added example of sensor fusion

08_Designing_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
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  "metadata": {
   "name": "",
-  "signature": "sha256:f8507f66fbeb31e50cdbfe859d03d60ef6247e894bbbd167d2e36a4581276772"
+  "signature": "sha256:7dc575216f053cfbbaa767cc727f9855cc796801287b64d216790b6fad40b35c"
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  "nbformat": 3,
  "nbformat_minor": 0,
@@ -257,13 +257,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 57,
+       "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7fc98829bac8>"
+        "<IPython.core.display.HTML at 0x7fe17eacd780>"
        ]
       }
      ],
-     "prompt_number": 57
+     "prompt_number": 1
     },
     {
      "cell_type": "heading",
@@ -325,7 +325,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 58
+     "prompt_number": 2
     },
     {
      "cell_type": "markdown",
@@ -355,13 +355,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e9f6208>"
+        "<matplotlib.figure.Figure at 0x7fe165c531d0>"
        ]
       }
      ],
-     "prompt_number": 59
+     "prompt_number": 3
     },
     {
      "cell_type": "markdown",
@@ -440,7 +440,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 60
+     "prompt_number": 4
     },
     {
      "cell_type": "markdown",
@@ -459,7 +459,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 61
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
@@ -526,7 +526,7 @@
        ]
       }
      ],
-     "prompt_number": 62
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -561,7 +561,7 @@
        ]
       }
      ],
-     "prompt_number": 63
+     "prompt_number": 7
     },
     {
      "cell_type": "markdown",
@@ -596,7 +596,7 @@
        ]
       }
      ],
-     "prompt_number": 64
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -640,7 +640,7 @@
        ]
       }
      ],
-     "prompt_number": 65
+     "prompt_number": 9
     },
     {
      "cell_type": "heading",
@@ -709,13 +709,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e9f60f0>"
+        "<matplotlib.figure.Figure at 0x7fe165c53240>"
        ]
       }
      ],
-     "prompt_number": 66
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -790,9 +790,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97eb3c9e8>"
+        "<matplotlib.figure.Figure at 0x7fe17eacd6a0>"
        ]
       },
       {
@@ -807,7 +807,7 @@
        ]
       }
      ],
-     "prompt_number": 67
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -844,7 +844,7 @@
        ]
       }
      ],
-     "prompt_number": 68
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -877,11 +877,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc9882c1ac8>"
+        "<matplotlib.figure.Figure at 0x7fe15a230320>"
        ]
       }
      ],
-     "prompt_number": 69
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -1000,11 +1000,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987ca0898>"
+        "<matplotlib.figure.Figure at 0x7fe1659569e8>"
        ]
       }
      ],
-     "prompt_number": 70
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -1061,7 +1061,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 71
+     "prompt_number": 15
     },
     {
      "cell_type": "heading",
@@ -1137,7 +1137,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 72
+     "prompt_number": 16
     },
     {
      "cell_type": "heading",
@@ -1207,7 +1207,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 73
+     "prompt_number": 17
     },
     {
      "cell_type": "heading",
@@ -1250,7 +1250,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 74
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -1283,7 +1283,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 75
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1315,7 +1315,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 76
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1347,11 +1347,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987cdb400>"
+        "<matplotlib.figure.Figure at 0x7fe15a24e8d0>"
        ]
       }
      ],
-     "prompt_number": 77
+     "prompt_number": 21
     },
     {
      "cell_type": "markdown",
@@ -1377,7 +1377,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 78
+     "prompt_number": 22
     },
     {
      "cell_type": "code",
@@ -1395,11 +1395,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97ea09438>"
+        "<matplotlib.figure.Figure at 0x7fe15a230a58>"
        ]
       }
      ],
-     "prompt_number": 79
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1426,11 +1426,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e8c49b0>"
