From f3e0d5bc2ea9f399a9c5b58914d3713c724fa2b5 Mon Sep 17 00:00:00 2001 From: Roger Labbe Date: Mon, 9 Mar 2015 21:44:47 -0700 Subject: [PATCH] Added interactive g-h filter Used IPython widgets to create an interactive chart of g-h filtering. --- 01_g-h_filter.ipynb | 38072 +++++++++++++++++++++--------------------- 1 file changed, 18855 insertions(+), 19217 deletions(-) diff --git a/01_g-h_filter.ipynb b/01_g-h_filter.ipynb index e58ffb8..09fe9b6 100644 --- a/01_g-h_filter.ipynb +++ b/01_g-h_filter.ipynb @@ -424,7 +424,7 @@ "GgAAkohrAABIIq4BACCJuAYAgCTiGgAAkohrAABI8v8BSJ+bTP4sBwsAAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -571,7 +571,7 @@ "ACT5/7HUkn4JFkA2AAAAAElFTkSuQmCC\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -709,7 +709,7 @@ "QBJxDQAASf4fTXV4ruVZZG0AAAAASUVORK5CYII=\n" ], "text/plain": [ - "" + "" ] }, "metadata": {}, @@ -749,7 +749,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Average of measurements is 165.0126\n" + "Average of measurements is 165.0365\n" ] } ], @@ -787,7 +787,7 @@ "name": "stdout", "output_type": "stream", "text": [ - "Average of measurements is 165.0040\n" + "Average of measurements is 164.9493\n" ] } ], @@ -827,515 +827,533 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAvwAAAGHCAYAAADMVYYQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd8VFX6x/HPmZn0hISE0KugFCFIUCz8KAJKcUVWxYKK\n", - "ldXFumIBy7Kr2NEVxHVFF6WoKKwNCygooGABJEoRFAUUkU56nzm/PyKJ40yUksxNJt/365WXuc89\n", - "984z18nwzJlzzzHWWouIiIiIiIQll9MJiIiIiIhI9VHBLyIiIiISxlTwi4iIiIiEMRX8IiIiIiJh\n", - "TAW/iIiIiEgYU8EvIiIiIhLGVPCLiIiIiIQxxwr+pUuXMnToUJo3b47L5WL69Ol++7Ozsxk9ejQt\n", - "WrQgNjaWDh068Pjjj/u16du3Ly6Xy+9nxIgRoXwaIiIiIiI1msepB87LyyMtLY1LL72UkSNHYozx\n", - "23/TTTexZMkSZs2aRZs2bViyZAmjRo2iQYMGXHzxxQAYY7jiiiu4//77y4+LiYkJ6fMQEREREanJ\n", - "HOvhHzx4MBMmTOCcc87B5QpMY8WKFYwcOZI+ffrQsmVLLrnkEk466SQ+//xzv3YxMTE0bNiw/Cch\n", - 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Eli Brookner in *Tracking and Kalman Filtering Made Easy* lists 11, and I am sure there are more. Not only that, but each type of filter has numerous subtypes. Each filter is differentiated by how *g* and *h* are chosen. So there is no 'one fits all' advice that I can give here. Some filters set *g* and *h* as constants, others vary them dynamically. The Kalman filter varies them dynamically at each step. Some filters allow *g* and *h* to take any value within a range, others constrain one to be dependent on the other by some function $f(\\dot{}), \\mbox{where }g = f(h)$.\n", + "The g-h filter is not one filter - it is a classification for a family of filters. Eli Brookner in *Tracking and Kalman Filtering Made Easy* lists 11, and I am sure there are more. Not only that, but each type of filter has numerous subtypes. Each filter is differentiated by how *g* and *h* are chosen. So there is no 'one size fits all' advice that I can give here. Some filters set *g* and *h* as constants, others vary them dynamically. The Kalman filter varies them dynamically at each step. Some filters allow *g* and *h* to take any value within a range, others constrain one to be dependent on the other by some function $f(\\dot{}), \\mbox{where }g = f(h)$.\n", "\n", - "The topic of this book is not the entire family of g-h filters; more importantly, we are interested in the *Bayesian* aspect of these filters, which I have not addressed yet. Therefore I will not cover selection of *g* and *h* in depth. Eli Brookner's book *Tracking and Kalman Filtering Made Easy* is an excellent resource for that topic, if it interests you. If this strikes you as an odd position for me to take, recognize that the typical formulation of the Kalman filter does not use *g* and *h* at all; the Kalman filter is a g-h filter because it mathematically reduces to this algorithm. When we design the Kalman filter we will be making a number of carefully considered choices to optimize it's performance, and those choices indirectly affect *g* and *h*. Don't worry if this is not too clear right now, it will be much clearer later after we develop the Kalman filter theory.