daviddoji/Kalman-and-Bayesian-Filters-in-Python
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Altered PDF generation.

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Added code to inject \appendix to the first appendix notebook
so that they are properly labelled. Started g-h chapter as chapter
one. There is probably a lot more I want to do now that I have
the basic idea of how to inject latex into the files before creating
the PDF, but this is a good start.
This commit is contained in:
Roger Labbe committed 2015-01-17 09:39:26 -08:00
1 parent dbcb166150
commit afe0b5626e
11 files changed
+278 -115
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05_Kalman_Filters/Kalman_Filters.ipynb
+173 -94
View File
@@ -1,7 +1,7 @@
{
"metadata": {
"name": "",
"signature": "sha256:2ead90ce1b4a67e39b677b6425cac07822bea68fe52acd7383dfb277392ba6ec"
"signature": "sha256:b01859212b570d5ecf1bb56f51a3f9aad5c617170e84aa4e71d3cb44fae8051f"
},
"nbformat": 3,
"nbformat_minor": 0,
@@ -259,7 +259,7 @@
"output_type": "pyout",
"prompt_number": 1,
"text": [
"<IPython.core.display.HTML at 0x7f29a9f40780>"
"<IPython.core.display.HTML at 0x7fc68143f748>"
]
}
],
@@ -296,7 +296,7 @@
"\n",
"Naturally, the sensor is not perfect. A reading of 23.4 could correspond to a real position of 23.7, or 23.0. However, it is very unlikely to correspond to a real position of say 47.6. Testing during the hackathon confirmed this result - the sensor is reasonably accurate, and while it had errors, the errors are small. Furthermore, the errors seemed to be evenly distributed on both sides of the measurement; a true position of 23m would be equally likely to be measured as 22.9 as 23.1.\n",
"\n",
"Implementing and/or robustly modeling an RFID system is beyond the scope of this book, so we will write a very simple model. We will start with a simulation of the dog moving from left to right at a constant speed with some random noise added. "
"Implementing and/or robustly modeling an RFID system is beyond the scope of this book, so we will write a very simple model. We will start with a simulation of the dog moving from left to right at a constant speed with some random noise added. We will talk about this in great detail later, but we need to model two kinds of noise. The *process noise* is the noise in the physical process. Something moving at a notionally 'constant' velocity will never maintain a perfectly constant velocity. Undulations on the ground, wind, and a host of other factors mean that there will always be slight variations in the velocity. The second noise we want to model is the noise in the measurement as no measurement is perfect."
]
},
{
@@ -310,29 +310,37 @@
"\n",
"class DogSensor(object):\n",
" \n",
" def __init__(self, x0=0, velocity=1, variance=0.0):\n",
" def __init__(self, x0=0, velocity=1, \n",
" measurement_variance=0.0, process_variance=0.0):\n",
" \"\"\" x0 - initial position\n",
" velocity - (+=right, -=left)\n",
" variance - variance in measurement\n",
" measurement_variance - variance in measurement\n",
" process_variance - variance in process (m/s)^2\n",
" \"\"\"\n",
" self.x = x0\n",
" self.velocity = velocity\n",
" self.noise = math.sqrt(variance)\n",
" self.noise = math.sqrt(measurement_variance)\n",
" self.pnoise = math.sqrt(process_variance)\n",
" self.constant_vel = velocity\n",
"\n",
" def sense_position(self):\n",
" pnoise = abs(random.rand() * self.pnoise)\n",
" if self.velocity > self.constant_vel:\n",
" pnoise = -pnoise\n",
" self.velocity += pnoise\n",
" self.x = self.x + self.velocity\n",
" return self.x + random.randn() * self.noise"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 2
"prompt_number": 26
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The constructor `_init()__` initializes the DogSensor class with an initial position `x0`, velocity `vel`, and an noise scaling factor. The `sense_position()` function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for `sense_position()` you will see a call to `numpy.random.randn()`. This returns a number sampled from a normal distribution with a mean of 0.0. and a standard deviation of 1.0. *Variance* is defined as the standard deviation squared, so in `__init()__` we take the square root of the variance to get the standard deviation. Therefore the expression `self.x + random.randn() * self.noise` computes a number with the variance that we desire. \n",
"The constructor `_init()__` initializes the DogSensor class with an initial position `x0`, velocity `vel`, and the variance in the measurement and noise. The `sense_position()` function has the dog move by the set velocity and returns its new position, with noise added. If you look at the code for `sense_position()` you will see a call to `numpy.random.randn()`. This returns a number sampled from a normal distribution with a mean of 0.0. and a standard deviation of 1.0. *Variance* is defined as the standard deviation squared, so in `__init()__` we take the square root of the variances to get the standard deviation. Therefore the expression `self.x + random.randn() * self.noise` computes a simulated measurement with the variance that we desire. \n",
"\n",
"Let's look at some example output for that."
