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In a time series dataset, demand from various customers are given on a daily basis. When the data is aggregated at month level for all the customers together it is easy to see the effect of quarterly and monthly seasonality. Higher volumes are seen in Q3 and to some extent in Q4. This is from simple cycle plots, STL decomposition, and ACF plots.

Many customers do not receive volumes at regular time intervals, i.e. there could be many days on which a customer does not receive any volumes.However, some customers may be getting daily shipments. The volumes also vary from one shipment to another.

Now, on similar analysis for a single customer, or even some groupings of customers, seasonal patterns are not evident.

What is a good explanation for having a clear seasonality in the total volumes shipped but not in the components? In other words, where is the seasonal pattern in the total coming from if it is not present at lower levels of aggregation (e.g. customer)?

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    $\begingroup$ These descriptions are vague, can you provide the visualizations or tabulations that are supporting these conclusions, e.g. "There are a dozen or so customers [whose] volumes ...are non-periodic."? $\endgroup$
    – AdamO
    Commented Mar 16, 2021 at 20:23
  • $\begingroup$ @AdamO Thanks for your comment. Unfortunately, I am not able to share data but I will clarify my post. $\endgroup$
    – PhatStats
    Commented Mar 16, 2021 at 21:13

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Most likely, seasonality is present at the lower levels. It's just that the signal to noise ratio is much lower at these levels. Aggregation improves this ratio to the point where seasonality becomes detectable.

A corollary: even if seasonality is present at lower levels, you may get better forecasts with non-seasonal models. These will be biased, but have lower variance, which is the bias-variance tradeoff.

As a matter of fact, I wrote a little paper that used exactly this phenomenon to illustrate why a simpler wrong model can yield better forecasts than a more complex correct one. Specifically, I simulated a lot of seasonal time series with sufficiently high levels of noise that fitting seasonality yielded worse forecasts - but the seasonality was blatantly obvious on aggregate levels. Take a look at Kolassa (2016, Foresight) if you are interested.

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  • $\begingroup$ Thanks. Can aggregating volumes for a customer or a customer group at weekly level provide a better chance of detecting seasonality? $\endgroup$
    – PhatStats
    Commented Mar 16, 2021 at 21:26
  • $\begingroup$ That sounds reasonable. Not least since there may be intra-weekly seasonality in the daily data, which you presumably are not interested in. $\endgroup$
    – Stephan Kolassa
    Commented Mar 16, 2021 at 21:28
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    $\begingroup$ Stephan Kolassa I wanted to ask, how can having a biased model ever be ok. I tend to think of bias as the cardinal sin of statistics. :) $\endgroup$
    – user54285
    Commented Mar 19, 2021 at 22:33
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    $\begingroup$ @user54285: that is a great question! The answer hinges on the difference between inference and prediction. If we want to do significance testing, then a misspecified model is indeed a major problem, because then all the theory about the p values goes out the window. ... $\endgroup$
    – Stephan Kolassa
    Commented Mar 20, 2021 at 9:06
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    $\begingroup$ ... But suppose we are interested in prediction/forecasting and want to minimize the mean squared error. By the bias-variance trade-off, we may (may!) be able to reduce the expected MSE if we reduce the variance of the model by more than we increase the bias, so accepting some bias in exchange for a large reduction in variance may lead to better predictions. The paper I cited may make for interesting reading; feel free to ask me for it via ResearchGate. $\endgroup$
    – Stephan Kolassa
    Commented Mar 20, 2021 at 9:07

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