+        "<matplotlib.figure.Figure at 0x7fe165928198>"
        ]
       }
      ],
-     "prompt_number": 80
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1459,11 +1459,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987c6bc50>"
+        "<matplotlib.figure.Figure at 0x7fe166a61a58>"
        ]
       }
      ],
-     "prompt_number": 81
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -1490,11 +1490,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987f1a9b0>"
+        "<matplotlib.figure.Figure at 0x7fe165c48550>"
        ]
       }
      ],
-     "prompt_number": 82
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",
@@ -1529,11 +1529,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987ca0d30>"
+        "<matplotlib.figure.Figure at 0x7fe1589ed668>"
        ]
       }
      ],
-     "prompt_number": 83
+     "prompt_number": 27
     },
     {
      "cell_type": "markdown",
@@ -1568,11 +1568,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97ea41630>"
+        "<matplotlib.figure.Figure at 0x7fe1589a3390>"
        ]
       }
      ],
-     "prompt_number": 84
+     "prompt_number": 28
     },
     {
      "cell_type": "markdown",
@@ -1606,11 +1606,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97ea96eb8>"
+        "<matplotlib.figure.Figure at 0x7fe165a123c8>"
        ]
       }
      ],
-     "prompt_number": 85
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1643,11 +1643,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987cda780>"
+        "<matplotlib.figure.Figure at 0x7fe1589e08d0>"
        ]
       }
      ],
-     "prompt_number": 86
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -1681,7 +1681,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987cf9390>"
+        "<matplotlib.figure.Figure at 0x7fe1659095c0>"
        ]
       },
       {
@@ -1689,11 +1689,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97ea28ef0>"
+        "<matplotlib.figure.Figure at 0x7fe165a081d0>"
        ]
       }
      ],
-     "prompt_number": 87
+     "prompt_number": 31
     },
     {
      "cell_type": "markdown",
@@ -1729,7 +1729,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987f24320>"
+        "<matplotlib.figure.Figure at 0x7fe158975748>"
        ]
       },
       {
@@ -1737,11 +1737,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987f247f0>"
+        "<matplotlib.figure.Figure at 0x7fe1658ae940>"
        ]
       }
      ],
-     "prompt_number": 88
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -1770,11 +1770,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e952828>"
+        "<matplotlib.figure.Figure at 0x7fe165c316a0>"
        ]
       }
      ],
-     "prompt_number": 89
+     "prompt_number": 33
     },
     {
      "cell_type": "markdown",
@@ -1838,11 +1838,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97eb19400>"
+        "<matplotlib.figure.Figure at 0x7fe165c862b0>"
        ]
       }
      ],
-     "prompt_number": 90
+     "prompt_number": 34
     },
     {
      "cell_type": "markdown",
@@ -1871,7 +1871,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987d0e128>"
+        "<matplotlib.figure.Figure at 0x7fe158975080>"
        ]
       },
       {
@@ -1879,11 +1879,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987ff0dd8>"
+        "<matplotlib.figure.Figure at 0x7fe16595b048>"
        ]
       }
      ],
-     "prompt_number": 91
+     "prompt_number": 35
     },
     {
      "cell_type": "markdown",
@@ -1914,7 +1914,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e96f908>"
+        "<matplotlib.figure.Figure at 0x7fe165c7ada0>"
        ]
       },
       {
@@ -1922,11 +1922,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e96f7f0>"
+        "<matplotlib.figure.Figure at 0x7fe1589b7c50>"
        ]
       }
      ],
-     "prompt_number": 92
+     "prompt_number": 36
     },
     {
      "cell_type": "markdown",
@@ -1955,7 +1955,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e96fb00>"
+        "<matplotlib.figure.Figure at 0x7fe15899b9e8>"
        ]
       },
       {
@@ -1963,11 +1963,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987fe8e10>"
+        "<matplotlib.figure.Figure at 0x7fe15899b9b0>"
        ]
       }
      ],
-     "prompt_number": 93
+     "prompt_number": 37
     },
     {
      "cell_type": "heading",
@@ -2088,17 +2088,186 @@
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97eaa5f60>"
+        "<matplotlib.figure.Figure at 0x7fe1659e25f8>"
        ]
       }
      ],
-     "prompt_number": 94
+     "prompt_number": 38
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "We can see the result of the Kalman filter in blue."