]
@@ -353,10 +361,10 @@
"output_type": "stream",
"stream": "stdout",
"text": [
" 0.5332\t-0.6438\t 1.0778\t-1.5408\t 0.7009\t\n",
"-0.1204\t-1.4358\t 1.0115\t-0.6912\t 0.5493\t\n",
" 0.7769\t 0.9842\t 0.8198\t 1.0959\t 0.7743\t\n",
"-1.2463\t 1.3107\t-0.8946\t 0.0512\t-1.9209\t\n"
" 1.6843\t 1.2912\t 1.1466\t 0.4007\t 0.4046\t\n",
"-1.3402\t 1.7754\t-0.7792\t-0.5521\t 0.0399\t\n",
" 0.4786\t-0.1534\t 0.2983\t-0.7419\t-3.0835\t\n",
"-0.3109\t 0.9006\t 1.8374\t 0.3517\t-0.2178\t\n"
]
}
],
@@ -378,7 +386,7 @@
"import matplotlib.pyplot as plt\n",
"import matplotlib.pylab as pylab\n",
"\n",
"dog = DogSensor(variance=0.0)\n",
"dog = DogSensor(measurement_variance=0.0)\n",
"xs = []\n",
"for i in range(10):\n",
" x = dog.sense_position()\n",
@@ -403,11 +411,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298d0a8ac8>"
"<matplotlib.figure.Figure at 0x7fc653369f28>"
]
}
],
"prompt_number": 4
"prompt_number": 27
},
{
"cell_type": "markdown",
@@ -422,8 +430,9 @@
"cell_type": "code",
"collapsed": false,
"input": [
"def test_sensor(variance):\n",
" dog = DogSensor(variance=variance)\n",
"def test_sensor(measurement_var, process_var=0.0):\n",
" dog = DogSensor(measurement_variance=measurement_var, \n",
" process_variance=process_var)\n",
"\n",
" xs = []\n",
" for i in range(100):\n",
@@ -431,15 +440,16 @@
" xs.append(x)\n",
"\n",
" plt.plot(xs, label='sensor')\n",
" plt.plot([0, 99], [1, 100], 'r--', label='actual')\n",
" plt.plot([0, 99], [1, 100], 'r--', label='ideal')\n",
" plt.xlabel('time')\n",
" plt.ylabel('pos')\n",
" plt.ylim([0,100])\n",
" plt.title('variance = ' + str(variance))\n",
" plt.title('variance = {}, process variance = {}'.format(\n",
" measurement_var, process_var))\n",
" plt.legend(loc='best')\n",
" plt.show()\n",
" \n",
"test_sensor(variance=4.0)"
"test_sensor(measurement_var=4.0)"
],
"language": "python",
"metadata": {},
@@ -447,13 +457,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298d05aa58>"
"<matplotlib.figure.Figure at 0x7fc6532d1b00>"
]
}
],
"prompt_number": 5
"prompt_number": 51
},
{
"cell_type": "markdown",
@@ -466,14 +476,14 @@
"\n",
"If you are running this in an interactive IPython Notebook, I strongly urge you to run the script several times in a row. You can do this by putting the cursor in the cell containing the Python code and pressing Ctrl+Enter. Each time it runs you should see a different jagged blue line wavering over the top of the dotted red line.\n",
"\n",
"I also urge you to adjust the noise setting to see the result of various values. However, since you may be reading this in a read only notebook, I will show two extreme examples. The first plot shows the noise set to 100.0, and the second shows noise set to 0.5."
