Added discrete Wiener process Acceleration model.

Got rid of the hand-wavey 'Q is hard' text and put in a discrete
Wiener process Acceleration model for Q.

Chapter06_Multivariate_Kalman_Filter/Multivariate_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -274,21 +274,33 @@
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      "source": [
       "The techniques in the last chapter are very powerful, but they only work in one dimension. The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n",
       "\n",
-      "In this chapter I am purposefully glossing over many aspects of the mathematics behind Kalman filters. If you are familiar with the topic you will read statements that you disagree with because they contain simplifications that do not necessarily hold in more general cases. If you are not familiar with the topic, expect some paragraphs to be somewhat 'magical' - it will not be clear how I derived a certain result. I prefer that you develop an intuition for how these filters work through several worked examples. If I started by presenting a rigorous mathematical formulation you would be left scratching your head about what all these terms mean and how you might apply them to your problem. In later chapters I will provide a more rigorous mathematical foundation, and at that time I will have to either correct approximations that I made in this chapter or provide additional information that I did not cover here. "
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-     "source": [
-      "The techniques in the last chapter are very powerful, but they only work in one dimension. The gaussians represent a mean and variance that are scalars - real numbers. They provide no way to represent multidimensional data, such as the position of a dog in a field. You may retort that you could use two Kalman filters for that case, one tracks the x coordinate and the other tracks the y coordinate. That does work in some cases, but put that thought aside, because soon you will see some enormous benefits to implementing the multidimensional case.\n",
+      "In this chapter I am purposefully glossing over many aspects of the mathematics behind Kalman filters. If you are familiar with the topic you will read statements that you disagree with because they contain simplifications that do not necessarily hold in more general cases. If you are not familiar with the topic, expect some paragraphs to be somewhat 'magical' - it will not be clear how I derived a certain result. I prefer that you develop an intuition for how these filters work through several worked examples. If I started by presenting a rigorous mathematical formulation you would be left scratching your head about what all these terms mean and how you might apply them to your problem. In later chapters I will provide a more rigorous mathematical foundation, and at that time I will have to either correct approximations that I made in this chapter or provide additional information that I did not cover here. \n",
       "\n",
-      "In this chapter I am purposefully glossing over many aspects of the mathematics behind Kalman filters. If you are familiar with the topic you will read statements that you disagree with because they contain simplifications that do not necessarily hold in more general cases. If you are not familiar with the topic, expect some paragraphs to be somewhat 'magical' - it will not be clear how I derived a certain result. I prefer that you develop an intuition for how these filters work through several worked examples. If I started by presenting a rigorous mathematical formulation you would be left scratching your head about what all these terms mean and how you might apply them to your problem. In later chapters I will provide a more rigorous mathematical foundation, and at that time I will have to either correct approximations that I made in this chapter or provide additional information that I did not cover here. "
+      "To make this possible we will restrict ourselves to a subset of problems which we can describe with Newton's equations of motion. In the literature these filters are called \n",
+      "*discretized continuous-time kinematic filters*. In the next chapter we will develop the math required for solving any kind of dynamic system. \n",
+      "\n",
+      "\n",
+      "In this chapter we are dealing with a simpler form that we can discuss in terms of Newton's equations of motion: given a constant velocity v we can compute distance exactly with:\n",
+      "\n",
+      "$$ x = vt + x_0$$\n",
