Cleaned up example for 1D vs 2D filter

The code was very unclear, including unused variables.

I also expanded the example.

06-Multivariate-Kalman-Filters.ipynb

@@ -428,7 +428,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x483cb00>"
+       "<matplotlib.figure.Figure at 0x435ca20>"
       ]
      },
      "metadata": {},
@@ -926,7 +926,7 @@
      "data": {
       "image/png": "iVBORw0KGgoAAAANSUhEUgAAAt4AAAEWCAYAAABRx5AbAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8115ac8>"
+       "<matplotlib.figure.Figure at 0x8096470>"
       ]
      },
      "metadata": {},
@@ -1392,9 +1392,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8159ef0>"
+       "<matplotlib.figure.Figure at 0x8020860>"
       ]
      },
      "metadata": {},
@@ -1402,9 +1402,9 @@
     },
     {
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x80ad278>"
+       "<matplotlib.figure.Figure at 0x79156d8>"
       ]
      },
      "metadata": {},
@@ -1453,7 +1453,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x82454a8>"
+       "<matplotlib.figure.Figure at 0x81ca7f0>"
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      },
      "metadata": {},
@@ -1541,7 +1541,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9511470>"
+       "<matplotlib.figure.Figure at 0x955f748>"
       ]
      },
      "metadata": {},
@@ -1590,7 +1590,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8189588>"
+       "<matplotlib.figure.Figure at 0x81a8860>"
       ]
      },
      "metadata": {},
@@ -1654,7 +1654,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9234a90>"
+       "<matplotlib.figure.Figure at 0x81914a8>"
       ]
      },
      "metadata": {},
@@ -1749,7 +1749,7 @@
     "\n",
     "$\\mathbf{x} = \\mathbf{x}^- + \\mathbf{Ky}$\n",
     "\n",
-    "We select our new state to be along the residual, scaled by the Kalman gain. The scaling is performed by $\\mathbf{Ky}$, which then needs to be added to the prior: $\\mathbf{x} =\\mathbf{x}^- + \\mathbf{Ky}$.\n",
+    "We select our new state to be along the residual, scaled by the Kalman gain. The scaling is performed by $\\mathbf{Ky}$, which both scales the residual and converts it back into state space. This is added to the prior, yielding the equation: $\\mathbf{x} =\\mathbf{x}^- + \\mathbf{Ky}$.\n",
     "\n",
     "$\\underline{\\texttt{Covariance Update}}$\n",
     "\n",
@@ -1836,14 +1836,15 @@
    "cell_type": "code",
    "execution_count": 30,
    "metadata": {
-    "collapsed": false
+    "collapsed": false,
+    "scrolled": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x95cf710>"
+       "<matplotlib.figure.Figure at 0x93fcf98>"
       ]
      },
      "metadata": {},
@@ -1851,52 +1852,56 @@
     }
    ],
    "source": [
+    "from math import sqrt\n",
     "from numpy.random import randn\n",
     "\n",
-    "def mkf_filter(R, Q):\n",
+    "def univariate_filter(x0, P, R, Q):\n",
     "    f = KalmanFilter(dim_x=1, dim_z=1, dim_u=1)\n",
-    "    f.P = 50.\n",
+    "    f.x = np.array([[x0]])\n",
+    "    f.P *= P\n",
     "    f.H = np.array([[1.]])\n",
     "    f.F = np.array([[1.]])\n",
     "    f.B = np.array([[1.]])\n",
-    "    f.Q = Q\n",
-    "    f.R = R   \n",
+    "    f.Q *= Q\n",
+    "    f.R *= R\n",
     "    return f\n",
     "\n",
-    "def plot_compare_pos_vel(x0, p0, R, Q, move):\n",
-    "    # storage for filter output\n",
-    "    x1, x2 = [], []\n",
-    "\n",
-    "    # initialize the filters\n",
-    "    f1 = mkf_filter(R, Q)\n",
-    "    f1.x[0] = 0.\n",
-    "    f1.P[0, 0] = p0\n",
-    "    \n",
-    "    f2 = pos_vel_filter(x=(0., 1.), P=p0, R=R, Q=Q)\n",
-    "    f2.x[0] = 0.\n",
-    "    f2.x[1] = 1.\n",
-    "    f2.P *= p0\n",
-    "    \n",
-    "    for i in range(50):\n",
-    "        u = move + randn()\n",
