Fixed description of fixed-point filter.
I mistakenly stated that it re-estimated the points between times j and k, when of course it recomputes only a single fixed point.
This commit is contained in:
Roger Labbe committed
1 parent
9ece51233b
commit
a1a1753c18
1 file changed
+32
-33
+32
-33
@@ -16,7 +16,7 @@
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@@ -262,7 +262,7 @@
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@@ -291,7 +291,7 @@
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@@ -300,7 +300,7 @@
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@@ -349,7 +349,7 @@
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@@ -381,7 +381,7 @@
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truncated
|
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"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7e82b38>"
|
||||
"<matplotlib.figure.Figure at 0x75a1470>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -421,33 +421,32 @@
|
||||
"\n",
|
||||
"There are three classes of Kalman smoothers that produce better tracking in these situations.\n",
|
||||
"\n",
|
||||
"* Fixed Interval Smoothing\n",
|
||||
"* Fixed-Interval Smoothing\n",
|
||||
"\n",
|
||||
"This is a batch processing based filter. This filter waits for all of the data to be collected before making any estimates. For example, you may be a scientist collecting data for an experiment, and don't need to know the result until the experiment is complete. A fixed interval smoother will collect all the data, then estimate the state at each measurement using all available previous and future measurements. If it is possible for you to run your Kalman filter in batch mode it is always recommended to use one of these filters a it will provide much better results than the recursive forms of the filter from the previous chapters.\n",
|
||||
"This is a batch processing based filter. This filter waits for all of the data to be collected before making any estimates. For example, you may be a scientist collecting data for an experiment, and don't need to know the result until the experiment is complete. A fixed-interval smoother will collect all the data, then estimate the state at each measurement using all available previous and future measurements. If it is possible for you to run your Kalman filter in batch mode it is always recommended to use one of these filters a it will provide much better results than the recursive forms of the filter from the previous chapters.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"* Fixed Lag Smoothing\n",
|
||||
"* Fixed-Lag Smoothing\n",
|
||||
"\n",
|
||||
"Fixed lag smoothers introduce latency into the output. Suppose we choose a lag of 4 steps. The filter will ingest the first 3 measurements but not output a filtered result. Then, when the 4th measurement comes in the filter will produce the output for measurement 1, taking measurements 1 through 4 into account. When the 5th measurement comes in, the filter will produce the result for measurement 2, taking measurements 2 through 5 into account. This is useful when you need recent data but can afford a bit of lag. For example, perhaps you are using machine vision to monitor a manufacturing process. If you can afford a few seconds delay in the estimate a fixed lag smoother will allow you to produce very accurate and smooth results.\n",
|
||||
"Fixed-lag smoothers introduce latency into the output. Suppose we choose a lag of 4 steps. The filter will ingest the first 3 measurements but not output a filtered result. Then, when the 4th measurement comes in the filter will produce the output for measurement 1, taking measurements 1 through 4 into account. When the 5th measurement comes in, the filter will produce the result for measurement 2, taking measurements 2 through 5 into account. This is useful when you need recent data but can afford a bit of lag. For example, perhaps you are using machine vision to monitor a manufacturing process. If you can afford a few seconds delay in the estimate a fixed-lag smoother will allow you to produce very accurate and smooth results.\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"* Fixed Point Smoothing\n",
|
||||
"* Fixed-Point Smoothing\n",
|
||||
"\n",
|
||||
"Fixed point smoothers start out as a normal Kalman filter. But once they get to measurement 4 (say) it then looks backwards and revises the filter output for the previous measurement(s). So, at step 5 the filter will produce a result for 5, and update the result for measurement 4 taking measurement 5 into account. When measurement 6 comes in the filter produces the result for 6, and then goes back and revises 4 using the measurements from 5 and 6. It will revise the output for measurement 5 as well. This process continues, with all previous outputs being revised each time a new input comes in. This filter can be used to fill in gaps. For example, suppose you did not get a sensor reading at step `k`. Instead of just using the prediction from step `k-1` you can continue using data collected at steps `k+1`, `k+2`, and so on to refine the estimate. \n",
|
||||
"A fixed-point filter operates as a normal Kalman filter, but also produces an estimate for the state at some fixed time $j$. Before the time $k$ reaches $j$ the filter operates as a normal filter. Once $k>j$ the filter estimates $x_k$ and then also updates its estimate for $x_j$ using all of the measurements between $j\\dots k$. This can be useful to estimate initial paramters for a system, or for producing the best estimate for an event that happened at a specific time. For example, you may have a robot that took a photograph at time $j$. You can use a fixed-point smoother to get the best possible pose information for the camera at time $j$ as the robot continues moving.\n",
|
||||
"\n",
|
||||
