# Least squares

## Homework Statement

Suppose that you are given a set of observations (tk,yk), k = 1,...,M.
You plot these points on a sheet & it seems that the relationship between (t,y) could be approximated with a second order polynomial.
a) Write down the model in the form y = Ax + c. Specify the vectors & matrices & give interpretation to all terms.
b) Write down the least squares estimate x(hat) for x.
c) Let the elements of x bear a physical interest. How could you assess the accuracy of the estimate x(hat)?
d) How would you assess the stability of the problem if max(k,j) |tk-tj| is very small? It may help if you draw a picture. Or better still, study the structure of the matrix A.

## The Attempt at a Solution

a) Not sure on this one.
b) Isn't that just using the definition of least squares.
c) Not sure on this one.
d) Not sure on this one.

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HallsofIvy
Homework Helper
If the relationship between t and y were a second degree polynomial, we would have $y_k= at_k^2+ bt_k+ c[/tex] for all x and y. That can be written as $$\begin{bmatrix}y_1 & y_2 & y_3 & \cdot\cdot\cdot\end{bmatrix}= \begin{bmatrix}a & b & c\end{bmatrix}\begin{bmatrix}x_1^2 & x_1 & 1\\ x_2^2 & x_2 & 1 \\ x_3^3 & x_3 & 1\\ \cdot\cdot\cdot & cdot\cdot\cdot & cdot\cdot\cdot \end{bmatrix}$$ vela Staff Emeritus Science Advisor Homework Helper Education Advisor Fixed your post up. If the relationship between t and y were a second degree polynomial, we would have [itex]y_k= at_k^2+ bt_k+ c$ for all t and y. That can be written as
$$\begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ \vdots\end{bmatrix}= \begin{bmatrix}t_1^2 & t_1 & 1\\ t_2^2 & t_2 & 1 \\ t_3^3 & t_3 & 1\\ \vdots & \vdots & \vdots \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix}$$

Thanks.
For b) is it just using the definition of least squares otherwise what do I do?

Still lost, any ideas.

What exactly does it mean write down the least squares estimate x(hat)?

vela
Staff Emeritus
Homework Helper
Do you know what the least squares method is used for?

Isn't it to fit a polynomial through a set of points where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model.
For c) does that mean assume x is real world data and if so how do you assess the accuracy of x(hat)?
For d) How do I assess the stability of the problem if maxk,j |tk - tj| is very small, not too sure on these problems, please help.

vela
Staff Emeritus
Homework Helper