Fix for probability text and covariance ellipses.
Fixed some text that was confusing probability with the probability density, and made the 3 covariance ellipses all use the same axis scales so they can be directly compared to each other.
This commit is contained in:
Roger Labbe committed
1 parent
1316a220b0
commit
e6868d91e2
1 file changed
+33
-53
@@ -1,7 +1,7 @@
|
||||
{
|
||||
"metadata": {
|
||||
"name": "",
|
||||
"signature": "sha256:8dd8877926c460a830c170bd1b1320400c48afa047d9fa4f200279e5f3c4a52d"
|
||||
"signature": "sha256:a173822504e2e8404015e6d858158001601adf2d500c7a1c4d4494238bb8fd25"
|
||||
},
|
||||
"nbformat": 3,
|
||||
"nbformat_minor": 0,
|
||||
@@ -257,13 +257,13 @@
|
||||
],
|
||||
"metadata": {},
|
||||
"output_type": "pyout",
|
||||
"prompt_number": 1,
|
||||
"prompt_number": 3,
|
||||
"text": [
|
||||
"<IPython.core.display.HTML at 0x7f56f45d49b0>"
|
||||
"<IPython.core.display.HTML at 0x7f5728987470>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 1
|
||||
"prompt_number": 3
|
||||
},
|
||||
{
|
||||
"cell_type": "heading",
|
||||
@@ -391,17 +391,17 @@
|
||||
"output_type": "display_data",
|
||||
"png": "iVBORw0KGgoAAAANSUhEUgAAAqsAAAFdCAYAAAAkOCRoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3XeYZGWV+PHve1PFzt2TAzkPGQYHERFYRAwormta07qr\nrq6uuLqiIqi7BlaXXTGsCoiIYf2ZEMFVRwTJeWBgmMAwsbtnuqdzd6Wbfn/UVE+HSt1d1RX6fJ7H\nR7rurVtv3alw6tzznldt2bLFRwghhBBCiCqkVXoAQgghhBBC5CLBqhBCCCGEqFoSrAohhBBCiKol\nwaoQQgghhKhaEqwKIYQQQoiqJcGqEEIIIYSoWka+jdFodL7GIYQQQgghFrDR0dGst0tmVQghhBBC\nVC0JVoUQQgghRNWSYFUIIYQQQlQtCVaFEEIIIUTVkmBVCCGEEEJULQlWhRBCCCFE1ZJgVQghhBBC\nVC0JVoUQQgghRNWSYFUIIYQQQlQtCVaFEEIIIUTVkmBVCCGEEEJULQlWhRBCCCFE1ZJgVQghhBBC\nVC0JVoUQQgghRNWSYFUIIYQQQlQtCVaFEEIIIUTVkmBVCCGEEEJULQlWhRBCCCFE1ZJgVQghhBBC\nVC0JVoUQQgghRNWSYFUIMW+UUpUeghBCiBpjVHoAQoj6pmna+P8cx8H3/UoPSQghRA2RYFUIUXJK\nKXRdR9O0abcLIYQQMyHBqhCiZDIBqqZpkkEVQghREhKsCiHmRNM0dF2flDWdGqj6vo/neXieN9/D\nE0IIUeMkWBVCzNjEOtRcfN8fD1InBq+5SgEkEyuEECIbCVaFEEXJVYc6Ua4ANXN/pRSu6+Y8fj4S\nzAohxMIkwaoQIq9CdaiZ27Jd5ldKoWnaeKCarwygUDCaL5iVQFYIIeqXBKtCiGnmWoeaCW5LOftf\nglkhhFiYJFgVQgBzv8w/MYNaiRZVEswKIUR9kmBViAUsE1jmmyxVqA51YpBazfIFpPNdL6uUkgBZ\nCCGKJMGqEAtQJoPq+z6u6+J53rRgNddl/okB7kwD1GoNaOeSlS3m/kIIIWZPglUhFohqrEOtFRLM\nCiFE5UiwKkQdK6YOFdLBlOM4VVeHWitmE8zm+9EghBDiEAlWhagzM50oNfG2zP1rpQ61VmSr9S1m\noYRc9xdCiIVEglUh6kQmQC00610u81cfKTMQQojcJFgVooZlq0OdauJM/lwZvkw2VlQnCWaFEAuZ\nBKtC1JhSLHs6MYOarVZV1BbpMSuEqGcSrApRA2YSoE6tRc3cP1sdqgQqC0MtBLPSe1YIkYsEq0JU\nMalDFfOhFoJZIcTCJcGqEFWm2DrUfMueZjKwEqSKUqim1b+EEAuPBKtCVIFS1qFKgCrmk0z+EkKU\nmwSrQlRIJsAsNAs/37KnEqCKajeTYDbb61iCWSGEBKtCzDOpQxXikImLUWQLTCUzK4SQYFWIeZAJ\nMGd7mX8+lj3N9FsVoppImYEQQoJVIcqkmMv8la5DrURwKi2KRClJMCtE/ZNgVYgSmhhguq6b84sw\nXx3q1Kb99aQen5OobhLMClH7JFgVogSm1qFmqzXNt+yp1KEKURm10GNWrkaIhU6CVSFmqdg61EwG\nVdpNSZZK1J5aCGaFqHcSrAoxAzPph5rhuu6k+y+0AHWihficRX2TYFaI8pNgVYgC5toPFeQyvxAL\nlaz+JcTcSbAqRA7F9kPNNZs/wzTNcg2xJDL1cNK6Soj5JQsmCFEcCVaFmGCu/VAzWViYfPlfCCFm\nShZMECJNglWx4BVbhwoUveypfEkIIcpN2nKJhUKCVbEgTaxDzdcWRupQhRC1SoJZUS8kWBULSrY6\n1KkfyKVe9lRqQSdnpoUQ1UGCWVErJFgVda9UdagzCVAXenCakW+lrmz7CiGqR60Es7JoQv2TYFXU\npWLqUGHhLntaToUy07k6J0g/SiFqi/SYFfNFglVRVzIBqqZpBetQZdnTtImtq+YiX+A/MTOd67Hk\ni0+I+iLvaVEqEqyKmjdxolTGTOpQF/qqUnMxn4G/NFcXor7Ie1oUS4JVUZNmUoeaCagmms8Atd4m\nWJV6AlqpxpSPZHCEqC0zfU9P/Vve1/VFglVRMzIBZjGrSlVDu6l6K/ov9jJ/NZJgVoj6MvF9me2z\ntpYys52dnaRSKQ4//PBKD6VqSbAqql6mDtV1XTzPy5pNLZTtW2h1qKWyUM6rXI4Uor7USicDgAce\neADf9yVYzUOCVVGVstWhZmQuq0sdannIeZ1MsrK1L+k59Nlx+lIx+uxD/3N9nzXRxZzWuJSIblV6\nmGIeVVMw29/fzxFHHFGy49UjCVZF1SimDhXSHxKZLOtECzGQKiU5r7Mz26xsvZWJVBPP93lyuItn\nR/dzwI4x6qZy7ru/f5Q/D+zgxEgHZzWtYEWgsape59U0loVkPoPZvr4+zjrrrKL3X4gkWBUVNZN+\nqLlWQarWy9GlaglVThPHN3Ws1Xpea0muf/vMa6OaLkXWiwE7zu09z7MzMTh+m640Wo0QbVaYNjNE\nmxmm3QwT9xweH+5kW6yPZ0b388zofhZbUc5qWs6a6GICWnV8RcrroPrMJZj1PI8vfOELNDQ00Nra\nSjAYpLOzk0gkQltbGy0tLRhGdbz2qoXasmVLzjMejUbncyxiASm2H2qhekmo3szDxBpbXdcrPZxx\nle6S4LouruuW5di1otis6kIqMZhrptnzfR4f7uSPfduxfZeIbnJx21GsDjbTZATR8pzLATvO48Od\nPDXcTcyzAQhoOqc3LOPCtiMxVP4f0+UkGfjCau0cDQ0Ncc011+TcrpSisbGR1tZW3vzmN7N48eKi\nj+26Lj/5yU/YunUrqVSKlStXcsUVV7BkyZK899u2bRvf+ta3sKxD5TBXXnnljB67FEZHR7PeLqG7\nmDdz7YeakcnGipnJ1yUhQ37NVxeZ+FWcqdnUE6OLeFX7MUXXobaYIS5uO4oLWo9g02gPjw13sicx\nxENDezhgx3jT4pMwNfnMEaXR0NDANddcQ39/P319ffz617/mpJNOor+/n/7+fgYHBxkaGmJoaGhS\n8FgM3/fp6OjgNa95DU1NTdxzzz3cdNNNfPrTny5436amJq699tpZPqvykm8mUValWPY0k+nLbK/W\nTGo1KiY7nalVlfNaW2TiVzqb+sRwJ384mE0Nayav7jiWE6KLZnU8Q2mc3LCEkxuWsCcxxE+6n2Fb\nrI8f73uGtyw5GUsCVlECmqbR0tJCS0sLRx55JNdddx1f/OIXx7e7rsvg4CD9/f00NTXN6NiGYXDJ\nJZeM/3322Wdz++23MzY2RiQSKdlzmG8SrIqSm2uAClIvOVcz6YlaL4GLmKzeg9lBO87tvZvZER8A\n4ITIIi7rKD6bWsjKYBPvWn4at3ZtYEd8gNu6N/DWpacQrJI6VlE/pr7fdF2nra2Ntra2OR97586d\nNDU1FRWojoyMcPXVVxMIBFi7di0XX3zxnB+/VORdJ0qmFHWo+VY/qrXgar7HK8G/mIlaLjHot2Pc\n1PkkY26KsGZyWcexnDjLbGo+i6wo71p2Ord2PcXuxBA/7NrA25eeQkg3S/5YYnZq/TPNtu2ylV/F\n43F+9atfcfnllxfcd8mSJVx11VW0t7fT2dnJjTfeSGNjI2vXri3L2GaqclXjoi5omoZpmliWNV6P\nmq0O1fM8HMfBcRxc1x3fJ5OFNQxjPNit9Q+f+TT13E4MVDVNwzCM8XMr51UUa+IkvGz/m/ijcur/\nym3USfHDrqcZc1McFmzmH1euLUugmtFuhXn38tNpNoJ0Jof5QddTjOVphSXmX6V/PM1Ff38/ra2t\nJT+u4zjcfPPNnH766Zx66qkF929oaKC9vR2A5cuXc9555/Hcc8+VfFyzJcGqmDGlFIZhYFkWhmHk\n/ILK1EJmC1AnBlISoM5cvnM7MfgvJoCotYy1qLx8gSxQtmA24Tnc1r2BASfO0kADb1l6MlGj/M38\nW8wQ715+Oq1miH2pUW7pfIoRJ1n2xxX1rxzBqud53HrrrXR0dHDppZeW9NiVIsGqKMrEANU0zZz1\nqBODKMn0lVaxWVQJ/kWlzSYrC/kv6Tq+x//u28i+1CitZoi3LTllXvugNhlB3r3sdDrMCL32GLd0\nPcWQk5i3xxf1qa+vryS1qRP97Gc/QynFG9/4xqzbb7jhBu64445Jt23bto2BgXT99759+3jggQc4\