3.10. Notation and formulas#
MUDE exam information
This overview with Notation and Formulas will be given on the MUDE exam.
Description |
Notation / formula |
|---|---|
Observables (random) |
\(Y=\begin{bmatrix} Y_1,Y_2,\ldots, Y_m \end{bmatrix}^T\) |
Observations (realization of \(Y\)) |
\(\mathrm{y}=\begin{bmatrix} y_1,y_2,\ldots, y_m \end{bmatrix}^T\) |
Unknown parameters (deterministic) |
\(\mathrm{x}=\begin{bmatrix} x_1,x_2,\ldots, x_n \end{bmatrix}^T\) |
Random errors |
\(\epsilon = \begin{bmatrix} \epsilon_1,\epsilon_2,\ldots, \epsilon_m \end{bmatrix}^T\) with \(\epsilon\sim N(0,\Sigma_{\epsilon})\) |
Functional model (linear) |
\(\mathbb{E}(Y) = \mathrm{A} \mathrm{x}\;\) or \(\;Y = \mathrm{A} \mathrm{x} + \epsilon\) |
Stochastic model |
\(\mathbb{D}(Y) = \Sigma_{Y}=\Sigma_{\epsilon}\) |
Estimator of \(\mathrm{x}\) |
\(\hat{X}\) |
Estimate of \(\mathrm{x}\) (realization of \(\hat{X}\)) |
\(\hat{\mathrm{x}}\) |
Adjusted (or predicted) observations |
\(\hat{\mathrm{y}}=\mathrm{A}\hat{\mathrm{x}}\) |
Residuals |
\(\hat{\epsilon}=\mathrm{y}-\mathrm{A}\hat{\mathrm{x}}\) |
Estimators of \(\mathrm{x}\)#
Estimators |
Formula |
|---|---|
Least-squares (LS) |
\(\hat{X}=\left(\mathrm{A}^T \mathrm{A} \right)^{-1} \mathrm{A}^T Y \) |
Weighted least-squares (WLS) |
\(\hat{X}=\left(\mathrm{A}^T\mathrm{WA} \right)^{-1} \mathrm{A}^T\mathrm{W} Y \) |
Best linear unbiased (BLU) |
\(\hat{X}=\left(\mathrm{A}^T\Sigma_Y^{-1} \mathrm{A} \right)^{-1} \mathrm{A}^T\Sigma_Y^{-1} Y \) |
Test statistics (with distribution under \(\mathcal{H}_0\))#
Test statistic |
Formula |
|---|---|
Generalized likelihood ratio test |
\(T_q = \hat{\epsilon}^T\Sigma_Y^{-1}\hat{\epsilon}-\hat{\epsilon}_a^T\Sigma_Y^{-1}\hat{\epsilon}_a\sim \chi^2(q,0)\) |
Overall model test |
\(T_{q=m-n} = \hat{\epsilon}^T\Sigma_Y^{-1}\hat{\epsilon}\sim \chi^2(q,0)\) |
w-test |
\(W = \frac{\mathrm{C}^T\Sigma_Y^{-1} \hat{\epsilon}}{\sqrt{\mathrm{C}^T\Sigma_Y^{-1}\Sigma_{\hat{\epsilon}}\Sigma_Y^{-1} \mathrm{C}}} \sim N(0,1)\) |
Linear propagation laws if \(\hat{X}=\mathrm{L^T} Y \)#
Propagation law of the … |
Formula |
|---|---|
… mean |
\(\mathbb{E}(\hat{X}) = \mathrm{L^T}\mathbb{E}(Y)\) |
… covariance matrix |
\(\Sigma_{\hat{X}} =\mathrm{L^T}\Sigma_Y \mathrm{L} \) |
Attribution
This chapter was written by Sandra Verhagen. Find out more here.