Started altering to use filterpy project.

Not fully tested, but the multidimensional chapter is working.

Designing_Nonlinear_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:c1819e9ca7da173dd60891f9141240e1487e969120eed3fa9b30207968d8eb4d"
+  "signature": "sha256:6e6749d931110e1edad026d2237ed80b25dbe6f60372d36194549b6f735220f0"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -41,7 +41,7 @@
         "        margin-right: auto;\n",
         "    }\n",
         "    div.text_cell code {\n",
-        "        background: #FFFFFF;\n",
+        "        background: transparent;\n",
         "        color: #000000;\n",
         "        font-weight: 600;\n",
         "        font-size: 13pt;\n",
@@ -248,13 +248,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 1,
+       "prompt_number": 4,
        "text": [
-        "<IPython.core.display.HTML at 0x2d74c90>"
+        "<IPython.core.display.HTML at 0x14a4490>"
        ]
       }
      ],
-     "prompt_number": 1
+     "prompt_number": 4
     },
     {
      "cell_type": "heading",
@@ -393,13 +393,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x44c8550>"
+        "<matplotlib.figure.Figure at 0x2793590>"
        ]
       }
      ],
-     "prompt_number": 113
+     "prompt_number": 5
     },
     {
      "cell_type": "markdown",
@@ -505,24 +505,24 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
+      "from math import sin, cos, atan2\n",
+      "\n",
       "def H_of (pos, pos_A, pos_B):\n",
       "    \"\"\" Given the position of our object at 'pos' in 2D, and two transmitters\n",
       "    A and B at positions 'pos_A' and 'pos_B', return the partial derivative\n",
       "    of H\n",
       "    \"\"\"\n",
-      "    \n",
-      "    from math import sin, cos, atan2\n",
       "\n",
       "    theta_a = atan2(pos_a[1]-pos[1], pos_a[0] - pos[0])\n",
       "    theta_b = atan2(pos_b[1]-pos[1], pos_b[0] - pos[0])\n",
       "\n",
-      "    return np.mat([[-cos(theta_a), 0, -sin(theta_a), 0],\n",
-      "                   [-cos(theta_b), 0, -sin(theta_b), 0]])"
+      "    return np.mat([[0, -cos(theta_a), 0, -sin(theta_a)],\n",
+      "                   [0, -cos(theta_b), 0, -sin(theta_b)]])"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 114
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -535,6 +535,8 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
+      "from numpy.random import randn\n",
+      "\n",
       "class DMESensor(object):\n",
       "    def __init__(self, pos_a, pos_b, noise_factor=1.0):\n",
       "        self.A = pos_a\n",
@@ -549,43 +551,13 @@
       "        ra = math.sqrt((self.A[0] - pos[0])**2 + (self.A[1] - pos[1])**2)\n",
       "        rb = math.sqrt((self.B[0] - pos[0])**2 + (self.B[1] - pos[1])**2)\n",
       "        \n",
-      "        return (ra + random.randn()*self.noise_factor, \n",
-      "                rb + random.randn()*self.noise_factor)"
+      "        return (ra + randn()*self.noise_factor, \n",
+      "                rb + randn()*self.noise_factor)"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 115
-    },
-    {
-     "cell_type": "markdown",
-     "metadata": {},
-     "source": [
-      "Now let's write the code to compute $\\frac{\\partial\\mathbf{h}}{\\partial\\mathbf{x}}$."
-     ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [
-      "def dist(a,b):\n",
-      "    return math.sqrt ((a[0]-b[0])**2 + (a[1]-b[1])**2)\n",
-      "\n",
-      "def H_of (pos, pos_A, pos_B):\n",
-      "    dist_a = dist(pos, pos_A)\n",
-      "    dist_b = dist(pos, pos_B)\n",
-      "    \n",
-      "    return np.mat([[0, (pos[0]-pos_A[0])/dist_a, 0, (pos[1]-pos_A[1])/dist_a],\n",
-      "                   [0, (pos[0]-pos_B[0])/dist_b, 0, (pos[1]-pos_B[1])/dist_b]])\n",
-      "    '''\n",
-      "    return np.mat([[(pos[0]-pos_A[0])/dist_a, 0, (pos[1]-pos_A[1])/dist_a,0],\n",
-      "                   [(pos[0]-pos_B[0])/dist_b, 0, (pos[1]-pos_B[1])/dist_b,0]])\n",
-      "    '''"
-     ],
-     "language": "python",
-     "metadata": {},
-     "outputs": [],
-     "prompt_number": 116
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -600,6 +572,8 @@
      "collapsed": false,
      "input": [
       "import math\n",
+      "from KalmanFilter import KalmanFilter\n",
+      "import numpy as np\n",
       "\n",
       "pos_a = (100,-20)\n",
       "pos_b = (-100, -20)\n",
@@ -638,7 +612,7 @@
       "    # compute the difference in range between the nominal track and measured \n",
       "    # ranges\n",
       "    ra,rb = d.range_of(pos)\n",
-      "    rx,ry = d.range_of((i+f1.x[0,0], i+f1.x[2,0]))\n",
+      "    rx,ry = d.range_of((pos[0]+f1.x[0,0], pos[1]+f1.x[2,0]))\n",
       "    z = np.mat([[ra-rx],[rb-ry]])\n",
       "\n",
       "    # compute linearized H for this time step\n",
@@ -658,8 +632,7 @@
       "p1, = plt.plot (xs, ys, 'r--')\n",
       "p2, = plt.plot (pxs, pys)\n",
       "plt.legend([p1,p2], ['filter', 'ideal'], 2)\n",
-      "plt.show()\n",
-      "\n"
+      "plt.show()"
      ],
      "language": "python",
      "metadata": {},
@@ -667,13 +640,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x4a48d50>"
+        "<matplotlib.figure.Figure at 0x2a57d10>"
        ]
       }
      ],
-     "prompt_number": 117
+     "prompt_number": 14
     },
     {
      "cell_type": "heading",

