Fundamentals of parameter estimation - Part III

This is the continuation to Fundamentals of parameter estimation - Part II. Exercise 3/8 from my optimal estimation course. The focus is on random vectors and unbiased linear MMSE estimation.

At the end of this exercise I should understand insights about the concept of covariance matrices and about unbiased linear MMSE estimation.

Context

Prior knowledge

I have a ship. The parameter vector to estimate is the position vector of that ship $\mathbf{x}=[xy{]}^T$. The prior knowledge, obtained via dead reckoning, is captured as a prior expectation ${\mu }_{\mathbf{x}}$ and a covariance matrix which expresses the prior uncertainty that we have about the position $C_{\mathbf{x}}$.

Measurement

In order to increase the accuracy, the navigator of the ship measures the direction $\phi$ of a beacon, e.g. lighthouse, relative to the ship as in the Figure below.

The beacon has a known reference position ${\mathbf{x}}_0$. The line of sight is defined by the position of the beacon and by the measured direction $\theta$. The compass reading gives $\theta =\phi +△\theta$. The following equation defines the line of sight in the $(\xi ,\eta )$ plane:

$$x_0\sin \theta -y_0\cos \theta =\xi \sin \theta -\eta \cos \theta$$

The measurement model

The relation between the ship’s true position $\mathbf{x}=(x,y)$ and the true bearing $\phi$ is:

$$x_0\sin \phi -y_0\cos \phi =x\sin \phi -y\cos \phi$$

or, by substituting $\phi =\theta -△\theta$

$$x_0\sin (\theta -\Delta \theta )-y_0\cos (\theta -\Delta \theta )=x\sin (\theta -\Delta \theta )-y\cos (\theta -\Delta \theta )$$

The relation between the ship’s true position $\mathbf{x}$ and the observed direction $\theta$ is nonlinear. To get a linear approximation, we apply a truncated Taylor series expansion to the sine and cosine functions:

$$\sin (\theta -\Delta \theta )\approx \sin \theta -\Delta \theta \cos \theta$$ $$\cos (\theta -\Delta \theta )\approx \cos \theta +\Delta \theta \sin \theta$$ $$\Downarrow \text{after some rearrangements}$$ $$x_0\sin \theta -y_0\cos \theta \approx x\sin \theta -y\cos \theta +\Delta \theta \left((x_0-x)\cos \theta +(y_0-y)\sin \theta \right)$$

Since $\theta \approx \phi$, the factor $(x_0-x)\cos \theta +(y_0-y)\sin \theta$ almost equals $(x_0-x)\cos \phi +(y_0-y)\sin \phi$. The latter equals the distance $d$ between beacon and ship. Therefore:

$$x_0\sin \theta -y_0\cos \theta \approx x\sin \theta -y\cos \theta +d\Delta \theta$$

This can be written in the form $z=H\mathbf{x}+v$ with the following definitions $\{\begin{array}{l}z=x_0\sin \theta -y_0\cos \theta \\ H=[\sin \theta -\cos \theta ]\\ v=d\Delta \theta \end{array}$

The distance $d$ is unknown, but can be estimated from prior knowledge ${\mu }_{\mathbf{x}}$ of the ship’s position and the position of the beacon: $d\approx ||{\mathbf{x}}_0-{\mu }_{\mathbf{x}}||$. Assuming the measurement of the bearing has an uncertainty of ${\sigma }_{\Delta \theta }$, the standard deviation of $v$ is ${\sigma }_v=d{\sigma }_{\Delta \theta }$ [radians].

The Case

$$\text{prior knowledge:}{\mu }_x=\left[\begin{array}{c}10\\ 20\end{array}\right]C_x=\left[\begin{array}{cc}25& -25\\ -25& 70\end{array}\right]$$ $$\text{measurement:}{\mathbf{x}}_0=\left[\begin{array}{c}100\\ 100\end{array}\right]\theta =35^{\circ }{\sigma }_{\Delta \theta }=1^{\circ }$$

Physical units are Nautical miles (Nm).

Uncertainty regions and principal axes

For normal distributions $p(\mathbf{x})=\frac{1}{\sqrt{(2\pi {)}^N|C_x|}}\exp \left(-\frac{(\mathbf{x}-{\mu }_x{)}^T{C}_x^{-1}(\mathbf{x}-{\mu }_x)}{2}\right)$, the equation for the contour simplifies to: $(\mathbf{x}-{\mu }_x{)}^T{C}_x^{-1}(\mathbf{x}-{\mu }_x)=k^2\text{ with }k=1,2,3$.

The eigenvectors and eigenvalues are solutions of $C_x\mathbf{v}=\lambda \mathbf{v}$ and the corresponding scaling factors are $a_m=\sqrt{{\lambda }_m}$

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First topic: Determine the eigenvalues and eigenvectors of $C_{\mathbf{x}}$ and draw the associated uncertainty region.

1.1 Generate a set of points on a circle with unit radius. The centre of the circle is positioned at the origin

1.2 Scale the $x$ and $y$ coordinates of these points in accordance with the scaling factors $a_0$ and $a_1$. The resulting points form an ellipse with the right shape, but not with the right orientation and position.

So based on the Figure 5 above, I need to extract the $a_0$ and $a_1$ scaling factors of the ellipse defined by $C_{\mathbf{x}}$. The corresponding scaling factors are $a_m=\sqrt{{\lambda }_m}$.

1.3 Rotate the set of points in accordance with the direction of the principal axes. The eigenvector-matrix is a rotation matrix.

This is really just a no-brainer, since the eigenvector-matrix is in itself the Rotation Matrix I need to apply.

1.4+1.5 Shift the whole set to the position determined by ${\mu }_{\mathbf{x}}$. Plot the curve defined by the resulting set of points.

Again, I simply add to each axis the values from ${\mu }_{\mathbf{x}}$.