n8cQTSzquuZCaVZFXMZfm52t5zoWYAazHpU9raayi9GazUILn+/yqZxM74gNEdIu3Lz11XjKqUzUY\ngfFJV/tTo9zatYH3rThLugSIWSt1ZrW/v59HHnkE0zS56qqrxm9/3/veN76k68DAwLQAee/evdx6\n660kk0kaGhpYt25d1dSrgiwKILLI1g91qko0lvd9H8dxADDN6p/gMJfxzmQ2fyXHORuZ7PBCVmtN\nzOdDrnPi+z6/O7CNR4f3ElA6715xBksDDVmPMV/nNO7a3Nz5JL32GGc1LueyjmPL9ljyWsmv1s/P\nLbfcQmtrK6997WsrPZSqkGtRACkDEMDs61DlUnTpyGV+Iaa7b3AXjw7vRUfx5qUns8SKzmriVynf\nMyHd5IrFJ6CheGy4kxdifSU7tlhYylEGUI8kWF3AZlKHWiiIMk1zXutQa/mX9EQz6ZQgAapYaJ4Y\n7uLu/hcBeMPiEzk81JJ3//nsYrAk0MAFrYcDcHvPZuKuPfMnKBa8cnUDqDcSrC5AExv258ugVlsQ\nVU/BWjGz+SWLKhayzWO9/LZ3MwCXtR9TkvZUsw1kc70Hz21ezYpAIyNukrsObJ3z+MTCI5nV4kiw\nukBM7Yeaka0nqgRRpTexvrfaLvNPfKx6yViL2tabGuPn+5/DB85vOYyzmlaU/TFnk5XVNYLine truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f56f45d4a58>"
|
||||
"<matplotlib.figure.Figure at 0x7f572897d940>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 2
|
||||
"prompt_number": 4
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Here we have plotted a two dimensional multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}8&0\\\\0&10\\end{smallmatrix}]$. The three dimensional shape shows the probability of for any value of (x,y) in the z-axis. I have projected just the variance for x and y onto the walls of the chart - you can see that they take on the normal Gaussian bell curve shape. You can also see that, as we might hope, that the curve for x is wider than the curve for y, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. Also, the highest point of the curve is centered over (2,17), the means for x and y. I hope this demystifies the equation for you. Any multivariate Gaussian will create this sort of shape. If we think of this as a the Gaussian for our dog's position in a two dimensional field, the z-value at each point of (x,y) indicates the probability of the dog being at that position. So, he has the highest probability of being at (2,17), a modest probability of being at (5,14), and a very low probability of being at (10,10).\n",
|
||||
"Here we have plotted a two dimensional multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}8&0\\\\0&10\\end{smallmatrix}]$. The three dimensional shape shows the probability of for any value of (x,y) in the z-axis. I have projected just the variance for x and y onto the walls of the chart - you can see that they take on the normal Gaussian bell curve shape. You can also see that, as we might hope, that the curve for x is wider than the curve for y, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. Also, the highest point of the curve is centered over (2,17), the means for x and y. I hope this demystifies the equation for you. Any multivariate Gaussian will create this sort of shape. If we think of this as a the Gaussian for our dog's position in a two dimensional field, the z-value at each point of (x,y) is the probability density forthe dog being at that position. So, he has the highest probability of being near (2,17), a modest probability of being near (5,14), and a very low probability of being near (10,10).\n",
