A lot of work getting baseball tracking working. Added long form for P in KF.

Designing_Kalman_Filters.ipynb

@@ -1,7 +1,7 @@
 {
  "metadata": {
   "name": "",
-  "signature": "sha256:63ae152874a0161848cb3170c24f73b9279fc0e24f7cb997ec958a1870ce9c18"
+  "signature": "sha256:1c5a1fcb5976bac18380450eaff48d09de6b44c73f5da10c88503a02faed3a7d"
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -27,100 +27,229 @@
       "import book_format\n",
       "book_format.load_style()\n",
       "\n",
-      "%install_ext https://raw.github.com/dpsanders/ipython_extensions/master/section_numbering/secnum.py\n",
-      "%load_ext secnum\n",
-      "%secnum"
+      "#%install_ext secnum.py\n",
+      "#%load_ext secnum\n",
+      "#%secnum"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
       {
-       "output_type": "stream",
-       "stream": "stdout",
-       "text": [
-        "Installed secnum.py. To use it, type:\n",
-        "  %load_ext secnum\n"
-       ]
-      },
-      {
-       "javascript": [
-        "console.log(\"Section numbering...\");\n",
-        "\n",
-        "function number_sections(threshold) {\n",
-        "\n",
-        "  var h1_number = 0;\n",
-        "  var h2_number = 0;\n",
-        "\n",
-        "  if (threshold === undefined) {\n",
-        "    threshold = 2;  // does nothing so far\n",
-        "  }\n",
-        "\n",
-        "  var cells = IPython.notebook.get_cells();\n",
-        "  \n",
-        "  for (var i=0; i < cells.length; i++) {\n",
-        "\n",
-        "    var cell = cells[i];\n",
-        "    if (cell.cell_type !== 'heading') continue;\n",
-        "    \n",
-        "    var level = cell.level;\n",
-        "    if (level > threshold) continue;\n",
-        "    \n",
-        "    if (level === 1) {\n",
-        "        \n",
-        "        h1_number ++;\n",
-        "        var h1_element = cell.element.find('h1');\n",
-        "        var h1_html = h1_element.html();\n",
-        "        \n",
-        "        console.log(\"h1_html: \" + h1_html);\n",
-        "\n",
-        "        var patt = /^[0-9]+\\.\\s(.*)/;   // section number at start of string\n",
-        "        var title = h1_html.match(patt);  // just the title\n",
-        "\n",
-        "        if (title != null) {  \n",
-        "          h1_element.html(h1_number + \". \" + title[1]);\n",
-        "        }\n",
-        "        else {\n",
-        "          h1_element.html(h1_number + \". \" + h1_html);\n",
-        "        }\n",
-        "        \n",
-        "        h2_number = 0;\n",
-        "        \n",
+       "html": [
+        "<style>\n",
+        "    div.cell{\n",
+        "        width: 850px;\n",
+        "        margin-left: 0% !important;\n",
+        "        margin-right: auto;\n",
         "    }\n",
-        "    \n",
-        "    if (level === 2) {\n",
-        "    \n",
-        "        h2_number ++;\n",
-        "        \n",
-        "        var h2_element = cell.element.find('h2');\n",
-        "        var h2_html = h2_element.html();\n",
-        "\n",
-        "        console.log(\"h2_html: \" + h2_html);\n",
-        "\n",
-        "        \n",
-        "        var patt = /^[0-9]+\\.[0-9]+\\.\\s/;\n",
-        "        var result = h2_html.match(patt);\n",
-        "\n",
-        "        if (result != null) {\n",
-        "          h2_html = h2_html.replace(result, \"\");\n",
-        "        }\n",
-        "\n",
-        "        h2_element.html(h1_number + \".\" + h2_number + \". \" + h2_html);\n",
-        "        \n",
+        "    div.text_cell code {\n",
+        "        background: #F6F6F9;\n",
+        "        color: #0000FF;\n",
+        "   }\n",
+        "    h1 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "\t}\n",
+        "\t\n",
+        "    div.input_area {\n",
+        "        background: #F6F6F9;\n",
+        "        border: 1px solid #586e75;\n",
         "    }\n",
-        "    \n",
-        "  }\n",
-        "  \n",
-        "}\n",
         "\n",
-        "number_sections();\n",
-        "\n",
-        "// $([IPython.evnts]).on('create.Cell', number_sections);\n",
-        "\n",
-        "$([IPython.events]).on('selected_cell_type_changed.Notebook', number_sections);\n",
-        "\n"
+        "    .text_cell_render h1 {\n",
+        "        font-weight: 200;\n",
+        "        font-size: 30pt;\n",
+        "        line-height: 100%;\n",
+        "        color:#c76c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 1em;\n",
+        "        display: block;\n",
+        "        white-space: wrap;\n",
+        "    } \n",
+        "    h2 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h2 {\n",
+        "        font-weight: 200;\n",
+        "        font-size: 20pt;\n",
+        "        font-style: italic;\n",
+        "        line-height: 100%;\n",
+        "        color:#c76c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 1.5em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    } \n",
+        "    h3 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h3 {\n",
+        "        font-weight: 300;\n",
+        "        font-size: 18pt;\n",
+        "        line-height: 100%;\n",
+        "        color:#d77c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 2em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    }\n",
+        "    h4 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h4 {\n",
+        "        font-weight: 300;\n",
+        "        font-size: 16pt;\n",
+        "        color:#d77c0c;\n",
+        "        margin-bottom: 0.5em;\n",
+        "        margin-top: 0.5em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    }\n",
+        "    h5 {\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "    }\n",
