Completed IMM description

The text for the IMM filter was incomplete, and wrong in a few
places.

14-Adaptive-Filtering.ipynb

@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 40,
+   "execution_count": 1,
    "metadata": {
     "collapsed": false
    },
@@ -289,7 +289,7 @@
        "<IPython.core.display.HTML object>"
       ]
      },
-     "execution_count": 40,
+     "execution_count": 1,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -336,7 +336,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 41,
+   "execution_count": 2,
    "metadata": {
     "collapsed": false
    },
@@ -406,7 +406,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 42,
+   "execution_count": 3,
    "metadata": {
     "collapsed": false
    },
@@ -435,16 +435,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 43,
+   "execution_count": 4,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvcAAAD+CAYAAAC+7ypaAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684e05a9e8>"
+       "<matplotlib.figure.Figure at 0x203116f7f60>"
       ]
      },
      "metadata": {},
@@ -500,7 +500,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 44,
+   "execution_count": 5,
    "metadata": {
     "collapsed": false
    },
@@ -541,16 +541,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 45,
+   "execution_count": 6,
    "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 0x2684e143390>"
+       "<matplotlib.figure.Figure at 0x203116d7e80>"
       ]
      },
      "metadata": {},
@@ -586,16 +586,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 46,
+   "execution_count": 7,
    "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 0x2684de02c88>"
+       "<matplotlib.figure.Figure at 0x203128270b8>"
       ]
      },
      "metadata": {},
@@ -629,16 +629,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 47,
+   "execution_count": 8,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684e1316a0>"
+       "<matplotlib.figure.Figure at 0x203116d7e48>"
       ]
      },
      "metadata": {},
@@ -668,16 +668,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 9,
    "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 0x2684def1550>"
+       "<matplotlib.figure.Figure at 0x203128aa278>"
       ]
      },
      "metadata": {},
@@ -708,7 +708,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 49,
+   "execution_count": 10,
    "metadata": {
     "collapsed": false
    },
@@ -733,16 +733,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 50,
+   "execution_count": 11,
    "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 0x2684e00b0f0>"
+       "<matplotlib.figure.Figure at 0x2031283a5c0>"
       ]
      },
      "metadata": {},
@@ -792,16 +792,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 51,
+   "execution_count": 12,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684de2cbe0>"
+       "<matplotlib.figure.Figure at 0x203128b31d0>"
       ]
      },
      "metadata": {},
@@ -822,16 +822,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 52,
+   "execution_count": 13,
    "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 0x2684cd204e0>"
+       "<matplotlib.figure.Figure at 0x20313ac0828>"
       ]
      },
      "metadata": {},
@@ -921,7 +921,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 53,
+   "execution_count": 14,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -929,9 +929,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684c5a1438>"
+       "<matplotlib.figure.Figure at 0x20312a2f588>"
       ]
      },
      "metadata": {},
@@ -972,16 +972,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 54,
+   "execution_count": 15,
    "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 0x2684c5de9b0>"
+       "<matplotlib.figure.Figure at 0x20312a34748>"
       ]
      },
      "metadata": {},
@@ -1056,7 +1056,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 55,
+   "execution_count": 16,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -1064,9 +1064,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684de24940>"
+       "<matplotlib.figure.Figure at 0x20311fbeba8>"
       ]
      },
      "metadata": {},
@@ -1155,16 +1155,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 56,
+   "execution_count": 17,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684deb1128>"
+       "<matplotlib.figure.Figure at 0x2031297e048>"
       ]
      },
      "metadata": {},
@@ -1184,16 +1184,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 57,
+   "execution_count": 18,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684cd999b0>"
+       "<matplotlib.figure.Figure at 0x2031295d470>"
       ]
      },
      "metadata": {},
