Notebook for Stochastic Modelling and Random Processes module

Notebooks will be uploaded for each support class.

Detail of each notebook:

Support Class 1.ipynb

• basic linear algebra in python
• simple random walk animation
• function SRW, a simple random walk with arguements p (probability steping up), tmax (when to terminate the walk) and N (number of replications)
• empirical distribution calculated at a fixed time (hist plot over possible states at time n)
• simulation with periodic boundary conditions (modulus L (L =10))
• simulation with closed boundary conditions (reflects at 10 and 0)

Support Class 2.ipynb

• Geometric random walk
• ergodic average
• empirical tail (1 - CDF)
• Wright-fisher model ( heatmaps)
• time to reach steady state
• Gershgorin disk theorem

Support Class 3.ipynb

• general plotting techniques in matplotlib http://matplotlib.org/1.5.3/api/pyplot_api.html#matplotlib.pyplot.plot

Support Class 4.ipynb

• Kingman's Coalescent
• CTMC (waiting times, exponentially distributed)

Support Class 5.ipynb

• Orenstein-Uhlenbeck Process
• simulated by finite difference approximation (taking the Weiner incremenet by sampling from normal distributioon with zero mean and dt vaiance)
• simulated using sdeint (python stochastic differential equations, numerical integration)

Support Class 6.ipynb

• Moran model (similar to wright-fisher but continuous time)
• CTMC with waiting times
• introduction to networks and using the networkx package in python
• degree distribution, clustering, transitivity, distance, largest component

Support Class 7.ipynb

• Erods-Renyi random graphs
• compare degree distribution to binomial distribution
• expected size of largest component for multiple realisations
• expected local clustering coefficient for multiple realisations
• Wigner semi-circle law

Support Class 8.ipynb

• compare degree distribution to poisson distribution
• Barabsai-albert model
• empirical tail distribution
• expected degree of nearest neighbour given node has degree k

Other

For Latex, I recommend you to use Overleaf (particularly useful for RSG as can have multiple people working on the same document at the same time - plus you can chat to each other on there): https://www.overleaf.com/

For GitHub help, check out https://www.gitkraken.com/

Steps to download GitKraken:

• Download the .deb file
• open terminal and go into your downloads (cd Downloads)
• sudo dpkg -i name.deb
• log in with you github

Useful links for the assignments:

• Log Normal Distribution: https://en.wikipedia.org/wiki/Log-normal_distribution
• Log Normal Scipy Documentstion: https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html
• KDE plot visualisation: https://mathisonian.github.io/kde/
• Fokker-Planck Equation: book Stochastic Processes in physics and chemistry - N.G. Van Kampen
• Ornstein-Uhlenbeck process: https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Fokker%E2%80%93Planck_equation_representation
• Colour options in matplotlib https://xkcd.com/color/rgb/