+        "    .text_cell_render h5 {\n",
+        "        font-weight: 300;\n",
+        "        font-style: normal;\n",
+        "        color: #1d3b84;\n",
+        "        font-size: 16pt;\n",
+        "        margin-bottom: 0em;\n",
+        "        margin-top: 1.5em;\n",
+        "        display: block;\n",
+        "        white-space: nowrap;\n",
+        "    }\n",
+        "    div.text_cell_render{\n",
+        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "        line-height: 135%;\n",
+        "        font-size: 110%;\n",
+        "        width:750px;\n",
+        "        margin-left:auto;\n",
+        "        margin-right:auto;\n",
+        "        text-align:justify;\n",
+        "        text-justify:inter-word;\n",
+        "    }\n",
+        "    div.output_subarea.output_text.output_pyout {\n",
+        "        overflow-x: auto;\n",
+        "        overflow-y: scroll;\n",
+        "        max-height: 300px;\n",
+        "    }\n",
+        "    div.output_subarea.output_stream.output_stdout.output_text {\n",
+        "        overflow-x: auto;\n",
+        "        overflow-y: scroll;\n",
+        "        max-height: 300px;\n",
+        "    }\n",
+        "    code{\n",
+        "      font-size: 70%;\n",
+        "    }\n",
+        "    .rendered_html code{\n",
+        "    background-color: transparent;\n",
+        "    }\n",
+        "    ul{\n",
+        "        margin: 2em;\n",
+        "    }\n",
+        "    ul li{\n",
+        "        padding-left: 0.5em; \n",
+        "        margin-bottom: 0.5em; \n",
+        "        margin-top: 0.5em; \n",
+        "    }\n",
+        "    ul li li{\n",
+        "        padding-left: 0.2em; \n",
+        "        margin-bottom: 0.2em; \n",
+        "        margin-top: 0.2em; \n",
+        "    }\n",
+        "    ol{\n",
+        "        margin: 2em;\n",
+        "    }\n",
+        "    ol li{\n",
+        "        padding-left: 0.5em; \n",
+        "        margin-bottom: 0.5em; \n",
+        "        margin-top: 0.5em; \n",
+        "    }\n",
+        "    ul li{\n",
+        "        padding-left: 0.5em; \n",
+        "        margin-bottom: 0.5em; \n",
+        "        margin-top: 0.2em; \n",
+        "    }\n",
+        "    a:link{\n",
+        "       font-weight: bold;\n",
+        "       color:#447adb;\n",
+        "    }\n",
+        "    a:visited{\n",
+        "       font-weight: bold;\n",
+        "       color: #1d3b84;\n",
+        "    }\n",
+        "    a:hover{\n",
+        "       font-weight: bold;\n",
+        "       color: #1d3b84;\n",
+        "    }\n",
+        "    a:focus{\n",
+        "       font-weight: bold;\n",
+        "       color:#447adb;\n",
+        "    }\n",
+        "    a:active{\n",
+        "       font-weight: bold;\n",
+        "       color:#447adb;\n",
+        "    }\n",
+        "    .rendered_html :link {\n",
+        "       text-decoration: underline; \n",
+        "    }\n",
+        "    .rendered_html :hover {\n",
+        "       text-decoration: none; \n",
+        "    }\n",
+        "    .rendered_html :visited {\n",
+        "      text-decoration: none;\n",
+        "    }\n",
+        "    .rendered_html :focus {\n",
+        "      text-decoration: none;\n",
+        "    }\n",
+        "    .rendered_html :active {\n",
+        "      text-decoration: none;\n",
+        "    }\n",
+        "    .warning{\n",
+        "        color: rgb( 240, 20, 20 )\n",
+        "    } \n",
+        "    hr {\n",
+        "      color: #f3f3f3;\n",
+        "      background-color: #f3f3f3;\n",
+        "      height: 1px;\n",
+        "    }\n",
+        "    blockquote{\n",
+        "      display:block;\n",
+        "      background: #fcfcfc;\n",
+        "      border-left: 5px solid #c76c0c;\n",
+        "      font-family: 'Open sans',verdana,arial,sans-serif;\n",
+        "      width:680px;\n",
+        "      padding: 10px 10px 10px 10px;\n",
+        "      text-align:justify;\n",
+        "      text-justify:inter-word;\n",
+        "      }\n",
+        "      blockquote p {\n",
+        "        margin-bottom: 0;\n",
+        "        line-height: 125%;\n",
+        "        font-size: 100%;\n",
+        "      }\n",
+        "</style>\n",
+        "<script>\n",
+        "    MathJax.Hub.Config({\n",
+        "                        TeX: {\n",
+        "                           extensions: [\"AMSmath.js\"]\n",
+        "                           },\n",
+        "                tex2jax: {\n",
+        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
+        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
+        "                },\n",
+        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
+        "                \"HTML-CSS\": {\n",
+        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
+        "                }\n",
+        "        });\n",
+        "</script>\n"
        ],
        "metadata": {},
-       "output_type": "display_data"
+       "output_type": "pyout",
+       "prompt_number": 1,
+       "text": [
+        "<IPython.core.display.HTML at 0x364f710>"
+       ]
       }
      ],
      "prompt_number": 1
@@ -211,9 +340,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2679a10>"
+        "<matplotlib.figure.Figure at 0x4191310>"
        ]
       }
      ],
@@ -547,9 +676,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x26f8210>"
+        "<matplotlib.figure.Figure at 0x4191350>"
        ]
       }
      ],
@@ -626,9 +755,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x2aa8950>"
+        "<matplotlib.figure.Figure at 0x4614b90>"
        ]
       },
       {
