Batch processing and RTS.
This belonged with the last check in. Added desciptions on batch processing and using the RTS smoothing algorithm.
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Roger Labbe committed
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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f093b034cf8>"
|
||||
"<matplotlib.figure.Figure at 0x7f37c14ba5f8>"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -428,18 +428,18 @@
|
||||
"\n",
|
||||
"The RTS smoother works by first running the Kalman filter in a batch mode, computing the filter output for each step. Given the filter output for each measurement along with the covariance matrix corresponding to each output the RTS runs over the data backwards, incorporating it's knowledge of the future into the past measurements. When it reaches the first measurement it is done, and the filtered output incorporates all of the information in a maximally optimal form.\n",
|
||||
"\n",
|
||||
"The equations for the RTS smoother are very straightforward and easy to implement.\n",
|
||||
"The equations for the RTS smoother are very straightforward and easy to implement. This derivation is for the linear Kalman filter. Similar derivations exist for the EKF and UKF. These steps are performed on the output of the batch processing, going backwards from the most recent in time back to the first estimate. Each iteration incorporates the knowledge of the future into the state estimate. Since the state estimate already incorporates all of the past measurements the result will be that each estimate will contain knowledge of all measurements in the past and future.\n",
|
||||
"\n",
|
||||
" Predict Step\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\mathbf{P}^- &= \\mathbf{FP}_k\\mathbf{F}^\\mathsf{T} + \\mathbf{Q }\n",
|
||||
"\\mathbf{P} &= \\mathbf{FP}_k\\mathbf{F}^\\mathsf{T} + \\mathbf{Q }\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
" Update Step\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\mathbf{K}_k &= \\mathbf{P}_k\\mathbf{F} \\hspace{2 mm}inv(\\mathbf{P}^-) \\\\\n",
|
||||
"\\mathbf{K}_k &= \\mathbf{P}_k\\mathbf{F} \\hspace{2 mm}\\mathbf{P}^{-1} \\\\\n",
|
||||
"\\mathbf{x}_k &= \\mathbf{x}_k + \\mathbf{K}_k(\\mathbf{x}_{x+1} - \\mathbf{FX}_k) \\\\\n",
|
||||
"\\mathbf{P}_k &= \\mathbf{P}_k + \\mathbf{K}_k(\\mathbf{P}_{K+1} - \\mathbf{P}^-)\\mathbf{K}_k^\\mathsf{T}\n",
|
||||
"\\mathbf{P}_k &= \\mathbf{P}_k + \\mathbf{K}_k(\\mathbf{P}_{K+1} - \\mathbf{P})\\mathbf{K}_k^\\mathsf{T}\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"As always, the hardest part of the implementation is correctly accounting for the subscripts. A basic implementation without comments or error checking would be:\n",
|
||||
@@ -471,20 +471,19 @@
|
||||
"import numpy as np\n",
|
||||
"from numpy import random\n",
|
||||
"import matplotlib.pyplot as plt\n",
|
||||
"from filterpy.kalman import KalmanFilter, rts_smoother\n",
|
||||
"from filterpy.kalman import KalmanFilter\n",
|
||||
"\n",
|
||||
"def plot_rts(noise):\n",
|
||||
" random.seed(123)\n",
|
||||
" fk = KalmanFilter(dim_x=2, dim_z=1)\n",
|
||||
"\n",
|
||||
" fk.x = np.array([[0.],\n",
|
||||
" [1.]]) # initial state (location and velocity)\n",
|
||||
" fk.x = np.array([0., 1.]) # initial state (location and velocity)\n",
|
||||
"\n",
|
||||
" fk.F = np.array([[1., 1.],\n",
|
||||
" [0., 1.]]) # state transition matrix\n",
|
||||
"\n",
|
||||
" fk.H = [[1., 0.]] # Measurement function\n",
|
||||
" fk.P = .01 # covariance matrix\n",
|
||||
" fk.P = 10. # covariance matrix\n",
|
||||
" fk.R = noise # state uncertainty\n",
|
||||
" fk.Q = 0.001 # process uncertainty\n",
|
||||
"\n",
|
||||
@@ -492,13 +491,13 @@
|
||||
" zs = [t + random.randn()*noise for t in range (40)]\n",
|
||||
"\n",
|
||||
" # filter data with Kalman filter, than run smoother on it\n",
|
||||
" mu, cov, _, _ = fk.batch_filter (zs)\n",
|
||||
" M,P,C = rts_smoother(mu, cov, fk.F, fk.Q)\n",
|
||||
" mu, cov, _, _ = fk.batch_filter(zs)\n",
|
||||
" M,P,C = fk.rts_smoother(mu, cov)\n",
|
||||
"\n",
|
||||
" # plot data\n",
|
||||
" plt.plot(zs,'r', marker='o', linewidth=1, alpha=0.5, label='measurement')\n",
|
||||
" plt.plot(M[:, 0], c='b', label='RTS')\n",
|
||||
" plt.plot(mu[:, 0, 0], c='g', linestyle='--', label='KF output')\n",
|
||||
" plt.plot(mu[:, 0], c='g', linestyle='--', label='KF output')\n",
|
||||
" plt.plot([0, len(zs)], [0, len(zs)], 'k', linewidth=1, label='ideal') \n",
|
||||
" plt.legend(loc=4)\n",
|
||||
"\n",
|
||||
@@ -512,19 +511,19 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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|
||||
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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f093afcd7f0>"
|
||||
"<matplotlib.figure.Figure at 0x7f37c1286320>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 5
|
||||
"prompt_number": 8
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"I've injected a lot of noise into the signal to allow you to visually distinguis the RTS output from the ideal output. In the graph above we can see that the Kalman filter, drawn as the green dotted line, is reasonably smooth compared to the input, but it still wanders from from the ideal line when several measurements in a row are biased towards one side of the line. In contrast, the RTS output is both extremely smooth and very close to the ideal output.\n",
|
||||
"I've injected a lot of noise into the signal to allow you to visually distinguish the RTS output from the ideal output. In the graph above we can see that the Kalman filter, drawn as the green dotted line, is reasonably smooth compared to the input, but it still wanders from from the ideal line when several measurements in a row are biased towards one side of the line. In contrast, the RTS output is both extremely smooth and very close to the ideal output.\n",
|
||||
"\n",
|
||||
"With a perhaps more reasonable amount of noise we can see that the RTS output nearly lies on the ideal output. The Kalman filter output, while much better, still varies by a far greater amount."
|
||||
]
|
||||
@@ -541,13 +540,13 @@
|
||||
{
|
||||
"metadata": {},
|
||||
"output_type": "display_data",
|
||||
"png": 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|
||||
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truncated
|
||||
"text": [
|
||||
"<matplotlib.figure.Figure at 0x7f093aff1eb8>"
|
||||
"<matplotlib.figure.Figure at 0x7f37ba2b7080>"
|
||||
]
|
||||
}
|
||||
],
|
||||
"prompt_number": 6
|
||||
"prompt_number": 9
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
|
||||
Reference in new issue
Block a user