Proof reading Chapter 6

- Fixed some typos
- Looked at issues https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/issues/395
- Looked at issue https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/issues/363
- Looked at issue https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/issues/354

06-Multivariate-Kalman-Filters.ipynb

@@ -155,7 +155,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 1100x300 with 1 Axes>"
       ]
@@ -836,7 +836,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -1125,15 +1125,15 @@
     "\n",
     "The *measurement noise matrix* models the noise in our sensors as a covariance matrix. In practice this can be difficult. A complicated system may have many sensors, the correlation between them might not be clear, and usually their noise is not a pure Gaussian. For example, a sensor might be biased to read high if the temperature is high, and so the noise is not distributed equally on both sides of the mean. We will learn to deal with these problems later.\n",
     "\n",
-    "The Kalman filter equations uses a covariance matrix $\\mathbf R$ for the measurement noise. The matrix will have dimension $m{\\times}m$, where $m$ is the number of sensors. It is a covariance matrix to account for correlations between the sensors. We have only 1 sensor so R is:\n",
+    "The Kalman filter equations uses a covariance matrix $\\mathbf R$ for the measurement noise. The matrix will have dimension $m{\\times}m$, where $m$ is the number of sensors. It is a covariance matrix to account for correlations between the sensors. We have only 1 sensor so $\\mathbf R$ is:\n",
     "\n",
-    "$$R = \\begin{bmatrix}\\sigma^2_z\\end{bmatrix}$$\n",
+    "$$\\mathbf R = \\begin{bmatrix}\\sigma^2_z\\end{bmatrix}$$\n",
     "\n",
-    "If $\\sigma^2_z$ is 5 meters squared we'd have $R = \\begin{bmatrix}5\\end{bmatrix}$. \n",
+    "If $\\sigma^2_z$ is 5 meters squared we'd have $\\mathbf R = \\begin{bmatrix}5\\end{bmatrix}$. \n",
     "\n",
     "If we had two position sensors, the first with a variance of 5 m$^2$, the second with a variance of 3 m$^2$, we would write\n",
     "\n",
-    "$$R = \\begin{bmatrix}5&0\\\\0&3\\end{bmatrix}$$\n",
+    "$$\\mathbf R = \\begin{bmatrix}5&0\\\\0&3\\end{bmatrix}$$\n",
     "\n",
     "We put the variances on the diagonal because this is a *covariance* matrix, where the variances lie on the diagonal, and the covariances, if any, lie in the off-diagonal elements. Here we assume there is no correlation in the noise between the two sensors, so the covariances are 0.\n",
     "\n",
@@ -1165,8 +1165,6 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "[[1. 0.]] [[680.587 301.175]\n",
-      " [301.175 502.35 ]] [[5.]]\n",
       "x = [ 1.085 -0.64 ]\n"
      ]
     }
@@ -1328,7 +1326,7 @@
        "likelihood = 2.2250738585072014e-308\n",
        "mahalanobis = 0.0\n",
        "alpha = 1.0\n",
-       "inv = <function inv at 0x00000296F726A948>"
+       "inv = <function inv at 0x000002ABA27DE280>"
       ]
      },
      "execution_count": 29,
@@ -1390,7 +1388,7 @@
    "source": [
     "This is the complete code for the filter, and most of it is boilerplate. I've made it flexible enough to support several uses in this chapter, so it is a bit verbose. Let's work through it line by line. \n",
     "\n",
-    "The first lines checks to see if you provided it with measurement data in `data`. If not, it creates the data using the `compute_dog_data` function we wrote earlier.\n",
+    "The first lines checks to see if you provided it with measurement data in `zs`. If not, it creates the data using the `compute_dog_data` function we wrote earlier.\n",
     "\n",
     "The next lines uses our helper function to create a Kalman filter.\n",
     "\n",
@@ -1409,7 +1407,7 @@
     "        kf.update(z)\n",
     "```\n",
     "\n",
-    "Each call to `predict` and `update` modifies the state variables `x` and `P`. Therefore, after the call to `predict`, `kf.x` contains the prior. After the call to update, `kf.x` contains the posterior. `data` contains the actual position and measurement of the dog, so we use `[:, 1]` to get an array of measurements.\n",
+    "Each call to `predict` and `update` modifies the state variables `x` and `P`. Therefore, after the call to `predict`, `kf.x` contains the prior. After the call to update, `kf.x` contains the posterior. The states and covariances are saved in `xs` and `cov`.\n",
     "\n",
