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Batch processing and RTS.

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This belonged with the last check in. Added desciptions on
batch processing and using the RTS smoothing algorithm.
This commit is contained in:
Roger Labbe committed 2014-12-13 20:05:06 -08:00
1 parent a794fdced2
commit 0c6999e25f
3 files changed
+408 -202
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06_Multivariate_Kalman_filter/Multivariate_Kalman_Filters.ipynb
+362 -179
View File
@@ -1,7 +1,7 @@
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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc20f160>"
"<matplotlib.figure.Figure at 0x7f32937da240>"
]
}
],
@@ -714,7 +714,7 @@
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc1a2908>"
"<matplotlib.figure.Figure at 0x7f3293a8ccf8>"
]
}
],
@@ -741,7 +741,7 @@
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"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc3f0d68>"
"<matplotlib.figure.Figure at 0x7f32939bf400>"
]
}
],
@@ -768,7 +768,7 @@
"output_type": "display_data",
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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc3b76d8>"
"<matplotlib.figure.Figure at 0x7f32937ec8d0>"
]
}
],
@@ -799,7 +799,7 @@
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc1bf860>"
"<matplotlib.figure.Figure at 0x7f3293a9b048>"
]
}
],
@@ -864,7 +864,7 @@
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc3d7c50>"
"<matplotlib.figure.Figure at 0x7f3293545cc0>"
]
}
],
@@ -1221,112 +1221,108 @@
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import numpy as np\n",
"import scipy.linalg as linalg\n",
"import matplotlib.pyplot as plt\n",
"import numpy.random as random\n",
"from numpy import dot\n",
"\n",
"class KalmanFilter:\n",
"\n",
" def __init__(self, dim_x, dim_z, dim_u=0):\n",
" \"\"\" Create a Kalman filter. You are responsible for setting \n",
" the various state variables to reasonable values; the defaults \n",
" below will not give you a functional filter.\n",
"\n",
" **Parameters**\n",
"\n",
" dim_x : int\n",
" Number of state variables for the Kalman filter. For \n",
" example, if you are tracking the position and velocity \n",
" of an object in two dimensions, dim_x would be 4.\n",
"\n",
" This is used to set the default size of P, Q, and u\n",
"\n",
" dim_z : int\n",
" Number of of measurement inputs. For example, if the sensor\n",
" provides you with position in (x,y), dim_z would be 2.\n",
"\n",
" dim_u : int (optional)\n",
" size of the control input, if it is being used.\n",
" Default value of 0 indicates it is not used.\n",
" \"\"\"\n",
"\n",
" self.x = np.zeros((dim_x,1)) # state\n",
" self.P = np.eye(dim_x) # uncertainty covariance\n",
" self.Q = np.eye(dim_x) # process uncertainty\n",
" self.u = np.zeros((dim_x,1)) # motion vector\n",
" self.B = 0 # control transition matrix\n",
" self.F = 0 # state transition matrix\n",
" self.H = 0 # Measurement function\n",
" self.R = np.eye(dim_z) # state uncertainty\n",
" \n",
" # identity matrix. Do not alter this. \n",
" self._I = np.eye(dim_x)\n",
" \n",
" if use_short_form:\n",
" self.update = self.update_short_form\n",
"\n",
"\n",
" def update(self, Z, R=None):\n",
" \"\"\"\n",
" Add a new measurement (Z) to the kalman filter. If Z is\n",
" None, nothing is changed.\n",
"\n",
" **Parameters**\n",
"\n",
" Z : np.array\n",
" measurement for this update.\n",
"\n",
" R : np.array, scalar, or None\n",
" Optionally provide R to override the measurement noise \n",
" for this one call, otherwise self.R will be used.\n",
" \"\"\"\n",
"\n",
" if Z is None:\n",
" return\n",
" \n",
" if R is None:\n",
" R = self.R\n",
" elif np.isscalar(R):\n",
" R = np.eye(self.dim_z) * R\n",
"\n",
" # error (residual) between measurement and prediction\n",
" y = Z - dot(H, x)\n",
" \n",
