Day 43 in math modeling

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"To achieve great things, two things are needed; a plan, and not quite enough time.": Leonard Bernstein

Arnold Bloom wrote a textbook, Global Climate Change: Convergence of Disciplines, in the preface of which he reports that his students call his course "Bloom and Doom". I thought of that when Matthew suggested that there's been a lot of gloom and doom....
I could lie to you, and tell you everything's all right, if you'd like. Here: "Everything's gonna be all right. Don't worry; be happy!" Music makes me happy. Life is full of really cool stuff, and laughter, and love, and we can't ever forget that. I've been in some grim situations, and it always helps if you're able to laugh at the tangled webs we weave.
A little good news, to start: A Smorgasbord of Solutions for Global Warming: One important source of guidance is Project Drawdown, a global coalition of researchers, scientists, economists and others, that, in recent years, has built a model to evaluate and rank the top active solutions to global warming, based on their actual impact on greenhouse gas emissions.
America's Top Scientists Reprimand Donald Trump (Again): More than five hundred and seventy members of the National Academy of Sciences published a statement on Monday decrying the Trump Administration's "denigration of scientific expertise and harassment of scientists."
From the statement: "Scientific evidence and research should be an important component of policymaking. We therefore call on the Federal Government to maintain scientific content on publicly accessible websites, to appoint qualified personnel to positions requiring scientific expertise, to cease censorship and intimidation of Government scientists, and to reverse the decision to withdraw the United States from the Paris Agreement."
And so it goes.... I could sugar-coat it, but that's what the world's top scientists are saying. I'm not even close to their league -- but I've studied the data myself -- and you've seen some of the data yourselves -- so you need to know that these people, these brilliant people, are extraordinarily concerned.
Could sprinkling sand save the Arctic's shrinking sea ice?:
In March, the National Snow and Ice Data Center reported that the Arctic had reached a seasonal maximum of 5.59m sq miles -- the second-lowest peak extent in the 39-year satellite record. Scientists forecast that Arctic sea ice could completely vanish in summertime by the 2040s.
As sea ice is lost, it locks the Arctic into a downward spiral. Less stark white reflective surface, or albedo, means more dark ocean, which absorbs sunlight rather than repels it, thus speeding up the warming process and spurring further melting. Older multi-year ice, 10ft thick in places, accounted for just 1% of the winter ice pack last year, meaning the sea ice that remains is now largely composed of newer, thinner and less reflective ice.

ael: The key process in this account is called feedback: one thing leads to another, which leads back to the first, which leads to another, etc.:
Warming, which leads to melting, leads to darker oceans, which leads to more
warming, which leads to melting, leads to darker oceans, which leads to more
warming, which leads to melting, leads to darker oceans, which leads to more
iterations... forever.
And you have the "spiral" referred to above. And the spiral begins to spin faster and faster, faster and faster, and soon you have a tornado.
Baby Bison Are Being Flown To Siberia To Try To Save The Permafrost:
The Pleistocene Park, in the wilds of Siberia, is trying to recreate the ecosystem of 20,000 years ago, in the hopes that the actions of grazing animals can help keep the ground from thawing and releasing billions of tons of carbon into the atmosphere.
Warming temperatures are beginning to melt the permafrost across the Arctic and sub-Arctic. And as the permafrost melts, it exacerbates climate change: The soil may hold as much as 1,500 billion tons of carbon. If it were released, it could have a bigger impact than burning all the world's remaining oil, gas, and coal.

ael: Wait -- what does the prefix "perma" in permafrost mean? So
atmospheric carbon, which causes warming, which thaws permafrost; which releases
atmospheric carbon, which causes warming, which thaws permafrost; which releases
atmospheric carbon, ....
What happened to winter? Vanishing ice convulses Alaskans' way of life:
The past winter was the warmest on record in the Arctic, putting a lifestyle that has endured for millennia at risk: "The magnitude of change is utterly unprecedented"

Today:
- The final exam comes in two parts:
  1. An in-class exam, which will focus mostly on interpretation and some simple computations, including material back to the beginning of the course but also including some material on Markov chains and other matrix models. You'll have two hours for that, but I intend to keep it to about the same length as the previous exam, and you'll be able to skip one (at least) question;
  2. a take-home, which, like the last one, is to be conducted entirely on your own (although you may ask me questions, which I may or may not answer!).
  I've decided that the final can't hurt you, it can only help. So if you're satisfied with where you're going to be after you successfully finish your final project, you can skip the final.
  On the other hand, if you've been blowing off homework (which some of you have), then the final is an opportunity to recover.
- A little more about Markov (your homework returned):
  - Random walks are really important. As mentioned in one of the on-line articles about long period climatic oscillations, and how they might be random walks rather than true oscillations.
  - You should have computed the $N$ and $B$ matrices to answer question four. The final distribution of states (and the number of transients visited to get to one) are given by the $N$ and $B$ matrices.
  - Life is like a Markov chain on a circle, with one absorbing state.
- Our Togo models:
  - Final models, and diagnostics
  - A couple of things from the presentations:
    1. It is not enough for a histogram to be roughly bell-shaped and symmetric for it to represent a normal distribution. A normal distribution is very specific! Only two parameters characterize it: its mean, and standard deviation. That completely specifies it.
    2. I'm sure that there were those present baffled by hearing that "Our-squared" (which they would have heard as "R-squared") was negative... A formula is a must in such a circumstance!
    3. Some of you were concerned because our rainfall estimates were negative; I'm not bothered. If our estimate is negative, call it zero. We're seeking the long-term trend, and don't mind if we occasionally err on the ridiculous side!
      However if you insist on maintaining positive estimates, it would be possible to model transformed data, and then back-transform, to keep the values positive. Just as we saw with the transformed data when we linearized, normalizing data means that when you back-transform, Confidence Intervals are no longer symmetric -- may be, in fact, very difficult to obtain (if you have many predictors in a model).
    4. If you're truly curious about why Togolese rainfall is shaped as it is, check out Rainfall Characteristics over Togo and their related Atmospheric circulation Anomalies, p. 37.
      They discuss the ITCZ -- inter-tropical convergence zone, and its significance.
- I think that we've done pretty well at accomplishing my GOALS OF THE COURSE:
  - To practice working in a group on realistic problems.
  - To study and apply the process of modeling.
  - To study basic types of models and how they have been applied.
  - To learn/review some basic properties of these models.
  - To study and learn some (perhaps new) mathematical techniques useful in modeling.
  - To practice the technical presentation of mathematics.
  There are several important problems that we've considered
  - Keeling Data
  - Togolese Temperatures
  - SIR and related models
  - Isle Royale Moose/Wolf predator-prey population dynamics
  - Markov processes: Tennis Tournaments, Random walks, Galla social structure
  - Chaotic population dynamics of flour beetles
  Just about every one of these problems made use of two things:
  - Linear or non-linear Regression
  - Matrix representations and operations
- Every one of you should take linear algebra, if you haven't already; and if you have, then take a second, more advanced course in linear algebra!
- So you'll be surprised by the secret of the universe....

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