@@ -1213,16 +1213,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 58,
+   "execution_count": 19,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684cd99cc0>"
+       "<matplotlib.figure.Figure at 0x20312a1e5c0>"
       ]
      },
      "metadata": {},
@@ -1230,9 +1230,9 @@
     },
     {
      "data": {
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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684e137e10>"
+       "<matplotlib.figure.Figure at 0x20312a1e320>"
       ]
      },
      "metadata": {},
@@ -1255,16 +1255,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 59,
+   "execution_count": 20,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684ded7e48>"
+       "<matplotlib.figure.Figure at 0x20313c81710>"
       ]
      },
      "metadata": {},
@@ -1272,9 +1272,9 @@
     },
     {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684ccf6cc0>"
+       "<matplotlib.figure.Figure at 0x203116d7630>"
       ]
      },
      "metadata": {},
@@ -1282,9 +1282,9 @@
     },
     {
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684df0c470>"
+       "<matplotlib.figure.Figure at 0x20313c4ce80>"
       ]
      },
      "metadata": {},
@@ -1292,9 +1292,9 @@
     },
     {
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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684ccfd080>"
+       "<matplotlib.figure.Figure at 0x20313abaef0>"
       ]
      },
      "metadata": {},
@@ -1302,9 +1302,9 @@
     },
     {
      "data": {
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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684cdeb080>"
+       "<matplotlib.figure.Figure at 0x20313b3ef98>"
       ]
      },
      "metadata": {},
@@ -1368,16 +1368,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 60,
+   "execution_count": 21,
    "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 0x2684e26af98>"
+       "<matplotlib.figure.Figure at 0x20313aefb38>"
       ]
      },
      "metadata": {},
@@ -1496,7 +1496,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 61,
+   "execution_count": 22,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -1504,9 +1504,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684bc67eb8>"
+       "<matplotlib.figure.Figure at 0x20311fc2780>"
       ]
      },
      "metadata": {},
@@ -1528,16 +1528,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 62,
+   "execution_count": 23,
    "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 0x2684cd91438>"
+       "<matplotlib.figure.Figure at 0x20313d2feb8>"
       ]
      },
      "metadata": {},
@@ -1584,7 +1584,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 63,
+   "execution_count": 24,
    "metadata": {
     "collapsed": false,
     "scrolled": true
@@ -1592,9 +1592,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684e20d1d0>"
+       "<matplotlib.figure.Figure at 0x20313d39f28>"
       ]
      },
      "metadata": {},
@@ -1624,7 +1624,7 @@
     "\n",
     "$$\\mathcal{L} = \\frac{1}{\\sqrt{2\\pi S}}\\exp [-\\frac{1}{2}\\mathbf{y}^\\mathsf{T}\\mathbf{S}^{-1}\\mathbf{y}]$$\n",
     "\n",
-    "which tells us how likely a filter is to be performing optimally given the inputs. We can use this to compute the probability that each filter is the best fit to the data. If we have N filters, we can compute the probability that filter i is correct in relation to the rest of the filters with\n",
+    "which tells us how likely a filter is to be performing optimally given the inputs. $\\mathbf y$ is the residual and $\\mathbf S$ is the system uncertainty (covariance in measurement space). This is just a Gaussian of the residual and the system uncertainty. A large residual will give a large uncertainty, and thus low likelihood that the measurement matches the filter's current state. We can use this to compute the probability that each filter is the best fit to the data. If we have N filters, we can compute the probability that filter i is correct in relation to the rest of the filters with\n",
     "\n",
     "$$p_k^i = \\frac{\\mathcal{L}_k^ip_{k-1}^i}{\\sum\\limits_{j=1}^N \\mathcal{L}_k^jp_{k-1}^j}$$\n",
     "\n",
@@ -1637,16 +1637,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 64,
+   "execution_count": 25,
    "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 0x2684cca2198>"
+       "<matplotlib.figure.Figure at 0x203117f9518>"
       ]
      },
      "metadata": {},
@@ -1716,16 +1716,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 65,
+   "execution_count": 26,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684e243a58>"
+       "<matplotlib.figure.Figure at 0x20312960da0>"
       ]
      },
      "metadata": {},
@@ -1761,22 +1761,67 @@
     "\n",