KalmanFilter.py

@@ -1,131 +0,0 @@
-# -*- coding: utf-8 -*-
-"""
-Created on Fri Oct 18 18:02:07 2013
-
-@author: rlabbe
-"""
-
-import numpy as np
-import scipy.linalg as linalg
-import matplotlib.pyplot as plt
-import numpy.random as random
-
-class KalmanFilter:
-
-    def __init__(self, dim_x, dim_z, use_short_form=False):
-        """ Create a Kalman filter of dimension 'dim', where dimension is the
-        number of state variables.
-        
-        use_short_form will force the filter to use the short form equation
-        for computing covariance: (I-KH)P. This is the equation that results
-        from deriving the Kalman filter equations. It is efficient to compute
-        but not numerically stable for very long runs. The long form,
-        implemented in update(), is very slightly suboptimal, and takes longer
-        to compute, but is stable. I have chosen to make the long form the
-        default behavior. If you really want the short form, either call
-        update_short_form() directly, or set 'use_short_form' to true. This
-        causes update() to use the short form. So, if you do it this way
-        you will never be able to use the long form again with this object.
-        """
-
-        self.x = 0 # state
-        self.P = np.matrix(np.eye(dim_x)) # uncertainty covariance
-        self.Q = np.matrix(np.eye(dim_x)) # process uncertainty
-        self.u = np.matrix(np.zeros((dim_x,1))) # motion vector
-        self.B = 0
-        self.F = 0 # state transition matrix
-        self.H = 0 # Measurement function (maps state to measurements)
-        self.R = np.matrix(np.eye(dim_z)) # state uncertainty
-        self.I = np.matrix(np.eye(dim_x))
-        
-        if use_short_form:
-            self.update = self.update_short_form
-
-
-    def update(self, Z):
-        """
-        Add a new measurement to the kalman filter.
-        """
-
-        # measurement update
-        y = Z - (self.H * self.x)                   # error (residual) between measurement and prediction
-        S = (self.H * self.P * self.H.T) + self.R   # project system uncertainty into measurment space + measurement noise(R)
-
-
-        K = self.P * self.H.T * linalg.inv(S) # map system uncertainty into kalman gain
-
-        self.x = self.x + (K*y)                # predict new x with residual scaled by the kalman gain
-
-        KH = K*self.H
-        I_KH = self.I - KH
-        self.P = I_KH*self.P * I_KH.T + K*self.R*K.T
-
-
-    def update_short_form(self, Z):
-        """
-        Add a new measurement to the kalman filter.
-        
-        Uses the 'short form' computation for P, which is mathematically
-        correct, but perhaps not that stable when dealing with large data
-        sets. But, it is fast to compute. Advice varies; some say to never
-        use this. My opinion - if the longer form in update() executes too
-        slowly to run in real time, what choice do you have. But that should
-        be a rare case, so the long form is the default use
-        """
-
-        # measurement update
-        y = Z - (self.H * self.x)                   # error (residual) between measurement and prediction
-        S = (self.H * self.P * self.H.T) + self.R   # project system uncertainty into measurment space + measurement noise(R)
-
-
-        K = self.P * self.H.T * linalg.inv(S) # map system uncertainty into kalman gain
-
-        self.x = self.x + (K*y)                # predict new x with residual scaled by the kalman gain
-
-        # and compute the new covariance
-        self.P = (self.I - (K*self.H))*self.P  # and compute the new covariance
-
-
-
-    def predict(self):
-        # prediction
-        self.x = (self.F*self.x) + self.B * self.u
-        self.P = self.F * self.P * self.F.T + self.Q
-
-
-if __name__ == "__main__":
-    f = KalmanFilter (dim_x=2, dim_z=2)
-
-    f.x = np.matrix([[2.],
-                     [0.]])       # initial state (location and velocity)
-
-    f.F = np.matrix([[1.,1.],
-                     [0.,1.]])    # state transition matrix
-
-    f.H = np.matrix([[1.,0.]])    # Measurement function
-    f.P *= 1000.                  # covariance matrix
-    f.R = 5                       # state uncertainty
-    f.Q *= 0.0001                 # process uncertainty
-
-    measurements = []
-    results = []
-    for t in range (100):
-        # create measurement = t plus white noise
-        z = t + random.randn()*20
-
-        # perform kalman filtering
-        f.measure (z)
-        f.predict()
-
-        # save data
-        results.append (f.x[0,0])
-        measurements.append(z)
-
-    # plot data
-    p1, = plt.plot(measurements,'r')
-    p2, = plt.plot (results,'b')
-    p3, = plt.plot ([0,100],[0,100], 'g') # perfect result
-    plt.legend([p1,p2, p3], ["noisy measurement", "KF output", "ideal"], 4)
-
-
-    plt.show()