@@ -713,7 +842,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x289cb10>"
+        "<matplotlib.figure.Figure at 0x41b2050>"
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@@ -778,7 +907,7 @@
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2730390>"
+        "<matplotlib.figure.Figure at 0x445c990>"
        ]
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      ],
@@ -1050,9 +1179,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x28b4050>"
+        "<matplotlib.figure.Figure at 0x4470510>"
        ]
       }
      ],
@@ -1151,7 +1280,7 @@
       "        self.y = self.y0 = 0\n",
       "        \n",
       "        self.v0_y = v0 * math.sin(self.theta)\n",
-      "        self.v0_x = v0*math.cos(self.theta)\n",
+      "        self.v0_x = v0 * math.cos(self.theta)\n",
       "        \n",
       "        self.noise_scale = noise_scale\n",
       "        \n",
@@ -1200,7 +1329,7 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x2898e90>"
+        "<matplotlib.figure.Figure at 0x42ecd10>"
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        "text": [
-        "<matplotlib.figure.Figure at 0x270ff10>"
+        "<matplotlib.figure.Figure at 0x4a46910>"
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@@ -1376,7 +1505,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x2e25950>"
+        "<matplotlib.figure.Figure at 0x42fd6d0>"
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        "text": [
-        "<matplotlib.figure.Figure at 0x2cd8710>"
+        "<matplotlib.figure.Figure at 0x42fd290>"
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      ],
@@ -1425,9 +1554,9 @@
       {
        "metadata": {},
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x33e3e50>"
+        "<matplotlib.figure.Figure at 0x46a6f50>"
        ]
       }
      ],
@@ -1469,7 +1598,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x2679950>"
+        "<matplotlib.figure.Figure at 0x430cb50>"
        ]
       }
      ],
@@ -1514,7 +1643,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 31
+     "prompt_number": 27
     },
     {
      "cell_type": "code",
@@ -1547,7 +1676,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 32
+     "prompt_number": 28
     },
     {
      "cell_type": "code",
@@ -1567,13 +1696,13 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
+       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x33e5910>"
+        "<matplotlib.figure.Figure at 0x4a69210>"
        ]
       }
      ],
-     "prompt_number": 33
+     "prompt_number": 29
     },
     {
      "cell_type": "markdown",
@@ -1595,25 +1724,478 @@
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "[[ 0.02143298  0.01351531  0.00364892  0.          0.          0.        ]\n",
-        " [ 0.01351531  0.01153516  0.00352809  0.          0.          0.        ]\n",
-        " [ 0.00364892  0.00352809  0.00115592  0.          0.          0.        ]\n",
-        " [ 0.          0.          0.          0.05456716  0.11122734  0.01865339]\n",
-        " [ 0.          0.          0.          0.11122734  0.5379301   0.07344247]\n",
-        " [ 0.          0.          0.          0.01865339  0.07344247  0.18002138]]\n"
+        "[[ 0.05261574  0.09937383  0.10362656  0.          0.          0.        ]\n",
+        " [ 0.09937383  0.29131298  0.41184609  0.          0.          0.        ]\n",
+        " [ 0.10362656  0.41184609  0.95895574  0.          0.          0.        ]\n",
+        " [ 0.          0.          0.          0.05261574  0.09937383  0.10362656]\n",
+        " [ 0.          0.          0.          0.09937383  0.29131298  0.41184609]\n",
+        " [ 0.          0.          0.          0.10362656  0.41184609  0.95895574]]\n"
        ]
       }
      ],
      "prompt_number": 30
     },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "now what if some measurements missing blah blah\n"
+     ]
+    },
     {
      "cell_type": "code",
      "collapsed": false,
-     "input": [],
+     "input": [
+      "def plot_ball_filter6_miss (zs, miss_ratio=0.2):\n",
+      "    xs, ys = [],[]\n",
+      "    pxs, pys = [],[]\n",
+      "        \n",
+      "    for i,z in enumerate(zs):\n",
+      "        m = np.mat([z[0], 0, 0, z[1], 0, 0]).T\n",
+      "\n",
+      "        f1.predict ()\n",
+      "        print (i, f1.x)   \n",
+      "        \n",
+      "        if i < 20 or i > 40:\n",
+      "        #if random.rand() >= miss_ratio:\n",
+      "            f1.update (m)\n",
+      "\n",
+      "\n",
+      "        xs.append (f1.x[0,0])\n",
+      "        ys.append (f1.x[3,0])\n",
+      "        pxs.append (z[0])\n",
+      "        pys.append(z[1])\n",
+      "        \n",
+      "        if i > 0 and z[1] < 0:\n",
+      "            break;\n",
+      "\n",
+      "    p1, = plt.plot (xs, ys, 'r--')\n",
+      "    p2, = plt.plot (pxs, pys)\n",
+      "    plt.axis('equal')\n",
+      "    plt.legend([p1,p2], ['filter', 'measurement'], 2)\n",
+      "    plt.show()"
+     ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 30
+     "prompt_number": 44
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "dt = 0.1\n",
+      "noise = 0.0\n",
+      "f1 = ball_filter6(dt, R=.16, Q=0.1)\n",
+      "\n",
+      "x,y = baseball.compute_trajectory(v_0_mph = 90., theta=65., dt=dt)\n",
+      "znoise = [(i+randn()*noise,j+randn()*noise) for (i,j) in zip(x,y)]\n",
+      "plot_ball_filter6_miss (znoise)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "0 [[ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]]\n",
+        "1 [[ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]\n",
+        " [ 0.]]\n",
+        "2 [[  2.44405979]\n",
+        " [ 10.68643825]\n",
+        " [  2.02487675]\n",
+        " [  5.24130313]\n",
