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--- gibson:teaching:fall-2014:math445:lecture6-diary [2014/09/18 11:27]
gibson
+++ gibson:teaching:fall-2014:math445:lecture6-diary [2014/09/18 12:21] (current)
gibson [Graphical data analysis of log-linear relations]
@@ Line 1: / Line 1: @@
-====== Math 445 lecture 6: logical array operations and log-linear relations ======
+====== Math 445 lecture 6 ======
+===== Finding twin primes using logical array operations =====
 <code matlab>
@@ Line 133: / Line 135: @@
 % Thus ends our homage to Yitang 'Tom' Zhang. I encourage you to
 % view Tom's video on the MacArthur Foundation website.
+</code>
+===== Summation of series examples =====
+<code matlab>
 % =======================================================================
 % Now a couple examples of summing series, as compactly as possible
@@ Line 182: / Line 187: @@
 ans =
+</code>
+===== Graphical data analysis of log-linear relations =====
-% =======================================================================
+==== example 1: a linear relationship ====
-% Ok. Let's move on the graphical data analysis
+<code matlab>
 % Load datafile 'data1.asc' into matlab with 'load' command
 >> D = load('data1.asc');
@@ Line 222: / Line 229: @@
 >> ylabel('y')
 >> grid on
+</code>
+{{ :gibson:teaching:fall-2014:math445:linearplot0.png?direct&400 |}}
+<code matlab>
 % It's a straight line on a linear plot, so the functional relation is y = mx + b.
 % Hmmm, the slope is negative, dropping about 1 in y for every 2 in x. So I'll
@@ Line 233: / Line 244: @@
 >> plot(xdata, ydata, 'mo-', x, y, 'b.-')
 >> xlabel('x'); ylabel('y')
+</code>
+{{ :gibson:teaching:fall-2014:math445:linearplot1.png?direct&400 |}}
+<code matlab>
 % Wow, that's a great guess! But the slope is a little too negative, and the
-% y-intercept a little low. Try y = -0.5 x + 18. And this time let's just put
+% y-intercept a little low. Try y = -0.49 x + 18.5. And this time let's just put
-% the expression for y into the plot command, so that we can regenerate the
+% the expression for y into the plot command, with the labeling commands following
-% plot with different constants in a single command.
+% on the same line, so that we can regenerate the plot with different constants
+% quickly by making small changes to a single command line.
->> plot(xdata, ydata, 'mo-', xg, -0.49 * xg + 18.5, 'b-')
+>> plot(xdata, ydata, 'mo-', x, -0.49 * x + 18.5, 'b-'); xlabel('x'); ylabel('y'); grid on;
+</code>
+{{ :gibson:teaching:fall-2014:math445:linearplot2.png?direct&400 |}}
->> plot(xdata, ydata, 'mo-', xg, -0.49 * xg + 19, 'b-')
+<code matlab>
+% Even better! The slope looks just right but our guess is still a little low.
+% Raise the y intercept up to 19.
+>> plot(xdata, ydata, 'mo-', x, -0.49 * x + 19, 'b-'); xlabel('x'); ylabel('y'); grid on;
+</code>
-plot(x,y, 'mo-', xg, -0.49 * xg + 18.5, 'b.-')
+{{ :gibson:teaching:fall-2014:math445:linearplot3.png?direct&400 |}}
-plot(x,y, 'mo-', xg, -0.49 * xg + 18.7, 'b.-')
-plot(x,y, 'mo-', xg, -0.49 * xg + 18.9, 'b.-')
-plot(x,y, 'mo-', xg,-0.49 * xg + 19, 'b.-')
+<code matlab>
+% Perfect! So the functional relation between y and x is y = -0.49 x + 19
+</code>
+==== example 2: a log-linear relationship ====
-% great! relation is y = -0.49 x + 19
+<code matlab>
+% Load the next data file and try to figure out its y = f(x) relation.
+>> D = load('data3.asc');
+>> xdata = D(:,1);
+>> ydata = D(:,2);
+>> plot(xdata, ydata,'mo-'); xlabel('x'); ylabel('y'); grid on
+</code>
+{{ :gibson:teaching:fall-2014:math445:logplot0.png?nolink&400 |}}
+<code matlab>
+% That looks exponential, so graph y logarithmically
+>> semilogy(xdata, ydata, 'mo-'); xlabel('x'); ylabel('y'); grid on
+</code>
+{{ :gibson:teaching:fall-2014:math445:logplot1.png?nolink&400 |}}
+<code matlab>
+% Great! It's a straight line with y graphed logatithmically, so the relation is
+% of the form
+%   log10 y = m x + b,   or equivalently
+%         y = 10^(mx+b), or equivalently
+%         y = c 10^(mx)
+% for some constants m and c. let's take rough guesses, judging from the plot.
+%
+% m is the slope in log10 y versus x. log10 y drops from 2 at x=10 to about 1 at x=20.
+% So m looks to be about -1/10, (rise of -1 over run of 10). You can get the constant c
+% by estimating the value of y at x=0. That looks to be about c=400. So let's give
+% y = 400 10^(-0.1 x) a try.
+>> x = linspace(-20, 50, 10);
+>> semilogy(xdata, ydata,'mo-', x, 400*10.^(-0.1*x)); xlabel('x'); ylabel('y'); grid on
+</code>
+{{ :gibson:teaching:fall-2014:math445:logplot2.png?nolink&400 |}}
+<code matlab>
+% Not too shabby. But the slope is a little too negative and y is too low at x=0.
+% A few iterations of adjusting the constants gives
+>> semilogy(xdata, ydata,'mo-', x, 700*10.^(-0.085*x)); xlabel('x'); ylabel('y'); grid on
+</code>
+{{ :gibson:teaching:fall-2014:math445:logplot3.png?nolink&400 |}}
+<code matlab>
+% so the functional form is y = 700 * 10^(-0.085 x).
-% try another
+% Don't ask me why Matlab keeps changing the grid lines on the logarithmic plots...
-D = load('data3.asc');
-x= D(:,1);
-y=D(:,2);
-plot(x,y,'mo-')
-% looks exponential, so graph y logarithmically
-semilogy(x,y,'mo-')
 </code>

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