|
||||
"\n",
|
||||
"We will discuss the mathematical description of covariances in the Kalman Filter math chapter. For this chapter we just need to understand the following.\n",
|
||||
"\n",
|
||||
@@ -443,7 +443,7 @@
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 3
|
||||
"prompt_number": 5
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -451,7 +451,7 @@
|
||||
"source": [
|
||||
"Let's use it to compute a few values just to make sure we know how to call and use the function, and then move on to more interesting things.\n",
|
||||
"\n",
|
||||
"First, let's find the probability for our dog being at (2.5, 7.3) if we believe he is at (2,7) with a variance of 8 for $x$ and a variance of 10 for $y$.\n",
|
||||
"First, let's find the probability density for our dog being at (2.5, 7.3) if we believe he is at (2,7) with a variance of 8 for $x$ and a variance of 10 for $y$.\n",
|
||||
"\n",
|
||||
"Start by setting $x$ to (2.5,7.3):"
|
||||
]
|
||||
@@ -465,7 +465,7 @@
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 4
|
||||
"prompt_number": 6
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -483,7 +483,7 @@
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 5
|
||||
"prompt_number": 7
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -502,7 +502,7 @@
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"prompt_number": 6
|
||||
"prompt_number": 8
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -528,42 +528,15 @@
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 7
|
||||
"prompt_number": 9
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"> Note that the function can accept lists, np.array, or np.matrix as arguments, or even scalars. Type `multivariate_gaussian?` in a cell and press ctrl-enter to get the help for this function. \n",
|
||||
"\n",
|
||||
"Let's check the probability for the dog being at exactly (2,7)"
|
||||
"> Note that the function can accept lists, np.array, or np.matrix as arguments, or even scalars. Type `multivariate_gaussian?` in a cell and press ctrl-enter to get the help for this function. \n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"from __future__ import print_function\n",
|
||||
"import numpy as np\n",
|
||||
"\n",
|
||||
"x = np.array([2,7]) # using array to show we can use lists and arrays\n",
|
||||
"prob = multivariate_gaussian(x,mu,P) * 100.\n",
|
||||
"\n",
|
||||
