Interim check in - adding more info about designing Q.

Multidimensional_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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@@ -262,7 +262,9 @@
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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."
+      "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 certaina 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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+        "<matplotlib.figure.Figure at 0x4471b10>"
        ]
       }
      ],
@@ -564,9 +566,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x29dafd0>"
+        "<matplotlib.figure.Figure at 0x420d190>"
        ]
       }
      ],
@@ -654,9 +656,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x283bbd0>"
+        "<matplotlib.figure.Figure at 0x4211550>"
        ]
       }
      ],
@@ -681,9 +683,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x29dadd0>"
+        "<matplotlib.figure.Figure at 0x4113750>"
        ]
       }
      ],
@@ -708,9 +710,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2cd1cd0>"
+        "<matplotlib.figure.Figure at 0x42115d0>"
        ]
       }
      ],
@@ -739,9 +741,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x2b0c410>"
+        "<matplotlib.figure.Figure at 0x46ecb50>"
        ]
       }
      ],
@@ -792,9 +794,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2ccd1d0>"
+        "<matplotlib.figure.Figure at 0x421de10>"
        ]
       }
      ],
@@ -1183,24 +1185,37 @@
       "    f.H = np.matrix([[1,0]])       # Measurement function\n",
       "    f.R = R                        # measurement uncertainty\n",
       "    f.P *= cov                     # covariance matrix \n",
-      "    f.Q = np.eye(2)*Q\n",
+      "    if np.isscalar(Q):\n",
+      "        f.Q = np.matrix([[0,0],[0,Q]])\n",
+      "    else:\n",
+      "        f.Q = Q\n",
       "    return f\n",
       "\n",
       "\n",
-      "def filter_dog(noise, count, R, Q=0):\n",
-      "    dog = DogSensor(velocity=1, noise=noise)\n",
+      "def filter_dog(noise=0, count=0, R=0, Q=0, data=None):\n",
+      "    \"\"\" Kalman filter 'count' readings from the DogSensor.\n",
+      "    'noise' is the noise scaling factor for the DogSensor.\n",
+      "    'data' provides the measurements. If set, noise will\n",
+      "    be ignored and data will not be generated for you.\n",
+      "    \n",
+      "    returns a tuple of (positions, measurements, covariance)\n",
+      "    \"\"\"\n",
+      "    if data is None:    \n",
+      "        dog = DogSensor(velocity=1, noise=noise)\n",
+      "        zs = [dog.sense() for t in range(count)]\n",
+      "    else:\n",
+      "        zs = data\n",
+      "\n",
       "    dog_filter = dog_tracking_filter(R=R, Q=Q, cov=500.)\n",
       "\n",
       "    pos  = [None] * count\n",
-      "    zs  = [None] * count\n",
       "    cov = [None] * count\n",
       "    \n",
-      "    for t in range (count):\n",
-      "        z = dog.sense()  # get the next measurement\n",
-      "        \n",
+      "    for t in range(count):\n",
+      "        z = zs[t]\n",
       "        pos[t] = dog_filter.x[0,0]\n",
       "        cov[t] = dog_filter.P\n",
-      "        zs[t]  = z\n",
+      "\n",
       "        \n",
       "        # perform the kalman filter steps\n",
       "        dog_filter.update (z)\n",
@@ -1251,9 +1266,9 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "def plot_track(noise, count, R, Q=0, plot_P=True, title='Kalman Filter'):\n",
+      "def plot_track(noise=None, count=0, R=0, Q=0, data=None, plot_P=True, title='Kalman Filter'):\n",
       "    \n",
-      "    ps, zs, cov = filter_dog(noise, count, R, Q)\n",
+      "    ps, zs, cov = filter_dog(noise=noise, data=data, count=count, R=R, Q=Q)\n",
       "    \n",
       "    p0, = plt.plot([0,count],[0,count],'g')\n",
       "    p1, = plt.plot(range(1,count+1),zs,c='r', linestyle='dashed')\n",
@@ -1300,17 +1315,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2bd7cd0>"
