Added section using Zarchan's adaptive filter.

15_Adaptive_Filtering/Adaptive_Filtering.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:5534535d5a4cd2ef165757afa8bff2a0400cc8d287ac8871b7f4380007c16895"
+  "signature": "sha256:86fc954574672293ddc089433553cec36487eac3099563fcbb3b3a65fc38906d"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -262,7 +262,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f5eb9994c18>"
+        "<IPython.core.display.HTML at 0x7f7ee9cb3240>"
        ]
       }
      ],
@@ -442,7 +442,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9cb227f0>"
+        "<matplotlib.figure.Figure at 0x7f7ecd826828>"
        ]
       }
      ],
@@ -471,9 +471,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9cb1f208>"
+        "<matplotlib.figure.Figure at 0x7f7ecd5304a8>"
        ]
       }
      ],
@@ -556,9 +556,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9c8fa4e0>"
+        "<matplotlib.figure.Figure at 0x7f7ecd5632b0>"
        ]
       }
      ],
@@ -596,9 +596,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9c8fedd8>"
+        "<matplotlib.figure.Figure at 0x7f7ecd530438>"
        ]
       }
      ],
@@ -633,9 +633,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9c8fef28>"
+        "<matplotlib.figure.Figure at 0x7f7ecd5307f0>"
        ]
       }
      ],
@@ -670,9 +670,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9ca7acf8>"
+        "<matplotlib.figure.Figure at 0x7f7ecd2f86d8>"
        ]
       }
      ],
@@ -722,9 +722,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9c837208>"
+        "<matplotlib.figure.Figure at 0x7f7ecd262710>"
        ]
       }
      ],
@@ -775,7 +775,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e950dc1d0>"
+        "<matplotlib.figure.Figure at 0x7f7ec91c4550>"
        ]
       }
      ],
@@ -824,13 +824,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9ca27630>"
+        "<matplotlib.figure.Figure at 0x7f7ebfeab630>"
        ]
       }
      ],
-     "prompt_number": 14
+     "prompt_number": 19
     },
     {
      "cell_type": "markdown",
@@ -868,7 +868,7 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The first method normalizes the square of the residual using the following equation:\n",
+      "The first method (from Bar-Shalom [1]) normalizes the square of the residual using the following equation:\n",
       "\n",
       "$$ \\epsilon = \\mathbf{y^\\mathsf{T}S}^{-1}\\mathbf{y}$$\n",
       "\n",
@@ -876,7 +876,6 @@
       "\n",
       "$$\\mathbf{S} = \\mathbf{HPH^\\mathsf{T}} + \\mathbf{R}$$\n",
       "\n",
-      "\n",
       "If the linear algebra used to compute this confuses you, recall that we can think of matrix inverses as division, so $\\epsilon = \\mathbf{y^\\mathsf{T}S}^{-1}\\mathbf{y}$ can be thought of as computing\n",
       "\n",
       "$$\\epsilon=\\frac{\\mathbf{y}^2}{\\mathbf{S}}$$\n",
@@ -916,13 +915,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e9c8a6978>"
+        "<matplotlib.figure.Figure at 0x7f7ecd2d3588>"
        ]
       }
      ],
-     "prompt_number": 15
+     "prompt_number": 20
     },
     {
      "cell_type": "markdown",
@@ -944,6 +943,7 @@
       "\n",
       "cvfilter.x = array([0., 0.])\n",
       "cvfilter.R = np.eye(1)*0.2\n",
+      "\n",
       "cvfilter.Q = Q_discrete_white_noise(dim=2, dt=dt, var=0.02)\n",
       "\n",
       "Q_scale_factor = 1000.\n",
