Added likelihood and orthogonal projections

Added the likelihood equations/form from the discrete bayes
chapter to better tie in that form of reasoning. then I converted
the 1d equations to the orthogonal projection form to show how
the Kalman gain is computed and where the residual comes from
computationally. This should make the full KF equations much more
approachable.

code/kf_internal.py

@@ -19,9 +19,9 @@ from __future__ import (absolute_import, division, print_function,
 import book_plots as bp
 import matplotlib.pyplot as plt
 
-def plot_dog_track(xs, measurement_var, process_var):
+def plot_dog_track(xs, dog, measurement_var, process_var):
     N = len(xs)
-    bp.plot_track([0, N-1], [1, N])
+    bp.plot_track(dog)
     bp.plot_measurements(xs, label='Sensor')
     bp.set_labels('variance = {}, process variance = {}'.format(
               measurement_var, process_var), 'time', 'pos')

code/mkf_internal.py

@@ -104,9 +104,11 @@ def show_residual_chart():
     plt.text (0.5, 159.6, "prediction", ha='center',va='top',fontsize=18,color='red')
     plt.text (1.0, 164.4, r"measurement ($z$)",ha='center',va='bottom',fontsize=18,color='blue')
     plt.text (0, 157.8, r"posterior ($x_{t-1}$)", ha='center', va='top',fontsize=18)
-    plt.text (1.02, est_y-1.5, "residual", ha='left', va='center',fontsize=18)
+    plt.text (1.02, est_y-1.5, "residual($y$)", ha='left', va='center',fontsize=18)
+    plt.text (1.02, est_y-2.2, r"$y=z-\bar x_t$", ha='left', va='center',fontsize=18)
     plt.text (0.9, est_y, "new estimate ($x_t$)", ha='right', va='center',fontsize=18)
     plt.text (0.8, est_y-0.5, "(posterior)", ha='right', va='center',fontsize=18)
+    plt.text (0.75, est_y-1.2, r"$\bar{x}_t + Ky$", ha='right', va='center',fontsize=18)
     plt.xlabel('time')
     ax.yaxis.set_label_position("right")
     plt.ylabel('state')