Added text for computation of velocities.

I explained how the covariances are computed by FPF',
and then how the velocity is computed using the kalman gain.

06-Multivariate-Kalman-Filters.ipynb

@@ -369,9 +369,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtYAAADlCAYAAAB3YbDsAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x4b2b198>"
+       "<matplotlib.figure.Figure at 0x1ee2938d518>"
       ]
      },
      "metadata": {},
@@ -862,9 +862,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x879d7f0>"
+       "<matplotlib.figure.Figure at 0x1ee2c3a48d0>"
       ]
      },
      "metadata": {},
@@ -1027,9 +1027,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x87b1198>"
+       "<matplotlib.figure.Figure at 0x1ee2c33ddd8>"
       ]
      },
      "metadata": {},
@@ -1368,9 +1368,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x88bc240>"
+       "<matplotlib.figure.Figure at 0x1ee2d4af320>"
       ]
      },
      "metadata": {},
@@ -1378,9 +1378,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9950f98>"
+       "<matplotlib.figure.Figure at 0x1ee2d3e4710>"
       ]
      },
      "metadata": {},
@@ -1429,9 +1429,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d5fc50>"
+       "<matplotlib.figure.Figure at 0x1ee2d5c9a90>"
       ]
      },
      "metadata": {},
@@ -1503,7 +1503,42 @@
     "\n",
     "Expressions in the form $\\mathbf{ABA}^\\mathsf{T}$ are common in linear algebra. You can think of it as *projecting* the middle term by the outer term. We will be using this many times in the rest of the book. \n",
     "\n",
-    "I admit this may be a 'magical' equation to you. If you have some experience with linear algebra and statistics, this may help. The covariance due to the prediction can be modeled as the expected value of the error in the prediction step, given by this equation. \n",
+    "I admit this may be a 'magical' equation to you. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "When we initialize $\\mathbf{P}$ with\n",
+    "\n",
+    "$$\\mathbf{P} = \\begin{bmatrix}\\sigma^2_x & 0 \\\\ 0 & \\sigma^2_v\\end{bmatrix}$$\n",
+    "\n",
+    "\n",
+    "the value for $\\mathbf{FPF}^\\mathsf{T}$ is:\n",
+    "\n",
+    "$$\\mathbf{FPF}^\\mathsf{T} = \\begin{bmatrix}1&\\Delta t\\\\0&1\\end{bmatrix}\n",
+    "\\begin{bmatrix}\\sigma^2_x & 0 \\\\  0 & \\sigma^2_{v}\\end{bmatrix}\n",
+    "\\begin{bmatrix}1&0\\\\\\Delta t&1\\end{bmatrix} \\\\\n",
+    "= \\begin{bmatrix}\\sigma^2_x&\\sigma_v^2\\Delta t\\\\  0 & \\sigma^2_{v}\\end{bmatrix}\n",
+    "\\begin{bmatrix}1&0\\\\\\Delta t&1\\end{bmatrix} \\\\\n",
+    "= \\begin{bmatrix}\\sigma^2_x +  \\sigma_v^2\\Delta t^2  &  \\sigma_v^2\\Delta t \\\\\n",
+    "\\sigma_v^2\\Delta t & \\sigma^2_{v}\\end{bmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The initial value for $\\mathbf{P}$ had no covariance between the position and velocity.  Position is computed as $\\dot{x}\\Delta t + x$, so there is a correlation between the position and velocity. The multiplication $\\mathbf{FPF}^\\mathsf{T}$ computes a covariance of $\\sigma_v^2 \\Delta t$. The exact value is not important; you just need to recognize that $\\mathbf{FPF}^\\mathsf{T}$ uses the process model to automatically compute the covariance between the position and velocity!"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "If you have some experience with linear algebra and statistics, this may help. The covariance due to the prediction can be modeled as the expected value of the error in the prediction step, given by this equation. \n",
     "\n",
     "$$\\begin{aligned}\n",
     "\\bar{\\mathbf{P}} &= E[(\\mathbf{Fx})(\\mathbf{Fx})^\\mathsf{T}]\\\\\n",
@@ -1513,8 +1548,13 @@
     "\n",
     "Of course, $E[\\mathbf{xx}^\\mathsf{T}]$ is just $\\mathbf{P}$, giving us\n",
     "\n",
-    "$$\\bar{\\mathbf{P}} = \\mathbf{FPF}^\\mathsf{T}$$\n",
-    "\n",
+    "$$\\bar{\\mathbf{P}} = \\mathbf{FPF}^\\mathsf{T}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "For now we will look at its effect. Here I use $\\mathbf{F}$ from our filter and project the state forward 6/10ths of a second. I do this five times so you can see how $\\mathbf{\\bar{P}}$ continues to change. "
