Interim check in.
dog_track_1d is just a file to test code before putting it into the Kalman filter chapter.
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Roger Labbe committed
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@@ -50,7 +50,7 @@
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"language": "python",
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"metadata": {},
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"outputs": [],
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"prompt_number": 106
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"prompt_number": 1
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},
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{
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"cell_type": "markdown",
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@@ -74,11 +74,11 @@
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"1.2396 0.5610 -0.5456 -0.8361 -0.8521 0.4101 -0.4417 0.0910 -0.4968 0.5187 -1.9393 -0.3410 0.0943 1.1800 0.1204 -1.0427 1.2128 -2.2642 -1.7840 -0.0621\n"
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]
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}
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],
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"prompt_number": 107
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"prompt_number": 2
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},
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{
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"cell_type": "markdown",
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@@ -117,7 +117,7 @@
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"output_type": "display_data",
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truncated
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"text": [
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"<matplotlib.figure.Figure at 0x30da390>"
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"<matplotlib.figure.Figure at 0x3138810>"
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]
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},
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{
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@@ -128,7 +128,7 @@
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]
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}
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],
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"prompt_number": 108
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"prompt_number": 3
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},
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{
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"cell_type": "markdown",
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@@ -167,13 +167,13 @@
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{
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"metadata": {},
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"output_type": "display_data",
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"png": 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||||
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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x37d5dd0>"
|
||||
"<matplotlib.figure.Figure at 0x31546d0>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 124
|
||||
"prompt_number": 4
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -200,13 +200,13 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x37c1850>"
|
||||
"<matplotlib.figure.Figure at 0x314ca10>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 123
|
||||
"prompt_number": 5
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
@@ -220,19 +220,135 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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|
||||
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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x36a4090>"
|
||||
"<matplotlib.figure.Figure at 0x316c190>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 113
|
||||
"prompt_number": 6
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"You may not have a full understanding of the exact *meaning* of a noise value of 100.0, but as it turns out if you multiply *randn()* with a number $n$, the result is just a normal distribution with $\\sigma = \\sqrt{n}$. So the example with noise = 100 is using the normal distribution $N(0,100)$. If the square root is confusing, recall that normal distributions use $\\sigma^2$. *dog_sensor().__init__* takes the square root of the noise setting so that the noise * randn() call properly computes the normal distribution."
|
||||
"You may not have a full understanding of the exact *meaning* of a noise value of 100.0, but as it turns out if you multiply *randn()* with a number $n$, the result is just a normal distribution with $\\sigma = \\sqrt{n}$. So the example with noise = 100 is using the normal distribution $N(0,100)$. Recall the notation for a normal distribution is $N(\\mu,\\sigma^2)$. If the square root is confusing, recall that normal distributions use $\\sigma^2$ for the variance, and $\\sigma$ is the standard deviation, which we do not use in this book. *dog_sensor.__init__()* takes the square root of the noise setting so that the *noise * randn()* call properly computes the normal distribution. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"#### Math with Gaussians\n",
|
||||
"\n",
|
||||
"Let's say we believe that our dog is at 23m, and the variance is 5 ($N(23,5)$). We can represent that in a plot:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"import gaussian\n",
|
||||