+      "\n",
+      "If we instead assume constant acceleration we get\n",
+      "\n",
+      "$$ x = \\frac{a}{2}t^2 + v_0 t + x_0$$\n",
+      "\n",
+      "And if we assume constant jerk we get\n",
+      "\n",
+      "$$ x = \\frac{j}{3}t^3 +  \\frac{a_0}{2}t^2 + v_0 t + x_0$$\n",
+      "\n",
+      "As a reminder, we can compute these equations using basic calculus. Given a constant velocity v we can compute the distance travelled over time with the equation\n",
+      "\n",
+      "$$ v = \\frac{dx}{dt}\\\\\n",
+      "x = \\int v \\, \\mathrm{d}t\\\\ x = vt + x_0$$"
      ]
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc5c06a9b0>"
+        "<matplotlib.figure.Figure at 0x7ff0f473f320>"
        ]
       }
      ],
@@ -572,9 +584,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b36ee10>"
+        "<matplotlib.figure.Figure at 0x7ff0f57b29b0>"
        ]
       }
      ],
@@ -619,9 +631,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b60b048>"
+        "<matplotlib.figure.Figure at 0x7ff0d13579e8>"
        ]
       }
      ],
@@ -709,9 +721,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b4cf4e0>"
+        "<matplotlib.figure.Figure at 0x7ff0d13695c0>"
        ]
       }
      ],
@@ -736,9 +748,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b497dd8>"
+        "<matplotlib.figure.Figure at 0x7ff0d15bd9e8>"
        ]
       }
      ],
@@ -763,9 +775,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b62ff60>"
+        "<matplotlib.figure.Figure at 0x7ff0d13f1780>"
        ]
       }
      ],
@@ -794,9 +806,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b4e91d0>"
+        "<matplotlib.figure.Figure at 0x7ff0d1259f60>"
        ]
       }
      ],
@@ -861,9 +873,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b3647b8>"
+        "<matplotlib.figure.Figure at 0x7ff0d1240978>"
        ]
       }
      ],
@@ -1609,15 +1621,65 @@
       "\n",
       "    sigma = sigma + movement_sigma\n",
       "    \n",
-      "$\\mathbf{P}$ is dimensioned $n\\times n$, so $\\mathbf{Q}$ must have the same dimensions. We are just adding matrices, so hopefully it is clear that each element in $\\mathbf{Q}$  specifies how much uncertainty is added to the system due to the process noise. We have not given the math for this yet, but if you suspect the math is difficult you would be correct. \n",
-      "\n",
-      "Fortunately you can get a long way with approximations. The filtered result will not be optimal, but in my opinion the promise of optimal results from Kalman filters is mostly wishful thinking. Consider, for example, tracking a car. In that problem the process noise would include things like potholes, wind gusts, changing drag due to turning, rolling down windows, and many more factors. We cannot realistically model that analytically, and so in practice we work out a simplified model, compute $\\mathbf{Q}$ based on that simplified model, and then add *a bit* to  $\\small\\mathbf{Q}$ in hopes of taking the uncalculable factors into account. Then we use a lot of simulations and trial runs to see if the filter behaves well; if it doesn't we adjust $\\small\\mathbf{Q}$ until the filter performs well. In this chapter we will focus on forming an intuitive understanding on how adjusting $\\small\\mathbf{Q}$ affects the output of the filter. In the Kalman Filter Math chapter we will discuss the analytic computation of $\\small\\mathbf{Q}$, and also provide code that will compute it automatically for you.\n",
-      "\n",
-      "For our first example, we will set the diagonal of $\\small\\mathbf{Q}$ to a small value, and the off-diagonal terms to zero. This is typically not a particularly good choice for a real filter, but it is enough for us to begin forming an understanding of the performance of the filter for various values of $\\small\\mathbf{Q}$. \n",
-      "\n",
-      "> Some books and papers use $\\small\\mathbf{R}$ for measurement noise and $\\small\\mathbf{Q}$ for the process noise. Others do the opposite, using $\\small\\mathbf{Q}$ for measurement noise and $\\small\\mathbf{R}$ for the process noise! Read carefully, and make sure you don't get confused. I use the following mnemonic. Radars are used to measure positions, and they have measurement error. So, for me, $\\small\\mathbf{R}$ is the **R**adar's measurement noise. I've read a lot of Kalman filter literature in the context of radar tracking, so it makes sense to me. I don't have a good one for $\\small\\mathbf{Q}$, other than to note that it alphabetically follows the p in **P**rocess."
