Fixed bulleted list

I was using HTML, which doesn't render in the PDF. Switched to
markdown and it is better.

06-Multivariate-Kalman-Filters.ipynb

@@ -364,7 +364,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9c7fa8710>"
+       "<matplotlib.figure.Figure at 0x18691f815f8>"
       ]
      },
      "metadata": {},
@@ -430,13 +430,11 @@
     "\n",
     "The details will be different than the univariate filter because these are vectors and matrices, but the concepts are exactly the same: \n",
     "\n",
-    "<ul>\n",
-    "    <li>Use a Gaussian to represent our estimate of the state and error<l/i>\n",
-    "    <li>Use a Gaussian to represent the measurement and its error<l/i>\n",
-    "    <li>Use a Gaussian to represent the process model<l/i>\n",
-    "    <li>Use the process model to predict the next state (the prior)<l/i>\n",
-    "    <li>Form an estimate part way between the measurement and the prior<l/i>\n",
-    "</ul>\n",
+    "-  Use a Gaussian to represent our estimate of the state and error\n",
+    "-  Use a Gaussian to represent the measurement and its error\n",
+    "-  Use a Gaussian to represent the process model\n",
+    "-  Use the process model to predict the next state (the prior)\n",
+    "-  Form an estimate part way between the measurement and the prior\n",
     "\n",
     "Your job as a designer will be to design the state $\\left(\\mathbf x, \\mathbf P\\right)$, the process $\\left(\\mathbf F, \\mathbf Q\\right)$, the measurement $\\left(\\mathbf z, \\mathbf R\\right)$, and the  measurement function $\\mathbf H$. If the system has control inputs, such as a robot, you will also design $\\mathbf B$ and $\\mathbf u$."
    ]
@@ -537,8 +535,8 @@
     {
      "data": {
       "text/plain": [
-       "array([[ 10.0],\n",
-       "       [ 4.5]])"
+       "array([[ 1000. ],\n",
+       "       [    4.5]])"
       ]
      },
      "execution_count": 4,
@@ -547,7 +545,7 @@
     }
    ],
    "source": [
-    "x = np.array([[10.0],\n",
+    "x = np.array([[1000.0],\n",
     "              [4.5]])\n",
     "x"
    ]
@@ -567,8 +565,8 @@
     {
      "data": {
       "text/plain": [
-       "array([[ 10.0],\n",
-       "       [ 4.5]])"
+       "array([[ 10. ],\n",
+       "       [  4.5]])"
       ]
      },
      "execution_count": 5,
@@ -596,7 +594,7 @@
     {
      "data": {
       "text/plain": [
-       "array([ 10.0,  4.5])"
+       "array([ 10. ,   4.5])"
       ]
      },
      "execution_count": 6,
@@ -624,7 +622,7 @@
     {
      "data": {
       "text/plain": [
-       "array([ 1.0,  0.0,  0.0,  0.0,  0.0,  0.0])"
+       "array([ 1.,  0.,  0.,  0.,  0.,  0.])"
       ]
      },
      "execution_count": 7,
@@ -654,13 +652,13 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 19.0]\n",
-      " [ 48.0]]\n",
+      "[[ 19.]\n",
+      " [ 48.]]\n",
       "\n",
-      "[[ 19.0]\n",
-      " [ 48.0]]\n",
+      "[[ 19.]\n",
+      " [ 48.]]\n",
       "\n",
-      "[ 19.0  48.0]\n"
+      "[ 19.  48.]\n"
      ]
     }
    ],
@@ -716,8 +714,8 @@
     {
      "data": {
       "text/plain": [
-       "array([[ 5e+02,  0.0],\n",
-       "       [ 0.0,  49.0]])"
+       "array([[ 500.,    0.],\n",
+       "       [   0.,   49.]])"