"I also urge you to adjust the noise setting to see the result of various values. However, since you may be reading this in a read only notebook, I will show several examples. The first plot shows the noise set to 100.0, and the second shows noise set to 0.5."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"test_sensor(variance=100.0)"
"test_sensor(measurement_var=100.0)"
],
"language": "python",
"metadata": {},
@@ -481,19 +491,19 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298d0a84a8>"
"<matplotlib.figure.Figure at 0x7fc65347bd68>"
]
}
],
"prompt_number": 6
"prompt_number": 52
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"test_sensor(variance=0.5)"
"test_sensor(measurement_var=0.5)"
],
"language": "python",
"metadata": {},
@@ -501,13 +511,73 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298cfcaeb8>"
"<matplotlib.figure.Figure at 0x7fc65339a278>"
]
}
],
"prompt_number": 7
"prompt_number": 62
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now lets see the effect of the process noise. My simulation is not meant to exactly model any given physical process. On each call to `sense_position()` I modify the current velocity by randomly generated process noise. However, I strive to keep the velocity close to the initial value, as I am assuming that there is a control mechanism in place trying to maintain the same speed. In this case, the control mechanism would be the dog's brain! For an automobile it could be a cruise control or the human driver.\n",
"\n",
"So let's first look at the plot with some process noise but no measurement noise to obscure the results."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import numpy as np\n",
"np.random.seed(1234)\n",
"test_sensor(measurement_var=0, process_var=0.5)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7fc66445a5c0>"
]
}
],
"prompt_number": 65
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can see that the position wanders slightly from the ideal track. You may have thought that the track would have wandered back and forth the ideal path like the measurement noise did, but recall that we are modifying velocity on each, not the position. So once the track has deviated, it will stay there until the random changes in velocity happen to result in the track going back to the original track. \n",
"\n",
"Finally, let's look at the combination of measurement noise and process noise."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"test_sensor(measurement_var=1, process_var=0.5)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7fc664424fd0>"
]
}
],
"prompt_number": 66
},
{
"cell_type": "heading",
@@ -539,11 +609,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298cf08e80>"
"<matplotlib.figure.Figure at 0x7fc66446af28>"
]
}
],
"prompt_number": 8
"prompt_number": 12
},
{
"cell_type": "markdown",
@@ -560,7 +630,8 @@
"cell_type": "code",
"collapsed": false,
"input": [
"dog = DogSensor(x0=23, velocity=0, variance=5)\n",
"dog = DogSensor(x0=23, velocity=0, \n",
" measurement_variance=5, process_variance=0.0)\n",
"xs = range(100)\n",
"ys = []\n",
"for i in xs:\n",
@@ -576,13 +647,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298d0a8ba8>"
"<matplotlib.figure.Figure at 0x7fc6533af358>"
]
}
],
"prompt_number": 9
"prompt_number": 69
},
{
"cell_type": "markdown",
@@ -681,11 +752,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f29841508d0>"
"<matplotlib.figure.Figure at 0x7fc653471b00>"
]
}
],
"prompt_number": 10
"prompt_number": 72
},
{
"cell_type": "markdown",
@@ -854,7 +925,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 13
"prompt_number": 71
},
{
"cell_type": "markdown",
@@ -879,7 +950,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"dog = DogSensor(velocity=0., variance=1.)\n",
"dog = DogSensor(velocity=0., measurement_variance=5, process_variance=0.0)\n",
"\n",
"pos,s = 2, 5\n",
"for i in range(20):\n",
@@ -895,30 +966,30 @@
"output_type": "stream",
"stream": "stdout",
"text": [
"time: 0 \tposition = 1.071 \tvariance = 2.500\n",
"time: 1 \tposition = 1.317 \tvariance = 1.667\n",
"time: 2 \tposition = 0.980 \tvariance = 1.250\n",
"time: 3 \tposition = 1.014 \tvariance = 1.000\n",
"time: 4 \tposition = 0.917 \tvariance = 0.833\n",
"time: 5 \tposition = 0.967 \tvariance = 0.714\n",