"The choice of these filters depends on your needs and how much memory and processing time you can spare. Fixed-point smoothing requires storage of all measurements, and is very costly to compute because the output is for every time step is recomputed for every measurement. On the other hand, the filter does produce a decent output for the current measurement, so this filter can be used for real time applications.\n",
|
||||
"\n",
|
||||
"The choice of these filters depends on your needs and how much memory and processing time you can spare. Fixed point smoothing requires storage of all measurements, and is very costly to compute because the output is for every time step is recomputed for every measurement. On the other hand, the filter does produce a decent output for the current measurement, so this filter can be used for real time applications.\n",
|
||||
"Fixed-lag smoothing only requires you to store a window of data, and processing requirements are modest because only that window is processed for each new measurement. The drawback is that the filter's output always lags the input, and the smoothing is not as pronounced as is possible with fixed-interval smoothing.\n",
|
||||
"\n",
|
||||
"Fixed lag smoothing only requires you to store a window of data, and processing requirements are modest because only that window is processed for each new measurement. The drawback is that the filter's output always lags the input, and the smoothing is not as pronounced as is possible with fixed interval smoothing.\n",
|
||||
"\n",
|
||||
"Fixed interval smoothing produces the most smoothed output at the cost of having to be batch processed. Most algorithms use some sort of forwards/backwards algorithm that is only twice as slow as a recursive Kalman filter. "
|
||||
"Fixed-interval smoothing produces the most smoothed output at the cost of having to be batch processed. Most algorithms use some sort of forwards/backwards algorithm that is only twice as slow as a recursive Kalman filter. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Fixed Point Smoothing"
|
||||
"## Fixed-Point Smoothing"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -461,14 +460,14 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Fixed Interval Smoothing"
|
||||
"## Fixed-Interval Smoothing"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"There are several fixed lag smoothers available in the literature. I have chosen to implement the smoother invented by Rauch, Tung, and Striebel because of its ease of implementation and efficiency of computation. This smoother is commonly known as an RTS smoother, and that is what we will call it\n",
|
||||
"There are many fixed-lag smoothers available in the literature. I have chosen to implement the smoother invented by Rauch, Tung, and Striebel because of its ease of implementation and efficiency of computation. It is also the smoother I have seen used most often in real applications. This smoother is commonly known as an RTS smoother.\n",
|
||||
"\n",
|
||||
"Derivation of the RTS smoother runs to several pages of densely packed math, and to be honest I have never read it through. I'm certainly not going to inflict it on you. I don't think anyone but thesis writers really need to understand the derivation. Instead I will briefly present the algorithm, equations, and then move directly to implementation and demonstration of the smoother.\n",
|
||||
"\n",
|
||||
@@ -523,9 +522,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7f67080>"
|
||||
"<matplotlib.figure.Figure at 0x783c6d8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -598,9 +597,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7eb5898>"
|
||||
"<matplotlib.figure.Figure at 0x75771d0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -627,9 +626,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8760ac8>"
|
||||
"<matplotlib.figure.Figure at 0x77e7860>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -666,9 +665,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x87b57f0>"
|
||||
"<matplotlib.figure.Figure at 0x7294ba8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -692,14 +691,14 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Fixed Lag Smoothing\n",
|
||||
"## Fixed-Lag Smoothing\n",
|
||||
"\n",
|
||||
"The RTS smoother presented above should always be your choice of algorithm if you can run in batch mode because it incorporates all available data into each estimate. Not all problems allow you to do that, but you may still be interested in receiving smoothed values for previous estimates. The number line below illustrates this concept."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"execution_count": 9,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
@@ -709,7 +708,7 @@
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAtoAAACeCAYAAAAMuK1gAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nTkSuQmCC\n",
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x87fa358>"
|
||||
"<matplotlib.figure.Figure at 0x883aba8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -750,7 +749,7 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"standard deviation fixed lag: 2.61624417392\n",
|
||||
"standard deviation fixed-lag: 2.61624417392\n",
|
||||
"standard deviation kalman: 3.56221928461\n"
|
||||
]
|
||||
},
|
||||
@@ -758,7 +757,7 @@
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAuYAAADaCAYAAADqvoesAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x99bca90>"
|
||||
"<matplotlib.figure.Figure at 0x99babe0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -809,7 +808,7 @@
|
||||
"plt.plot(kf_x[:, 0], label='KF', ls='--')\n",
|
||||
"plt.legend(loc=4)\n",
|
||||
"\n",
|
||||
"print('standard deviation fixed lag:', np.mean(fls_res))\n",
|
||||
"print('standard deviation fixed-lag:', np.mean(fls_res))\n",
|
||||
"print('standard deviation kalman:', np.mean(kf_res))"
|
||||
]
|
||||
},
|
||||
|
||||
Reference in new issue
Block a user