Problems with seasonality removal

Question

My problem is similar to this one from stack overflow: https://stackoverflow.com/questions/23568275/cannot-remove-time-series-seasonality

I'll provide some data and make it more detailed. Please keep in mind that I am a beginner in time series, so do not overcomplicate my problem.

[1] 7.857481 8.122074 8.074026 8.042699 8.231110 8.346642 8.561210 8.395252
[9] 8.276903 8.246958 8.205492 8.241440 7.734121 7.984122 8.001355 8.201386
[17] 8.342364 8.504108 8.633731 8.604105 8.195334 7.971776 8.243283 7.770223
[25] 7.770223 7.828436 8.032360 8.223627 8.331105 8.372399 8.527935 8.544419
[33] 8.104703 8.162801 8.240121 8.144969 7.662468 7.833204 8.012018 8.091015
[41] 8.248529 8.300777 8.393895 8.566935 8.296297 8.030735 8.150468 8.220941
[49] 7.577634 7.864804 8.195885 7.950150 8.304495 8.184235 8.348775 8.416931
[57] 7.969704 7.964156 8.137396 8.058011 7.650169 7.687539 7.909857 7.990915
[65] 8.316789 8.151910 8.291547 8.352083 7.956827 7.971431 8.141190 8.103797
[73] 7.556951 7.702556 7.749322 7.906179 8.197814 8.100161 8.384119 8.398860
[81] 7.928046 7.951559 7.947325 8.231642 7.383989 7.584773 7.717796 8.090096
[89] 8.059592 7.996990 8.304247 8.132119 7.971776 7.845808 8.024862 8.220672
[97] 7.470224 7.693026 7.695303 7.929126 8.031385 8.128290 8.350194 8.044947
[105] 7.833600 7.731931 7.959276 8.161090 7.351158 7.570443 7.707962 7.892078
[113] 8.053569 8.078378 8.322880 8.144969 7.816820 7.761745 8.001690 8.059276
[121] 7.409136 7.452982 7.871693 7.747597 7.998335 8.090402 8.281724 7.978311
[129] 7.774856 7.807103 7.966587 7.855157 7.192934 7.430114 7.806696 7.829630
[137] 7.949444 7.921898 8.138857 8.008366 7.770223 7.609367 7.826842 8.060856
[145] 7.342779 7.404279 7.655391 7.789455 7.959276 7.945201 8.089482 7.865572
[153] 7.724888 7.718685 7.841886 7.957177 7.096721 7.252762 7.582738 7.609862
[161] 7.753194 7.886081 7.980366 7.873217 7.664816 7.479864 7.834392 7.921173
[169] 7.050989 7.357556 7.583248 7.885705 7.763871 7.860185 7.906179 7.738052
[177] 7.789869 7.608871 7.937017 7.810758
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I uploaded two types of the data. They are exactly the same. One is copy from R directly, and the latter is export.csv from R.

First, I plot acf for data and get this:

Well, I cannot post images. But you can get the image by a simple acf command in R.

Then, I difference lag 12 by datadif <- data[13:180] - data[1:168], and the new act plot is:

From the second figure, you can see that lag 12 is still significant. Even I try diff for datadif, I fail to remove seasonality.

What is this phenomenon? Briefly, how can we deal with it? I did not find the answer in the textbook.

Thanks in advance.

Answer 1

Just taking seasonal differences may not be enough or even necessary to deal with seasonality. There may be some further stationary seasonal cycles.

The framework of ARIMA time series models is a possible approach to analyze your data. For example, in R you can use the function forecast::auto.arima to find an ARIMA model based on the BIC criterion. You can call this function from the tsoutliers::tso interface in order to account for the presence of possible additive outliers or level shifts.