+      "We can see the result of the Kalman filter in blue.\n",
+      "\n",
+      "It may be somewhat difficult to understand the previous example at an intuitive level. Let's look at a different problem. Suppose we are tracking an object in 2D space, and have two radar systems at different positions. Each radar system gives us a range and bearing to the target. How do the readings from each data affect the results?\n",
+      "\n",
+      "This is a nonlinear problem because we need to use a trigonometry  to compute coordinates from a range and bearing, and we have not yet learned how to solve nonlinear problems with Kalman filters. So for this problem ignore the code that I used and just concentrate on the charts that the code outputs. We will revisit this problem in subsequent chapters and learn how to write this code.\n",
+      "\n",
+      "I will position the target at (100,100). The first radar will be at (50,50), and the second radar at (150, 50). This will cause the first radar to measure a bearing of 45 degrees, and the second will report 135 dgrees.\n",
+      "\n",
+      "I will create the Kalman filter first, and then plot its initial covariance matrix. I am using what is called an Unscented Kalman filter, which is the subject of a later chapter."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from filterpy.kalman import  UnscentedKalmanFilter as UKF\n",
+      "\n",
+      "def hx(x):\n",
+      "    \"\"\" compute measurements corresponding to state x\"\"\"\n",
+      "    dx = x[0] - hx.radar_pos[0]\n",
+      "    dy = x[1] - hx.radar_pos[1]    \n",
+      "    return np.array([atan2(dy,dx), (dx**2 + dy**2)**.5])\n",
+      "    \n",
+      "def fx(x,dt):\n",
+      "    \"\"\" predict state x at 'dt' time in the future\"\"\"\n",
+      "    return x\n",
+      "\n",
+      "# create unscented Kalman filter with large initial uncertainty\n",
+      "kf = UKF(2, 2, dt=0.1, hx=hx, fx=fx, kappa=2.)\n",
+      "kf.x = np.array([100, 100.])\n",
+      "kf.P *= 40\n",
+      "\n",
+      "x0 = kf.x.copy()\n",
+      "p0 = kf.P.copy()\n",
+      "\n",
+      "stats.plot_covariance_ellipse(\n",
+      "       x0, cov=p0,\n",
+      "       facecolor='y', edgecolor=None, alpha=0.6)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fe165bfa470>"
+       ]
+      }
+     ],
+     "prompt_number": 39
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We are equally uncertain about the position in x and y, so the covariance is circular.\n",
+      "\n",
+      "Now we will update the Kalman filter with a reading from the first radar. I will set the standard deviation of the bearing error at 0.5$^\\circ$, and the standard deviation of the distance error at 3."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from math import radians, atan2\n",
+      "\n",
+      "# set the error of the radar's bearing and distance\n",
+      "kf.R[0,0] = radians (.5)**2\n",
+      "kf.R[1,1] = 3.**2\n",
+      "\n",
+      "# compute position and covariance from first radar station\n",
+      "hx.radar_pos = (50, 50)\n",
+      "dist = (50**2 + 50**2) ** 0.5\n",
+      "kf.predict()\n",
+      "kf.update([radians(45), dist])\n",
+      "\n",
+      "\n",
+      "# plot the results\n",
+      "x1 = kf.x.copy()\n",
+      "p1 = kf.P.copy()\n",
+      "\n",
+      "stats.plot_covariance_ellipse(\n",
+      "       x0, cov=p0,\n",
