Various changes. Main thing is removed the slow gaussian class from

the kalman filters. It was annoying anyway - have to pack and unpack
it everywhere when calling around.

Kalman Filters.ipynb

@@ -50,7 +50,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 3
+     "prompt_number": 1
     },
     {
      "cell_type": "markdown",
@@ -74,11 +74,11 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "-0.9868 -0.7268 -1.2716 1.8108 -0.0831 0.5857 0.3981 -0.2151 0.5696 0.7693 -2.0554 0.0504 -0.4504 0.2174 -0.0658 -0.7274 -1.7664 -0.1899 1.8342 1.9822\n"
+        "-0.1102 0.0360 -0.4926 0.7730 1.2450 0.4095 -1.9778 -0.1399 1.1690 1.4289 1.4885 -0.7673 -2.8012 1.3191 -0.5105 2.1412 -0.1252 -2.1175 -0.1274 -0.5646\n"
        ]
       }
      ],
-     "prompt_number": 4
+     "prompt_number": 2
     },
     {
      "cell_type": "markdown",
@@ -117,7 +117,7 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAW4AAAEACAYAAACTXJylAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3998a50>"
+        "<matplotlib.figure.Figure at 0x2c90810>"
        ]
       },
       {
@@ -128,7 +128,7 @@
        ]
       }
      ],
-     "prompt_number": 5
+     "prompt_number": 3
     },
     {
      "cell_type": "markdown",
@@ -167,13 +167,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAAEZCAYAAACJjGL9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x39e8dd0>"
+        "<matplotlib.figure.Figure at 0x2ca7410>"
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 4
     },
     {
      "cell_type": "markdown",
@@ -200,13 +200,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x39aa8d0>"
+        "<matplotlib.figure.Figure at 0x2c9ffd0>"
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 5
     },
     {
      "cell_type": "code",
@@ -220,13 +220,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x39b62d0>"
+        "<matplotlib.figure.Figure at 0x2cbf3d0>"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 6
     },
     {
      "cell_type": "markdown",
@@ -259,11 +259,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x39b14d0>"
+        "<matplotlib.figure.Figure at 0x2eef290>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 7
     },
     {
      "cell_type": "markdown",
@@ -295,13 +295,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3998a90>"
+        "<matplotlib.figure.Figure at 0x2c9f410>"
        ]
       }
      ],
-     "prompt_number": 10
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -379,11 +379,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3a5e410>"
+        "<matplotlib.figure.Figure at 0x2f08a50>"
        ]
       }
      ],
-     "prompt_number": 11
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -424,11 +424,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x38ebfd0>"
+        "<matplotlib.figure.Figure at 0x2f14950>"
        ]
       }
      ],
-     "prompt_number": 12
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -464,7 +464,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 13
+     "prompt_number": 11
     },
     {
      "cell_type": "markdown",
@@ -483,7 +483,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 14
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
@@ -513,30 +513,30 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "time: 0 position =  1.439 variance =  2.500\n",
-        "time: 1 position =  1.151 variance =  1.667\n",
-        "time: 2 position =  1.143 variance =  1.250\n",
-        "time: 3 position =  1.171 variance =  1.000\n",
-        "time: 4 position =  0.915 variance =  0.833\n",
-        "time: 5 position =  1.102 variance =  0.714\n",
-        "time: 6 position =  1.095 variance =  0.625\n",
-        "time: 7 position =  0.984 variance =  0.556\n",
-        "time: 8 position =  0.943 variance =  0.500\n",
-        "time: 9 position =  0.957 variance =  0.455\n",
-        "time: 10 position =  0.804 variance =  0.417\n",
-        "time: 11 position =  0.907 variance =  0.385\n",
-        "time: 12 position =  0.605 variance =  0.357\n",
-        "time: 13 position =  0.626 variance =  0.333\n",
-        "time: 14 position =  0.612 variance =  0.312\n",
-        "time: 15 position =  0.610 variance =  0.294\n",
-        "time: 16 position =  0.718 variance =  0.278\n",
-        "time: 17 position =  0.648 variance =  0.263\n",
-        "time: 18 position =  0.589 variance =  0.250\n",
-        "time: 19 position =  0.564 variance =  0.238\n"
+        "time: 0 position =  0.929 variance =  2.500\n",
+        "time: 1 position =  0.612 variance =  1.667\n",
+        "time: 2 position =  0.826 variance =  1.250\n",
+        "time: 3 position =  0.710 variance =  1.000\n",
+        "time: 4 position =  0.333 variance =  0.833\n",
+        "time: 5 position =  0.233 variance =  0.714\n",
+        "time: 6 position =  0.145 variance =  0.625\n",
+        "time: 7 position =  0.140 variance =  0.556\n",
+        "time: 8 position =  0.005 variance =  0.500\n",
+        "time: 9 position =  0.109 variance =  0.455\n",
+        "time: 10 position =  -0.026 variance =  0.417\n",
