Explicit piecewise linear approximation of a function of 4 variables

Question

I have a table of numbers for fixed values of 4 parameters $x, y, z, t$, at this $x$ belongs to finite set of natural numbers, $y\in\{1;2\}$, $z\in\{5;10;15;20;25\}$ and $t\in\{1,2,3\}$. Is there a method to construct such a function $f$ that inserting particular values from the table, $r=f(x,y,z,t)$ will coincide with my data.

The data are of this type, for example, $f(27,1,20,3)=46.08$, $f(35,1,20,3)=63.9$, $f(32,1,5,3)=17.24$, etc.

Update

I need explicit piecewise linear approximation formula for $f(x,y,z,t)$ on the basis of known $f_i$, $i\in\{1;1240\}$. The formula for functions with 3 variables will of help as well.

Answer 1

For fixed $t$, the $x$ range is fixed for both $y$s, therefore $$ f(x,y,z)\sim\sum_{i=1}^n\sum_{j=1}^5\sum_{k=1}^2f(x_i,y_j,z_k)L_i(x)L_j(y)L_k(z), $$ where $L_i$ is Lagrange`s polynomial: $$ L_i(h)=\prod_{l=1,~ l\neq i}^n\frac{h-h_l}{h_i-h_l}. $$