How to predict item purchase price with sparse purchase history I want to predict the price of the next item a user purchases based on the prices of the items they have purchased in the past.  The caveat is that most users have less than 3 previous purchases, so I am wondering what would be a good approach for predicting a purchase when the historical data is so sparse?
 A: I would use a multiple regression with the following variables:


*

*$y_{i,j}$, the price paid by person-profile $i$ for good $j$

*$x_j$, a categorical variable for the product category (e.g. car, motorbike, boat etc.)

*$c_i$, a categorical variable of customers after they have been clustered based on their purchasing habits


The model becomes $\hat{y}_{i,j} = \beta_0+\beta_1 x_j + \beta_2 c_i$.
Controlling for the product is easy to explain: different product naturally hold different values (compare a house to a doll house).
Clustering the customers addresses the sample size somewhat. I.e. you now have larger, homogenous groups of customers. These will need to represent customer's price sensitivity (high, medium, low).
A: EDIT. I just finished a book on time series analysis so i am updating this with the best found answer.
The main class of models for doing time series modeling are called auto-regressive (AR) models.  The most commonly used are ARIMA models (Auto-Regressive Integrated Moving Average) which also incorporate noise terms.  
These models are of the form:
$$f(x_t) = \alpha_1 x_{t-1} + ... + \alpha_p x_{t-p} + w_t + b_1 w_{t-1} + ... + b_q*w_{t-q}$$
Where the $w_{i}$'s are noise values, the $x_{i}$'s are previously seen values in the sequence.
The constants of the model would be learned by minimizing the error between the predicted value of the model and the observed value:
$ \alpha_1,..,\alpha_p,\beta_1,..,\beta_q = \text{argmin}_{\alpha_1',..,\alpha_p',\beta_1',..,\beta_q'}  \sum_{i=1}^{N}\sum_{t=max(p,q)}^T (x^{(i)}_t - f(x^{(i)}_t))^2 $
