Day 6 in math modeling

• Read 2.3.
• Exercises pp. 90-91, #1, 3, and 4.
• Continue working your choice of either Project 1.6 or 1.7, pp. 43-45. Your solution to Project 1.6 or 1.7 will be due on February 7th. This is something that you may work on with another student in the class, as a team.
• Problem notebooks due on Wednesday, 1/31, with all assignments up to and including your results for Project 1.2 or 1.3.

• how to characterize and interpret noisy data
• how to build models with stochastic components (incorporating randomness)

• Are deterministic models "meaningless"? (p. 47)
  • events happen in a somewhat unpredictable manner;
  • on average, however, things may be very predictable.
• The deterministic model breaks down:
  • What is the sense of "stochastic squirrels" and how do M&Ms model them?
  • Is the noisy solution (i.e. time series of squirrel population) a result of a noisy R?
  • What becomes of R in the first-order growth model for the stochastic squirrel simulations?
• Statistical issue: R (and/or any other parameter) is no longer treated as a constant:
  • R belongs to a distribution, which is characterized (not necessarily completely!) by measures of
    • central tendency, and
    • spread of the distribution.
    We might call this a population of R values, but you must be aware that we use population for several other "animals" in these sections and chapter.
  • R may be sampled from a population: e.g. this particular deer population (there's that word again!) is subjected to an R of 3, whereas another is subjected to an R of 4. The environment may be considered a source of stochasticity in R.
  • Sampling R over and over and generating multiple solutions (or time series) gives us a population of solutions (e.g. 51). (Simulation studies can be very useful.)
• Statistical issue: spread and central tendency
  • central tendency:
    • Mean - sensitive by outliers (but what is an outlier?)
    • Median - resistant to outliers
  • spread:
    • Standard deviation
    • Range
    • Inter-quartile Range
• Characterizing or representing stochastic data
  • "Binning" (summarizing by grouping)
  • Percentiles, Quartiles
  • Frequency Distributions
  • Histograms (glimpse of distribution?)
  • Boxplot (five-point summary: min, first quartile, second quartile (median), third quartile, max)

This will allow us to get started on exercises pp. 90-91, #1, 3, and 4. You should be able to "hack" the Mathematica code to produce the results you need for these exercises. Feel free to use any other program, however, if you would be more comfortable doing so.

Notes:

• They demonstrate how to define a function when they create the function SingleBoxplot.
• One of their data values for the season1 data was wrong in Chapter2.nb), so the results may differ between what they show in the appendix and what Mathematica produces for you.