-    "        f1.predict(u=u)\n",
-    "        f2.predict(u=u)\n",
-    "        \n",
-    "        z = i*move + randn()\n",
-    "        f1.update(z)\n",
-    "        f2.update(z)\n",
-    "        \n",
-    "        x1.append(f1.x[0, 0])\n",
-    "        x2.append(f2.x[0])\n",
-    "\n",
-    "    plt.plot(x1, label='1D Filter')\n",
-    "    plt.scatter(range(len(x2)), x2, c='r', label='2D Filter')\n",
+    "def plot_1d_2d(xs, xs1d, xs2d):\n",
+    "    plt.plot(xs1d, label='1D Filter')\n",
+    "    plt.scatter(range(len(xs2d)), xs2d, c='r', label='2D Filter')\n",
+    "    plt.plot(xs, ls='--', color='k', lw=1, label='track')\n",
     "    plt.title('State')\n",
     "    plt.legend(loc=4)\n",
-    "    plt.xlim([0,50])\n",
     "    plt.show()\n",
     "    \n",
-    "plot_compare_pos_vel(x0=0., p0=50., R=5., Q=.2, move=1.)    "
+    "def compare_1D_2D(x0, P, R, Q, vel, u=0):\n",
+    "    # storage for filter output\n",
+    "    xs, xs1, xs2 = [], [], []\n",
+    "\n",
+    "    # 1d KalmanFilter\n",
+    "    f1D = univariate_filter(x0, P, R, Q)\n",
+    "\n",
+    "    #2D Kalman filter\n",
+    "    f2D = pos_vel_filter(x=(x0, vel), P=P, R=R, Q=0)\n",
+    "    \n",
+    "    pos = 0 # true position\n",
+    "    for i in range(100):\n",
+    "        pos += vel\n",
+    "        xs.append(pos)\n",
+    "\n",
+    "        # control input u - discussed below\n",
+    "        f1D.predict(u=u)\n",
+    "        f2D.predict()\n",
+    "        \n",
+    "        z = pos + randn()*sqrt(R) # measurement\n",
+    "        f1D.update(z)\n",
+    "        f2D.update(z)\n",
+    "        \n",
+    "        xs1.append(f1D.x[0])\n",
+    "        xs2.append(f2D.x[0])\n",
+    "    plot_1d_2d(xs, xs1, xs2)\n",
+    "\n",
+    "compare_1D_2D(x0=0, P=50., R=5., Q=.02, vel=1.) "
    ]
   },
   {
@@ -1905,15 +1910,75 @@
    "source": [
     "### Discussion\n",
     "\n",
-    "The output of the filter that incorporates velocity into the state produces much better output than the filter that only tracks position - the output is much closer to a straight line. We've already discussed why hidden variables increase the precision of the filter, so I will not repeat that explanation here. But this exercise is intended to trigger a train of thought:\n",
+    "The output of the filter that incorporates velocity into the state produces much better output than the filter that only tracks position. The univariate filter has no way to estimate the velocity or change in position, so it lags the tracked object. \n",
     "\n",
-    "1. The equations in this chapter are mathematically equivalent to the equations in the last chapter when we are only tracking one state variable.\n",
-    "    \n",
-    "2. Therefore, the simple Bayesian reasoning we used in the last chapter applies to this chapter as well.\n",
+    "In the univarate Kalman filter chapter we had a control input `u` to the predict equation:\n",
     "\n",
-    "3. Therefore, the equations in this chapter might 'look ugly', but they really implement multiplying and addition of Gaussians.\n",
+    "```python\n",
+    "    def predict(self, u=0.0):\n",
+    "        self.x += u\n",
+    "        self.P += self.Q\n",
+    "```\n",
     "\n",
-    "> The above might not seem worth emphasizing, but as we continue in the book the mathematical demands will increase significantly. It is easy to get lost in a thicket of linear algebra equations when you read a book or paper on optimal estimation. Any time you start getting lost, go back to the basics of the predict/update cycle based on residuals between measurements and predictions and the meaning of the math will usually be much clearer. The math *looks* daunting, and can sometimes be very hard to solve analytically, but the concepts are quite simple."
