# Coursework

**Deadline: 3/1/2025** (please note that this is a hard deadline and I cannot provide extensions as the assessment has to be finalised by the beginning of the following week. Given the release date, I strongly suggest you to finalize your submission before Christmas).

**Feedback**: I will provide feedback on your coursework by 10/1/2025. This will be a brief feedback on the content and the presentation of your work. If you would like to discuss your feedback in more detail, please let me know.

## Task

1. Consider a multivariate linear-Gaussian state-space model of your choice. This could be a model that you have seen in the lectures, or a model that you have encountered in your research. Give the details of this model, its parameters, and its use case. Simulate $y_{1:T}$ (data) from this model and fix this data for the rest of the coursework (as well as SSM parameters).

2. Implement the Kalman filter for this model. Demonstrate the performance of the Kalman filter by plotting the true state, the observations, and the filtered state estimates. Discuss the results.

3. Implement the particle filter for this model. Demonstrate the performance of the particle filter by plotting the true state, the observations, and the filtered state estimates. Compare the results (in particular, mean estimates) to the Kalman filter.

4. Expand your Kalman filtering implementation to provide the marginal likelihood estimate of the state-space model. Report the result.

5. Implement the marginal likelihood estimator of the SSM using the particle filter. This is a stochastic estimator of the original value you found in the previous step. To compare your result, run the particle filter multiple times (Monte Carlo runs) and report the mean and the standard deviation of the estimates.

6. Vary the previous step for different numbers of particles and Monte Carlo runs and build box plots for each case. Discuss the results.

## Submission

Page limit: 10 pages, recommended length around 6-8 pages, use appendices if you need to go beyond page limits

Submit by email to name.surname at imperial.ac.uk

Here name = deniz and surname = akyildiz

Email subject: **MFC2025CW**

Please make sure that you set the email subject as above, otherwise your submission may be missed.