+        " [ 22.91714079]\n",
+        " [  4.34236221]]\n",
+        "3 [[  4.66270121]\n",
+        " [ 15.0959457 ]\n",
+        " [  3.41998384]\n",
+        " [  9.8787654 ]\n",
+        " [ 31.92489774]\n",
+        " [  7.1854626 ]]\n",
+        "4 [[  6.53889664]\n",
+        " [ 16.51155164]\n",
+        " [  4.13016625]\n",
+        " [ 13.68152078]\n",
+        " [ 34.38243413]\n",
+        " [  8.33579   ]]\n",
+        "5 [[  8.26612083]\n",
+        " [ 17.03075509]\n",
+        " [  4.25071537]\n",
+        " [ 17.06352352]\n",
+        " [ 34.78964978]\n",
+        " [  7.8517589 ]]\n",
+        "6 [[  9.89364089]\n",
+        " [ 17.09199018]\n",
+        " [  3.69648655]\n",
+        " [ 20.11627647]\n",
+        " [ 34.01938789]\n",
+        " [  5.48237438]]\n",
+        "7 [[ 11.42611493]\n",
+        " [ 16.79051327]\n",
+        " [  2.55972089]\n",
+        " [ 22.84083372]\n",
+        " [ 32.26802293]\n",
+        " [  1.5872627 ]]\n",
+        "8 [[ 12.87107224]\n",
+        " [ 16.23334243]\n",
+        " [  1.17838404]\n",
+        " [ 25.26137529]\n",
+        " [ 29.85249228]\n",
+        " [ -2.78578886]]\n",
+        "9 [[ 14.24807549]\n",
+        " [ 15.56446867]\n",
+        " [ -0.09987618]\n",
+        " [ 27.43868368]\n",
+        " [ 27.19628665]\n",
+        " [ -6.64898788]]\n",
+        "10 [[ 15.57762857]\n",
+        " [ 14.89469275]\n",
+        " [ -1.10589939]\n",
+        " [ 29.43283218]\n",
+        " [ 24.60662615]\n",
+        " [ -9.58380039]]\n",
+        "11 [[ 16.87332115]\n",
+        " [ 14.27675526]\n",
+        " [ -1.82766582]\n",
+        " [ 31.28127693]\n",
+        " [ 22.21442747]\n",
+        " [-11.6235898 ]]\n",
+        "12 [[ 18.14212805]\n",
+        " [ 13.7239801 ]\n",
+        " [ -2.31896548]\n",
+        " [ 33.00164543]\n",
+        " [ 20.0392546 ]\n",
+        " [-12.9684103 ]]\n",
+        "13 [[ 19.3872636 ]\n",
+        " [ 13.23249429]\n",
+        " [ -2.64174192]\n",
+        " [ 34.60092379]\n",
+        " [ 18.05657215]\n",
+        " [-13.82119383]]\n",
+        "14 [[ 20.61032298]\n",
+        " [ 12.79363675]\n",
+        " [ -2.84612871]\n",
+        " [ 36.08182295]\n",
+        " [ 16.23273747]\n",
+        " [-14.33801914]]\n",
+        "15 [[ 21.81236356]\n",
+        " [ 12.39889874]\n",
+        " [ -2.96807894]\n",
+        " [ 37.44581783]\n",
+        " [ 14.53772217]\n",
+        " [-14.62728128]]\n",
+        "16 [[ 22.99434735]\n",
+        " [ 12.0413384 ]\n",
+        " [ -3.03216273]\n",
+        " [ 38.69430853]\n",
+        " [ 12.94798303]\n",
+        " [-14.7612569 ]]\n",
+        "17 [[ 24.15728912]\n",
+        " [ 11.71569135]\n",
+        " [ -3.05485765]\n",
+        " [ 39.82896024]\n",
+        " [ 11.44599382]\n",
+        " [-14.78742499]]\n",
+        "18 [[ 25.30228828]\n",
+        " [ 11.41810935]\n",
+        " [ -3.04713115]\n",
+        " [ 40.851739  ]\n",
+        " [ 10.01900867]\n",
+        " [-14.73682841]]\n",
+        "19 [[ 26.43052127]\n",
+        " [ 11.14585251]\n",
+        " [ -3.01624667]\n",
+        " [ 41.76485573]\n",
+        " [  8.65788025]\n",
+        " [-14.6297248 ]]\n",
+        "20 [[ 27.54322312]\n",
+        " [ 10.89702311]\n",
+        " [ -2.96698642]\n",
+        " [ 42.57069332]\n",
+        " [  7.35611195]\n",
+        " [-14.47931437]]\n",
+        "21 [[ 28.60325556]\n",
+        " [ 10.60032446]\n",
+        " [ -2.96698642]\n",
+        " [ 43.16151138]\n",
+        " [  5.90818051]\n",
+        " [-14.47931437]]\n",
+        "22 [[ 29.63361814]\n",
+        " [ 10.30362582]\n",
+        " [ -2.96698642]\n",
+        " [ 43.60753628]\n",
+        " [  4.46024908]\n",
+        " [-14.47931437]]\n",
+        "23 [[ 30.63431086]\n",
+        " [ 10.00692718]\n",
+        " [ -2.96698642]\n",
+        " [ 43.90876805]\n",
+        " [  3.01231764]\n",
+        " [-14.47931437]]\n",
+        "24 [[ 31.60533372]\n",
+        " [  9.71022854]\n",
+        " [ -2.96698642]\n",
+        " [ 44.06520667]\n",
+        " [  1.5643862 ]\n",
+        " [-14.47931437]]\n",
+        "25 [[ 32.54668671]\n",
+        " [  9.4135299 ]\n",
+        " [ -2.96698642]\n",
+        " [ 44.07685214]\n",
+        " [  0.11645477]\n",
+        " [-14.47931437]]\n",
+        "26 [[ 33.45836983]\n",
+        " [  9.11683125]\n",
+        " [ -2.96698642]\n",
+        " [ 43.94370448]\n",
+        " [ -1.33147667]\n",
+        " [-14.47931437]]\n",
+        "27 [[ 34.34038309]\n",
+        " [  8.82013261]\n",
+        " [ -2.96698642]\n",
+        " [ 43.66576367]\n",
+        " [ -2.77940811]\n",
+        " [-14.47931437]]\n",
+        "28 [[ 35.19272649]\n",
+        " [  8.52343397]\n",
+        " [ -2.96698642]\n",
+        " [ 43.24302971]\n",
+        " [ -4.22733955]\n",
+        " [-14.47931437]]\n",
+        "29 [[ 36.01540002]\n",
+        " [  8.22673533]\n",
+        " [ -2.96698642]\n",
+        " [ 42.67550261]\n",
+        " [ -5.67527098]\n",
+        " [-14.47931437]]\n",
+        "30 [[ 36.80840369]\n",
+        " [  7.93003669]\n",
+        " [ -2.96698642]\n",
+        " [ 41.96318237]\n",
+        " [ -7.12320242]\n",
+        " [-14.47931437]]\n",
+        "31 [[ 37.57173749]\n",
+        " [  7.63333804]\n",
+        " [ -2.96698642]\n",
+        " [ 41.10606899]\n",
+        " [ -8.57113386]\n",
+        " [-14.47931437]]\n",
+        "32 [[ 38.30540143]\n",
+        " [  7.3366394 ]\n",
+        " [ -2.96698642]\n",
+        " [ 40.10416246]\n",
+        " [-10.01906529]\n",
+        " [-14.47931437]]\n",
+        "33 [[ 39.00939551]\n",
+        " [  7.03994076]\n",
+        " [ -2.96698642]\n",
+        " [ 38.95746278]\n",
+        " [-11.46699673]\n",
+        " [-14.47931437]]\n",