+        "time: 11 position =  -0.178 variance =  0.385\n",
+        "time: 12 position =  -0.199 variance =  0.357\n",
+        "time: 13 position =  -0.111 variance =  0.333\n",
+        "time: 14 position =  -0.057 variance =  0.312\n",
+        "time: 15 position =  -0.073 variance =  0.294\n",
+        "time: 16 position =  -0.017 variance =  0.278\n",
+        "time: 17 position =  -0.086 variance =  0.263\n",
+        "time: 18 position =  -0.024 variance =  0.250\n",
+        "time: 19 position =  -0.061 variance =  0.238\n"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 13
     },
     {
      "cell_type": "markdown",
@@ -595,7 +595,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 16
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -645,47 +645,47 @@
        "stream": "stdout",
        "text": [
         "UPDATE: 1.0000 , 502.0000\n",
-        "SENSE: 1.8930 , 9.8047\n",
+        "SENSE: -1.5928 , 9.8047\n",
         "\n",
-        "UPDATE: 2.8930 , 11.8047\n",
-        "SENSE: 2.6048 , 5.4138\n",
+        "UPDATE: -0.5928 , 11.8047\n",
+        "SENSE: 2.6835 , 5.4138\n",
         "\n",
-        "UPDATE: 3.6048 , 7.4138\n",
-        "SENSE: 3.7046 , 4.2574\n",
+        "UPDATE: 3.6835 , 7.4138\n",
+        "SENSE: 3.7262 , 4.2574\n",
         "\n",
-        "UPDATE: 4.7046 , 6.2574\n",
-        "SENSE: 6.3120 , 3.8490\n",
+        "UPDATE: 4.7262 , 6.2574\n",
+        "SENSE: 5.0089 , 3.8490\n",
         "\n",
-        "UPDATE: 7.3120 , 5.8490\n",
-        "SENSE: 6.4078 , 3.6904\n",
+        "UPDATE: 6.0089 , 5.8490\n",
+        "SENSE: 5.4084 , 3.6904\n",
         "\n",
-        "UPDATE: 7.4078 , 5.6904\n",
-        "SENSE: 7.5106 , 3.6267\n",
+        "UPDATE: 6.4084 , 5.6904\n",
+        "SENSE: 4.3071 , 3.6267\n",
         "\n",
-        "UPDATE: 8.5106 , 5.6267\n",
-        "SENSE: 7.1136 , 3.6007\n",
+        "UPDATE: 5.3071 , 5.6267\n",
+        "SENSE: 6.1013 , 3.6007\n",
         "\n",
-        "UPDATE: 8.1136 , 5.6007\n",
-        "SENSE: 6.3304 , 3.5900\n",
+        "UPDATE: 7.1013 , 5.6007\n",
+        "SENSE: 5.9480 , 3.5900\n",
         "\n",
-        "UPDATE: 7.3304 , 5.5900\n",
-        "SENSE: 7.6277 , 3.5856\n",
+        "UPDATE: 6.9480 , 5.5900\n",
+        "SENSE: 8.1846 , 3.5856\n",
         "\n",
-        "UPDATE: 8.6277 , 5.5856\n",
-        "SENSE: 8.4180 , 3.5838\n",
+        "UPDATE: 9.1846 , 5.5856\n",
+        "SENSE: 8.8179 , 3.5838\n",
         "\n"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x39f0810>"
+        "<matplotlib.figure.Figure at 0x3184c90>"
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -797,13 +797,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3a57a50>"
+        "<matplotlib.figure.Figure at 0x2f10650>"
        ]
       }
      ],
-     "prompt_number": 18
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -848,13 +848,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3a57b50>"
+        "<matplotlib.figure.Figure at 0x319b710>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -899,13 +899,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3a83590>"
+        "<matplotlib.figure.Figure at 0x3174d50>"
        ]
       }
      ],
-     "prompt_number": 20
+     "prompt_number": 18
     },
     {
      "cell_type": "markdown",
@@ -951,13 +951,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3d75410>"
+        "<matplotlib.figure.Figure at 0x2f180d0>"
        ]
       }
      ],
-     "prompt_number": 21
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -1011,13 +1011,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAYYAAAEACAYAAAC3adEgAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x39f0050>"
+        "<matplotlib.figure.Figure at 0x317a390>"
        ]
       }
      ],
-     "prompt_number": 22
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -1050,14 +1050,14 @@
      "outputs": [
       {
        "ename": "SyntaxError",
-       "evalue": "invalid syntax (<ipython-input-23-1e7762f04e9a>, line 1)",
+       "evalue": "invalid syntax (<ipython-input-21-1e7762f04e9a>, line 1)",
        "output_type": "pyerr",
        "traceback": [
-        "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-23-1e7762f04e9a>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    author notes:\u001b[0m\n\u001b[1;37m               ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
+        "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-21-1e7762f04e9a>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    author notes:\u001b[0m\n\u001b[1;37m               ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
        ]
       }
      ],
-     "prompt_number": 23
+     "prompt_number": 21
     },
     {
      "cell_type": "code",
@@ -1094,11 +1094,19 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAX0AAAEACAYAAABfxaZOAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3a7ab50>"
+        "<matplotlib.figure.Figure at 0x31912d0>"
        ]
       }
      ],
-     "prompt_number": 27
+     "prompt_number": 22
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": []
     }
    ],
    "metadata": {}

