Comparison of 1D and 2D filter.

Added exercises comparing 1D filter using Gaussians vs 1D filter using KalmanFilter class, and comparing a 1D filter vs a filter incorporating the unobserved derivative of x.
2014-08-30 00:08:27 -07:00
1 parent 87d4d973ac
commit 6ab8c5da92
@@ -1,7 +1,7 @@
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@@ -930,7 +930,7 @@
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     "source": [
-      "Implementing Updates"
+      "Implementing Predictions"
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@@ -1,7 +1,7 @@
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     "input": [
      "import mkf_internal\n",
+      "import numpy as np\n",
      "mkf_internal.plot_3d_covariance((2,17), np.array([[10.,0],[0,4.]]))"
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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3aff6d110>"
+        "<matplotlib.figure.Figure at 0x7fa14b619a50>"
       ]
      }
     ],
-     "prompt_number": 8
+     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
@@ -375,11 +600,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afc1b490>"
+        "<matplotlib.figure.Figure at 0x7fa14b38ca10>"
       ]
      }
     ],
-     "prompt_number": 76
+     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
@@ -463,11 +688,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af75be50>"
+        "<matplotlib.figure.Figure at 0x7fa14b56f4d0>"
       ]
      }
     ],
-     "prompt_number": 10
+     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
@@ -490,11 +715,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afa2c7d0>"
+        "<matplotlib.figure.Figure at 0x7fa14b45a1d0>"
       ]
      }
     ],
-     "prompt_number": 11
+     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
@@ -517,11 +742,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afa84750>"
+        "<matplotlib.figure.Figure at 0x7fa14b4facd0>"
       ]
      }
     ],
-     "prompt_number": 12
+     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
@@ -548,11 +773,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af74b310>"
+        "<matplotlib.figure.Figure at 0x7fa14b404a90>"
       ]
      }
     ],
-     "prompt_number": 13
+     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
@@ -615,11 +840,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afe96190>"
+        "<matplotlib.figure.Figure at 0x7fa14b470150>"
       ]
      }
     ],
-     "prompt_number": 14
+     "prompt_number": 15
    },
    {
     "cell_type": "heading",
@@ -1020,7 +1245,7 @@
     "language": "python",
     "metadata": {},
     "outputs": [],
-     "prompt_number": 15
+     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
@@ -1045,7 +1270,7 @@
     "language": "python",
     "metadata": {},
     "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
@@ -1141,7 +1366,7 @@
     "language": "python",
     "metadata": {},
     "outputs": [],
-     "prompt_number": 17
+     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
@@ -1211,7 +1436,7 @@
     "language": "python",
     "metadata": {},
     "outputs": [],
-     "prompt_number": 18
+     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
@@ -1232,9 +1457,9 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
+       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af7a7e10>"
+        "<matplotlib.figure.Figure at 0x7fa14b4c2810>"
       ]
      },
      {
@@ -1242,11 +1467,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af985450>"
+        "<matplotlib.figure.Figure at 0x7fa14b66c810>"
       ]
      }
     ],
-     "prompt_number": 19
+     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
@@ -1540,7 +1765,292 @@
     "language": "python",
     "metadata": {},
     "outputs": [],
-     "prompt_number": 20
+     "prompt_number": 21
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "#####Exercise: Compare to Previous Chapter\n",
+      "\n",
+      "The equations in this chapter look very different from the equations in the last chapter, yet I claimed the last chapter implemented a full 1-D Kalman filter. \n",
+      "\n",
+      "Recall that the 1-D equations for the update step are:\n",
+      "$$\n",
+      "\\mu =\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}, \n",
+      "\\sigma = \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
+      "$$\n",
+      "\n",
+      "and that the 1-D equations for the predict step are:\n",
+      "$$ \\mu = \\mu_1+\\mu_2, \\sigma^2 = \\sigma_1^2 + \\sigma_2^2$$\n",
+      "\n",
+      "Implement a simple 1-D kalman filter using the Kalman filter from this chapter, and compare it's output to the kalman filter from the previous chapter by plotting it."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# Your code here"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##### Solution\n",
+      "\n",
+      "We will use a simple model of tracking an object that starts at x=0 and moves by 1 at each step. We will assume the arbitrary value 5 for the measurement noise and .02 for the process noise.\n",
+      "\n",
+      "First, let's implement the filter from the last chapter:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from __future__ import division\n",
+      "import numpy as np\n",
+      "from numpy.random import randn\n",
+      "from filterpy.kalman import KalmanFilter\n",
+      "\n",
+      "# 1-D Kalman filter equations\n",
+      "def predict(pos, variance, movement, movement_variance):\n",
+      "    return (pos + movement, variance + movement_variance)\n",
+      "\n",
+      "\n",
+      "def update (mu1, var1, mu2, var2):\n",
+      "    mean = (var1*mu2 + var2*mu1) / (var1+var2)\n",
+      "    variance = 1 / (1/var1 + 1/var2)\n",
+      "    return (mean, variance)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 117
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now, let's implement the same thing using the kalman filter. I will implement it as a function that returns a KalmanFilter object so that you can run the analysis code several times with the KalmanFilter initialized to the same starting conditions each time."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from filterpy.kalman import KalmanFilter\n",