+      "$\\mathbf{P}$ is dimensioned $n\\times n$, so $\\mathbf{Q}$ must have the same dimensions. We are just adding matrices, so hopefully it is clear that each element in $\\mathbf{Q}$  specifies how much uncertainty is added to the system due to the process noise. We have not given the math for this yet, but if you suspect the math is simetimes difficult you would be correct. "
      ]
     },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "However, for the class of problems we are solving in this chapter (*discretized continuous-time kinematic filters*), where we can directly compute the state equations for moving objects by using Newton's equations.\n",
+      "\n",
+      "For these kinds of problems we can rely on precomputed forms for $\\mathbf{Q}$. We will learn how to perform the computation in the next chapter. For now I present them without proof. If we assume that for each time period the acceleration due to process noise is constant and uncorrelated, we get the following.\n",
+      "\n",
+      "For constant velocity the form is\n",
+      " $$\\begin{bmatrix}\n",
+      " \\frac{1}{4}{\\Delta t}^4 & \\frac{1}{2}{\\Delta t}^3 \\\\\n",
+      " \\frac{1}{2}{\\Delta t}^3 & \\Delta t\n",
+      " \\end{bmatrix}\\sigma^2\n",
+      " $$\n",
+      "\n",
+      "and for constant acceleration we have\n",
+      "\n",
+      " $$\\begin{bmatrix}\n",
+      " \\frac{1}{4}{\\Delta t}^4 & \\frac{1}{2}{\\Delta t}^3 & \\frac{1}{2}{\\Delta t}^2 \\\\\n",
+      " \\frac{1}{2}{\\Delta t}^3 & {\\Delta t}^2 & \\Delta t \\\\\n",
+      " \\frac{1}{2}{\\Delta t}^2 & \\Delta t & 1\n",
+      " \\end{bmatrix} \\sigma^2\n",
+      " $$\n",
+      "\n",
+      "It is in general not true that acceleration will be constant and uncorrelated, but this is still a useful approximation for moderate time period, and will suffice for this chapter. Fortunately you can get a long way with approximations and simulation. Let's think about what these matrices are implying. We are trying to model the effects of *process noise*, such as the wind buffeting the flight of a thrown ball. Variations in wind will cause changes in acceleration, and so the effect on the acceleration is large. However, the effects on velocity and position are proportionally smaller. In the matrices, the acceleration term is in the lower right, and this is the largest value. A good rule of thumb is to set $\\sigma$ somewhere from $\\frac{1}{2}\\Delta a$ to $\\Delta a$, where $\\Delta a$ is the maximum amount that the acceleration will change between sample periods. In practice we pick a number, run simulations on data, and choose a value that works well. \n",
+      "\n",
+      "The filtered result will not be optimal, but in my opinion the promise of optimal results from Kalman filters is mostly wishful thinking. Consider, for example, tracking a car. In that problem the process noise would include things like potholes, wind gusts, changing drag due to turning, rolling down windows, and many more factors. We cannot realistically model that analytically, and so in practice we work out a simplified model, compute $\\mathbf{Q}$ based on that simplified model, and then add *a bit* to  $\\small\\mathbf{Q}$ in hopes of taking the uncalculable factors into account. Then we use a lot of simulations and trial runs to see if the filter behaves well; if it doesn't we adjust $\\small\\mathbf{Q}$ until the filter performs well. In this chapter we will focus on forming an intuitive understanding on how adjusting $\\small\\mathbf{Q}$ affects the output of the filter. In the Kalman Filter Math chapter we will discuss the analytic computation of $\\small\\mathbf{Q}$, and also provide code that will compute it automatically for you.\n",
+      "\n",
+      "For now we will implement a short Python function that produces this matrix for you. In the literature this model is called the *Discrete Wiener Process Acceleration Model*, which is normally abbreviated **DWPA**, and we will use that nomenclature here."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def Q_DWPA(dim, dt=1., sigma=1.):\n",
+      "    \"\"\" Returns the Q matrix for the Discrete Wiener Process Acceleration Model.\n",
+      "    dim may be either 2 or 3, dt is the time step, and sigma is the variance in \n",
+      "    the noise\"\"\"\n",
+      "    \n",
+      "    assert dim == 2 or dim == 3\n",
+      "    if dim == 2:\n",
+      "        Q = np.array([[.25*dt**4, .5*dt**3],\n",
+      "                      [ .5*dt**3,    dt**2]], dtype=float)\n",
+      "    else:\n",
+      "        Q = np.array([[.25*dt**4, .5*dt**3, .5*dt**2],\n",
+      "                      [ .5*dt**3,    dt**2,       dt],\n",
+      "                      [ .5*dt**2,       dt,        1]], dtype=float)\n",