     "It really cannot get much simpler than that. As we tackle more complicated problems this code will remain largely the same; all of the work goes into setting up the `KalmanFilter` matrices; executing the filter is trivial.\n",
     "\n",
@@ -1425,7 +1423,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -1435,7 +1433,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 2 Axes>"
       ]
@@ -1473,7 +1471,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x500 with 1 Axes>"
       ]
@@ -1634,7 +1632,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -1771,7 +1769,7 @@
     "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",
     "$$\\begin{aligned}\n",
-    "\\bar{\\mathbf P} &= \\mathbb E[(\\mathbf{Fx - \\bar \\mu})(\\mathbf{Fx - \\bar\\mu})^\\mathsf T]\\\\\n",
+    "\\bar{\\mathbf P} &= \\mathbb E[(\\mathbf{Fx - F\\bar \\mu})(\\mathbf{Fx - F\\bar\\mu})^\\mathsf T]\\\\\n",
     " &= \\mathbf F\\, \\mathbb E[\\mathbf{(x- \\bar\\mu)(x- \\bar\\mu)}^\\mathsf T]\\, \\mathbf F^\\mathsf T\n",
     "\\end{aligned}$$\n",
     "\n",
@@ -1789,7 +1787,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -1834,7 +1832,7 @@
     {
      "data": {
       "application/vnd.jupyter.widget-view+json": {
-       "model_id": "a828518321f448be916ab551fe70519e",
+       "model_id": "8f04127ee78d437784e419afaf11eeb3",
        "version_major": 2,
        "version_minor": 0
       },
@@ -1948,7 +1946,7 @@
     "For the multivariate Kalman filter $\\mathbf K$ is a vector, not a scalar. Here is the equation again: $\\mathbf K = \\mathbf{\\bar PH}^\\mathsf T \\mathbf{S}^{-1}$. Is this a *ratio*? We can think of the inverse of a matrix as linear algebra's way of finding the reciprocal. Division is not defined for matrices, but it is useful to think of it in this way. So we can read the equation for $\\textbf{K}$ as meaning\n",
     "\n",
     "$$\\begin{aligned} \\mathbf K &\\approx \\frac{\\mathbf{\\bar P}\\mathbf H^\\mathsf T}{\\mathbf{S}} \\\\\n",
-    "\\mathbf K &\\approx \\frac{\\mathsf{uncertainty}_\\mathsf{prediction}}{\\mathsf{uncertainty}_\\mathsf{prediction} + \\mathsf{uncertainty}_\\mathsf{measurement}}\\mathbf H^\\mathsf T\n",
+    "\\mathbf K &\\approx \\frac{\\mathsf{uncertainty}_\\mathsf{prediction}}{\\mathsf{uncertainty}_\\mathsf{measurement}}\\mathbf H^\\mathsf T\n",
     "\\end{aligned}$$\n",
     "\n",
     "The Kalman gain equation computes a ratio based on how much we trust the prediction vs the measurement. We did the same thing in every prior chapter. The equation is complicated because we are doing this in multiple dimensions via matrices, but the concept is simple. The $\\mathbf H^\\mathsf T$ term is less clear, I'll explain it soon. If you ignore that term the equation for the Kalman gain is the same as the univariate case: divide the uncertainty of the prior with the of the sum of the uncertainty of the prior and measurement.\n",
@@ -1974,7 +1972,7 @@
     "$$\\begin{aligned}\n",
     "\\mathbf x &= \\mathbf{\\bar x} + \\mathbf{Ky} \\\\\n",
     "&= \\mathbf{\\bar x} + \\mathbf{\\bar PH}^\\mathsf T \\mathbf{S}^{-1}\\mathbf y \\\\\n",
-    "&\\approx \\mathbf{\\bar x} + \\frac{\\mathsf{uncertainty}_\\mathsf{prediction}}{\\mathsf{uncertainty}_\\mathsf{measurement}}\\mathbf H^\\mathsf T\\mathbf y\n",
+    "&\\approx \\mathbf{\\bar x} + \\frac{\\mathsf{uncertainty}_\\mathsf{prediction}}{\\mathsf{uncertainty}_\\mathsf{measurement}}\\mathbf H^\\mathsf T \\mathbf y\n",
     "\\end{aligned}$$\n",
     "\n",
     "Perhaps a better way to *see* the ratio is to rewrite the estimate equation:\n",
@@ -2033,7 +2031,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2115,7 +2113,7 @@
     "\\\\\\end{aligned}\n",
     "$$\n",
     "\n",
-    "This notation uses the Bayesian $a\\mid b$ notation, which means $a$ given the evidence of $b$. The hat means estimate. Thus $\\hat{\\mathbf x}_{k\\mid k}$ means the estimate of the state $\\mathbf x$ at step $k$ (the first k) given the evidence from step $k$ (the second k). The posterior, in other words. $\\hat{\\mathbf x}_{k\\mid k-1}$ means the estimate for the state $\\mathbf x$ at step $k$ given the estimate from step $k - 1$. The prior, in other words. \n",
+    "This notation uses the Bayesian $a\\mid b$ notation, which means $a$ given the evidence of $b$. The hat means estimate. Thus $\\hat{\\mathbf x}_{k\\mid k}$ means the estimate of the state $\\mathbf x$ at step $k$ (the first $k$) given the evidence from step $k$ (the second $k$). The posterior, in other words. $\\hat{\\mathbf x}_{k\\mid k-1}$ means the estimate for the state $\\mathbf x$ at step $k$ given the estimate from step $k - 1$. The prior, in other words. \n",
