Day 24 in math modeling

1. First of all, you remain in charge of your town, with your group. Whatever final model is concocted, your group will analyze it with respect to your town.
2. You will be responsible for providing some climate-related information regarding your town, to the climatological department.
3. Secondly, you will have a second role within the publication process.

For the rest of you, you're all (or almost all) in your first or second choice.

Here are your specific roles in the paper (I'll say more about these later; but let's keep our focus on the exam at the moment.):

• Division of Data Management and Quality: Jacob E., Head
  • Department of Data Integrity (to sift out MIsleading, Fraudulent, Fouled, or otherwise Egly Data -- MIFFED): Lizzy, Head; Patrick

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

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

    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.).

    Matthew, Clayton, 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

• Division of Research and Writing: Donna, Head
  • 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

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

    Leah, Head; Jacob K.

1. An in-class portion (Friday, for which you may use your notebooks with anything you've written in them).
2. A take-home portion (which you are to do without discussion with anyone but me, and with me only if you have any questions which I feel I may legitimately answer).

   This portion is a very straight-forward set of tasks, so I don't think that you'll have any questions; but it will involve software.

However, I criticized his suggestion to fit the data of page 19 (and Figure 10) with an exponential model: so we talked about how to "do it right" (IMHO), using a logistic growth model, and non-linear regression.

We can see the weight asymptoting by week 20. It's clearly NOT appropriate to model this with an exponential, as Alexander curiously suggests (see Figure 12, p. 23), although he may have been focused on the short term, relatively unbridled growth at the beginning. If so, he should not have used the later data points in the regression....

So the regression serves the purpose of providing parameter estimates for a general (differential equations) model for corn seedling growth.

This is not "typical", in some sense, of logistic growth. For example, "carrying capacity" is the wrong terminology: and the initial value is never greater than the "carrying capacity" -- the corn seedling doesn't start out monstrous, and then shrivel to a stalk...:)

This model is appropriate if the growth ($dw/dt$) in the weight $w(t)$ of the seedling is

1. proportional to the weight, $bw(t)$, and
2. proportional to a substrate which is ultimately exhausted, and itself proportional to $(K-w(t))$.

1. Adam (and Donna)
   • Is your tangent line straight? Does it intersect the axis at the first guess provided by Newton's method?
   • Adam computed the exact solution; some of you used Mathematica's NSolve, which is a numerical solver. They might well be using Newton's method to find their "solution"!

2. Donna -- how to compare models? Putting the residuals together was a good idea, on the same axes....
3. Jacob E.
   • Alyssa commented that "it is always best to start with the simplest model, adding in biological features only when necessary and one at a time."
   • The "sleek" SIR

• The modeling process: UPCE
• Simple linear regression:
  • How the equations are derived -- multivariate calculus
  • How the equations are derived -- linear algebra
  • Properties of the equations
  • Simple calculations
  • Interpreting results (parameter estimates, t-statistics, p-values, CIs, residuals, R^2)
  • Comparing models
  • Incorporating sinusoidal variation
    • Keeling $CO_2$ Data
    • Togolese Temperature Data
    • Using (pairs of) trig functions to capture periodic behavior in time series regression
      • Why do we need pairs of sines and cosines?
      • Amplitude and Phase

• Approaching the non-linear via the linear:
  • Solving non-linear problems with linear ones (e.g. Newton's method)
  • Non-linear regression
    • via linearization (e.g. power or exponential models);
    • as iterated linear regression;
    • using Mathematica's native non-linear regression command....:)

• Properties of models under shifts in time
• Compartmental models and differential equations
  • SIR model dynamics
  • Lotka-Volterra model dynamics
  • Finding equilibria of systems of differential equations
  • Making models more realistic; incorporating model parameters or forms obtained from field data
  • Know how to use InsightMaker.

• You should be familiar with the various data sets we've studied (Keeling, Togo, Ad Expenditures, Sea Anemones, Moose/Wolves, Cadavers, Corn Seedlings) and procedures we've carried out with each.

• You might have a look at some of the sections of my "mathematical review" -- 9-12 in particular.