+    "It may strike you as 'cheating' that I did not use it here. Let's try that:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x93d8898>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "compare_1D_2D(x0=0, P=50., R=5., Q=.02, vel=1., u=1.) "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here the performance of the two filters are very close, and perhaps the univariate filter is tracking more cloesly. But let's see what happens when the actual velocity `vel` is different from the control input `u`:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x94dbc18>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "compare_1D_2D(x0=0, P=50., R=5., Q=.02, vel=-2., u=1.) "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "If we are tracking a robot which we are controlling the univariate filter can do a very good job because the control input allows the filter to make an accurate prediction. But if we are tracking passively the control input is not much help unless we can make an accurate *apriori* guess as to the velocity. This is rarely possible."
    ]
   },
   {
@@ -1936,7 +2001,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 33,
    "metadata": {
     "collapsed": false,
     "scrolled": false
@@ -1944,9 +2009,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x92bdb70>"
+       "<matplotlib.figure.Figure at 0x81de160>"
       ]
      },
      "metadata": {},
@@ -1954,9 +2019,9 @@
     },
     {
      "data": {
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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAv0AAAE/CAYAAADVDnw2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x95b4b38>"
+       "<matplotlib.figure.Figure at 0x7bee0b8>"
       ]
      },
      "metadata": {},
@@ -1996,7 +2061,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 32,
+   "execution_count": 34,
    "metadata": {
     "collapsed": false,
     "scrolled": false
@@ -2004,9 +2069,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAv0AAAE/CAYAAADVDnw2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9467ac8>"
+       "<matplotlib.figure.Figure at 0x7929be0>"
       ]
      },
      "metadata": {},
@@ -2015,28 +2080,28 @@
    ],
    "source": [
     "run(data=data, R=10000, Q=.2, P=P, plot_P=False, \n",
-    "    title='R=$10,000\\, m^2$, Q=$20\\, m^2$');"
+    "    title='R=$10,000\\, m^2$, Q=$.2\\, m^2$');"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The effect of this can be subtle. We have created an suboptimal filter because the actual measurement noise variance is 225 $m^2$, not 10,000 $m^2$. By setting the filter's noise variance so high we force the filter to favor the prediction over the measurement. This can lead to apparently very smooth and good looking results. In the chart above the track may look extremely good to you since it follows the ideal path very closely. But, the 'great' behavior at the start should give you pause - the filter has not converged yet ($\\mathbf{P}$ is still large) so it should not be able to be so close to the actual position. We can see that $\\mathbf{P}$ has not converged because the entire chart is colored with the yellow background denoting the size of $\\mathbf{P}$. Let's make $\\mathbf{R}$ larger yet. Let's see the result of a bad initial guess for the position by guessing the initial position to be 20 m and the initial velocity to be 1 m/s."
+    "The effect of this can be subtle. We have created an suboptimal filter because the actual measurement noise variance is 225 $m^2$, not 10,000 $m^2$. By setting the filter's noise variance so high we force the filter to favor the prediction over the measurement. This can lead to apparently very smooth and good looking results. In the chart above the track may look extremely good to you since it follows the ideal path very closely. But, the 'great' behavior at the start should give you pause - the filter has not converged yet ($\\mathbf{P}$ is still large) so it should not be able to be so close to the actual position. We can see that $\\mathbf{P}$ has not converged because the entire chart is colored with the yellow background denoting the size of $\\mathbf{P}$. Let's see the result of a bad initial guess for the position by guessing the initial position to be 20 m and the initial velocity to be 1 m/s."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 33,
+   "execution_count": 35,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAv0AAAE/CAYAAADVDnw2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x80e2198>"
+       "<matplotlib.figure.Figure at 0x812cb38>"
       ]
      },
      "metadata": {},
@@ -2044,7 +2109,7 @@
     }
    ],
    "source": [
-    "run(data=data, R=10000, Q=.02, P=P, plot_P=False,\n",
+    "run(data=data, R=10000, Q=.2, P=P, plot_P=False,\n",
     "    x0=np.array([20., 1.]), \n",
     "    title='R=$10,000\\, m^2$, Q=$.2\\, m^2$');"
    ]
@@ -2058,16 +2123,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 34,
+   "execution_count": 36,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x81150b8>"
+       "<matplotlib.figure.Figure at 0x94d0eb8>"
       ]
      },
      "metadata": {},
