Week 15
Synthesis: Markov Models
Business
The start of the week email. No need to reread it if you received it. It's just linked here in case anyone accidentally deleted it (it's happened before 🥺).
Monday: Markov models
The last topic of the course(!) is Markov Models, which are probabilistic models of the state transitions in a system. Today's resources develop the idea of Markov Models by working through a case study of how the percentage of women in the workforce changed in the second half of the 20th century.
The resources are listed below in the order they would have been presented in the classroom, but you might want to watch the summary, or peek at the pictures of the final marked up pages, first if you like to begin with an idea of where everything is heading.
- 📹Probabilities as Model Pencast video 7:15 | 📘 PDF
- The number of women in the workforce increased dramatically in the second half of the 20th century. How to understand this societal transformation? One answer is by a change in the probabilities that daughters would follow in their mothers' footsteps.
- 📹Graph as Model Pencast video 9:11 | 📘 PDF
- The conditional probabilities developed in the 1st pencast suggest using a tree, but the trees branches cross and want to join each other as it grows. We can straighten this out either by using a lattice which still grows like a tree to the right, or a small graph.
- 📹Matrix as Model Pencast video 9:41 | 📘 PDF
- If we can represent the model as a graph, then we can represent it as a matrix, and that lets us use software to
do the repetitive calculations.
There is an error in this pencast! Read the error messsage above the video before watching it. - 📹Excel Helps Out Screencast video 4:44
- Excel makes it easy to simulate the development of the system, and suggests its final destination.
- 📹Finding the System Equilibrium Pencast video 8:11 | 📘 PDF
- The Excel simulation suggested where the system was heading, but we can find out exactly where by solving a few simultaneous equations.
- 📹Summary: Markov Models Pencast video 5:02 | 📘 PDF
- Markov Model A collection states and the transitions between them.
Three representations:
• A set of conditional probabilities describing the state transitions.
• A graph where the vertices are the states and the edge weights are the probability of moving from one state to another.
• A transition matrix where the entry at row r and column c is the probability of moving from state r to state c.
The time course of the system can be found by simulating its development using Excel.
The final equilibrium state of the system can be found by solving a set of simultaneous equations based on the transition matrix.
- 📹Fine Tuning our Model Screencast video 5:59
- At the moment our model captures two important qualitative aspects of the data, that the proportion of women in the workforce grows and that it levels off, but it has it grow faster than it does in the data, and it levels off too low (60% vs 83%). Can it be adjusted to model more closely? (Spoiler: Of course it can!)
Supplementary resources:
Chapter 10 Markov chains in Applied finite mathematics by Sekhon and Bloom.
Tuesday: Examples (+ it's the last day of classes! 🤪)
- 📘Four examples PDF 6 pages
- Four smallish examples of Markov Models that show how to:
- draw the graph given a state transition matrix,
- write the matrix given a Markov graph,
- find the equilibrium state of a Markov system, and
- model the time course of a Markov system using Excel.
- 📹Example 1: Matrix→Diagram + Equilibrium Pencast 7:45
- The first example shows how to go from a transition matrix to a system diagram, and to determine the equilibrium distribution.
- 📹Example 2: Diagram→Matrix + Equilibrium Pencast 9:33
- This example shows how to go from a system diagram to a transition matrix, and then to determine the equilibrium distribution.
- 📹Example: Disease Outbreak in a House Finch System Pencast 10:36
- An example of applying Markov modelling to a pandemic—among house finches.
- 📘Example: Disease Outbreak in a House Finch System PDF 2 pages
Supplementary resources:
As above: Chapter 10 Markov chains in Applied finite mathematics by Sekhon and Bloom.
The week's exercise.
There's an exercise for you to do to confirm your understanding of Markov models, but not to be handed in.
Exercise Set 14.1: Markov Models Problems.
Exercise Set 14.2: Course improvement survey. The course evaluations are open, but I have some more specific questions about this course I would appreciate your views on.
Assignment
No assignment this week. The term is ending 🥲. There will probably be an exam question on Markov models though.