"print(\"Probability dog is at (2,7) is {:.3}%\".format(prob))"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"stream": "stdout",
|
||||
"text": [
|
||||
"Probability dog is at (2,7) is 1.78%\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 8
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
@@ -586,17 +559,17 @@
|
||||
"output_type": "display_data",
|
||||
"png": "iVBORw0KGgoAAAANSUhEUgAAAqsAAAFdCAYAAAAkOCRoAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f56f1553320>"
|
||||
"<matplotlib.figure.Figure at 0x7f5706f9dcf8>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 9
|
||||
"prompt_number": 10
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The result is clearly a 3D bell shaped curve. We can see that the gaussian is centered around (2,7), and that the probability quickly drops away in all directions. On the sides of the plot I have drawn the Gaussians for $x$ in greens and for $y$ in orange.\n",
|
||||
"The result is clearly a 3D bell shaped curve. We can see that the gaussian is centered around (2,7), and that the probability density quickly drops away in all directions. On the sides of the plot I have drawn the Gaussians for $x$ in greens and for $y$ in orange.\n",
|
||||
"\n",
|
||||
"As beautiful as this is, it is perhaps a bit hard to get useful information. For example, it is not easy to tell if $x$ and $y$ both have the same variance or not. So for most of the rest of this book we will display multidimensional Gaussian using contour plots. I will use some helper functions in `stats.py` to plot them. If you are interested in linear algebra go ahead and look at the code used to produce these contours, otherwise feel free to ignore it."
|
||||
]
|
||||
@@ -610,17 +583,24 @@
|
||||
"P = np.array([[2,0],[0,2]])\n",
|
||||
"plt.subplot(131)\n",
|
||||
"stats.plot_covariance_ellipse((2,7), cov=P, facecolor='g', alpha=0.2, \n",
|
||||
" title='|2 0|\\n|0 2|')\n",
|
||||
" title='|2 0|\\n|0 2|', axis_equal=False)\n",
|
||||
"plt.ylim((4,10))\n",
|
||||
"plt.gca().set_aspect('equal', adjustable='box')\n",
|
||||
"\n",
|
||||
"plt.subplot(132)\n",
|
||||
"P = np.array([[2,0],[0,9]])\n",
|
||||
"plt.ylim((4,10))\n",
|
||||
"plt.gca().set_aspect('equal', adjustable='box')\n",
|
||||
"stats.plot_covariance_ellipse((2,7), P, facecolor='g', alpha=0.2, \n",
|
||||
" title='|2 0|\\n|0 9|')\n",
|
||||
" axis_equal=False, title='|2 0|\\n|0 9|')\n",
|
||||
"\n",
|
||||
"plt.subplot(133)\n",
|
||||
"P = np.array([[2,1.2],[1.2,2]])\n",
|
||||
"plt.ylim((4,10))\n",
|
||||
"plt.gca().set_aspect('equal', adjustable='box')\n",
|
||||
"stats.plot_covariance_ellipse((2,7), P, facecolor='g', alpha=0.2, \n",
|
||||
" title='|2 1.2|\\n|1.2 2|')\n",
|
||||