+      "\n",
+      "def mkf_filter(R, Q):\n",
+      "    f = KalmanFilter(dim_x=1, dim_z=1, dim_u=1)\n",
+      "    f.P = 500.\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",
+      "    \n",
+      "    return f"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 105
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Finally, let's compare the two. I will plot the data from the 1-D Kalman filter as a blue line, and the output of the filter from this chapter as red dots. I wrote it as a function so you can easily modify the parameters and regenerate the plots."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def plot_kf_compare(x0, p0, R, Q, move):\n",
+      "    # storage for filter output\n",
+      "    x1 = []\n",
+      "    x2 = []\n",
+      "\n",
+      "    p1 = []\n",
+      "    p2 = []\n",
+      "\n",
+      "    # initialize the filters\n",
+      "    f = mkf_filter(R, Q)\n",
+      "    f.x[0,0] = 0.\n",
+      "    f.P[0,0] = p0\n",
+      "    pos = (x0, p0)\n",
+      "    for i in range(50):\n",
+      "        z = i*move + randn()\n",
+      "        pos = update(pos[0], pos[1], z, R)\n",
+      "        f.update(z)\n",
+      "\n",
+      "        x1.append(pos[0])\n",
+      "        x2.append(f.x[0,0])\n",
+      "\n",
+      "        p1.append(pos[1])\n",
+      "        p2.append(f.P[0,0])\n",
+      "\n",
+      "        u = move + randn()\n",
+      "        pos = predict(pos[0], pos[1], u, Q)\n",
+      "        f.predict(u=u)\n",
+      "\n",
+      "    plt.plot(x1, 'b')\n",
+      "    plt.scatter(range(len(x2)), x2, c='r')\n",
+      "    plt.title('State')\n",
+      "    plt.figure()\n",
+      "    plt.plot(p1)\n",
+      "    plt.scatter(range(len(x2)), p2, c='r')\n",
+      "    plt.title('Variance')\n",
+      "    plt.show()\n",
+      "    \n",
+      "plot_kf_compare(x0=0., p0=500., R=5., Q=.2, move=1.)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa14b4b2690>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa14986a550>"
+       ]
+      }
+     ],
+     "prompt_number": 136
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##### Discussion\n",
+      "\n",
+      "As you can see, both filters produce the same results. Feel free to vary the initial guess, the measurement noise, and the process noise; so long as you make the same changes to both filters the output should be the same. This is a solid demonstration, albeit not a rigorous proof, that both filters in fact implement the same math for the 1-D case. \n",
+      "\n",
+      "If you are inclined to do so, and have the linear algebra background, try to prove that the equations are identical for the 1-D case. This is well beyond the scope of this chapter, so I will not present the solution."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##### Exercise: Compare to a Filter That Incorporates Velocity\n",
+      "\n",
+      "The last example did not use one of the fundamental insights of this chapter, unobserved variables. In this example velocity would the the unobserved variable. Write a Kalman filter that uses the state $\\mathbf{x}=\\begin{bmatrix}x & \\dot{x}\\end{bmatrix}^T$ and compare it against the filter in the last exercise which used the state $\\mathbf{x}=\\begin{bmatrix}x\\end{bmatrix}$."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# your code here"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "##### Solution\n",
+      "\n",
+      "We've already implemented a Kalman filter for position and velocity, so I will provide the code without much comment, and then plot the result."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def pos_vel_filter(R,Q):\n",
+      "    f = KalmanFilter(dim_x=2, dim_z=1)\n",
+      "    f.R = R\n",
+      "    f.Q = Q\n",
+      "   \n",
+      "    f.F = np.array([[1,1],\n",
+      "                    [0,1]])  # state transition matrix\n",
+      "    f.H = np.array([[1,0]])  # Measurement function\n",
+      "    return f\n",
+      "\n",
+      "def plot_compare_pos_vel(x0, p0, R, Q, move):\n",
+      "    # storage for filter output\n",
+      "    x1 = []\n",
+      "    x2 = []\n",
+      "\n",
+      "    # initialize the filters\n",
+      "    f1 = mkf_filter(R, Q)\n",
+      "    f.x[0,0] = 0.\n",
+      "    f.P[0,0] = p0\n",
+      "    \n",
+      "    f2 = pos_vel_filter(R, Q)\n",
+      "    f2.x[0,0] = 0.\n",
+      "    f2.x[1,0] = 1.\n",
+      "    f.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[0,0])\n",
+      "\n",
+      "    p1, = plt.plot(x1, 'b')\n",
+      "    p2 = plt.scatter(range(len(x2)), x2, c='r')\n",
+      "    plt.title('State')\n",
+      "    plt.legend([p1,p2], ['1D filter', '2D filter'], 4)\n",
+      "    plt.show()\n",
+      "    \n",
+      "plot_compare_pos_vel(x0=0., p0=500., R=5., Q=.2, move=1.)    "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+       "text": [
+        "<matplotlib.figure.Figure at 0x7fa14b30de10>"
+       ]
+      }
+     ],
+     "prompt_number": 146
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "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 unobserved variables increase the precision of the filter, so I will not repeat that explanation here. But the last exercise and this one is intended to trigger a train of thought:\n",
+      "\n",
+      "1. The equations in this chapter are mathematically equivelent 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",
+      "\n",
+      "3. Therefore, the equations in this chapter might 'look ugly', but they really are just implementing multiplying and addition of Gaussians.\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 algrebra equations when you read a book or paper on optimal estimation. Any time you start getting lost, just 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."
