# Forecasting daily time series with many zeros

I need to forecast a univariate time-series of sales data with the following characterica.

• It is a daily time-series
• Around 70-80 % of the date nothing is sold ($x_t = 0$)
• At the 20-30 % remaining days there is a positive integer numberof sales
• The days during which nothing is sold are not always at the sameay day of the week

Until now I tried the croston-method (croston() from the forecast package in R).

Is the croston-method appropriate? Are there any suitable alternatives?

I am also grateful for code in R.

Edit:

My data looks similar to the data below:

0,0,1,0,0,0,0,2,0,0,0,0,0,0, 0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0

• Please post your data and I will try to help you ... AUTOBOX has a number of special control files which might be useful . Please specify the starting date and country. – IrishStat Dec 5 '17 at 9:32
• Thank you. Finally I had problems running AUTOBOX on my OS. I added the data. – Ferdi Dec 5 '17 at 15:46
• Is that all you have 6 weeks and 42 values ? ... If you have a longer series please post it as 1 column – IrishStat Dec 5 '17 at 15:58
• Thank you for your answer. I have 14 weeks, but the original data is confidential. – Ferdi Dec 5 '17 at 16:13
• Why not try a zero-inflated Poisson model with autocorrelated errors? Maybe I am doing a gross over-simplification here but you can model the "hurdle mechanism" with a binomial and then then amplitude mechanism with a truncated Poisson/Negative Binomial. – usεr11852 says Reinstate Monic Dec 5 '17 at 19:32

(This answer is based on experience with the business side of sales forecasting, more so than on rigorous statistical/mathematical knowledge)

Looking at your data, it makes more sense to forecast it at a weekly level than at a daily level. At at daily level it is too sparse, but at a weekly level you would have a more meaningful times series.

week 1: 0,0,1,0,0,0,0

week 2: 2,0,0,0,0,0,0

week 3: 0,0,0,1,0,0,0

week 4: 1,0,1,0,0,0,0

week 5: 0,0,0,0,0,0,0

week 6: 1,0,0,2,0,0,0


Any forecasting method you would use at a daily level, would give a fractional value per day. This doesn't really help, since these are sales units, so a forecast value of ~ 0.14 doesn't mean much, unless you interpret it as a probability (and I don't know enough math to help in that case, but others might know better how to treat that).

If you aggregate the data by week, you get:

week 1: 1

week 2: 2

week 3: 1

week 4: 2

week 5: 0

week 6: 3


You can then simply average that value over all the weeks you have, or maybe use a moving average. You would then get an average of 3 units sold per two weeks.

Keep in mind that this is a sales forecast: What is the purpose of a sales forecast? To make sure that you have enough inventory to satisfy customers' demand. Based on the method I described above, you would know that you need to ship/order 3 units of inventory every 2 weeks to satisfy the demand for that product - without going into ARIMA or Exponential smoothing or some other more involved time series analysis.

Croston's method is definitely an appropriate choice for this case. Its basic idea is to estimate non-zero demand and inter-demand interval separately. But note that its output is actually "demand rate", not actual demand units (e.g. a forecast of 0.1 means a demand of 1 unit over 10 periods). The exact timing of the demand is actually not provided.

tsintermittent package provides some alternatives for intermittent time series forecasting, including iMAPA and Teunter-Syntetos-Babai method. This package also lets you use some adjustments to deal with the bias of Croston's method, like Syntetos-Boylan approximation.