@@ -2084,7 +2149,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Here we see that the filter initially struggles for several iterations to acquire the track, but then it accurately tracks our dog. In fact, this is nearly optimum - we have not designed $\\mathbf{Q}$ optimally, but $\\mathbf{R}$ is optimal. Recall that our rule of thumb for the variance of $\\mathbf{Q}$ was to set it between $\\frac{1}{2}\\Delta a$ to $\\Delta a$, where $\\Delta a$ is the maximum amount that the acceleration will change between sample period. This only applies for the assumption we are making in this chapter - that acceleration is constant and uncorrelated between each time period. In the Kalman Math chapter we will discuss several different ways of designing $\\mathbf{Q}$.\n",
+    "Here we see that the filter initially struggles for several iterations to acquire the track, but then it accurately tracks our dog. In fact, this is nearly optimum - we have not designed $\\mathbf{Q}$ optimally, but $\\mathbf{R}$ is optimal. A rule of thumb for $\\mathbf{Q}$ is to set it between $\\frac{1}{2}\\Delta a$ to $\\Delta a$, where $\\Delta a$ is the maximum amount that the acceleration will change between sample period. This only applies for the assumption we are making in this chapter - that acceleration is constant and uncorrelated between each time period. In the Kalman Math chapter we will discuss several different ways of designing $\\mathbf{Q}$.\n",
     "\n",
     "To some extent you can get similar looking output by varying either ${\\mathbf{R}}$ or ${\\mathbf{Q}}$, but I urge you to not 'magically' alter these until you get output that you like. Always think about the physical implications of these assignments, and vary ${\\mathbf{R}}$ and/or ${\\mathbf{Q}}$ based on your knowledge of the system you are filtering."
    ]
@@ -2128,7 +2193,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 35,
+   "execution_count": 37,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -2138,7 +2203,7 @@
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x92418d0>"
+       "<matplotlib.figure.Figure at 0x939cb70>"
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      "metadata": {},
@@ -2163,7 +2228,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 38,
    "metadata": {
     "collapsed": false
    },
@@ -2172,7 +2237,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x94efeb8>"
+       "<matplotlib.figure.Figure at 0x954f390>"
       ]
      },
      "metadata": {},
@@ -2226,7 +2291,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 37,
+   "execution_count": 39,
    "metadata": {
     "collapsed": false
    },
@@ -2235,7 +2300,7 @@
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x92461d0>"
+       "<matplotlib.figure.Figure at 0x958d860>"
       ]
      },
      "metadata": {},
@@ -2243,9 +2308,9 @@
     },
     {
      "data": {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c8ee10>"
+       "<matplotlib.figure.Figure at 0x94f4b38>"
       ]
      },
      "metadata": {},
@@ -2271,7 +2336,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 38,
+   "execution_count": 40,
    "metadata": {
     "collapsed": false
    },
@@ -2280,7 +2345,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8135588>"
+       "<matplotlib.figure.Figure at 0x954dbe0>"
       ]
      },
      "metadata": {},
@@ -2288,9 +2353,9 @@
     },
     {
      "data": {
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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x81b59b0>"
+       "<matplotlib.figure.Figure at 0x95ce160>"
       ]
      },
      "metadata": {},
@@ -2315,7 +2380,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 39,
+   "execution_count": 41,
    "metadata": {
     "collapsed": false
    },
@@ -2324,7 +2389,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a2ab00>"
+       "<matplotlib.figure.Figure at 0x92270b8>"
       ]
      },
      "metadata": {},
@@ -2349,7 +2414,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 40,
+   "execution_count": 42,
    "metadata": {
     "collapsed": false
    },
@@ -2358,7 +2423,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x92d5eb8>"
+       "<matplotlib.figure.Figure at 0x94da3c8>"
       ]
      },
      "metadata": {},
@@ -2388,16 +2453,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 43,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x926e780>"
+       "<matplotlib.figure.Figure at 0x79913c8>"
       ]
      },
      "metadata": {},
@@ -2405,9 +2470,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x81e4fd0>"
+       "<matplotlib.figure.Figure at 0x8163630>"
       ]
      },
      "metadata": {},
@@ -2451,7 +2516,7 @@
     "\n",
     "Think about what this means physically. The x-axis of the ellipse denotes our uncertainty in position, and the y-axis our uncertainty in velocity. So, an ellipse that is taller than it is wide signifies that we are more uncertain about the velocity than the position. Conversely, a wide, narrow ellipse shows high uncertainty in position and low uncertainty in velocity. Finally, the amount of tilt shows the amount of correlation between the two variables. \n",
     "\n",
-    "The first plot, with $R=5 m^2$, finishes up with an ellipse that is wider than it is tall. If that is not clear I have printed out the variances for the last ellipse in the lower right hand corner. The variance for position is 3.85 $m^2$, and the variance for velocity is 3.0 $m^2$. \n",
