Can we fit a regression model when the dependent variable is poorly correlated with the independent variables? I have a requirement, I need to predict Y from 2 X_variables, I plotted two scattered plots ie Y vs X1, Y vs X2, As you can see the below pics, the plots are sparced. There is poor correlation between Y and X_variables, I also plotted a correlation plot, which also shows a poor correlation between X_variables and Y. I'm asked to predict Y from X1 and X2, can I fit any regression model which will work for this dataset? Is it possible to fit a regression model for the below data? If yes, then how and which regression model? Should I make any data transformations? If yes then which transformations I need to apply. I attached scattered plots, correlation plot and distribution plots below. I also tried polynomial regression, It didn't work well. I scaled data with StandardScaler and MinMax scaler, they didn't work either. There are negative values in X2 so I couldn't apply log transformation. Please refer below Images. If you need more information I can add it.



 A: Relative to your explicit question, there is no minimum correlation that is required to fit a multiple regression model.  (You could argue that there is a maximum correlation between the $X$ variables, viz., if $r_{x_1, x_2} = 1.0$ it would be rank deficient, but that's not a problem here.)  If the correlations between the $x$'s and $Y$ were $0$, the regression would fit just fine and return a flat plane.  That wouldn't help you make future predictions, but the regression would still 'work' and would have shown you exactly what it should have.
Looking at your plots, it is clear that there are patterns in the data that are not accounted for by just looking at $X_1$ and $X_2$.  There are different ranges within each variable in which very different things are happening.  You should work hard to find out what's going on there.  In the absence of that information, you won't be able to do much with these data.
In terms of what is possible, I gather you have a very large $N$.  If so, you could fit splines with a fine grid of many knots evenly spaced across each predictor.  It will take up a large number of degrees of freedom, but if your $N$ is sufficient, it might yield some useful information.
A: On the question: Should I make any data transformations? Yes, take a subsample and apply an analysis of transformation routine (research, for example, the Box-Cox method here).
As to whether you should use poor explanatory variables to improve your predictions, my answer is it depends on whether say variables are understandably fundamental to the nature of the dependent variable and there is a good explanation as to the current poor correlation. If not, just work with the dependent variable possibly in a transformed state (and then reverse the transform albeit interpretation issues may arise). Speaking somewhat generally, this likely will, with time, produce a lower average error measure.
I hope this helps.
