Added lying to filter, and filterpy sections

2015-07-14 10:36:48 -07:00
1 parent aba6a9dcf7
commit cd6a2e9d42
@@ -329,7 +329,7 @@
     "data": {
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      "text/plain": [
-       "<matplotlib.figure.Figure at 0x53f72b0>"
+       "<matplotlib.figure.Figure at 0x5a630f0>"
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     },
     "metadata": {},
@@ -364,7 +364,7 @@
     "data": {
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      "text/plain": [
-       "<matplotlib.figure.Figure at 0x72bb8d0>"
+       "<matplotlib.figure.Figure at 0x78ee710>"
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     },
     "metadata": {},
@@ -396,7 +396,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a858d0>"
+       "<matplotlib.figure.Figure at 0x80574e0>"
      ]
     },
     "metadata": {},
@@ -432,7 +432,7 @@
     "name": "stdout",
     "output_type": "stream",
     "text": [
-      "Average of measurements is 165.0558\n"
+      "Average of measurements is 165.0055\n"
     ]
    }
   ],
@@ -463,7 +463,7 @@
     "name": "stdout",
     "output_type": "stream",
     "text": [
-      "Average of measurements is 164.9520\n"
+      "Average of measurements is 164.9421\n"
     ]
    }
   ],
@@ -496,7 +496,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x53ebeb8>"
+       "<matplotlib.figure.Figure at 0x5a4bef0>"
      ]
     },
     "metadata": {},
@@ -534,7 +534,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x53f70b8>"
+       "<matplotlib.figure.Figure at 0x5a4b8d0>"
      ]
     },
     "metadata": {},
@@ -563,7 +563,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c20748>"
+       "<matplotlib.figure.Figure at 0x824aeb8>"
      ]
     },
     "metadata": {},
@@ -594,7 +594,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7bc6780>"
+       "<matplotlib.figure.Figure at 0x82c7cf8>"
      ]
     },
     "metadata": {},
@@ -626,7 +626,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c449b0>"
+       "<matplotlib.figure.Figure at 0x81f1d30>"
      ]
     },
     "metadata": {},
@@ -661,7 +661,7 @@
     "data": {
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      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a427f0>"
+       "<matplotlib.figure.Figure at 0x8137b70>"
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     },
     "metadata": {},
@@ -690,7 +690,7 @@
     "data": {
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      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a9a400>"
+       "<matplotlib.figure.Figure at 0x81a4400>"
      ]
     },
     "metadata": {},
@@ -732,7 +732,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a39208>"
+       "<matplotlib.figure.Figure at 0x80e6588>"
      ]
     },
     "metadata": {},
@@ -790,7 +790,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7cf3e10>"
+       "<matplotlib.figure.Figure at 0x83149e8>"
      ]
     },
     "metadata": {},
@@ -867,7 +867,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f4f278>"
+       "<matplotlib.figure.Figure at 0x856e160>"
      ]
     },
     "metadata": {},
@@ -909,7 +909,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a88550>"
+       "<matplotlib.figure.Figure at 0x7972160>"
      ]
     },
     "metadata": {},
@@ -979,16 +979,16 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "This algorithm is known as the g-h filter. *g* and *h* refer to the two scaling factors that we used in our example. *g* is the scaling we used for the measurement (weight in our example), and *h* is the scaling for the change in measurement over time (lbs/day in our example)."
+    "This algorithm is known as the g-h filter. $g$ and $h$ refer to the two scaling factors that we used in our example. $g$ is the scaling we used for the measurement (weight in our example), and $h$ is the scaling for the change in measurement over time (lbs/day in our example)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "This filter is the basis for a huge number of filters, including the Kalman filter. In other words, the Kalman filter is a form of the g-h filter, which I will prove later in the book. So is the Least Squares filter, which you may have heard of, and so is the Benedict-Bordner filter, which you probably have not. Each filter has a different way of assigning values to *g* and *h*, but otherwise the algorithms are identical. For example, the $\\alpha$-$\\beta$ filter assigns a constant to *g* and *h*, constrained to a certain range of values. Other filters such as the Kalman will vary *g* and *h* dynamically at each time step.\n",
+    "This filter is the basis for a huge number of filters, including the Kalman filter. In other words, the Kalman filter is a form of the g-h filter, which I will prove later in the book. So is the Least Squares filter, which you may have heard of, and so is the Benedict-Bordner filter, which you probably have not. Each filter has a different way of assigning values to $g$ and $h$, but otherwise the algorithms are identical. For example, the $\\alpha$-$\\beta$ filter assigns a constant to $g$ and $h$, constrained to a certain range of values. Other filters such as the Kalman will vary $g$ and $h$ dynamically at each time step.\n",
    "\n",
-    "**Let me repeat the key points as they are so important**. If you do not understand these you will not understand the rest of the book. If you do understand them, then the rest of the book will unfold naturally for you as mathematically elaborations to various 'what if' questions we will ask about *g* and *h*. The math may look profoundly different, but the algorithm will be exactly the same.\n",
+    "**Let me repeat the key points as they are so important**. If you do not understand these you will not understand the rest of the book. If you do understand them, then the rest of the book will unfold naturally for you as mathematically elaborations to various 'what if' questions we will ask about $g$ and $h$. The math may look profoundly different, but the algorithm will be exactly the same.\n",
