DebugConversion.tex

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    \title{DebugConversion}
    
    
    

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    Note that the error is measured, and the purpose is to fined those
parameters that better match the errors (in this LSM application the
error plays the role of \(f(x)\))

\[
E = \frac{1}{2}\sum_{(i, j)\in I} \left[ E_{ij} - \left(\alpha_0 + \alpha_1\left(R_i + \Delta R_i + \beta^{-1}\Delta V_j\right) - T_r\right)\right]^2
\]

where \(I = \{1, ..., n\}\times \{1, ..., m\}\). By definition \(R_i\)
is the resistance the sensor would have in absence of deviations, that
is \(T_r = \alpha_0 + \alpha_1 R_i\). Therefore equation above becomes

\[
E = \frac{1}{2}\sum_{i=1}^n\sum_{j=1}^m \left[E_{ij} - \alpha_1\Delta R_i  - \alpha_1 \beta^{-1}\Delta V_j \right]^2
\]

    Taking the derivatives with respect of the unknown parameters
\(\Delta R_i\) and \(\Delta V_j\) we obtain

\begin{equation}
\frac{\partial E}{\partial \Delta R_i} = -\sum_{j=1}^m [E_{ij} - \alpha_1\Delta R_i  - \alpha_1\beta^{-1}\Delta V_j ] \alpha_1  
\end{equation}

and

\begin{equation}
\frac{\partial E}{\partial \Delta V_j} = -\displaystyle\sum_{i=1}^n [E_{ij} - \alpha_1\Delta R_i  - \alpha_1\beta^{-1}\Delta V_j ] \beta^{-1}\alpha_1  
\end{equation}

from the equations
\(\frac{\partial E}{\partial \Delta R_i} = 0\ (i\in\{1, ..., n\})\) and
\(\frac{\partial E}{\partial \Delta V_j} = 0\ (j\in\{1, ..., m\})\), the
essence of the LSM, the following system of equations can be derived:

\begin{equation}
\begin{bmatrix}
m\alpha_1 & \cdots & 0 & \alpha_1\beta^{-1} & \cdots & \alpha_1\beta^{-1}\\
 & \ddots & & \vdots & \ddots & \vdots\\
0 & \cdots  & m\alpha_1 & \alpha_1\beta^{-1} & \cdots & \alpha_1\beta^{-1}\\
\alpha_1 & \cdots & \alpha_1 & n\alpha_1\beta^{-1} & \cdots & 0\\
\vdots & \ddots & \vdots &  &  \ddots & \\
\alpha_1 & \cdots & \alpha_1 & 0 & \cdots & n\alpha_1\beta^{-1}
\end{bmatrix}
\begin{bmatrix}
\Delta R_1 \\ \vdots \\ \Delta R_n \\ \Delta V_1 \\ \vdots \\ \Delta V_m
\end{bmatrix} = 
\begin{bmatrix}
\sum_{j=1}^m E_{1j} \\ \vdots \\ \sum_{i=1}^m E_{nj} \\ \sum_{i=1}^n E_{i1} \\ \vdots \\ \sum_{i=1}^n E_{im}
\end{bmatrix}
\end{equation}

The deviations \(\Delta R_i\) and \(\Delta V_j\) can be obtained solving
the above linear system.

    \begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}10}]:} \PY{k+kn}{from} \PY{n+nn}{numpy} \PY{k+kn}{import} \PY{n}{array}\PY{p}{,} \PY{n}{hstack}\PY{p}{,} \PY{n}{vstack}\PY{p}{,} \PY{n}{eye}\PY{p}{,} \PY{n}{ones}\PY{p}{,} \PY{n}{zeros}\PY{p}{,} \PY{n}{mean}
         \PY{k+kn}{from} \PY{n+nn}{numpy.random} \PY{k+kn}{import} \PY{n}{rand}
         \PY{k+kn}{from} \PY{n+nn}{numpy.linalg} \PY{k+kn}{import} \PY{n}{solve}\PY{p}{,} \PY{n}{inv}
         \PY{k+kn}{from} \PY{n+nn}{matplotlib.pyplot} \PY{k+kn}{import} \PY{n}{matshow}\PY{p}{,} \PY{n}{colorbar}\PY{p}{,} \PY{n}{title}\PY{p}{,} \PY{n}{xlabel}\PY{p}{,} \PY{n}{ylabel}
         