    ]
   },
@@ -1527,9 +1567,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAuwAAADqCAYAAAD55hDIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7b45eb8>"
+       "<matplotlib.figure.Figure at 0x1ee2d60a1d0>"
       ]
      },
      "metadata": {},
@@ -1556,7 +1596,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "You can see that with a velocity of 5 the position correctly moves 3 units in each 6/10ths of a second step. At each step the width of the ellipse is larger, indicating that we have lost information about the position due to adding $\\dot{x}\\Delta t$ to x at each step. The height has not changed - our system model says the velocity does not change, so the belief we have about the velocity cannot change. As time continues you can see that the ellipse becomes more and more tilted. Recall that a tilt indicates *correlation*. $\\mathbf{F}$ linearly correlates $x$ with $\\dot{x}$ with the expression $\\bar{x} = \\dot{x} \\Delta t + x$. The $\\mathbf{FPF}^\\mathsf{T}$ computation correctly incorporates this correlation into the covariance matrix!\n",
+    "You can see that with a velocity of 5 the position correctly moves 3 units in each 6/10ths of a second step. At each step the width of the ellipse is larger, indicating that we have lost information about the position due to adding $\\dot{x}\\Delta t$ to x at each step. The height has not changed - our system model says the velocity does not change, so the belief we have about the velocity cannot change. As time continues you can see that the ellipse becomes more and more tilted. Recall that a tilt indicates *correlation*. $\\mathbf{F}$ linearly correlates $x$ with $\\dot{x}$ with the expression $\\bar{x} = \\dot{x} \\Delta t + x$. The $\\mathbf{FPF}^\\mathsf{T}$ computation correctly incorporates this correlation into the covariance matrix.\n",
     "\n",
     "Here is an animation of this equation that allows you to change the design of $\\mathbf{F}$ to see how it affects shape of $\\mathbf{P}$. The `F00` slider affects the value of F[0, 0]. `covar` sets the intial covariance between the position and velocity($\\sigma_x\\sigma_{\\dot{x}}$). I recommend answering these questions at a minimum\n",
     "\n",
@@ -1576,9 +1616,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAu8AAAEUCAYAAACbPtgWAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x7d00cf8>"
+       "<matplotlib.figure.Figure at 0x1ee2c2c8710>"
       ]
      },
      "metadata": {},
@@ -1640,9 +1680,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a1a0b8>"
+       "<matplotlib.figure.Figure at 0x1ee2d5cd908>"
       ]
      },
      "metadata": {},
@@ -1792,9 +1832,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a6e390>"
+       "<matplotlib.figure.Figure at 0x1ee2d4af710>"
       ]
      },
      "metadata": {},
@@ -1802,9 +1842,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a8a8d0>"
+       "<matplotlib.figure.Figure at 0x1ee2bb2f978>"
       ]
      },
      "metadata": {},
@@ -1940,9 +1980,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x8830c88>"
+       "<matplotlib.figure.Figure at 0x1ee2d88ef98>"
       ]
      },
      "metadata": {},
@@ -2030,9 +2070,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x997d978>"
+       "<matplotlib.figure.Figure at 0x1ee2d645198>"
       ]
      },
      "metadata": {},
@@ -2059,9 +2099,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x997af60>"
+       "<matplotlib.figure.Figure at 0x1ee2bdb5fd0>"
       ]
      },
      "metadata": {},
@@ -2079,6 +2119,83 @@
     "If we are tracking a robot which we are also controlling the univariate filter can do a very good job because the control input allows the filter to make an accurate prediction. But if we are tracking passively the control input is not much help unless we can make an accurate *apriori* guess as to the velocity. This is rarely possible."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## How Velocity is Calculated\n",
+    "\n",
+    "I haven't really explained how the filter computes the velocity, or any hidden variable. If we plug in the values we calculated for each of the filter's matrices we can see what happens.\n",