"time: 6 \tposition = 0.887 \tvariance = 0.625\n",
"time: 7 \tposition = 0.866 \tvariance = 0.556\n",
"time: 8 \tposition = 0.791 \tvariance = 0.500\n",
"time: 9 \tposition = 0.737 \tvariance = 0.455\n",
"time: 10 \tposition = 0.544 \tvariance = 0.417\n",
"time: 11 \tposition = 0.555 \tvariance = 0.385\n",
"time: 12 \tposition = 0.512 \tvariance = 0.357\n",
"time: 13 \tposition = 0.473 \tvariance = 0.333\n",
"time: 14 \tposition = 0.467 \tvariance = 0.312\n",
"time: 15 \tposition = 0.428 \tvariance = 0.294\n",
"time: 16 \tposition = 0.343 \tvariance = 0.278\n",
"time: 17 \tposition = 0.286 \tvariance = 0.263\n",
"time: 18 \tposition = 0.242 \tvariance = 0.250\n",
"time: 19 \tposition = 0.242 \tvariance = 0.238\n"
"time: 0 \tposition = -0.009 \tvariance = 2.500\n",
"time: 1 \tposition = 0.837 \tvariance = 1.667\n",
"time: 2 \tposition = -0.287 \tvariance = 1.250\n",
"time: 3 \tposition = 0.048 \tvariance = 1.000\n",
"time: 4 \tposition = -0.004 \tvariance = 0.833\n",
"time: 5 \tposition = 0.302 \tvariance = 0.714\n",
"time: 6 \tposition = -0.011 \tvariance = 0.625\n",
"time: 7 \tposition = -0.244 \tvariance = 0.556\n",
"time: 8 \tposition = -0.323 \tvariance = 0.500\n",
"time: 9 \tposition = -0.304 \tvariance = 0.455\n",
"time: 10 \tposition = -0.155 \tvariance = 0.417\n",
"time: 11 \tposition = -0.235 \tvariance = 0.385\n",
"time: 12 \tposition = -0.007 \tvariance = 0.357\n",
"time: 13 \tposition = 0.041 \tvariance = 0.333\n",
"time: 14 \tposition = 0.171 \tvariance = 0.312\n",
"time: 15 \tposition = -0.007 \tvariance = 0.294\n",
"time: 16 \tposition = -0.072 \tvariance = 0.278\n",
"time: 17 \tposition = -0.007 \tvariance = 0.263\n",
"time: 18 \tposition = 0.272 \tvariance = 0.250\n",
"time: 19 \tposition = 0.159 \tvariance = 0.238\n"
]
}
],
"prompt_number": 14
"prompt_number": 73
},
{
"cell_type": "markdown",
@@ -985,7 +1056,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 15
"prompt_number": 74
},
{
"cell_type": "markdown",
@@ -1004,7 +1075,9 @@
"sensor_variance = 10\n",
"pos = (0, 500) # gaussian N(0,500)\n",
"\n",
"dog = DogSensor(pos[0], velocity=movement, variance=sensor_variance)\n",
"dog = DogSensor(pos[0], velocity=movement, \n",
" measurement_variance=sensor_variance, \n",
" process_variance=sensor_variance)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1034,28 +1107,28 @@
"output_type": "stream",
"stream": "stdout",
"text": [
"PREDICT: 1.0000 502.0000\tUPDATE: 5.9904 9.8047\n",
"PREDICT: 6.9904 11.8047\tUPDATE: 2.7834 5.4138\n",
"PREDICT: 3.7834 7.4138\tUPDATE: 4.0761 4.2574\n",
"PREDICT: 5.0761 6.2574\tUPDATE: 5.6457 3.8490\n",
"PREDICT: 6.6457 5.8490\tUPDATE: 4.4591 3.6904\n",
"PREDICT: 5.4591 5.6904\tUPDATE: 7.7529 3.6267\n",
"PREDICT: 8.7529 5.6267\tUPDATE: 9.4817 3.6007\n",
"PREDICT: 10.4817 5.6007\tUPDATE: 9.3966 3.5900\n",
"PREDICT: 10.3966 5.5900\tUPDATE: 10.0462 3.5856\n",
"PREDICT: 11.0462 5.5856\tUPDATE: 13.3613 3.5838\n"
"PREDICT: 1.0000 502.0000\tUPDATE: 2.1518 9.8047\n",
"PREDICT: 3.1518 11.8047\tUPDATE: 5.0312 5.4138\n",
"PREDICT: 6.0312 7.4138\tUPDATE: 5.7636 4.2574\n",
"PREDICT: 6.7636 6.2574\tUPDATE: 5.5107 3.8490\n",
"PREDICT: 6.5107 5.8490\tUPDATE: 5.0846 3.6904\n",
"PREDICT: 6.0846 5.6904\tUPDATE: 6.8312 3.6267\n",
"PREDICT: 7.8312 5.6267\tUPDATE: 8.6453 3.6007\n",
"PREDICT: 9.6453 5.6007\tUPDATE: 10.8448 3.5900\n",
"PREDICT: 11.8448 5.5900\tUPDATE: 13.2680 3.5856\n",
"PREDICT: 14.2680 5.5856\tUPDATE: 12.3199 3.5838\n"
]
},
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f297bed0128>"
"<matplotlib.figure.Figure at 0x7fc66450cf28>"
]
}
],
"prompt_number": 16
"prompt_number": 76
},
{
"cell_type": "markdown",
@@ -1085,7 +1158,9 @@
"source": [
"Next we initialize the RFID simulator with\n",
"\n",
" dog = DogSensor(pos[0], velocity=movement, noise=sensor_variance)\n",
" dog = DogSensor(pos[0], velocity=movement, \n",
" measurement_variance=sensor_variance, \n",
" process_variance=sensor_variance)\n",
"\n",
"It may seem very 'convenient' to set the simulator to the same position as our guess, and it is. Do not fret. In the next example we will see the effect of a wildly inaccurate guess for the dog's initial position.\n",
"\n",
@@ -1155,7 +1230,9 @@
"sensor_variance = 4.5\n",
"pos = (0, 100) # gaussian N(0,100)\n",
"\n",
"dog = DogSensor(pos[0], velocity=movement, variance=sensor_variance)\n",
