Method of Least Squares (Approximation Theory)/Examples
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Examples of Method of Least Squares
Arbitrary Example
Let $B$ be a false balance.
$2$ items are weighed on $B$: first individually and then together.
The recorded weights are:
- $17 \, \mathrm g$ and $25 \, \mathrm g$ for the separate items
- $40 \, \mathrm g$ for the combined weight.
The least squares estimates of the true weights are the values of $\hat {w_1}$ and $\hat {w_2}$ that minimize:
- $L = \paren {w_1 - 25}^2 + \paren {w_1 - 17}^2 + \paren {w_1 + w_2 - 40}^2$
Differentiating with respect to $w_1$ and $w_2$ and equating the derivatives to zero, gives us:
\(\ds \hat {w_1}\) | \(=\) | \(\ds 16.33\) | ||||||||||||
\(\ds \hat {w_2}\) | \(=\) | \(\ds 24.33\) |