+    "\n",
+    "First we need to compute the system uncertainty."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\\begin{aligned}\n",
+    "\\textbf{S} &= \\mathbf{H\\bar{P}H}^\\mathsf{T} + \\mathbf{R} \\\\\n",
+    "&= \\begin{bmatrix} 1 & 0\\end{bmatrix}\n",
+    "\\begin{bmatrix}\\sigma^2_x & \\sigma_{xv} \\\\ \\sigma_{xv} & \\sigma^2_v\\end{bmatrix}\n",
+    "\\begin{bmatrix} 1 \\\\ 0\\end{bmatrix} + \\begin{bmatrix}\\sigma_z^2\\end{bmatrix}\\\\\n",
+    "&= \\begin{bmatrix}\\sigma_x^2 & \\sigma_{xv}\\end{bmatrix}\\begin{bmatrix} 1 \\\\ 0\\end{bmatrix}+ \\begin{bmatrix}\\sigma_z^2\\end{bmatrix} \\\\\n",
+    "&= \\begin{bmatrix}\\sigma_x^2 +\\sigma_z^2\\end{bmatrix}\n",
+    "\\end{aligned}$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now that we have $\\mathbf S$ we can find the value for the Kalman gain:\n",
+    "$$\\begin{aligned}\n",
+    "\\mathbf{K} &= \\mathbf{\\bar{P}H}^\\mathsf{T} \\mathbf{S}^{-1} \\\\\n",
+    "&= \\begin{bmatrix}\\sigma^2_x & \\sigma_{xv} \\\\ \\sigma_{xv} & \\sigma^2_v\\end{bmatrix}\n",
+    "\\begin{bmatrix} 1 \\\\ 0\\end{bmatrix}\n",
+    "\\begin{bmatrix}\\frac{1}{\\sigma_x^2 +\\sigma_z^2}\\end{bmatrix} \\\\\n",
+    "&= \\begin{bmatrix}\\sigma^2_x  \\\\ \\sigma_{xv}\\end{bmatrix}\n",
+    "\\begin{bmatrix}\\frac{1}{\\sigma_x^2 +\\sigma_z^2}\\end{bmatrix} \\\\\n",
+    "&= \\begin{bmatrix}\\sigma^2_x/(\\sigma_x^2 +\\sigma_z^2)  \\\\ \\sigma_{xv}/(\\sigma_x^2 +\\sigma_z^2)\\end{bmatrix}\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "\n",
+    "In other words, the Kalman gain for $x$ is \n",
+    "\n",
+    "$$K_x = \\frac{VAR(x)}{VAR(x)+VAR(z)}$$\n",
+    "\n",
+    "This should be very familiar to you from the univariate case. \n",
+    "\n",
+    "The Kalman gain for $\\dot x$ is\n",
+    "$$K_{\\dot x} = \\frac{COV(x, \\dot x)}{VAR(x)+VAR(z)}$$\n",
+    "\n",
+    "What is the effect of this? Recall that we compute the state as "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\\begin{aligned}\\mathbf x \n",
+    "&=\\mathbf{\\bar x}+\\mathbf{K}(z-\\mathbf{Hx)} \\\\\n",
+    "&= \\mathbf{\\bar x}+\\mathbf Ky\\end{aligned}$$\n",
+    "\n",
+    "Here the residual $y$ is a scalar. Therefore it is multiplied into each element of $\\mathbf K$. Therefore we have\n",
+    "\n",
+    "$$\\begin{bmatrix}x \\\\ \\dot x\\end{bmatrix}=\\begin{bmatrix}x \\\\ \\dot x\\end{bmatrix}^- + \\begin{bmatrix}K_x \\\\ K_{\\dot x}\\end{bmatrix}y$$\n",
+    "\n",
+    "Which gives this system of equations: \n",
+    "\n",
+    "$$x = \\bar x + yK_x\\\\\n",
+    "\\dot x = \\bar{\\dot x} + yK_{\\dot x}$$\n",
+    "\n",
+    "The prediction $\\bar x$ was computed as $x + \\bar x \\Delta t$. If the prediction was perfect then the residual will be $y=0$ (ignoring noise in the measurement) and the velocity estimate will be unchanged. On the other hand, if the velocity estimate was very bad then the prediction will be very bad, and the residual will be large: $y >> 0$. In this case we update the velocity estimate with $yK_{\\dot x}$. $K_{\\dot x}$ is proportional to $COV(x,\\dot x)$. Therefore the velocity is updated by the error in the position times the a value proportional to the covariance between the position and velocity. The higher the correlation the larger the correction. \n",
+    "\n",
+    "To bring this full circle, $COV(x,\\dot x)$ is the off-diagonal elements of $\\mathbf P$. Recall that those values were computed with $\\mathbf{FPF}^\\mathsf{T}$. So the covariance of position and velocity is computed during the predict step. The Kalman gain for the velocity is proportional to this covariance, and we adjust the velocity estimate based on how inaccurate it was during the last epoch times a value proportional to this covariance. \n",
+    "\n",
+    "In summary, these linear algebra equations may be very unfamiliar to you, but computation is actually very simple. It is essentially the same computation that we performed in the g-h filter. Our constants are different in this chapter because we take the noise in the process model and sensors into account, but the math is the same."
