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I have a error signal which I want to minimize using MSE. This error signal at time $k$ is a vector of length $3$: $e_k = C^{T} R_k - B^{T} A_k = [c_0, \ldots, c_{N_c-1}] \begin{bmatrix} r_{2k}\\ \vdots \\ r_{2k-N_c+1} \end{bmatrix} - [B_0, B_1, B_2]\begin{bmatrix} a_{k-d}\\ a_{k-d-1}\\ a_{k-d-2} \end{bmatrix} = [C^{T} - B^{T}] \begin{bmatrix} R_k\\ A_k \end{bmatrix} = W^{T} V_k$

where each $c_i, 0\leq i \leq N_c-1$ is $3 \times 3$, $r_{2k},\ldots, r_{2k-N_c+1}$ is $3 \times 1$, each $B_i, 0\leq i \leq 2$ is $3 \times 9$, and $a_{k-d}, a_{k-d-1}, a_{k-d-2}$ are $3 \times 1$ matrices.

Therefore,$\nabla_W \mathrm{MSE} = \nabla_W E(e_k^{T} e_k) = 2R_k W = 0$. Unfortunately this gives me $W=0$.

Any idea how else I can find my weight matrix $W$ which minimizes the error?

Michael Hardy
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Elnaz
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  • What are data and are what are unknowns here? Is the number of unknowns less than the number of data? – PA6OTA May 22 '14 at 18:23
  • \ldots and \vdots exist. See my edits to your question. When you use "..." in ordinary TeX or LaTeX rather than MathJax, you'll see $a,\text{...},b$ rather than $a,\ldots,b$, and the former is considered incorrect in standard usage. Here you see \vdots: $\begin{bmatrix} a \ \vdots \ b \end{bmatrix}$. ${}\qquad{}$ – Michael Hardy May 27 '14 at 21:51

1 Answers1

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There is a certain defect in your setting. Of course if you model your error vector is a liner function of parameters $W$ the best solution will be $W=0$. The typical way to implement MSE is 1. Specify you measurement observable vector $y$. 2. Specify your model explaining $y$ as $W^T V$, where $V$ are some explanatory variables. 3. Minimize error $|y-W^TV|$.

Otherwise you cannot apply MSE.

Alexander Vigodner
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  • Well, I have to jointly optimize for $C$ and $B$ and my error vector is in the form of $W^{T}V$. Can't we solve this in any way? – Elnaz May 22 '14 at 18:11
  • Like I said, something is wrong here. You measure a signal $y$. You can not model the signal $y=Wv$ and after that minimize it's norm. In your final function of minimazation there is no real measurement - only the model. Where is the real data that you measure? – Alexander Vigodner May 23 '14 at 16:23