" # project system uncertainty into measurement space \n",
" S = dot(H, P).dot(H.T) + R\n",
"\n",
" # map system uncertainty into kalman gain\n",
" K = dot(P, H.T).dot(linalg.inv(S))\n",
"\n",
" # predict new x with residual scaled by the kalman gain\n",
" self.x += dot(K, y)\n",
"\n",
" I_KH = self._I - dot(K, H)\n",
" self.P = dot(I_KH).dot(P).dot(I_KH.T) + dot(K, R).dot(K.T)\n",
"\n",
"\n",
" def predict(self, u=0):\n",
" \"\"\" Predict next position.\n",
" \n",
" **Parameters**\n",
"\n",
" u : np.array\n",
" Optional control vector. If non-zero, it is multiplied\n",
" by B to create the control input into the system.\n",
" \"\"\"\n",
" \n",
" self.x = dot(self.F, self.x) + dot(self.B, u)\n",
" self.P = self.F.dot(self.P).dot(self.F.T) + self.Q"
],
"language": "python",
"cell_type": "markdown",
"metadata": {},
"outputs": [],
"prompt_number": 21
"source": [
" import numpy as np\n",
" import scipy.linalg as linalg\n",
" import matplotlib.pyplot as plt\n",
" import numpy.random as random\n",
" from numpy import dot\n",
"\n",
" class KalmanFilter:\n",
"\n",
" def __init__(self, dim_x, dim_z, dim_u=0):\n",
" \"\"\" Create a Kalman filter. You are responsible for setting \n",
" the various state variables to reasonable values; the defaults \n",
" below will not give you a functional filter.\n",
"\n",
" **Parameters**\n",
"\n",
" dim_x : int\n",
" Number of state variables for the Kalman filter. For \n",
" example, if you are tracking the position and velocity \n",
" of an object in two dimensions, dim_x would be 4.\n",
"\n",
" This is used to set the default size of P, Q, and u\n",
"\n",
" dim_z : int\n",
" Number of of measurement inputs. For example, if the sensor\n",
" provides you with position in (x,y), dim_z would be 2.\n",
"\n",
" dim_u : int (optional)\n",
" size of the control input, if it is being used.\n",
" Default value of 0 indicates it is not used.\n",
" \"\"\"\n",
"\n",
" self.x = np.zeros((dim_x,1)) # state\n",
" self.P = np.eye(dim_x) # uncertainty covariance\n",
" self.Q = np.eye(dim_x) # process uncertainty\n",
" self.u = np.zeros((dim_x,1)) # motion vector\n",
" self.B = 0 # control transition matrix\n",
" self.F = 0 # state transition matrix\n",
" self.H = 0 # Measurement function\n",
" self.R = np.eye(dim_z) # state uncertainty\n",
"\n",
" # identity matrix. Do not alter this. \n",
" self._I = np.eye(dim_x)\n",
"\n",
" if use_short_form:\n",
" self.update = self.update_short_form\n",
"\n",
"\n",
" def update(self, Z, R=None):\n",
" \"\"\"\n",
" Add a new measurement (Z) to the kalman filter. If Z is\n",
" None, nothing is changed.\n",
"\n",
" **Parameters**\n",
"\n",
" Z : np.array\n",
" measurement for this update.\n",
"\n",
" R : np.array, scalar, or None\n",
" Optionally provide R to override the measurement noise \n",
" for this one call, otherwise self.R will be used.\n",
" \"\"\"\n",
"\n",
" if Z is None:\n",
" return\n",
"\n",
" if R is None:\n",
" R = self.R\n",
" elif np.isscalar(R):\n",
" R = np.eye(self.dim_z) * R\n",
"\n",
" # error (residual) between measurement and prediction\n",
" y = Z - dot(H, x)\n",
"\n",
" # project system uncertainty into measurement space \n",
" S = dot(H, P).dot(H.T) + R\n",
"\n",
" # map system uncertainty into kalman gain\n",
" K = dot(P, H.T).dot(linalg.inv(S))\n",
"\n",
" # predict new x with residual scaled by the kalman gain\n",
" self.x += dot(K, y)\n",
"\n",
" I_KH = self._I - dot(K, H)\n",
" self.P = dot(I_KH).dot(P).dot(I_KH.T) + dot(K, R).dot(K.T)\n",
"\n",
"\n",
" def predict(self, u=0):\n",
" \"\"\" Predict next position.\n",
"\n",
" **Parameters**\n",
"\n",
" u : np.array\n",
" Optional control vector. If non-zero, it is multiplied\n",
" by B to create the control input into the system.\n",
" \"\"\"\n",
"\n",
" self.x = dot(self.F, self.x) + dot(self.B, u)\n",
" self.P = self.F.dot(self.P).dot(self.F.T) + self.Q"