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -2110,9 +2227,9 @@
    "outputs": [
     {
      "data": {
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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x880b860>"
+       "<matplotlib.figure.Figure at 0x1ee2d5905c0>"
       ]
      },
      "metadata": {},
@@ -2120,9 +2237,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9aaa470>"
+       "<matplotlib.figure.Figure at 0x1ee2d41a550>"
       ]
      },
      "metadata": {},
@@ -2170,9 +2287,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9eb76a0>"
+       "<matplotlib.figure.Figure at 0x1ee2edb9240>"
       ]
      },
      "metadata": {},
@@ -2200,9 +2317,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x880f5f8>"
+       "<matplotlib.figure.Figure at 0x1ee2d5a13c8>"
       ]
      },
      "metadata": {},
@@ -2231,9 +2348,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x997dd68>"
+       "<matplotlib.figure.Figure at 0x1ee2d857978>"
       ]
      },
      "metadata": {},
@@ -2278,9 +2395,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a706d8>"
+       "<matplotlib.figure.Figure at 0x1ee2f0b1128>"
       ]
      },
      "metadata": {},
@@ -2288,9 +2405,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9994e80>"
+       "<matplotlib.figure.Figure at 0x1ee2d5c9518>"
       ]
      },
      "metadata": {},
@@ -2332,9 +2449,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a6ac88>"
+       "<matplotlib.figure.Figure at 0x1ee2d3a94a8>"
       ]
      },
      "metadata": {},
@@ -2342,9 +2459,9 @@
     },
     {
      "data": {
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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0xaa252b0>"
+       "<matplotlib.figure.Figure at 0x1ee2bb41ef0>"
       ]
      },
      "metadata": {},
@@ -2376,9 +2493,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x88122e8>"
+       "<matplotlib.figure.Figure at 0x1ee2eeef208>"
       ]
      },
      "metadata": {},
@@ -2410,9 +2527,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x99152e8>"
+       "<matplotlib.figure.Figure at 0x1ee2d4e5898>"
       ]
      },
      "metadata": {},
@@ -2448,9 +2565,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a1fc88>"
+       "<matplotlib.figure.Figure at 0x1ee2f096438>"
       ]
      },
      "metadata": {},
@@ -2458,9 +2575,9 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9aebda0>"
+       "<matplotlib.figure.Figure at 0x1ee2d3aebe0>"
       ]
      },
      "metadata": {},
@@ -2550,9 +2667,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9eae978>"
+       "<matplotlib.figure.Figure at 0x1ee2bdbfe10>"
       ]
      },
      "metadata": {},
@@ -2687,9 +2804,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x88bb748>"
+       "<matplotlib.figure.Figure at 0x1ee2eef43c8>"
       ]
      },
      "metadata": {},
@@ -2738,9 +2855,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9971358>"
+       "<matplotlib.figure.Figure at 0x1ee2d5d4588>"
       ]
      },
      "metadata": {},
@@ -2781,9 +2898,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9963ef0>"
+       "<matplotlib.figure.Figure at 0x1ee2bdd6d68>"
       ]
      },
      "metadata": {},
@@ -2840,9 +2957,9 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
-       "<matplotlib.figure.Figure at 0x9a6b7b8>"
+       "<matplotlib.figure.Figure at 0x1ee2c3b79e8>"
       ]
      },
      "metadata": {},