+      "        \n",
+      "    return Q * sigma"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 24
+    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -1645,7 +1707,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 24
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -1664,13 +1726,13 @@
       "dog_filter.F = np.array([[1,1], [0,1]]) # state transition matrix\n",
       "dog_filter.H = np.array([[1,0]])        # Measurement function\n",
       "dog_filter.R *= 5                       # measurement noise\n",
-      "dog_filter.Q *= 0                       # process noise\n",
+      "dog_filter.Q = Q_DWPA(2, sigma=0.)      # process noise\n",
       "dog_filter.P *= 500.                    # covariance matrix "
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 25
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",
@@ -1724,8 +1786,7 @@
       "    dog_filter.R *= R                 # measurement uncertainty\n",
       "    dog_filter.P *= cov               # covariance matrix \n",
       "    if np.isscalar(Q):\n",
-      "        dog_filter.Q = np.array([[0,0],\n",
-      "                                 [0,Q]])\n",
+      "        dog_filter.Q = Q_DWPA(2, sigma=Q)\n",
       "    else:\n",
       "        dog_filter.Q = Q\n",
       "    return dog_filter\n",
@@ -1766,7 +1827,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 26
+     "prompt_number": 27
     },
     {
      "cell_type": "markdown",
@@ -1815,7 +1876,7 @@
       "    p0, = plt.plot([0,count],[0,count])\n",
       "    p1, = plt.plot(range(1,count+1),zs, linestyle='dashed')\n",
       "    p2, = plt.plot(range(1,count+1),ps)\n",
-      "    plt.legend([p0,p1,p2], ['actual','measurement', 'filter'], 2)\n",
+      "    plt.legend([p0,p1,p2], ['actual','measurement', 'filter'], loc=2)\n",
       "    plt.ylim((0-10,count+10))\n",
       "    plt.title(title)\n",
       "    plt.show()\n",
@@ -1836,7 +1897,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 27
+     "prompt_number": 28
     },
     {
      "cell_type": "markdown",
@@ -1857,21 +1918,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b38fcf8>"
+        "<matplotlib.figure.Figure at 0x7ff0d141aa20>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc5c06a6d8>"
+        "<matplotlib.figure.Figure at 0x7ff0d1485c88>"
        ]
       }
      ],
-     "prompt_number": 28
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1942,7 +2003,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 29
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -1971,7 +2032,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 30
+     "prompt_number": 31
     },
     {
      "cell_type": "markdown",
@@ -2029,21 +2090,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b634160>"
+        "<matplotlib.figure.Figure at 0x7ff0d13c11d0>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b3517f0>"
+        "<matplotlib.figure.Figure at 0x7ff0d1267ef0>"
        ]
       }
      ],
-     "prompt_number": 31
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -2205,7 +2266,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 32
+     "prompt_number": 33
     },
     {
      "cell_type": "markdown",
@@ -2223,7 +2284,7 @@
       "def pos_vel_filter(R,Q):\n",
       "    f = KalmanFilter(dim_x=2, dim_z=1)\n",
       "    f.R = R\n",
-      "    f.Q = Q\n",
+      "    f.Q = Q_DWPA(2, sigma=Q)\n",
       "   \n",
       "    f.F = np.array([[1,1],\n",
       "                    [0,1]])  # state transition matrix\n",
@@ -2271,13 +2332,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b5c55f8>"
+        "<matplotlib.figure.Figure at 0x7ff0d127cda0>"
        ]
       }
      ],
-     "prompt_number": 33
+     "prompt_number": 34
     },
     {
      "cell_type": "markdown",
@@ -2331,21 +2392,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b53da20>"
+        "<matplotlib.figure.Figure at 0x7ff0d14995c0>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b53d978>"
+        "<matplotlib.figure.Figure at 0x7ff0d125e978>"
        ]
       }
      ],
-     "prompt_number": 34
+     "prompt_number": 35
     },
     {
      "cell_type": "markdown",
@@ -2376,6 +2437,8 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
+      "**author's note: text needs to move to kalman math chapter. leaving here for now. Do not read **\n",
+      "\n",
       "But what does \"quite large\" and \"quite small\" mean, and what should $\\textbf{Q}$ contain? The numbers in the $\\textbf{Q}$ matrix are not arbitrary, but the variances of the process noise. This means that they have the same units as the rest of the system. So, suppose the noise of our sensor has a standard deviation of $0.5m$, and the rest of our system is specified in meters as well. Variance is the standard deviation squared, so if $\\sigma=0.5$, then $\\sigma^2 = 0.25$.\n",
