Added algebra to prove equivalence 1D filters.

Added text that represented the multivariate KF equations with scalars
for the univariate case to illustrate how simple the equations actually
are. Then to complete the section I proved the equivelence of the last
chapter's equations with this one using algebra.

Multidimensional_Kalman_Filters.ipynb

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        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b38ca10>"
+        "<matplotlib.figure.Figure at 0x7f7144e55a10>"
        ]
       }
      ],
@@ -688,7 +688,7 @@
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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b56f4d0>"
+        "<matplotlib.figure.Figure at 0x7f71450394d0>"
        ]
       }
      ],
@@ -715,7 +715,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b45a1d0>"
+        "<matplotlib.figure.Figure at 0x7f7144f24d90>"
        ]
       }
      ],
@@ -742,7 +742,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b4facd0>"
+        "<matplotlib.figure.Figure at 0x7f7144fc46d0>"
        ]
       }
      ],
@@ -773,7 +773,7 @@
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        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b404a90>"
+        "<matplotlib.figure.Figure at 0x7f7144f74a90>"
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       }
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@@ -840,7 +840,7 @@
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        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b470150>"
+        "<matplotlib.figure.Figure at 0x7f7144e72b90>"
        ]
       }
      ],
@@ -1457,9 +1457,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
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        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b4c2810>"
+        "<matplotlib.figure.Figure at 0x7f71450a4b10>"
        ]
       },
       {
@@ -1467,7 +1467,7 @@
        "output_type": "display_data",
        "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b66c810>"
+        "<matplotlib.figure.Figure at 0x7f7144e80910>"
        ]
       }
      ],
@@ -1767,43 +1767,30 @@
      "outputs": [],
      "prompt_number": 21
     },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Compare to Single Dimensional Filter"
+     ]
+    },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "#####Exercise: Compare to Previous Chapter\n",
-      "\n",
       "The equations in this chapter look very different from the equations in the last chapter, yet I claimed the last chapter implemented a full 1-D Kalman filter. \n",
       "\n",
       "Recall that the 1-D equations for the update step are:\n",
       "$$\n",
       "\\mu =\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}, \n",
-      "\\sigma = \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
+      "\\sigma^2 = \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
       "$$\n",
       "\n",
       "and that the 1-D equations for the predict step are:\n",
       "$$ \\mu = \\mu_1+\\mu_2, \\sigma^2 = \\sigma_1^2 + \\sigma_2^2$$\n",
       "\n",
-      "Implement a simple 1-D kalman filter using the Kalman filter from this chapter, and compare it's output to the kalman filter from the previous chapter by plotting it."
-     ]
-    },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [
-      "# Your code here"
-     ],
-     "language": "python",
-     "metadata": {},
-     "outputs": []
-    },
-    {
-     "cell_type": "markdown",
-     "metadata": {},
-     "source": [
-      "##### Solution\n",
-      "\n",
-      "We will use a simple model of tracking an object that starts at x=0 and moves by 1 at each step. We will assume the arbitrary value 5 for the measurement noise and .02 for the process noise.\n",
+      "Let's implement a simple 1-D kalman filter using the Kalman filter from this chapter, and compare it's output to the kalman filter from the previous chapter by plotting it. We will use a simple model of tracking an object that starts at x=0 and moves by 1 at each step. We will assume the arbitrary value 5 for the measurement noise and .02 for the process noise.\n",
       "\n",
       "First, let's implement the filter from the last chapter:"
      ]
@@ -1830,7 +1817,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 117
+     "prompt_number": 22
     },
     {
      "cell_type": "markdown",
@@ -1859,7 +1846,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 105
+     "prompt_number": 23
     },
     {
      "cell_type": "markdown",
@@ -1917,9 +1904,9 @@
       {
        "metadata": {},
        "output_type": "display_data",
-       "png": 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truncated
        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b4b2690>"
+        "<matplotlib.figure.Figure at 0x7f7144ed22d0>"
        ]
       },
       {
@@ -1927,11 +1914,11 @@
        "output_type": "display_data",
        "png": 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        "text": [
-        "<matplotlib.figure.Figure at 0x7fa14986a550>"
+        "<matplotlib.figure.Figure at 0x7f7144e80b90>"
        ]
       }
      ],
-     "prompt_number": 136
+     "prompt_number": 24
     },
     {
      "cell_type": "markdown",
@@ -1939,9 +1926,140 @@
      "source": [
       "##### Discussion\n",
       "\n",
-      "As you can see, both filters produce the same results. Feel free to vary the initial guess, the measurement noise, and the process noise; so long as you make the same changes to both filters the output should be the same. This is a solid demonstration, albeit not a rigorous proof, that both filters in fact implement the same math for the 1-D case. \n",
+      "As you can see, both filters produce the same results. Feel free to vary the initial guess, the measurement noise, and the process noise; so long as you make the same changes to both filters the output should be the same. This is a solid demonstration, albeit not a rigorous proof, that both filters in fact implement the same math for the 1-D case. "
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Converting the Multivariate Equations to the Univariate Case"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
       "\n",
-      "If you are inclined to do so, and have the linear algebra background, try to prove that the equations are identical for the 1-D case. This is well beyond the scope of this chapter, so I will not present the solution."