KalmanFilter1D.py

@@ -4,32 +4,32 @@ Created on Thu May  1 19:48:54 2014
 
 @author: rlabbe
 """
-from __future__ import division
-import gauss as g;
+
 import math
 import matplotlib.pyplot as plt
 import noise
-import numpy.random as random
-
+import numba
 
 class KalmanFilter1D(object):
 
-    def __init__ (self, est_0=g.gaussian(0,1000)):
-        self.estimate = est_0
+    def __init__ (self, x0, var):
+        self.mean = x0
+        self.variance = var
+    @jit
+    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 update_estimate(self, Z):
-        self.estimate = self.estimate * Z
-
-    def project_ahead(self, U):
-        self.estimate = self.estimate + U
+    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):
-    mu = 0
+def _fixed_error_kf(measurement_error, motion_error, noise_factor = 1.0):
+    mean = 0
     sig = 1000
-    measurements = [x+5 for x in range(100)]
-    f = KalmanFilter1D (g.gaussian(mu,sig))
+    measurements = [x for x in range(100)]
+    f = KalmanFilter1D (mean,sig)
 
     ys = []
     errs = []
@@ -37,14 +37,14 @@ def fixed_error_kf(measurement_error, motion_error, noise_factor = 1.0):
 
     for i in range(len(measurements)):
         r = noise.white_noise (noise_factor)
-        m = measurements[i] + r
-        f.update_estimate (g.gaussian(m, measurement_error))
+        z = measurements[i] + r
+        f.estimate (z, measurement_error)
 
-        xs.append(m)
-        ys.append(f.estimate.mu)
-        errs.append (f.estimate.sigma)
+        xs.append(z)
+        ys.append(f.mean)
+        errs.append (f.variance)
 
-        f.project_ahead (g.gaussian(20, motion_error))
+        f.project (1, motion_error)
 
     plt.clf()
 
@@ -55,30 +55,28 @@ def fixed_error_kf(measurement_error, motion_error, noise_factor = 1.0):
     #plt.errorbar (x=range(len(ys)), color='b', y=ys, yerr=errs)
     plt.show()
 
-def varying_error_kf(noise_factor=1.0):
+def _varying_error_kf(noise_factor=1.0):
     motion_sig = 2.
-    mu = 0
+    mean = 0
     sig = 1000
 
+    measurements = [x for x in range(100)]
 
-    f = KF1D (mu,sig)
+    f = KalmanFilter1D (mean,sig)
     ys = []
-    us = []
     errs = []
     xs = []
 
     for i in range(len(measurements)):
         r = random.randn() * noise_factor
         m = measurements[i] + r
-        print (r)
-        f.update (m, abs(r*10))
-        xs.append(m)
-        #print ("measure:" + str(f.estimate))
-        ys.append(f.estimate.mu)
-        errs.append (f.estimate.sigma)
 
-        f.predict (1.0, motion_sig)
-        #print ("predict:" + str(f.estimate))
+        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')
@@ -87,29 +85,29 @@ def varying_error_kf(noise_factor=1.0):
     plt.show()
 
 
-def _test_foo ():
+
+def _test_sin ():
     sensor_error = 1
     movement = .1
     movement_error = .1
-    pos = g.gaussian(1,500)
+    pos = (1,500)
 
     zs = []
     ps = []
-    filter_ = KalmanFilter1D(pos)
-    m_1 = filter_.estimate.mu
+    filter_ = KalmanFilter1D(pos[0],pos[1])
+    m_1 = filter_.mean
 
     for i in range(300):
-        filter_.update_estimate(g.gaussian(movement, movement_error))
+        filter_.project(movement, movement_error)
 
         Z = math.sin(i/12.) + math.sqrt(abs(noise.white_noise(.02)))
-        movement = filter_.estimate.mu - m_1
-        m_1 = filter_.estimate.mu
+        movement = filter_.mean - m_1
+        m_1 = filter_.mean
 
-        print movement, filter_.estimate.sigma
         zs.append(Z)
 
-        filter_.project_ahead (g.gaussian(Z, sensor_error))
-        ps.append(filter_.estimate[0])
+        filter_.estimate (Z, sensor_error)
+        ps.append(filter_.mean)
 
 
     p1, = plt.plot(zs,c='r', linestyle='dashed')
@@ -119,11 +117,19 @@ def _test_foo ():
 
 if __name__ == "__main__":
 
-    if 1:
+    if 0:
         # use same seed to get repeatable results
         random.seed(10)
 
-    fixed_error_kf(measurement_error=100.,
-                   motion_error=.1,
-                   noise_factor=50)
-    #_test_foo()
+    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