+        "34 [[ 39.68371972]\n",
+        " [  6.74324212]\n",
+        " [ -2.96698642]\n",
+        " [ 37.66596997]\n",
+        " [-12.91492817]\n",
+        " [-14.47931437]]\n",
+        "35 [[ 40.32837407]\n",
+        " [  6.44654348]\n",
+        " [ -2.96698642]\n",
+        " [ 36.22968401]\n",
+        " [-14.3628596 ]\n",
+        " [-14.47931437]]\n",
+        "36 [[ 40.94335855]\n",
+        " [  6.14984483]\n",
+        " [ -2.96698642]\n",
+        " [ 34.6486049 ]\n",
+        " [-15.81079104]\n",
+        " [-14.47931437]]\n",
+        "37 [[ 41.52867317]\n",
+        " [  5.85314619]\n",
+        " [ -2.96698642]\n",
+        " [ 32.92273265]\n",
+        " [-17.25872248]\n",
+        " [-14.47931437]]\n",
+        "38 [[ 42.08431793]\n",
+        " [  5.55644755]\n",
+        " [ -2.96698642]\n",
+        " [ 31.05206726]\n",
+        " [-18.70665391]\n",
+        " [-14.47931437]]\n",
+        "39 [[ 42.61029282]\n",
+        " [  5.25974891]\n",
+        " [ -2.96698642]\n",
+        " [ 29.03660873]\n",
+        " [-20.15458535]\n",
+        " [-14.47931437]]\n",
+        "40 [[ 43.10659784]\n",
+        " [  4.96305027]\n",
+        " [ -2.96698642]\n",
+        " [ 26.87635705]\n",
+        " [-21.60251679]\n",
+        " [-14.47931437]]\n",
+        "41 [[ 43.57323301]\n",
+        " [  4.66635162]\n",
+        " [ -2.96698642]\n",
+        " [ 24.57131223]\n",
+        " [-23.05044822]\n",
+        " [-14.47931437]]\n",
+        "42 [[ 49.53875091]\n",
+        " [  8.41022586]\n",
+        " [ -1.35829677]\n",
+        " [ 36.25872343]\n",
+        " [-14.16609388]\n",
+        " [-10.36567957]]\n",
+        "43 [[ 50.41875369]\n",
+        " [  8.33134456]\n",
+        " [ -1.32141671]\n",
+        " [ 34.91649773]\n",
+        " [-15.00976788]\n",
+        " [-10.24076045]]\n",
+        "44 [[ 51.27664835]\n",
+        " [  8.25959907]\n",
+        " [ -1.27392093]\n",
+        " [ 33.47134349]\n",
+        " [-15.78218742]\n",
+        " [-10.04285686]]\n",
+        "45 [[ 52.12583444]\n",
+        " [  8.20292466]\n",
+        " [ -1.21554564]\n",
+        " [ 31.96934394]\n",
+        " [-16.43913584]\n",
+        " [ -9.75614053]]\n",
+        "46 [[ 52.96740558]\n",
+        " [  8.15248606]\n",
+        " [ -1.1568817 ]\n",
+        " [ 30.42208773]\n",
+        " [-17.00163737]\n",
+        " [ -9.41536186]]"
+       ]
+      },
+      {
+       "output_type": "stream",
+       "stream": "stdout",
+       "text": [
+        "\n",
+        "47 [[ 53.79863088]\n",
+        " [  8.09573592]\n",
+        " [ -1.10900586]\n",
+        " [ 28.82446485]\n",
+        " [-17.52231427]\n",
+        " [ -9.07349441]]\n",
+        "48 [[ 54.61635331]\n",
+        " [  8.02488892]\n",
+        " [ -1.07699429]\n",
+        " [ 27.16568144]\n",
+        " [-18.04621201]\n",
+        " [ -8.76703269]]\n",
+        "49 [[ 55.41860877]\n",
+        " [  7.93827355]\n",
+        " [ -1.06032042]\n",
+        " [ 25.43776715]\n",
+        " [-18.59398868]\n",
+        " [ -8.50690811]]\n",
+        "50 [[ 56.20461371]\n",
+        " [  7.83758198]\n",
+        " [ -1.05610405]\n",
+        " [ 23.63768075]\n",
+        " [-19.16739559]\n",
+        " [ -8.2879896 ]]\n",
+        "51 [[ 56.97422095]\n",
+        " [  7.72532752]\n",
+        " [ -1.0613927 ]\n",
+        " [ 21.76593979]\n",
+        " [-19.75952268]\n",
+        " [ -8.09959974]]\n",
+        "52 [[ 57.72749988]\n",
+        " [  7.60367152]\n",
+        " [ -1.07395753]\n",
+        " [ 19.82490448]\n",
+        " [-20.36130989]\n",
+        " [ -7.93102968]]\n",
+        "53 [[ 58.46453621]\n",
+        " [  7.47415459]\n",
+        " [ -1.09230102]\n",
+        " [ 17.81769808]\n",
+        " [-20.96425829]\n",
+        " [ -7.77334825]]\n",
+        "54 [[ 59.18537127]\n",
+        " [  7.3377984 ]\n",
+        " [ -1.11541441]\n",
+        " [ 15.74768189]\n",
+        " [-21.56116738]\n",
+        " [ -7.61961252]]\n",
+        "55 [[ 59.89000815]\n",
+        " [  7.19530697]\n",
+        " [ -1.14251657]\n",
+        " [ 13.61822836]\n",
+        " [-22.14615781]\n",
+        " [ -7.46464341]]\n",
+        "56 [[ 60.5784447 ]\n",
+        " [  7.04726395]\n",
+        " [ -1.17284942]\n",
+        " [ 11.43261659]\n",
+        " [-22.71455044]\n",
+        " [ -7.30479717]]\n",
+        "57 [[ 61.25071319]\n",
+        " [  6.89428876]\n",
+        " [ -1.20555065]\n",
+        " [  9.19396376]\n",
+        " [-23.26278579]\n",
+        " [ -7.13781556]]\n",
+        "58 [[ 61.9069146 ]\n",
+        " [  6.73713538]\n",
+        " [ -1.23960771]\n",
+        " [  6.90516348]\n",
+        " [-23.78839568]\n",
+        " [ -6.96270983]]\n",
+        "59 [[ 62.54723957]\n",
+        " [  6.57672729]\n",
+        " [ -1.27388497]\n",
+        " [  4.56883211]\n",
+        " [-24.28997646]\n",
+        " [ -6.77961145]]\n",
+        "60 [[ 63.17197196]\n",
+        " [  6.41413318]\n",
+        " [ -1.30720331]\n",
+        " [  2.18727552]\n",
+        " [-24.7671111 ]\n",
+        " [ -6.58955065]]\n",
+        "61 [[ 63.781476  ]\n",
+        " [  6.25050063]\n",
+        " [ -1.33844179]\n",
+        " [ -0.23751324]\n",
+        " [-25.22021619]\n",
+        " [ -6.39416907]]\n",
+        "62 [[ 64.37617233]\n",
+        " [  6.0869722 ]\n",
+        " [ -1.36663031]\n",
+        " [ -2.70382451]\n",
+        " [-25.65032694]\n",
+        " [ -6.19540949]]\n"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAsAAAAFyCAYAAAD7xWwMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+       "text": [
+        "<matplotlib.figure.Figure at 0x4308850>"
+       ]
+      }
+     ],
+     "prompt_number": 46
     }
    ],
    "metadata": {}