     "This naive approach leads to combinatorial explosion. At step 1 we generate $N$ hypotheses, or 1 per filter. At step 2 we generate another $N$ hypotheses which then need to be combined with the prior $N$ hypotheses, which yields $N^2$ hypothesis. Many different schemes have been tried which either cull unlikely hypotheses or merge similar ones, but the algorithms still suffered from computational expense and/or poor performance. I will not cover these in this book, but prominent examples in the literature are the generalized pseudo Bayes (GPB) algorithms.\n",
     "\n",
-    "What if we take a Bayesian approach to the problem? It's worked so far. Lets assume the target can be in one of $N$ modes. For example, we might have two modes: going straight, or turning. Each mode $i$ can be represented by a model $M_i$.  We can implement the model with a Kalman filter. For example, the model for going straight could be a constant velocity model.\n",
+    "The *Interacting Multiple Models* (IMM) algorithm was invented by Blom[5] to solve the combinatorial explosion problem of multiple models. A subsequent paper by Blom and Bar-Shalom is the most cited paper [6]. The idea is to have 1 filter for each possible mode of behavior of the system. At each epoch we we let the filters *interact* with each other. The more likely filters modify the estimates of the less likely filters so they more nearly represent the current state of the sytem. This blending is done probabilistically, so the unlikely filters also modify the likely filters, but by a much smaller amount. \n",
     "\n",
+    "For example, suppose we have two modes: going straight, or turning. Each mode is represented by a Kalman filter, maybe a first order and second order filter. Now say the target it turning. The second order filter will produce a good estimate, and the first order filter will lag the signal. The likelihood function of each tells us which of the filters is most probable. The first order filter will have low likelihood, so we adjust its estimate greatly with the second order filter. The the second order filter is very likely, so its estimate will only be changed slightly by the first order Kalman filter. \n",
+    "\n",
+    "Now suppose the target stops turning. Because we have been revising the first order filter's estimate with the second order estimate it will not have been lagging the signal by very much. within just a few epochs it will be producing very good (high likelihood) estimates and be the most probable filter. It will then start contributing heavily to the estimate of the second order filter. Recall that a second order filter mistakes measurement noise for acceleration. This adjustment insures reduces this effect greatly."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "### Mode Probabilities\n",
     "\n",
-    "We can assign a probability that the target is in any given mode. This gives us a vector of *mode probabilities*: a list containing the probability that the system is in any given mode. In the example above we have two modes straight and turn, so we will have a vector of two elements. If we think that there is a 70% chance that the target is going straight we can say\n",
+    "We define a set of modes for our system, $m$, and assume that the target is always in one of these modes. In the discussion above we have the modes straight and turn, so $m=\\{\\text{straight},\\ \\text{turn}\\}$.\n",
+    "\n",
+    "\n",
+    "We assign a probability that the target is in any given mode. This gives us a vector of *mode probabilities* with one probability for each possible mode. $m$ has two modes, so we will have a vector of two probabilities. If we think that there is a 70% chance that the target is going straight we can say\n",
     "\n",
     "$$\\mu = \\begin{bmatrix} 0.7 & 0.3\\end{bmatrix}$$\n",
     "\n",
-    "0.3 for the turn is required because we are assuming that the target is in one of the modes in $M$. $\\mu$ is typically but not universally used as the symbol for the mode probabilities, so I will use it. Do not confuse it with the mean.\n",
+    "We get 0.3 for the turn because the probabilities must sum to one. $\\mu$ is typically but not universally used as the symbol for the mode probabilities, so I will use it. Do not confuse it with the mean.\n",
     "\n",
-    "We can formalize it by saying that the prior probability that $M_i$ is correct (the maneuvering object is in mode $i$), given the prior measurements $Z$, is\n",
-    "\n",
-    "$$\\mu_i = P(M_i|Z)$$\n",
-    "\n",
-    "We will learn how to calculate this soon.\n",
+    "In Python we can implement this as"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([ 0.7,  0.3])"
