====== Math 445 Lab 3: graphical data analysis  ======

For this lab you will deduce the functional relationship between variables in data sets using graphical analysis. 
The data sets are given as // N x 2// matrices with //x// as the first column and //y// as the second. 
For each data set, you will find a function //y(x)// that fits the data, using the following steps:

  - Cut & paste the data set to a text file with an appropriate name, e.g. ''earthquakes.asc'' for problem 1.
  - Load the dataset to Matlab with ''load''.
  - Extract the two columns of the loaded data into two appropriately named vectors, e.g. //R// and //N// for problem 1. For the remaining generic instructions I'll use the names ''x'' and ''y''.
  - Experiment with ''plot'', ''semilogy'', ''semilogx'', and ''loglog'' to determine the functional relationship between ''y'' and ''x''.
  - Estimate the constants in the log-linear relationship graphically to determine the function. 
  - Plot the estimated function and the data together, and fine-tune your function by adjusting the constants until there is a good fit between the function and the data.

Once you have good fit between the data and the function, make a plot that shows

  * the data set's //y// versus //x// as red circles
  * your function //y(x)// as a solid blue line
  * a legend indicating the meaning of each plotting symbol
  * appropriate labels for each axis and a title

For each data set, turn in your plots and your estimate of the function //y(x)//.

**Problem 1: The distribution of earthquake magnitudes, by Moment Magnitude scale.** Big earthquakes are rare, and 
little earthquakes are frequent. In fact, there is a very clean empirical law that governs how many earthquakes of a 
given magnitude typically occur world-wide in a given year. Your job is to deduce that law from the following 
historical data.

<code>
% M N
8 2
7 18
6 120
5 800
4 6200
3 49000
2 365000
1 2920000
</code>

The first column is the [[http://en.wikipedia.org/wiki/Moment_magnitude_scale | moment magnitude]] //M//, and the second column is the number of earthquakes //N// of that magnitude that occur, on average, in a year. The last two entries are estimates, since it's impossible to detect every small earthquake around the world. The data are obtained from [[http://www.earthquake.ethz.ch/education/NDK/NDK|Earthquake Statistics and Earthquake Prediction Research]] by Stefan Wiemer, Institute of Geophysics, Zurich.

Using Matlab plotting commands, deduce the form of the functional relationship //N(M)//. Estimate  the constants in the relationship by estimating the slope and the //y//-intercept, and then fine-tuning by matching the plot of your estimate against the plot of the data. 

**Problem 2: The distribution of earthquake magnitudes, by energy.** The moment magnitude scale is logarithmic, in that an earthquake of magnitude //M+1// releases about 32 times energy than an earthquake of magnitude //M//. The following data
set gives the number //N// of earthquakes in a given year of energy //E// measured in Joules. 
<code>
% E  N
6e16 2
2e15 18
6e13 120
2e12 800
6e10 6200
2e09 49000
6e07 365000
1e06 2920000
</code>

Deduce the form of the functional relation //E(N)// using Matlab plotting, then estimate and fine-tune the constants
in the relation, just as in problem 1.

**Problem 3: World population.** The following data set provides the human population //P// of the earth at a given 
time //t//, measured in years A.D.

<code>
% t   P
1927 2e09
1960 3e09
1974 4e09
1987 5e09
1999 6e09
2011 7e09
</code>
 
Deduce the form of the functional relation //P(t)// and determine the constants graphically. 

Assume that the formula you derived for //P(t)// is valid indefinitely into the future and the past. What year will 
the population of the earth reach one trillion? What year were the first humans born? Do you believe these answers?
If not, why not?