How to define the symbolic expression $V = \frac{1}{N} \sum_{i=1}^N ( y_i(g) - z_i)^2$ in Mathematica?

Question

I want to define this expression symbolically:

$V = \frac{1}{N} \sum_{i=1}^N ( y_i(g) - z_i)^2$

in which $\{y_i(g)\}_{i=1}^N$ is to be understood as a symbolic sequence that depends on a vector $g$. That is $y(g) = \{ y_1(g), \dots y_N(g)\}$

The motivation is that I would like to compute the derivative of V with respect to $g$.

Define $\{u_i\}_{i=1}^N$ and $\{z_i\}_{i=1}^N$ to be mutually independent (and independent from the rest) symbolic sequences of size $N$. Define $\{g\}_{k=1}^n$ as a finite sequence of size $n$.

We should also define $G = \sum_{k=1}^n g_k q^{-k}$ in which $q^-k$ is the backward shift operator, applied on sequences: $q^{-k} \;u_i = u_{i-k}$

Furthermore, define $C(G)$ to be some rational function in $G$.

Then $y(g) = \frac{ C(G) G} {1+ C(G) G} u$. This should return the finite sequence $\{y_i(g)\}_{i=1}^N$.

Then, we should ask for $\frac{\text{d} V}{\text{d} g}$.

Answer 1

If I understand you correctly, the main idea of the procedure can be outlined as follows.

Call the vector gg and let it have k components g[i]:

gg[k_] := Array[g, k]

Then your expression can be written as

V[n_, k_] := Sum[(y[i, gg[k]] - z[i])^2, {i, 1, n}]

Example n = 3, k = 2

V[3, 2]

(* Out[32]= (y[1, {g[1], g[2]}] - z[1])^2 + (y[2, {g[1], g[2]}] - 
   z[2])^2 + (y[3, {g[1], g[2]}] - z[3])^2 *)

Now the drivatives with respect to gg, i.e. to the g[i] become

With[{n = 3, k = 2},
 Table[{i, D[V[n, k], g[i]]}, {i, 1, k}]]

$\left\{\left\{1,2 (y[1,\{g[1],g[2]\}]-z[1]) y^{(0,\{1,0\})}[1,\{g[1],g[2]\}]+2 (y[2,\{g[1],g[2]\}]-z[2]) y^{(0,\{1,0\})}[2,\{g[1],g[2]\}]+2 (y[3,\{g[1],g[2]\}]-z[3]) y^{(0,\{1,0\})}[3,\{g[1],g[2]\}]\right\},\left\{2,2 (y[1,\{g[1],g[2]\}]-z[1]) y^{(0,\{0,1\})}[1,\{g[1],g[2]\}]+2 (y[2,\{g[1],g[2]\}]-z[2]) y^{(0,\{0,1\})}[2,\{g[1],g[2]\}]+2 (y[3,\{g[1],g[2]\}]-z[3]) y^{(0,\{0,1\})}[3,\{g[1],g[2]\}]\right\}\right\}$

If you define clearly the function y[i,gg] then MMA will perform the differentiation explicitly. I suggest to start with a simple example for y[i,gg].

Regards,
Wolfgang