KalmanFilter1D.py

@@ -1,136 +0,0 @@
-# -*- coding: utf-8 -*-
-"""
-Created on Thu May  1 19:48:54 2014
-
-@author: rlabbe
-"""
-
-import math
-import matplotlib.pyplot as plt
-import noise
-import numba
-import numpy.random as random
-
-class KalmanFilter1D(object):
-
-    def __init__ (self, x0, var):
-        self.mean = x0
-        self.variance = var
-    
-    def estimate(self, z, z_variance):
-        self.mean = (self.variance*z + z_variance*self.mean) / (self.variance + z_variance)
-        self.variance = 1. / (1./self.variance + 1./z_variance)
-
-    def project(self, u, u_variance):
-        self.mean += u
-        self.variance += u_variance
-
-
-def _fixed_error_kf(measurement_error, motion_error, noise_factor = 1.0):
-    mean = 0
-    sig = 1000
-    measurements = [x for x in range(100)]
-    f = KalmanFilter1D (mean,sig)
-
-    ys = []
-    errs = []
-    xs = []
-
-    for i in range(len(measurements)):
-        r = noise.white_noise (noise_factor)
-        z = measurements[i] + r
-        f.estimate (z, measurement_error)
-
-        xs.append(z)
-        ys.append(f.mean)
-        errs.append (f.variance)
-
-        f.project (1, motion_error)
-
-    plt.clf()
-
-    p1, = plt.plot (measurements, 'r')
-    p2, = plt.plot (xs,'g')
-    p3, = plt.plot (ys, 'b')
-    plt.legend ([p1,p2,p3],['actual', 'measurement', 'filter'], 2)
-    #plt.errorbar (x=range(len(ys)), color='b', y=ys, yerr=errs)
-    plt.show()
-
-def _varying_error_kf(noise_factor=1.0):
-    motion_sig = 2.
-    mean = 0
-    sig = 1000
-
-    measurements = [x for x in range(100)]
-
-    f = KalmanFilter1D (mean,sig)
-    ys = []
-    errs = []
-    xs = []
-
-    for i in range(len(measurements)):
-        r = random.randn() * noise_factor
-        m = measurements[i] + r
-
-        f.estimate (m, abs(r*10))
-        xs.append(m)
-        ys.append(f.mean)
-        errs.append (f.variance)
-
-        f.project (1.0, motion_sig)
-
-    plt.clf()
-    plt.plot (measurements, 'r')
-    plt.plot (xs,'g')
-    plt.errorbar (x=range(len(ys)), color='b', y=ys, yerr=errs)
-    plt.show()
-
-
-
-def _test_sin ():
-    sensor_error = 1
-    movement = .1
-    movement_error = .1
-    pos = (1,500)
-
-    zs = []
-    ps = []
-    filter_ = KalmanFilter1D(pos[0],pos[1])
-    m_1 = filter_.mean
-
-    for i in range(300):
-        filter_.project(movement, movement_error)
-
-        Z = math.sin(i/12.) + math.sqrt(abs(noise.white_noise(.02)))
-        movement = filter_.mean - m_1
-        m_1 = filter_.mean
-
-        zs.append(Z)
-
-        filter_.estimate (Z, sensor_error)
-        ps.append(filter_.mean)
-
-
-    p1, = plt.plot(zs,c='r', linestyle='dashed')
-    p2, = plt.plot(ps, c='b')
-    plt.legend([p1,p2], ['measurement', 'filter'], 2)
-    plt.show()
-
-if __name__ == "__main__":
-
-    if 0:
-        # use same seed to get repeatable results
-        random.seed(10)
-
-    if 1:
-        plt.figure()
-        _varying_error_kf ()
-    if 1:
-        plt.figure()
-        _fixed_error_kf(measurement_error=1.,
-                        motion_error=.1,
-                        noise_factor=50)
-
-    if 1:
-        plt.figure()
-        _test_sin()

