# Multivariate Linear Regression with exponential trend-line

The dataset that I'm working with has 2 independent variables (qty, volume) and 1 dependent variable (cost). When I plot individual X with Y, it turns out qty vs cost gives an exponential decay trend line while volume vs cost gives a linear relationship.

I'm trying to come up with a linear model (since it's beginner-friendly) to predict the cost when a new input, volume set is given. I tried to excel and while the statistic output looks reasonable, the predicted values are off quite a lot. I read up on some articles and there is something regarding log transformation so that both qty and volume have a linear relationship with cost. I'm not sure if log transformation applicable in this situation. What would be the best way to approach this problem?

I plan to use R instead of excel next.

Here are some outputs of what I've done

Note: Thickness, width, and height essentially become volume so it doesn't affect the data much

## 1 Answer

I would suggest transforming "qty" into the log space. Even, you can do this using Excel. You can make a new column (for example qty_log), which is equal to the log of "qty". Then, you can fit a linear regression as follows:

$$cost = a1 * log(qty) + a_2 * volume + a_0$$

You will get much better results in this case.

This equation simply means that try to fit a model using volume and log of qty for predicting cost. You did so before but you used qty instead of log(qty)

• Thank you, can you explain in your equation what the term after a1 is? It looks like qtylog, not sure what it means. Jun 17, 2020 at 16:04
• I add two more lines. Just try to transform qty to log space as I explained using excel. Then, fit a model as you did before. You just need to use the new column (which is log of qty) instead of using qty directly. Jun 17, 2020 at 17:08
• I tried it and the predicted values come out much closer to the expected. There's one issue with taking the log(qty) however is that if qty is 1, then log(qty) becomes 0 and it throws off the prediction. I read a discussion saying to do log(qty+1) instead however it gives higher SE. Do you have any suggestions? Jun 17, 2020 at 20:27
• It depends on your data. In most cases, log(qty+1) is the best solution. Jun 17, 2020 at 23:18