# Analyzing online sales where data are only produced when the sales is made

I am trying to model data on the number of online sales are made within a fixed sale period of 3 days. Data are generated only when the sale is made. I think for this kind of data I will be using a type of time series model for count data with inflated zero samples. But this dataset adds one more layer of complexity of the time limit. I don’t know any time series statistical model that limits the length of time. I may consider the stochastic process as a (Poisson like) draw of the limited number of trials: if it’s hourly then 72, if it’s by minutes then 4320 trials. By then, if it’s by minutes, the data ends up with too many zero observations, which ruins the effectiveness of the model. Do you have any better idea?

The data is structured like this:

 2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-02 19:15:39        45
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-02 22:32:06        46
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-02 23:39:05        47
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-02 23:20:09        48
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 01:32:09        49
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 03:33:11        50
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 02:47:07        51
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 04:05:05        52
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 04:01:08        53
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 06:53:09        55
2011-10-31 07:01:55 2011-11-03 06:59:59  2011-11-03 07:00:54        56


The first and second times (1st and 2nd column) are the start and end times of the deal, which is about 3-day apart as the deal lasts for 3 days, and of course fixed for each deal. The third time (in the 3rd column) is increasing as the deal goes by with the cummulative number of items sold on the last columne as a positive integer in the 4th columne. The data is generated only when the sale is made, so the time between succesive sales is irregular, which could be a key term to search for the appropirate model. After further search for the appropriate model, I found integer autoregressive model (INAR) may be used, but I am not sure... Since the sale is recorded by the second, I may think the data series with T=259200 observations having many (too many!) zero observations. I may transform the data by minutes by summing over the sales numbers withine one minute to reduce to T=4320. This will reduce the ratio of zero-total observations smaller, and apply INAR or some kind of zero-inflated stochastic model (truncated poisson or something like that). Could you give me any good ideas?

• Why do you want time series as opposed to a multi-level model or even just summing the 3 days? Are you interested in day-of-week effects? – Peter Flom Aug 17 '12 at 6:50
• Could you give some more information on why you are modelling this and what is of interest? Is it the time into the deal you are interested in, the time between sales, just the number of sales, or what... Also do you have just one three day period or lots of them? – Peter Ellis Aug 18 '12 at 4:16
• The ultimate goal is to estimate the total number of items sold at the end of the deal. The data has a lot of deals data: 126350 deals. Each deal has the data for the number of items sold on what day and at what time between a 3-day period. Some deals have more than 60 sale occurances, and some deals have only 1 or 2 sale occurances, but I ignore those, and focus on the deals that has many sales and total item sold are more than 1000. I thought it would be better to model the process first, and the total sales can be calculated by figuring out the expected values of the sum of counts. – Ikuyasu Aug 18 '12 at 17:12
• Somehow, I am not given any covariates other than above. I included dates manually: Monday, Tuesday,... because I think dates are important for people to do shopping online. If I can assume the same interval of time such as every minute or every second, then I can have a balanced panel for a subset of those 126350 deals. But I don't think each deal data behave any way similar, so it doesn't make sense to use panel count data model unless I have more covariates to control the enough heteorgneity across the deals. I don't even have data for the original price of the deal. – Ikuyasu Aug 18 '12 at 17:20
• ALL sales data looks like this, because in every case a sale occurs at a particular time. Usually it is aggregated to a reasonable time unit, e.g. number of sales in an hour, day, or a week, with zeroes put in where appropriate. A very simple analysis to start might be "I'm 15 hours into a 36 hour deal and have sold 200 items. Looking at past 36 hour deals, what proportion of total sales are made in the first 15 hours." [assuming all 36 hour deals have same daily seasonality, e.g. all are Friday through Sunday] – zbicyclist Aug 18 '12 at 22:17