Minor layout changes, typos.

15_Adaptive_Filtering/Adaptive_Filtering.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:0adb3177d0f465a38bed1a6cd92dbad50949762c108800f0867d362caa213ab5"
+  "signature": "sha256:5534535d5a4cd2ef165757afa8bff2a0400cc8d287ac8871b7f4380007c16895"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -262,7 +262,7 @@
        "output_type": "pyout",
        "prompt_number": 1,
        "text": [
-        "<IPython.core.display.HTML at 0x7f14d0924240>"
+        "<IPython.core.display.HTML at 0x7f5eb9994c18>"
        ]
       }
      ],
@@ -442,7 +442,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b44c8860>"
+        "<matplotlib.figure.Figure at 0x7f5e9cb227f0>"
        ]
       }
      ],
@@ -471,9 +471,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b41e37b8>"
+        "<matplotlib.figure.Figure at 0x7f5e9cb1f208>"
        ]
       }
      ],
@@ -556,9 +556,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b41ee668>"
+        "<matplotlib.figure.Figure at 0x7f5e9c8fa4e0>"
        ]
       }
      ],
@@ -596,13 +596,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14ac3eb5f8>"
+        "<matplotlib.figure.Figure at 0x7f5e9c8fedd8>"
        ]
       }
      ],
-     "prompt_number": 39
+     "prompt_number": 8
     },
     {
      "cell_type": "markdown",
@@ -633,13 +633,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b44c8f98>"
+        "<matplotlib.figure.Figure at 0x7f5e9c8fef28>"
        ]
       }
      ],
-     "prompt_number": 40
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
@@ -670,13 +670,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAxkAAAGdCAYAAAB+Xb4YAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14ac4cb9b0>"
+        "<matplotlib.figure.Figure at 0x7f5e9ca7acf8>"
        ]
       }
      ],
-     "prompt_number": 41
+     "prompt_number": 10
     },
     {
      "cell_type": "markdown",
@@ -703,7 +703,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 42
+     "prompt_number": 11
     },
     {
      "cell_type": "code",
@@ -722,19 +722,19 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b41eeda0>"
+        "<matplotlib.figure.Figure at 0x7f5e9c837208>"
        ]
       }
      ],
-     "prompt_number": 43
+     "prompt_number": 12
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "The constant acceleration model is able to track the manuever with no lag, but at the cost of very noisy output during the state state portion of the filter. The noisy output is due to the filter being unable to distinguish between the beginning of an  manuever and just noise in the signal. Noise in the signal implies an acceleration, and so the acceleration term of the filter tracks it. \n",
+      "The constant acceleration model is able to track the manuever with no lag, but at the cost of very noisy output during the steady state behavior. The noisy output is due to the filter being unable to distinguish between the beginning of an  manuever and just noise in the signal. Noise in the signal implies an acceleration, and so the acceleration term of the filter tracks it. \n",
       "\n",
       "It seems we cannot win. A constant velocity filter cannot react quickly when the target is accelerating, but a constant acceleration filter misinterprets noise during zero acceleration regimes as acceleration.\n",
       "\n",
@@ -775,7 +775,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14afea5b38>"
+        "<matplotlib.figure.Figure at 0x7f5e950dc1d0>"
        ]
       }
      ],
@@ -808,9 +808,9 @@
       "    res.append(cvfilter.y[0])\n",
       "xs = np.asarray(xs)\n",
       "plt.subplot(121)\n",
-      "plt.plot(t, z_xs2, label='measurement')\n",
+      "plt.plot(t, z_xs2, label='z')\n",
       "plt.plot(t, xs, label='filter')\n",
-      "plt.legend(loc=4)\n",
+      "plt.legend(loc=2)\n",
       "plt.subplot(122)\n",
       "# plot twice so it has the same color as the plot to the left!\n",
       "plt.plot(t, res)\n",
@@ -824,13 +824,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b41b5f98>"
+        "<matplotlib.figure.Figure at 0x7f5e9ca27630>"
        ]
       }
      ],
-     "prompt_number": 45
+     "prompt_number": 14
     },
     {
      "cell_type": "markdown",
@@ -916,13 +916,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14ac40c0f0>"
+        "<matplotlib.figure.Figure at 0x7f5e9c8a6978>"
        ]
       }
      ],
-     "prompt_number": 46
+     "prompt_number": 15
     },
     {
      "cell_type": "markdown",
@@ -968,11 +968,11 @@
       "        cvfilter.Q /= Q_scale_factor\n",
       "        count -= 1\n",
       "        \n",
-      "plt.plot(t, z_xs2, label='measurement')\n",
+      "plt.plot(t, z_xs2, label='z')\n",
       "plt.plot(t, xs, label='filter')\n",
       "plt.ylabel('$\\epsilon$')\n",