]
},
{
"cell_type": "markdown",
@@ -1350,7 +1346,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 22
"prompt_number": 21
},
{
"cell_type": "heading",
@@ -1710,7 +1706,7 @@
]
}
],
"prompt_number": 23
"prompt_number": 22
},
{
"cell_type": "code",
@@ -1725,14 +1721,14 @@
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 24,
"prompt_number": 23,
"text": [
"array([[ 5.87500000e-05, 1.17500000e-03],\n",
" [ 1.17500000e-03, 2.35000000e-02]])"
]
}
],
"prompt_number": 24
"prompt_number": 23
},
{
"cell_type": "heading",
@@ -1793,7 +1789,7 @@
]
}
],
"prompt_number": 25
"prompt_number": 24
},
{
"cell_type": "markdown",
@@ -1808,7 +1804,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"dog_filter.x = np.array([[0], [0]]) # state (location and velocity)\n",
"dog_filter.x = np.array([0., 0.]) # state (location and velocity)\n",
"dog_filter.F = np.array([[1, 1], [0, 1]]) # state transition matrix\n",
"dog_filter.H = np.array([[1, 0]]) # Measurement function\n",
"dog_filter.R *= 5 # measurement noise\n",
@@ -1818,7 +1814,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 26
"prompt_number": 25
},
{
"cell_type": "markdown",
@@ -1853,8 +1849,7 @@
"\n",
"def dog_tracking_filter(R, Q=0, cov=1.):\n",
" dog_filter = KalmanFilter(dim_x=2, dim_z=1)\n",
" dog_filter.x = np.array([[0], \n",
" [0]]) # (location and velocity)\n",
" dog_filter.x = np.array([0., 0.]) # (location and velocity)\n",
" dog_filter.F = np.array([[1, 1],\n",
" [0, 1]]) # state transition matrix\n",
" dog_filter.H = np.array([[1, 0]]) # Measurement function\n",
@@ -1891,7 +1886,7 @@
" \n",
" for t in range(count):\n",
" z = zs[t]\n",
" pos[t] = dog_filter.x[0, 0]\n",
" pos[t] = dog_filter.x[0]\n",
" cov[t] = dog_filter.P\n",
" \n",
" # perform the kalman filter steps\n",
@@ -1903,7 +1898,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 27
"prompt_number": 26
},
{
"cell_type": "markdown",
@@ -1986,7 +1981,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 28
"prompt_number": 27
},
{
"cell_type": "markdown",
@@ -2007,9 +2002,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": "iVBORw0KGgoAAAANSUhEUgAAAsoAAAGDCAYAAAAyKTZ5AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc50f828>"
"<matplotlib.figure.Figure at 0x7f32937a8e10>"
]
},
{
@@ -2017,11 +2012,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc4bc518>"
"<matplotlib.figure.Figure at 0x7f3293a2b668>"
]
}
],
"prompt_number": 29
"prompt_number": 28
},
{
"cell_type": "markdown",
@@ -2097,7 +2092,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 30
"prompt_number": 29
},
{
"cell_type": "markdown",
@@ -2126,7 +2121,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 31
"prompt_number": 30
},
{
"cell_type": "markdown",
@@ -2146,7 +2141,7 @@
"\n",
" # initialize the filters\n",
" f = mkf_filter(R, Q)\n",
" f.x[0, 0] = 0.\n",
" f.x[0] = 0.\n",
" f.P[0, 0] = p0\n",
" pos = (x0, p0)\n",
" for i in range(50):\n",
@@ -2155,7 +2150,7 @@
" f.update(z)\n",
"\n",
" x1.append(pos[0])\n",
" x2.append(f.x[0, 0])\n",
" x2.append(f.x[0])\n",
"\n",
" p1.append(pos[1])\n",
" p2.append(f.P[0, 0])\n",
@@ -2184,9 +2179,9 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc50fc88>"
"<matplotlib.figure.Figure at 0x7f32937a8080>"
]
},
{
@@ -2194,11 +2189,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebbfc1588>"
"<matplotlib.figure.Figure at 0x7f329357c400>"
]
}
],
"prompt_number": 32
"prompt_number": 31
},
{
"cell_type": "markdown",
@@ -2362,7 +2357,7 @@
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 33
"prompt_number": 32
},
{
"cell_type": "markdown",
@@ -2393,12 +2388,12 @@
"\n",
" # initialize the filters\n",
" f1 = mkf_filter(R, Q)\n",
" f1.x[0, 0] = 0.\n",
" f1.x[0] = 0.\n",
" f1.P[0, 0] = p0\n",
" \n",
" f2 = pos_vel_filter(R, Q)\n",
" f2.x[0, 0] = 0.\n",
" f2.x[1, 0] = 1.\n",
" f2.x[0] = 0.\n",
" f2.x[1] = 1.\n",
" f2.P *= p0\n",