"dog = DogSensor(pos[0], velocity=movement, \n",
" measurement_variance=sensor_variance, \n",
" process_variance=0.5)\n",
"\n",
"zs, ps, vs = [], [], []\n",
"for i in range(50):\n",
@@ -1168,7 +1245,7 @@
" ps.append(pos[0])\n",
" \n",
"#plt.subplot(121) \n",
"plt.plot(zs,c='r', linestyle='dashed', label='measurement')\n",
"plt.plot(zs, linestyle='dashed', label='measurement')\n",
"plt.plot(ps, label='filter')\n",
"plt.legend(loc=2)\n",
"plt.show()\n",
@@ -1186,9 +1263,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f297bec9438>"
"<matplotlib.figure.Figure at 0x7fc6645c4da0>"
]
},
{
@@ -1196,7 +1273,7 @@
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f298cff7048>"
"<matplotlib.figure.Figure at 0x7fc653439438>"
]
},
{
@@ -1216,7 +1293,7 @@
]
}
],
"prompt_number": 17
"prompt_number": 92
},
{
"cell_type": "markdown",
@@ -1283,7 +1360,7 @@
" self.measurement_variance = measurement_variance\n",
" self.movement_variance = movement_variance\n",
"\n",
"That works, but I am going to use different naming. The Kalman filter literature has settled on one letter notations for each of these concepts, and so you might as well start getting exposed to it now. At first it seems impossiblely terse, but as you become familiar with the nomenclature you'll see that the math formulas in the textbooks will have an exact one-to-one correspondance with the code. Unfortunately there is not a lot of meaning behind the names chosen; you will just have to memorize them.\n",
"That works, but I am going to use different naming. The Kalman filter literature has settled on one letter notations for each of these concepts, and so you might as well start getting exposed to it now. At first it seems impossiblely terse, but as you become familiar with the nomenclature you'll see that the math formulas in the textbooks will have an exact one-to-one correspondance with the code. Unfortunately there is not a lot of meaning behind the names chosen; you will just have to memorize them. If you do not make this effort you will never be able to read the Kalman filter literature.\n",
"\n",
"So, we use `x` for the state (estimated value of the filter) and `P` for the variance of the state. `R` is the measurement error, and `Q` is the movement error. This gives us:\n",
"\n",
@@ -1331,7 +1408,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 18
"prompt_number": 89
},
{
"cell_type": "heading",
@@ -1464,7 +1541,7 @@
"\n",
"> For example, I just randomly looked up a data sheet for an airflow sensor. There is a field *Repeatability*, with the value $\\pm 0.50\\%$. Is this a Gaussian? Is there a bias? For example, perhaps the repeatability is nearly 0.0% at low temperatures, and always nearly +0.50 at high temperatures. Data sheets for electrical components often contain a section of \"Typical Performance Characteristics\". These are used to capture information that cannot be easily conveyed in a table. For example, I am looking at a chart showing output voltage vs current for a LM555 timer. There are three curves showing the performance at different temperatures. The response is ideally linear, but all three lines are curved. This clarifies that errors in voltage outputs are probably not Gaussian - in this chip's case higher temperatures leads to lower voltage output, and the voltage output is quite nonlinear if the input current is very high. \n",
"\n",
"> As you might guess, modeling the performance of your sensors is one of the harder parts of creating good Kalman filter. "
"> As you might guess, modeling the performance of your sensors is one of the harder parts of creating a Kalman filter that performs well. "
]
},
{
@@ -1698,7 +1775,9 @@
"movement_sensor = 30\n",
"pos = (0,500)\n",
"\n",
"dog = DogSensor(0, velocity=movement, variance=sensor_variance)\n",
"dog = DogSensor(0, velocity=movement, \n",
" measurement_variance=sensor_variance,\n",
" process_variance=0.5)\n",
"\n",
"zs = []\n",
"ps = []\n",
@@ -1728,21 +1807,21 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f297bf18978>"
"<matplotlib.figure.Figure at 0x7fc6533816d8>"
]
},
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f297be162e8>"
"<matplotlib.figure.Figure at 0x7fc653418cf8>"
]
}
],
"prompt_number": 23
"prompt_number": 91
},
{
"cell_type": "markdown",