+      ]
+     },
+     "execution_count": 27,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "mu = np.array([0.7, 0.3])\n",
+    "mu"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can formalize it by saying that the prior probability that $m_i$ is correct (the maneuvering object is in mode $i$), given the prior measurements $Z$, is\n",
     "\n",
+    "$$\\mu_i = P(m_i|Z)$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "### Mode Transitions\n",
     "\n",
     "Next we have to consider that this is a maneuvering target. It will go straight, then turn, then go straight again. We can model the transition between these modes as a [*Markov chain*](https://en.wikipedia.org/wiki/Markov_chain), as in this illustration:"
@@ -1784,7 +1829,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 66,
+   "execution_count": 28,
    "metadata": {
     "collapsed": false
    },
@@ -1793,7 +1838,7 @@
      "data": {
       "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684e0c25f8>"
+       "<matplotlib.figure.Figure at 0x20312852c88>"
       ]
      },
      "metadata": {},
@@ -1813,35 +1858,137 @@
     "\n",
     "The algorithm is not sensitive to the exact numbers, and you will typically use simulation or trials to choose appropriate values. However, these values are quite representative.\n",
     "\n",
-    "We represent Markov chains with a [*transition probability matrix*](https://en.wikipedia.org/wiki/Stochastic_matrix), which we will call $\\mathbf M$. The literature sometimes calls this $p$. For the Markov chain in the illustration we would write\n",
+    "We represent Markov chains with a [*transition probability matrix*](https://en.wikipedia.org/wiki/Stochastic_matrix), which we will call $\\mathbf M$. For the Markov chain in the illustration we would write\n",
     "\n",
     "$$\\mathbf M = \\begin{bmatrix}.97 & .03\\\\.05 & .95\\end{bmatrix}$$\n",
     "\n",
-    "This should be clear, but formally $\\mathbf M_{ij}$ is the probability of mode being $i$ given that the last mode was $j$. In this example the probability of the mode currently being straight $(i=0)$ given that the last mode was a turn $(j=1)$ is $M[0,1] = 0.05$"
+    "In other words $\\mathbf M[i, j]$ is the probability of mode being $i$ given that the last mode was $j$. In this example the probability of the mode currently being straight $(i=0)$ given that the last mode was a turn $(j=1)$ is $\\mathbf M[0,1] = 0.05$. In Python we'd write:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 0.97,  0.03],\n",
+       "       [ 0.05,  0.95]])"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "M = np.array([[.97, .03], [.05, .95]])\n",
+    "M"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Compute Mode Probabilities\n",
+    "This allows us to compute the new mode probabilities based on the probability of a transition. Let's compute the probability of the mode being straight after a transition. There are two ways for us to be moving straight. We could have been moving straight, and then continued straight, or we could have been turning, but then went straight. The former probability is calculated with $(0.7\\times 0.97)$ and the latter with $(0.3\\times 0.05)$. We are multiplying the mode probability with the relevant probability from the Markov Chain. The *total probability* is the sum of the two, or $(0.7)(0.97) + (0.3)(0.05) = 0.694$. \n",
     "\n",
-    "At each epoch we need to compute the mode probabilities. In simple terms, we are computing a weight based on how likely each model is. A high probability means it is very likely this is the correct model.\n",
+    "Recall the [*total probability theorem*](https://en.wikipedia.org/wiki/Law_of_total_probability) from the second chapter. It states that the probability of several distinct events is\n",
     "\n",
-    "The Kalman filter computes the *likelihood function*. This computes the likelihood of the measurements given the current state of the filter. The **Designing Kalman Filters** chapter describes this computation. The `Kalmanfilter` class computes the likelihood when you call `update()`. You won't need to remember this, but for review the equation is:\n",
+    "$$P(A) = \\sum P(A\\mid B)\\, P(B)$$\n",
+    "\n",
+    "Here $P(A\\mid B)$ is the transition matrix $\\mathbf M$ and $P(B)$ is $\\mu$. We are using arrays and matrices, and so we take advantage of the fact that a vector times a matrix computes the sum of products: \n",
+    "\n",
+    "$$\\begin{bmatrix}\\mu_1 & \\mu_2 \\end{bmatrix}\\begin{bmatrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{bmatrix} = \\begin{bmatrix}\\mu_1 m_{11} + \\mu_2 m_{21} & \\mu_1 m_{12} + \\mu_2 m_{22}\\end{bmatrix}$$\n",
