Questions

You might find the following questions useful starting points for testing your understanding of point process models.

1. How you would simulate one realisation from a homogeneous Poisson process on the one dimensional interval \([t_\text{min}, t_{max}]\)?

2. Describe how you would simulate one realisation from a Poisson process in one dimension with a quartic polynomial intensity function over the interval \([t_\text{min}, t_{max}]\) by:

• generating inter-arrival times;
• rejection sampling;
• using a piecewise constant approximation of \(\lambda(t)\).

What other methods might you use?

3. Can you write pseudo-code for each of the approaches in question 2? Have you written R or Python code for any/all of the approaches in question 2?

4. Suppose you have observed the point pattern \((x_1,...,x_n)\) on the interval \([x_\text{min}, x_{max}]\) and you wish to model the intensity function \(\lambda(x)\) as a polynomial function. Write down the likelihood and log-likelihood of the observed point pattern when modelling the intensity using a cubic polynomial.

5. How would your answer to 4 change if you instead modelled \(\lambda(t)\) as a quadratic polynomial?

6. How could you test whether a cubic term improves the fit of your model?

7. Generate a point pattern with a cubic intensity function. Find point estimates and confidence intervals for the polynomial coefficients.

8. How might you visually represent the uncertainty in your estimated intensity function?

9. How could you measure whether a point pattern is more or less clustered than a Poisson process?

10. How could you test whether a point pattern is Poisson, more clustered or more regular?