Multidimensional_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:cacd3ef5d9830b08d539ac3933475779ace5170a2903b1fce9e7afb6e87e4362"
+  "signature": "sha256:8308d6b4e959bab21d85e3e32855ed8611242c5db237c72712607673761dd677"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -248,13 +248,13 @@
        ],
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 5,
+       "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x20207d0>"
+        "<IPython.core.display.HTML at 0x2dddcd0>"
        ]
       }
      ],
-     "prompt_number": 5
+     "prompt_number": 1
     },
     {
      "cell_type": "heading",
@@ -528,7 +528,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x28053d0>"
+        "<matplotlib.figure.Figure at 0x4450250>"
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        "text": [
-        "<matplotlib.figure.Figure at 0x27f34d0>"
+        "<matplotlib.figure.Figure at 0x4661350>"
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@@ -660,7 +660,7 @@
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        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2a5c7d0>"
+        "<matplotlib.figure.Figure at 0x4450110>"
        ]
       }
      ],
@@ -687,7 +687,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x29831d0>"
+        "<matplotlib.figure.Figure at 0x46d4b90>"
        ]
       }
      ],
@@ -714,7 +714,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x2924e10>"
+        "<matplotlib.figure.Figure at 0x4688310>"
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2b7e110>"
+        "<matplotlib.figure.Figure at 0x4653b50>"
        ]
       }
      ],
@@ -798,7 +798,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x29aba10>"
+        "<matplotlib.figure.Figure at 0x4682310>"
        ]
       }
      ],
@@ -1101,7 +1101,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "As promised, the Kalman filter equations are already programmed for you. In many circumstances you will never have to write your own Kalman filter equations. We will look at the code later, but for now we will just import the code and use it. I have placed it in `KalmanFilter.py`, so let's start by importing it and creating a filter."
+      "As promised, the Kalman filter equations are already programmed for you. In many circumstances you will never have to write your own Kalman filter equations. We will look at the code later, but for now we will just import the code and use it. It lives in `filterpy.kalman` so let's start by importing it and creating a filter."
      ]
     },
     {
@@ -1109,13 +1109,13 @@
      "collapsed": false,
      "input": [
       "import numpy as np\n",
-      "from KalmanFilter import KalmanFilter\n",
-      "dog_filter = KalmanFilter (dim_x=2, dim_z=2)"
+      "import filterpy.kalman as kf\n",
+      "dog_filter = kf.KalmanFilter (dim_x=2, dim_z=2)"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -1130,12 +1130,12 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "dog_filter.x = np.matrix([[0], [0]])    # initial state (location and velocity)\n",
-      "dog_filter.F = np.matrix([[1,1], [0,1]]) # state transition matrix\n",
-      "dog_filter.H = np.matrix([[1,0]])       # Measurement function\n",
-      "dog_filter.R  = 5                       # measurement noise\n",
-      "dog_filter.Q = 0                        # process noise\n",
-      "dog_filter.P *= 500.                    # covariance matrix \n"
+      "dog_filter.x = np.array([[0], [0]])     # initial 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",
+      "dog_filter.Q *= 0                       # process noise\n",
+      "dog_filter.P *= 500.                    # covariance matrix "
      ],
      "language": "python",
      "metadata": {},
@@ -1181,19 +1181,20 @@
      "collapsed": false,
      "input": [
       "from DogSensor import DogSensor\n",
+      "import filterpy.kalman as kf\n",
       "\n",
       "def dog_tracking_filter(R,Q=0,cov=1.):\n",
-      "    dog_filter = KalmanFilter (dim_x=2, dim_z=2)\n",
-      "    dog_filter.x = np.matrix([[0], \n",
-      "                              [0]])   # initial state (location and velocity)\n",
-      "    dog_filter.F = np.matrix([[1,1],\n",
-      "                              [0,1]]) # state transition matrix\n",
-      "    dog_filter.H = np.matrix([[1,0]]) # Measurement function\n",
-      "    dog_filter.R = R                  # measurement uncertainty\n",
+      "    dog_filter = kf.KalmanFilter (dim_x=2, dim_z=1)\n",
+      "    dog_filter.x = np.array([[0], \n",
+      "                             [0]])   # initial state (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",
+      "    dog_filter.R *= R                 # measurement uncertainty\n",
       "    dog_filter.P *= cov               # covariance matrix \n",
       "    if np.isscalar(Q):\n",
-      "        dog_filter.Q = np.matrix([[0,0],\n",
-      "                                  [0,Q]])\n",
+      "        dog_filter.Q = np.array([[0,0],\n",
+      "                                 [0,Q]])\n",
       "    else:\n",
       "        dog_filter.Q = Q\n",
       "    return dog_filter\n",
@@ -1232,7 +1233,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 18
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1300,7 +1301,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 19
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1321,9 +1322,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2976710>"
+        "<matplotlib.figure.Figure at 0x4a10610>"
        ]
       },
       {
@@ -1331,11 +1332,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2989550>"
+        "<matplotlib.figure.Figure at 0x4534b50>"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1362,52 +1363,90 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
+      "# -*- coding: utf-8 -*-\n",
+      "\"\"\"\n",
+      "Created on Fri Oct 18 18:02:07 2013\n",
+      "\n",
+      "@author: rlabbe\n",
+      "\"\"\"\n",
+      "\n",
       "import numpy as np\n",
       "import scipy.linalg as linalg\n",
+      "import matplotlib.pyplot as plt\n",
+      "import numpy.random as random\n",
       "\n",
       "class KalmanFilter:\n",
       "\n",
-      "    def __init__(self, dim_x, dim_z):\n",
-      "        \"\"\" Create a Kalman filter of dimension 'dim'\"\"\"\n",
-      "        \n",
+      "    def __init__(self, dim_x, dim_z, use_short_form=False):\n",
+      "        \"\"\" Create a Kalman filter with 'dim_x' state variables, and\n",
+      "        'dim_z' measurements. You are responsible for setting the various\n",
+      "        state variables to reasonable values; the defaults below will\n",
+      "        not give you a functional filter.\n",
+      "        \"\"\"\n",
+      "\n",
       "        self.x = 0 # state\n",
-      "        self.P = np.matrix(np.eye(dim_x)) # uncertainty covariance\n",
-      "        self.Q = np.matrix(np.eye(dim_x)) # process uncertainty\n",
-      "        self.u = np.matrix(np.zeros((dim_x,1))) # motion vector\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\n",
       "        self.F = 0 # state transition matrix\n",
       "        self.H = 0 # Measurement function (maps state to measurements)\n",
-      "        self.R = np.matrix(np.eye(dim_z)) # state uncertainty\n",
-      "        self.I = np.matrix(np.eye(dim_x))\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):\n",
+      "    def update(self, Z, R=None):\n",
       "        \"\"\"\n",
-      "        Add a new measurement to the kalman filter.\n",
+      "        Add a new measurement (Z) to the kalman filter. \n",
+      "        \n",
+      "        Optionally provide R to override the measurement noise for this \n",
+      "        one call, otherwise  self.R will be used.\n",
+      "        \n",
+      "        self.residual, self.S, and self.K are stored in case you want to\n",
+      "        inspect these variables. Strictly speaking they are not part of the\n",
+      "        output of the Kalman filter, however, it is often useful to know\n",
+      "        what these values are in various scenarios.\n",
       "        \"\"\"\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",
-      "        # measurement update\n",
-      "        y = Z - (self.H * self.x)                   # error (residual) between measurement \n",
-      "                                                    # and prediction\n",
-      "        S = (self.H * self.P * self.H.T) + self.R   # project system uncertainty into \n",
-      "                                                    # measurment space + measurement noise(R)\n",
+      "        # error (residual) between measurement and prediction\n",
+      "        self.residual = Z - self.H.dot(self.x)\n",
+      "        \n",
+      "        # project system uncertainty into measurement space \n",
+      "        self.S = self.H.dot(self.P).dot(self.H.T) + R   \n",
       "\n",
+      "        # map system uncertainty into kalman gain\n",
+      "        self.K = self.P.dot(self.H.T).dot(linalg.inv(self.S)) \n",
       "\n",
-      "        K = self.P * self.H.T * linalg.inv(S) # map system uncertainty into kalman gain\n",
+      "        # predict new x with residual scaled by the kalman gain\n",
+      "        self.x = self.x + self.K.dot(self.residual)                \n",
+      "\n",
+      "        KH = self.K.dot(self.H)\n",
+      "        I_KH = self._I - KH\n",
+      "        self.P = (I_KH.dot(self.P.dot(I_KH.T)) + \n",
+      "                 self.K.dot(self.R.dot(self.K.T)))\n",
       "\n",
-      "        self.x = self.x + (K*y)                # predict new x with residual scaled \n",
-      "                                               #by the kalman gain\n",
-      "        self.P = (self.I - (K*self.H))*self.P  # and compute the new covariance\n",
       "\n",
       "    def predict(self):\n",
-      "        # prediction\n",
-      "        self.x = (self.F*self.x) + self.u\n",
-      "        self.P = self.F * self.P * self.F.T + self.Q"
+      "        \"\"\" predict next position \"\"\"\n",
+      "    \n",
+      "        self.x = self.F.dot(self.x)\n",
+      "        if self.B != 0:\n",
+      "           self.x += self.B.dot(self.u)\n",
+      "        self.P = self.F.dot(self.P).dot(self.F.T) + self.Q"
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [],
-     "prompt_number": 21
+     "outputs": []
     },
     {
      "cell_type": "heading",
@@ -1440,25 +1479,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x2e8ae90>"
-       ]
-      },
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x2805190>"
-       ]
-      }
-     ],
-     "prompt_number": 22
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -1563,17 +1584,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x28f9950>"
-       ]
-      }
-     ],
-     "prompt_number": 23
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -1632,17 +1643,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x29773d0>"
-       ]
-      }
-     ],
-     "prompt_number": 24
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -1660,17 +1661,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x29ab8d0>"
-       ]
-      }
-     ],
-     "prompt_number": 25
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -1751,25 +1742,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x2e93f50>"
-       ]
-      },
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x2ee3bd0>"
-       ]
-      }
-     ],
-     "prompt_number": 26
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -1826,17 +1799,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": "iVBORw0KGgoAAAANSUhEUgAAA3wAAAF9CAYAAABf12quAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x2e86150>"
-       ]
-      }
-     ],
-     "prompt_number": 27
+     "outputs": []
     },
     {
      "cell_type": "markdown",

