Least squares

Read LA 4.3

1) Main Points

4.3 continues with the concept of using vectors to approximate solutions. The book gives a real life example dealing with approximating a constant that works for two specimens. To calculate this approximation we can use vectors like we did in 4.2 to, knowing that the approximate vector will be distanced from the desired vector by r, a residual vector. We know that for the closest approximation the new multiplied vector and the residual vector will be perpendicular meaning their dot product will be equal to zero.
In equations we are dealing with mx + r = y and x dot r = 0. We can solve this by taking the dot product of both sides of mx + r = y, this way we can find m which is the best approximation for a constant of the original problem. This approximation is where lines of best fit come from.

4.3.2 Applies this method to a more complicated situation. if you are given a table of numbers, these as we have learned can be written in vector form. With this more complicated way of thinking about more three dimensional vectors we can conclude that mu + bv + r = s, and u dot r = 0 and v dot r = 0. This can be solved by taking the dot product of u and the equation, then v and the equation. This gives a solvable system of equations which gives the values of m and b needed to make the closest estimate.

This method is called the fundamental problem of linear modeling and is recapped very well in the end of the LA packet.

2) Challenges

I feel pretty good on the two dimensional stuff, but when it gets the vectors with more elements I foresee some difficulty. Overall though I understand the concept I just need to make sure I print the packet and review/apply the concepts

3) Reflections

I now finally understand where a line of best fit comes from, which is very useful in statistical modeling. Right on!