Functions Functions for Modeling

Strategy for presentation:
I thought we would lead the participants through this material, so that there is a common basis for later.

General Features of Functions

What are the inputs?

Models should be intended for a purpose. So it matters what is the input and what is the output, since you typically want to transform a particular quantity into another type of quantity.
Snow Tree Cricket (from Chapter 1 of Hughes-Hallett). See Math 135 notes from 25 Jan. 2012

Difference between traditional math and computer notation for functions.

\( y = mx + b \), \( f(x) = mx + b \), \( \int x dx \), \( \sin(x) \), \( \sqrt{x} \), and so on.

Parameters.

Where do parameters get introduced in Stewart? Why would we ever write \( \sin x \)?

Specific Functions for Modeling

Straight-Line functions \( f(x) = mx + b \)

Parameterization

Students already know a lot about these: slope and intercept. Emphasize that these are parameters. The parameters specify a particular function out of the whole set of functions.

Other parameterizations (e.g. point-slope, two-points, point-angle) can be converted to slope-intercept.

Applications

This is the general purpose modeling function. All simple relationships can and are represented this way.
1. World record swim times versus year. swim100m.csv
2. Wage versus education. cps.csv
3. Natural gas usage versus temperature. utilities.csv
See more details on these examples

Exponentials

Proportional versus constant change

  1. Atmosphere pressure, water pressure
  2. Population of Mexico (Math 135 27 Jan 2012)
  3. Exponential models of data Income and housing variables

Parameterization of exponentials

  1. Half-life, interest rate, exponential time constant. Which is best depends on what information you have.
  2. Conversion: Rule of 72.

Models

  1. Half-lives of meds. How often should you take the pill to keep the drug level within 10% over the long run?
  2. Activity on doubling and half-lives
  3. Sums of exponentials. Why is cooling not a simple exponential? Fitting Stan's data

Logarithms

Examples

  1. Happiness on a log scale
  2. Profanity and log axes

Activities

  1. Log axes — how to interpret. Example of log axes. Activity: Plot intervals on these scale to indicate absolute and relative precision on both log and linear axes, for instance, 10% precision on measurements of 1 and of 100 on a log axis.

Modeling project

  1. Prices versus log prices in the stock market. Which generates a better simulation of stock variation: a random walk on \( \Delta \) prices or a random walk on \( \Delta \) log prices?

Splines

  1. Squeezing Orange Juice … measuring the cumulative. (Make a video of the scale when squeezing orange juice.)

Power Laws

  1. Random walks. How does the typical distance scale? Explanation in terms of random vectors being perpendicular. (Later application: Explaining the power law: integration of the square distance compared to integration of the absolute distance)
  2. Chest circumference versus weight, tree height and thickness. (Later example in multivariate: relationships in the internal combustion engine)

Sines and Cosines

Parameterization

Amplitude, time lag, offset.

Models

  1. Day length as a function of the day of the year. (For later example, as a function of longitude.) Where does the model break down (near the arctic circle)?
  2. Tone and period. Model a musical scale. How does the period change from note to note? Model a chord.
  3. Modeling tide data. Example on Hawaiian tides
  4. Kepler and the planets. (See kepler.Rnw)