       "\n",
       "If we have $m$ state variables then $\\textbf{Q}$ will be an $m{\\times}m$ matrix. $\\textbf{Q}$ is a covariance matrix for the state variables, so it will contain the variances and covariances for the process noise for each state variable. \n",
@@ -2422,9 +2485,7 @@
       "\n",
       "So white noise has the variance $U_n$ and a mean of 0, which we notate as $w \\sim \\mathcal{N}(0,Q)$. \n",
       "\n",
-      "Finding an analytic value for $\\textbf{Q}$ in a simple problem like this is not difficult, but it quickly becomes difficult to impossible as the number of state variables increase. So in this chapter we will use the simplification that only the variance of the last term is important. In the Kalman filter math chapter we will discuss finding an analytic solution for $\\textbf{Q}$, and then present C. F. van Loan's extremely useful numerical technique for finding $\\textbf{Q}$, which is what you will typically use in practice. \n",
-      "\n",
-      "**author's note: text needs to move to kalman math chapter. leaving here for now **"
+      "Finding an analytic value for $\\textbf{Q}$ in a simple problem like this is not difficult, but it quickly becomes difficult to impossible as the number of state variables increase. So in this chapter we will use the simplification that only the variance of the last term is important. In the Kalman filter math chapter we will discuss finding an analytic solution for $\\textbf{Q}$, and then present C. F. van Loan's extremely useful numerical technique for finding $\\textbf{Q}$, which is what you will typically use in practice. "
      ]
     },
     {
@@ -2454,13 +2515,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b608438>"
+        "<matplotlib.figure.Figure at 0x7ff0d12633c8>"
        ]
       }
      ],
-     "prompt_number": 35
+     "prompt_number": 36
     },
     {
      "cell_type": "markdown",
@@ -2524,13 +2585,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b4cf0b8>"
+        "<matplotlib.figure.Figure at 0x7ff0d12f9da0>"
        ]
       }
      ],
-     "prompt_number": 36
+     "prompt_number": 37
     },
     {
      "cell_type": "markdown",
@@ -2553,13 +2614,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b6345f8>"
+        "<matplotlib.figure.Figure at 0x7ff0d1446fd0>"
        ]
       }
      ],
-     "prompt_number": 37
+     "prompt_number": 38
     },
     {
      "cell_type": "markdown",
@@ -2623,13 +2684,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b4f0470>"
+        "<matplotlib.figure.Figure at 0x7ff0d12b6898>"
        ]
       }
      ],
-     "prompt_number": 38
+     "prompt_number": 39
     },
     {
      "cell_type": "markdown",
@@ -2652,13 +2713,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b4fdf98>"
+        "<matplotlib.figure.Figure at 0x7ff0d156e748>"
        ]
       }
      ],
-     "prompt_number": 39
+     "prompt_number": 40
     },
     {
      "cell_type": "markdown",
@@ -2686,13 +2747,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b61c0b8>"
+        "<matplotlib.figure.Figure at 0x7ff0d14e08d0>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 41
     },
     {
      "cell_type": "markdown",
@@ -2717,13 +2778,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b5084a8>"
+        "<matplotlib.figure.Figure at 0x7ff0d1231630>"
        ]
       }
      ],
-     "prompt_number": 41
+     "prompt_number": 42
     },
     {
      "cell_type": "markdown",
@@ -2789,21 +2850,21 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b5506a0>"
+        "<matplotlib.figure.Figure at 0x7ff0d12e9860>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b550a20>"
+        "<matplotlib.figure.Figure at 0x7ff0d1459828>"
        ]
       }
      ],
-     "prompt_number": 42
+     "prompt_number": 43
     },
     {
      "cell_type": "markdown",
@@ -2864,13 +2925,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fcc3b4e9dd8>"
+        "<matplotlib.figure.Figure at 0x7ff0d14327f0>"
        ]
       }
      ],
-     "prompt_number": 43
+     "prompt_number": 44
     },
     {
      "cell_type": "markdown",
@@ -2900,7 +2961,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "** author's note: this code is somewhat old. This section needs to be edited **\n",
+      "** author's note: this code is somewhat old. This section needs to be edited; I would not pay a lot of attention to it right now. **\n",
       "\n",
       "The kalman filter code that we are using is implemented in my Python library `filterpy`. If you are interested in the full implementation of the filter you should look in `filterpy\\kalman\\kalman_filter.py`. In the following I will present a simplified implementation of the same code. The code in the library handles issues that are beyond the scope of this chapter, such as numerical stability and support for the extended Kalman filter, subject of a later chapter. \n",