UKF.py

@@ -1,99 +0,0 @@
-# -*- coding: utf-8 -*-
-"""
-Created on Wed May 28 17:40:19 2014
-
-@author: rlabbe
-"""
-
-from numpy import matrix, zeros, asmatrix, size
-from  numpy.linalg import cholesky
-
-
-def sigma_points (mean, covariance, kappa):
-    """ Computes the sigma points and weights for an unscented Kalman filter.
-    xm are the means, and P is the covariance. kappa is an arbitrary constant
-    constant. Returns tuple of the sigma points and weights.
-
-    This is the original algorithm as published by Julier and Uhlmann.
-    Later algorithms introduce more parameters - alpha, beta,
-
-    Works with both scalar and array inputs:
-    sigma_points (5, 9, 2) # mean 5, covariance 9
-    sigma_points ([5, 2], 9*eye(2), 2) # means 5 and 2, covariance 9I
-    """
-
-    mean = asmatrix(mean)
-    covariance = asmatrix(covariance)
-
-    n = size(mu)
-
-    # initialize to zero
-    Xi = asmatrix (zeros((n,2*n+1)))
-    W  = asmatrix (zeros(2*n+1))
-
-
-    # all weights are 1/ 2(n+kappa)) except the first one.
-    W[0,1:] = 1. / (2*(n+kappa))
-    W[0,0] = float(kappa) / (n + kappa)
-
-
-    # use cholesky to find matrix square root of (n+kappa)*cov
-    #  U'*U = (n+kappa)*P
-    U = asmatrix (cholesky((n+kappa)*covariance))
-
-    # mean is in location 0.
-    Xi[:,0] = mean
-
-    for col in range (n):
-        Xi[:, col+1] = mean + U[:, col]
-
-    for k in range (n):
-        Xi[:, n+col+1] = mean - U[:, col]
-
-    return (Xi, W)
-
-
-
-def unscented_transform (Xi, W, NoiseCov=None):
-    """ computes the unscented transform of a set of sigma points 'X'
-    and weights 'W'.
-
-    W should be in the form:
-
-       [w0, w1, w2,...wn].T
-
-    where w0 is the mean,
-
-    Xi should be in the form:
-       [X_00,
-
-
-    returns the mean and covariance in a tuple '(mean, cov)'
-    """
-
-    W  = asmatrix(W)
-    Xi = asmatrix(Xi)
-
-    n, kmax = Xi.shape
-
-    # initialize results to 0
-    mean = matrix (zeros((n,1)))
-    cov  = matrix (zeros((n,n)))
-
-    for k in range (kmax):
-        mean += W[0,k] * Xi[:,k]
-
-    for k in range (kmax):
-        cov += W[0,k]*(Xi[:,k]-mu) * (Xi[:,k]-mu).T
-
-    return (mean, cov)
-
-
-if __name__ == "__main__":
-
-    xi = matrix ([[1,2,3,4,5,6,7],[1,2,3,4,5,6,7],[1,2,3,4,5,6,7]])
-    mu = matrix ([1,2,3,4,5,6,7])
-
-    m,c = unscented_transform(xi, mu)
-    print m
-    print c