@@ -981,13 +981,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e94e126a0>"
+        "<matplotlib.figure.Figure at 0x7f7ebfde1240>"
        ]
       }
      ],
-     "prompt_number": 16
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -996,6 +996,242 @@
       "The performance of this filter is markedly better than the constant velocity filter. The constant velocity filter took roughly 10 seconds to reacquire the signal after the start of the manuever. The adaptive filter takes under a second to do the same. "
      ]
     },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Continuous Adjustment - Standard Deviation Version"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Another, very similar method from Zarchan [2] sets the limit based on the standard deviation of the measurement error covariance. Here the equations are:\n",
+      "\n",
+      "$$ \\begin{aligned}\n",
+      "std &= \\sqrt{\\mathbf{HPH}^\\mathsf{T} + \\mathbf{R}} \\\\\n",
+      "&= \\sqrt{\\mathbf{S}}\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "\n",
+      "If the absolute value of the residual is more than some multiple of the standard deviation computed above we increase the process noise by a fixed amount, recompute Q, and continue."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from filterpy.common import dot3\n",
+      "from numpy.linalg import inv\n",
+      "from math import sqrt\n",
+      "\n",
+      "\n",
+      "def zarchan_adaptive_filter(Q_scale_factor, std_scale):\n",
+      "    cvfilter.x = array([0., 0.])\n",
+      "    cvfilter.R = np.eye(1)*0.2\n",
+      "\n",
+      "    phi = 0.02\n",
+      "    cvfilter.Q = Q_discrete_white_noise(dim=2, dt=dt, var=phi)\n",
+      "    xs = []\n",
+      "    ys = []\n",
+      "    count = 0\n",
+      "    for i,z in zip(t,z_xs2):\n",
+      "        cvfilter.predict()\n",
+      "        cvfilter.update([z])\n",
+      "        y = cvfilter.y\n",
+      "        S = cvfilter.S\n",
+      "        std = sqrt(S)\n",
+      "\n",
+      "        xs.append(cvfilter.x)\n",
+      "        ys.append(y)\n",
+      "\n",
+      "        if abs(y[0]) > std_scale*std:\n",
+      "            phi += Q_scale_factor\n",
+      "            cvfilter.Q = Q_discrete_white_noise(dim=2, dt=dt, var=phi)\n",
+      "            count += 1\n",
+      "        elif count > 0:\n",
+      "            phi -= Q_scale_factor\n",
+      "            cvfilter.Q = Q_discrete_white_noise(dim=2, dt=dt, var=phi)\n",
+      "            count -= 1\n",
+      "\n",
+      "    xs = np.asarray(xs)\n",
+      "    plt.subplot(121)\n",
+      "    plt.plot(t, z_xs2, label='z')\n",
+      "    plt.plot(t, xs[:,0], label='filter')\n",
+      "    plt.ylabel('$\\epsilon$')\n",
+      "    plt.xlabel('time (sec)')\n",
+      "    plt.legend(loc=2)\n",
+      "    plt.title('position')\n",
+      "    plt.subplot(122)\n",
+      "    plt.plot(t, xs[:,1])\n",
+      "    plt.xlabel('time (sec)')\n",
+      "    plt.title('velocity')\n",
+      "    plt.show()\n",
+      "    \n",
