# Forecasting daily data with trend, yearly, day of the week, and moving holiday effects

I'm expanding a question I posed earlier because I think it was lacking detail.

I'm attempting to forecast daily demand for a restaurant that sells take away food, primarily to office workers on their lunch breaks. They are located in the downtown core of a major city.

They are only open on workdays - no holidays, no weekends. I'm familiar with models that take into account seasonality and trend - Holt-Winters triple exponential smoothing, for example. I'm also familiar with models that take into account complex seasonality and trend - the TBATS package for R, for example.

My problem is that I've identified 8 components that determine sales on a given day:

1. The yearly seasonal component. Sales are lower in the summer, for example, when many office workers are on vacation.
2. The weekly component. Sales very obviously peak on Thursdays (in the absence of other effects - see below)
3. The Friday-Long-Weekend-Effect. The weekly sales pattern changes if the coming Friday is a holiday. Wednesday will typically have higher sales, for example.
4. The Post-Friday-Long-Weekend-Effect. The weekly sales pattern changes if the week before was shortened due to the Friday being a holiday.
5. The Monday-Long-Weekend-Effect. The weekly sales pattern changes in week $t$ if the Monday in week $t+1$ is a holiday. For example, sales are much lower on Fridays preceding Monday-Long-Weekends. Presumably people are leaving the office early and skipping lunch.
6. The Post-Monday-Long-Weekend-Effect. The weekly sales pattern changes if the week is shortened due to the Monday being a holiday.
7. The trend component.
8. The noise component.

If holidays fell on the same date every year, then the "long-weekend-effects" would be captured in the yearly seasonal component. However, they don't.

My first thought was to include dummy variables. For example, let $X_{M+1}$ be the "Monday-long-weekend-effect" component, and $\beta_{M+1}$ be the associated coefficient, for a given day. Then for the Friday preceding a Monday-Long-Weekend, $X_{M+1}=1$, and for a Friday not preceding a Monday-Long-Weekend, $X=0$.

I'm only using three years of data, so it would be easy for me to change the $X_{M+1}$ values to 0 or 1 by hand for each year. However, I don't know how to include such dummy variables in models like those that I've mentioned.

Any input as to a model that can take into account the components I've mentioned would be greatly appreciated. It seems like I need to capture moving-holiday-effects, day-of-the-week effects, seasonal patterns, and trend, all in one.

Question: Is there a model I can use that can be implemented in R and take into account the components I've listed?

My background: I'm a forth year mathematics and economics student. I've also taken statistics classes, and I'm using R to perform my analysis. This is for a final report for a forth year data analysis class.

• Because this is for school work, your question should bear the self-study tag. – Patrick Coulombe Mar 10 '14 at 21:57
• [Searches](stats.stackexchange.com/search?q=[time-series]+trend+seasonal+holiday) can often turn up useful posts. I see a couple there that seem to be relevant. – Glen_b Mar 10 '14 at 22:58
• Respectfully: please include questions when submitting "questions." – Jack Ryan Mar 11 '14 at 12:35
• Do you really need a daily forecast, or might a weekly forecast do? Also, you've identified 4 long-weekend effects in only three years of data? That's not much data for events that happen 10 times a year (with other confounding factors as well). – Wayne Mar 11 '14 at 20:27