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====== Math 445 Lab 8: log-linear relationships ====== 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. Data 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 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.
