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We can solve this using the method of least squares: $$\overline X = \frac{\sum_{n=1}^{8}x_i}{8} = 2011.5\\\overline Y = \frac{\sum_{n=1}^{8}y_i}{8} = 974.375$$ The equation of the line of best fit is $y= mx + b$ where $$m = \frac{\sum_{n=1}^{8}(x_i-\overline X)(y_i-\overline Y)}{\sum_{n=1}^{8}(x_i-\overline X)^2}\approx 28.1622024$$ and $$b = \overline Y ... 2 You may observe that$$ \hat{\beta}_1 =\frac{\sum_{i=1}^n x_i y_i - n \bar{x}\bar{y}}{\sum_{i=1}^n x_i^2 -n\bar{x}^2} =\frac{\frac1n\sum_{i=1}^n x_i y_i - \bar{x}\bar{y}}{\frac1n\sum_{i=1}^n x_i^2 -\bar{x}^2}.\tag1 $$Then you may prove that$$ \frac1n\sum_{i=1}^n \left(x_i -\bar{x}\right)^2=\frac1n\sum_{i=1}^n x_i^2 -\bar{x}^2 \tag2 $$and that$$ ...

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It turns out there is some literature available on this subject that I didn't see before. One trick is to first fit a cylinder to the data, then assume the helix lies on that cylinder. For all future helix-fitters out there, check out these links. http://www.geometrictools.com/Documentation/HelixFitting.pdf ...

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Approximation: As you have observed the problem is just fulfilling the sign constraints. The partial derivatives are: $\frac{\partial f}{\partial x}=cbx^{c-1}$ and $\frac{\partial f}{\partial y}=dey^{d-1}$. Since $x$ and $y$ are positive the derivatives are positive if $cb > 0$ and $de>0$. To make sure that $f>0$ as well, you actually need all ...

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