" axis_equal=False, title='|2 1.2|\\n|1.2 2|')\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"plt.tight_layout()\n",
|
||||
"plt.show()"
|
||||
@@ -631,13 +611,13 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": "iVBORw0KGgoAAAANSUhEUgAAA04AAAGmCAYAAABLKNRHAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3XuQJWV5P/Bv3879Nte9BwWBxawxiEWUEpZFEVQ0sCUJ\nKYuohKxIJKIiIgkk+a0CpVYQRcqCMlimguIFFisXUqBVSyFeUAJawC6yy+7szvVc+3T36Xv374/Z\nHRl2YW7nTJ/T5/upotg5s9P99Ow5b79Pv+/7vMLevXtDEBERERER0asSow6AiIiIiIio2zFxIiIi\nIiIiWgATJyIiIiIiogUwcSIiIiIiIloAEyciIiIiIqIFMHEiIiIiIiJaABOnmPv617+OzZs3Rx3G\ngjZv3ow777wz6jCIaJnY1hDRSvVKO9Juv/zlL7F582Y8+eSTUYdCC5CjDoA6TxCE476uqiq++93v\n4ic/+QnGxsYAAG94wxuwY8cObN26ddnnc10Xd955Jx588EGoqopTTjkFn/rUp3DWWWct+5hE1P3Y\n1hDRSq12O7Jr1y7s3r0bTz31FKanp/H+978fX/7yl5d9vJ///Of44Q9/iKeffhqVSgVr167F2Wef\njWuuuQbFYnHZx6XuwBGnPhCGx9/j+JlnnsE3vvENbNiwAX//93+PT3ziE7AsCx/72Mdw3333Lft8\nX/jCF3D33Xfj3e9+N/7hH/4BALBjxw4888wzyz4mEXU/tjVEtFKr3Y58+9vfxhNPPIE3vvGNAF49\ncVus22+/Hc8++yze//7346abbsK5556L+++/H5deeil0XV/RsSl6HHHqY294wxvwyCOPYO3atXOv\nXXrppbjgggtwxx134LLLLoMoLi23PnToEL7//e/jQx/6EP7xH/8RAPDe974X7373u/HVr34V9957\nb1uvgYi6H9saIlqpTrQjAHDXXXdh/fr1ANCWaYI33HAD3vKWt8x7bcOGDbjlllvwox/9CB/+8IdX\nfA6KDkec+tj69evnNUAAkEql8OY3vxmqqqJWqy35mD/96U8RhiG2b98+91oul8OFF16IX/3qV9A0\nbcVxE1FvYVtDRCvViXbk6HHb6ZVJEwC8/e1vBwDs37+/reei1cfEiY5RrVaRTCaXNRd37969kCQJ\np5566rzXTzvtNPi+j9///vftCpOIehzbGiJaqZW0I6ulUqkAAEZHRyOOhFaKiRPN8/zzz+Opp57C\ne9/7XiiKsuSfL5fLKBaLkCQJ3/zmN/HWt74Vv/rVrzA0NATgD40HEfU3tjVEtFIrbUdWy3333QdZ\nlnHRRRdFHQqtEBMnmmOaJq6//noMDg7iM5/5zLKO4TjOXOM1MTEBwzBQrVaRSCQAAJZltS1eIupN\nbGuIaKXa0Y6shl27duGRRx7Bjh07cMIJJ0QdDq0QEycCAARBgM997nM4ePAg7rjjDgwPDy/rOIlE\nAo7jAAD++Z//GY899hje8573zL2WSqXaFjMR9R62NUS0Uu1qRzrtmWeewc0334ytW7fimmuuiToc\nagNW1SMAwG233YZHHnkEX/7yl/HWt7512ccZGRmBqqrwfR+SJGFkZATA7BxkAF3buBHR6mBbQ0Qr\n1a52pJMOHjyIq666CieffDK++tWvrrjMOXUHjjgR/u3f/g3f+c53cMMNN6x4/u2pp56KIAiwZ8+e\nea8/99xzEEURJ5988oqOT0S9i20NEa1UO9uRTqlWq7jyyitRKBRwzz33IJ1ORx0StQkTpz73n//5\nn/jyl7+Mq666qi17C5x33nkQRREPPPDA3Gu6ruPhhx/GmWeeiXw+v+JzEFHvYVtDRCvV7nZkOR55\n5BFceOGF+NznPnfc7xuGgR07dsBxHNx7770YHBxc5QipkzhVr4/99re/xQ033IDXve51OPHEE/HQ\nQw/N+/473vGOuQpVi7Vp0yZ88IMfxH333QdBEHDyySfjhz/8ITRNw7XXXtvO8ImoR7CtIaKV6kQ7\nAszuCffykesXXngBd911FwDgne985zFbHmiahgMHDrxqafHPfvazePbZZ7Fjxw48+eST8743MjKC\ns846a8kxUvdg4tTH9u3bB8/zcODAAVx//fXzvicIAr7zne8sqxG66aabMDAwgAcffBDf//73ceqp\np+Luu+/Gn/7pn7YrdCLqIWxriGilOtWOPPLII3jwwQfnjrN3717s2bMHgiBg/fr1xyROC61VOvqz\nd9999zHfO/PMM5k49TgmTn3skksuwSWXXNL24yqKgk996lP41Kc+1fZjE1HvYVtDRCvVqXbk1ltv\nxa233tq2OH7605+2IyzqUlzjREREREREtAAmTkRERERERAtg4hRzgiBw7wAi6ji2NUS0Uv3cjvTr\ndfcaYe/evWHUQRAREREREXUzjjgREREREREtgIkTERERERHRApg4xdzXv/51bN68OeowFrR582bc\