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UPDATED for clarity (originally I used the words "missing" and "censored" data interchangeably, whereas only "censored" is accurate in this case).

I am modeling a collection of timeseries. These timeseries are periodic sales for different products over the product's lifecycle. Some products have been launched more recently and have not yet reached maturity in their life cycle. Thus, the data is right censored to a varying degree, depending on how recent was the product launch (about 30% of the data is censored overall).

There are some general trends in the data, specifically, most individual time-series show a growth period until a peak sales level, followed by an exponential decay to zero, and/or a sharp drop to zero, if the product is suddenly discontinued. The amplitude, growth/decay rates, timing of the switch from growth to decay all differ between individual time-series.

I tried a few different models based on MCMC framework, including a hierarchical parametric model and a Multivariate Gaussian Random Walk Model. However, I can’t seem to get a good fit for the data.

The flat parametric model works well on most individual time series but the hierarchical model fit is poor. I am not sure why, but I am guessing that there is too much variability in the data for a multi-level model.

The Multivariate Gaussian Random Walk model does not forecast the censored data correctly, i.e. it forecasts that the timeseries process continues around the level of the last observation, whereas it should be going up until some timepoint, t, and then decaying to zero. Another issue with the Gaussian model is that it forecasts the censored data as negative, which does not make sense for the real-world application. I tried constraining the model to force the likelihood to be located above zero but that slows the sampling down and makes the fit worse.

At this point, I am contemplating fitting many flat parametric models for individual timeseries. However, that might be computationally and logistically difficult, as I would be looking at fitting ~2k models. The second issue is that I would like to use the timeseries with less right-censoring to forecast the ones with more censored data (either via their parameters, autocorrelations, or directly from the data), and I am not quite sure how to do that.

Does someone have a suggestion on how I can try modeling this data? Any tips on what have I not tried / done wrong would be greatly appreciated!

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  • $\begingroup$ Since censoring and missingness are different things (and tend to require different statistical procedures), and yet you appear to use these terms synonymously, could you please tell us what you mean by them? $\endgroup$ – whuber Mar 10 at 16:28
  • $\begingroup$ Thanks @whuber. Updated accordingly $\endgroup$ – nijshar28 Mar 11 at 20:40
  • $\begingroup$ Okay. Now that's cleared up, could you tell us a little more about the nature of the censoring? (There are several ways it can occur and their differences can matter.) For instance, are you measuring a response using an instrument with a fixed upper threshold? Or maybe the threshold varies? Or perhaps even that threshold varies according to the values of preceding observations in time? $\endgroup$ – whuber Mar 11 at 20:43
  • $\begingroup$ Thanks, @whuber. I updated the post again. Briefly, these timeseries are periodic sales for different products over the product's lifecycle. Some products have been launched more recently and have not yet reached maturity in their life cycle. Thus, the data is right censored to a varying degree, depending on how recent was the product launch (about 30% of the data is censored overall). $\endgroup$ – nijshar28 Mar 12 at 23:52

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