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Suppose I collected a time-series data (e.g. drug prescription on every month for 12 years). I have no reason to believe that my data is influenced by a seasonal factor (e.g. drug consumption is not influenced by month; in other words there is no reason to believe that a certain drug is more often prescripted on January than on July). By the way, I need a method to assess if my hypothesis is true. Consider this question an explanatory analysis.

Here is the data I collected (number of drug prescriptions on 12 Months x 12 Years):

mydata.ts1 <- structure(c(71000, 71800, 81575, 67920, 84794, 82861, 83325, 
76457, 84856, 85239, 83462, 80482, 90160, 75600, 85240, 78780, 
87520, 89590, 87000, 72760, 72910, 77120, 74520, 67970, 82934, 
64658, 69454, 72709, 81635, 75456, 81548, 61695, 70815, 73175, 
67669, 65700, 74196, 59196, 71878, 64313, 71462, 67512, 72989, 
54281, 70563, 67742, 60547, 67140, 66069, 60681, 65806, 63598, 
63433, 60693, 72621, 56640, 63742, 62789, 63145, 62732, 61949, 
56494, 62700, 59506, 63133, 63900, 62332, 56182, 62930, 58961, 
57935, 60002, 63017, 53995, 60337, 54123, 62467, 60154, 60349, 
53672, 59569, 56979, 57853, 55614, 59170, 53160, 57427, 53174, 
61753, 56018, 58341, 48843, 53526, 57569, 54203, 48811, 57320, 
48970, 48280, 50640, 53340, 51970, 56190, 45590, 49840, 50850, 
46040, 50590, 51710, 45550, 48980, 47890, 48550, 51360, 53340, 
42800, 48170, 49720, 46260, 49400, 46960, 44070, 50440, 46300, 
46020, 50410, 50500, 41290, 46560, 45820, 44780, 48410, 45220, 
43350, 47370, 43800, 46910, 47960, 46750, 42110, 45740, 46070, 
46870, 47170), .Tsp = c(1, 12.9166666666667, 12), class = "ts")

Simply by plotting my data, I can see that a certain kind of "seasonality" is present in my data (against my initial hypothesis). stl plotting show a decreasing trend on drug prescriptions over years, with some effect of the variable "month".

plot(stl(mydata.ts1, s.window=12))

enter image description here

The summary of stl function is quite ambiguous to me, since it gives to me no information about how much my data is influenced by the seasonal component :

summary(stl(mydata.ts1, s.window=12))
 Call:
 stl(x = mydata.ts1, s.window = 12)

 Time.series components:
    seasonal             trend            remainder        
 Min.   :-6818.566   Min.   :45844.41   Min.   :-6419.665  
 1st Qu.:-1833.162   1st Qu.:49439.62   1st Qu.:-1084.186  
 Median :  804.645   Median :59426.75   Median :  350.938  
 Mean   :    7.947   Mean   :60599.00   Mean   :   49.549  
 3rd Qu.: 2071.084   3rd Qu.:69002.22   3rd Qu.: 1553.892  
 Max.   : 5778.352   Max.   :83116.20   Max.   : 6619.956  
 IQR:
     STL.seasonal STL.trend STL.remainder data 
      3904        19563      2638         18531
   %  21.1        105.6      14.2         100.0

 Weights: all == 1

 Other components: List of 5
 $ win  : Named num [1:3] 12 21 13
 $ deg  : Named int [1:3] 0 1 1
 $ jump : Named num [1:3] 2 3 2
 $ inner: int 2
 $ outer: int 0

Finally, tslm function gives information to me that a clearly seasonal effect is present in my data:

library(forecast)
summary(tslm(mydata.ts1 ~ trend + season))

Call:
lm(formula = formula, data = "mydata.ts1", na.action = na.exclude)

Residuals:
     Min       1Q   Median       3Q      Max 
-11254.6  -2329.6   -249.4   1767.3  11613.6 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 82529.062   1223.421  67.458  < 2e-16 ***
trend        -274.433      7.725 -35.525  < 2e-16 ***
season2     -7407.317   1567.737  -4.725 5.85e-06 ***
season3     -1135.968   1567.794  -0.725  0.47001    
season4     -4756.036   1567.889  -3.033  0.00292 ** 
season5      1207.063   1568.022   0.770  0.44280    
season6       387.079   1568.194   0.247  0.80543    
season7      2944.928   1568.403   1.878  0.06265 .  
season8     -7861.056   1568.650  -5.011 1.72e-06 ***
season9     -1178.206   1568.935  -0.751  0.45402    
season10     -669.357   1569.259  -0.427  0.67041    
season11    -2790.758   1569.620  -1.778  0.07773 .  
season12    -2454.909   1570.019  -1.564  0.12032    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3840 on 131 degrees of freedom
Multiple R-squared:  0.9125,    Adjusted R-squared:  0.9045 
F-statistic: 113.8 on 12 and 131 DF,  p-value: < 2.2e-16

My final question is: Given a certain time series, how can I assess if the seasonality effect is influencing my data? How much is my data influenced by the seasonal effect?

