[likelihood] improved figure for line fit

likelihood/lecture/likelihood.tex

@@ -228,7 +228,12 @@ maximized respectively.
 \begin{figure}[t]
   \includegraphics[width=1\textwidth]{mlepropline}
   \titlecaption{\label{mleproplinefig} Maximum likelihood estimation
-    of the slope of line through the origin.}{}
+    of the slope of line through the origin.}{The data (blue and
+    left histogram) originate from a straight line $y=mx$ trough the origin
+    (red). The maximum-likelihood estimation of the slope $m$ of the
+    regression line (orange), \eqnref{mleslope}, is close to the true
+    one. The residuals, the data minus the estimated line (right), reveal
+    the normal distribution of the data around the line (right histogram).}
 \end{figure}
 
 
@@ -282,10 +287,11 @@ To see what this expression is, we need to standardize the data. We
 make the data mean free and normalize them to their standard
 deviation, i.e. $x \mapsto (x - \bar x)/\sigma_x$. The resulting
 numbers are also called \enterm{$z$-values} or $z$-scores and they
-have the property $\bar x = 0$ and $\sigma_x = 1$.  If this is the
+have the property $\bar x = 0$ and $\sigma_x = 1$. $z$-scores are
-case, the variance
+often used in Biology to make quantities that differ in their units
+comparable. For standardized data the variance
 \[ \sigma_x^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 = 1 \]
-is the mean squared data and equals one.
+is given by the mean squared data and equals one.
 The covariance between $x$ and $y$ also simplifies to
 \[ \text{cov}(x, y) = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)(y_i -
 \bar y) =\frac{1}{n} \sum_{i=1}^n x_i y_i \]

likelihood/lecture/mlepropline.py

@@ -1,16 +1,17 @@
 import numpy as np
+import scipy.stats as st
 import matplotlib.pyplot as plt
 
 plt.xkcd()
-fig = plt.figure( figsize=(6,3.5) )
+fig = plt.figure(figsize=(6, 3))
 
 # the line:
 slope = 2.0
 xx = np.arange(0.0, 4.1, 0.1)
 yy = slope*xx
 # the data:
-n = 80
+n = 40
-rng = np.random.RandomState(218)
+rng = np.random.RandomState(5218)
 sigma = 1.5
 x = 4.0*rng.rand(n)
 y = slope*x+rng.randn(n)*sigma
@@ -19,7 +20,7 @@ slopef = np.sum(x*y)/np.sum(x*x)
 yf = slopef*xx
 
 # plot it:
-ax = fig.add_subplot( 1, 1, 1 )
+ax = fig.add_axes([0.09, 0.02, 0.33, 0.9])
 ax.spines['left'].set_position('zero')
 ax.spines['bottom'].set_position('zero')
 ax.spines['right'].set_visible(False)
@@ -30,20 +31,60 @@ ax.yaxis.set_ticks_position('left')
 ax.xaxis.set_ticks_position('bottom')
 ax.set_xticks(np.arange(0.0, 4.1))
 ax.set_xlim(0.0, 4.2)
-#ax.set_ylim(-1, 5)
+ax.set_ylim(-4.0, 12.0)
-#ax.set_xticks( np.arange(0, 5))
-#ax.set_yticks( np.arange(0, 0.9, 0.2))
 ax.set_xlabel('x')
 ax.set_ylabel('y')
-#ax.annotate('', xy=(mu, 0.02), xycoords='data',
+ax.scatter(x, y, label='data', s=40, zorder=10)
-#            xytext=(mu, 0.75), textcoords='data',
+ax.plot(xx, yy, 'r', lw=5.0, color='#ff0000', label='original', zorder=5)
-#            arrowprops=dict(arrowstyle="->", relpos=(0.5,0.5),
+ax.plot(xx, yf, '--', lw=1.0, color='#ffcc00', label='fit', zorder=7)
-#            connectionstyle=cs), zorder=1 )
+ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.15), frameon=False)
-ax.scatter(x, y, label='data', s=50, zorder=10)
+
-ax.plot(xx, yy, 'r', lw=6.0, color='#ff0000', label='original', zorder=5)
+ax = fig.add_axes([0.42, 0.02, 0.07, 0.9])
-ax.plot(xx, yf, '--', lw=2.0, color='#ffcc00', label='fit', zorder=7)
+ax.spines['left'].set_position('zero')
-ax.legend(loc='upper left', frameon=False)
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.spines['bottom'].set_visible(False)
+ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
+ax.yaxis.set_ticks_position('left')
+ax.set_xticks([])
+ax.set_ylim(-4.0, 12.0)
+ax.set_yticks([])
+bins = np.arange(-4.0, 12.1, 0.75)
+ax.hist(y, bins, orientation='horizontal', zorder=10)
+
+ax = fig.add_axes([0.6, 0.02, 0.33, 0.9])
+ax.spines['left'].set_position('zero')
+ax.spines['bottom'].set_position('zero')
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.get_xaxis().set_tick_params(direction='inout', length=10, width=2)
+ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
+ax.yaxis.set_ticks_position('left')
+ax.xaxis.set_ticks_position('bottom')
+ax.set_xticks(np.arange(0.0, 4.1))
+ax.set_xlim(0.0, 4.2)
+ax.set_ylim(-4.0, 12.0)
+ax.set_xlabel('x')
+ax.set_ylabel('y - mx')
+ax.scatter(x, y - slopef*x, label='residuals', s=40, zorder=10)
+#ax.legend(loc='upper left', bbox_to_anchor=(0.0, 1.0), frameon=False)
+
+ax = fig.add_axes([0.93, 0.02, 0.07, 0.9])
+ax.spines['left'].set_position('zero')
+ax.spines['right'].set_visible(False)
+ax.spines['top'].set_visible(False)
+ax.spines['bottom'].set_visible(False)
+ax.get_yaxis().set_tick_params(direction='inout', length=10, width=2)
+ax.yaxis.set_ticks_position('left')
+ax.set_xlim(0.0, 11.0)
+ax.set_xticks([])
+ax.set_ylim(-4.0, 12.0)
+ax.set_yticks([])
+r = y - slopef*x
+ax.hist(r, bins, orientation='horizontal', zorder=10)
+gx = np.arange(-4.0, 12.1, 0.1)
+gy = st.norm.pdf(gx, np.mean(r), np.std(r))
+ax.plot(1.0+gy*29.0, gx, 'r', lw=2, zorder=5)
 
-plt.tight_layout();
 plt.savefig('mlepropline.pdf')
 #plt.show();