discrete_bayes.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:a7f391a6729b3d01b0ccfdf3825a3ab878f59e1d4937e5135a8b6c0f64c253a2"
+  "signature": "sha256:4596c3be2efa3686f3281d6c9e9e897bfa114ca6945b5da108745cb466846335"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -41,7 +41,7 @@
         "        margin-right: auto;\n",
         "    }\n",
         "    div.text_cell code {\n",
-        "        background: #FFFFFF;\n",
+        "        background: transparent;\n",
         "        color: #000000;\n",
         "        font-weight: 600;\n",
         "        font-size: 13pt;\n",
@@ -250,7 +250,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x25adb90>"
+        "<IPython.core.display.HTML at 0x2919d50>"
        ]
       }
      ],
@@ -344,9 +344,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsMAAAFyCAYAAAAQ8tcPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x364c910>"
+        "<matplotlib.figure.Figure at 0x2919890>"
        ]
       }
      ],
@@ -705,9 +705,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x374c650>"
+        "<matplotlib.figure.Figure at 0x3933e90>"
        ]
       }
      ],
@@ -743,9 +743,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x36598d0>"
+        "<matplotlib.figure.Figure at 0x3918690>"
        ]
       }
      ],
@@ -782,9 +782,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3731e10>"
+        "<matplotlib.figure.Figure at 0x392f690>"
        ]
       }
      ],
@@ -838,9 +838,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3749050>"
+        "<matplotlib.figure.Figure at 0x3928e90>"
        ]
       }
      ],
@@ -875,9 +875,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3748bd0>"
+        "<matplotlib.figure.Figure at 0x394e710>"
        ]
       }
      ],
@@ -904,9 +904,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x374fa90>"
+        "<matplotlib.figure.Figure at 0x3b56d50>"
        ]
       }
      ],
@@ -933,9 +933,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3760290>"
+        "<matplotlib.figure.Figure at 0x3916690>"
        ]
       }
      ],
@@ -985,9 +985,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3746fd0>"
+        "<matplotlib.figure.Figure at 0x391fa50>"
        ]
       },
       {
@@ -1022,9 +1022,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x373a910>"
+        "<matplotlib.figure.Figure at 0x3b58150>"
        ]
       }
      ],
@@ -1054,9 +1054,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x390ff50>"
+        "<matplotlib.figure.Figure at 0x3b59c10>"
        ]
       }
      ],
@@ -1108,9 +1108,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x37655d0>"
+        "<matplotlib.figure.Figure at 0x3b56110>"
        ]
       }
      ],

