MTH 227-228 Sequence: Calculus and Probability for the Life Sciences

Additional Notes for MTH 227.

  1. On Discrete Probability
  2. On Linear Difference Equations and Pharmacokinetics

Extra Credit Project

I developed an extra credit project for the Spring 2023 MTH 228 class at Oregon State University involving the calculation of the best-fit allometric model

\[y = a x^{k} \quad\text{ for constants } a \text{ and } k\]

given some allometric dataset \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\) using matrix operations. The calculation involves linearizing the allometric model by application of \(\log(-)\) as follows:

\[\log(y) = \log(a) + k \log(x)\]

Here, we assume that the best-fit line (i.e. doing linear regression) for the equation above yields the best-fit allometric model. This best-fit line can be determined by solving the following normal matrix equation:

\[M^T Mx = M^T b\]

where \(M\) and \(b\) are matrices containing the relevant allometric data and \(x\) is a column vector containing the coefficients of the best-fit line.

  • The PDF of the handout/project description given to students is available here.
  • This project is based on the paper “The Allometry of Brain Size in Mammals” by Burger, George, Leadbetter, and Shaikh (April 2019). The paper is available via Oxford Academic, linked here, and a copy of the pdf is linked here. We asked students to recreate the results given in Table 1 for specific orders.
  • The dataset is posted in Dryad, linked here. The relevant csv file is also linked here.
  • An example calculation of linear regression by matrices is given here