neeedUYdBRMvEtoaIVqpX2pF2++Uvf4nNmzcfs2EudR/u49QHFlpw+NBDD+Gee+7B2NgY1q5di8sv\nvxyXX375ss61b98+fO9738MTTzyBiYkJFAoF/Mmf/AmuvfZanHTSScs6JhH1htVsawDgxRdfxFe+\n8hX85je/geu6OOWUU/B3f/d32Lp167KPSUTReq125Ne//jV+/OMf46mnnsKLL76INWvWYPfu3cs+\nVyf6LD//+c/xwx/+EE8//TQqlQrWrl2Ls88+G9dccw2KxeKyY6XuwBGnPhCGr17/47//+7/xuc99\nDuvWrcNNN92EN73pTfjiF7+Ie++9d1nnevDBB/HjH/8Yf/Znf4Ybb7wRf/VXf4X/+7//w/bt2/Hs\ns88u9xKIqAesZlszPT2ND33oQ/jtb3+LK6+8Etdddx0cx8HHP/5xPPHEE8u9BCKK2Gu1I//7v/+L\nhx56CAMDAxgeHl5xJbpO9Fluv/12PPvss3j/+9+Pm266Ceeeey7uv/9+XHrppdB1fUXxUvQ44tTH\nwjDEV77yFWzevBl33303BEHApZdeClVV8Y1vfAOXXXYZ0un0ko753ve+F5/4xCeQSqXmXjv33HNx\n8cUX46677sI3vvGNdl8GEXW5TrQ13/ve96CqKv7jP/4DZ5xxBgBg+/bteNe73oU77rgDZ511Vicu\nhYgi9Ld/+7e4/vrroSgKLr/8chw6dGhFx+tEn+WGG27AW97ylnmvbdiwAbfccgt+9KMf4cMf/vCK\nYqZoccSpj+3duxcTExO4+OKL5z21+eAHPwhd1/GLX/xiycd84xvfOK8BAmbXFJRKJezfv3/FMRNR\n7+lEW/Pcc88hnU7PJU0AkMlk8Pa3vx2//e1vUavV2hI7EXWP0dFRKIrStuN1os/yyqQJAN7+9rcD\nAPtBMcDEqY/t2bMHAHDaaafNe/3o13v37m3LeSzLgmEYGB0dbcvxiKi3dKKtEQQBiUTimNdTqRTC\nMGQHhYiWpRN9lkqlAgDsB8UAE6c+Vi6XAQDDw8P4+c9/jre+9a245557MDw8DOAPH/SV+sEPfgDX\ndXHxxRe35XhE1Fs60da8/vWvh6qqmJiYmHstDEM888wzAMARJyJalk70We677z7IsoyLLrqobcek\naDBx6mOO4wAAFEVBtVqFruuYmJiYe4prWdaKz7Fv3z7867/+K84880xccsklKz4eEfWeTrQ1H/zg\nByFJEm644QY899xzeOmll7Bz507MzMwAAGzbbt8FEFFf6ESfZdeuXXjkkUewY8cOnHDCCW05JkWH\nxSH62NFOi+u6uOiii3DmmWdidHR0rurLK+f9LlW9XsfHP/5xDAwM4Pbbb19xvETUmzrR1px00kn4\np3/6J9x2223Yvn07AODEE0/E1Vdfjdtuu23JxSaIqL91os/yzDPP4Oabb8bWrVtxzTXXtOWYFC2O\nOPWxkZERAMfOva1WqwAwN41mOSzLwsc//nGoqop77rkHQ0NDK4yWiHpVp9qav/iLv8DPfvYz3H//\n/XjggQfwX//1X3PfW7du3UpCJqI+0ok+y8GDB3HVVVfh5JNPxle/+tUVl06n7sDEqY+deuqpAGar\nU73c0a+Pfn+pgiDAZz7zGezduxff/OY3ufEtUZ/rVFsDAOl0Gm9+85vxxje+EYIg4Gc/+xkymQxO\nPvnk5QdMRH2jE32WarWKK6+8EoVCAffccw9HwGOEiVMfO+2007B+/Xrs2rULQRDMvf6jH/0I+Xwe\nb3vb25Z13H/5l3/B7t27cccdd+D0009vV7hE1KM61da80pNPPonHH38cF1100XEr7hFRf5mensaF\nF16I97znPa/6d9rdZzEMAzt27IDjOLj33nsxODi44mNS9+Aapz736U9/Gtdddx127NiBCy64AL/4\nxS/w+OOP4/rrr1/WE5Jvf/vbuP/++/HOd74T9XodDz300Lzv//mf/3m7QieiHtLutgYA/uZv/gZv\nfvObsW7dOuzfvx/f/e53sW7dOlx77bVtjp6IusGePXvw05/+FAAwPj4OXddx1113AZh9QLNt27Z5\nf991XRw4cOBVp8l1os/y2c9+Fs8++yx27NiBJ598ct73RkZGuDl3j2Pi1OcuuugieJ6He+65B//v\n//0/rFu3DjfeeCP++q//elnH27t3LwRBwE9+8hP85Cc/mfc9QRCYOBH1qXa3NcDsRpW7du1CuVxG\nqVTCRRddhE9+8pN8wksUU88//zy+9rWvAcBcMvS1r30NgiDg4osvPiZxWkgn+ix79uyBIAi4++67\nj/nemWeeycSpxzFxIlx88cVt26/g1ltvxa233tqWYxFRvLSzrQFmn+x+9rOfbdvxiKi7XXLJJUsq\nE75x48a5DbiPpxN9lqMjYhRPXONERERERES0ACZOREREREREC2DiFHOCIHDvACLqOLY1RLLine truncated
|
||||
"png": "iVBORw0KGgoAAAANSUhEUgAAA1QAAAG3CAYAAACzLhRYAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f56d0432a58>"
|
||||
"<matplotlib.figure.Figure at 0x7f57070672b0>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 10
