Math Modeling: Day 21

Just a few more remarks....

• In the course of that modeling exercise we saw that we can perform non-linear regression by iteratively solving linear problems, to approximate a solution to the non-linear problem.

  I tried to indicate how this is a generalization of Newton's method.

• We take a look at some corn seedling growth data, and weight in grams against age in weeks (use this Mathematica file). The age goes to 20 weeks, so these are corn stalks by the time 20 weeks rolls around.

  We can see the weight asymptoting by week 20. It's clearly NOT appropriate to model this with an exponential, as John Alexander curiously suggests in Curve Fitting via the Criterion of Least Squares (see Figure 12, p. 23).

• So a logistic curve seems appropriate (more on that in a moment). But, if we do that, we can actually say more: we can use the logistic regression to propose an ordinary differential growth equation for corn.

  So the regression serves the purpose of providing parameters for a general (differential equation) model for corn seedling growth. This is an example of a structural model: we tie an empirical model into a structural model, and learn something (or incorporate something) about the nature of plant 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 in the weight of the seedling is proportional to its weight, and proportional to a substrate which is ultimately exhausted, and itself proportional to $(K-w(t))$. So "carrying capacity" in this case is sort of the maximal size of the plant that the soil can support.

  Plants are notorius for having "switches", however -- at some point they switch from producing leaves to producing flowers to producing fruit.... Lots of ugly non-linearities.