"gaussian.norm_plot (23, 5)"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x315c150>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 9
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This corresponds to a fairly inexact belief. While we believe that the dog is at 23, note that roughly 21 to 25 are quite likely as well. Let's assume for the moment our dog is standing still, and we query the sensor again. This time it returns 23.2 as the position. Can we use this additional information to improve our estimate of the dog's position.\n",
|
||||
"\n",
|
||||
"Intuition suggests 'yes'. Consider: if we read the sensor 100 times and each time it returned a value between 21 and 25, all centered around 23, we should be very confident that the dog is somewhere very near 23. Of course, a different physical interpertation is possible. Perhaps our dog was randomly wandering back and forth in a way that exactly emulated a normal distribution. But that seems extremely unlikely - I certainly have never seen a dog do that. So the only reasonable assumption is that the dog was mostly standing still at 23.0.\n",
|
||||
"\n",
|
||||
"Let's look at this in a plot:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"dog = dog_sensor (23, 0, 5)\n",
|
||||
"ys = range (100)\n",
|
||||
"xs = []\n",
|
||||
"for i in ys:\n",
|
||||
" xs.append (dog.sense())\n",
|
||||
" \n",
|
||||
"plot (xs,ys)\n",
|
||||
"show()"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x37cb050>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 18
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Eyeballing this confirms our intuition - no dog moves like this. However, noisy sensor data certainly looks like this. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"#### Implementing Sensing\n",
|
||||
"\n",
|
||||
"Recall the histogram filter uses a numpy array to encode our belief about the position of our dog at any time. That array stored our belief that the dog was in any position in the hallway using 10 positions. This was very crude, because with a 100m hallway that corresponded to positions 10m apart. It would have been trivial to expand the number of positions to say 1,000, and that is what we would do if using it for a real problem. But the problem remains that the distribution is discrete and multimodal - it can express strong belief that the dog is in two positions at the same time.\n",
|
||||
"\n",
|
||||
"Therefore, let us use a single gaussian to reflect our current belief of the dog's position. Gaussians extend to infinity on both sides of the mean, so the single gaussian will cover the entire hallway. They are unimodal, and seem to reflect the behavior of real-world sensors - most errors are small and clustered around the mean. \n",
|
||||
"\n",
|
||||
"So let us implement the sensing function, but using gaussians instead of the histogram array. First, here is the histogram code for reference:\n",
|
||||
"\n",
|
||||
" def sense (pos, measure, p_hit, p_miss):\n",
|
||||
" q = array(pos, dtype=float)\n",
|
||||
" for i in range(len(hallway)):\n",
|
||||
" if hallway[i] == measure:\n",
|
||||
" q[i] = pos[i] * p_hit\n",
|
||||
" else:\n",
|
||||
" q[i] = pos[i] * p_miss\n",
|
||||
" normalize(q)\n",
|
||||
" return q\n",
|
||||
" \n",
|
||||
"Note the algorithm is essentially computing:\n",
|
||||
"\n",
|
||||
" new_belief = old_belief * measurement * sensor_error\n",
|
||||
" \n",
|
||||
"The measurement term might not be clear, but recall that measurement in this case was always 1 or 0, and so it was left out for convience. \n",
|
||||
" \n",
|
||||
"If we are implementing this with gaussians, we might expect it to be implemented as:\n",
|
||||
"\n",
|
||||
" new_gaussian = measurement * old_gaussian\n",
|
||||
" \n",
|
||||
"where measurement is a gaussian returned from the sensor. But does that make sense? Can we multiply gaussians? If we multiply a gaussian with a gaussing is the result another gaussian, or something else?\n",
|
||||
"\n",
|
||||
"\n",
|
||||
" \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -242,7 +358,7 @@
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 111
|
||||
"prompt_number": 7
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
|
||||
@@ -0,0 +1,35 @@
|
||||
# -*- coding: utf-8 -*-
|
||||
"""
|
||||
Created on Wed Apr 30 10:35:19 2014
|
||||
|
||||
@author: rlabbe
|
||||
"""
|
||||
import numpy.random as random
|
||||
|
||||
class dog_sensor(object):
|
||||
def __init__(self, x0 = 0, motion=1, noise=0.0):
|
||||
self.x = x0
|
||||
self.motion = motion
|
||||
self.noise = math.sqrt(noise)
|
||||
|
||||
def sense(self):
|
||||
self.x = self.x + self.motion
|
||||
self.x += random.randn() * self.noise
|
||||
return self.x
|
||||
|
||||
|
||||
def measure_dog ():
|
||||
if not hasattr(measure_dog, "x"):
|
||||
measure_dog.x = 0
|
||||
measure_dog.motion = 1
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
|
||||
dog = dog_sensor(noise = 1)
|
||||
for i in range(10):
|
||||
print (dog.sense())
|
||||
|
||||
|
||||
|
||||
|
||||
+9
-1
@@ -1,6 +1,6 @@
|
||||
import numpy as np
|
||||
import math
|
||||
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
def _to_array(x):
|
||||
""" returns any of a scalar, matrix, or array as a 1D numpy array
|
||||
@@ -63,6 +63,14 @@ def multivariate_gaussian (x, mu, cov):
|
||||
fprime = (x - mu)**2
|
||||
return frac * np.exp(-0.5*np.dot(fprime, 1./np.diag(cov)))
|
||||
|
||||
def norm_plot (mean, var):
|
||||
min_x = mean - var * 1.5
|
||||
max_x = mean + var * 1.5
|
||||
|
||||
xs = np.arange (min_x, max_x, 0.1)
|
||||
ys = [gaussian (x,23,5) for x in xs]
|
||||
plt.plot (xs,ys)
|
||||
plt.show()
|
||||
|
||||
if __name__ == '__main__':
|
||||
from scipy.stats import norm
|
||||
|
||||
Reference in new issue
Block a user