" \n",
" for i in range(50):\n",
@@ -2428,13 +2423,13 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc4bc1d0>"
"<matplotlib.figure.Figure at 0x7f32939e2e48>"
]
}
],
"prompt_number": 34
"prompt_number": 33
},
{
"cell_type": "markdown",
@@ -2500,7 +2495,7 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc19c0f0>"
"<matplotlib.figure.Figure at 0x7f3293871278>"
]
},
{
@@ -2508,11 +2503,11 @@
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc28e550>"
"<matplotlib.figure.Figure at 0x7f32938710b8>"
]
}
],
"prompt_number": 35
"prompt_number": 34
},
{
"cell_type": "markdown",
@@ -2551,11 +2546,11 @@
"output_type": "display_data",
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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc4e2ac8>"
"<matplotlib.figure.Figure at 0x7f329357ce48>"
]
}
],
"prompt_number": 36
"prompt_number": 35
},
{
"cell_type": "markdown",
@@ -2569,7 +2564,7 @@
"collapsed": false,
"input": [
"plot_track(data=zs, R_var=10000., Q_var=.2, count=50, \n",
" initial_x=np.array([[20], [1]]), plot_P=False, \n",
" initial_x=np.array([20., 1.]), plot_P=False, \n",
" title='R_var = 10000 m, Q_var = 0.2 m')"
],
"language": "python",
@@ -2580,11 +2575,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc543fd0>"
"<matplotlib.figure.Figure at 0x7f32939cd940>"
]
}
],
"prompt_number": 37
"prompt_number": 36
},
{
"cell_type": "markdown",
@@ -2598,7 +2593,7 @@
"collapsed": false,
"input": [
"plot_track(data=zs, R_var=30, Q_var=.2, count=50, \n",
" initial_x=np.array([[20], [1]]), plot_P=False, \n",
" initial_x=np.array([20., 1.]), plot_P=False, \n",
" title='R_var = 30 m, Q_var = 0.2 m')"
],
"language": "python",
@@ -2609,11 +2604,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc534390>"
"<matplotlib.figure.Figure at 0x7f32938369e8>"
]
}
],
"prompt_number": 38
"prompt_number": 37
},
{
"cell_type": "markdown",
@@ -2678,11 +2673,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc0040f0>"
"<matplotlib.figure.Figure at 0x7f32937c5b38>"
]
}
],
"prompt_number": 39
"prompt_number": 38
},
{
"cell_type": "markdown",
@@ -2708,11 +2703,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc427be0>"
"<matplotlib.figure.Figure at 0x7f32939145f8>"
]
}
],
"prompt_number": 40
"prompt_number": 39
},
{
"cell_type": "markdown",
@@ -2780,11 +2775,11 @@
"output_type": "display_data",
"png": 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"text": [
"<matplotlib.figure.Figure at 0x7f4ebc33a278>"
"<matplotlib.figure.Figure at 0x7f32938ee518>"
]
}
],
"prompt_number": 41
"prompt_number": 40
},
{
"cell_type": "markdown",
@@ -2810,11 +2805,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc20e320>"
"<matplotlib.figure.Figure at 0x7f32938ee940>"
]
}
],
"prompt_number": 42
"prompt_number": 41
},
{
"cell_type": "markdown",
@@ -2831,7 +2826,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"x = np.array([[100., 0.]]).T\n",
"x = np.array([100., 0.]).T\n",
"plot_track(data=zs, R_var=50., Q_var=.2, P=1., initial_x=x,\n",
" count=len(zs), plot_P=False, title='P=1 m')"
],
@@ -2843,11 +2838,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc534ef0>"
"<matplotlib.figure.Figure at 0x7f329389bba8>"
]
}
],
"prompt_number": 43
"prompt_number": 42
},
{
"cell_type": "markdown",
@@ -2862,7 +2857,7 @@
"cell_type": "code",
"collapsed": false,
"input": [
"x = np.array([[100, 0]]).T\n",
"x = np.array([100, 0]).T\n",
"plot_track(data=zs, R_var=50, Q_var=.02, P=500., initial_x=x,\n",
" count=len(zs), plot_P=False, title='P=500')"
],
@@ -2874,11 +2869,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc262ac8>"
"<matplotlib.figure.Figure at 0x7f3293ae8cc0>"
]
}
],
"prompt_number": 44
"prompt_number": 43
},
{
"cell_type": "markdown",
@@ -2911,7 +2906,7 @@
" for t in range (count):\n",
" z = dog.sense()\n",
" f.update(z)\n",
" ps.append(f.x[0, 0])\n",
" ps.append(f.x[0])\n",
" cov.append(f.P)\n",