+      "       facecolor='y', edgecolor=None, alpha=0.6)\n",
+      "\n",
+      "stats.plot_covariance_ellipse(\n",
+      "       x1, cov=p1,\n",
+      "       facecolor='g', edgecolor='k', alpha=0.6)\n",
+      "\n",
+      "plt.scatter([100], [100], c='y', label='Initial')\n",
+      "plt.scatter([100], [100], c='g', label='1st station')\n",
+      "plt.legend(scatterpoints=1, markerscale=3)\n",
+      "plt.plot([92,100],[92,100], c='g', lw=2, ls='--')\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fe16593f5c0>"
+       ]
+      }
+     ],
+     "prompt_number": 40
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see the effect of the errors on the geometry of the problem. The radar station is to the lower left of the target. The bearing measurement is extremely accurate at $\\sigma=0.5^\\circ$, but the distance error is inaccurate at $\\sigma=3$. I've shown the radar reading with the dotted green line. We can easily see the effect of the accurate bearing and inaccurate distance in the shape of the covariance ellipse. \n",
+      "\n",
+      "Now we can incorporate the second radar station's measurement. The second radar is at (150,50), which is below and to the right of the target. Before you go on, think about how you think the covariance will change when we incorporate this new reading."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "# compute position and covariance from first radar station\n",
+      "hx.radar_pos = (150, 50)\n",
+      "kf.predict()\n",
+      "kf.update([radians(135), dist])\n",
+      "\n",
+      "stats.plot_covariance_ellipse(\n",
+      "       x0, cov=p0,\n",
+      "       facecolor='y', edgecolor='k', alpha=0.6)\n",
+      "\n",
+      "stats.plot_covariance_ellipse(\n",
+      "       x1, cov=p1,\n",
+      "       facecolor='g', edgecolor='k', alpha=0.6)\n",
+      "\n",
+      "stats.plot_covariance_ellipse(\n",
+      "       kf.x, cov=kf.P,\n",
+      "       facecolor='b', edgecolor='k', alpha=0.6)\n",
+      "\n",
+      "plt.scatter([100], [100], c='y', label='Initial')\n",
+      "plt.scatter([100], [100], c='g', label='1st station')\n",
+      "plt.scatter([100], [100], c='b', label='2nd station')\n",
+      "plt.legend(scatterpoints=1, markerscale=3)\n",
+      "plt.plot([92,100],[92,100], c='g', lw=2, ls='--')\n",
+      "plt.plot([108,100],[92,100], c='b', lw=2, ls='--')\n",
+      "plt.show()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7fe1588fd160>"
+       ]
+      }
+     ],
+     "prompt_number": 41
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see how the second radar measurement altered the covariance. The angle to the target is orthogonal to the first radar station, so the effects of the error in the bearing and range are swapped. So the angle of the covariance matrix switches to match the direction to the second station. It is important to note that the direction did not mearly change; the size of the covariance matrix became much smaller as well.\n",
+      "\n",
+      "The covariance will always incorporate all of the information available, including the effects of the geometry of the problem. This formulation makes it particularly easy to see what is happening, but the same thing occurs if one sensor gives you position and a second sensor gives you velocity, or if two sensors provide measurements of position."
