Day 38 in math modeling

• Thanks to Alyssa, Sally, and Rachel for their presentations on Friday.

• This Friday: more Greatest Hits of the KYMAA

• Today's climate news:

  Good news, from Chris! Thanks, Chris!

  Young people worldwide take up climate activism: '"It's quite empowering to be something in your global community," said Melanie Mattauch, the European communications coordinator for 350.org, a group working to build a global grassroots movement to demand action on climate change.'

  A bit of history:

  • In 1988, James Hansen testified before Congress that climate change could now be detected in the data. He said that "The greenhouse effect has been detected, and it is changing our climate now."

  • In 2008, Bill McKibben asked James Hansen what level of CO₂ would be safe; Hansen said 350, and 350.org was founded.

  • Hansen has since said we've got to get down below 350.

  • Hansen is now helping a case wend through the courts, JULIANA v. U.S. - CLIMATE LAWSUIT. Hansen's granddaughter is one of the plaintiffs.

    "Their complaint asserts that, through the government's affirmative actions that cause climate change, it has violated the youngest generation's constitutional rights to life, liberty, and property, as well as failed to protect essential public trust resources."

    (Interestingly enough, in today's news is this from The Guardian: The courts are deciding who's to blame for climate change. "Oil companies? The government? The public? All of the above share the blame." In the article they declare that "[i]n short, Our Children's Trust is correct to allege that the American government has failed to protect the coming generations from the threats and damages of climate change.")

  • I'm proud to say that I was arrested along with James Hansen in Washington, D.C. protesting an allied issue -- Mountaintop Removal, a damnably dirty process for the extraction of dirty coal.

  More on the Atlantic ocean circulation pattern known as AMOC (Atlantic Meridional Overturning Circulation):

  Avoid Gulf stream disruption at all costs, scientists warn: "How close the world is to a catastrophic collapse of giant ocean currents is unknown, making halting global warming more critical than ever, scientists say"

  

  'The new research worries scientists because of the huge impact global warming has already had on the currents and the unpredictability of a future "tipping point".'

  That phrase "tipping point" is very important: more on that this week, as we study chaotic beetles.

• Today's stereopticon of the day: more fun!

• The final project, updates:
  • Thank you for your city reviews.

  • Questions have been sent. I've heard back from Binao with some hints that answers may be slow in coming (colleagues are out of the country, etc.). Since they don't have remote access to the data, they can't do the checks we've requested right away.

    Binao did mention that they're nervous about their data being out of their hands. I've told them that it's been locked down. At the moment, with the exception of one file that I've got open for today, I think that I've gotten everything locked down on my website. If you have any files out on public websites, please remove them, or lock them down.

    If you notice that I still have anything out and available, please let me know. So you should access any data you need off of the course website under the control of our data quality team.

    That being said, could the data quality team please place copies of the original data we were provided on the website? That's really v0....

  • I've proposed a few models to the modeling group, and am awaiting word on whether the modelers have something better that they'd like to recommend:

    • TogoTempAndRainfallFullModels.nb. This is my suggestion, at this point.

    • One could use the model to do imputations of missing values. If you're planning to do imputations, an interesting comparison would be between methods.

    • TogoTempAndRainfallWithENSO.nb: this was a test of whether the ENSO (El Nino/La Nina) was important to the models -- turns out that, although significant, it's not very important; so we won't use it. One problem is our ability to project into the future. At the moment, with location (latitude, longitude), and time, we can estimate maximum temperature, minimum temperature, and rainfall. If we use ENSO, then we'd need a model for that, as well.

      I'm sure that there are models already available, however, out of the box....

• Now finish up an interesting application where Markov chains prove to be pretty successful for Olinick: tennis matches.

  We'll apply these ideas, and actually test the Markovian nature of tennis tournament finals!

I'd like to do an example of a classic Markov chain for homework: the random walk. Random walks are extraordinarily important -- not because we randomly walk, but because molecules do, say, and pollen grains on the surface of a glass of water....

So I'll have you all take a crack at the following problem (you will probably want to reuse the Mathematica code for the Tennis example):

This is to be handed in next Friday, 4/20. You may work on it together.

1. Draw the State Graph.

2. What is the transition matrix? (Arrange it with absorbing states first, and then transients, in order from lowest to highest).

3. Starting from position 1, what is the probability that it will have been absorbed in three steps?

   1. Draw a tree diagram

   2. Compute the third power of the matrix, and check that you get the same answer!

4. If you start from 2,

   1. how long on average will the particle spend in transient states?

   2. what is the distribution of final states?

   3. Do these answers change if the probability of going left becomes 1/3, and right 2/3?

More on that later.

To set the stage, we're looking at flour beetles which have a variety of age classes, and are cannibalistic; that turns out to be one of the keys to the "chaotic" part.

We're going to be working, as usual, with matrices, which represent transition between three classes:

1. larvae (L),

2. pupae (P), and

3. adults (A).

• This Mathematica file gives us our introduction to the problem.

• A Leslie matrix is a matrix similar to a Markov matrix: a Markov matrix can be a Leslie Matrix (but it need not be); similarly a Leslie matrix may be a Markov matrix, but need not be.
  \[ \left [ \begin{array}{ccccc} {f_{11}}&{f_{12}}&{\ldots}&{f_{1,m-1}}&{f_{1,m-1}}\cr {t_{21}}&{0}&{\ldots}&{0}&{0}\cr {0}&{t_{32}}&{\ldots}&{0}&{0}\cr {\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\cr {0}&{0}&{\ldots}&{t_{m,m-1}}&{0} \end{array} \right ] \] Unlike in a Markov chain, the transition components are not confined to probabilities: the Leslie matrix is not a stochastic matrix, in general (otherwise the sum of rows would be necessarily 1, and all transition components would have to be 1!).

• A generalization of the Leslie matrix is an Usher matrix:
  \[ \left [ \begin{array}{ccccc} {f_{11}}&{f_{12}}&{\ldots}&{f_{1,m-1}}&{f_{1,m-1}}\cr {t_{21}}&{t_{22}}&{\ldots}&{0}&{0}\cr {0}&{t_{32}}&{\ldots}&{0}&{0}\cr {\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\cr {0}&{0}&{\ldots}&{t_{m,m-1}}&{t_{mm}} \end{array} \right ] \] This is appropriate when individuals are in increasing size classes, and may not return to a smaller class (but may "graduate" to the next largest).

• In the end, it is a type of generalization of an Usher matrix that we will use to study the flour beetle.

• I first implemented the essentials in this Mathematica file

• This insightmaker model allows us to perform the same analysis. Let's check that the Mathematica code and the InsightMaker model give the same results in our first case.