Exercise 14 (last)

14.1 CRLB identity

Show the following equality, where for simplicity it is assumed that  $\theta$ is a scalar parameter

$E\left(\left(\frac{dl(y,\theta)}{d\theta}\right)^2\right) = -E\left(\frac{d^2l(y,\theta)}{d\theta^2}\right)$.

Here $l$ denotes the log-likelihood function, i.e. $l(y,\theta) = \log p(y;\theta)$
(You can make appropriate assumptions of regularity of functions if needed.)

14.2 CRLB for exponential distribution

The pdf for an exponentional distribution is given by

$p(y;\theta) = \frac{1}{\theta} exp(-y/\theta), \; y\geq 0$

It is easy to check that $E(y)=\theta$ and $\textrm{cov}(y) = \theta^2$. Assume we have $N$ data points $y_1,\ldots,y_N$drawn from $p(y;\theta)$. 

a) Show that $\widehat{\theta} = \frac{1}{N}\sum_{k=1}^N y_k$ is a bias-free estimator of $\theta$ and determine the variance of $\widehat{\theta}$.

b) Calculate the CRLB, and show that the estimator in a) is efficient, i.e. achieves the Cramer Rao lower bound.

14.3 Kalman Filter for improved GPS positioning

Download the matlabb code gps.m  Download gps.m 

and study how the motion model described on the lecture is implemented and how it improves tracking performance. 

a) Change sigma2 from 1e-1 to 1e-3 and describe what happens to the position estimates

b) The code contains the following lines

kalman1 = ss(A-A*M*C,A*M,eye(4),zeros(4,2),h);
kalman0 = ss(A-A*M*C,A*M,eye(4)-M*C,M,h);

Explain why this gives the kalman filter without and with direct term (predictor vs filter).

14.4 Particle filter

Study the code PF.m Download PF.m

and identify where the height measurement is used to update the weights.

The code assumes the height sensor has a normally distributed error. How should the code on lines 80 and 83 be updated if the height sensor error was instead uniformly distributed in the interval [-a,a] (the parameter a is assumed known). You do not have to change the code.

We will also talk about the exam.

Solutions:

(Please disregard the different problem numbering in these videos)

ex14_1.mp4 Download ex14_1.mp4Play media comment.

(Note: I made a slight error at the end, missing to move the minus sign over before defining $I(\theta)$)

ex14_2.mp4 Download ex14_2.mp4Play media comment.

(Note: And here I missed a transpose after 5 min, the formula should be $\mathrm{Var}(Ax) = A\mathrm{Var}(x) A^T$)

ex14_3.mp4 Download ex14_3.mp4Play media comment.

ex14_4.mp4 Download ex14_4.mp4Play media comment.