       "plt.xlabel('time (sec)')\n",
-      "plt.legend(loc=4)\n",
+      "plt.legend(loc=2)\n",
       "plt.show()"
      ],
      "language": "python",
@@ -981,13 +981,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14b4175550>"
+        "<matplotlib.figure.Figure at 0x7f5e94e126a0>"
        ]
       }
      ],
-     "prompt_number": 48
+     "prompt_number": 16
     },
     {
      "cell_type": "markdown",
@@ -1079,7 +1079,7 @@
       "xs = np.asarray(xs)\n",
       "\n",
       "plt.subplot(222)\n",
-      "plt.plot(t[0:100], z_xs2, label='measurement', alpha=0.5)\n",
+      "plt.plot(t[0:100], z_xs2, label='z', alpha=0.5)\n",
       "plt.plot(t[0:100], xs, label='filter')\n",
       "plt.legend(loc=2)\n",
       "plt.title('$\\\\mathbf{Q}=20$')\n",
@@ -1098,7 +1098,7 @@
       "    res.append(cvfilter.y[0])\n",
       "xs = np.asarray(xs)\n",
       "plt.subplot(223)\n",
-      "plt.plot(t[0:100], z_xs2, label='measurement', alpha=0.5)\n",
+      "plt.plot(t[0:100], z_xs2, label='z', alpha=0.5)\n",
       "plt.plot(t[0:100], xs, label='filter')\n",
       "plt.legend(loc=2)\n",
       "plt.title('Fading Memory ($\\\\alpha$ = 1.02)')\n",
@@ -1116,7 +1116,7 @@
       "    res.append(cvfilter.y[0])\n",
       "xs = np.asarray(xs)\n",
       "plt.subplot(224)\n",
-      "plt.plot(t[0:100], z_xs2, label='measurement', alpha=0.5)\n",
+      "plt.plot(t[0:100], z_xs2, label='z', alpha=0.5)\n",
       "plt.plot(t[0:100], xs, label='filter')\n",
       "plt.legend(loc=2)\n",
       "plt.title('Fading Memory ($\\\\alpha$ = 1.05)')\n",
@@ -1129,13 +1129,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7f14aa800d30>"
+        "<matplotlib.figure.Figure at 0x7f5e94b3b278>"
        ]
       }
      ],
-     "prompt_number": 79
+     "prompt_number": 17
     },
     {
      "cell_type": "markdown",
@@ -1146,6 +1146,74 @@
       "This is quite good performance for such a small change in code! Note that there is no 'correct' choice here. You will need to design your filter based on your needs and the characteristics of the measurement noise, process noise, and manuevering behavior of the target."
      ]
     },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Noise Level Switching"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "To be written"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Variable State Dimension"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "To be written - vary state order based on whether a manuever is happening."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Multiple Model Estimation"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The example I have been using in this chapter entails a target moving in a steady state, performing a manuever, and then returning to a steate state. We have been thinking of this as two models - a constant velocity model, and a constant acceleration model. Whenever you can describe the system as obeying one of a finite set of models you can use *Multiple Model (MM) Estimation*. This means what it sounds like. We use a bank multiple filters, each using a different process to describe the system, and either switch between them or blend them based on the dynamics of the tracked object. \n",
+      "\n",
+      "As you might imagine this is a broad topic, and there are many ways of designing and implementing MM estimators. But consider a simple approach for the target we have been tracking in this chapter. One idea would be to simultaneously run a constant velocity and a constant acceleration filter, and to switch between their outputs when we detect a manuever by inspecting the residuals. Even this choice gives us many options. Consider the dynamics of a turning object. For example, an automobile turns on a wheelbase - the front wheels turn, and the car pivots around the rear wheels. This is a nonlinear process, so for best results we would want to use some type of nonlinear filter (EKF, UKF, etc) to model the turns. On the other hand, a linear constant velocity filter would perform fine for the steady state portions of the travel. So our bank of filters might consist of a linear KF and an EKF filter for the turns. However, neither is particularly well suited for modelling behaviors such as accelerating and braking. So a highly performing MM estimator might contain a bank of many filters, each designed to perform best for a certain performance envelope of the tracked object.\n",
+      "\n",
+      "I trust the idea of switching between filters is clear, but what mathematical foundations should we use to implement it? The problem that we face is trying to detect via noisy meaurements when a change in regime should result in a change in model. \n",
+      "\n"
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