.17226595 -0.18465123 -0.19072933 -0.22043396
1930 -0.24735028 -0.27386782 -0.29193707 -0.29925459 -0.30039372 -0.28014958 -0.23551136 -0.19511701 -0.18006660 -0.18282789 -0.20113355 -0.22095253
1931 -0.24903438 -0.27439043 -0.29219506 -0.30312159 -0.30557600 -0.28180333 -0.22676008 -0.19048014 -0.18982644 -0.18459638 -0.19550196 -0.22127202
1932 -0.25870503 -0.27650825 -0.28521052 -0.30685609 -0.30896898 -0.28378619 -0.23614859 -0.18945699 -0.17575919 -0.17820312 -0.19620912 -0.21774873
1933 -0.24187599 -0.25575287 -0.28325644 -0.29554461 -0.29018996 -0.27040369 -0.23514812 -0.19935749 -0.18732198 -0.18606057 -0.19327237 -0.22321366
1934 -0.24793807 -0.26986056 -0.29217378 -0.30479126 -0.30199154 -0.27574924 -0.24097380 -0.18560708 -0.18643606 -0.18501770 -0.19375478 -0.22418002
1935 -0.25587642 -0.27805131 -0.29239104 -0.30784907 -0.30459449 -0.28216514 -0.23839965 -0.20137460 -0.18619998 -0.18328896 -0.20121286 -0.22869388
1936 -0.25322320 -0.28025116 -0.29713940 -0.30800346 -0.31177201 -0.28473251 -0.23552472 -0.20313945 -0.18251793 -0.18383941 -0.20554430 -0.23061875
1937 -0.26268769 -0.28529769 -0.30230641 -0.31107806 -0.30183547 -0.28324508 -0.23840574 -0.19862786 -0.19297314 -0.19392849 -0.19603212 -0.22877177
1938 -0.25445601 -0.28160871 -0.29837676 -0.29879519 -0.30328832 -0.28288226 -0.23577573 -0.19521124 -0.18393512 -0.19039895 -0.20537533 -0.21924241
1939 -0.25180969 -0.28199995 -0.29601764 -0.30147945 -0.30372884 -0.27837795 -0.23720063 -0.19929773 -0.18770674 -0.19341142 -0.20753282 -0.22484697
1940 -0.15145157 -0.16596690 -0.17572643 -0.18225920 -0.18823836 -0.17504012 -0.16019626 -0.12920340 -0.12369614 -0.12024704 -0.12891992 -0.14234080
1941 -0.10045275 -0.11095497 -0.11585389 -0.11932455 -0.11976700 -0.11653216 -0.10259231 -0.08271703 -0.07621320 -0.07184160 -0.07284514 -0.07385666
1942  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1943  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1944  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1945  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1946  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1947  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1948  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1949  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1950  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1951  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1952  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1953  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1954  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1955  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1956  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1957  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1958  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1959  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1960  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000
1961  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000

I am running a band pass filter using the bkfilter function in the mFilter package

t.bk <- bkfilter(t, pl=9.7,pu=16)

The first plot shows the 'raw' time series t while the second plot shows the cycle component of t.bk.

The red line indicates where t goes to zero and where also the t:Cycle should go to zero. However, the latter after the red line is still wobbling.. Any help?

Answer 1

This is to be expected. Are you familiar with Gibbs Phenomenon, as it relates to BK-filters (or any finite length filter, for that matter)? They oscillate around the power transfer function of the filter.

Here is a paper that discusses a modification to the standard BK-filter to reduce this oscillation, though with more limited input response, of course: http://www.gla.ac.uk/media/media_219052_en.pdf

Fortunately for you the mFilter package source code is available on the CRAN website: https://cran.r-project.org/web/packages/mFilter/index.html The file you would modify is written in R--not C like some packages--and is found in mFilter/R/bkfilter.R. The part of the function you would modify is here:

if(type=="fixed")
{
    bb = matrix(0,2*nfix+1,1)
    bb[(nfix+1):(2*nfix+1)] = B[1:(nfix+1)]
    bb[nfix:1] = B[2:(nfix+1)]
    bb = bb-sum(bb)/(2*nfix+1)

    for(i in (nfix+1):(n-nfix))
        AA[i,(i-nfix):(i+nfix)] = t(bb)
}

Compiling and installing packages you have modified is simple. Go to the directory where the mFilter directory is and enter from the shell command line: R CMD INSTALL mFilter. The next time you go into R the mFilter package will use your modified bkFilter() function.