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MathModelingNotes2

# MathModelingNotes2 - Least square and beyond Lecture notes...

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4/8/2011 1 Least square and beyond Lecture notes taken by Michael Bird on 4/7/2011 in Math Modeling I When we have data we seek to determine whether there is a relation. With these plotted points we try to fit a line to best fit all of the data ? 𝑖 , ? 𝑖 One way we can do this is to find k and m such that ? 𝑖 = ?? 𝑖 + ? for 𝑖 = 1,2,3, … , ? Writing out all of the equations for each i we see ? 1 = ?? 1 + ? ? 2 = ?? 2 + ? ? 3 = ?? 3 + ? ? ? = ?? ? + ?

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4/8/2011 2 ? 1 = ?? 1 + ? ? 2 = ?? 2 + ? ? 3 = ?? 3 + ? ? ? = ?? ? + ? These equations can also be represented in matrix notation ? 1 ? 2 ? ? = ? 1 1 ? 2 ? ? 1 1 ? ? b A x given Need to find Thus our problem can be seen as 𝐴? = ? What we want is to find what our vector ? is so let’s set our equation equal to zero 𝐴? − ? = 0 Unfortunately, it is impossible to solve exactly for 0. Therefore, the plan of attack is to find k and m such that the LHS is approximately equal to the RHS or 𝐴? − ? ≈ 0 How do we solve this though? One popular method is by minimizing the ? 2 -norm 𝐴? − ? 2 This particular method is known as the least square method
4/8/2011 3 𝐴? − ?

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MathModelingNotes2 - Least square and beyond Lecture notes...

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