       "\n",

Chapter07_Kalman_Filter_Math/Kalman_Filter_Math.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:4f0493fb7c7a8813b0cda0b31e92c7dc575f7d5eb07061f42ba36214d17920ec"
+  "signature": "sha256:5e618e97b591e89d2e34bd50626e7a9afaab2de1c0ab60da49a9119975fdd7aa"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -466,13 +466,26 @@
      ]
     },
     {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
+     "cell_type": "markdown",
      "metadata": {},
-     "outputs": [],
-     "prompt_number": 2
+     "source": [
+      "*equations of continuous Wiener process acceleration model - just a cut paste being saved for later use when I write this*\n",
+      "\n",
+      " $$\\begin{bmatrix}\n",
+      " \\frac{1}{3}{\\Delta t}^3 & \\frac{1}{2}{\\Delta t}^2 \\\\\n",
+      " \\frac{1}{2}{\\Delta t}^2 & \\Delta t\n",
+      " \\end{bmatrix}\\tilde{q}\n",
+      " $$\n",
+      "\n",
+      "\n",
+      " $$\\begin{bmatrix}\n",
+      " \\frac{1}{20}{\\Delta t}^5 & \\frac{1}{8}{\\Delta t}^4 & \\frac{1}{6}{\\Delta t}^3 \\\\\n",
+      " \\frac{1}{8}{\\Delta t}^8 & \\frac{1}{3}{\\Delta t}^3 & \\frac{1}{2}{\\Delta t}^2  \\\\\n",
+      " \\frac{1}{6}{\\Delta t}^3 & \\frac{1}{2}{\\Delta t}^2 & \\Delta t\n",
+      " \\end{bmatrix} \\tilde{q}\n",
+      " $$\n",
+      "    "
+     ]
     }
    ],
    "metadata": {}

Chapter09_Extended_Kalman_Filters/Extended_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:8639f3d09166c8c1ae2b55e044e278c8b7e31e19bd72545341522e3c86eb9f7b"
+  "signature": "sha256:19938ddfba204e20c435b33acc3d723467354126675c5aa78dc8200036994fad"
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@@ -269,6 +269,8 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
+      "**Author's note: this is still being heavily edited - there is a lot of duplicate material, incomplete material, and so on**\n",
+      "\n",
       "The Kalman filter that we have developed to this point is extremely good, but it is also limited. Its derivation is in the linear space, and hence it only works for linear problems. Let's be a bit more rigorous here. You can, and we have in this book, apply the Kalman filter to nonlinear problems. For example, in the g-h filter chapter we explored using a g-h filter in a problem with constant acceleration. It 'worked', in that it remained numerically stable and the filtered output did track the input, but there was always a lag. It is easy to prove that there will always be a lag when $\\mathbf{\\ddot{x}}>0$. The filter no longer produces an optimal result. If we make our time step arbitrarily small we can still handle many problems, but typically we are using Kalman filters with physical sensors and solving real-time problems. Either fast enough sensors do not exist, are prohibitively expensive, or the computation time required is excessive. It is not a workable solution.\n",
       "\n",
       "The early adopters of Kalman filters were the radar people, and this fact was not lost on them. Radar is inherently nonlinear. Radars measure the slant range to an object, and we are typically interested in the aircraft's position over the ground. We invoke Pythagoras and get the nonlinear equation:\n",
@@ -613,7 +615,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We want a linear appoximation of this function so that we can use it in the Kalman filter. We will see how it is used in the Kalman filter in the next section, so don't worry about that yet. We can see that there is no single linear function (line) that gives a close approximation of this function. However, during each innovation of the Kalman filter we know it's current state, so if we linearize the function at that value we will have a close approximation. For example, suppose our current state is $x=1.5$. What would be a good linearization for this function?\n",
+      "We want a linear appoximation of this function so that we can use it in the Kalman filter. We will see how it is used in the Kalman filter in the next section, so don't worry about that yet. We can see that there is no single linear function (line) that gives a close approximation of this function. However, during each innovation (update) of the Kalman filter we know its current state, so if we linearize the function at that value we will have a close approximation. For example, suppose our current state is $x=1.5$. What would be a good linearization for this function?\n",
       "\n",
       "We can use any linear function that passes through the curve at (1.5,-0.75). For example, consider using f(x)=8x\u221212.75 as the linearization, as in the plot below."
