[simulations] added exercise randomnumbers

simulations/code/normaldata.m

@@ -1,8 +1,3 @@
-% getting familiar with the randn() function:
-randn(1, 3)
-randn(3, 1)
-randn(2, 4)
-
 % simulate tiger weights:
 mu = 220.0;       % mean and ...
 sigma = 40.0;     % ... standard deviation of the tigers in kg

simulations/code/normaldata.out

@@ -1,21 +1,5 @@
 >> normaldata
 
-ans =
-
-  -0.89120   1.19863   0.95487
-
-ans =
-
-  -1.10001
-   0.79473
-   0.85979
-
-ans =
-
-  -1.19206   0.58278   1.70286  -1.28122
-  -0.19966  -1.85623   0.17962  -0.19272
-
-
 n=100:
   m=218kg, std= 39kg
   m=223kg, std= 39kg

simulations/code/randomnumbers.m

@@ -0,0 +1,17 @@
+% getting familiar with the rand() function:
+rand(1, 3)
+rand(3, 1)
+rand(2, 4)
+
+% three times the same sequence of 10 random numbers:
+n = 10;
+for k = 1:3
+    rand(1, n)
+end
+
+% serial corraltion at lag 1:
+n = 10000;
+x = rand(n, 1);
+r1 = corr(x(1:end-1), x(2:end));
+fprintf('correlation between subsequent random numbers: %.3f\n', r1);
+

simulations/code/randomnumbers.out

@@ -0,0 +1,30 @@
+>> randomnumbers
+
+ans =
+
+   0.740875   0.193576   0.064584
+
+ans =
+
+   0.061028
+   0.695705
+   0.177097
+
+ans =
+
+   0.707430   0.404868   0.550246   0.393093
+   0.087565   0.473358   0.247850   0.161137
+
+ans =
+
+   0.350969   0.340726   0.145924   0.769714   0.203317   0.066427   0.451685   0.959766   0.850558   0.642769
+
+ans =
+
+   0.145262   0.175168   0.462693   0.089379   0.706870   0.353830   0.604305   0.405531   0.804180   0.253496
+
+ans =
+
+   0.647119   0.468534   0.484289   0.586001   0.851326   0.972554   0.014812   0.906628   0.982962   0.575003
+
+correlation between subsequent random numbers: 0.003

simulations/lecture/normaldata.py

@@ -2,7 +2,7 @@ import numpy as np
 import scipy.stats as st
 import matplotlib.pyplot as plt
 import matplotlib.gridspec as gridspec
-from plotstyle import colors, cm_size, show_spines, set_xlabel, set_ylabel, bar_fac
+from plotstyle import *
 
 if __name__ == "__main__":
     # wikipedia:
@@ -21,7 +21,7 @@ if __name__ == "__main__":
     ax1 = fig.add_subplot(spec[0, 0])
     show_spines(ax1, 'lb')
     ax1.scatter(indices, data, c=colors['blue'], edgecolor='white', s=50)
-    set_xlabel(ax1, 'index')
+    set_xlabel(ax1, 'Index')
     set_ylabel(ax1, 'Weight', 'kg')
     ax1.set_xlim(-10, 310)
     ax1.set_ylim(0, 370)
@@ -35,7 +35,7 @@ if __name__ == "__main__":
     bw = 20.0
     h, b = np.histogram(data, np.arange(0, 401, bw))
     ax2.barh(b[:-1], h/np.sum(h)/(b[1]-b[0]), fc=colors['yellow'], height=bar_fac*bw, align='edge')
-    set_xlabel(ax2, 'pdf', '1/kg')
+    set_xlabel(ax2, 'Pdf', '1/kg')
     ax2.set_xlim(0, 0.012)
     ax2.set_xticks([0, 0.005, 0.01])
     ax2.set_xticklabels(['0', '0.005', '0.01'])

simulations/lecture/randomnumbers.py

@@ -1,7 +1,7 @@
 import numpy as np
 import matplotlib.pyplot as plt
 import matplotlib.gridspec as gridspec
-from plotstyle import colors, cm_size, show_spines
+from plotstyle import *
 
 if __name__ == "__main__":
     n = 21

simulations/lecture/simulations.tex

@@ -51,8 +51,11 @@ exactly the same sequence of noise values.  This is useful for plots
 that involve some random numbers but should look the same whenever the
 script is run.
 
-\begin{exercise}{}{}
+\begin{exercise}{randomnumbers.m}{randomnumbers.out}
-  Generate three times the same sequence of 20 uniformly distributed
+  First, read the documentation of the \varcode{rand()} function and
+  check its output for some (small) input arguments.
+
+  Generate three times the same sequence of 10 uniformly distributed
   numbers using the \code{rand()} and \code{rng()} functions.
 