+      "As it turns out the Kalman filter equations are quite easy to deal with in one dimension, so let's do the mathematical proof. \n",
+      "\n",
+      "> **Important** This section will provide you with a strong intuition into what the Kalman filter equations are actually doing. I strong recommend reading this section carefully as it should make this material much easier to understand. It is not merely a proof of correctness that you would normally want to skip past!\n",
+      "\n",
+      "Let'start with the predict step, which is slightly easier. Here are the multivariate equations. \n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "\\mathbf{x}^- &:= \\mathbf{F x} + \\mathbf{B u} \\\\\n",
+      "\\mathbf{P} &:= \\mathbf{FP{F}}^T + \\mathbf{Q}\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "\n",
+      "The state $mathbf{x}$ only has one variable, so it is a $1\\times 1$ matrix. Our motion $\\mathbf{u}$ is also be a $1\\times 1$ matrix. Therefore, $\\mathbf{F}$ and $\\mathbf{B}$ must also be $1\\times 1$ matrices. That means that they are all scalars, and we can write\n",
+      "\n",
+      "$$x = Fx + Bu$$\n",
+      "\n",
+      "Here the variables are not bold, denoting that they are just variables. \n",
+      "\n",
+      "Our state transition is simple - the next state is the same as this state, so $F=1$. The same holds for the motion transistion, so, $B=1$. Thus we have\n",
+      "$$x = x + u$$\n",
+      "\n",
+      "which is equivelant to the Gaussian equation from the last chapter\n",
+      "\n",
+      "$$ \\mu = \\mu_1+\\mu_2$$\n",
+      "\n",
+      "Hopefully the general process is clear, so now I will go a bit faster on the rest. Our other equation for the predict step is\n",
+      "\n",
+      "$$\\mathbf{P} := \\mathbf{FP{F}}^T + \\mathbf{Q}$$\n",
+      "\n",
+      "Again, since our state only has one variable $\\mathbf{P}$ and $\\mathbf{Q}$ must also be $1\\times 1$ matrix, which we can treat as scalars, yielding  \n",
+      "$$P := FPF^T + Q$$\n",
+      "\n",
+      "We already know $F=1$. The transpose of a scalar is the scalar, so $F^T = 1$. This yields\n",
+      "\n",
+      "$$P := P + Q$$\n",
+      "\n",
+      "which is equivalent to the Gaussian equation of \n",
+      "$$\\sigma^2 = \\sigma_1^2 + \\sigma_2^2$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Here our our multivariate Kalman filter equations for the update step.\n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "\\textbf{y} &:= \\mathbf{z} - \\mathbf{H x}\\\\\n",
+      "\\mathbf{K}&:= \\mathbf{PH}^T (\\mathbf{HPH}^T + \\mathbf{R})^{-1} \\\\\n",
+      "\\mathbf{x}&:=\\mathbf{x}^- +\\mathbf{K\\textbf{y}} \\\\\n",
+      "\\mathbf{P}&:= (\\mathbf{I}-\\mathbf{KH})\\mathbf{P}\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "\n",
+      "As above, all of the matrices become scalars. $H$ defines how we convert from a position to a measurement. Both are positions, so there is no conversion, and thus $H=1$. Let's substitute in our known values and convert to scalar in one step. One final thing you need to know - division is scalar's analogous operation for matrix inversion, so we will convert the matrix inversion to division.\n",
+      "\n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "y &:= z - x\\\\\n",
+      "K &:=P / (P + R) \\\\\n",
+      "x &:=x +Ky \\\\\n",
+      "P &:= (1-K)P\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "\n",