" zs.append(z)\n",
" f.predict()\n",
@@ -2945,21 +2940,21 @@
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc28ec88>"
"<matplotlib.figure.Figure at 0x7f329358c4e0>"
]
},
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc1a7978>"
"<matplotlib.figure.Figure at 0x7f3293a21f98>"
]
}
],
"prompt_number": 45
"prompt_number": 44
},
{
"cell_type": "markdown",
@@ -3030,11 +3025,11 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f4ebc1aed68>"
"<matplotlib.figure.Figure at 0x7f32937949b0>"
]
}
],
"prompt_number": 46
"prompt_number": 45
},
{
"cell_type": "markdown",
@@ -3052,6 +3047,194 @@
"> **sidebar**: when plotting covariance ellipses, make sure to always use *plt.axis('equal')* in your code. If the axis use different scales the ellipses will be drawn distorted. For example, the ellipse may be drawn as being taller than it is wide, but it may actually be wider than tall."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Batch Processing\n",
"\n",
"The Kalman filter is designed as a recursive algorithm - as new measurements come in we immediately create a new estimate. But it is very common to have a set of data that have been already collected which we want to filter. Kalman filters can always be run in a *batch* mode, where all of the measurements are filtered at once. We have implemented this in `KalmanFilter.batch_filter()`. Internally, all the function does is loop over the measurements and collect the resulting state and covariance estimates in arrays. It may seem a bit trivial, but you will almost always want to do this if you can for several reasons. First, it will execute a bit quicker than if you implement the loop yourself. Second, the logic in your code will be a bit cleaner, and you will have a reduced chance of bugs. Third, and most importantly, it you batch process your data you can then use an extremely powerful techique to generate far smoother results than you have seen so far. We will present that technique in the next section; here I will show you how to use `KalmanFilter.batch_filter()`.\n",
"\n",
"All you have to do is collect your measurements into an array or list. Maybe it is in a CSV file, for example.\n",
"\n",
" zs = read_altitude_from_csv()\n",
"\n",
"Or maybe you will generate it using a generator:\n",
"\n",
" zs = [some_func(i) for i in range(1000)]\n",
" \n",
"Just do whatever it takes to get your measurements in an array.\n",
"\n",
"Now we will just call the `batch_filter()` method.\n",
"\n",
" Xs, Ps, Xs_pred, Ps_pred = kfilter.batch_filter(zs)\n",
" \n",
"The funtion takes the list/array of measurements, filters it, and returns a list of state estimates (Xs), covariance matrices (Ps), and the predictions for the same (Xs_pred, Ps_pred).\n",
"\n",
"Here is a complete example drawing from the code above"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dog = DogSensor(velocity=1, noise_var=10.)\n",
"f = dog_tracking_filter(R=3., Q=.02, cov=P)\n",
"count = 50\n",
"zs = [dog.sense() for t in range(count)]\n",
"Xs, _, _, _ = f.batch_filter(zs)\n",
"\n",
"plt.plot([0, count], [0, count], 'g', label='actual')\n",
"plt.plot(range(1,count+1), zs, c='r', linestyle='dashed', \n",
" label='measurement')\n",
"plt.plot(range(1,count+1), Xs[:,0], c='b', label='filter')\n",
"plt.legend(loc='best')\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f3293674da0>"
]
}
],
"prompt_number": 46
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Smoothing the Results\n",
"\n",
"I have an entire chapter on using the Kalman filter to smooth data; I will not repeat the chapter's information here. However, it is so easy to use, and offers such a profoundly improved output that I will tease you will a few examples. The smoothing chapter is not especially difficult; you are sufficiently prepared to read it now.\n",
"\n",