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Another way to explore possible seasonality is a cycle plot. Here is one:

enter image description here

This rearranges your series in blocks, in each this case blocks of months. So all Januaries are plotted together, etc. The month labelled 1 is the month that comes first in your data. I am guessing it is January. The horizontal lines are the means for each month.

As with your plot of the raw series it shows that trend is dominant, but there is some seasonal structure. February is low (shorter month is part of the answer); August is low (people on holidays, feeling better in the summer sun if this is Northern Hemisphere)?

I have bundled together some references for this plot, including other names used in the literature:

Becker, R. A., J. M. Chambers, and A. R. Wilks. 1988. The New S Language: A Programming Environment for Data Analysis and Graphics. Pacific Grove, CA: Wadsworth & Brooks/Cole, pp. 508-509. [month plot]

Cleveland, R. B., W. S. Cleveland, J. E. McRae, and I. Terpenning. 1990. STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics 6: 3-73. [cycle-subseries plot]

Cleveland, W. S. 1993. Visualizing Data. Summit, NJ: Hobart Press, pp. 164-165. [cycle plot]

Cleveland, W. S. 1994. The Elements of Graphing Data. Summit, NJ: Hobart Press, pp. 186-187. [cycle plot]

Cleveland, W. S., and S. J. Devlin. 1980. Calendar effects in monthly time series: Detection by spectrum analysis and graphical methods. Journal of the American Statistical Association 75: 487-496. [seasonal-by-month plot]

Cleveland, W. S., A. E. Freeny, and T. E. Graedel. 1983. The seasonal component of atmospheric CO2: Information from new approaches to the decomposition of seasonal time series. Journal of Geophysical Research 88: 10934-10946. [seasonal subseries plot]

Cleveland, W. S., and I. J. Terpenning. 1982. Graphical methods for seasonal adjustment. Journal of the American Statistical Association 77: 52-62. [seasonal subseries plot]

Cox, N. J. 2006. Graphs for all seasons. Stata Journal 6: 397-419. [cycle plot] [http://www.stata-journal.com/sjpdf.html?articlenum=gr0025 ]

McRae, J. E., and T. E. Graedel. 1979. Carbon dioxide in the urban atmosphere: Dependencies and trends. Journal of Geophysical Research 84: 5011-5017.

Robbins, N. B. 2005. Creating More Effective Graphs. Hoboken, NJ: Wiley. [month plot, cycle plot] (reissued 2013)

Robbins, N. B. 2008. Introduction to cycle plots. Perceptual Edge Visual Business Intelligence Newsletter January .pdf original here

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    $\begingroup$ Great answer and excellent bibliography. By the way, except from a visual plot analysis, is there a formal method to assess the presence of a seasonal factor? On scientific papers, usually, visual plot analysis is not considered good enough $\endgroup$ – Tommaso Jan 24 '16 at 20:40
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    $\begingroup$ Indeed. You need a model. @Tom Reilly's answer indicates one direction. My own impression is that structural time series models would be a serious alternative. $\endgroup$ – Nick Cox Jan 24 '16 at 23:07
  • $\begingroup$ what about this article: Autoregression as a means of assessing the strength of seasonality in a time series? Is something similar to the last example of the this lesson? $\endgroup$ – Tommaso Jan 25 '16 at 23:27
  • $\begingroup$ @NickCox it would be worth adding an R code generating this plot $\endgroup$ – Kasia Kulma Dec 4 '18 at 15:10
  • $\begingroup$ @Kasia Kulma I can't believe it doesn't exist already. Cleveland worked in S back in the day. if it doesn't, well, perhaps R people should note that Stata people sometimes get ahead too. $\endgroup$ – Nick Cox Dec 4 '18 at 15:17
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Why aren't you looking at the ACF/PACF? The PACF has a spike at period 12 signalling seasonality. The initial model is a constant, an AR1 and an AR12. However, the parameters change over time so (using the Chow test) there is break identified at period 101. If you proceed using just the last 44 observations, you will still find that there is an AR 12, a constant an outlier at period 123 as that month isn't usually that high.

Y(T) = 42728.
+[X1(T)][(+ 2532.0 )] :PULSE 11/ 3 123 + [(1- .726B** 12)]**-1 [A(T)]

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    $\begingroup$ Since I'm not really into time-series analysis: saying that AR12 is a good model for my data, is sufficient enough to accept the hypothesis that a seasonal effect is present in my data? $\endgroup$ – Tommaso Jan 24 '16 at 20:41
  • $\begingroup$ Yes, your data has seasonality. $\endgroup$ – Tom Reilly Jan 29 '16 at 13:04

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