@@ -63,7 +63,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 1
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -71,7 +71,7 @@
      "source": [
       "Let's use it to compute a few values just to make sure we know how to call and use the function, and then move on to more interesting things.\n",
       "\n",
-      "First, let's find the probability for our dog being at (2.5, 7.3) if we believe he is at (2,7) with a variance of 4 for x and a variance of 5 for y. This function requires us to pass everything in as numpy arrays (we will soon provide a more robust version that works with numpy matrices, numpy arrays, and/or scalars in any combinations. That code contains a lot of boilerplate which obscures the algorithm).\n",
+      "First, let's find the probability for our dog being at (2.5, 7.3) if we believe he is at (2,7) with a variance of 8 for x and a variance of 10 for y. This function requires us to pass everything in as numpy arrays (we will soon provide a more robust version that works with numpy matrices, numpy arrays, and/or scalars in any combinations. That code contains a lot of boilerplate which obscures the algorithm).\n",
       "\n",
       "So we set x to (2.5,7.3)"
      ]
@@ -85,7 +85,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 2
+     "prompt_number": 30
     },
     {
      "cell_type": "markdown",
@@ -103,7 +103,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 3
+     "prompt_number": 31
     },
     {
      "cell_type": "markdown",
@@ -116,12 +116,12 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "cov = np.array([[4,0],[0,5]])"
+      "cov = np.array([[8.,0],[0,10.]])"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 4
+     "prompt_number": 32
     },
     {
      "cell_type": "markdown",
@@ -143,11 +143,11 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "0.0341841496392\n"
+        "0.0174395374407\n"
        ]
       }
      ],
-     "prompt_number": 5
+     "prompt_number": 33
     },
     {
      "cell_type": "markdown",
@@ -170,11 +170,11 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "0.0355881271709\n"
+        "0.0177940635854\n"
        ]
       }
      ],
-     "prompt_number": 6
+     "prompt_number": 34
     },
     {
      "cell_type": "markdown",
@@ -188,15 +188,19 @@
      "collapsed": false,
      "input": [
       "from mpl_toolkits.mplot3d import Axes3D\n",
-      "xs, ys = arange(-8, 13, .1), arange(-8, 20, .1)\n",
+      "pylab.rcParams['figure.figsize'] = 12,6\n",
+      "\n",
+      "xs, ys = arange(-8, 13, .75), arange(-8, 20, .75)\n",
       "xv, yv = meshgrid (xs, ys)\n",
       "\n",
-      "zs = np.array([multivariate_gaussian(np.array([x,y]),mu,cov) for x,y in zip(np.ravel(xv), np.ravel(yv))])\n",
+      "zs = np.array([multivariate_gaussian(np.array([x,y]),mu,cov) \n",
+      "               for x,y in zip(np.ravel(xv), np.ravel(yv))])\n",
       "zv = zs.reshape(xv.shape)\n",
       "\n",
       "ax = plt.figure().add_subplot(111, projection='3d')\n",
-      "ax.plot_surface(xv, yv, zv)\n",
-      "show()"
+      "ax.plot_wireframe(xv, yv, zv)\n",
+      "show()\n",
+      "pylab.rcParams['figure.figsize'] = 6,4"
      ],
      "language": "python",
      "metadata": {},
@@ -204,13 +208,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": "iVBORw0KGgoAAAANSUhEUgAAAV0AAADtCAYAAAAcNaZ2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3106550>"
+        "<matplotlib.figure.Figure at 0x4068f50>"
        ]
       }
      ],
-     "prompt_number": 7
+     "prompt_number": 35
     },
     {
      "cell_type": "markdown",
@@ -226,21 +230,25 @@
      "collapsed": false,
      "input": [
       "import gaussian as g\n",
+      "pylab.rcParams['figure.figsize'] = 12,4\n",
       "\n",
       "cov = array([[2,0],[0,2]])\n",
       "e = g.sigma_ellipse (cov, 2, 7)\n",
-      "g.plot_sigma_ellipse(e, '\"2 0; 0 2\"')\n",
-      "show()\n",
+      "subplot(131)\n",
+      "g.plot_sigma_ellipse(e, '|2 0|\\n|0 2|')\n",
       "\n",
-      "cov = array([[2,0],[0,12]])\n",
-      "e = g.sigma_ellipse (cov, 2, 7)\n",
-      "g.plot_sigma_ellipse(e, '\"2 0; 0 12\"')\n",
-      "show()\n",
       "\n",
-      "cov = array([[2,3],[1,2.5]])\n",
+      "cov = array([[2,0],[0,9]])\n",
       "e = g.sigma_ellipse (cov, 2, 7)\n",
-      "g.plot_sigma_ellipse(e,'\"2 3; 1 2.5\"')\n",
-      "show()"
+      "subplot(132)\n",
+      "g.plot_sigma_ellipse(e, '|2 0|\\n|0 9|')\n",
+      "\n",
+      "subplot(133)\n",
+      "cov = array([[2,3],[1,2]])\n",
+      "e = g.sigma_ellipse (cov, 2, 7)\n",
+      "g.plot_sigma_ellipse(e,'|2 3|\\n|1 2|')\n",
+      "show()\n",
+      "pylab.rcParams['figure.figsize'] = 6,4"