KalmanFilter.py

@@ -12,8 +12,21 @@ import numpy.random as random
 
 class KalmanFilter:
 
-    def __init__(self, dim):
-        """ Create a Kalman filter of dimension 'dim'"""
+    def __init__(self, dim, 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)) # uncertainty covariance
@@ -24,6 +37,9 @@ class KalmanFilter:
         self.H = 0 # Measurement function (maps state to measurements)
         self.R = np.matrix(np.eye(1)) # state uncertainty
         self.I = np.matrix(np.eye(dim))
+        
+        if use_short_form:
+            self.update = self.update_short_form
 
 
     def update(self, Z):
@@ -39,11 +55,41 @@ class KalmanFilter:
         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.u
+        self.x = (self.F*self.x) + self.B * self.u
         self.P = self.F * self.P * self.F.T + self.Q
 
 

baseball.py

@@ -1,14 +1,7 @@
 # -*- coding: utf-8 -*-
-"""
-Spyder Editor
-
-This temporary script file is located here:
-/home/rlabbe/.spyder2/.temp.py
-"""
 
 
 """ 
-
 Computes the trajectory of a stitched baseball with air drag.
 Takes altitude into account (higher altitude means further travel) and the
 stitching on the baseball influencing drag. 
@@ -20,9 +13,11 @@ LOWER the higher its velocity, which is somewhat counterintuitive.
 """
 
 
-
+from __future__ import division
 import math
 import matplotlib.pyplot as plt
+from numpy.random import randn
+import numpy as np
 
 def a_drag (vel, altitude):
     """ returns the drag coefficient of a baseball at a given velocity (m/s)
@@ -38,13 +33,45 @@ def a_drag (vel, altitude):
     
     return val
 
-
+def compute_trajectory_vacuum (v_0_mph, 
+                               theta, 
+                               dt=0.01, 
+                               noise_scale=0.0, 
+                               x0=0., y0 = 0.):
+    theta = math.radians(theta)
+        
+    x = x0
+    y = y0
+    
+    v0_y = v_0_mph * math.sin(theta)
+    v0_x = v_0_mph * math.cos(theta)    
+    
+    xs = []
+    ys = []
+    
+    t = dt
+    while y >= 0:
+        x = v0_x*t + x0
+        y = -0.5*9.8*t**2 + v0_y*t + y0
+        
+        xs.append (x + randn() * noise_scale)        
+        ys.append (y + randn() * noise_scale)
+        
+        t += dt
+        
+    return (xs, ys)
+        
+        
 
 def compute_trajectory(v_0_mph, theta, v_wind_mph=0, alt_ft = 0.0, dt = 0.01):
+    g = 9.8
+    
     ### comput
     theta = math.radians(theta)
-    
+    # initial velocity in direction of travel
     v_0 = v_0_mph * 0.447 # mph to m/s
+    
+    # velocity components in x and y
     v_x = v_0 * math.cos(theta)
     v_y = v_0 * math.sin(theta)
    
@@ -58,7 +85,6 @@ def compute_trajectory(v_0_mph, theta, v_wind_mph=0, alt_ft = 0.0, dt = 0.01):
     
     i = 0
     while y[i] >= ground_level:
-        g = 9.8
         
         v = math.sqrt((v_x - v_wind) **2+ v_y**2)
         
@@ -69,7 +95,37 @@ def compute_trajectory(v_0_mph, theta, v_wind_mph=0, alt_ft = 0.0, dt = 0.01):
         v_y = v_y - a_drag(v, altitude) * v * v_y * dt - g * dt
         i += 1
         