+    "\n",
+    "The IMM literature expresses this as\n",
+    "\n",
+    "$$\\bar c_j = \\sum\\limits_{i=1}^{N} \\mu_i M_{ij}$$\n",
+    "\n",
+    "We use NumPy's `dot` function to compute this for us:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([ 0.694,  0.306])"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "cbar = np.dot(mu, M)\n",
+    "cbar"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Computing the Mode Probabilities\n",
     "\n",
-    "$$\n",
-    "\\mathcal{L} = \\frac{1}{\\sqrt{2\\pi S}}\\exp [-\\frac{1}{2}\\mathbf y^\\mathsf T\\mathbf S^{-1}\\mathbf y]$$\n",
     "\n",
     "We will compute the new mode probabilities using Bayes theorem. Recall that Bayes theorem states\n",
     "\n",
     "$$\\text{posterior} = \\frac{\\text{prior} \\cdot \\text{likelihood}}{\\text{normalization factor}}$$\n",
     "\n",
-    "In notation, to compute the mode probability:\n",
+    "Here the prior is the total probability computation we performed in the last section. The Kalman filter computes the *likelihood*, which is the likelihood of the measurements given the current state of the filter. For review the equation is:\n",
     "\n",
-    "$$\\mu_i = \\| \\mu_{i}\\mathcal{L}_i \\|$$\n",
+    "$$\n",
+    "\\mathcal{L} = \\frac{1}{\\sqrt{2\\pi \\mathbf S}}\\exp [-\\frac{1}{2}\\mathbf y^\\mathsf T\\mathbf S^{-1}\\mathbf y]$$\n",
     "\n",
-    "In words, for each Kalman filter (mode) we compute the mode probability as the prior mode probability times the likelihood that this is the correct mode. Then we normalize all of the probabilities so they sum to one. It couldn't be easier. "
+    "In mathematical notation the updated mode probability is:\n",
+    "\n",
+    "$$\\mu_i = \\| \\mathcal{L}_i {\\bar c}_{i}\\|$$\n",
+    "\n",
+    "In words, for each Kalman filter (mode) we compute the mode probability as the probability of the current mode taking the possible transition into account times the likelihood that this is the correct mode. Then we normalize all of the probabilities so they sum to one.\n",
+    "\n",
+    "This is trivial to compute in Python. I'll introduce the variable `L` to store the likelihoods. Likelihoods are computed by the `KalmanFilter.update()` step, and in the code snippet below I just hard coded values for `L` since we haven't created the Kalman filters yet:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([ 0.802,  0.198])"
+      ]
+     },
+     "execution_count": 31,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "# L = [kf0.L, kf1.L]  # get likelihoods from Kalman filters\n",
+    "L = [0.000134, 0.0000748] \n",
+    "mu = cbar * L\n",
+    "mu /= sum(mu) # normalize\n",
+    "mu"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Here you can see that the relatively strong likelihood for the straight filter pushed the probability for the straight mode from 70% to 80.2%."
    ]
   },
   {
@@ -1850,44 +1997,99 @@
    "source": [
     "## Mixing Probabilities\n",
     "\n",
-    "At this point we could use the mode transitiions to compute the probabilities for all possible choices. If $\\mu = \\begin{bmatrix} 0.63 & 0.27\\end{bmatrix}$, then we can use the transition probability matrix to compute all possible outcomes. In other words, if the current mode is straight $(\\mu=0.63)$, we can compute two new probabilities based on whether the target keeps moving straight, or turns. We do the same for the turning mode $(\\mu=0.27)$. We will have gone from 2 mode probabilities to 4. At the next step 4 will turn into 8, and so on. It's computationally exact, but infeasible in practice.  After only 30 epochs you'd require 8GB of memory to store the mode probabilities in double precision.\n",
+    "At this point we could use the mode transitions to compute the probabilities for all possible choices. If $\\mu = \\begin{bmatrix} 0.63 & 0.27\\end{bmatrix}$, then we can use the transition probability matrix to compute all possible outcomes. In other words, if the current mode is straight $(\\mu=0.63)$, we can compute two new probabilities based on whether the target keeps moving straight, or turns. We do the same for the turning mode $(\\mu=0.27)$. We will have gone from 2 mode probabilities to 4. At the next step 4 will turn into 8, and so on. It's computationally exact, but infeasible in practice.  After only 30 epochs you'd require 8GB of memory to store the mode probabilities in double precision.\n",
     "\n",