      ],
      "language": "python",
      "metadata": {},
@@ -248,29 +256,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x3795710>"
+        "<matplotlib.figure.Figure at 0x4054b90>"
        ]
       }
      ],
-     "prompt_number": 8
+     "prompt_number": 36
     },
     {
      "cell_type": "markdown",
@@ -282,11 +274,11 @@
       "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&2\\end{bmatrix}$. Let this be our current belief about the position of our dog in a field. In other words, we believe that he is positioned at (2,7) with a variance of $\\sigma^2=2$ for both x and y. The contour plot shows where we believe the dog is located with the '+' in the center of the ellipse. The ellipse shows the boundary for the $1\\sigma^2$ probability - points where the dog is quite likely to be based on our current knowledge. Of course, the dog might be very far from this point, as Gaussians allow the mean to be any value. For example, the dog could be at (3234.76,189989.62), but that has vanishing low probability of being true. Generally speaking displaying  the $1\\sigma^2$ to $2\\sigma^2$ contour captures the most likely values for the distribution. An equivelent way of thinking about this is the circle/ellipse shows us the amount of error in our belief. A tiny circle would indicate that we have a very small error, and a very large circle indicates a lot of error in our belief. We will use this throughout the rest of the book to display and evaluate the accuracy of our filters at any point in time. \n",
       "\n",
       "The second plot uses mean and the covariance matrices $\n",
-      "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&12\\end{bmatrix}$. This time we use a different variance for x (2) vs y (12). The result is an ellipse. When we look at it we can immediately tell that we have a lot more uncertainty in the y value vs the x value. Our belief that the value is (2,7) is the same in both cases, but errors are different. This sort of thing happens naturally as we track objects in the world - one sensor has a better view of the object, or is closer, than another sensor, and so we end up with different error rates in the different axis.\n",
+      "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&0\\\\0&9\\end{bmatrix}$. This time we use a different variance for x (2) vs y (9). The result is an ellipse. When we look at it we can immediately tell that we have a lot more uncertainty in the y value vs the x value. Our belief that the value is (2,7) is the same in both cases, but errors are different. This sort of thing happens naturally as we track objects in the world - one sensor has a better view of the object, or is closer, than another sensor, and so we end up with different error rates in the different axis.\n",
       "\n",
       "\n",
       "The third plot uses mean and the covariance matrices $\n",
-      "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&3\\\\1&2.5\\end{bmatrix}$. This is the first contour that has values in the off-diagonal elements of $cov$, and this is the first contour plot with a slanted ellipse. This is not a coincidence. The two facts are telling use the same thing. A slanted ellipse tells us that the x and y values are somehow **correlated**. We denote that in the covariance matrix with values off the diagonal. What does this mean in physical terms? Think of trying to park your car in a parking spot. You can not pull up beside the spot and then move sideways into the space because most cars cannot go purely sideways. $x$ and $y$ are not independent. This is a consequence of the steering system in a car. When your tires are turned the car rotates around its rear axle while moving forward. Or think of a horse attached to a pivoting exercise bar in a corral. The horse can only walk in circles, he cannot vary $x$ and $y$ independently, which means he cannot walk straight forward to to the side. If $x$ changes, $y$ must also change in a defined way. \n",
+      "\\mu =\\begin{bmatrix}2\\\\7\\end{bmatrix}, cov = \\begin{bmatrix}2&3\\\\1&2\\end{bmatrix}$. This is the first contour that has values in the off-diagonal elements of $cov$, and this is the first contour plot with a slanted ellipse. This is not a coincidence. The two facts are telling use the same thing. A slanted ellipse tells us that the x and y values are somehow **correlated**. We denote that in the covariance matrix with values off the diagonal. What does this mean in physical terms? Think of trying to park your car in a parking spot. You can not pull up beside the spot and then move sideways into the space because most cars cannot go purely sideways. $x$ and $y$ are not independent. This is a consequence of the steering system in a car. When your tires are turned the car rotates around its rear axle while moving forward. Or think of a horse attached to a pivoting exercise bar in a corral. The horse can only walk in circles, he cannot vary $x$ and $y$ independently, which means he cannot walk straight forward to to the side. If $x$ changes, $y$ must also change in a defined way. \n",