+        "<matplotlib.figure.Figure at 0x4209550>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x283e650>"
+        "<matplotlib.figure.Figure at 0x3de2c90>"
        ]
       }
      ],
@@ -1467,8 +1482,11 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "plot_track (noise=30, R=5, Q=10,count=30, plot_P=False, title='R = 5, Q = 10*I')\n",
-      "plot_track (noise=30, R=5, Q=0.1,count=30, plot_P=False, title='R = 5, Q = 0.1*I')"
+      "dog = DogSensor(velocity=1, noise=30)\n",
+      "zs = [dog.sense() for t in range(30)]\n",
+      "\n",
+      "plot_track (data=zs, R=5, Q=10,count=30, plot_P=False, title='R = 5, Q = 10')\n",
+      "plot_track (data=zs, R=5, Q=.02,count=30, plot_P=False, title='R = 5, Q = 0.02')"
      ],
      "language": "python",
      "metadata": {},
@@ -1476,17 +1494,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2854e10>"
+        "<matplotlib.figure.Figure at 0x46e1790>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x2cdfcd0>"
+        "<matplotlib.figure.Figure at 0x4218250>"
        ]
       }
      ],
@@ -1502,10 +1520,76 @@
       "\n",
       "Let's remind ourselves of what the term *process uncertainty* means. Consider the problem of tracking a ball. We can accurately model its behavior in statid air with math, but if there is any wind our model will diverge from reality. \n",
       "\n",
-      "In the first case we set ${\\mathbf{Q}}=100$, which is quite large. In physical terms this is telling the filter \"I don't trust my motion prediction step\". Strictly speaking, we are telling the filter there is a lot of external noise that we are not modeling with $\\small{\\mathbf{F}}$, but the upshot of that is to not trust the motion prediction step. So the filter will be computing velocity ($\\dot{x}$), but then mostly ignoring it because we are telling the filter that the computation is extremely suspect. Therefore the filter has nothing to use but the measurements, and thus it follows the measurements closely. \n",
+      "In the first case we set ${\\mathbf{Q}}=10$, which is quite large. In physical terms this is telling the filter \"I don't trust my motion prediction step\". Strictly speaking, we are telling the filter there is a lot of external noise that we are not modeling with $\\small{\\mathbf{F}}$, but the upshot of that is to not trust the motion prediction step. So the filter will be computing velocity ($\\dot{x}$), but then mostly ignoring it because we are telling the filter that the computation is extremely suspect. Therefore the filter has nothing to use but the measurements, and thus it follows the measurements closely. \n",
       "\n",
-      "In the second case we set ${\\mathbf{Q}}=0.1$, which is quite small. In physical terms we are telling the filter \"trust the motion computation, it is really good!\". Again, more strictly this actually says there is very small amounts of process noise, so the motion computation will be accurate. So the filter ends up ignoring some of the measurement as it jumps up and down, because the variation in the measurement does not match our trustworthy velocity prediction. \n",
+      "In the second case we set ${\\mathbf{Q}}=0.02$, which is quite small. In physical terms we are telling the filter \"trust the motion computation, it is really good!\". Again, more strictly this actually says there is very small amounts of process noise, so the motion computation will be accurate. So the filter ends up ignoring some of the measurement as it jumps up and down, because the variation in the measurement does not match our trustworthy velocity prediction.\n",
       "\n",
+      "**AUTHOR'S NOTE: move covariance matrix coverage here, then do R, then Q. Order as below is confusing.**"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Designing $\\textbf{Q}$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "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",
+      "\n",
+      "Let's make this concrete. Assume our state variables are $x=\\begin{bmatrix}x&\\dot{x}&\\end{bmatrix}^T$. (**note**: it is customary to use this transpose form of writing an matrix in text. It is how we denote that $x$ is a column matrix without taking up a lot of line space). Then $\\textbf{Q}$ will contain:\n",