"Briefly, the recursive form of Kalman filtering that we have been using up to now use information from the past to generate the current estimate. Recall the last section, where we used batch processing to filter a collection of measurements. This implies that we know now only the past, but the future! This is a key insight.\n",
"\n",
"Let's assume that we are tracking a car. Suppose we get a noisy measurement that implies that the car is starting to turn to the left, but the state function has predicted that the car is moving straight. The Kalman filter has no choice but to move the state estimate somewhat towards the noisy measurement, as it cannot judge whether this is just a particularly noisy measurement or the true start of a turn. \n",
"\n",
"However, if we have measurements from the future we can often figure out if a turn was made or not. Suppose the subsequent measurements all continue turning left. We can then be sure that the measurement was not very noisy, but instead a true indication that a turn was initiated. On the other hand, if the subsequent measurements continued on in a straight line we would know that the measurement was noisy and should be mostly ignored.\n",
"\n",
"We will not develop the math or algorithm here, I will just show you how to call the algorithm in `filterpy`. The algorithm that we have implemented is called an *RTS smoother*, after the three inventors of the algorithm: Rauch, Tung, and Striebel. The routine is `KalmanFilter.rts_smoother()`. Using it is trivial; we pass in the means and covariances computed from the `batch_filter` step, and receive back the smoothed means, covariances, and Kalman gain."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from numpy.random import seed\n",
"dog = DogSensor(velocity=1, noise_var=40.)\n",
"f = dog_tracking_filter(R=3., Q=.2, cov=P)\n",
"count = 50\n",
"seed(8923)\n",
"zs = [dog.sense() for t in range(count)]\n",
"Xs, Covs, _, _ = f.batch_filter(zs)\n",
"Ms, Ps, _ = f.rts_smoother(Xs, Covs)\n",
"\n",
"#plt.plot([0, count], [0, count], 'g', label='actual', alpha=.7)\n",
"plt.plot(range(1,count+1), zs, c='r', linestyle='dashed', \n",
" alpha=.5, label='measurement')\n",
"plt.plot(range(1,count+1), Xs[:,0], c='b', label='filter', alpha=.5)\n",
"plt.plot(range(1,count+1), Ms[:,0], c='k', label='RTS')\n",
"plt.legend(loc='best')\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f329357c2b0>"
]
}
],
"prompt_number": 85
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This output is fantastic!. Two things are very apparent to me in this chart. First, the RTS output is much, much smoother than the KF output. Second, it is almost always more accurate than the KF output (we will examine this claim in detail in the **Smoothing** chapter).\n",
"\n",
"While you can usually find isolated sections where the KF produces a more accurate estimate than the RTS smoother, the RTS almost always significantly outperforms the KF. If you are able to batch process your data, there is no reason to not use the RTS algorithm.\n",
"\n",
"We cannot always wait to collect all of our data before we filter it. If so, that means that we cannot use the RTS filter, but we may still have options. The **Smoothing** chapter discusses various strategies that you can use, and it is a mandatory (and easy) chapter if you want to achieve the best possible results from your filtering.\n",
"\n",
"Let's wrap this up by looking at the velocity estimates of the two filters."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dog = DogSensor(velocity=1, noise_var=40.)\n",
"f = dog_tracking_filter(R=3., Q=.2, cov=P)\n",
"count = 50\n",
"seed(8923)\n",
"zs = [dog.sense() for t in range(count)]\n",
"Xs, Covs, _, _ = f.batch_filter(zs)\n",
"Ms, Ps, _ = f.rts_smoother(Xs, Covs)\n",
"plt.plot(Xs[:,1], c='b', label='filter', alpha=.5)\n",