    "\n",
    "* Multiple data points are more accurate than one data point, so throw nothing away no matter how inaccurate it is.\n",
    "* Always choose a number part way between two data points to create a more accurate estimate.\n",
@@ -1010,7 +1010,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x72f17b8>"
+       "<matplotlib.figure.Figure at 0x82fbeb8>"
      ]
     },
     "metadata": {},
@@ -1031,7 +1031,7 @@
    "\n",
    "Now consider trying to track a child's balloon in a hurricane. We have no legitimate model that would allow us to predict the balloon's behavior except over very brief time scales (we know the balloon cannot go 10 miles in 1 second, for example). In this case we would design a filter that emphasized the measurements over the predictions.\n",
    "\n",
-    "Most of this book is devoted to expressing the concerns in the last three paragraphs mathematically, which then allows us to find an optimal solution. In this chapter we will merely be assigning different values to *g* and *h* in a more intuitive, and thus less optimal way. But the fundamental idea is to blend somewhat inaccurate measurements with somewhat inaccurate models of how the systems behaves to get a filtered estimate that is better than either information source by itself.\n",
+    "Most of this book is devoted to expressing the concerns in the last three paragraphs mathematically, which then allows us to find an optimal solution. In this chapter we will merely be assigning different values to $g$ and $h$ in a more intuitive, and thus less optimal way. But the fundamental idea is to blend somewhat inaccurate measurements with somewhat inaccurate models of how the systems behaves to get a filtered estimate that is better than either information source by itself.\n",
    "\n",
    "We can express this as an algorithm:\n",
    "\n",
@@ -1417,7 +1417,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7b1e828>"
+       "<matplotlib.figure.Figure at 0x81cac88>"
      ]
     },
     "metadata": {},
@@ -1460,11 +1460,11 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "The g-h filter is not one filter - it is a classification for a family of filters. Eli Brookner in *Tracking and Kalman Filtering Made Easy* lists 11, and I am sure there are more. Not only that, but each type of filter has numerous subtypes. Each filter is differentiated by how *g* and *h* are chosen. So there is no 'one size fits all' advice that I can give here. Some filters set *g* and *h* as constants, others vary them dynamically. The Kalman filter varies them dynamically at each step. Some filters allow *g* and *h* to take any value within a range, others constrain one to be dependent on the other by some function $f(\\dot{}), \\mbox{where }g = f(h)$.\n",
+    "The g-h filter is not one filter - it is a classification for a family of filters. Eli Brookner in *Tracking and Kalman Filtering Made Easy* lists 11, and I am sure there are more. Not only that, but each type of filter has numerous subtypes. Each filter is differentiated by how $g$ and $h$ are chosen. So there is no 'one size fits all' advice that I can give here. Some filters set $g$ and $h$ as constants, others vary them dynamically. The Kalman filter varies them dynamically at each step. Some filters allow $g$ and $h$ to take any value within a range, others constrain one to be dependent on the other by some function $f(\\dot{}), \\mbox{where }g = f(h)$.\n",
    "\n",
-    "The topic of this book is not the entire family of g-h filters; more importantly, we are interested in the *Bayesian* aspect of these filters, which I have not addressed yet. Therefore I will not cover selection of *g* and *h* in depth. Eli Brookner's book *Tracking and Kalman Filtering Made Easy* is an excellent resource for that topic, if it interests you. If this strikes you as an odd position for me to take, recognize that the typical formulation of the Kalman filter does not use *g* and *h* at all; the Kalman filter is a g-h filter because it mathematically reduces to this algorithm. When we design the Kalman filter we will be making a number of carefully considered choices to optimize it's performance, and those choices indirectly affect *g* and *h*, but you will not be choosing *g* and *h* directly. Don't worry if this is not too clear right now, it will be much clearer later after we develop the Kalman filter theory.\n",
+    "The topic of this book is not the entire family of g-h filters; more importantly, we are interested in the *Bayesian* aspect of these filters, which I have not addressed yet. Therefore I will not cover selection of $g$ and $h$ in depth. Eli Brookner's book *Tracking and Kalman Filtering Made Easy* is an excellent resource for that topic, if it interests you. If this strikes you as an odd position for me to take, recognize that the typical formulation of the Kalman filter does not use $g$ and $h$ at all; the Kalman filter is a g-h filter because it mathematically reduces to this algorithm. When we design the Kalman filter we will be making a number of carefully considered choices to optimize it's performance, and those choices indirectly affect $g$ and $h$, but you will not be choosing $g$ and $h$ directly. Don't worry if this is not too clear right now, it will be much clearer later after we develop the Kalman filter theory.\n",
    "\n",
-    "However, it is worth seeing how varying *g* and *h* affects the results, so we will work through some examples. This will give us strong insight into the fundamental strengths and limitations of this type of filter, and help us understand the behavior of the rather more sophisticated Kalman filter."
+    "However, it is worth seeing how varying $g$ and $h$ affects the results, so we will work through some examples. This will give us strong insight into the fundamental strengths and limitations of this type of filter, and help us understand the behavior of the rather more sophisticated Kalman filter."
