Parabolic Interpolation with three data points and measurement noise

Solution 1:

I denote vectors by small letters. The estimated coefficients are given by \begin{align} \hat{\mathbf{a}}=\mathbf{X}^{-1}\mathbf{z}, \end{align} whereas the true coefficients are given by \begin{align} \mathbf{a}=\mathbf{X}^{-1}(\mathbf{z}-\mathbf{w}), \end{align} where \begin{align} \mathbf{X}=\left[\begin{array}{ccc}x_1^2&x_1&1\\x_2^2&x_2&1\\x_3^2&x_3&1\end{array}\right],~ \mathbf{a}=\left[\begin{array}{c}a\\b\\c\end{array}\right],~ \mathbf{z}=\left[\begin{array}{c}z_1\\z_2\\z_3\end{array}\right],~ \mathbf{w}=\left[\begin{array}{c}w_1\\w_2\\w_3\end{array}\right]. \end{align} We see that $\text{var}(\mathbf{z})=\text{var}(\mathbf{w})$. A variance of the obtained parameters $\hat{\mathbf{a}}$ is given by \begin{align} \text{var}(\hat{\mathbf{a}})=\text{var}(\mathbf{X}^{-1}\mathbf{z})=\mathbf{X}^{-1}\text{var}(\mathbf{z})\mathbf{X}^{-\text{T}}=\mathbf{X}^{-1}\underbrace{\text{var}(\mathbf{w})}_{=P\mathbf{I}}\mathbf{X}^{-\text{T}}=P\mathbf{X}^{-1}\mathbf{X}^{-\text{T}}, \end{align} where we have used formula $\text{var}(\mathbf{Ay})=\mathbf{A}\text{var}(\mathbf{y})\mathbf{A}^{\text{T}}$ and $\text{var}(\mathbf{w})=P$. Therefore we get \begin{align} \text{var}(a)&=P(\mathbf{X}^{-1}\mathbf{X}^{-\text{T}})_{11},\\ \text{var}(b)&=P(\mathbf{X}^{-1}\mathbf{X}^{-\text{T}})_{22}. \end{align} Finally, we would like to evaluate $\text{var}\left(-\frac{2a}{b}\right)$, which is not possible directly, but only approximately. By the use of the first order Taylor expansion $-\frac{2a}{b}\approx -\frac{2a_0}{b_0}-\frac{2}{b_0}a+\frac{2a_0}{b_0^2}b$ and the formula $\text{var}(a\mathbf{x}+b)=a^2\text{var}(\mathbf{x})$ we obtain the following approximation \begin{align} \text{var}(\hat{x}_{\text{m}})=\text{var}\left(-\frac{2a}{b}\right)\approx\,& \text{var}\left(-\frac{2}{b_0}a+\frac{2a_0}{b_0^2}b\right)=\frac{4}{b_0^2}\text{var}(a)+\frac{4a_0^2}{b_0^4}\text{var}(b)= \nonumber\\ &\frac{4P}{b_0^2}\left[(\mathbf{X}^{-1}\mathbf{X}^{-\text{T}})_{11}+\frac{a_0^2}{b_0^2}(\mathbf{X}^{-1}\mathbf{X}^{-\text{T}})_{22}\right]. \end{align}