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--- gibson:teaching:fall-2014:math445:lecture6-diary [2014/09/18 11:35]
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 223: / Line 230: @@
 >> grid on
 </code>
+{{ :gibson:teaching:fall-2014:math445:linearplot0.png?direct&400 |}}
 <code matlab>
@@ Line 236: / Line 245: @@
 >> xlabel('x'); ylabel('y')
 </code>
+{{ :gibson:teaching:fall-2014:math445:linearplot1.png?direct&400 |}}
 <code matlab>
@@ Line 246: / Line 257: @@
 >> 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 |}}
 <code matlab>
@@ Line 253: / Line 266: @@
 >> plot(xdata, ydata, 'mo-', x, -0.49 * x + 19, 'b-'); xlabel('x'); ylabel('y'); grid on;
 </code>
+{{ :gibson:teaching:fall-2014:math445:linearplot3.png?direct&400 |}}
 <code matlab>
-% Perfect! The functional relation between y and x is y = -0.49 x + 19
+% Perfect! So the functional relation between y and x is y = -0.49 x + 19
+</code>
+==== example 2: a log-linear relationship ====
+<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).
-% Ok, let's move on the the next data file and try to figure out its y = f(x) relation.
+% 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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