+    "The first plot, with $R=5 m^2$, finishes up with an ellipse that is wider than it is tall. If that is not clear I have printed out the variances for the last ellipse in the lower right hand corner.\n",
     "\n",
     "In contrast, the second plot, with `R=0.5` $m^2$, has a final ellipse that is taller than wide. The ellipses in the second plot are all much smaller than the ellipses in the first plot. This stands to reason because a small $\\small\\mathbf{R}$ implies a small amount of noise in our measurements. Small noise means accurate predictions, and thus a strong belief in our position. "
    ]
@@ -2490,16 +2555,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 44,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x815c1d0>"
+       "<matplotlib.figure.Figure at 0x8014c50>"
       ]
      },
      "metadata": {},
@@ -2550,7 +2615,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 45,
    "metadata": {
     "collapsed": false
    },
@@ -2625,21 +2690,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 46,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
-     "ename": "NameError",
-     "evalue": "name 'bp' is not defined",
-     "output_type": "error",
-     "traceback": [
-      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
-      "\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
-      "\u001b[1;32m<ipython-input-44-f857681b333d>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[0mxs\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0m_\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0m_\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0m_\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mbatch_filter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mzs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      6\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 7\u001b[1;33m \u001b[0mbp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot_track\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcount\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcount\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      8\u001b[0m \u001b[0mbp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot_measurements\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcount\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mzs\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      9\u001b[0m \u001b[0mbp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mplot_filter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mrange\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mcount\u001b[0m \u001b[1;33m+\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mxs\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
-      "\u001b[1;31mNameError\u001b[0m: name 'bp' is not defined"
-     ]
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x948d390>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     }
    ],
    "source": [
@@ -2649,9 +2713,9 @@
     "f = pos_vel_filter(x=(0., 0.), R=3., Q=.02, P=P)\n",
     "xs, _, _, _ = f.batch_filter(zs)\n",
     "\n",
-    "bp.plot_track([0, count], [0, count])\n",
-    "bp.plot_measurements(range(1, count + 1), zs)\n",
-    "bp.plot_filter(range(1, count + 1), xs[:, 0])\n",
+    "book_plots.plot_track([0, count], [0, count])\n",
+    "book_plots.plot_measurements(range(1, count + 1), zs)\n",
+    "book_plots.plot_filter(range(1, count + 1), xs[:, 0])\n",
     "plt.legend(loc='best');"
    ]
   },
@@ -2677,11 +2741,22 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 47,
    "metadata": {
     "collapsed": false
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAuYAAAEWCAYAAAAq41LXAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x9580278>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "from numpy.random import seed\n",
     "count = 50\n",
@@ -2707,11 +2782,22 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 48,
    "metadata": {
     "collapsed": false
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAuoAAAEWCAYAAAAw37JZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x92272e8>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "plt.plot(xs[:, 1], ls='--', label='Kalman Velocity')\n",
     "plt.plot(Ms[:, 1], label='RTS Velocity')\n",
@@ -2737,7 +2823,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 49,
    "metadata": {
     "collapsed": true
    },
@@ -2755,11 +2841,22 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 50,
    "metadata": {
     "collapsed": false
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAt0AAAEWCAYAAAC68CsYAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x948ee10>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "dx = np.diff(xs[:, 0], axis=0)\n",
     "plt.scatter(range(1,count), dx, facecolor='none', edgecolor='k',\n",

code/mkf_internal.py

@@ -483,7 +483,7 @@ def plot_track(ps, actual, zs, cov, std_scale=1,
         ax.set_title("$\sigma^2_x$ (pos variance)")
         plot_covariance(cov, (0, 0))
         ax = plt.subplot(122)
-        ax.set_title("$\sigma^2_y$ (vel variance)")
+        ax.set_title("$\sigma^2_\dot{x}$ (vel variance)")
         plot_covariance(cov, (1, 1))
         plt.show()