+      "Before we continue with the proof, I want you to look at those equations to recognize what a simple concept these equations implement. The residual $y$ is nothing more than the measurement minus the previous state. The gain $K$ is scaled based on how certain we are about the last prediction vs how certain we are about the measurement. We choose a new state $x$ based on the old value of $x$ plus the scaled value of the residual. Finally, we update the uncertainty based on how certain we are about the measurement. Algorithmically this should sound exactly like what we did in the last chapter.\n",
+      "\n",
+      "So let's finish off the algebra to prove that. It's straightforward, and not at all  necessary for you to learn unless you are interested. Feel free to skim past if you like - it will not help you with Kalman filtering.\n",
+      "\n",
+      "Recall that the univariate equations for the update step are:\n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "\\mu &=\\frac{\\sigma_1^2 \\mu_2 + \\sigma_2^2 \\mu_1} {\\sigma_1^2 + \\sigma_2^2}, \\\\\n",
+      "\\sigma^2 &= \\frac{1}{\\frac{1}{\\sigma_1^2} + \\frac{1}{\\sigma_2^2}}\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "\n",
+      "Here we will say that $\\mu_1$ is the state $x$, and $\\mu_2$ is the measurement $z$. That is entirely arbitrary, we could have chosen the opposite assigment. \n",
+      "Thus it follows that that $\\sigma_1^2$ is the state uncertainty $P$, and $\\sigma_2^2$ is the measurement noise $R$. Let's substitute those in.\n",
+      "\n",
+      "$$ \\mu = \\frac{Pz + Rx}{P+R} \\\\\n",
+      "\\sigma^2 = \\frac{1}{\\frac{1}{P} + \\frac{1}{R}}\n",
+      "$$\n",
+      "\n",
+      "I will handle $\\mu$ first. The corresponding equation in the multivariate case is\n",
+      "\n",
+      "$$\n",
+      "\\begin{aligned}\n",
+      "x &= x + Ky \\\\\n",
+      " &= x + \\frac{P}{P+R}(z-x) \\\\\n",
+      " &= \\frac{P+R}{P+R}x + \\frac{Pz - Px}{P+R} \\\\\n",
+      " &= \\frac{Px + Rx + Pz - Px}{P+R} \\\\\n",
+      " &= \\frac{Pz + Rx}{P+R} \\quad \\blacksquare\n",
+      "\\end{aligned}\n",
+      "$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Now let's look at $\\sigma^2$. The corresponding equation in the multivarate case is\n",
+      "\n",
+      "$$ \n",
+      "\\begin{aligned}\n",
+      "P &= (1-K)P \\\\\n",
+      " &= (1-\\frac{P}{P+R})P \\\\\n",
+      " &= (\\frac{P+R}{P+R}-\\frac{P}{P+R})P \\\\\n",
+      "  &= (\\frac{P+R-P}{P+R})P \\\\\n",
+      "  &= \\frac{RP}{P+R}\\\\\n",
+      "  &= \\frac{1}{\\frac{P+R}{RP}}\\\\\n",
+      "  &= \\frac{1}{\\frac{R}{RP} + \\frac{P}{RP}} \\\\\n",
+      "  &= \\frac{1}{\\frac{1}{P} + \\frac{1}{R}}\n",
+      " \\quad\\blacksquare\n",
+      "\\end{aligned}\n",
+      "$$\n",
+      "\n",
+      "So we have proven that the multivariate equations are equivalent to the univariate equations when we only have one state variable. I'll close this section by recognizing one quibble - I handwaved my assertion that $H=1$ and $F=1$. In general we know this is not true. For example, a digital therometer may provide measurement in volts, and we need to convert that to temperature, and we use $H$ to do that conversion. I left that issue out of the last chapter to keep the explanation as simple and streamlined as possible. It is very straightforward to add that generalization to the equations of the last chapter, redo the algebra above, and still have the same results. In practice we do not use the equations in the last chapter to perform Kalman filtering due to the material in the next section which demonstrates how much better the Kalman filter performs when we include unobserved variables. So I prefer to leave the equations from the last chapter in their simplest form so that they economically represent our central ideas without any extra complications."