       "\n",
       "So when we see this ellipse we know that $x$ and $y$ are correlated, and that the correlation is \"strong\". I will not prove it here, but a 45 $^{\\circ}$ angle denotes complete correlation between $x$ and $y$, whereas $0$ and $90$ denote no correlation at all. Those who are familiar with this math will be objecting quite strongly, as this is actually quite sloppy language that does not adress all of the mathematical issues. They are right, but for now this is a good first approximation to understanding these ellipses from a physical interpretation point of view. The size of the ellipse shows how much error we have in each axis, and the slant shows how strongly correlated the values are.\n",
       "\n",
@@ -319,11 +311,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3106410>"
+        "<matplotlib.figure.Figure at 0x404e910>"
        ]
       }
      ],
-     "prompt_number": 9
+     "prompt_number": 37
     },
     {
      "cell_type": "markdown",
@@ -349,11 +341,11 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x325fad0>"
+        "<matplotlib.figure.Figure at 0x407b310>"
        ]
       }
      ],
-     "prompt_number": 10
+     "prompt_number": 38
     },
     {
      "cell_type": "markdown",
@@ -383,11 +375,11 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAD7CAYAAACYLnSTAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJztnXmQXNV1/789M0K7NFpn0L7vSBoWCWP80xA2Q1BCjELA\nNmBjbCeV4EC5yi5SriC5sCKXndgidtlOUkWSMoWDSWEIqJQyywBGrEFik0AIaSSNNKN1NItGs7/f\nH5fDO+/2fd09PdPvvZn+fqqm5nX3m9atq+7vO+/cc8835XmeB0IIIYOakrgHQAghpP9QzAkhZAhA\nMSeEkCEAxZwQQoYAFHNCCBkCUMwJIWQIUFaoN169ejXefvvtQr09IYQMSVatWoVdu3b1+e8KFpm/\n/fbb8Dwv8T/3339/7GPgODlGjpPjlJ98g2CmWQghZAhAMSeEkCFA0Yt5dXV13EPICY5z4BgMYwQ4\nzoFmsIwzX1Ke5xWkN0sqlUKB3poQQoYs+Wpn0UfmhBAyFKCYE0LIEIBiTgghQwCKOSGEDAEo5oQQ\nMgSgmBNCyBCAYk4IIUMAijkhhAwBKOaEEDIEyCjmd955JyoqKnDBBRd8+tzp06dx9dVXY9GiRbjm\nmmtw5syZgg+SEEJIZjKK+Ve/+lVs37498NyWLVtw9dVXY+/evbjyyiuxZcuWgg6QEEJIdrL2Zqmt\nrcX69evx7rvvAgCWLFmCF154ARUVFWhoaEB1dTU++OCD9DdmbxZCCOkzkfVmOXbsGCoqKgAAFRUV\nOHbsWJ//UUIIIQNLvxZAU6kUUqnUQI2FEEJInvTZA1TSK5WVlaivr8fUqVNDz924ceOnx9XV1UO+\nnzAhxUZXF9DRAYwZE/dIBi81NTWoqanp9/v0OWf+ne98B5MmTcJ3v/tdbNmyBWfOnHEugjJnTsjQ\npqkJ+Oxngdpa4He/A666Ku4RDQ3y1c6MYn7rrbfihRdewMmTJ1FRUYHvf//7+NM//VPcfPPNOHTo\nEObMmYNHH30U5eXlAzYgQsjg4L77gIYG4NZbgb/6K2DvXqC0NO5RDX4KIub9gWJOyNClqwuYMQN4\n+WVg/nzg0kuB++8Hrr8+7pENfug0RAiJjJoaYO5cYMECIJUCbrkFePzxuEdV3FDMCSF95ve/Bz7/\nef/x+vXAtm0Ab8bjg2JOCOkzzz8PXHml/3j+fCPktbWxDanooZgTQvpEezuwezdw8cX+c6kUcPnl\nwB/+EN+4ih2KOSGkT7z9NrBoETByZPD5iy8G3nornjERijkhpI/s3AlceGH686tXA7t2RT8eYqCY\nE0L6xHvvAaor9qeImHMRNB4o5oSQPvHee8CKFenPT51qNg2x9148UMwJIX1izx5g6VL3a0uWAI6O\n2CQCKOaEkJxpbgbOngWmTXO/TjGPD4o5ISRn9u0zNeVhna8XLjTnkOihmBNCcmbfPiPYYcybB+zf\nH914iA/FnBCSM/v3m54sYcydCxw4EN14iA/FnBCSMwcPAnPmhL8ukTnLE6OHYk4IyZmDB4HZs8Nf\nnzAB6O01xhUkWijmhJCcySbmqRQwcyZw+HB0YyIGijkhJGcOHQJmzcp8DsU8HijmpKj47W+Byy4D\nnn467pEMPpqbTS583LjM51HM44FiToqGQ4eAv/xL4K67gDvuAE6ejHtEg4sjR4Dp08NrzAWKeTxQ\nzEnR8MtfGhG/807gxhuBX/0q7hENLkTMszFtGlBfX/jxkCAUc1IUeB7w8MPAV75iHn/lK8Cjj8Y5\nosHHkSPh2/g1559PMY8DijkpCqRfiLRu/cxnjOAcPBjfmAYbDQ1GqLNBMY8HijkpCp59FrjqKj/f\nW1oKrFsHvPhivOMaTDQ0AJWV2c+jmMcDxZwUBa+8YjwqNZdfDrz0UjzjGYwcOwZUVGQ/b+pU4NQp\noKen8GMiPhRzUhS8/jqwZk3wubVrgTffjGc8g5Fjx3KLzMvKgPJyI+gkOijmZMjT1GRu+5csCT6/\ncqXJpXd2xjOuwUZDQ26ROWCi8+PHCzseEoRiToY8778PLFtm8uSaUaPM1nSaKeTG8eNGpHOBYh49\nFHMy5AnzrATM87t3RzuewUhPD9DYCEyalNv5FPPooZiTIc/77wPLl7tfo81ZbjQ2AuPHm3x4LlDM\no4diToY8H30ELFrkfo1inhsnTgCTJ+d+/uTJbJcQNRRzMuT56CNgwQL3a4sXAx9+GO14BiMnTwJT\npuR+/qRJrGaJGoo5GdJ0dZmmT/PmuV+nM05u9DUyp5hHT95i/g//8A9Yvnw5LrjgAnzxi19ER0fH\nQI6LkAHh8GFTTjd8uPv1CRPMrtDGxmjHNdg4fTp98dPzgFtuMf3NbRNnplmiJy8xr62txb/+67/i\nrbfewrvvvouenh785je/GeixEdJvsnlWplLGhJiO8plxiflTT5kU1e23A9/7XvA1RubRk+PadJBx\n48Zh2LBhaGtrQ2lpKdra2jA9l96YhERMNpszwIh5bS1w8cWRDGlQcvo0MHFi8Ll//3fg7ruBL3zB\nROctLcDYsea1SZMYmUdNXpH5xIkT8e1vfxuzZs3CtGnTUF5ejquuumqgx0ZIv6mtzS7mNFPIji3m\nXV2medn69Wbr/qWXmsfCxIlMXUVNXpH5xx9/jJ/+9Keora3F+PHj8ed//ud4+OGH8aUvfSlw3saN\nGz89rq6uRnV1dX/GSkifOXTItLvNBMU8O7aY79xpLpJS4VJdbTpQ3nijeTxmDNDRYUR/2LDIhzuo\nqKmpQU1NTb/fJy8xf/PNN3HZZZdh0idJtC984QvYsWNHRjEn+dHeDvz858BFF5kvDOkbR44AM2Zk\nPmfmTOC116IZz2Dl9GmzWCy8/rppVCZcdhnwd3/nP06lTMTe2Jh7C4BixQ50N23alNf75JVmWbJk\nCV599VWcO3cOnufhmWeewbJly/IaAMnM3XcD27cDN98MvP123KMZfORidcbIPDtnzgTF/P/+zwQY\nwurVwDvvBNveTpjAVEuU5CXmq1atwu23346LL74YK1euBAB84xvfGNCBEZPvffxx4LHHgPvuAzZv\njntEg49cxHzaNODo0WjGM1g5c8ZE2sL77/uuTYB5bfLkYFVQebn5OxINKc8rzHaJVCqFAr110fDA\nA6Z1689/btq4zpwJ1NUB48bFPbLBQVubyfOeO5fZUb693fQdaW/P7jxfrEycCOzbZ3739prP4JEj\nZt6E664D/vqvgRtuMI+vvRa4917g85+PZ8yDlXy1kztAE8z//A9w003mePx444yzfXu8YxpM1Ncb\nC7NsAj1ihFmwY120m95eoLnZDyKOHjXzpYUcMH1u9uzxHzMyjxaKeUJpbDRfjM9+1n/ummuA556L\nb0yDjVw9KwEj+ky1uGltNb3fpWPivn3AwoXp5y1caF4Txo83d5QkGijmCeXVV80mFr0NnZ6VfSNX\nz0qAJsSZsPPlH38MzJ+ffp70uREo5tFCMU8odukXYCoGDh7kFyRX+hKZV1Sw/3YYTU3BdZraWneL\nBIp5vFDME8qbbwKXXBJ8rqzMOOO88048Yxps9CUyp5lCOM3Nwfx42K7aWbPMAn1vr3lMMY8WinlC\nefddYNWq9OdXrwZ27Yp+PIMRivnAoBc/AbOr1iXmI0YYAZd5pJhHC8U8gTQ3m/7Rc+emv7ZypRF6\nkp2+GCpMmWLmnKRji3ldXfiuWr0Bi2IeLRTzBLJ7tynzKnH87yxZQmecXDlxIncxZ2QejhZzzzNV\nP2EbsWQvBGA6KDY3RzNGQjFPJB9+aETbBT0rc6cv7jhTplDMw9Bifvq0qbAaPdp9rq4KGjvWtMUl\n0UAxTyBhdbyA+bKcO8eeF7nQlzQLzRTC0WJeX2/aH4RBMY8PinkCyWRAnEqZGl8642Smt9dc8Gx3\nnDAmT6aYh9HcbHZ8AtnLPSsr/c1XFPNooZgnkH37wsUcSK/nJemcOWNSAWVWk+edO43Nmd52DpjF\nurNnTf9tEqS11XcQylYhVFlpzgEo5lFDMU8gBw+6K1kEelZmx+VZ2dMD/MVfAJ2dwK23msU8oaTE\n7HI8fTracQ4GbDHPFJnrzVdjxpiUoNSdk8JCMU8YZ8+aL0+mhv7z5gEHDkQ3psGIy7PymWdMBP7I\nI8YF59VXg68z1eKmtdVPsxw/nnkdYupUv8SzpMT0dGltLfwYCcU8cRw8aHbSZer0N3Om2bhBwnGJ\n+RNPGJOPVArYsAH43e+Cr0+cyMjcRUuLL+bZyj3tEs8xYyjmUUExTxhhu+s0dMbJjkvMn3sOuPpq\nc3zVVYBtu0hnHDc6zZJNzEePNmmVs2f9x3JMCgvFPGFk2l0nUMyzc+pUUMwbG42ZgrjjrFljdtKe\nO+efQzF3o9MsJ09mr93X6aoxYyjmUUExTxi52JxNnGgW8VgpEE5jY7pnZVUVUFpqHo8cCSxaBLz3\nnn8OxdyNLebZyj11zf7o0UyzRAXFPGHkIuaplIneaaYQTlNTsAe3q3GZ3bSMYu7m7Fl/x6erSshm\n8mQj+gAj8yihmCeMXMQcoJlCNlwGxMuXB89ZvjxYb04xd3P2rKlK6e0182qvRdjoyJwLoNFBMU8Y\nR49m3i4t0OYsM7aYf/BBer+bxYuDfW7oWZlOV5cR8fPOMztBXRuxbPRFkWmW6KCYJ4yGBiPU2WBk\nnhlbzF39bhYvDnagZMvWdNrajCCnUibFotchwrDFvK2tsGMkBop5gujpyb05VGUlxTwTWsxbWsyP\nfccze7apHurpMY8p5ulIigUwc5qLmJeX+2I+ahTFPCoo5gni1CnzRRg2LPu555/v98Ag6TQ1+VZn\nYnNmb8QaMcJscpH+2+PHs/+2jV78tO92wtCROcU8OijmCaKhgTZnA4X2rQzzrASMMbG0RmLine truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x371e050>"
+        "<matplotlib.figure.Figure at 0x3f8d690>"
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 39
     },
     {
      "cell_type": "markdown",
@@ -423,11 +415,11 @@
        "output_type": "display_data",
        "png": "iVBORw0KGgoAAAANSUhEUgAAAXMAAAD7CAYAAACYLnSTAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x3719990>"
+        "<matplotlib.figure.Figure at 0x40514d0>"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 40
     },
     {
      "cell_type": "markdown",
@@ -461,18 +453,15 @@
      "metadata": {},
      "outputs": [
       {
-       "ename": "NameError",
-       "evalue": "global name 'xlim' is not defined",
-       "output_type": "pyerr",
-       "traceback": [
-        "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
-        "\u001b[1;32m<ipython-input-19-cd0500ac6f58>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mfrom\u001b[0m \u001b[0mmkf_internal\u001b[0m \u001b[1;32mimport\u001b[0m \u001b[1;33m*\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0mshow_residual_chart\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
-        "\u001b[1;32m/var/host/media/removable/chrome/CS373/book/mkf_internal.py\u001b[0m in \u001b[0;36mshow_residual_chart\u001b[1;34m()\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      8\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mshow_residual_chart\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 9\u001b[1;33m     \u001b[0mxlim\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0.9\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2.5\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m;\u001b[0m\u001b[0mylim\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0.5\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2.5\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     10\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     11\u001b[0m     \u001b[0mscatter\u001b[0m \u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m1.3\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
-        "\u001b[1;31mNameError\u001b[0m: global name 'xlim' is not defined"
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAW8AAAEKCAYAAADdBdT9AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x4674f10>"
        ]
       }
      ],
-     "prompt_number": 19
+     "prompt_number": 41
     },
     {
      "cell_type": "code",
@@ -481,7 +470,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 13
+     "prompt_number": 41
     }
    ],
    "metadata": {}