         \PY{n}{n} \PY{o}{=} \PY{l+m+mi}{6}\PY{p}{;} \PY{c}{\PYZsh{} Number of sensors}
         \PY{n}{m} \PY{o}{=} \PY{l+m+mi}{8}\PY{p}{;} \PY{c}{\PYZsh{} Number of channels}
         
         \PY{c}{\PYZsh{} Pls change these virtual data with actual data }
         \PY{c}{\PYZsh{}E = (rand(n, m) \PYZhy{} 0.2) \PYZsh{} °C}
         
         \PY{c}{\PYZsh{} Here you are! :)}
         \PY{n}{E} \PY{o}{=} \PY{n}{array}\PY{p}{(}\PY{p}{(}\PY{p}{(}\PY{l+m+mf}{0.417}\PY{p}{,} \PY{l+m+mf}{0.557}\PY{p}{,} \PY{l+m+mf}{0.423}\PY{p}{,} \PY{l+m+mf}{0.559}\PY{p}{,} \PY{l+m+mf}{0.393}\PY{p}{,} \PY{l+m+mf}{0.454}\PY{p}{,} \PY{l+m+mf}{0.495}\PY{p}{,} \PY{l+m+mf}{0.598}\PY{p}{)}\PY{p}{,}
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         \PY{p}{(}\PY{l+m+mf}{0.532}\PY{p}{,} \PY{l+m+mf}{0.699}\PY{p}{,} \PY{l+m+mf}{0.558}\PY{p}{,} \PY{l+m+mf}{0.68}\PY{p}{,} \PY{l+m+mf}{0.551}\PY{p}{,} \PY{l+m+mf}{0.598}\PY{p}{,} \PY{l+m+mf}{0.639}\PY{p}{,} \PY{l+m+mf}{0.763}\PY{p}{)}\PY{p}{,}
         \PY{p}{(}\PY{l+m+mf}{0.435}\PY{p}{,} \PY{l+m+mf}{0.536}\PY{p}{,} \PY{l+m+mf}{0.383}\PY{p}{,} \PY{l+m+mf}{0.504}\PY{p}{,} \PY{l+m+mf}{0.363}\PY{p}{,} \PY{l+m+mf}{0.424}\PY{p}{,} \PY{l+m+mf}{0.721}\PY{p}{,} \PY{l+m+mf}{0.574}\PY{p}{)}\PY{p}{,}
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         \PY{n}{matshow}\PY{p}{(}\PY{n}{E}\PY{p}{)}\PY{p}{;}
         \PY{n}{title}\PY{p}{(}\PY{l+s}{\PYZdq{}}\PY{l+s}{Dev. from reference temperature \PYZdl{}T\PYZus{}\PYZob{}ij\PYZcb{} \PYZhy{} T\PYZus{}r\PYZdl{}}\PY{l+s}{\PYZdq{}}\PY{p}{)}
         \PY{n}{xlabel}\PY{p}{(}\PY{l+s}{\PYZdq{}}\PY{l+s}{Channel index}\PY{l+s}{\PYZdq{}}\PY{p}{)}
         \PY{n}{ylabel}\PY{p}{(}\PY{l+s}{\PYZdq{}}\PY{l+s}{Sensor index}\PY{l+s}{\PYZdq{}}\PY{p}{)}
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         \PY{n}{alpha1} \PY{o}{=} \PY{l+m+mf}{0.255866} \PY{c}{\PYZsh{} [K/Ohm]}
         \PY{n}{beta} \PY{o}{=} \PY{l+m+mf}{300e\PYZhy{}6}\PY{o}{*}\PY{l+m+mi}{1000}\PY{p}{;} \PY{c}{\PYZsh{} 300 uA and conversion factor (1000) to express the number in mV}
         
