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:
earthquakes.ascfor problem 1.
loglogto determine the functional relationship between
Once you have good fit between the data and the function, make a plot that shows
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.
% M N 8 2 7 18 6 120 5 800 4 6200 3 49000 2 365000 1 2920000
The first column is the 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 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.
% E N 6e16 2 2e15 18 6e13 120 2e12 800 6e10 6200 2e09 49000 6e07 365000 1e06 2920000
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.
% t P 1927 2e09 1960 3e09 1974 4e09 1987 5e09 1999 6e09 2011 7e09
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?