Untitled0.ipynb

@@ -92,7 +92,8 @@
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "print numpy.array(3)"
+      "print numpy.array(3)\n",
+      "print pylab.rcParams['figure.figsize']"
      ],
      "language": "python",
      "metadata": {},
@@ -101,11 +102,58 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "3\n"
+        "3\n",
+        "(6.0, 4.0)\n"
        ]
       }
      ],
-     "prompt_number": 17
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "class KalmanFilter1D(object):\n",
+      "\n",
+      "    def __init__ (self, x0, var):\n",
+      "        self.mean = x0\n",
+      "        self.variance = var\n",
+      "\n",
+      "    def estimate(self, z, z_variance):\n",
+      "        self.mean = \\\n",
+      "        (self.variance*z + z_variance*self.mean) / \\\n",
+      "        self.variance + z_variance\n",
+      "        self.variance = 1. / (1./self.variance+ 1./z_variance)\n",
+      "\n",
+      "f = KalmanFilter1D(1,2)\n",
+      "f.estimate(z = 2,\n",
+      "           z_variance = 3)\n",
+      "\n",
+      "print f.mean, f.variance"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "6 1.2\n"
+       ]
+      }
+     ],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import scipy.interpolate\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 9
     }
    ],
    "metadata": {}

gauss.py

@@ -11,32 +11,49 @@ import numpy.random as random
 
 
 class gaussian(object):
-    def __init__ (self,m,s):
-        self.mu    = float(m)
-        self.sigma = float(s)
+    def __init__(self, mean, variance):
+        try:
+            self.mean     = float(mean)
+            self.variance = float(variance)
 
-    def __add__ (a,b):
-       return gaussian (a.mu + b.mu, a.sigma + b.sigma)
+        except:
+            self.mean     = mean
+            self.variance = variance
 
-    def __mul__ (a,b):
-        m = (a.sigma*b.mu + b.sigma*a.mu) / (a.sigma + b.sigma)
-        s = 1. / (1./a.sigma + 1./b.sigma)
-        return gaussian (m,s)
+    def __add__ (a, b):
+       return gaussian (a.mean + b.mean, a.variance + b.variance)
+
+    def __mul__ (a, b):
+        m = (a.variance*b.mean + b.variance*a.mean) / (a.variance + b.variance)
+        v = 1. / (1./a.variance + 1./b.variance)
+        return gaussian (m, v)
+
+    def __call__(self, x):
+        """ Impl
+        """
+        return math.exp (-0.5 * (x-self.mean)**2 / self.variance) / \
+               math.sqrt(2.*math.pi*self.variance)
 
-    def __call__(self,x):
-        return math.exp (-0.5 * (x-self.mu)**2 / self.sigma) / \
-        math.sqrt(2.*math.pi*self.sigma)
 
     def __str__(self):
-        return "(%f, %f)" %(self.mu, self.sigma)
+        return "(%f, %f)" %(self.mean, self.sigma)
+
+    def stddev(self):
+        return math.sqrt (self.variance)
+
+    def as_tuple(self):
+        return (self.mean, self.variance)
+
+    def __tuple__(self):
+        return (self.mean, self.variance)
 
     def __getitem__ (self,index):
         """ maybe silly, allows you to access obect as a tuple:
         a = gaussian(3,4)
         print (tuple(a))
         """
-        if index == 0:   return self.mu
-        elif index == 1: return self.sigma
+        if index == 0:   return self.mean
+        elif index == 1: return self.variance
         else:            raise StopIteration
 
 class KF1D(object):
@@ -52,7 +69,10 @@ class KF1D(object):
 
 
 
-
+def mul2 (a, b):
+    m = (a['variance']*b['mean'] + b['variance']*a['mean']) / (a['variance'] + b['variance'])
+    v = 1. / (1./a['variance'] + 1./b['variance'])
+    return gaussian (m, v)
 
 
 #varying_error_kf( noise_factor=100)

gaussian.py

@@ -13,6 +13,14 @@ def gaussian(x, mean, var):
     """
     return math.exp((-0.5*(x-mean)**2)/var) / math.sqrt(_two_pi*var)
 
+def mul (a_mu, a_var, b_mu, b_var):
+    m = (a_var*b_mu + b_var*a_mu) / (a_var + b_var)
+    v = 1. / (1./a_var + 1./b_var)
+    return (m, v)
+
+def add (a_mu, a_var, b_mu, b_var):
+    return (a_mu+b_mu, a_var+b_var)
+
 
 def multivariate_gaussian(x, mu, cov):
     """ This is designed to work the same as scipy.stats.multivariate_normal

mkf_internal.py

@@ -4,12 +4,14 @@ Created on Thu May  1 16:56:49 2014
 
 @author: rlabbe
 """
+import matplotlib.pyplot as plt
 
 def show_residual_chart():
-    xlim([0.9,2.5]);ylim([0.5,2.5])
+    plt.xlim([0.9,2.5])
+    plt.ylim([0.5,2.5])
 
-    scatter ([1,2,2],[1,2,1.3])
-    scatter ([2],[1.8],marker='o')
+    plt.scatter ([1,2,2],[1,2,1.3])
+    plt.scatter ([2],[1.8],marker='o')
     ax = plt.axes()
     ax.annotate('', xy=(2,2), xytext=(1,1),
                 arrowprops=dict(arrowstyle='->', ec='b',shrinkA=3, shrinkB=4))
@@ -23,5 +25,5 @@ def show_residual_chart():
                 arrowprops=dict(arrowstyle="<->",
                                 ec="r",
                                 shrinkA=5, shrinkB=5))
-    title("Kalman Filter Prediction Update Step")
-    show()
+    plt.title("Kalman Filter Prediction Update Step")
+    plt.show()