       ]
      },
      "execution_count": 9,
@@ -745,8 +743,8 @@
     {
      "data": {
       "text/plain": [
-       "array([[ 5e+02,  0.0],\n",
-       "       [ 0.0,  49.0]])"
+       "array([[ 500.,    0.],\n",
+       "       [   0.,   49.]])"
       ]
      },
      "execution_count": 10,
@@ -779,11 +777,12 @@
     "\n",
     "$$ x = v \\Delta t + x_0$$\n",
     "\n",
-    "We implemented this as follows, where `pos` and `movement` are Gaussian tuples:\n",
+    "We implemented this as follows:\n",
     "\n",
     "```python\n",
     "def predict(pos, movement):\n",
-    "    return (pos[0] + movement[0], pos[1] + movement[1])\n",
+    "    return gaussian(pos.mean + movement.mean, \n",
+    "                    movement.var + movement.var)\n",
     "```\n",
     "\n",
     "We will do the same thing in this chapter, using multivariate Gaussians instead of univariate Gaussians. You might imagine this sort of implementation:\n",
@@ -822,8 +821,8 @@
     {
      "data": {
       "text/plain": [
-       "array([[ 1.0],\n",
-       "       [ 2.0]])"
+       "array([[ 1.],\n",
+       "       [ 2.]])"
       ]
      },
      "execution_count": 11,
@@ -847,6 +846,10 @@
     "\n",
     "$$\\mathbf{\\bar x} = \\mathbf{Fx}$$\n",
     "\n",
+    "which we can make explicit as\n",
+    "\n",
+    "$$\\begin{bmatrix} \\bar x \\\\ \\dot{\\bar x}\\end{bmatrix} = \\begin{bmatrix}? & ? \\\\? & ?\\end{bmatrix}\\begin{bmatrix}x\\\\\\dot x\\end{bmatrix}$$\n",
+    "\n",
     "Our job as Kalman filters designers is to specify $\\mathbf F$ such that $\\bar{\\mathbf x}  = \\mathbf{Fx}$ performs the innovation (prediction) for our system. To do this we need one equation for each state variable. In our problem $\\mathbf x = \\begin{bmatrix}x & \\dot x\\end{bmatrix}^\\mathtt{T}$, so we need one equation to compute the position $x$ and another to compute the velocity $\\dot x$ . We already know the equation for the position innovation:\n",
     "\n",
     "$$\\bar x = x + \\dot x \\Delta t$$\n",
@@ -880,7 +883,7 @@
     "\\mathbf{\\bar x} &= \\mathbf{Fx}\n",
     "\\end{aligned}$$\n",
     "\n",
-    "$\\mathbf F$ is called the *state transition function*."
+    "$\\mathbf F$ is called the *state transition function* or the *state transition matrix*. In later chapters it will be a true function, not a  matrix, so calling it a function is a bit more general."
    ]
   },
   {
@@ -891,8 +894,8 @@
     {
      "data": {
       "text/plain": [
-       "array([[ 1.0,  0.1],\n",
-       "       [ 0.0,  1.0]])"
+       "array([[ 1. ,  0.1],\n",
+       "       [ 0. ,  1. ]])"
       ]
      },
      "execution_count": 12,
@@ -923,7 +926,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "x = [ 10.4  4.5]\n"
+      "x = [ 10.45   4.5 ]\n"
      ]
     }
    ],
@@ -957,10 +960,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "x = [ 10.9  4.5]\n",
-      "x = [ 11.3  4.5]\n",
-      "x = [ 11.8  4.5]\n",
-      "x = [ 12.2  4.5]\n"
+      "x = [ 10.9   4.5]\n",
+      "x = [ 11.35   4.5 ]\n",
+      "x = [ 11.8   4.5]\n",
+      "x = [ 12.25   4.5 ]\n"
      ]
     }
    ],
@@ -986,8 +989,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 5.12e+02  24.5]\n",
-      " [ 24.5  49.0]]\n"
+      "[[ 512.25   24.5 ]\n",
+      " [  24.5    49.  ]]\n"
      ]
     }
    ],
@@ -1015,7 +1018,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9ca12b358>"
+       "<matplotlib.figure.Figure at 0x186940eb390>"
       ]
      },
      "metadata": {},
@@ -1075,8 +1078,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[ 0.588  1.18]\n",
-      " [ 1.18  2.35]]\n"
+      "[[ 0.588  1.175]\n",
+      " [ 1.175  2.35 ]]\n"
      ]
     }
    ],