   ]
  },
  {
@@ -1508,9 +1508,9 @@
   "outputs": [
    {
     "data": {
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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5EAAAEPCAYAAADbIUClAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c7e080>"
+       "<matplotlib.figure.Figure at 0x806f390>"
      ]
     },
     "metadata": {},
@@ -1568,9 +1568,9 @@
   "outputs": [
    {
     "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5kAAAEPCAYAAADI9gBRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5kAAAEPCAYAAADI9gBRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c4d668>"
+       "<matplotlib.figure.Figure at 0x8230550>"
      ]
     },
     "metadata": {},
@@ -1631,9 +1631,9 @@
   "outputs": [
    {
     "data": {
-      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA6UAAAEPCAYAAAC6OWLnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
+      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9109978>"
+       "<matplotlib.figure.Figure at 0x9739a58>"
      ]
     },
     "metadata": {},
@@ -1698,7 +1698,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x912b5c0>"
+       "<matplotlib.figure.Figure at 0x974bac8>"
      ]
     },
     "metadata": {},
@@ -1776,7 +1776,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x919e438>"
+       "<matplotlib.figure.Figure at 0x97c4358>"
      ]
     },
     "metadata": {},
@@ -1817,7 +1817,7 @@
     "data": {
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x91f6b00>"
+       "<matplotlib.figure.Figure at 0x9dd1400>"
      ]
     },
     "metadata": {},
@@ -1862,7 +1862,7 @@
     "data": {
      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ac76a0>"
+       "<matplotlib.figure.Figure at 0x95ce2b0>"
      ]
     },
     "metadata": {},
@@ -1909,7 +1909,7 @@
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA5gAAAFLCAYAAACz/SMVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAIABJREFUeJzs3XdclfX7x/HXfQAZDsDEvTLcWporcy9AKXNmmlvTrExz\n21ApR2rZMDWxwizt5ygtS0Uzv5o5QLP0m5qmOb6uXOBiKOf+/XHkJIKCevAc4P18PHicm/t87vu+\n7sOdcXF9hmGapomIiIiIiIjIPbI4OwARERERERHJHpRgioiIiIiIiEMowRQRERERERGHUIIpIiIi\nIiIiDqEEU0RERERERBzC/X5fMDY29n5fUkRERETEYXx9fZ0dgojLUgVTREREREREHEIJpoiIiIiI\niDjEfe8ie6Os3r3g3AWTvpNg2Qbb9/3bQINSqxjwWiSfTQmhfdsQAEzT5NhpiN4DUbttr9F74OIV\n23scGgKlp2EYBqZpUif3EDavfg/DMJx4d65j27ZtANSsWdPJkUhWoOdF7pSeGblTemZyJg3zEskY\npyaYWdkvO026jIOjp8A3D4SPhI5NDUwzhLU/RtKuTbC9rWEYFC8IxQtC20a2fVaryZ9HIGq3wVeL\ng1kTvRrTNxhiIqnSIETJpYiIiIiIZDnqInuHkpJMJnxu0vglW3JZpxL8GmFLLsGWTH4yO/3qo8Vi\nULG0QY9WBis/C6FWkVW2amZMJJ/+HMSU+eb9uB0RERERERGHUYJ5B06cMQl+Bd4Ih6QkGNEVNsyC\nB4umTCbvtPpoGAbDBgST7/QQ+vcMwWIxGDUTxswxbUmniIiIiIhIFqAEM4NWbTGp1gN+2g4F/WHV\nNHh7gIGHu2O6srZvG0KHJjBrQjCfvwFubjB+LgydjpJMERERERHJEjQG8ya9+o3k4ElPDGyJo9WE\nv4+b/O9UAkapt2leE+aNgcIPOHaM5I1da7sGQ24vk2fGwPsL4XI8zBxq4uamcZkiIiIiIuK6lGDe\nJDSkMT0nGFzx+neSHjNuFRY/g/H9YWRX2/jJzHBj19q2jQy+nWzSbjTM+RauxEHEaybuDqqYioiI\niIiIOJq6yN6kfdsQquRfZe+WapomuS5HsmFBEKO7G5mWXKYl5DGDldMgjzfMXw2d3oCERHWXFRER\nERER16QE8yaGYfBS32CM2NUAuF+IJHxSCPUeds5H1ai6wZoPwC8vLN0AbUbBlXglmSIiIiIi4noy\nLWuaNGkSFouFgQMHZtYlMs3huGDM87YqZo0ikXTvHJz+QZmoTmWDddMhwA8it0KroXDhspJMERER\nERFxLZmSYG7ZsoU5c+bw8MMP3/GSHc52NtZk6gID/ILxOTmE4S+EuMQ9PFLWYP1MKFoANvwGQYPh\n3AUlmSIiIiIi4jocPslPbGwsXbt2JSIignHjxjn69Jnu7S/gwmUICgqm+JVI2rVxbvXyRhVKGWyY\nadJiMGxdOYqS6z15ONAg1/WfoolJmcIJRIRPdm6gIiIiIiKSIzk8wezXrx8dO3akUaNGWW79xv/9\nY/LR17btic8bPFr+PZeoXt6oTDGD9TNM6nRozPEzBlti/02AfeJX8XIv14pXRERERERyDod2kZ0z\nZw4HDx5k/PjxAC6XnKUn7DNISISnm0KNCobLxl+8oMGvy4LxiU85222V/K5VcRURERERkZzFYRXM\nP//8k9dee42NGzfi5uYG2JKe21Uxt23b5qjL37NDpzyJ+L4ybhZ4uu4fbNuW4OyQ0jWsd3nGfxWJ\n6ReC5cIq2rQty/bt250dVqZxpedFXJ+eF7lTembkTumZyVnKli3r7BBEsgSHVTA3b97MmTNnqFy5\nMh4eHnh4eLBhwwZmzpxJrly5uHr1qqMulSk+/qEYVtOg9WNnKFnQ9ZNLgCeCa1Iuz1JM08R6bjVn\nLaHODklERERERHIww3TQQMnY2FiOHTtm/940TXr16kW5cuV49dVXqVSpkr1dMl9fX0dc+p5F7TZ5\n7Dnw9oT9C6FogGt2jU3Lkm9W0n1oJHHeIeQuEszvn8NDxbNO/BmR/BfimjVrOjkSyQr0vMid0jMj\nd0rPTM7kir/Dirgih3WR9fX1TfUfm4+PD/7+/vbk0hWZpsnoWbbtlztmreQSoH3bEFauiuRK4SAW\nroU+k+Cn6SYWS9a6DxERERERyfoyZR3MZIbhuhPlJPsxGtb9Cn55YcSzzo7mzhmGwSez3+OjoQaF\n8tvWyJzxtbOjEhERERGRnChTE8x169bx4YcfZuYl7onVajL6Y9v2qG7gn8+1k+FbMQyDB3wNZg23\nfT/6Y/jrf1lriRgREREREcn6MjXBdHVL1sGvf0LRAvBSe2dHc+/aNDTo0gKuxEOfibYEWkRERERE\n5H7JsQnm1Wsmr4fbtsf0Bh+vrFm9vNkHr0Ch/PDz7zB9ibOjERERERGRnCTHJpiffQ9//Q/KlYDe\n2Wh1jwd8DT4eYdt+9WPYf1RVTBERERERuT9yZIJ5Jd7kzc9s22/1A3f37FG9TPZUA4NngyAuQV1l\nRURERETk/smRCeaHi+HEWahRHto3dnY0meODV6DwA7Bxp+1+RUREREREMluOSzDPXzCZMt+2PWkA\n2Xa9yPz5DGbf0FV23xFVMUVEREREJHPluATz7S8h5iI0qwnNa2XP5DLZk/UNuoVAfCL0nghJSUoy\nRUREREQk8+SoBPPYaZPp17uLTnzeubHcL+8PgiIPwKZd8IG6yoqIiIiISCbKUQnmmxG2al6HJlCr\nYvauXibzz2cwe6Rt+/XZ8OdhVTFFRERERCRz5JgE88/DJp99D25u8NZzzo7m/nqinkF3dZUVERER\nEZFMlmMSzDGfQFIS9AqF8qVyRvXyRu8NgqIFYPN/4f1Fzo5GRERERESyoxyRYG7bY7L4J/DKBWN6\nOTsa5/DPZxCe3FU2HPaqq6yIiIiIiDiYu7MDyCy9+o3k4ElPDAx+/wvMi/BAQZM3Xk8gInyys8Nz\nilaPG/RsZRIxaxS1gz2pXtbAuF7MNTEpUzjnfjYiIiIiInLvsm2CGRrSmJ4TDK54BYM/GP5wLm4V\nT7TMed1jbzTtZfh2eWPOXTD4+Vywfb9P/Cpe7pWzPxsREREREbk32baLbPu2IVTJvwrTtHUFNU2T\nqg9E0q5NcDpHZm9+eQ2+mBYMMSk/myr59dmIiIiIiMi9ybYJpmEYPNc9GGJWA+ATH8nwF0IwDFXp\nWj1uoXHTfz+b3An6bERERERE5N5l2wQTwPT9t1Kn6mVKi6YH43bB9tmU9NRnIyIiIiIi9y5bJ5jf\nbTTALxiv40NUobtJgL+Fju2C4dAQEnOHYGpSWRERERERuUfZNsG8dMVkTTQY/sG0aYgqdGmY83Yw\n3p5w4HIQS9Y5OxoREREREcnqsm2CuWorJCTC41UNFnz+nqqXacjjY+G996ZhGAavzYar11TGFBER\nERGRu5dtE8xvN9hen2qAksvb6POEQbkScOAYhH/r7GhERERERCQry5YJ5tVrJt9vsm23aejcWFyd\nu7vBxOdt229FwMXLqmKKiIiIiMjdyZYJ5vodEHsJKj8IZUuoepmeto3gscrwz3mYttDZ0YiIiIiI\nSFaVLRPMZcndY1W9zBDDMHj7Bdv2Owvg1DlVMUVERERE5M5luwTTajX59mfbdpsGzo0lK2lYzeCJ\nenA5ztZVVkRERERE5E5luwRz+59w7DQULwg1Kjg7mqxl4vNgsdgm+/nrf6piioiIiIjIncl2CeYy\nzR5716qUMejeEq4lwRvhzo5GRERERESymmyXYCYvT6LZY+9OWB/wygUL10L0HlUxRUREREQk47JV\ngrnviMnuQ+CXFxpWc3Y0WVOJQgYDO9q2R80E01SSKSIiIiIiGZOtEsxl1yf3eeJx8HBX99i7Naqr\nLUlf9yusjnJ2NCIiIiIiklVkqwTz2xvGX8rd889nMLqbbXvUTNvMvCIiIiIiIunJNgnmiTMmW/6w\njR8MeczZ0WR9AztAiULw+1+wYI2zoxERERERkawg2ySYy38B04QWtSC3t7rH3isvT4OwPrbtN8Ih\nIVFVTBERERERub1sk2DalyfR7LEO0y0EqpSBwydh5jfOjkZERERERFxdtkgwL1w2WbsNLBZ4sp6z\no8k+3NwMJg2wbU/4HGIvqYopIiIiIiK3li0SzJWb4eo1qFcVAvzVPdaRWtW1Lfly7gJMme/saERE\nRERExJVliwTz2+vLk7RR91iHMwyDt69XMd9fCMdOq4opIiIiIiJpy/IJZkKiyQ+bbNtaniRzPFbF\noH1jiEuAsM+cHY2IiIiIiLgqd2cHcK/W/QoXr8DDgVCmmLrHZpbx/eDreaOY85Env6018PGy7Tcx\nKVM4gYjwyc4NUEREREREnC7LJ5j22WNVvcxU5UsZNGvamB+jDbZdDobLtv0+8at4uZcSexERERER\nyeJdZK1Wk+822rbbavxlpvv8nWAssaswTds4TNM0qZI/knZtgp0cmYiIiIiIuIIsnWBG7YaTZ6FU\nYXikrLOjyf6KBlho+1QwxKwGIHdCJMNfCMEwVMEUEREREZEsnmAuuz577FMNUJJzn8x5Oxi3C7Yq\nZmkvVS9FRERERORfDkswZ8yYwSOPPIKvry++vr48/vjjrFixwlGnT8U0TZatt21reZL7xz+fhTat\ng+HQEDwKqHopIiIiIiL/cliCWaJECaZMmcKOHTvYvn07TZs2pU2bNvz++++OukQKew/DvqPwgC/U\nfzhTLiG38PHEYNzd4LfTQUTt1rqYIiIiIiJi47AEs3Xr1gQHB1OmTBkCAwMZP348efPmJSoqylGX\nSCF59tgn64G7u6po91MBPwtDRk/DMAzGz3V2NCIiIiIi4ioyZQxmUlIS//d//0d8fLine truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c8d7f0>"
+       "<matplotlib.figure.Figure at 0x81c0390>"
      ]
     },
     "metadata": {},
@@ -1943,7 +1943,7 @@
    "If you really want to test yourself, read the next paragraph and try to predict the results before you move the sliders. \n",
    "\n",
    "Some things to try include setting $g$  and $h$ to their miminum values. See how perfectly the filter tracks the data! This is only because we are perfectly predicting the weight gain. Adjust $\\dot{x}$ to larger or smaller than 5. The filter should diverge from the data and never reacquire it. Start adding back either $g$ or $h$ and see how the filter snaps back to the data. See what the difference in the line is when you add only $g$ vs only $h$. Can you explain the reason for the difference? Then try setting $g$ greater than 1. Can you explain the results? Put $g$ back to a reasonable value (such as 0.1), and then make $h$ very large. Can you explain these results? Finally, set both $g$ and $h$ to their largest values. \n",