exp/KalmanFilter.py

@@ -1,131 +0,0 @@
-# -*- coding: utf-8 -*-
-"""
-Created on Fri Oct 18 18:02:07 2013
-
-@author: rlabbe
-"""
-
-import numpy as np
-import scipy.linalg as linalg
-import matplotlib.pyplot as plt
-import numpy.random as random
-
-class KalmanFilter:
-
-    def __init__(self, dim_x, dim_z, use_short_form=False):
-        """ Create a Kalman filter of dimension 'dim', where dimension is the
-        number of state variables.
-        
-        use_short_form will force the filter to use the short form equation
-        for computing covariance: (I-KH)P. This is the equation that results
-        from deriving the Kalman filter equations. It is efficient to compute
-        but not numerically stable for very long runs. The long form,
-        implemented in update(), is very slightly suboptimal, and takes longer
-        to compute, but is stable. I have chosen to make the long form the
-        default behavior. If you really want the short form, either call
-        update_short_form() directly, or set 'use_short_form' to true. This
-        causes update() to use the short form. So, if you do it this way
-        you will never be able to use the long form again with this object.
-        """
-
-        self.x = 0 # state
-        self.P = np.matrix(np.eye(dim_x)) # uncertainty covariance
-        self.Q = np.matrix(np.eye(dim_x)) # process uncertainty
-        self.u = np.matrix(np.zeros((dim_x,1))) # motion vector
-        self.B = 0
-        self.F = 0 # state transition matrix
-        self.H = 0 # Measurement function (maps state to measurements)
-        self.R = np.matrix(np.eye(dim_z)) # state uncertainty
-        self.I = np.matrix(np.eye(dim_x))
-        
-        if use_short_form:
-            self.update = self.update_short_form
-
-
-    def update(self, Z):
-        """
-        Add a new measurement to the kalman filter.
-        """
-
-        # measurement update
-        y = Z - (self.H * self.x)                   # error (residual) between measurement and prediction
-        S = (self.H * self.P * self.H.T) + self.R   # project system uncertainty into measurment space + measurement noise(R)
-
-
-        K = self.P * self.H.T * linalg.inv(S) # map system uncertainty into kalman gain
-
-        self.x = self.x + (K*y)                # predict new x with residual scaled by the kalman gain
-
-        KH = K*self.H
-        I_KH = self.I - KH
-        self.P = I_KH*self.P * I_KH.T + K*self.R*K.T
-
-
-    def update_short_form(self, Z):
-        """
-        Add a new measurement to the kalman filter.
-        
-        Uses the 'short form' computation for P, which is mathematically
-        correct, but perhaps not that stable when dealing with large data
-        sets. But, it is fast to compute. Advice varies; some say to never
-        use this. My opinion - if the longer form in update() executes too
-        slowly to run in real time, what choice do you have. But that should
-        be a rare case, so the long form is the default use
-        """
-
-        # measurement update
-        y = Z - (self.H * self.x)                   # error (residual) between measurement and prediction
-        S = (self.H * self.P * self.H.T) + self.R   # project system uncertainty into measurment space + measurement noise(R)
-
-
-        K = self.P * self.H.T * linalg.inv(S) # map system uncertainty into kalman gain
-
-        self.x = self.x + (K*y)                # predict new x with residual scaled by the kalman gain
-
-        # and compute the new covariance
-        self.P = (self.I - (K*self.H))*self.P  # and compute the new covariance
-
-
-
-    def predict(self):
-        # prediction
-        self.x = (self.F*self.x) + self.B * self.u
-        self.P = self.F * self.P * self.F.T + self.Q
-
-
-if __name__ == "__main__":
-    f = KalmanFilter (dim_x=2, dim_z=2)
-
-    f.x = np.matrix([[2.],
-                     [0.]])       # initial state (location and velocity)
-
-    f.F = np.matrix([[1.,1.],
-                     [0.,1.]])    # state transition matrix
-
-    f.H = np.matrix([[1.,0.]])    # Measurement function
-    f.P *= 1000.                  # covariance matrix
-    f.R = 5                       # state uncertainty
-    f.Q *= 0.0001                 # process uncertainty
-
-    measurements = []
-    results = []
-    for t in range (100):
-        # create measurement = t plus white noise
-        z = t + random.randn()*20
-
-        # perform kalman filtering
-        f.measure (z)
-        f.predict()
-
-        # save data
-        results.append (f.x[0,0])
-        measurements.append(z)
-
-    # plot data
-    p1, = plt.plot(measurements,'r')
-    p2, = plt.plot (results,'b')
-    p3, = plt.plot ([0,100],[0,100], 'g') # perfect result
-    plt.legend([p1,p2, p3], ["noisy measurement", "KF output", "ideal"], 4)
-
-
-    plt.show()

exp/dme.py

@@ -0,0 +1,108 @@
+# -*- coding: utf-8 -*-
+"""
+Spyder Editor
+
+This is a temporary script file.
+"""
+
+from KalmanFilter import *
+from math import cos, sin, sqrt, atan2
+
+
+def H_of (pos, pos_A, pos_B):
+    """ Given the position of our object at 'pos' in 2D, and two transmitters
+    A and B at positions 'pos_A' and 'pos_B', return the partial derivative
+    of H
+    """
+
+    theta_a = atan2(pos_a[1]-pos[1], pos_a[0] - pos[0])
+    theta_b = atan2(pos_b[1]-pos[1], pos_b[0] - pos[0])
+
+    if False:
+        return np.mat([[0, -cos(theta_a), 0, -sin(theta_a)],
+                       [0, -cos(theta_b), 0, -sin(theta_b)]])
+    else:                  
+        return np.mat([[-cos(theta_a), 0, -sin(theta_a), 0],
+                       [-cos(theta_b), 0, -sin(theta_b), 0]])
+
+class DMESensor(object):
+    def __init__(self, pos_a, pos_b, noise_factor=1.0):
+        self.A = pos_a
+        self.B = pos_b
+        self.noise_factor = noise_factor
+
+    def range_of (self, pos):
+        """ returns tuple containing noisy range data to A and B
+        given a position 'pos'
+        """
+
+        ra = sqrt((self.A[0] - pos[0])**2 + (self.A[1] - pos[1])**2)
+        rb = sqrt((self.B[0] - pos[0])**2 + (self.B[1] - pos[1])**2)
+
+        return (ra + random.randn()*self.noise_factor,
+                rb + random.randn()*self.noise_factor)
+
+
+pos_a = (100,-20)
+pos_b = (-100, -20)
+
+f1 = KalmanFilter(dim_x=4, dim_z=2)
+
+f1.F = np.mat ([[0, 1, 0, 0],
+                [0, 0, 0, 0],
+                [0, 0, 0, 1],
+                [0, 0, 0, 0]])
+
+f1.B = 0.
+
+f1.R *= 1.
+f1.Q *= .1
+
+f1.x = np.mat([1,0,1,0]).T
+f1.P = np.eye(4) * 5.
+
+# initialize storage and other variables for the run
+count = 30
+xs, ys = [],[]
+pxs, pys = [],[]
+
+# create the simulated sensor
+d = DMESensor (pos_a, pos_b, noise_factor=1.)
+
+# pos will contain our nominal position since the filter does not
+# maintain position.
+pos = [0,0]
+
+for i in range(count):
+    # move (1,1) each step, so just use i
+    pos = [i,i]
+    
+    # compute the difference in range between the nominal track and measured 
+    # ranges
+    ra,rb = d.range_of(pos)
+    rx,ry = d.range_of((i+f1.x[0,0], i+f1.x[2,0]))
+    
+    print ra, rb
+    print rx,ry
+    z = np.mat([[ra-rx],[rb-ry]])
+    print z.T
+
+    # compute linearized H for this time step
+    f1.H = H_of (pos, pos_a, pos_b)
+
+    # store stuff so we can plot it later
+    xs.append (f1.x[0,0]+i)
+    ys.append (f1.x[2,0]+i)
+    pxs.append (pos[0])
+    pys.append(pos[1])
+    
+    # perform the Kalman filter steps
+    f1.predict ()
+    f1.update(z)
+
+
+p1, = plt.plot (xs, ys, 'r--')
+p2, = plt.plot (pxs, pys)
+plt.legend([p1,p2], ['filter', 'ideal'], 2)
+plt.show()  
+    