+     ]
    },
    {
     "cell_type": "heading",
@@ -1577,21 +2087,21 @@
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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afa04390>"
+        "<matplotlib.figure.Figure at 0x7fa14b5bb350>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af9dae50>"
+        "<matplotlib.figure.Figure at 0x7fa16d154b90>"
       ]
      }
     ],
-     "prompt_number": 21
+     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
@@ -1700,13 +2210,13 @@
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afb50fd0>"
+        "<matplotlib.figure.Figure at 0x7fa14b614850>"
       ]
      }
     ],
-     "prompt_number": 22
+     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
@@ -1771,11 +2281,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af74b490>"
+        "<matplotlib.figure.Figure at 0x7fa14b655e50>"
       ]
      }
     ],
-     "prompt_number": 23
+     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
@@ -1799,11 +2309,11 @@
       "output_type": "display_data",
       "png": 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       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af7a0bd0>"
+        "<matplotlib.figure.Figure at 0x7fa14be68cd0>"
       ]
      }
     ],
-     "prompt_number": 24
+     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
@@ -1869,11 +2379,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3af95de90>"
+        "<matplotlib.figure.Figure at 0x7fa14b62cf50>"
       ]
      }
     ],
-     "prompt_number": 25
+     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
@@ -1898,11 +2408,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afa18f10>"
+        "<matplotlib.figure.Figure at 0x7fa15333fad0>"
       ]
      }
     ],
-     "prompt_number": 26
+     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
@@ -1932,11 +2442,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afec89d0>"
+        "<matplotlib.figure.Figure at 0x7fa15333fa90>"
       ]
      }
     ],
-     "prompt_number": 27
+     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
@@ -1963,11 +2473,11 @@
       "output_type": "display_data",
       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afec8450>"
+        "<matplotlib.figure.Figure at 0x7fa14b632a10>"
       ]
      }
     ],
-     "prompt_number": 28
+     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
@@ -2034,21 +2544,21 @@
      {
       "metadata": {},
       "output_type": "display_data",
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truncated
+       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afa89a90>"
+        "<matplotlib.figure.Figure at 0x7fa14b5fdf10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afa4f9d0>"
+        "<matplotlib.figure.Figure at 0x7fa14b6c0710>"
       ]
      }
     ],
-     "prompt_number": 29
+     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
@@ -2111,11 +2621,11 @@
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAA3wAAAF9CAYAAABf12quAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text": [
-        "<matplotlib.figure.Figure at 0x7fa3afb65c90>"
+        "<matplotlib.figure.Figure at 0x7fa14b662590>"
       ]
      }
     ],
-     "prompt_number": 30
+     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
@@ -1,7 +1,7 @@
 {
 "metadata": {
  "name": "",
-  "signature": "sha256:65ed87658245046683e9bc29217fbb80b0e626bf1af1d8084589c08a10b413cc"
+  "signature": "sha256:ce40207f3bc97f093a6d0e943ce3efb3506ade3873886bdf36d74e785611b3e0"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
@@ -44,7 +44,7 @@
     "language": "python",
     "metadata": {},
     "outputs": [],
-     "prompt_number": 1
+     "prompt_number": 20
    },
    {
     "cell_type": "code",
@@ -81,110 +81,110 @@
       "output_type": "stream",
       "stream": "stdout",
       "text": [
-        "0.829799081522 0.829799081522\n",
-        "1.11657328836 1.11657328836\n",
-        "2.60484331656 2.60484331656\n",
-        "4.49991898992 4.49991898992\n",
-        "4.96959734671 4.96959734671\n",
-        "5.62090599924 5.62090599924\n",
-        "6.62676551801 6.62676551801\n",
-        "7.17411383707 7.17411383707\n",
-        "8.22589645236 8.22589645236\n",
-        "9.68407847845 9.68407847845\n",
-        "10.7846369056 10.7846369056\n",
-        "10.9474107429 10.9474107429\n",
-        "13.1938820521 13.1938820521\n",
-        "14.5982855965 14.5982855965\n",
-        "16.0869457423 16.0869457423\n",
-        "16.9075718464 16.9075718464\n",
-        "17.4098040813 17.4098040813\n",
-        "19.0869215468 19.0869215468\n",
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