I started to learn about time series, and I am having difficulty in trying to identify some things just looking in the graph.

enter image description here

What I realized is:

  • Trend: No increase or decrease trend.

  • Seasonality: I think that is not possible say nothing about that, since the data is divided at intervals of 100 years. Seasonality is only linked to the calendar?

  • Outliers: I believe that there is no outliers. Around the year 1700, the series reaches the value 1000, but looking at the years near this, this value does not appear to be discrepant. The same thing happens at the end of the series when the price is below 200.

  • long-run cycle: What I realized is that at intervals of 100 years, at certain times there are peaks.

  • constant variance: There is no constant variance. The variance in the interval 1300-1600 is much lower than in 1800-2000.

  • abrupt changes: I think there is a drastic change if we look at what happened after 1800.

Is there something wrong with what I visualized? Is there anything more?

I tried to plot this time series in R, but it was not good.


The results was it

enter image description here

The dataset has two variables: Year and Price.

[The data are available as the series wheat in the R package fma, and should also be available at datamarket.com]

  • $\begingroup$ You have quite a few inter-linked questions! To start, I would suggest re-considering your first two bullets: trend & seasonality. The term "trend" is not necessarily limited to a persistent linear trend over the entire time series. And the term "seasonality" refers to any cyclical variation, so the period does not have to be yearly (I just learned this term recently myself). So in the big picture: Would you say the time series has a longer timescale smooth variation (trend) with a super-imposed oscillatory variation (posssibly "seasonality")? $\endgroup$
    – GeoMatt22
    Commented Sep 11, 2016 at 23:56
  • $\begingroup$ @GeoMatt22 What I found recently about seasonality is "A seasonal pattern exists when a series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week). Seasonality is always of a fixed and known period." and about cyclic "A cyclic pattern exists when data exhibit rises and falls that are not of fixed period. The duration of these fluctuations is usually of at least 2 years." It is quite confused the difference between cycle and seasonality. $\endgroup$
    – user72621
    Commented Sep 12, 2016 at 0:25
  • $\begingroup$ Data is also here (Ch. 9, Fig. 6) $\endgroup$
    – GeoMatt22
    Commented Sep 12, 2016 at 2:10

1 Answer 1


I see little to indicate any real trend.

What do I mean by that? Well, look at these time series plots:

three Bernoulli random walks

Would you say they indicate trend? Many people would, but in fact all three are fair Bernoulli random walks -- each observation is the previous one $\pm 1$ (with equal chance of either going up or down).

(plot is taken from this post)

I would not be inclined to say that those show a real trend because none of them are actually going anywhere -- the best guess at the next observation is always the current observation. If we look at the first differences, $Z_t=B_t-B_{t-1}$ we get zero-mean Bernoulli "noise" -- $\pm 1$ with equal chance of either.

That idea of also plotting first differences of a series as well as the series itself can often be quite informative.

So let us return to the wheat series. If you plot first differences,

Time series plot of wheat series and first differences

there's little to clearly suggest much beyond a random walk or perhaps random-walk-with-noise. The mean is nonzero but very close to it -- close enough that I wouldn't think to include a drift term in a random-walk model.

[However, that random walk doesn't quite capture all the structure -- if we look at an ACF/PACF of first differences these would indicate some remaining serial correlation]

If by seasonality you're asking about periodicity across multiple years, I see no clear indication of any cycles with a constant period in the data (e.g. ACF/PACF of first differences doesn't indicate anything). There's a mild effect at multiples of 5 year lag but it's quite weak and I'd be inclined to regard it as just noise.

  • $\begingroup$ Can you explain the difference between seasonality and cycle? From what I read seasonality is only related to behaviors within a fixed and known period of time. $\endgroup$
    – user72621
    Commented Sep 12, 2016 at 0:40
  • 1
    $\begingroup$ @PRAGAKHAM You have annual data, so you won't have effects due to seasons themselves (as you would often see with monthly or quarterly data -- e.g. with wheat, it would historically tend to be cheaper right after harvest time though good storage techniques and the ability to ship food worldwide mitigates those kinds of effects. But you can sometimes see periodic cycles of the same kind as seasonality (and I still call it that, though some people object to it) where the period is longer than a year; these things are more rare but can occur for various reasons. ... ctd $\endgroup$
    – Glen_b
    Commented Sep 12, 2016 at 0:47
  • 1
    $\begingroup$ ctd....On the other hand the term cyclical can refer to a varying-length but somewhat regular cycle - e.g. where there's a cycle of up and down around a the trend with an average length of some number of years (say 10, for example) but the actual time between peaks might vary from 8 to 13 years $\endgroup$
    – Glen_b
    Commented Sep 12, 2016 at 0:47
  • $\begingroup$ @PRAG However, if you want a good explanation of the difference, why not ask a question about the difference is? You might quote some of those things you say you have read (with proper references and links where available) for context $\endgroup$
    – Glen_b
    Commented Sep 12, 2016 at 0:50
  • $\begingroup$ @Glen_b I disagree. There appears to be a downward trend starting at about the second half of the 19th Century. This is probably a "real trend" related to the changes in the production process. I'm no agricultural expert, but examples can be found here and here. A key point is that relevant information often lies beyond the data series. $\endgroup$ Commented Sep 12, 2016 at 0:55

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