+      "zarchan_adaptive_filter(1000, 2)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAzAAAAGxCAYAAAC9e1NPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3Xd4FHX+B/D3zPb0QhoJpEgoiVIkYASBAGIB5VDPcqg0\nFUEseCpKDTUUEWzoccIJyiGC9xO75hQEEUFyROkhSExIAklIL7ubLfP7I8mSkEICSWY3+349Dw8w\nO7Pz3g3s7Ge+TZAkSQIREREREZEDEOUOQERERERE1FwsYIiIiIiIyGGwgCEiIiIiIofBAoaIiIiI\niBwGCxgiIiIiInIYLGCIiIiIiMhhsIAhagebNm2CKIrYvHlzi44TRRHDhw9vo1RERGTP4uLiIIrt\n+1UtLCwM4eHh7XpOopZiAUPUDgRBsP2qTRTFK14oLj+GiIicR3tfAxq6Vi1cuPCqbsIRtRWl3AGI\nnME999yDm2++GYGBgfUea+ridOrUKbi4uLRlNCIiIptdu3Y1+hhvqJG9YAFD1A48PDzg4eHR4uO6\nd+/eBmmIiIga1lSvAEmS2jEJUePYhYw6rD///NM2hiQrKwsPP/ww/Pz84OLiggEDBmD79u31jpEk\nCRs2bEBsbCzc3d3h6uqKG2+8EWvWrIHZbK63/5EjRzB+/HiEh4dDp9PBz88Pffr0wYwZM1BSUmLb\n7/IxMD/++KOtX3NNzppfkydPth3X2BiY0tJSzJs3Dz179oROp4O3tzdGjhyJzz//vMn3IT8/H1On\nTkVQUBC0Wi2uv/56bNq0qcXvLRGRsztw4ABEUcTYsWMb3ScmJgYKhQLp6em2bbt378bYsWPh5+cH\njUaDsLAwzJgxAzk5Oc0+d0uvVQCQlZWFmTNnonv37nBxcYGPjw9iYmIQHx9f55jLx8DExcVh8eLF\nAIDJkyfXuV6lp6dj9uzZEEURH3zwQYPnPXXqFERRxJAhQ5r9+oiuhC0w1OEVFhbilltugY+PDx5/\n/HEUFBRg+/bteOihh5CdnY2ZM2fa9p04cSK2bNmCkJAQPPbYY1CpVPj888/x4osvIjExEV999RUU\nCgWAquLlpptugkKhwF133YXrrrsOZWVlOHv2LDZv3owXXnihXqtLTfN7eHg44uPjsWjRInh6euL5\n55+37dO3b98Gj6lRXFyMW265BcePH8eNN96ImTNnorCwEDt27MC4ceOwaNEizJ8/v977UFRUhMGD\nB0Oj0eCBBx6A0WjE9u3bMWXKFIiiiAkTJlzbG01E5ERiY2PRs2dPfPvtt8jLy4Ofn1+dx48fP47D\nhw8jLi4OoaGhAICVK1di9uzZ8PX1xZgxYxAYGIjff/8d7777Lj7//HMcOHAAwcHBVzx3S65VAJCU\nlIQ77rgDBQUFGDp0KO69914YDAacPHkSK1asqHe9qn3dmTx5MgRBwJ49ezBu3Lg61ygvLy9Mnz4d\nr776KtavX9/gdWT9+vUAgOnTpzfznSVqBomog0pLS5MEQZAEQZAeeuihOo/98ccfkpeXl6TRaKT0\n9HRJkiRp27ZtkiAIUt++faXS0lLbvpWVldLIkSMlQRCk1atX27b//e9/lwRBkD777LN65y4rK5OM\nRqPt7++//74kCIK0efPmOvsJgiCFh4c3+hoEQZCGDx9eZ9u0adMkQRCkxx57rM72zMxMKSgoSBJF\nUTp06FCD78MTTzwhWa1W22MnTpyQlEqlFBUV1WgGIiJq2MqVKyVBEKS1a9fWe+yll16q87m/Z88e\nSRAEadCgQVJxcXGdfT/88ENJEATpvvvuq7N92LBhkiiKdba19FplNBqlsLAwSRRF6cMPP6yXMycn\nRzKbzba/h4aG1rsuxcfHN3gNq3H33XdLgiBIR44cqbNdr9dL3t7ekp+fn1RZWdngsURXg13IqMNT\nKpVYsWJFnW0RERF46qmnUFlZiS1btgAANmzYAABYvnw53NzcbPuqVCqsXbsWAPDee+/Ve36tVltv\nm6urK9Rqdau9hhomkwkffPABXF1dsWrVqjqPBQcHY86cObauBQ1lWrNmTZ07a7169cKgQYNw6tQp\nVFRUtHpeIqKO7NFHH4VCoag3O5fFYsGWLVvg5uaGv/71rwCAN954A0BVi8TlrfOPPPII+vbti88+\n+wxlZWVNnrOl16ovvvgC6enpGD16NB555JF6z+fv71+nteZqPPXUUwAutbbU2L59O4qKijBp0iSo\nVKprOgdRbexCRh1e165dbc33tQ0dOhTLly/Hb7/9BgA4fPgwBEFocMzJDTfcAD8/P6SmpqKiogIu\nLi546KGH8Oabb2LcuHG47777MGLECNx8883o2bNnm72WU6dOQa/XIzY2Fj4+PvUev/XWWwEAycnJ\n9R6LjIysc7Gr0aVLF0iShMLCQs54RkTUAkFBQRg1ahS+/fZb/P777+jTpw8AIDExERcuXMCkSZNs\nn6s///wzlEol/vOf/+CTTz6p91xGoxEWiwWnT5/GjTfe2Og5W3qtOnDgAADgzjvvbI2X3KA77rgD\nERER2LJlC1atWmV7zevXr4coinjyySfb7NzknFjAUIcXEBDQ5Pbi4mLb756entBoNA3uHxQUhIsX\nL6K4uNg2EcC+ffuQkJCATz/9FP/+978BVA2AfPnll9vkA7sma0PTMdfeXlRUVO8xLy+vBo9RKqs+\nBiwWS2tEJCJyKpMnT8a3336LzZs3Y82aNQBga5GZNGmSbb/8/HxYLBYsWrSo0ecSBAHl5eVNnq+l\n16qa60FzxtZci2nTpmHWrFn46KOP8Nhjj+Ho0aP45ZdfcOutt+K6665r03OT82EXMurwGpvZpWa7\np6en7ffi4mIYjcYG9z9//nyd/QHgpptuwmeffYaioiIcOHAAS5YsgV6vx/Tp0/Hhhx+25suoc+4L\nFy40OyMREbWdv/zlL/D29sbWrVthtVpRVFSEzz77DBERERg6dKhtP09PT3h4eMBqtTb6y2KxXHG2\nrpZeq2puXmVmZrbGy23UlClToNVqbd3Ian6fNm1am56XnBMLGOrwMjIy6kxhWWPPnj0AgH79+gEA\n+vfvD0mSsHv37nr7Hjt2DHl5ebbpJy+nVCoxcOBAzJ071zaV5M6dO6+YTRCEFrV89OrVCzqdDkeP\nHkV+fn69x3/44QfbayEioranVqvx0EMPITc3F1999RU+/vhjGI3GejNyDRo0CCUlJThy5Mg1na+l\n16qbb74ZAPDNN99c9Tlrxsg0db3y8fHBgw8+iKSkJOzbtw9btmxBUFAQxo0bd9XnJWoMCxjq8Mxm\nM15++eU6C3D98ccfeOedd6BWq/Hwww8DAB577DEAwJw5c+o04ZtMJvz9738HADz++OO27fv374fB\nYKh3vpq7X80ZT+Lr64u8vLwGn6chSqUSEyZMQHl5OWbPnl3nsezsbCxfvhyiKGLKlCnNej4iIrp2\nNet3bd68GZs3b4Yoipg4cWKdfWquI1OnTkVWVla95zAYDNi3b98Vz9XSa9Xdd9+NsLAwfP3117ZJ\na2rLycm54o00X19fAGjwZmBtNYP5//a3v6GkpASPPfaYbc0zotbEMTDU4fXu3RsHDx5ETEwMRo0a\nZVsHprS0FGvWrEHXrl0BAA8++CC++OILbN26FVFRURg3bhxUKhW++OILpKam4tZbb62zXsuqVauw\na9cuDBkyBGFhYfDw8MDp06fx5ZdfwsXFpc76Mo257bbbsHXrVtxxxx0YMmQINBoN+vbti7vuuqvR\nY1asWIGffvoJGzZsQHJyMkaOHImioiLs2LEDRUVFWLBgAQYMGHDtbxwRETVLTEwMoqOj8fnnn8Ns\nNtdZ+6VGXFwcXn31Vbz88suIjIzE6NGjER4eDr1ej4yMDOzduxcRERE4fPhwneNq33wDWn6tUqlU\n2LFjB26//XZMmDABGzZswE033YTKykqkpKTghx9+QF5eXr2Z0WobOXIkRFHE66+/jvz8fNsY0mef\nfbbOcQMGDED//v3xv//9DwqFAlOnTr3q95SoSXLO4UzUlmrWPxk+fLiUlZUlPfzww5Kfn5+k0+mk\nmJgY6eOPP653jNVqldavXy8NHDhQcnV1lXQ6ndS3b19p9erVkslkqrNvYmKiNGXKFCk6Olry8vKS\nXFxcpO7du0tTp06VUlJS6uy7adMmSRTFenPo5+XlSRMmTJCCgoIkhUIhiaIoTZ482fZ4Q+vASJIk\nFRcXS3PmzJF69OghaTQaydPTUxo+fLj06aefNvk+NGTSpEmSKIq29XCIOpI9e/ZId999txQcHCwJ\ngiBt2rSp3j7x8fFS586dJZ1OJ8XFxUnHjx+XISk5utWrV0uCIDT4WV/bgQMHpL/97W9SSEiIpFar\npU6dOkm9e/eWnn76aemnn36qs29cXFy9dWAkqWXXqhrnzp2Tnn76aSkiIkLSaDSSr6+vNGDAAGnR\nokV1jgkLC2twfbKPPvpI6t+/v+Ti4mJ7nQ1dN9atWycJgiDdfffdjb4HRNdKkKTLSnuiDuLPP/9E\nREQE4uLisGvXLrnjEJEMvvnmG/z888/o168fJkyYgHfffbfO2ISVK1di2bJl2Lx5M7p3747Fixdj\n3759SElJaXDacSJq2hNPPIGNGzfiyy+/xOjRo+WOQx0UCxjqsFjAEFFt7u7uWLduna2AkSQJnTt3\nxrPPPmsbU2YwGODv74/Vq1ez+wtRC2VlZaFbt24ICQlBamqq3HGoA+PIKiIickppaWnIycnBbbfd\nZtum1WoxdOhQ7N+/X8ZkRI5l69atWLRoEW699VZUVlZi8eLFckeiDq5DDOKvWdyPqLaSkhIAVbOQ\n8d8I0dXrqOsK1ayndPlit/7+/sjOzq6zjZ8hRI179913sX//fnTu3BmLFi3C6NGj+X+GWqLine truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ecd55a748>"
+       ]
+      }
+     ],
+     "prompt_number": 65
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "So I chose to use 1000 as the scaling factor for the noise, and 2 as the standard deviation limit. Why these numbers? Well, first, let's look at the difference between 2 and 3 standard deviations. "
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "zarchan_adaptive_filter(1000, 2)\n",
+      "zarchan_adaptive_filter(1000, 3)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAzAAAAGxCAYAAAC9e1NPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ec91cb630>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ecd2b8048>"
+       ]
+      }
+     ],
+     "prompt_number": 63
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can see from the charts that the filter output for the position is very similar regardless of weather we use 2 standard deviations or three. But the computation of the velocity is a different matter. Let's explore this further. First, lets make the standard deviation very small."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "zarchan_adaptive_filter(1000, .1)\n",
+      "zarchan_adaptive_filter(1000, 1)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAzAAAAGxCAYAAAC9e1NPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ecd4fd668>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ecd4fdeb8>"
+       ]
+      }
+     ],
+     "prompt_number": 66
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "As the standard deviation limit gets smaller the computation of the velocity gets worse. Think about why this is so. If we start varying the filter so that it perfers the measurement over the prediction as soon as the residual deviates even slightly from the prediction we very quickly be giving almost all the weight towards the measurment. With no weight for the prediction we have no information from which to create the unobserved variables. So, when the limit is 0.1 std you can see that the velocity is swamped by the noise in the measurement. On the other hand, because we are favoring the measurements so much the position follows the manuver almost perfectly.\n",
