# Theoretical reason for 1 linear model not being able to model a bimodal distribution?

Recently, I got a set of data where I try to predict the label (a continuous variable between 1000 - 3500) given 13 feature variables. By applying the kernel density approximation on the label (shown below), I was able to see a bi-modal distribution. I assume that this is because there are two "operating conditions". Each corresponding to each distribution.

After some research, I decided to separate the 2 operating conditions using a clustering technique such as k-means, GMM, DBSCAN, etc. After separating the 2 operating conditions, I was going to build a separate model for each distribution. During live implementation, depending on the distance of the new data to each cluster, I was going to use a mixed model to do the prediction on the output.

My questions are:
1. Is the above a right way to even tackle the problem?
2. What is the theoretical reason behind why one linear model cannot fit a bi-modal distribution. I assumed that by building a well generalized model, it should be able to predict all values within my label with adequate accuracy. And that the density of my labels should not really affect accuracy.

Thanks!

• Could you please explain what you mean by a "linear model" being used to "fit a ... distribution"? – whuber Mar 28 at 16:40
• Sorry for the confusing question. I mean I am using a least squares linear regression with 13 parameters (one for each feature) to predict for the labels. But the labels seem to be from a bimodal distribution, rather than a normal distribution. Therefore, some other researchers are saying it is better to build 2 least squares linear regression models. One to predict for each distribution. So during implementation, given the distribution that the features statistically belong to, you would use one of the models to predict the label. – Rui Nian Mar 28 at 17:09
• Added clarification: I would first separate my data into two sets. Ideally, the distribution of the label in each set should be normally distributed. Then I would build two models. – Rui Nian Mar 28 at 17:11