Day 26 in math modeling

• Your take home exams are due now.
  • Your in-class midterm exam (60% of your exam grade) is returned. As I told you on Friday, you may redo any problem on which you missed points, for half the points back.
  • You can even use this key, made up of good responses by your peers.
  • I'll hope to have your take-home exams graded by Wednesday (40% of your exam grade).

• Some comments of my own regarding the in-class portion:
  1. Problem 1 is straight-forward calculation. I tried to make things easy. For example, the data is odd -- do you see why that means that the linear model has to be odd, and why it therefore has to be a line through the origin?

     So $(\overline{x},\overline{y})=(0,0)$. That makes the rest of the calculations pretty easy (e.g. $a=0-b0=0$ -- so it's a line through the origin!)

  2. I essentially ask you to do what you should have done in the exam-prep homework -- but to explain the connection between the non-linear problem ($\ln(x)=0$) and the linear problem ($f(x_0)+f'(x_0)(x-x_0)=0$), and that one proceeds by iteration until satisfied.

  3. Three regressions:
     1. I see a smiley face in the residuals.
     2. This "linear regression line" can't possibly be that obtained by simple linear regression, because it doesn't go through the center of mass of the point cloud.

        Furthermore, a linear model for happiness as a function of hours of sleep doesn't seem at all reasonable when you get to the high end of sleep (often associated with depression or illness).

     3. The cubic is quite a stretch, literally, to that last value; it is true that increasing moose density means increasing pressure, and perhaps it's so bad that, at that particular value of density, population plummets.

        But I'd argue for a qualitatively different model at that point, something like falling off a cliff, not a smooth cubic transition.

  4. Data we've known....
     1. Keeling involved sines and cosines of matching periods of a year; this provides phase and amplitude information (how much the earth's forests are breathing, and when they're inhaling and exhaling).
     2. The sea anemone data was an example of structural modeling, assuming $x$ was proportional to area, and that response time was proportional to length (radius). So this suggested a square root.

        The square root is rejected by our results, since 0.5 is not in the interval for the power.

  5. Moose and Wolves. You knew that they'd be on here somewhere!

     I want you to understand how we reason as we go about modeling. That's the first part -- replacing the unrealistic exponential growth with bounded logistic growth (which we perceive as related to sustainability).

     The second part relates to the dynamic behavior we expect in the system as a whole. As we discovered, the oscillations disappear after this change, and we settled down to constant (asymptotically) wolf and moose populations.

  6. This was pretty much right out of the notes about linearization. Sorry about the mix-up of variables: $y(x)=a x^b$ is what I meant!

  7. This was a test of your calc 1 skills, but also your ability to navigate summation notation. We've done a lot of summations, and you've got to get a good handle on them. They're really important.

• Your roles for the final project: I don't have requests from some of you, so you will not have been assigned. See me, if not.

  This is essentially our "org chart". You work within your unit, and then report up the line.

  Here are your special roles in producing the paper, and we'll talk more about specific objectives today:

  • Let's start by looking over the paper so far...., and see how specific sections will be divied up (at least as things stand at the moment).

    Donna, et al: this was produced using latex, and I would like to use that. So this is an opportunity to use the best mathematical typesetting software available... but may involve a learning curve. This format is the one provided by the American Geophysical Union (AGU) and their line of journals. That includes their bibliography style (using BibTex).

  • Division of Data Management and Quality: Jacob E., Head

    This group will "own" the data, and Jacob, we need to implement a "check out" procedure (so that the modeling group is always using the correct data).

    That's the number one business item.

    • Department of Data Integrity (to sift out MIsleading, Fraudulent, Fouled, or otherwise Egly Data -- MIFFED): Lizzy, Head; Patrick

      You're in charge of Appendix 1

      You'll need to investigate every suspicious, troublesome data point. First thing is to solicit proposed data problems from all cities.

    • Department of Outlier Detection (statistical egliness): Sally, Head; Chris, Terra

      You're in charge of Appendix 2.

      You need to develop a consistent, sensible strategy for temperature and rainfall, then implement it across the data set.

    Jacob meets with Lizzy and Sally.

  • Division of Modeling: Alyssa, Head

    I've made some suggestions about how we move forward, but your hands are not tied -- and if you have better ideas, let's try them!

    The model I've got in mind for you (linear in time, with appropriate periodic oscillations, latitude, longitude, and elevation) would likely produce the results that I speak of in the title. It might be a good place to start. But you're not bound to my model.

    • Department of Modeling (Temperature, and possibly other variables, e.g. rainfall). Matthew, Head.

      We'll all be part of this, to some extent; but we need a team to make sure that it gets the attention it deserves. This involves residual analysis, parameter assessment (significance, inclusion, etc.).

      Joey, Clayton, Austin H., Parker (Head Ringer!:)

    • Department of Computer Modeling and Graphics (to implement the models, create graphics, etc.)
      • R-Coding: Alli, Head; Rachel attached, but code editor generally
      • Mathematica-Coding: Maria R., Head; Adam

      These groups should begin thinking about graphics to produce, etc. We have ten cities to compare. How to do it intelligently?

      We'll be providing your files to the journal as supplementary material (although it's likely that we won't be able to include the data). So how can one make use of these even without the Togolese data?

      We could provide simulated data.

    Alyssa meets with Matthew, Alli, and Maria.

  • Division of Research and Writing: Donna, Head

    Donna: one of your first jobs will be to investigate any style requirements of this journal. Latex is an option, but they don't have their own style file, that I found.

    I don't mind sticking with AGU style, as a default.

    • Department of Background Info, Geography and Climate of Togo, and References and Citations.

      This unit will have sole responsibility for interacting with the Togolese, to answer our questions.

      So someone who is interested in diplomacy would be good to have in charge!:)

      Maria M., Head; Connor

      This group is in charge of the introduction, and the material for the bibliography. Other groups may be digging up references, but these should be passed along to this group.

      You might begin by going over past reports, and amassing and distilling the questions people had for the Togolese. Then we'll send those along.

    • Department of Style, NARrative, and Color (SNARC). Principal writing unit.

      Leah, Head; Jacob K.

      This group is in charge of the writing, generally: making sure that it's consistent, etc.

      You'll want to investigate latex. Here's the "code" for our paper, so far.

    Donna meets with Leah and Maria.

  • Finally, anyone can help out anywhere! Suggestions and help are welcome! Just don't interfere with a group trying to do its job.

We are about to begin the back end of the course talking about stochasticity and probabilistic models; then I'd like to talk a little about chaos and/or fractals, just because they're fun. The first examples will be Markov Chains, and we'll use one in a familiar context -- an SIR -- to get acquainted.

We'll do that, however, next time: today I want us to get together in our groups, to begin to plot out our strategy for the final project.

• You will make a presentation in the last week of class, detailing your efforts. We'll use two class sessions for this, so we'll have about 30 minutes per division.

  I'll be inviting some faculty in to see these presentations, because they'll be interested!:)

• You need to organize and divide up any work.
• You need to complete your work on your section(s).