Sample R code to apply this idea (for ease of reproducibility the output from dput(x) is posted at the end of this answer):

require(forecast)
require(tsoutliers)
fit <- tso(x, tsmethod = "auto.arima", args.tsmethod = list(ic = "bic"))
resids <- residuals(fit$fit)
fit
# Series: x 
# ARIMA(0,0,0)(1,1,1)[12] with drift         
# Coefficients:
#         sar1     sma1    drift     AO24
#       0.0483  -0.7080  -0.0034  -0.4777
# s.e.  0.1311   0.1046   0.0002   0.0922
# sigma^2 estimated as 0.009102:  log likelihood=147.47
# AIC=-284.93   AICc=-284.56   BIC=-269.31
# Outliers:
#   type ind time coefhat  tstat
# 1   AO  24 2:12 -0.4777 -5.181

You can see that the chosen model includes an autoregressive and seasonal moving average coefficients or seasonal order. An additive outliers is also detected but for the illustration of dealing with seasonality should not make much difference in this case.

As shown in the plot of the ACF and PACF of the residuals, the autocorrelations of seasonal order lie well within the confidence bands:

par(mfrow=c(2,1), mar = c(3,4.2,3,3))
acf(resids, lag.max = 48)
pacf(resids, lag.max = 48)

I don't claim that this is the best model or the only approach to analyze your data. The point that I want to show you is that just by taking seasonal differences, seasonality may not be fully removed or addressed properly.

---

x <- structure(c(7.85748078694253, 8.12207437536222, 8.07402621612406, 8.04269949689764, 8.23110984032815, 8.3466420902212, 8.56121007683301, 8.39525152061099, 8.27690348126706, 8.24695803256818, 8.20549161312024, 8.24143968982973, 7.7341213033283, 7.98412195870293, 8.0013550258267, 
8.20138595523861, 8.34236350038058, 8.50410795186758, 8.63373100766419, 8.60410456340553, 8.19533366716287, 7.97177612288063, 8.24328252304838, 7.77022320415879, 7.77022320415879, 7.82843635915759, 8.0323601479245, 
8.22362717580548, 8.33110454805304, 8.372398606513, 8.52793528794814, 8.54441917766983, 8.10470346837111, 8.16280135349207, 8.24012129807647, 8.14496941708788, 7.66246781520024, 7.83320394864106, 8.01201823915906, 8.09101504171053, 8.24852912480022, 8.30077696085145, 8.39389497507174, 
8.56693528331105, 8.29629711264251, 8.03073492409854, 8.150467911624, 8.22094116828139, 7.57763383260273, 7.86480400332846, 8.1958853913148, 7.95014988765202, 8.30449489796357, 8.18423477409482, 8.34877453979127, 
8.41693076947784, 7.96970358327866, 7.96415571884094, 8.13739583005665, 8.05801080080209, 7.650168700845, 7.68753876620163, 7.9098566672694, 7.99091546309133, 8.31678912707152, 8.15190987294091, 8.29154650988391, 
8.35208267135264, 7.95682712209011, 7.97143099776935, 8.14118979345769, 8.10379671298179, 7.5569505720129, 7.70255611326858, 7.74932246466036, 7.90617884039481, 8.1978140322212, 8.10016144693661, 8.3841188371909, 8.39886000445437, 7.92804560087478, 7.95155933115525, 7.94732502701646, 
8.23164217997341, 7.38398945797851, 7.5847730776122, 7.71779621101358, 8.09009578318096, 8.05959232888755, 7.99699040583765, 8.30424746507847, 8.13211877295581, 7.97177612288063, 7.8458075026378, 8.02486215028641, 8.22067217029725, 7.47022413589997, 7.69302574841789, 7.69530313496357, 
7.9291264873068, 8.03138533062553, 8.12829017160705, 8.35019365072007, 8.04494704961772, 7.8336002236611, 7.73193072194849, 7.9592759601164, 8.1610895128458, 7.35115822643069, 7.57044325205737, 7.70796153183549, 7.89207842124812, 8.05356916913454, 8.07837810362652, 8.3228800217699, 
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7.87321705486274, 7.66481578528574, 7.47986413116503, 7.83439230291044, 7.92117272158701, 7.05098944706805, 7.35755620091035, 7.58324752430336, 7.88570539124302, 7.76387128782022, 7.86018505747217, 7.90617884039481, 7.73805229768932, 7.78986855905471, 7.60887062919126, 7.93701748951545, 
7.81075811652936), .Tsp = c(1, 15.9166666666667, 12), class = "ts")