Predicting amount of purchases

Question

I have a problem that requires the prediction of the number of customers who will buy a product every month segmented on the date they came into the site. For example, my data looks like:

Customer | Start Date | Purchase Date | Predictors ...
   A       2010-01-11     2010-02-01     
   B       2010-01-11        NULL
   C       2010-01-11        NULL
   D       2010-01-11     2010-01-15
   E       2010-01-12     2010-01-18
   F       2010-01-12        NULL
   G       2010-01-12     2010-03-02
   H       2010-01-13        NULL
   I       2010-01-13        NULL
   ...

In the example, 2 customers converted from 2010-01-11 segment whereas, no customers in the 2010-01-13 segment bought the product.

I think there are multiple ways to approach this problem but cannot decide on a method. My question is three-fold.

1) Survival Analysis/Poisson Regression: I could use survival analysis to predict purchase but this would not help solve the time to purchase problem for customers who did purchase. Instead, I could try Poisson regression but I'm not sure how this would answer the purchase/non-purchase problem. Would a nested logistic model be appropriate in this case?

2) Improving prediction: I would like my predictor to keep getting more accurate as every month passes because the cumulative number of customers predicted should never decrease month-on-month. Would any model be more suitable to this assumption?

3) Time-Series for date segment: Instead of predicting for a customer, could I predict the number of customers who do purchase by date segment by month? Similar to obtaining multiple time-series and bootstrapping.

Update: It seems that a hurdle model approach is suited well to this problem. However, I have a problem.

1. My data does not have any 0 purchase dates but NULL which represents not purchasing. This means that the time to purchase for a nonpurchasing user is NULL. How can I transform my data so that there is a 0 hurdle?

Answer 1

It sounds like you would like to make a "two-stage" prediction about each customer:

1. Does the customer buy the product at least once after signing up?
2. If so, in how many months does the customer buy the product?

This is a perfect use case for a "hurdle model", which has a two-stage specification that mirrors your problem statement. Let $Y$ denote the number of months in which a user buys the product after signing up. Then a hurdle model has the following general specification: $$\begin{align} Y &\sim \operatorname{Bernoulli}\left(1 - \pi\right) \\ Y=y\ \vert\ Y \neq 0 &\sim D\left(\theta\right) \end{align}$$

for some distribution $D$ parameterized by $\theta$. In your case, it makes sense to use a Poisson hurdle model, in which $D$ is a Poisson distribution with parameter $\lambda$.

McDowell (2003) provides an excellent introduction to hurdle models, and specifically Poisson hurdle models, so I won't go into more detail here. There is also a lot of useful information in What is the difference between zero-inflated and hurdle distributions (models)? and its answers. Among some of the gems you'll find in there are a link to an R package called pscl which contains functions for fitting hurdle models.

Reference: McDowell, Allen. (2003). "From the help desk: hurdle models." The Stata Journal, 3(2), 178-184. Available online at http://ageconsearch.umn.edu/bitstream/116071/2/sjart_st0040.pdf