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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.** 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> % magnitude number/year 8 2 7 18 6 120 5 800 4 6200 3 49000 2 365000 1 2920000 </code> The first column is the Richter magnitude //R//, 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.) Deduce a functional relationship //N(R)//.