         \PY{c}{\PYZsh{} Construction of matrix A}
         \PY{n}{Arr} \PY{o}{=} \PY{n}{eye}\PY{p}{(}\PY{n}{n}\PY{p}{)}\PY{o}{*}\PY{n}{m}\PY{o}{*}\PY{n}{alpha1}\PY{p}{;}
         \PY{n}{Arc} \PY{o}{=} \PY{n}{ones}\PY{p}{(}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{m}\PY{p}{)}\PY{p}{)}\PY{o}{*}\PY{n}{alpha1}\PY{o}{/}\PY{n}{beta}\PY{p}{;}
         \PY{n}{Acc} \PY{o}{=} \PY{n}{eye}\PY{p}{(}\PY{n}{m}\PY{p}{)}\PY{o}{*}\PY{n}{n}\PY{o}{*}\PY{n}{alpha1}\PY{o}{/}\PY{n}{beta}\PY{p}{;}
         \PY{n}{Acr} \PY{o}{=} \PY{n}{ones}\PY{p}{(}\PY{p}{(}\PY{n}{m}\PY{p}{,} \PY{n}{n}\PY{p}{)}\PY{p}{)}\PY{o}{*}\PY{n}{alpha1}\PY{p}{;}
         
         \PY{n}{A} \PY{o}{=} \PY{n}{hstack}\PY{p}{(}\PY{p}{(}\PY{n}{vstack}\PY{p}{(}\PY{p}{(}\PY{n}{Arr}\PY{p}{,} \PY{n}{Acr}\PY{p}{)}\PY{p}{)}\PY{p}{,} \PY{n}{vstack}\PY{p}{(}\PY{p}{(}\PY{n}{Arc}\PY{p}{,} \PY{n}{Acc}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{p}{;}
         
         \PY{c}{\PYZsh{} Construction of vector b (A*x = b)}
         \PY{n}{b} \PY{o}{=} \PY{n}{zeros}\PY{p}{(}\PY{n}{n} \PY{o}{+} \PY{n}{m}\PY{p}{)}\PY{p}{;}
         
         \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{n}\PY{p}{)}\PY{p}{:}
             \PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{m}\PY{p}{)}\PY{p}{:}
                 \PY{n}{b}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{+}\PY{o}{=} \PY{n}{E}\PY{p}{[}\PY{n}{i}\PY{p}{,} \PY{n}{j}\PY{p}{]}\PY{p}{;}
                 
         \PY{k}{for} \PY{n}{j} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{m}\PY{p}{)}\PY{p}{:}
             \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{n}\PY{p}{)}\PY{p}{:}
                 \PY{n}{b}\PY{p}{[}\PY{n}{n} \PY{o}{+} \PY{n}{j}\PY{p}{]} \PY{o}{+}\PY{o}{=} \PY{n}{E}\PY{p}{[}\PY{n}{i}\PY{p}{,} \PY{n}{j}\PY{p}{]}\PY{p}{;}
         
         \PY{c}{\PYZsh{} Computations of the results x = A\PYZca{}\PYZob{}\PYZhy{}1\PYZcb{}b}
         \PY{n}{x} \PY{o}{=} \PY{n}{solve}\PY{p}{(}\PY{n}{A}\PY{p}{,} \PY{n}{b}\PY{p}{)}\PY{p}{;}
         
         \PY{k}{print}\PY{p}{(}\PY{n}{x}\PY{p}{)}
\end{Verbatim}

    \begin{Verbatim}[commandchars=\\\{\}]
[ 1.90224293  1.57736585  0.66463223  2.45135848  1.92373856  2.62527764
 -0.10144533  0.07102776 -0.04583029  0.06164785 -0.11612098 -0.04504863
  0.0491413   0.12926136]
    \end{Verbatim}

    \begin{center}
    \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{DebugConversion_files/DebugConversion_2_1.png}
    \end{center}
    { \hspace*{\fill} \\}
    

    % Add a bibliography block to the postdoc
    
    
    
    \end{document}