checkpoint; in midst of big re-org.

This is not a 'useful'checkin. I'm heavily reorganizing teh
multivariate chapter (spliltting in two) and I am getting
worried about losing edits.

06_Kalman_Filter_Math.ipynb

@@ -881,7 +881,22 @@
     "The final equation is:\n",
     "$$\\mathbf{P} = \\mathbf{FPF}^\\mathsf{T} + \\mathbf{Q}\\tag{2}$$\n",
     "\n",
-    "$\\mathbf{FPF}^\\mathsf{T}$ is the way we put $\\mathbf{P}$ into the process space using linear algebra so that we can add the process noise $\\mathbf{Q}$ to it."
+    "$\\mathbf{FPF}^\\mathsf{T}$ is the way we put $\\mathbf{P}$ into the process space using linear algebra so that we can add the process noise $\\mathbf{Q}$ to it.\n",
+    "\n",
+    "\n",
+    "**needs edit**\n",
+    "\n",
+    "I admit this may be a 'magical' equation to you. If you have some experience with linear algebra and statistics, this may help. The covariance due to the prediction can be modeled as the expected value of the error in the prediction step, given by this equation. \n",
+    "\n",
+    "$$\\begin{aligned}\n",
+    "\\mathbf{P}^- &= E[(\\mathbf{Fx})(\\mathbf{Fx})^\\mathsf{T}]\\\\\n",
+    " &= E[\\mathbf{Fxx}^\\mathsf{T}\\mathbf{F}^\\mathsf{T}] \\\\\n",
+    " &= \\mathbf{F}\\, E[\\mathbf{xx}^\\mathsf{T}]\\, \\mathbf{F}^\\mathsf{T}\n",
+    "\\end{aligned}$$\n",
+    "\n",
+    "Of course, $E[\\mathbf{xx}^\\mathsf{T}]$ is just $\\mathbf{P}$, giving us\n",
+    "\n",
+    "$$\\mathbf{P}^- = \\mathbf{FPF}^\\mathsf{T}$$"
    ]
   },
   {

Interactions.ipynb

@@ -38,7 +38,7 @@
     "\n",
     "The Kalman filter uses the equation $P^- = FPF^\\mathsf{T}$ to compute the prior of the covariance matrix during the prediction step, where P is the covariance matrix and F is the system transistion function. For a Newtonian system $x = \\dot{x}\\Delta t + x_0$ F might look like\n",
     "\n",
-    "$$F = \\begin{bmatrix}1 & \\Delta t\\\\0 1\\end{bmatrix}$$\n",
+    "$$F = \\begin{bmatrix}1 & \\Delta t\\\\0 & 1\\end{bmatrix}$$\n",
     "\n",
     "$FPF^\\mathsf{T}$ alters P by taking the correlation between the position ($x$) and velocity ($\\dot{x}$). This interactive plot lets you see the effect of different designs of F has on this value. For example,\n",
     "\n",

code/mkf_internal.py

@@ -355,14 +355,16 @@ def plot_correlation_covariance():
 
 
 import book_plots as bp
-def plot_track(ps, zs, cov,
+def plot_track(ps, zs, cov, std_scale=1,
                plot_P=True, y_lim=None,
                title='Kalman Filter'):
 
+    print(std_scale)
+
     count = len(zs)
     actual = np.linspace(0, count - 1, count)
     cov = np.asarray(cov)
-    std = np.sqrt(cov[:,0,0])
+    std = std_scale*np.sqrt(cov[:,0,0])
     std_top = np.minimum(actual+std, [count + 10])
     std_btm = np.maximum(actual-std, [-50])
 

pdf/build_book

@@ -1,6 +1,6 @@
 #! /bin/bash
 
-echo "merging book..."
+#echo "merging book..."
 python merge_book.py
 
 echo "creating pdf..."

pdf/merge_book.py

@@ -39,6 +39,7 @@ if __name__ == '__main__':
          '../02_Discrete_Bayes.ipynb',
          '../03_Gaussians.ipynb',
          '../04_One_Dimensional_Kalman_Filters.ipynb',
+         '../05_Multivariate_Gaussians.ipynb',
          '../05_Multivariate_Kalman_Filters.ipynb',
          '../06_Kalman_Filter_Math.ipynb',
          '../07_Designing_Kalman_Filters.ipynb',