Day 15 in math modeling

• I appreciate your very frank assessments, both of your own projects, and of your peers' projects. Some of you were too hard on yourselves.

  My main observation is that, by looking at others' work, you have identified ways in which you can improve your own. That's a very positive thing!

• Chris noted that linear models were used in 14 of 20 cases. Thanks for that summary, Chris. That's helpful information.

• You'll be doing a second round of reviews with this project. Just a head's-up.

If you need any help, please come and see me. Feel free to email with questions, as well.

A couple of things to point out (here's my annotated pdf):

• First of all, their results rely on the conclusions from a regression analysis.

• decadal variability in Terrestrial Water Storage (p. 1 of 4). We noticed "decadal scale" variability in the Keeling data, and we may detect it in our temperatures for Togo, as well.

  The "PDO" -- The Pacific Decadal Oscillation -- is a well-known source of long time-scale variability:

  

  There is also an AMO (guess what that means?:). It might be more relevant for Togo:

  Nathan felt that the 24 year periodic sine his group fit to the Niamtougou data might have been "ridiculous", but there's nothing ridiculous about it at all! There are big things out there lurking, some of which we are unaware of.

  Adam observed something that I did, too: the Sotouboua minima seems to have a "U or V shaped curve every twenty years or so, which could possibly be modeled with a periodic function."

  So Nathan, we don't think that you're being ridiculous!:)

• Global sea level rise is accelerating, acceleration measured by the parameter multiplying the quadratic term in a quadratic fit to trend (p. 2 of 4). Interpretation of parameters is critical. The model $SLR(t)$ (sea level rise as a function of $t$) has, as its trend's second derivative, the value $SLR''(t)=2\beta$, where $\beta$ is the quadratic term's parameter. Interpreting parameters is important!
• Estimate is given as a box -- a confidence interval (p. 2 of 4): "Their joint consideration yields a final acceleration estimate of $0.084 \pm 0.025 mm/y^2$."
• p-values are mentioned in an interesting way (p. 2 of 4): "The probability that the acceleration is actually zero is less than 1%."
• While the degree of sea level rise they suggest is imminent (around 65 cm by 2100) may sound alarming, perhaps even more alarming is their statement on p. 3: "If sea level begins changing more rapidly, for example due to rapid changes in ice sheet dynamics, then this simple extrapolation will likely represent a conservative lower bound on future sea-level change. In contrast, few potential processes exist to suggest that this estimate is too high."

  My young friends, in all honesty, this is a truly conservative lower estimate: sea level is likely to far exceed this amount.

  My personal experience in this area of climate change research is that "worst case scenarios" have been exceeded over and over. That's not supposed to happen, but it illustrates the non-linear nature of the phenomena involved, and our lack of understanding of the interconnectedness of nature.

  Our National Intelligence agencies are worried (as they should be): Intelligence Agencies Warn of Climate Risks in Worldwide Threat Assessment: While top Trump administration officials deny climate change, the intelligence agencies warn global warming can fuel disasters and violent conflicts.

Here's the Mathematica file.

One question you may have is how to interpret the parameters?

• The model to the cumulative data was \[ cumulative(t)=ae^\frac{-b}{t^p} \]

  where all parameters should be positive. $a$ is the asymptote, $p$ was designed to control how flat the curve starts out, and $b$ was intended to be related to growth rate of the curve, inflection, etc.

  As it turned out, $b$ and $p$ are a little confounded.

• Cumulative was in number of bodies, so in the end, I wanted to write this as \[ CDF(t)=\frac{cumulative(t)}{a}=e^\frac{-b}{t^p} \] That is, I'm thinking of this as a Cumulative Density Function (CDF). The derivative of the CDF is the PDF, the Probability Density Function: \[ PDF(t)=CDF'(t)=\frac{bp}{t^{p+1}}e^\frac{-b}{t^p} \] which hits its peak at $x=\left( \frac{bp}{1+p} \right)^\frac{1}{p}$. This corresponds to the point of inflection in the model.

• These features are illustrated in a modified version of the cadaver file, that includes a Manipulate command to help in getting a start on the good guess, as well as an illustration of the CDF and PDF.

We've been mostly dealing with empirical models -- we notice a relationship between two variables, and we construct a model of one as a function of the other -- without any belief, necessarily that one is causative. We study correlation, rather than causation.

Exceptions include that sea anemone data we looked at: response duration could be considered related to the square root of area, because the greater the area, the longer it takes a signal to travel out from the center; or the periodic oscillations on the Keeling data -- those sinusoidal oscillations we understand as the "breathing of the Earth's forests".

But now we want to do some modeling where we build in the relationships between the variables, like we're building some structure in your back yard. These are structural models.

I asked you to read a short introduction to modeling by some Canadian math/biology types: pages 3-8 (the 7th-12th pages of the pdf) of this intro to modeling, by De Vries, et al. (which I've marked up).

It gives us a tiny introduction to differential equations, difference equations, growth, and SIR infection models -- things we'll be studying in the week(s) ahead.

Let's take a look at that. In particular, we want to look at page 5, which outlines two different ways of thinking about modeling infectivity.