\n", + "The topic of this book is not the entire family of g-h filters; more importantly, we are interested in the *Bayesian* aspect of these filters, which I have not addressed yet. Therefore I will not cover selection of *g* and *h* in depth. Eli Brookner's book *Tracking and Kalman Filtering Made Easy* is an excellent resource for that topic, if it interests you. If this strikes you as an odd position for me to take, recognize that the typical formulation of the Kalman filter does not use *g* and *h* at all; the Kalman filter is a g-h filter because it mathematically reduces to this algorithm. When we design the Kalman filter we will be making a number of carefully considered choices to optimize it's performance, and those choices indirectly affect *g* and *h*, but you will not be choosing *g* and *h* directly. Don't worry if this is not too clear right now, it will be much clearer later after we develop the Kalman filter theory.\n", "\n", "However, it is worth seeing how varying *g* and *h* affects the results, so we will work through some examples. This will give us strong insight into the fundamental strengths and limitations of this type of filter, and help us understand the behavior of the rather more sophisticated Kalman filter." ] @@ -8671,414 +8604,403 @@ { "data": { "image/png": [ - "iVBORw0KGgoAAAANSUhEUgAAAtsAAAFsCAYAAAAdV3X4AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4VFX+x/H3zKT3kJACBAgQem8CKh2kiXSlig0L9s6u\n", - "S5C1r2vZn4grinQFFRGVKiAYUCAgvfea0JMQ0uf+/pgFxMykkEzq5/U8Po/M+Z57z3gFPhzOPcdk\n", - "GIaBiIiIiIgUOnNxD0BEREREpKxS2BYRERERcRKFbRERERERJ1HYFhERERFxEoVtEREREREnUdgW\n", - "EREREXEShW0RERERESfJMWxPmjSJJk2a4O/vj7+/P+3atWPRokU31EyYMIHKlSvj5eVFp06d2LVr\n", - "l1MHLCIiIiJSWphyOtRm4cKFuLu7ExUVhdVqZdq0abzzzjts3LiRJk2a8Pbbb/P6668zffp0ateu\n", - "zcSJE4mJiWHv3r34+PjccK2EhASnfxkREREREWfx9/fPd58cw7Y9QUFBvPXWWzz44INUqlSJJ598\n", - "knHjxgGQmppKSEgI7777LmPGjLmhn8K2iIiIiJRmNxO287xmOysri6+++orU1FTat2/P4cOHiY+P\n", - "p3v37tdqPDw8aN++PevWrcv3QEREREREyhqX3Aq2b99O27ZtSUtLw9PTk3nz5lGnTp1rgTo0NPSG\n", 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version of the filter so you can see the effect of changing $\\dot{x}$, $g$ and $h$ in real time. As you adjust the sliders for $\\dot{x}$, $g$ and $h$ the date will be refiltered and the results plotted for you.\n", + "\n", + "If you really want to test yourself, read the next paragraph and try to predict the results before you move the sliders. \n", + "\n", + "Some things to try include setting $g$ and $h$ to their miminum values. See how perfectly the filter tracks the data! But now adjust $\\dot{x}$ to larger or smaller than 5. The filter should diverge from the data and never reacquire it. Start adding back either $g$ or $h$ and see how the filter snaps back to the data. See what the difference in the line is when you add only $g$ vs only $h$. Can you explain the reason for the difference? Then try setting $g$ greater than 1. Can you explain the results? Put $g$ back to a reasonable value (such as 0.1), and then make $h$ very large. Can you explain these results? Finally, set both $g$ and $h$ to their largest values. \n", + "\n", + "If you want to explore with this more, change the value of the array `zs` to the values used in