"plt.plot(Ms[:,1], c='k', linestyle=':', label='RTS')\n",
"plt.legend(loc='best')\n",
"plt.gca().axhline(1, c='k')\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f329357cc50>"
]
}
],
"prompt_number": 87
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The improvement in the velocity, which is an unobserved variable, is even more dramatic. \n",
"\n",
"Since we are plotting velocies let's look at what the the 'raw' velocity is, which we can compute by subtracting subsequent measurements. We see below that the noise swamps the signal, causing the computed values to be essentially worthless. I show this to reiterate the importance of using Kalman filters to compute velocities, accelerations, and even higher order values. I use a Kalman filter even when my measurements are so accurate that I am willing to use them unfiltered if I am also interested in the velocities and/or accelerations. Even a very small error in a measurement gets magnified when computing velocities. "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dx = np.diff(zs, axis=0)\n",
"plt.plot(range(1,count), dx, c='r',ls='--', label='raw velocity')\n",
"plt.plot(Xs[:,1], c='b', label='filter', alpha=.5)\n",
"plt.plot(Ms[:,1], c='k', linestyle=':', label='RTS')\n",
"plt.legend(loc='best')\n",
"plt.show()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f329b700cf8>"
]
}
],
"prompt_number": 88
},
{
"cell_type": "heading",
"level": 2,
14_Smoothing/Smoothing.ipynb
+22 -23
View File
@@ -1,7 +1,7 @@
{
"metadata": {
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"signature": "sha256:977e85323406c88caf44809425339ba81920c38d8fdba4011a544e4cea573690"
"signature": "sha256:492530ff0ddd97db5e24c23a8bdc358b1994c47f912d2c7ad0fe4b85f11ec641"
},
"nbformat": 3,
"nbformat_minor": 0,
@@ -261,7 +261,7 @@
"output_type": "pyout",
"prompt_number": 1,
"text": [
"<IPython.core.display.HTML at 0x7f09578a54e0>"
"<IPython.core.display.HTML at 0x7f37de3cbf98>"
]
}
],
@@ -294,7 +294,7 @@
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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f09578a5b70>"
"<matplotlib.figure.Figure at 0x7f37de3d54e0>"
]
}
],
@@ -335,7 +335,7 @@
"output_type": "display_data",
"png": 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truncated
"text": [
"<matplotlib.figure.Figure at 0x7f0957895be0>"
"<matplotlib.figure.Figure at 0x7f37c155d208>"
]
}
],
@@ -371,7 +371,7 @@
"output_type": "display_data",
"png": 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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": {},
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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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"text": [
"<matplotlib.figure.Figure at 0x7f093aff1eb8>"
"<matplotlib.figure.Figure at 0x7f37ba2b7080>"
]
}
],
"prompt_number": 6
"prompt_number": 9
},
{
"cell_type": "markdown",
code/book_format.py
+24
View File
@@ -3,10 +3,34 @@ from IPython.core.display import HTML
import matplotlib.pylab as pylab
import matplotlib.pyplot as plt
def test_filterpy_version():
import filterpy
min_version = [0,0,10]
v = filterpy.__version__
tokens = v.split('.')
for i,v in enumerate(tokens):
if int(v) > min_version[i]:
return
i = len(tokens) - 1
if min_version[i] > int(tokens[i]):
raise Exception("Minimum FilterPy version supported is {}.{}.{}.\n"
"Please install a more recent version." .format(
*min_version))
v = int(tokens[0]*1000)
def load_style(name='../styles/custom2.css'):
styles = open(name, 'r').read()
return HTML(styles)
# ensure that we have the correct filterpy loaded. This is
# called when this module is imported at the top of each book
# chapter so the reader can see that they need to update FilterPy.
test_filterpy_version()
pylab.rcParams['lines.linewidth'] = 2
pylab.rcParams['lines.antialiased'] = True
pylab.rcParams['patch.linewidth'] = 0.5