Seeking regression modeling strategies for predicting prices based on categorical variables (one of which is ordered) I have a question similar to this one, which never received an answer.
Let's say I have widgets that have different quality ratings $q\in\{0,1,\dots,N_q\}$ and which are in different regions $R\in\{R_1,R_2,\dots,R_{N_R}\}$. Suppose that the price of these widgets is based on, in a time-varying manner, both objective quality (for which quality ratings are imperfect proxies) and regional effects. Prices are known to fluctuate a little and offer somewhat noisy signals. Finally, let's say that I have an amazing black-box model for the time-varying prices of medium quality ($q=5$) widgets in $R_{Canada}$ and want to find a simple way to adjust this model to work for different qualities and regions.
In practice it seems that when there is enough data for a given quality/regional pair that a simple multiplicative adjustment is not bad. But how can I impute this adjustment for quality/region pairs for which I don't have much data?
If these were continuous, rather than categorical values, imputation isn't necessary as you could compute the covariance of price on the various variables (perhaps after a transformation) and then back out the multiplicative adjustment based on the Canadian 5-quality model predictions. But here although quality is ordered, region is not. I have heard that factor analysis can be useful here, but I don't have the slightest idea how.
Does anyone have some regression modeling strategies that I might explore here? I've considered hierarchical models as a commentator on the above-linked post considered, but though I think I have used hierarchical models before I don't see how they would extend to this problem.
What strategies might I explore here?
 A: One approach is to share information across individual (region,quality) problems. 
For example, you could use a multi-task learning framework to couple the individual adjustments (at the quality level) of the Canada-5 model. For each region independently, fit a linear adjustment by minimizing some loss function like this one:
$L_R(\alpha,\beta) = \sum_{q} (\alpha_{q}+\beta_{q} C_5-Y_{q})^2+\lambda \sum_{q>q^\prime}H(\alpha_{q}+\beta_{q} C_5-\alpha_{q^\prime}-\beta_{q^\prime} C_5)$,
where $C_5$ is the output of the Canada-5 model, and $H$ is the hinge-loss function $H(x)=\left\lbrace \begin{array}{cc} -x, & x<0\\ 0, & \text{otherwise} \end{array}\right.$
This is just $L_2$ loss with a hinge-loss penalty on different quality models. This would penalize $\alpha$'s and $\beta$'s in such a way as to encourage the predictions of higher quality models to be higher than lower quality models in the same region. 

Another approach is to use transfer learning to train a new model for each (region,quality) pair using the Canada-5 dataset as the source domain. One such method is outlined in Want et al. (2014). Active learning under Model Shift. The model shift method is to 


*

*Fit a model to predict $y_{source}=f(x_{source})$. (You've done this, it's your Canada-5 model)

*Apply the model to the target domain to get the offsets $z=y_{target}-f(x_{target})$.

*Fit another model to the offset $z=g(x_{target})$.

*Apply the offset model to the source domain to get $\hat{y}_{source}=g(x_{source})+y_{source}$. 

*Train the target domain model on the data set $(x_{target},y_{target})\cup(x_{source},\hat{y}_{source})$.


In the paper, the authors use Gaussian processes to model the offsets and such, but it seems to me the approach will admit other types of models.

Both ideas could be combined; one could conceive of a multi-task transfer learning problem. 
