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Given the input-output relation:

$ \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} =p_1 \begin{pmatrix} p_2 & p_3 \\ p_4 & p_4 \end{pmatrix} * \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} p_5 \\ p_6 \end{pmatrix} $

with $p_1-p_6$ parameters.I want to minimize the least square error using Gauss-Newton method. Suppose we have 100 measurements. My question is about calculation and size of residual vector.

$ r_i = output - f(input,parameters) \\ \begin{pmatrix} r_1 \\ r_2 \end{pmatrix} = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} - \Bigg( p_1 \begin{pmatrix} p_2 & p_3 \\ p_4 & p_4 \end{pmatrix} * \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} p_5 \\ p_6 \end{pmatrix}\Bigg) $

In order to calculate minimised-error parameters, we have:

$ p_{i+1}=p_i+\Delta \\\Delta=(J_f^TJ^T)^{-1}J_f^Tr_i $

the size of each is as follows:

$ input vector :100*2\\ output vector :100*2\\ r:\quad\quad\quad\quad\quad100*2\\ J_f : \quad\quad\quad\quad100*6\\ p_i:\quad\quad\quad\quad\quad6*1 (six \quad parameters) $

As you can see, the size of $\Delta$ would be 6x2 that seems not consistent with $p_{i+1}$

Now is my residual vector calculation process right? if yes how can I compute parameter's vector? And if not what is the correct answer?

Thank you so much.

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1 Answer 1

Let me correct your Gauss-Newton step:

$$\Delta=-\left(J_{f}^{T}J_{f}\right)^{-1}J_{f}^{T}r$$

In terms of matrix sizes:

$$\left([100\times 6]^{T}[100\times 6]\right)^{-1}[100\times 6]^{T}[100\times 1]=\\ \left([6\times100][100\times 6]\right)^{-1}[6\times 100][100\times 1]=\\ ([6\times6])^{-1}[6\times 1]=\\ [6\times 6][6\times 1]=\\ [6\times 1]$$

Hence the $\Delta$ is a vector with 6 rows which is of the same size as $p_{i}$ and you can compute $p_{i+1}=p_{i}+\Delta$.

You can take a look here for derivation of the Gauss-Newton step.

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