I guess, this might even be high-school level - but if it is, I definitely forgot how this operation is called, and all my online searches are fruitless, so I thought I'd ask here.
Here, I've constructed some functions:
Now, let's assume I know the analytical forms of $y_1(x)$, $y_2(x)$ and $y_3(x)$; and I have a recording of $y_4(x)$ in the domain shown on the plot, but I do not have its analytical form.
At this point, if I make the assumption, that $y_4(x)$ is a linear combination of $y_1(x)$, $y_2(x)$ and $y_3(x)$:
$$ y_4(x) = a + b\cdot y_1(x) + c\cdot y_2(x) + d\cdot y_3(x)$$
... is there a procedure, that would take in the data of all four functions ( $y_1(x)$, $y_2(x)$, $y_3(x)$ and $y_4(x)$) in a given domain of $x$, and allow me to calculate the linear coefficients $a$, $b$, $c$ and $d$ (at least one set of such values)? If so, what is this procedure/operation called?
What do you mean with "a recording” of $y_4(x)$ in the domain of the plot? If you meant that you can find out the value of $y_4(x)$ for those values of $x$, then all you need to do it is choose four values for $x$, let's say $2$, $4$, $6$ and $8$, feed that values to the expression $y_4(x) = a + b\cdot y_1(x) + c\cdot y_2(x) + d\cdot y_3(x)$ to get four linear equations in $a$, $b$, $c$ and $d$ and solve that linear system of equations to find $a$, $b$, $c$ and $d$.
Edit:
I you have the data points, then you can choose a pair $(u,v)$ from those thousands of points. Set $x=u$ in the expression to get $v = a+ b\cdot y_1(u) + c\cdot y_2(u) +d \cdot y_3(u)$, a linear equation in $a$, $b$, $c$ and $d$. Do that four times to get a linear system of equations and solve that system. That would allow you to find the values of $a$, $b$, $c$ and $d$. Then you'll have an analytic form of $y_4$