Linear Algebra

Topic: Line of Best Fit 269
Topic: Line of Best Fit
This Topic requires the formulas from the subsections on Orthogonal Projection
IntoaLine,andProjectionIntoaSubspace.
Scientists are often presented with a system that has no solution and they
must find an answer anyway, that is, they must find a value that is as close as
possible to being an answer. An often-encountered example is in finding a line
that, as closely as possible, passes through experimental data.
For instance, suppose that we have a coin to flip, and want to know: is it
fair? This question means that a coin has some proportion m of heads to flips,
determined by how it is balanced beween the two sides, and we want to know
if m =1/2. We can get experimental information about it by flipping the coin
many times. This is the result a penny experiment, including some intermediate
numbers.
number of flips 30 60 90
number of heads 16 34 51
Naturally, because of randomness, the exact proportion is not found with this
sample — indeed, there is no solution to this system.
30m =16
60m =34
90m =51
That is, the vector of experimental data is not in the subspace of solutions.
⎛ ⎞ ⎛ ⎞
16 30
⎝34⎠
�∈ {m ⎝60⎠
51 90
� � m ∈ R}
However, as described above, we expect that there is an m that nearly works.
An orthogonal projection of the data vector into the line subspace gives our best
guess at m.
⎛ ⎞ ⎛ ⎞
16 30
⎝34⎠
⎝60⎠
⎛ ⎞
51 90
30
⎛ ⎞ ⎛ ⎞ · ⎝60⎠
=
30 30
90
⎝60⎠
⎝60⎠
90 90
7110
12600 ·
⎛ ⎞
30
⎝60⎠
90
The estimate (m = 7110/12600 ≈ 0.56) is higher than 1/2, but not by much, so
probably the penny is fair enough for flipping purposes.
The line with the slope m ≈ 0.56 is called the line of best fit for this data.
heads
60
30
��
��
��
30 60 90
flips