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README.md

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@@ -7,7 +7,7 @@ We assume that the dynamics are **nonlinear** and, specifically, that
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<a href="https://www.codecogs.com/eqnedit.php?latex=y&space;=&space;f(\underline{x};\underline{\theta})&space;&plus;&space;\epsilon" target="_blank"><img src="https://latex.codecogs.com/gif.latex?y&space;=&space;f(\underline{x};\underline{\theta})&space;&plus;&space;\epsilon" title="y = f(\underline{x};\underline{\theta}) + \epsilon" /></a>
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where <a href="https://www.codecogs.com/eqnedit.php?latex=\underline{\theta}&space;\in&space;\mathbb{R}^k" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\underline{\theta}&space;\in&space;\mathbb{R}^k" title="\underline{\theta} \in \mathbb{R}^k" /></a> is a vector of unknown real parameters, _f_ is a known deterministic function nonlinear in <ins>&theta;</ins> and &epsilon; is a random noise with distribution <a href="https://www.codecogs.com/eqnedit.php?latex=\epsilon&space;\sim&space;N(0,&space;\sigma^2)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\epsilon&space;\sim&space;N(0,&space;\sigma^2)" title="\epsilon \sim N(0, \sigma^2)" /></a> for some positive value of &sigma;.
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where <a href="https://www.codecogs.com/eqnedit.php?latex=\underline{\theta}&space;\in&space;\mathbb{R}^k" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\underline{\theta}&space;\in&space;\mathbb{R}^k" title="\underline{\theta} \in \mathbb{R}^k" /></a> is a vector of unknown real parameters, _f_ is a known deterministic function nonlinear in <ins>&theta;</ins> and &epsilon; is a random noise with distribution <a href="https://www.codecogs.com/eqnedit.php?latex=\epsilon&space;\sim&space;N(0,&space;\sigma^2)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\epsilon&space;\sim&space;N(0,&space;\sigma^2)" title="\epsilon \sim N(0, \sigma^2)" /></a> for some positive and unknown value of &sigma;.
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If we have N independent observations <a href="https://www.codecogs.com/eqnedit.php?latex=(\underline{x}_1,&space;y_1),&space;...,&space;(\underline{x}_N,&space;y_N)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?(\underline{x}_1,&space;y_1),&space;...,&space;(\underline{x}_N,&space;y_N)" title="(\underline{x}_1, y_1), ..., (\underline{x}_N, y_N)" /></a>, we can estimate the value of <ins>&theta;</ins> by maximizing the log-likelihood. We can optionally choose to weight some observations more or less that others by choosing weights <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{100}&space;w_i,&space;i&space;=&space;1,&space;...,&space;n" target="_blank"><img src="https://latex.codecogs.com/svg.latex?\dpi{100}&space;w_i,&space;i&space;=&space;1,&space;...,&space;n" title="w_i, i = 1, ..., n" /></a> and assuming that <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{100}&space;\small&space;y_i&space;\sim&space;N(f(\underline{x}_i,&space;\underline{\theta}),&space;\frac{\sigma^2}{w_i})" target="_blank"><img src="https://latex.codecogs.com/svg.latex?\dpi{100}&space;\small&space;y_i&space;\sim&space;N(f(\underline{x}_i,&space;\underline{\theta}),&space;\frac{\sigma^2}{w_i})" title="\small y_i \sim N(f(\underline{x}_i, \underline{\theta}), \frac{\sigma^2}{w_i})" /></a> for all i (where &sigma; is unknown).
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