@@ -1118,9 +1121,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "x = [ 12.7  4.5]\n",
-      "P = [[ 6.81e+02  3.01e+02]\n",
-      " [ 3.01e+02  5.02e+02]]\n"
+      "x = [ 12.7   4.5]\n",
+      "P = [[ 680.587  301.175]\n",
+      " [ 301.175  502.35 ]]\n"
      ]
     }
    ],
@@ -1346,7 +1349,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "x = [ 1.09  -0.64]\n"
+      "x = [ 1.085 -0.64 ]\n"
      ]
     }
    ],
@@ -1389,11 +1392,11 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "x =  [[ 0.0  0.0]]\n",
-      "R =  [[ 1.0]]\n",
+      "x =  [[ 0.  0.]]\n",
+      "R =  [[ 1.]]\n",
       "Q = \n",
-      " [[ 1.0  0.0]\n",
-      " [ 0.0  1.0]]\n"
+      " [[ 1.  0.]\n",
+      " [ 0.  1.]]\n"
      ]
     }
    ],
@@ -1556,9 +1559,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9ca194b70>"
+       "<matplotlib.figure.Figure at 0x18693befb00>"
       ]
      },
      "metadata": {},
@@ -1568,7 +1571,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb2026d8>"
+       "<matplotlib.figure.Figure at 0x186951a06a0>"
       ]
      },
      "metadata": {},
@@ -1608,7 +1611,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb272ba8>"
+       "<matplotlib.figure.Figure at 0x18695222630>"
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@@ -1760,7 +1763,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cbe4f5c0>"
+       "<matplotlib.figure.Figure at 0x18694144160>"
       ]
      },
      "metadata": {},
@@ -1805,7 +1808,7 @@
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
-       "model_id": "e1ad782948d04785983c0c5b1e922df9",
+       "model_id": "c377c5e3efb74ffa94016da4d50d9f2d",
        "version_major": 2,
        "version_minor": 0
       },
@@ -1998,7 +2001,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb2091d0>"
+       "<matplotlib.figure.Figure at 0x1869518cbe0>"
       ]
      },
      "metadata": {},
@@ -2130,7 +2133,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb229f28>"
+       "<matplotlib.figure.Figure at 0x1869518c630>"
       ]
      },
      "metadata": {},
@@ -2219,7 +2222,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb3629e8>"
+       "<matplotlib.figure.Figure at 0x18695288ef0>"
       ]
      },
      "metadata": {},
@@ -2246,7 +2249,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb637940>"
+       "<matplotlib.figure.Figure at 0x1869530d470>"
       ]
      },
      "metadata": {},
@@ -2348,7 +2351,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb600128>"
+       "<matplotlib.figure.Figure at 0x186955ca6d8>"
       ]
      },
      "metadata": {},
@@ -2358,7 +2361,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cba3b898>"
+       "<matplotlib.figure.Figure at 0x18695e0c320>"
       ]
      },
      "metadata": {},
@@ -2402,7 +2405,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb1f9828>"
+       "<matplotlib.figure.Figure at 0x186955c2320>"
       ]
      },
      "metadata": {},
@@ -2430,7 +2433,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb237668>"
+       "<matplotlib.figure.Figure at 0x186938c0a20>"
       ]
      },
      "metadata": {},
@@ -2459,7 +2462,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb202a20>"
+       "<matplotlib.figure.Figure at 0x18695219cf8>"
       ]
      },
      "metadata": {},
@@ -2499,7 +2502,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb27f748>"
+       "<matplotlib.figure.Figure at 0x186952e2f28>"
       ]
      },
      "metadata": {},
@@ -2509,7 +2512,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb38f080>"
+       "<matplotlib.figure.Figure at 0x186952f4b00>"
       ]
      },
      "metadata": {},
@@ -2552,7 +2555,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cbbb7320>"