-    "\n",
+    " \n",
    "If you want to explore with this more, change the value of the array `zs` to the values used in any of the charts above and rerun the cell to see the result."
   ]
  },
@@ -1956,19 +1956,9 @@
   "outputs": [
    {
     "data": {
+      "image/png": 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truncated
      "text/plain": [
-       "<function __main__.interactive_gh>"
-      ]
-     },
-     "execution_count": 42,
-     "metadata": {},
-     "output_type": "execute_result"
-    },
-    {
-     "data": {
-      "image/png": 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truncated
-      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7312668>"
+       "<matplotlib.figure.Figure at 0x80c9d30>"
      ]
     },
     "metadata": {},
@@ -1981,14 +1971,89 @@
    "\n",
    "zs = gen_data(x0=5, dx=5, count=100, noise_factor=50)\n",
    "\n",
-    "def interactive_gh(dx, g, h):\n",
-    "    data = g_h_filter(data=zs, x0=0., dx=dx, dt=1.,g=g, h=h)\n",
-    "    plot_g_h_results(zs, data)\n",
+    "def interactive_gh(x, dx, g, h):\n",
+    "    data = g_h_filter(data=zs, x0=x, dx=dx, dt=1., g=g, h=h)\n",
+    "    plt.plot(zs, color='r')\n",
+    "    plt.plot(data, color='k')\n",
+    "    plt.show()\n",
    "\n",
-    "interact (interactive_gh, \n",
-    "          dx=widgets.FloatSliderWidget(value=5., min=1., max=10.), \n",
-    "          g=widgets.FloatSliderWidget(value=0.5, min=0.01, max=2), \n",
-    "          h=widgets.FloatSliderWidget(value=0.02, min=0.0, max=0.5, step=0.01))"
+    "interact (interactive_gh,           \n",
+    "          x=widgets.FloatSliderWidget(value=0., min=-50, max=50.), \n",
+    "          dx=widgets.FloatSliderWidget(value=5., min=-50., max=50.), \n",
+    "          g=widgets.FloatSliderWidget(value=0.1, min=0.01, max=2, step=.02), \n",
+    "          h=widgets.FloatSliderWidget(value=0.02, min=0.0, max=0.5, step=0.01));"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Don't Lie to the Filter\n",
+    "\n",
+    "You are free to set $g$ and $h$ to any value. Here is a filter that performs perfectly despite extreme noise."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 43,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x97c6588>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "zs = gen_data(x0=5, dx=.2, count=100, noise_factor=100)\n",
+    "data = g_h_filter(data=zs, x0=5, dx=.2, dt=1., g=0, h=0)\n",
+    "with book_format.figsize(y=2.):\n",
+    "    book_plots.plot_measurements(zs)\n",
+    "    book_plots.plot_filter(data, label='filter', lw=2)\n",
+    "    book_plots.show_legend()"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "I brilliantly extracted a straight line out of very noisy data! Maybe I shouldn't try to collect my Fields Medal in mathematics just yet. I did this by setting both $g$ and $h$ to 0. What does this do? It makes the filter ignore the measurements, and so for each update it computes the new position as $x + \\Delta x \\Delta t$. Of course the result is a straight line if we ignore the measurements. But a filter that ignores the measurements is useless.\n",
+    "\n",
+    "You would never set $g$ and $h$ to zero as that takes a special kind of brilliance that only I possess, but I promise that if you are not careful you will set them lower than they should be. You can always make great looking results from test data. When you try your filter on different data you will be disappointed in the results because you finely tuned the constants for a specific data set. $g$ and $h$ must reflect the real world behavior of the system you are filtering, not the behavior of one specific data set. In later chapters we will learn a lot about how to do that. For now I can only say be careful, or you will be getting perfect results with your test data, but results like this once you switch to real data:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 44,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x831d1d0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "zs = gen_data(x0=5, dx=-2, count=100, noise_factor=5)\n",