      ]
     },
     {
@@ -1961,7 +2079,8 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 25
     },
     {
      "cell_type": "markdown",
@@ -2025,15 +2144,18 @@
      "metadata": {},
      "outputs": [
       {
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-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b30de10>"
+       "ename": "NameError",
+       "evalue": "global name 'f' is not defined",
+       "output_type": "pyerr",
+       "traceback": [
+        "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[1;31mNameError\u001b[0m                                 Traceback (most recent call last)",
+        "\u001b[1;32m<ipython-input-26-b2a3166033b0>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m     42\u001b[0m     \u001b[0mplt\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mshow\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     43\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 44\u001b[1;33m \u001b[0mplot_compare_pos_vel\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m0.\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mp0\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m500.\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mR\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m5.\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mQ\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m.2\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mmove\u001b[0m\u001b[1;33m=\u001b[0m\u001b[1;36m1.\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
+        "\u001b[1;32m<ipython-input-26-b2a3166033b0>\u001b[0m in \u001b[0;36mplot_compare_pos_vel\u001b[1;34m(x0, p0, R, Q, move)\u001b[0m\n\u001b[0;32m     16\u001b[0m     \u001b[1;31m# initialize the filters\u001b[0m\u001b[1;33m\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     17\u001b[0m     \u001b[0mf1\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mmkf_filter\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mR\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mQ\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 18\u001b[1;33m     \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0.\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     19\u001b[0m     \u001b[0mf\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mP\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m0\u001b[0m\u001b[1;33m]\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mp0\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     20\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
+        "\u001b[1;31mNameError\u001b[0m: global name 'f' is not defined"
        ]
       }
      ],
-     "prompt_number": 146
+     "prompt_number": 26
     },
     {
      "cell_type": "markdown",
@@ -2083,25 +2205,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b5bb350>"
-       ]
-      },
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa16d154b90>"
-       ]
-      }
-     ],
-     "prompt_number": 22
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2206,17 +2310,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b614850>"
-       ]
-      }
-     ],
-     "prompt_number": 23
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2275,17 +2369,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b655e50>"
-       ]
-      }
-     ],
-     "prompt_number": 24
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2303,17 +2387,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14be68cd0>"
-       ]
-      }
-     ],
-     "prompt_number": 25
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2373,17 +2447,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b62cf50>"
-       ]
-      }
-     ],
-     "prompt_number": 26
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2402,17 +2466,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa15333fad0>"
-       ]
-      }
-     ],
-     "prompt_number": 27
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2436,17 +2490,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa15333fa90>"
-       ]
-      }
-     ],
-     "prompt_number": 28
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2467,17 +2511,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b632a10>"
-       ]
-      }
-     ],
-     "prompt_number": 29
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2540,25 +2574,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b5fdf10>"
-       ]
-      },
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b6c0710>"
-       ]
-      }
-     ],
-     "prompt_number": 30
+     "outputs": []
     },
     {
      "cell_type": "markdown",
@@ -2615,17 +2631,7 @@
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [
-      {
-       "metadata": {},
-       "output_type": "display_data",
-       "png": 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truncated
-       "text": [
-        "<matplotlib.figure.Figure at 0x7fa14b662590>"
-       ]
-      }
-     ],
-     "prompt_number": 31
+     "outputs": []
     },
     {
      "cell_type": "markdown",