      ]
@@ -691,7 +693,7 @@
      "source": [
       "Here we can see that this linearization is much better. It is still exactly correct at $x=1.5$, but the errors are very small as x varies. Compare the tiny error at $x=1.4$ vs the very large error at $x=1.4$ in the previous plot. This does not constitute a formal proof of correctness, but this sort of geometric depiction should be fairly convincing. Certainly it is easy to see that in this case if the line had any other slope the errors would accumulate more quickly. \n",
       "\n",
-      "To implement the extended Kalman filter we will leave the linear equations as they are, and use partial derivatives to evaluate the system matrix $\\mathbf{F}$ and the measurement matrix $\\mathbf{H}$ at the state at time t ($\\mathbf{x}_t$). Since $\\mathbf{F}$ also depends on the control input vector $\\mathbf{u}$m we will need to include that term:\n",
+      "To implement the extended Kalman filter we will leave the linear equations as they are, and use partial derivatives to evaluate the system matrix $\\mathbf{F}$ and the measurement matrix $\\mathbf{H}$ at the state at time t ($\\mathbf{x}_t$). Since $\\mathbf{F}$ also depends on the control input vector $\\mathbf{u}$ we will need to include that term:\n",
       "\n",
       "$$\n",
       "\\begin{aligned}\n",
@@ -795,7 +797,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "As we can see there are two minor changes to the Kalman filter equations. The first change replaces the equation $\\mathbf{x} = \\mathbf{Fx}$ with $\\mathbf{x} = f(\\mathbf{x})$. In the Kalman filter, $\\mathbf{Fx}$ is how we compute the new state based on the old state. However, in a nonlinear system we cannot use linear algebra to compute this transition. So instead we hypothesize a nonlinear function $f()$ which performs this function. Likewise, in the Kalman filter we convert the state to a measurement with the linear function $\\mathbf{Hx}$. For the extended Kalman filter we replace this with a nonlinear function $h()$, giving $\\mathbf{z}_x = h(\\mathbf{x})$.\n",
+      "As we can see there are two minor changes to the Kalman filter equations, which I have underlined. The first change replaces the equation $\\mathbf{x} = \\mathbf{Fx}$ with $\\mathbf{x} = f(\\mathbf{x})$. In the Kalman filter, $\\mathbf{Fx}$ is how we compute the new state based on the old state. However, in a nonlinear system we cannot use linear algebra to compute this transition. So instead we hypothesize a nonlinear function $f()$ which performs this function. Likewise, in the Kalman filter we convert the state to a measurement with the linear function $\\mathbf{Hx}$. For the extended Kalman filter we replace this with a nonlinear function $h()$, giving $\\mathbf{z}_x = h(\\mathbf{x})$.\n",
       "\n",
       "The only question left is how do we implement use $f()$ and $h()$ in the Kalman filter if they are nonlinear? We reach for the single tool that we have available for solving nonlinear equations - we linearize them at the point we want to evaluate the system.  For example, consider the function $f(x) = x^2 -2x$\n",
       "\n",