-    "We need a better, albeit approximate way. IMMs resolve this by computing *mixing probabilities*. The idea is simple. Let's say the first mode (straight) is currently very likely, and the second mode (turn) is unlikely. Instead of having the Kalman filter for the straight mode compute the state assuming the target is moving straight, we will compute the state *weighted* by how likely the state transition is. That is, we will incorporate the small chance that the target began to turn. (** not quite right - state is weighted by states of the other KF**)\n",
+    "We need a better, albeit approximate way. IMMs resolve this by computing *mixing probabilities*. The idea is simple. Let's say the first mode (straight) is currently very likely, and the second mode (turn) is unlikely. Instead of having the Kalman filter for the straight mode compute its state as the weighted average of all of the filters in the filter bank. Filters with high probability of matching the target's mode get weighted more than filters with lower probability. The result is that the information from the probable filters improve the accuracy of the filters that are improbable. This is the crux of the algorithm.\n",
     "\n",
-    "That may strike you as a bad idea, but think it through. Based on the mode probabilities we think we are going straight, and the Kalman filter for going straight incorporates a bit of the state from the Kalman filter that assumes the target is turning. If the target is not turning this Kalman filter because just a bit less accurate. That doesn't seem good. But the state for the turning Kalman filter will be weighted by the state of the straight Kalman filter. That state is very likely, so the state of the turning Kalman filter will incorporate the information from the high probability filter. In other words we take the information from the filters that match the target's mode and feed it to the filters that don't match the target mode. Thus all filters will contain a pretty good estimate. This is obviously an approximation, but it avoids combinatorial explosion, and does work well in practice.\n",
-    "\n",
-    "What we need to do is very simple. Each Kalman filter performs the update step, computing a new mean and covariance. But then we compute a new mean and covariance for each filter as a weighted sum of these means and covariances according to *mixing probabilities* which we call $\\omega$. Likely filters will be slightly adjusted by the unlikely filters, and the unlikely filters will be strongly adjusted by the likely ones. The literature calls these adjusted means and covariances the *mixed initial conditions*, and uses the notation $\\mathbf x^{0(j)}, \\mathbf P^{0(j)}$ for the mixed initial condition of the $j^{th}$ matrix. The equations for the mixed initial conditions are:\n",
+    "What we need to do is very simple. Each Kalman filter performs the update step, computing a new mean and covariance. But then we compute a new mean and covariance for each filter as a weighted sum of these means and covariances according to *mixing probabilities* which we call $\\omega$. Likely filters will be slightly adjusted by the unlikely filters, and the unlikely filters will be strongly adjusted by the likely ones. The literature calls these adjusted means and covariances either the *mixed conditions* or *mixed initial conditions*. I use the notation $\\mathbf x^m_j$ for the mixed state, and  $\\mathbf P^m_j $ for the mixed covariance. The equations are:\n",
     "\n",
     "$$\\begin{aligned}\n",
-    "\\mathbf x^{0(j)} &= \\sum_{i=1}^N \\omega^{i \\mid j} \\mathbf x^i \\\\\n",
-    "\\mathbf P^{0(j)} &= \\sum_{i=1}^N \\omega^{i \\mid j}\\left[\\mathbf P^i + (\\mathbf x^i - \\mathbf  x^{0(i)}) (\\mathbf x^i - \\mathbf x^{0(i)})^\\mathsf T\\right]\n",
-    "\\end{aligned}$$\n",
+    "\\mathbf x^m_j &= \\sum_{i=1}^N \\omega_{ij} \\mathbf x_i \\\\\n",
+    "\\mathbf P^m_j &= \\sum_{i=1}^N \\omega_{ij}\\left[(\\mathbf x^i - \\mathbf  x^m_i) (\\mathbf x^i - \\mathbf  x^m_i)^\\mathsf T + \\mathbf P_i\\right]\n",
+    "\\end{aligned}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Just think of the subscripts as indexes into arrays. Putting it in pseudo-Python we can write this as:\n",
     "\n",
-    " The superscript $0$ stands for the *initial condition*. $\\omega^{i \\mid j}$ means the mixing probability we will use from the $i^{th}$ filter given the current filter is the  $j^{th}$ filter. They are all superscripts because there is an implied $k$ subscript, denoting the $k^{th}$ step. $\\mathbf x^{0(j)}_k$ would be the complete notation for the initial mean of the $j^{th}$ filter at step $k$.\n",
-    " \n",
-    "Don't let the notation confuse what is a simple idea: perform the Kalman filter update step, then weight all of them based on how likely each model is. That's all these equations are doing.\n",
-    " \n",