   Generate 10\,000 uniformly distributed random numbers and compute
@@ -109,17 +112,15 @@ mean we just add the desired mean $\mu$ to the random numbers:
 \end{figure}
 
 \begin{exercise}{normaldata.m}{normaldata.out}
-  First, read the documentation of the \varcode{randn()} function and
+  Write a little script that generates $n=100$ normally distributed
-  check its output for some (small) input arguments.  Write a little
+  data simulating the weight of Bengal tiger males with mean 220\,kg
-  script that generates $n=100$ normally distributed data simulating
+  and standard deviation 40\,kg. Check the actual mean and standard
-  the weight of Bengal tiger males with mean 220\,kg and standard
+  deviation of the generated data. Do this, let's say, five times
-  deviation 40\,kg. Check the actual mean and standard deviation of
+  using a for-loop. Then increase $n$ to 10\,000 and run the code
-  the generated data. Do this, let's say, five times using a
+  again. It is so simple to measure the weight of 10\,000 tigers for
-  for-loop. Then increase $n$ to 10\,000 and run the code again. It is
+  getting a really good estimate of their mean weight, isn't it?
-  so simple to measure the weight of 10\,000 tigers for getting a
+  Finally plot from the last generated data set of tiger weights the
-  really good estimate of their mean weight, isn't it?  Finally plot
+  first 1000 values as a function of their index.
-  from the last generated data set of tiger weights the first 1000
-  values as a function of their index.
 \end{exercise}
 
 \subsection{Other probability densities}
@@ -136,12 +137,12 @@ gamma
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{staticnonlinearity}
   \titlecaption{\label{staticnonlinearityfig} Generating data
-    fluctuating around a function.}{The open probability of the
+    fluctuating around a function.}{The conductance of the
-    mechontransducer channel in hair cells of the inner ear is a
+    mechontransducer channels in hair cells of the inner ear is a
-    saturating function of the deflection of hairs (left, red line).
+    saturating function of the deflection of their hairs (left, red
-    Measured data will fluctuate around this function (blue dots).
+    line).  Measured data will fluctuate around this function (blue
-    Ideally the residuals (yellow histogram) are normally distributed
+    dots).  Ideally the residuals (yellow histogram) are normally
-    (right, red line).}
+    distributed (right, red line).}
 \end{figure}
 
 Example: mechanotransduciton!

simulations/lecture/staticnonlinearity.py

@@ -2,10 +2,10 @@ import numpy as np
 import scipy.stats as st
 import matplotlib.pyplot as plt
 import matplotlib.gridspec as gridspec
-from plotstyle import colors, cm_size, show_spines, set_xlabel, set_ylabel, bar_fac
+from plotstyle import *
 
 def boltzmann(x, x0, k):
-    return 1.0/(1.0+np.exp(-k*(x-x0)))
+    return 8.0/(1.0+np.exp(-k*(x-x0)))
 
 if __name__ == "__main__":
     n = 50
@@ -13,7 +13,7 @@ if __name__ == "__main__":
     xmax = 18.0
     x0 = 2.0
     k = 0.25
-    sigma = 0.08
+    sigma = 0.6
     rng = np.random.RandomState(15281)
     x = np.linspace(xmin, xmax, n)
     y = boltzmann(x, x0, k) + sigma*rng.randn(len(x))
@@ -28,28 +28,25 @@ if __name__ == "__main__":
     ax1.plot(xx, yy, colors['red'], lw=2)
     ax1.scatter(x, y, c=colors['blue'], edgecolor='white', s=50)
     set_xlabel(ax1, 'Hair deflection', 'nm')
-    set_ylabel(ax1, 'Open probability')
+    set_ylabel(ax1, 'Conductance', 'nS')
     ax1.set_xlim(-20, 20)
-    ax1.set_ylim(-0.2, 1.17)
+    ax1.set_ylim(-1.5, 9.5)
     ax1.set_xticks(np.arange(-20.0, 21.0, 10.0))
-    ax1.set_yticks(np.arange(-0.2, 1.1, 0.2))
+    ax1.set_yticks(np.arange(0, 9, 2))
     
     ax2 = fig.add_subplot(spec[0, 1])
     show_spines(ax2, 'lb')
-    xg = np.linspace(-1.0, 1.01, 200)
+    xg = np.linspace(-3.0, 3.01, 200)
     yg = st.norm.pdf(xg, 0.0, sigma)
     ax2.plot(xg, yg, colors['red'], lw=2)
-    bw = 0.05
+    bw = 0.25
-    h, b = np.histogram(y-boltzmann(x, x0, k), np.arange(-1.0, 1.01, bw))
+    h, b = np.histogram(y-boltzmann(x, x0, k), np.arange(-3.0, 3.01, bw))
     ax2.bar(b[:-1], h/np.sum(h)/(b[1]-b[0]), fc=colors['yellow'], width=bar_fac*bw, align='edge')
-    set_xlabel(ax2, 'residuals', 'nm')
+    set_xlabel(ax2, 'Residuals', 'nS')
-    set_ylabel(ax2, 'pdf')
+    set_ylabel(ax2, 'Pdf', '1/nS')
-    ax2.set_xlim(-0.3, 0.3)
+    ax2.set_xlim(-2.5, 2.5)
-    ax2.set_ylim(0, 5.05)
+    ax2.set_ylim(0, 0.75)
-    #ax2.set_xticks([0, 0.005, 0.01])
+    ax2.set_yticks(np.arange(0, 0.75, 0.2))
-    #ax2.set_xticklabels(['0', '0.005', '0.01'])
-    #ax2.set_yticks(np.arange(0, 351, 100))
-    #ax2.set_yticklabels([])
 
     fig.savefig("staticnonlinearity.pdf")
     plt.close()