+    # meters to yards
+    np.multiply (x, 1.09361)
+    np.multiply (y, 1.09361)
+    
     return (x,y)
+    
+
+def predict (x0, y0, x1, y1, dt, time):
+    g = 10.724*dt*dt
+    
+    v_x = x1-x0
+    v_y = y1-y0    
+    v = math.sqrt(v_x**2 + v_y**2)
+    
+    x = x1
+    y = y1
+    
+    
+    while time > 0:
+        v_x = v_x - a_drag(v, y) * v * v_x
+        v_y = v_y - a_drag(v, y) * v * v_y - g
+       
+        x = x + v_x
+        y = y + v_y
+
+        time -= dt
+        
+    return (x,y)
+
+        
+    
  
 if __name__ == "__main__":
     x,y = compute_trajectory(v_0_mph = 110., theta=35., v_wind_mph = 0., alt_ft=5000.)

bb_test.py

@@ -0,0 +1,218 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Wed Jun  4 12:33:38 2014
+
+@author: rlabbe
+"""
+
+from __future__ import print_function,division
+from KalmanFilter import KalmanFilter
+import numpy as np
+import matplotlib.pyplot as plt
+import baseball
+from numpy.random import randn
+
+def ball_filter6(dt,R=1., Q = 0.1):
+    f1 = KalmanFilter(dim=6)
+    g = 10
+
+    f1.F = np.mat ([[1., dt, dt**2,  0,       0,  0],
+                    [0,  1., dt,     0,       0,  0],
+                    [0,  0,  1.,     0,       0,  0],
+                    [0,  0,  0.,    1., dt, -0.5*dt*dt*g],
+                    [0,  0,  0,      0, 1.,      -g*dt],
+                    [0,  0,  0,      0, 0.,      1.]])
+
+    f1.H = np.mat([[1,0,0,0,0,0],
+                   [0,0,0,0,0,0],
+                   [0,0,0,0,0,0],
+                   [0,0,0,1,0,0],
+                   [0,0,0,0,0,0],
+                   [0,0,0,0,0,0]])
+
+
+    f1.R = np.mat(np.eye(6)) * R 
+    
+    f1.Q = np.zeros((6,6))
+    f1.Q[2,2] = Q    
+    f1.Q[5,5] = Q
+    f1.x = np.mat([0, 0 , 0, 0, 0, 0]).T
+    f1.P = np.eye(6) * 50.
+    f1.B = 0.
+    f1.u = 0
+    
+    return f1
+    
+    
+def ball_filter4(dt,R=1., Q = 0.1):
+    f1 = KalmanFilter(dim=4)
+    g = 10
+
+    f1.F = np.mat ([[1., dt,  0, 0,],
+                    [0,  1.,  0, 0],
+                    [0,  0,  1., dt],
+                    [0,  0,  0.,  1.]])
+
+    f1.H = np.mat([[1,0,0,0],
+                   [0,0,0,0],
+                   [0,0,1,0],
+                   [0,0,0,0]])
+  
+
+
+    f1.B = np.mat([[0,0,0,0],
+                   [0,0,0,0],
+                   [0,0,1.,0],
+                   [0,0,0,1.]])
+                   
+    f1.u = np.mat([[0],
+                   [0],
+                   [-0.5*g*dt**2],
+                   [-g*dt]])
+                   
+    f1.R = np.mat(np.eye(4)) * R 
+    
+    f1.Q = np.zeros((4,4))
+    f1.Q[1,1] = Q    
+    f1.Q[3,3] = Q
+    f1.x = np.mat([0, 0 , 0, 0]).T
+    f1.P = np.eye(4) * 50.    
+    return f1
+    
+
+def plot_ball_filter6 (f1, zs, skip_start=-1, skip_end=-1):
+    xs, ys = [],[]
+    pxs, pys = [],[]
+        
+    for i,z in enumerate(zs):
+        m = np.mat([z[0], 0, 0, z[1], 0, 0]).T
+
+        f1.predict ()
+                    
+        if i == skip_start:
+            x2 = xs[-2]
+            x1 = xs[-1]
+            y2 = ys[-2]
+            y1 = ys[-1]
+            
+        if i >= skip_start and i <= skip_end:
+            #x,y = baseball.predict (x2, y2, x1, y1, 1/30, 1/30* (i-skip_start+1))
+            x,y = baseball.predict (xs[-2], ys[-2], xs[-1], ys[-1], 1/30, 1/30)
+
+            m[0] = x
+            m[3] = y
+        #print ('skip', i, f1.x[2],f1.x[5])
+
+        f1.update_long_form (m)
+
+        
+        '''
+        if i >= skip_start and i <= skip_end:
+            #f1.x[2] = -0.1
+            #f1.x[5] = -17.
+            pass
+        else:
+            f1.update (m)
+
+        '''       
+
+        xs.append (f1.x[0,0])
+        ys.append (f1.x[3,0])
+        pxs.append (z[0])
+        pys.append(z[1])
+        
+        if i > 0 and z[1] < 0:
+            break;
+'''
+    p1, = plt.plot (xs, ys, 'r--')
+    p2, = plt.plot (pxs, pys)
+    plt.axis('equal')
+    plt.legend([p1,p2], ['filter', 'measurement'], 2)
+    plt.xlim([0,xs[-1]])
+    plt.show()
+'''    
+
+
+def plot_ball_filter4 (f1, zs, skip_start=-1, skip_end=-1):
+    xs, ys = [],[]
+    pxs, pys = [],[]
+        
+    for i,z in enumerate(zs):
+        m = np.mat([z[0], 0, z[1], 0]).T
+
+        f1.predict ()
+                    
+        if i == skip_start:
+            x2 = xs[-2]
+            x1 = xs[-1]
+            y2 = ys[-2]
+            y1 = ys[-1]
+            
+        if i >= skip_start and i <= skip_end:
+            #x,y = baseball.predict (x2, y2, x1, y1, 1/30, 1/30* (i-skip_start+1))
+            x,y = baseball.predict (xs[-2], ys[-2], xs[-1], ys[-1], 1/30, 1/30)
+
+            m[0] = x
+            m[2] = y
+ 
+        f1.update_long_form (m)
+
+        
+        '''
+        if i >= skip_start and i <= skip_end:
+            #f1.x[2] = -0.1
+            #f1.x[5] = -17.
+            pass
+        else:
+            f1.update (m)
+
+        '''       
+
+        xs.append (f1.x[0,0])
+        ys.append (f1.x[2,0])
+        pxs.append (z[0])
+        pys.append(z[1])
+        
+        if i > 0 and z[1] < 0:
+            break;
+'''
+    plt.plot (xs, ys, 'r--')
+    plt.plot (pxs, pys)
+    plt.axis('equal')
+    plt.legend([p1,p2], ['filter', 'measurement'], 2)
+    plt.xlim([0,xs[-1]])
+    plt.show()
+'''
+
+start_skip = 20
+end_skip = 60
+
+def run_6():  
+    dt = 1/30
+    noise = 0.0
+    f1 = ball_filter6(dt, R=.16, Q=0.1)
+    #plt.cla()
+    
+    x,y = baseball.compute_trajectory(v_0_mph = 100., theta=50., dt=dt)
+    
+    
+    znoise = [(i+randn()*noise,j+randn()*noise) for (i,j) in zip(x,y)]
+    
+    plot_ball_filter6 (f1, znoise, start_skip, end_skip)
+
+
+def run_4():  
+    dt = 1/30
+    noise = 0.0
+    f1 = ball_filter4(dt, R=.16, Q=0.1)
+    plt.cla()
+    
+    x,y = baseball.compute_trajectory(v_0_mph = 100., theta=50., dt=dt)
+    
+    
+    znoise = [(i+randn()*noise,j+randn()*noise) for (i,j) in zip(x,y)]
+    
+    plot_ball_filter4 (f1, znoise, start_skip, end_skip)
+    
+    
+run_6()