any of the charts above and rerun the cell to see the result." + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "image/png": [ + "iVBORw0KGgoAAAANSUhEUgAAAugAAAFpCAYAAADOXt1wAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", + "AAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xl8VNX5+PHPzGTfF7KRsCQhQAgQlgBhEWRXiyB1RW3V\n", + "Ym2/rtSqdWmrrVZrtRYR/bWlRREXFOuCiBpEEJAthCUQ1pAEyL6Qfc/M/P64ySQ3M0kmySSTwPN+\n", + "vfKa3HPOvfeEjO0zJ899jsZoNBoRQgghhBBC9Alae09ACCGEEEII0UwCdCGEEEIIIfoQCdCFEEII\n", + "IYToQyRAF0IIIYQQog+RAF0IIYQQQog+xMHeE2iptLTU3lMQQgghhBCiy7y9vbt9DVlBF0IIIYQQ\n", + "og+RAF0IIYQQQog+pE+luLRkiz8PiMvTwYMHAYiLi7PzTERfJu8TYQ15n4iOyHvkMpWeDpMng1YL\n", + "586Bh0e3LmfrNG1ZQRdCCCGEEFeGnBw4dQqGDIHISMjPh9dft/eszEiALoQQQgghrgzvvAPR0fD4\n", + 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"\n", + "interact (interactive_gh, \n", + " dx=widgets.FloatSliderWidget(value=5., min=1., max=10.), \n", + " g=widgets.FloatSliderWidget(value=0.5, min=0.01, max=2), \n", + " h=widgets.FloatSliderWidget(value=0.02, min=0.0, max=0.5, step=0.01))" + ] + }, { "cell_type": "markdown", "metadata": {}, @@ -17052,7 +17582,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "Now let's try a practical example. Earlier in the chapter we talked about tracking a train. Trains are heavy and slow, thus they cannot change speed quickly. They are on a track, so they cannot change direction except by slowing to a stop and then reversing course. Hence, we can conclude that if we already know the train's approximate position and velocity then we can predict its position in the near future with a great deal of accuracy. The train just cannot change its velocity much in a second or two. \n", + "Now let's try a practical example. Earlier in the chapter we talked about tracking a train. Trains are heavy and slow, thus they cannot change speed quickly. They are on a track, so they cannot change direction except by slowing to a stop and then reversing course. Hence, we can conclude that if we already know the train's approximate position and velocity then we can predict its position in the near future with a great deal of accuracy. A train just cannot change its velocity much in a second or two. \n", "\n", "So let's write a filter for a train. Its position is expressed as its position on the track in relation to some fixed point which we say is 0 km. I.e., a position of 1 means that the train is 1 km away from the fixed point. Velocity is expresses as meters per second. We perform measurement of position once per second, and the error is $\\pm$ 500 meters. How should we implement our filter?\n", "\n", @@ -17076,7 +17606,7 @@ }, { "cell_type": "code", - "execution_count": 33, + "execution_count": 34, "metadata": { "collapsed": false }, @@ -17085,925 +17615,843 @@ "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAwMAAAGUCAYAAABp3v7xAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", - "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXl0I9WZ9/8tWZIly7a8u+12d9u90gtL0ws0Q9iaAMMS\n", - "AhNIyGQY8psACZCwnDeTNzMDzZtz5sf8kjBvMkwCmclM0mxhm6aBQCBAb/QCvdOLe2/b7X23ZUuy\n", - "1vr9YSTr3ltVqtJu+/mc0+d0lUpVV1JZus99vt/nkWRZlkEQBEEQBEEQxLTDlO0BEARBEARBEASR\n", - "HSgYIAiCIAiCIIhpCgUDBEEQBEEQBDFNoWCAIAiCIAiCIKYpFAwQBEEQBEEQxDSFggGCIAiCIAiC\n", - "mKZQMEAQBDHFMJlMuPrqq7M9jIxQX1+PhoYGQ8958sknYTKZsHXr1jSNiiAIYvJAwQBBEESKMZlM\n", - "hv6tX78+5WOQJCnl54zQ3NwsvAaLxYKqqipcd911eO2119J2bR5JkoTXGpnsq72vkeek8z0iCIKY\n", - "LJizPQCCIIipxrp164SJ5u9+9zu0tLTgnnvuQX19PfPY8uXLU3r948ePo6CgIKXnVKKkpASPPPII\n", - "AMDn8+HYsWN455138NFHH2Hv3r346U9/mvYxbNq0SfUxtcn+Qw89hLvuuguzZs1K17AIgiAmDRQM\n", - 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