+       "<matplotlib.figure.Figure at 0x186960693c8>"
       ]
      },
      "metadata": {},
@@ -2562,7 +2565,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb9b5358>"
+       "<matplotlib.figure.Figure at 0x1869606d9e8>"
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      "metadata": {},
@@ -2594,7 +2597,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9ca138c18>"
+       "<matplotlib.figure.Figure at 0x18693957828>"
       ]
      },
      "metadata": {},
@@ -2625,7 +2628,7 @@
      "data": {
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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb1e3ac8>"
+       "<matplotlib.figure.Figure at 0x1869517f860>"
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      },
      "metadata": {},
@@ -2658,7 +2661,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb30e198>"
+       "<matplotlib.figure.Figure at 0x186951e2550>"
       ]
      },
      "metadata": {},
@@ -2739,7 +2742,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cb63fcf8>"
+       "<matplotlib.figure.Figure at 0x1869593bba8>"
       ]
      },
      "metadata": {},
@@ -2793,7 +2796,7 @@
      "output_type": "stream",
      "text": [
       "[[ 3.2]\n",
-      " [ 0.0]]\n"
+      " [ 0. ]]\n"
      ]
     }
    ],
@@ -2861,7 +2864,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cbbc2ba8>"
+       "<matplotlib.figure.Figure at 0x1869579c048>"
       ]
      },
      "metadata": {},
@@ -2904,7 +2907,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9cbbc26a0>"
+       "<matplotlib.figure.Figure at 0x18695981f60>"
       ]
      },
      "metadata": {},
@@ -2944,7 +2947,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2b9ca15edd8>"
+       "<matplotlib.figure.Figure at 0x18695dfa358>"
       ]
      },
      "metadata": {},
@@ -3001,7 +3004,7 @@
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@@ -3115,131 +3203,65 @@
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        "outputs": [
         {
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truncated
-          "text/plain": "<matplotlib.figure.Figure at 0x2298ad62c18>"
+          "text/plain": "<matplotlib.figure.Figure at 0x18694177940>"
          },
          "metadata": {},
          "output_type": "display_data"
@@ -3247,36 +3269,17 @@
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      },
-     "d1e392d588e44a5eaea51baa05b6ea03": {
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       "model_module": "@jupyter-widgets/controls",
       "model_module_version": "1.0.0",
       "model_name": "FloatSliderModel",
       "state": {
        "continuous_update": false,
        "description": "F01(dt)",
-       "layout": "IPY_MODEL_51c29b009b674369955d3826055fe394",
+       "layout": "IPY_MODEL_1adb400d9d5e4ba7b379a0c942ce4ba3",
        "max": 2,
        "step": 0.1,
-       "style": "IPY_MODEL_2eac8b466c2d447f81e0120bf7a5ff99",
-       "value": 1
-      }
-     },
-     "d3c5171f35ae4ae8a6d69cf5275a10e3": {
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-      "model_module_version": "1.0.0",
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-     "dfacfab461cb4988a358e03b62a0e33c": {
-      "model_module": "@jupyter-widgets/controls",
-      "model_module_version": "1.0.0",
-      "model_name": "IntSliderModel",
-      "state": {
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-       "description": "F00",
-       "layout": "IPY_MODEL_d3c5171f35ae4ae8a6d69cf5275a10e3",
-       "max": 2,
-       "style": "IPY_MODEL_7b4df903319543819f1212e905599f44",
+       "style": "IPY_MODEL_7c951421433244f38f2e8bd4e0bb0676",
        "value": 1
       }
      }