Fitting a Second Order Polynomial to Data Points

[squenshl]

Homework Statement

Suppose that you are given a set of observations (t_k,y_k), 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) |t_k-t_j| is very small? It may help if you draw a picture. Or better still, study the structure of the matrix A.

Homework Equations

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.

 

[HallsofIvy]

If the relationship between t and y were a second degree polynomial, we would have [itex]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]

Fixed your post up.

If the relationship between t and y were a second degree polynomial, we would have $y_k= at_k^2+ bt_k+ c$ for all t and y. That can be written as
$$\begin{bmatrix}y_1 \\ y_2 \\ y_3 \\ \vdots\end{bmatrix}= \begin{bmatrix}t_1^2 & t_1 & 1\\ t_2^2 & t_2 & 1 \\ t_3^3 & t_3 & 1\\ \vdots & \vdots & \vdots \end{bmatrix}\begin{bmatrix}a \\ b \\ c\end{bmatrix}$$

 

[squenshl]

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

 

[squenshl]

Still lost, any ideas.

 

[squenshl]

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

 

[vela]

Do you know what the least squares method is used for?

 

[squenshl]

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 max_k,j |t_k - t_j| is very small, not too sure on these problems, please help.

 

[vela]

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.

So in this problem, the method is used to calculate what exactly?

For c) does that mean assume x is real world data and if so how do you assess the accuracy of x(hat)?

Could you clarify what you mean by "real world data"? The data in this problem are the pairs (t_k, y_k).

For d) How do I assess the stability of the problem if max_k,j |t_k - t_j| is very small, not too sure on these problems, please help.

Surely, the topic of stability must have been covered in your book or lecture. What do you know about it?

So far, all you've done is ask for the answers to the question. You need to show some effort that you've tried to figure out the problems on your own.