      ]
     },
     {
@@ -2125,7 +2294,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 95
+     "prompt_number": 42
     },
     {
      "cell_type": "heading",
@@ -2200,7 +2369,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987fb8748>"
+        "<matplotlib.figure.Figure at 0x7fe158941240>"
        ]
       },
       {
@@ -2208,7 +2377,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987d71860>"
+        "<matplotlib.figure.Figure at 0x7fe15897cc88>"
        ]
       },
       {
@@ -2216,11 +2385,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e8c0eb8>"
+        "<matplotlib.figure.Figure at 0x7fe15890ec50>"
        ]
       }
      ],
-     "prompt_number": 96
+     "prompt_number": 43
     },
     {
      "cell_type": "markdown",
@@ -2247,7 +2416,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97ea06860>"
+        "<matplotlib.figure.Figure at 0x7fe15886cd30>"
        ]
       },
       {
@@ -2255,7 +2424,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97ea06978>"
+        "<matplotlib.figure.Figure at 0x7fe15886ce10>"
        ]
       },
       {
@@ -2263,11 +2432,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e98e128>"
+        "<matplotlib.figure.Figure at 0x7fe1589ba320>"
        ]
       }
      ],
-     "prompt_number": 97
+     "prompt_number": 44
     },
     {
      "cell_type": "markdown",
@@ -2314,7 +2483,7 @@
        ]
       }
      ],
-     "prompt_number": 98
+     "prompt_number": 45
     },
     {
      "cell_type": "markdown",
@@ -2394,11 +2563,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e86a6a0>"
+        "<matplotlib.figure.Figure at 0x7fe165c2ef98>"
        ]
       }
      ],
-     "prompt_number": 99
+     "prompt_number": 46
     },
     {
      "cell_type": "markdown",
@@ -2433,7 +2602,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 100
+     "prompt_number": 47
     },
     {
      "cell_type": "heading",
@@ -2558,11 +2727,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e9d49b0>"
+        "<matplotlib.figure.Figure at 0x7fe1589accf8>"
        ]
       }
      ],
-     "prompt_number": 101
+     "prompt_number": 48
     },
     {
      "cell_type": "markdown",
@@ -2669,7 +2838,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 102
+     "prompt_number": 49
     },
     {
      "cell_type": "markdown",
@@ -2714,7 +2883,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987f62518>"
+        "<matplotlib.figure.Figure at 0x7fe15898beb8>"
        ]
       },
       {
@@ -2722,11 +2891,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987f626d8>"
+        "<matplotlib.figure.Figure at 0x7fe15898b2b0>"
        ]
       }
      ],
-     "prompt_number": 103
+     "prompt_number": 50
     },
     {
      "cell_type": "markdown",
@@ -2863,7 +3032,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 104
+     "prompt_number": 51
     },
     {
      "cell_type": "markdown",
@@ -2907,11 +3076,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987d3c438>"
+        "<matplotlib.figure.Figure at 0x7fe158926f60>"
        ]
       }
      ],
-     "prompt_number": 105
+     "prompt_number": 52
     },
     {
      "cell_type": "markdown",
@@ -3052,7 +3221,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 106
+     "prompt_number": 53
     },
     {
      "cell_type": "markdown",
@@ -3103,11 +3272,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e864898>"
+        "<matplotlib.figure.Figure at 0x7fe1658a5780>"
        ]
       }
      ],
-     "prompt_number": 107
+     "prompt_number": 54
     },
     {
      "cell_type": "markdown",
@@ -3246,11 +3415,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e82d4a8>"
+        "<matplotlib.figure.Figure at 0x7fe165c0a6d8>"
        ]
       }
      ],
-     "prompt_number": 108
+     "prompt_number": 55
     },
     {
      "cell_type": "markdown",
@@ -3324,7 +3493,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 109
+     "prompt_number": 56
     },
     {
      "cell_type": "markdown",
@@ -3384,11 +3553,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e952f28>"
+        "<matplotlib.figure.Figure at 0x7fe16595c0f0>"
        ]
       }
      ],
-     "prompt_number": 110
+     "prompt_number": 57
     },
     {
      "cell_type": "markdown",
@@ -3452,7 +3621,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97eb509b0>"
+        "<matplotlib.figure.Figure at 0x7fe165c830f0>"
        ]
       },
       {
@@ -3460,11 +3629,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987cc8a90>"
+        "<matplotlib.figure.Figure at 0x7fe165be2438>"
        ]
       }
      ],
-     "prompt_number": 111
+     "prompt_number": 58
     },
     {
      "cell_type": "markdown",
@@ -3490,7 +3659,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc987c67128>"
+        "<matplotlib.figure.Figure at 0x7fe165c220b8>"
        ]
       },
       {
@@ -3498,11 +3667,11 @@
        "output_type": "display_data",
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fc97e9772e8>"
+        "<matplotlib.figure.Figure at 0x7fe1587be940>"
        ]
       }
      ],
-     "prompt_number": 112
+     "prompt_number": 59
     },
     {
      "cell_type": "markdown",