+    "data = g_h_filter(data=zs, x0=5, dx=2., dt=1., g=.005, h=0.001)\n",
+    "with book_format.figsize(y=2.):\n",
+    "    book_plots.plot_measurements(zs)\n",
+    "    book_plots.plot_filter(data, label='filter', lw=2)\n",
+    "    book_plots.show_legend()"
   ]
  },
  {
@@ -2002,7 +2067,7 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "Now let's try a practical example. Earlier in the chapter we talked about tracking a train. Trains are heavy and slow, thus they cannot change speed quickly. They are on a track, so they cannot change direction except by slowing to a stop and then reversing course. Hence, we can conclude that if we already know the train's approximate position and velocity then we can predict its position in the near future with a great deal of accuracy. A train cannot change its velocity much in a second or two. \n",
+    "We are ready for a practical example. Earlier in the chapter we talked about tracking a train. Trains are heavy and slow, thus they cannot change speed quickly. They are on a track, so they cannot change direction except by slowing to a stop and then reversing course. Hence, we can conclude that if we already know the train's approximate position and velocity then we can predict its position in the near future with a great deal of accuracy. A train cannot change its velocity much in a second or two. \n",
    "\n",
    "So let's write a filter for a train. Its position is expressed as its position on the track in relation to some fixed point which we say is 0 km. I.e., a position of 1 means that the train is 1 km away from the fixed point. Velocity is expressed as meters per second. We perform measurement of position once per second, and the error is $\\pm$ 500 meters. How should we implement our filter?\n",
    "\n",
@@ -2026,16 +2091,16 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAwMAAAE6CAYAAABKyI5vAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7ac1cc0>"
+       "<matplotlib.figure.Figure at 0x8100f98>"
      ]
     },
     "metadata": {},
@@ -2072,21 +2137,21 @@
   "source": [
    "We can see from the chart how poor the measurements are. No real train could ever move like that. \n",
    "\n",
-    "So what should we set *g* and *h* to if we want to filter this data? We have not developed the theory for this, but let's try to get a reasonable answer by the seat of our pants. We know that the measurements are very inaccurate, so we don't want to give them much weight at all. To do this we need to choose a very small *g*. We also know that trains can not accelerate or decelerate quickly, so we also want a very small *h*. For example:"
+    "So what should we set $g$ and $h$ to if we want to filter this data? We have not developed the theory for this, but let's try to get a reasonable answer by the seat of our pants. We know that the measurements are very inaccurate, so we don't want to give them much weight at all. To do this we need to choose a very small $g$. We also know that trains can not accelerate or decelerate quickly, so we also want a very small $h$. For example:"
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x91a7128>"
+       "<matplotlib.figure.Figure at 0x97c84e0>"
      ]
     },
     "metadata": {},
@@ -2103,21 +2168,21 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "That is pretty good for an initial guess. Lets make *g* larger to see the effect."
+    "That is pretty good for an initial guess. Lets make $g$ larger to see the effect."