|
||||
"prompt_number": 23
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
@@ -651,9 +631,9 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"From a mathematical perspective these display the values that the multivariate gaussian takes for a specific standard deviation (in this case $\\sigma=1$). Think of it as taking a horizontal slice through the 3D surface plot we did above. However, thinking about the physical interpretation of these plots clarifies their meaning.\n",
|
||||
"From a mathematical perspective these display the values that the multivariate gaussian takes for a specific standard deviation (in this case $\\sigma=1$). \n",
|
||||
"\n",
|
||||
"The first plot uses the mean and covariance matrices of\n",
|
||||
"Thinking about the physical interpretation of these plots clarifies their meaning. The first plot uses the mean and covariance matrices of\n",
|
||||
"$$\n",
|
||||
"\\begin{aligned}\n",
|
||||
"\\mathbf{\\mu} &= \\begin{bmatrix}2\\\\7\\end{bmatrix} \\\\\n",
|
||||
@@ -661,7 +641,7 @@
|
||||
"\\end{aligned}\n",
|
||||
"$$ \n",
|
||||
"\n",
|
||||
"Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma^2$ probability - points where the dog is quite likely to be based on our current knowledge. Of course, the dog might be very far from this point, as Gaussians allow the mean to be any value. For example, the dog could be at (3234.76,189989.62), but that has vanishing low probability of being true. Generally speaking displaying the $1\\sigma$ to $2\\sigma$ contour captures the most likely values for the distribution. Recall from the Gaussians chapter the the 68-95-99.7 rule - 68% of all values will fall within 1 standard deviation ($1\\sigma$), 95% within $2\\sigma$, and 99.7% within $3\\sigma$.\n",
|
||||
"Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma$ probability, where $1\\sigma$ is one *standard deviation*. In other words, for this Gaussian 68% of the data will fall within this ellipse. Recall from the Gaussians chapter the the 68-95-99.7 rule - 68% of all values will fall within 1 standard deviation ($1\\sigma$), 95% within $2\\sigma$, and 99.7% within $3\\sigma$. The dog could be at (356443,58483), but values that far away from the mean are infitestimally small.\n",
|
||||
"\n",
|
||||
"An equivalent way of thinking about this is the circle/ellipse shows us the amount of error in our belief. A tiny circle would indicate that we have a very small error, and a very large circle indicates a lot of error in our belief. We will use this throughout the rest of the book to display and evaluate the accuracy of our filters at any point in time. "
|
||||
]
|
||||
|
||||
Reference in new issue
Block a user