+      "\n",
+      "$$Q = \\begin{bmatrix}\n",
+      "\\sigma_x^2 & p\\sigma_x\\sigma_{\\dot{x}} \\\\\n",
+      "p\\sigma_x\\sigma_{\\dot{x}}& \\sigma_{\\dot{x}}^2\n",
+      "\\end{bmatrix}\n",
+      "$$\n",
+      "\n",
+      "But again, what does this *mean*? This is a one dimensional problem, where our variables are $x$ and $\\dot{x}$. Assume we are tracking a person walking in 1D, so $x$ is their position, and  $\\dot{x}$ is their velocity. We have no state variable for $\\ddot{x}$, so we are assuming aceleration is zero, and thus their velocity is constant. No one walks with constant velocity, even if they are trying to do so. The *process noise* specifies how much variance there is in each state variable due to the changes in velocity that inevitably happen.\n",
+      "\n",
+      "You will very typically see Q expressed in this form (indeed, this is what the code immediately above does):\n",
+      "\n",
+      "$$\\begin{bmatrix}\n",
+      "0&0 \\\\\n",
+      "0&0.1\n",
+      "\\end{bmatrix}\n",
+      "$$\n",
+      "\n",
+      "Why all zeros but in the last row and column. This is a useful approximation that we can use for $\\textbf{Q}$ under certain circumstances. Think about the person. As they accelerate, that will also alter their velocity, and eventually their position. But if the acceleration is small compared to our time sample rate, then the changes to distance will small. This follows from the Newtonian equations:\n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "v&=a\\Delta t \\\\\n",
+      "d&=\\frac{a}{2}{\\Delta t}^2\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "If t is small and a is small than the contribution of $\\frac{a}{2}{\\Delta t}^2$ will be extremely small. In this case it is safe to set all of the terms in $\\mathbf{Q}$ to 0 except the variance for the last term. "
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "On the other hand, let's suppose this is not the case. How should Q be designed? Our design of the system is:\n",
+      "\n",
+      "$$ X_{n+1} = \\Phi X_n + U_n$$\n",
+      "\n",
+      "where $\\Phi X_n$ is our state transition, which computes $\\mathbf{x}$ at time $n+1$ using Newtonan equations, and $U_n$ is the white noise associated with the process. For a walking human, based on the equations above we get\n",
+      "\n",
+      "$$U_n = \\begin{bmatrix}\\frac{a{\\Delta t}^2}{2} \\\\ \\Delta t\\end{bmatrix}$$\n",
+      "\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"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
       "Now let's leave ${\\mathbf{Q}}=0.1$, but bump ${\\mathbf{R}}$ up to $1000$. This is telling the filter that the measurement noise is very large. "
      ]
     },
@@ -1513,7 +1597,7 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "plot_track (noise=30, R=1000, Q=0.1,count=50, plot_P=False, title='R = 1000, Q = 0.1')"
+      "plot_track (data=zs, R=1000, Q=0.1,count=30, plot_P=False, title='R = 1000, Q = 0.1')"
      ],
      "language": "python",
      "metadata": {},
@@ -1521,9 +1605,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2cabad0>"
+        "<matplotlib.figure.Figure at 0x3de2c90>"
        ]
       }
      ],
@@ -1591,9 +1675,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x22e5350>"
+        "<matplotlib.figure.Figure at 0x421d410>"
        ]
       }
      ],
@@ -1620,9 +1704,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x2853350>"
+        "<matplotlib.figure.Figure at 0x41fe0d0>"
        ]
       }
      ],
@@ -1711,17 +1795,17 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2851510>"
+        "<matplotlib.figure.Figure at 0x410e810>"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x2bd8110>"
+        "<matplotlib.figure.Figure at 0x46ffe90>"
        ]
       }
      ],
@@ -1786,9 +1870,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2589110>"
+        "<matplotlib.figure.Figure at 0x4b22d10>"
        ]
       }
      ],