exp/temp.py

@@ -0,0 +1,9 @@
+# -*- coding: utf-8 -*-
+"""
+Spyder Editor
+
+This is a temporary script file.
+"""
+
+import KalmanFilter
+

g-h_filter.ipynb

@@ -1414,7 +1414,12 @@
       "\n",
       "I assumed that my weight gain was constant at 1 lb/day, and so when I tried to fit a polynomial of $n=1$, which is a line, the result very closely matched the actual weight gain. But, of course, no one consistantly only gains or loses weight. We flucuate. Using 'polyfit()' for a longer series of data would yield poor results. In contrast, the g-h filter reacts to changes in the rate - the $h$ term controls how quickly  the filter reacts to these changes. If we gain weight, hold steady for awhile, then lose weight, the filter will track that change automatically. 'polyfit()' would not be able to do that unless the gain and loss could be well represented by a polynomial.\n",
       "\n",
-      "Another advantage of this form of filter, even if the data fits a *n*-degree polynomial, is that it is *recursive*. That is, we can compute the estimate for this time period knowing nothing more than the estimate and rate from the last time period. In contrast, if you dig into the implementation for `polyfit()` you will see that it needs all of the data before it can produce an answer. Therefore algorithms like `polyfit()` are not well suited for real-time data filtering. In the 60's when the Kalman filter was developed computers were very slow and had extremely limited memory. They were utterly unable to store, for example, thousands of readings from an aircraft's inertial navigation system, nor could they process all of that data in the short period of time needed to provide accurate and up-to-date navigation information. The Kalman filter only needs to store the last estimate and a few related parameters, and requires only a relatively small number of computations to generate the next estimate. Today we have so much memory and processing power that this advantage is somewhat less important, but at the time the Kalman filter was a major breakthrough not just because of the mathematical properties, but because it could (barely) run on the hardware of the day. \n",
+      "Another advantage of this form of filter, even if the data fits a *n*-degree polynomial, is that it is *recursive*. That is, we can compute the estimate for this time period knowing nothing more than the estimate and rate from the last time period. In contrast, if you dig into the implementation for `polyfit()` you will see that it needs all of the data before it can produce an answer. Therefore algorithms like `polyfit()` are not well suited for real-time data filtering. In the 60's when the Kalman filter was developed computers were very slow and had extremely limited memory. They were utterly unable to store, for example, thousands of readings from an aircraft's inertial navigation system, nor could they process all of that data in the short period of time needed to provide accurate and up-to-date navigation information. \n",
+      "\n",
+      "\n",
+      "Up until the mid 20th century various forms of Least Squares Estimation was used for this type of filtering. For example, for NASA's Apollo program had a ground network for tracking the Command and Service Model (CSM) and the Lunar Module (LM). They took measurements over many minutes, batched the data together, and slowly computed an answer. In 1960 Stanley Schmidt at NASA Ames recognized the utility of Rudolf Kalman's seminal paper and invited him to Ames. Schmidt applied Kalman's work to the onboard navigation sytems on the CSM and LM, and called it the \"Kalman filter\".[1] Soon after, the world moved to this faster, recursive filter.\n",
+      "\n",
+      "The Kalman filter only needs to store the last estimate and a few related parameters, and requires only a relatively small number of computations to generate the next estimate. Today we have so much memory and processing power that this advantage is somewhat less important, but at the time the Kalman filter was a major breakthrough not just because of the mathematical properties, but because it could (barely) run on the hardware of the day. \n",
       "\n",
       "This subject is much deeper than this short discussion suggests. We will consider these topics many more times throughout the book."
      ]
@@ -1443,12 +1448,19 @@
       "There is another feature of these filters we have barely touched upon - Bayesian statistics. You will note that the term 'Bayesian' is in the title of this book; this is not a coincidence! For the time being we will leave $g$ and $h$ behind, largely unexplored, and develop a very powerful form of probabilistic reasoning about filtering. Yet suddenly this same g-h filter algorithm will appear, this time with a formal mathematical edifice that allows us to create filters from multiple sensors, to accurately estimate the amount of error in our solution, and to control robots."
      ]
     },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "References"
+     ]
+    },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "missing: properties - recursive - markov - random variable\n",
-      "should we fit bayes into here, or just leave to the next chapter"
+      "[1] http://nescacademy.nasa.gov/review/downloadfile.php?file=FundamentalsofKalmanFiltering_Presentation.pdf&id=199&distr=Public"
      ]
     }
    ],