     "\n",
     "This notation, copied from [Wikipedia](https://en.wikipedia.org/wiki/Kalman_filter#Details) [[1]](#[wiki_article]), allows a mathematician to express himself exactly. In formal publications presenting new results this precision is necessary. As a programmer I find it fairly unreadable. I am used to thinking about variables changing state as a program runs, and do not use a different variable name for each new computation. There is no agreed upon format in the literature, so each author makes different choices. I find it challenging to switch quickly between books and papers, and so have adopted my admittedly less precise notation. Mathematicians may write scathing emails to me, but I hope programmers and students will rejoice at my simplified notation.\n",
     "\n",
@@ -2162,7 +2160,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2252,7 +2250,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2269,7 +2267,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Here the performance of the two filters are similar, and perhaps the univariate filter is tracking more cloesly. But let's see what happens when the actual velocity `vel` is different from the control input `u`:"
+    "Here the performance of the two filters are similar, and perhaps the univariate filter is tracking more closely. But let's see what happens when the actual velocity `vel` is different from the control input `u`:"
    ]
   },
   {
@@ -2279,7 +2277,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2379,7 +2377,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2389,7 +2387,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2431,7 +2429,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2459,7 +2457,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2488,7 +2486,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2528,7 +2526,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2538,7 +2536,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 900x400 with 2 Axes>"
       ]
@@ -2581,7 +2579,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2591,7 +2589,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 2 Axes>"
       ]
@@ -2623,7 +2621,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2654,7 +2652,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2685,7 +2683,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 1000x600 with 2 Axes>"
       ]
@@ -2714,7 +2712,7 @@
     "plt.subplot(121)\n",
     "plot_covariances(R=5, Q=.02, count=20, title='$R = 5\\, m^2$')\n",
     "plt.subplot(122)\n",
-    "plot_covariances(R=.1, Q=.02, count=20, title='$R = 0.5\\, m^2$')"
+    "plot_covariances(R=.5, Q=.02, count=20, title='$R = 0.5\\, m^2$')"
    ]
   },
   {
@@ -2733,7 +2731,7 @@
     "\n",
     "Think about what this means physically. The x-axis of the ellipse denotes our uncertainty in position, and the y-axis our uncertainty in velocity. So, an ellipse that is taller than it is wide signifies that we are more uncertain about the velocity than the position. Conversely, a wide, narrow ellipse shows high uncertainty in position and low uncertainty in velocity. Finally, the amount of tilt shows the amount of correlation between the two variables. \n",
     "\n",
-    "The first plot, with $R=5 m^2$, finishes up with an ellipse that is wider than it is tall. If that is not clear I have printed out the variances for the last ellipse in the lower right hand corner.\n",
+    "The first plot, with `R=5` $m^2$, finishes up with an ellipse that is wider than it is tall. If that is not clear I have printed out the variances for the last ellipse in the lower right hand corner.\n",
     "\n",
     "In contrast, the second plot, with `R=0.5` $m^2$, has a final ellipse that is taller than wide. The ellipses in the second plot are all much smaller than the ellipses in the first plot. This stands to reason because a small $\\small\\mathbf R$ implies a small amount of noise in our measurements. Small noise means accurate predictions, and thus a strong belief in our position. "
    ]
@@ -2767,7 +2765,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2846,7 +2844,7 @@
     "\n",
     "$$\\mathbf P = \\begin{bmatrix}\\mathbf R_0 & 0 \\\\0 & vel_{max}^2\\end{bmatrix}$$\n",
     "\n",
-    "The diagonal of $\\mathbf P$ contains the variance of each state variable, so we populate it with reasonable values. $\\mathbf R$ is a reasonable variance for the position, and the maximum velocity squared is a reasonable variance for the velocity. It is squared because variance is squared: $\\sigma^2$.\n",
