# Alternatives to Holt-Winters models when the seasonality pattern has changed

I am forecasting a series of daily volumes in terms of units processed for a particular time period (the period around Christmas).

Historically, I have used a Holt-Winters model, with the minor modification that time period t-364, rather than t-365, is used for the seasonality component and parameter estimation, since the thing I am forecasting has a nested seasonality cycle with a week-within-year cycle and a day-within-week cycle. This has performed pretty well and generally produced unbiased estimators.

However, last year, there was a change in the intra-week seasonality pattern which was caused by a special-cause factor which has remained in place. (The change happened early enough in the year that the values from last Christmas were post-change). Accordingly, I now only have one year's values with which to perform the estimates.

If I understand correctly, this means that the only option I have (if using H-W) is to set the seasonality smoothing parameter to 1, so that the seasonality component of the projections is based exclusively on the corresponding value from one year ago, rather than on a smoothed average from the equivalent seasonal period in all years. (I should point out that, in previous years, the optimised value of the smoothing constant - derived from SSE minimisation) has been 0.9 or greater.

Obviously, this doesn't sound particularly great, but I don't know if there's any alternative forecasting approach I could use which would avoid this problem. I have been looking into possibly trying to fit an alternative ARIMA model, but I don't think that would resolve the basic problem, which is not really to do with how well the HW model fits as it is to do with the absence of data from more than one year, and I don't think an alternative ARIMA model would resolve this. Am I correct in this assumption? And if so, are there any models or approaches which might be better suited for dealing with this problem?

Finally, I assume that it will still be valid to estimate the coefficients for the level and trend parameters using SSE based on historical values - is this the case?

• Your intuition is right that with essentially just one year of post-change data none of the methods would reliably estimate seasonality. The estimates of the level and trend should be fine if they have not changed together with seasonality last year. Commented Oct 2, 2017 at 18:48
• Do you mean "intra-week seasonality pattern" or "inter-week seasonality pattern"? The former would refer to the 7-day pattern over the week, in which case having 40 or so data points (from whenever the change occurred to now) would likely be sufficient for good estimation. The latter would refer to the annual seasonality pattern, which seems to be what you are describing as being affected. Commented Oct 2, 2017 at 19:46
• @jbowman I mean intra-week seasonality. As in, the relative volume of (e.g.) Thursday relative to Friday. That is what has changed, although as mentioned, we also have an inter-week seasonality pattern - one is nested inside the other. I have never examined (quantitatively) the extent to which there is an interaction between week and day, so that, for the sake of the argument, Friday Week 49 is higher, relative to Wednesday Week 49, than Friday Week 49 is relative to Wednesday Week 49. Any such interaction would be captured by the Seasonality component of the Holt-Winters model. Commented Oct 3, 2017 at 14:53
• @jbowman However, I have obviously analysed this qualitatively. Last year, recognising the pattern change, I took the approach of: Using the HW model to forecast volumes for each day Taking the average for each week, and multiplying this by fractions representing the percentage of weekly volume accounted for by each weekday since the intra-week pattern change (based on averages) Qualitatively analysing the specific week/day interaction (which is limited) with subject matter experts and then modifying the numbers as appropriate. Ugly and arguably pseudo-statistical, but it worked well. Commented Oct 3, 2017 at 15:20
• @RichardHardy so am I basically out of luck then? Any alternative time series (or other) model I could use which would circumvent the problem? I'm wondering if using some alternative kind of exotic model to resolve the problem, but I doubt this is possible? Commented Oct 3, 2017 at 15:42