-    "How do we compute the mixing probabilities? Think about it, and try to give a reasonable answer before reading on. We have mode probabilities which describe the current probabilitiy of each mode, and then transition probabilities describing how likely we are to change modes. How do we compute the new probability?\n",
+    "```python\n",
+    "for j in N:\n",
+    "    x0[j] = sum_over_i(w[i,j] * x[i])\n",
+    "    P0[j] = sum_over_i(w[i, j] * (P[i] + np.outer(x[i] - x0[i]))) \n",
+    "```\n",
+    "\n",
+    "Don't let the notation confuse what is a simple idea: incorporate estimates from the probable filters into the estimates of the improbable filters, ensuring all have a good estimate."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "How do we compute the mixing probabilities? Think about it, and try to give a reasonable answer before reading on. We have mode probabilities which describe the current probability of each mode, and then transition probabilities describing how likely we are to change modes. How do we compute the new probability?\n",
     "\n",
     "Bayes theorem, of course! Prior times the likelihood, normalized. The prior is the mode probability, and the likelihood comes from the Markov chain, which we store in the matrix $\\mathbf M$. \n",
     "\n",
-    "$$\\omega^{i|j} = \\| \\mu^i \\cdot \\mathbf M^{i,j}\\|$$\n",
+    "$$\\boldsymbol\\omega_{ij} = \\| \\mu_i \\cdot \\mathbf M_{ij}\\|$$\n",
     "\n",
-    "The Kalman filters need to perform the prediction step to compute the new prior. They use the initial conditions:\n",
+    "We can compute this as follows. I computed the update of $\\mu$ and $\\bar c$ out of order above (ou must compute $\\bar c$ incorporating the transition probability matrix into $\\mu$), so I'll need to correct that here:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 0.987,  0.114],\n",
+       "       [ 0.0125,  0.886]])"
+      ]
+     },
+     "execution_count": 32,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "cbar = np.dot(mu, M) #compute total probability that target is in mode j\n",
+    "\n",
+    "omega = np.zeros((2, 2))\n",
+    "for i in range(2):\n",
+    "    for j in range(2):\n",
+    "        omega[i, j] = (M[i, j] * mu[i]) / cbar[j]\n",
+    "omega"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The Kalman filters need to perform the prediction step to compute the new prior. They use the mixed estimates:\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
-    "\\bar{\\mathbf x}^j &= \\mathbf F^j\\mathbf x^{0(j)}\\\\\n",
-    "\\bar{\\mathbf P}^j &= \\mathbf F^j\\mathbf P^{0(j)}\\mathbf F^{j\\mathsf T} + \\mathbf Q^j\n",
-    "\\end{aligned}$$\n",
-    "\n",
+    "\\bar{\\mathbf x}_j &= \\mathbf F_j\\mathbf x^m_j\\\\\n",
+    "\\bar{\\mathbf P}_j &= \\mathbf F_j\\mathbf P^m_j\\mathbf F_j^\\mathsf T + \\mathbf Q_j\n",
+    "\\end{aligned}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "### IMM Estimate\n",
     "\n",
-    "Now we need a final state estimate from the bank of filters. How do we do that? Just weight the initial condition from each Kalman filter that we just computed:\n",
+    "Now we need a final state estimate from the bank of filters. How do we do that? Just weight the mixed estimate from each Kalman filter:\n",
     "\n",
     "$$\\begin{aligned}\n",
-    "\\mathbf x &= \\sum_{j=1}^N \\mu^j\\bar{\\mathbf x}^j\\\\\n",
-    "\\mathbf P &= \\sum_{j=1}^N \\mu^j\\left[({\\mathbf x}^j - \\mathbf x)({\\mathbf x}^j - \\mathbf x)^\\mathsf T + \\mathbf P^j\\right]\n",
+    "\\mathbf x &= \\sum_{j=1}^N \\mu_j{\\mathbf x}_j\\\\\n",
+    "\\mathbf P &= \\sum_{j=1}^N \\mu_j\\left[({\\mathbf x}_j - \\mathbf x)({\\mathbf x}_j - \\mathbf x)^\\mathsf T + \\mathbf P_j\\right]\n",
     "\\end{aligned}$$"
    ]
   },
@@ -1897,21 +2099,21 @@
    "source": [
     "### Tracking Maneuvering Target with the IMM\n",
     "\n",
-    "Let's work an example. Crassidis[4] is one of the few texts with a worked example, so I have chosen his example. "
+    "Let's work an example. Crassidis[4] is one of the few texts with a worked example, so I have chosen his example. He tracks a moving target for 600 seconds. The target starts off moving straight, and then a control input is injected starting at 400 seconds, causing the target to make a 90 degree turn. He uses two constant acceleration Kalman filters. One filter assumes no process noise, and the other assumes process noise with spectral density $10^{-3}\\mathbf I$. He assumes very good initialization of the filters, setting $\\mathbf P =10^{-12}$ for both filters. My implementation follows:"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 67,
+   "execution_count": 33,
