Curve fitting
- Write the formula for the second order Lagrange Polynomial
- For the second order polynomial $y=\alpha x^2 + \beta x + \gamma$:
- write the equation for the sum square error.
- write the linear system $Ax=b$ to solve for the coefficients $\alpha$, $\beta$, and $\gamma$, that is, write $A$, $x$, and $b$ with the forumula for each element shown.
- Why do we minimize the sum square error and not the sum of the error?
- T/F: the sum absolute error could be used instead of the sum square error.
- What norms corresponds to the sum absolute error and the sum square error?
- For the data $C_i(t_i)$ given, code and solve for the least squares best fit parameters $C_0$, $k$ for the model $\frac{1}{C} - \frac{1}{C_0} = kt$. Do this for a linearized form of the model.
| t_i |
C_i |
| 0 |
998 |
| 100 |
440 |
| 200 |
300 |
| 300 |
300 |
| 400 |
190 |
- A model for linear least squares, for some linear model $y_i(x_i)$, can be written as $A\eta = b$, where the columns of $A$ are the additive terms of the model, and each row of $A$ corresponds to a separate $x$ data point. To minimize the sum square error, we solve $\hat{A}\eta=\hat{b}$. Define $\hat{A}$ and $\hat{b}$.