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 45,
+   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": 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+      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7f6a1d0>"
+       "<matplotlib.figure.Figure at 0x806b780>"
      ]
     },
     "metadata": {},
@@ -2136,21 +2201,21 @@
   "source": [
    "We made `g=0.2` and we can see that while the train's position is smoothed, the estimated position (and hence velocity) fluctuates a lot in a very tiny frame, far more than a real train can do. So empirically we know that we want `g<<0.2`.\n",
    "\n",
-    "Now let's see the effect of a poor choice for *h*."
+    "Now let's see the effect of a poor choice for $h$."
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 46,
+   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7c83898>"
+       "<matplotlib.figure.Figure at 0x97660f0>"
      ]
     },
     "metadata": {},
@@ -2167,14 +2232,14 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "Here the position changes smoothly thanks to the small *g*, but the large *h* makes the filter very reactive to the measurements. This happens because in the course of a few seconds the rapidly changing measurement implies a very large velocity change, and a large *h* tells the filter to react to those changes quickly. Trains cannot change velocity quickly, so the filter is not doing a good job of filtering the data - the filter is changing velocity faster than a train can.\n",
+    "Here the position changes smoothly thanks to the small $g$, but the large $h$ makes the filter very reactive to the measurements. This happens because in the course of a few seconds the rapidly changing measurement implies a very large velocity change, and a large $h$ tells the filter to react to those changes quickly. Trains cannot change velocity quickly, so the filter is not doing a good job of filtering the data - the filter is changing velocity faster than a train can.\n",
    "\n",
    "Finally, let's add some acceleration to the train. I don't know how fast a train can actually accelerate, but let's say it is accelerating at 0.2 m/sec^2. "
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 47,
+   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
@@ -2191,16 +2256,16 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x7a5d470>"
+       "<matplotlib.figure.Figure at 0x976e128>"
      ]
     },
     "metadata": {},
@@ -2217,21 +2282,21 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "Here we see that the filter is not quite tracking the train anymore due to the acceleration. We can fiddle with *h* to let it track better, at the expense of a less smooth filtered estimate."
+    "Here we see that the filter is not quite tracking the train anymore due to the acceleration. We can fiddle with $h$ to let it track better, at the expense of a less smooth filtered estimate."
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
-      "image/png": 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truncated
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truncated
      "text/plain": [
-       "<matplotlib.figure.Figure at 0x9152978>"
+       "<matplotlib.figure.Figure at 0x80f7d68>"
      ]
     },
     "metadata": {},
@@ -2248,9 +2313,9 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "There are two lessons to be learned here. First, use the *h* term to respond to changes in velocity that you are not modeling. But, far more importantly, there is a trade off here between responding quickly and accurately to changes in behavior and producing ideal output for when the system is in a steady state that you have. If the train never changes velocity we would make *h* extremely small to avoid having the filtered estimate unduly affected by the noise in the measurement. But in an interesting problem there are almost always changes in state, and we want to react to them quickly. The more quickly we react to them, the more we are affected by the noise in the sensors. \n",
+    "There are two lessons to be learned here. First, use the $h$ term to respond to changes in velocity that you are not modeling. But, far more importantly, there is a trade off here between responding quickly and accurately to changes in behavior and producing ideal output for when the system is in a steady state that you have. If the train never changes velocity we would make $h$ extremely small to avoid having the filtered estimate unduly affected by the noise in the measurement. But in an interesting problem there are almost always changes in state, and we want to react to them quickly. The more quickly we react to them, the more we are affected by the noise in the sensors. \n",
    "\n",
-    "I could go on, but my aim is not to develop g-h filter theory here so much as to build insight into how combining measurements and predictions leads to a filtered solution, so I will stop here. Do understand that there is extensive literature on choosing *g* and *h* for problems such as this, and that there are optimal ways of choosing them to achieve various goals. In the subsequent chapters we will learn how the Kalman filter solves this problem in the same basic manner, but with far more sophisticated mathematics. "
+    "I could go on, but my aim is not to develop g-h filter theory here so much as to build insight into how combining measurements and predictions leads to a filtered solution. Tthere is extensive literature on choosing $g$ and $h$ for problems such as this, and there are optimal ways of choosing them to achieve various goals. As I explained earlier it is easy to 'lie' to the filter when experimenting with test data like this. In the subsequent chapters we will learn how the Kalman filter solves this problem in the same basic manner, but with far more sophisticated mathematics. "
   ]
  },
  {
@@ -2264,31 +2329,187 @@
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "## Final Thoughts"
+    "FilterPy is an open source filtering library that I wrote. It has all of the filters in this book, along with others. It is rather easy to program your own g-h filter, but as we progress we will rely on FilterPy more. As a quick introduction, let's look at the g-h filter in FilterPy.\n",
+    "\n",
+    "If you do not have FilterPy installed just issue the following command from the command line.\n",
+    "\n",
+    "    pip install filterpy\n",
+    "    \n",
+    "Read Appendix A for more information on installing or downloding FilterPy from GitHub.\n",
+    "\n",
+    "To use the g-h filter import it and create an object from the class `GHFilter`. "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 52,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "from filterpy.gh import GHFilter\n",