+    "The diagonal of $\\mathbf P$ contains the variance of each state variable, so we populate it with reasonable values. $\\mathbf R_0$ is a reasonable variance for the position, and the maximum velocity squared is a reasonable variance for the velocity. It is squared because variance is squared: $\\sigma^2$.\n",
     "\n",
     "You really need to understand the domain in which you are working and initialize your filter on the best available information. For example, suppose we were trying to track horses in a horse race. The initial measurements might be very bad, and provide you with a position far from the starting gate. We know that the horse must start at the starting gate; initializing the filter to the initial measurement would lead to suboptimal results. In this scenario we would want to always initialize the Kalman filter with the starting gate position of the horse."
    ]
@@ -2889,7 +2887,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2924,7 +2922,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -2965,7 +2963,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -3005,7 +3003,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -3060,7 +3058,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
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truncated
       "text/plain": [
        "<Figure size 900x400 with 1 Axes>"
       ]
@@ -3095,7 +3093,7 @@
     "\n",
     "Multivariate Gaussians allow us to simultaneously handle multiple dimensions, both spacial and others (velocity, etc). We made a key insight: hidden variables have the ability to significantly increase the accuracy of the filter. This is possible because the hidden variables are correlated with the observed variables. \n",
     "\n",
-    "I gave an intuitive definition of *observability*. Observability was invented by Dr. Kalman for linear systems, and there is a fair amount of theory behind it. It answers the question of whether a system state can be determined by observing the system's output. For our problems this has been easy to determine, but more complicated systems may require rigorous analysis. Wikipedia's [Observability](https://en.wikipedia.org/wiki/Observability) article has an overview; if you need to learn the topic [[Grewal2008]](#References) is a good source. \n",
+    "I gave an intuitive definition of *observability*. Observability was invented by Dr. Kalman for linear systems, and there is a fair amount of theory behind it. It answers the question of whether a system state can be determined by observing the system's output. For our problems this has been easy to determine, but more complicated systems may require rigorous analysis. Wikipedia's [Observability](https://en.wikipedia.org/wiki/Observability) article has an overview; if you need to learn the topic [[2]](#References) is a good source. \n",
     "\n",
     "There is one important caveat about hidden variables. It is easy to construct a filter that produces estimates for hidden variables. I could write a filter that estimates the color of a tracked car. But there is no way to compute car color from positions, so the estimate for the color will be nonsense. The designer must verify that these variables are being estimated correctly. If you do not have a velocity sensor and yet are estimating velocity, you will need to test that the velocity estimates are correct.; do not trust that they are. For example, suppose the velocity has a periodic component to it - it looks like a sine wave. If your sample time is less than 2 times the frequency you will not be able to accurately estimate the velocity (due to Nyquist's Theorem). Imagine that the sample period is equal to the frequency of the velocity. The filter will report that the velocity is constant because it samples the system at the same point on the sin wave. \n",
     "\n",
@@ -3128,7 +3126,7 @@
     "- <A name=\"[wiki_article]\">[1]</A> 'Kalman Filters'. Wikipedia\n",
     "https://en.wikipedia.org/wiki/Kalman_filter#Details\n",
     "\n",
-    "* **[Grewal2008]** Grewal, Mohinder S., Andrews, Angus P. *Kalman Filtering: Theory and Practice Using MATLAB*. Third Edition. John Wiley & Sons. 2008."
+    "- [2] Grewal, Mohinder S., Andrews, Angus P. *Kalman Filtering: Theory and Practice Using MATLAB*. Third Edition. John Wiley & Sons. 2008."
    ]
   }
  ],