+      "\n",
+      "Now let's look at the effect of various increments for the process noise. Here I have held the standard deviation limit to 2 std, and varied the increment from 1 to 10,000."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "zarchan_adaptive_filter(1, 2)\n",
+      "zarchan_adaptive_filter(10, 2)\n",
+      "zarchan_adaptive_filter(100, 2)\n",
+      "zarchan_adaptive_filter(1000, 2)\n",
+      "zarchan_adaptive_filter(10000, 2)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ebfdb44e0>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ec91d2400>"
+       ]
+      },
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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ebfccf390>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ecd2d8e10>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x7f7ecd2d8048>"
+       ]
+      }
+     ],
+     "prompt_number": 68
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Here we can see that the position estimate gets marginally better as the increment factor increases, but that the velocity estimate starts to create a large overshoot. \n",
+      "\n",
+      "It isn't possible for me to tell you which of these is 'correct'. You will need to test your filter's performance against real and simulated data, and choose the design that best matches the performance you need for each of the state variables."
+     ]
+    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -1016,7 +1252,7 @@
       "\n",
       "The fading memory filter accounts for this problem by giving less weight to older measurements, and greater weight to the more recent measurements. \n",
       "\n",
-      "There are many formulations for the fading memory Filter; I use the one provided by Dan simon in *Optimal State Estimation* [1]. I will not go through his derivation, but only provide the results.\n",
+      "There are many formulations for the fading memory Filter; I use the one provided by Dan simon in *Optimal State Estimation* [3]. I will not go through his derivation, but only provide the results.\n",
       "\n",
       "The Kalman filter equation for the covariances of the estimation error is\n",
       "\n",
@@ -1129,9 +1365,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f5e94b3b278>"
+        "<matplotlib.figure.Figure at 0x7f7ec9170be0>"
        ]
       }
      ],
@@ -1205,15 +1441,6 @@
      "outputs": [],
      "prompt_number": 17
     },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
-     "metadata": {},
-     "outputs": [],
-     "prompt_number": 17
-    },
     {
      "cell_type": "heading",
      "level": 2,
@@ -1226,7 +1453,11 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "* [1] Dan Simon. \"Optimal State Estimation.\" John Wiley & Sons. p. 208-212. (2006)"
+      "* [1] Bar-Shalom, Yaakov, Xiao-Rong Li, and Thiagalingam Kirubarajan. *Estimation with Applications to Tracking and Navigation*. New York: Wiley, p. 424, 2001. \n",
+      "\n",
+      "* [2] Zarchan, Paul, and Howard Musoff. *Fundamentals of Kalman Filtering: A Practical Approach*. Reston, VA: American Institute of Aeronautics and Astronautics, 2000. Print.\n",
+      "\n",
+      "* [3] Simon, Dan. Optimal State Estimation: Kalman, H and Nonlinear Approaches. Hoboken, NJ: Wiley-Interscience, p. 208-212, 2006"
      ]
     }
    ],