+    "f = GHFilter (x=0., dx=0., dt=1., g=.8, h=.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
-    "*author's note*: the following few paragraphs really belong in the not yet written least squares chapter.\n",
+    "When you construct the object you specify the initial value and rate of change for the signal (`x` and 'dx'), the time step between updates(`dt`) and the two filter parameter (`g` and `h`). `dx` must have the same units of `x`/`dt` - if `x` is in meters and `dt` is in seconds then `dx` must be in meters per second.\n",
    "\n",
-    "Near the beginning of the chapter I used `numpy.polyfit()` to fit a straight line to the weight measurements. It fits a n-th degree polynomial to the data using a 'least squared fit'. How does this differ from the g-h filter?\n",
+    "To run the filter call update, passing the measurement in the parameter `z`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(0.96, 0.24)"
+      ]
+     },
+     "execution_count": 53,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "f.update(z=1.2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "`update()` returns the new value of `x` and `dx` in a tuple, but you can also access them from the object."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 54,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0.96 0.24\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(f.x, f.dx)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can dynamically alter `g` and `h`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 55,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(1.965, 0.375)"
+      ]
+     },
+     "execution_count": 55,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "f.update(z=2.1, g=.85, h=.15)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can filter a sequence of measurements in a batch."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 56,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 1.965 ,  0.375 ],\n",
+       "       [ 2.868 ,  0.507 ],\n",
+       "       [ 3.875 ,  0.632 ],\n",
+       "       [ 4.9014,  0.7306]])"
+      ]
+     },
+     "execution_count": 56,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "f.batch_filter([3., 4., 5.])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "You can filter multiple independent variables. If you are tracking an aircraft you'll need to track it in 3D space. Use NumPy arrays for `x`, `dx`, and the measurements."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(array([   3.8 ,   13.2 ,  101.64]), array([ 8.2 ,  9.8 ,  0.56]))"
+      ]
+     },
+     "execution_count": 57,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "x_0  = np.array([1., 10., 100.])\n",
+    "dx_0 = np.array([10., 12., .2])\n",
+    "               \n",
+    "f_air = GHFilter (x=x_0, dx=dx_0, dt=1., g=.8, h=.2)\n",
+    "f_air.update(z=np.array((2., 11., 102.)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The class `GHFilterOrder` allows you to create a filter of order 0, 1, or 2. A g-h filter is order 1. The g-h-k filter, which we haven't talked about, also tracks accelerations. Both classes have functionality required by real applications, such as computing the Variance Reduction Factor (VRF), which we haven't discussed in this chapter. I could fill a book just on the theory and applications of g-h filters, but we have other goals in this book. If you are interested, explore the FilterPy code and do some further reading.\n",
    "\n",
-    "Well, it depends. We will eventually learn that the Kalman filter is optimal from a least squared fit perspective under certain conditions. However, `polyfit()` fits a polynomial to the data, not an arbitrary curve, by minimizing the value of this formula:\n",
-    "\n",
-    "$$E = \\sum_{j=0}^k |p(x_j) - y_j|^2$$\n",
-    "\n",
-    "I assumed that my weight gain was constant at 1 lb/day, and so when I tried to fit a polynomial of $n=1$, which is a line, the result very closely matched the actual weight gain. But, of course, no one consistently only gains or loses weight. We fluctuate. Using 'polyfit()' for a longer series of data would yield poor results. In contrast, the g-h filter reacts to changes in the rate - the $h$ term controls how quickly  the filter reacts to these changes. If we gain weight, hold steady for awhile, then lose weight, the filter will track that change automatically. 'polyfit()' would not be able to do that unless the gain and loss could be well represented by a polynomial.\n",
-    "\n",
-    "Another advantage of this form of filter, even if the data fits a *n*-degree polynomial, is that it is *recursive*. That is, we can compute the estimate for this time period knowing nothing more than the estimate and rate from the last time period. In contrast, if you dig into the implementation for `polyfit()` you will see that it needs all of the data before it can produce an answer. Therefore algorithms like `polyfit()` are not well suited for real-time data filtering. In the 60's when the Kalman filter was developed computers were very slow and had extremely limited memory. They were utterly unable to store, for example, thousands of readings from an aircraft's inertial navigation system, nor could they process all of that data in the short period of time needed to provide accurate and up-to-date navigation information. \n",
-    "\n",
-    "\n",
-    "Up until the mid 20th century various forms of Least Squares Estimation was used for this type of filtering. For example, for NASA's Apollo program had a ground network for tracking the Command and Service Model (CSM) and the Lunar Module (LM). They took measurements over many minutes, batched the data together, and slowly computed an answer. In 1960 Stanley Schmidt at NASA Ames recognized the utility of Rudolf Kalman's seminal paper and invited him to Ames. Schmidt applied Kalman's work to the on board navigation systems on the CSM and LM, and called it the \"Kalman filter\".[1] Soon after, the world moved to this faster, recursive filter.\n",
-    "\n",
-    "The Kalman filter only needs to store the last estimate and a few related parameters, and requires only a relatively small number of computations to generate the next estimate. Today we have so much memory and processing power that this advantage is somewhat less important, but at the time the Kalman filter was a major breakthrough not just because of the mathematical properties, but because it could (barely) run on the hardware of the day. \n",
-    "\n",
-    "This subject is much deeper than this short discussion suggests. We will consider these topics many more times throughout the book."
+    "The documentation for FilterPy is at https://filterpy.readthedocs.org/."
   ]
  },
  {