secnum.js

@@ -0,0 +1,74 @@
+console.log("Section numbering...");
+
+function number_sections(threshold) {
+
+  var h1_number = 0;
+  var h2_number = 0;
+
+  if (threshold === undefined) {
+    threshold = 2;  // does nothing so far
+  }
+
+  var cells = IPython.notebook.get_cells();
+  
+  for (var i=0; i < cells.length; i++) {
+
+    var cell = cells[i];
+    if (cell.cell_type !== 'heading') continue;
+    
+    var level = cell.level;
+    if (level > threshold) continue;
+    
+    if (level === 1) {
+        
+        h1_number ++;
+        var h1_element = cell.element.find('h1');
+        var h1_html = h1_element.html();
+        
+        console.log("h1_html: " + h1_html);
+
+        var patt = /^[0-9]+\.\s(.*)/;   // section number at start of string
+        var title = h1_html.match(patt);  // just the title
+
+        if (title != null) {  
+          h1_element.html(h1_number + ". " + title[1]);
+        }
+        else {
+          h1_element.html(h1_number + ". " + h1_html);
+        }
+        
+        h2_number = 0;
+        
+    }
+    
+    if (level === 2) {
+    
+        h2_number ++;
+        
+        var h2_element = cell.element.find('h2');
+        var h2_html = h2_element.html();
+
+        console.log("h2_html: " + h2_html);
+
+        
+        var patt = /^[0-9]+\.[0-9]+\.\s/;
+        var result = h2_html.match(patt);
+
+        if (result != null) {
+          h2_html = h2_html.replace(result, "");
+        }
+
+        h2_element.html(h1_number + "." + h2_number + ". " + h2_html);
+        
+    }
+    
+  }
+  
+}
+
+number_sections();
+
+// $([IPython.evnts]).on('create.Cell', number_sections);
+
+$([IPython.events]).on('selected_cell_type_changed.Notebook', number_sections);
+

secnum.py

@@ -0,0 +1,79 @@
+"""
+IPython Notebook magic command for automatically numbering sections of a notebook "interactively"
+
+Use:
+    %install_ext https://raw.github.com/dpsanders/ipython_extensions/master/section_numbering/secnum.py
+
+once to install, and then
+
+    %load_ext secnum
+    %secnum
+
+to get automatic section numbering, which is updated when any cell changes type.
+(This could, of course, be more efficient.)
+
+This is based, completely and shamelessly, on MinRK's `nbtoc` magic
+<https://github.com/minrk/ipython_extensions>
+
+Thanks also to:
+- Clayton Davis, for a great explanation of HTML, CSS and JavaScript
+- Brian Granger, for a crucial hint about extracting a JQuery object from a notebook cell object
+- Fernando Perez, for an inspiring talk at SciPy 2013 (and, of course, for creating IPython)
+
+"""
+
+import io
+import os
+import urllib2
+
+from IPython.display import display_html, display_javascript
+
+here = os.path.abspath(os.path.dirname(__file__))
+secnum_js = ""
+
+def download(fname, redownload=False):
+    """download a file
+    
+    if redownload=False, the file will not be downloaded if it already exists.
+    """
+    dest = os.path.join(here, fname)
+    if os.path.exists(dest) and not redownload:
+        return
+    url = 'https://raw.github.com/dpsanders/ipython_extensions/master/section_numbering/' + fname
+    print("Downloading %s to %s" % (url, dest))
+    
+    filein  = urllib2.urlopen(url)
+    fileout = open(dest, "wb")
+    chunk = filein.read(1024)
+    while chunk:
+        fileout.write(chunk)
+        chunk = filein.read(1024)
+    filein.close()
+    fileout.close()
+
+def load_file(fname, redownload=False):
+    """load global variable from a file"""
+    download(fname, redownload)
+    with io.open(os.path.join(here, fname)) as f:
+        globals()[fname.replace('.', '_')] = f.read()
+
+
+load_file('secnum.js')
+
+
+
+def secnum(line):
+    """add a table of contents to an IPython Notebook"""
+    display_javascript(secnum_js, raw=True)
+
+
+def update_secnum(line):
+    """download the latest version of the nbtoc extension from GitHub"""
+    download('secnum.py', True)
+    download('secnum.js', True)
+    get_ipython().extension_manager.reload_extension("secnum")
+    
+def load_ipython_extension(ip):
+    ip.magics_manager.register_function(secnum)
+    ip.magics_manager.register_function(update_secnum)
+