I'm trying to fit a multiple linear regression model to my data with couple of input parameters, say 3.
\begin{align} F(x) &= Ax_1 + Bx_2 + Cx_3 + d \tag{i} \\ &\text{or} \\ F(x) &= (A\ B\ C)^T (x_1\ x_2\ x_3) + d \tag{ii} \end{align}
How do I explain and visualize this model? I could think of the following options:
Mention the regression equation as described in $(i)$ (coefficients, constant) along with standard deviation and then a residual error plot to show the accuracy of this model.
Pairwise plots of independent and dependent variables, like this:
Once the coefficients are known, can the data points used to obtain equation $(i)$ be condensed to their real values. That is, the training data have new values, in the form $x$ instead of $x_1$, $x_2$, $x_3$, $\ldots$ where each of independent variable is multiplied by its respective coefficient. Then this simplified version can be visually shown as a simple regression as this:
I'm confused on this in spite of going through appropriate material on this topic. Can someone please explain to me how to "explain" a multiple linear regression model and how to visually show it.