    "metadata": {
     "collapsed": false
    },
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAv0AAADnCAYAAACNFOnAAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x2684cd58278>"
+       "<matplotlib.figure.Figure at 0x203117f9978>"
       ]
      },
      "metadata": {},
@@ -1979,7 +2181,7 @@
     "probs = np.array(probs)\n",
     "plt.subplot(121)\n",
     "plt.plot(xs[:, 0], xs[:, 3], 'k')\n",
-    "plt.scatter(zs[:, 0], zs[:, 1], marker='+', alpha=0.2)\n",
+    "plt.scatter(zs[:, 0], zs[:, 1], marker='+')\n",
     "\n",
     "plt.subplot(122)\n",
     "plt.plot(probs[:, 0])\n",
@@ -1992,9 +2194,61 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### History of Development\n",
+    "It is rather hard to see the performance of the filter, so let's look at the performance just as the turn starts. I've swapped the $x$ and $y$ axis to let us zoom in closely. In the chart below the turn starts at $Y=4000$. If you look very closely you can see that the estimate wavers slightly after the turn is initiated, but the filter tracks the measurement without lag and soon tracks smoothly."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": false,
+    "scrolled": true
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(4200.0, 3200.0, 1980.0, 2120.0)"
+      ]
+     },
+     "execution_count": 34,
+     "metadata": {},
+     "output_type": "execute_result"
+    },
+    {
+     "data": {
+      "image/png": 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truncated
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x20312a340b8>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "plt.plot(xs[390:450, 3], xs[390:450, 0], 'k')\n",
+    "plt.scatter(zs[390:450, 1], zs[390:450, 0], marker='+', s=100);\n",
+    "plt.xlabel('Y'); plt.ylabel('X')\n",
+    "plt.gca().invert_xaxis()\n",
+    "plt.axis('equal')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Summary\n",
     "\n",
-    "The *Interacting Multiple Models* (IMM) algorithm was invented by Blom[5] to solve the combinatorial explosion problem of multiple models. A subsequent paper by Blom and Bar-Shalom is the most cited paper [6]. It is often called *concise*, but with the treatment above I think you will find it readable. "
+    "This chapter contains some of the more challenging material in this book. However, it is the gateway to implementing realistic Kalman filters. If we are controlling a robot we know its process model, and it is easy to construct a Kalman filter for it. Far more commonly we are given a set of time series data and asked to make sense of it. The process model is largely unknown to us. We use the techniques in this chapter to *learn* (in a machine learning sense) how to parameterize our models. The models change over time as the target maneuver, so our filters must be adaptive. \n",
+    "\n",
+    "Finding an optimal answer involves combinatorial explosion, and is impractical in practice. The IMM algorithm has become the standard algorithm because of its good performance and computational tractability.\n",
+    "\n",
+    "A real filter bank usually involves more than two filters. It is common to have many filters. As the target's regime changes some filters become infinitesimally likely. Most adaptive filters implement an algorithm that kills off extremely unlikely filters and replaces them with filters that more closely match the current regime. This is highly specific to your problem space, and is usually very ad-hoc. You will need to devise schemes for killing and creating filters and test them against simulated or real data. \n",
+    "\n",
+    "Despite the complexity of the algorithms, I hope you recognize the underlying ideas are very simple. We use the same two tools that we learned in the second chapter: Bayes theorem and the total probability theorem. We incorporate new information using Bayes theorem, and compute the effect of the process models using the total probability theorem.\n",
+    "\n",
+    "For me, this chapter underscores the beauty of the Bayesian formulation of Kalman filters. I don't much care if you learn the details of the IMM algorithm. I do hope that you see that very simple probabilistic reasoning led to these results. The linear algebra equations of the Kalman filter that Dr. Kalman derived came from a different form of reasoning called *orthogonal projection*. It is beautiful in its own way, and I urge you to read his paper. But I'm not sure I find them intuitive to use, and it is not at all clear how to devise new, non-optimal filters such as the IMM using those techniques. In contrast, Bayes theorem lets us handle these problems with ease. "
    ]
   },
   {