Typos. github issue #93.

02-Discrete-Bayes.ipynb

@@ -4709,7 +4709,7 @@
     " \n",
     "One cycle of prediction and updating with a measurement is called the state or system *evolution*, which is short for *time evolution* [7]. Another term is *system propogation*. It refers to how the state of the system changes over time. For filters, time is usually a discrete step, such as 1 second. For our dog tracker the system state is the position of the dog, and the state evolution is the position after a discrete amount of time has passed.\n",
     "\n",
-    "We model the system behavior with the *process model*. Here, our process model is that the dog moves one or more positions at each time step. This is not a particularly accurate model of how dog's behave. The error in the model is called the *system error* or *process error*. \n",
+    "We model the system behavior with the *process model*. Here, our process model is that the dog moves one or more positions at each time step. This is not a particularly accurate model of how dogs behave. The error in the model is called the *system error* or *process error*. \n",
     "\n",
     "The prediction is our new *prior*. Time has moved forward and we made a prediction without benefit of knowing the measurements. \n",
     "\n",
@@ -12892,7 +12892,7 @@
     "\n",
     "    1. Get a measurement and associated belief about its accuracy\n",
     "    2. Compute residual between estimated state and measurement\n",
-    "    3. Determine whether whether the measurement matches each state\n",
+    "    3. Determine whether the measurement matches each state\n",
     "    4. Update state belief if it matches the measurement\n",
     "\n",
     "When we cover the Kalman filter we will use this exact same algorithm; only the details of the computation will differ. \n",

03-Gaussians.ipynb

@@ -5596,7 +5596,7 @@
    "source": [
     "The y-axis depicts the *probability density* — the relative amount of cars that are going the speed at the corresponding x-axis.\n",
     "\n",
-    "You may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $-\\infty$. This is true, but this is a common limitation of mathematical modeling. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternative. \n",
+    "You may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $-\\infty$. This is true, but this is a common limitation of mathematical modeling. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternatives. \n",
     "\n",
     "You will see these distributions called *Gaussian distributions* or *normal distributions*.  *Gaussian* and *normal* both mean the same thing in this context, and are used interchangeably. I will use both throughout this book as different sources will use either term, and I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and talk about a *Gaussian* or *normal* — these are both typical shortcut names for the *Gaussian distribution*. "
    ]

04-One-Dimensional-Kalman-Filters.ipynb

@@ -4626,7 +4626,7 @@
    "source": [
     "If you ask two people to measure the distance of a table from a wall, and one gets 10.2 meters, and the other got 9.7 meters, your best guess must be the average, 9.95 meters if you trust the skills of both equally.\n",
     "\n",
-    "Recall the g-h filter. We agreed that if I weighed myself on two scales, and the first read 160 lbs while the second read 170 lbs, and both were equally accurate, the best estimate was 165 lbs. Furthermore I should be a bit more confident about 165 lbs vs 160 lbs or 170 lbs because I know have two readings, both near this estimate, increasing my confidence that neither is wildly wrong. \n",
+    "Recall the g-h filter. We agreed that if I weighed myself on two scales, and the first read 160 lbs while the second read 170 lbs, and both were equally accurate, the best estimate was 165 lbs. Furthermore I should be a bit more confident about 165 lbs vs 160 lbs or 170 lbs because I now have two readings, both near this estimate, increasing my confidence that neither is wildly wrong. \n",
     "\n",
     "This becomes counter-intuitive in more complicated situations, so let's consider it further. Perhaps a more reasonable assumption would be that one person made a mistake, and the true distance is either 10.2 or 9.7, but certainly not 9.95. Surely that is possible. But we know we have noisy measurements, so we have no reason to think one of the measurements has no noise, or that one person made a gross error that allows us to discard their measurement. Given all available information, the best estimate must be 9.95.\n",
     "\n",
@@ -8951,7 +8951,7 @@
    "source": [
     "### Exercise: Modify Variance Values\n",
     "\n",
-    "Modify the values of `process_var` and `sensor_var` and note the effect on the filter and on the variance. Which has a larger effect on the variance convergence?. For example, which results in a smaller variance:\n",
+    "Modify the values of `process_var` and `sensor_var` and note the effect on the filter and on the variance. Which has a larger effect on the variance convergence? For example, which results in a smaller variance:\n",
     "\n",
     "```python\n",
     "process_var = 40\n",
@@ -8977,7 +8977,7 @@
     "\n",
     "The top plot shows the output of the filter in green, and the measurements with a dashed red line. The bottom plot shows the Gaussian at each step. \n",
     "\n",
-    "When the track first starts you can see that the measurements varies quite a bit from the initial prediction. At this point the Gaussian probability is small (the curve is low and wide) so the filter does not trust its prediction. As a result, the filter adjusts its estimate a large amount. As the filter innovates you can see that as the Gaussian becomes taller, indicating greater certainty in the estimate, the filter's output becomes very close to a straight line. At `x = 15` and greater you can see that there is a large amount of noise in the measurement, but the filter does not react much to it compared to how much it changed for the firs noisy measurement."
+    "When the track first starts you can see that the measurements varies quite a bit from the initial prediction. At this point the Gaussian probability is small (the curve is low and wide) so the filter does not trust its prediction. As a result, the filter adjusts its estimate a large amount. As the filter innovates you can see that as the Gaussian becomes taller, indicating greater certainty in the estimate, the filter's output becomes very close to a straight line. At `x = 15` and greater you can see that there is a large amount of noise in the measurement, but the filter does not react much to it compared to how much it changed for the first noisy measurement."
    ]
   },
   {
@@ -10703,7 +10703,7 @@
    "source": [
     "## Introduction to Designing a Filter\n",
     "\n",
-    "So far we have developed filters for a position sensor. We are used to this problem by now, and may feel ill-equipped to implement a Kalman filter for a different problem. To be honest, there is still quite a bit of information missing from this presentation. Following chapters will fill in the gaps. Still, lets get a feel for it by designing and implementing a Kalman filter for a thermometer. The sensor for the thermometer outputs a voltage that corresponds to the temperature that is being measured. We have read the manufacturer's specifications for the sensor, and it tells us that the sensor exhibits white noise with a standard deviation of 0.13 volts.\n",
+    "So far we have developed filters for a position sensor. We are used to this problem by now, and may feel ill-equipped to implement a Kalman filter for a different problem. To be honest, there is still quite a bit of information missing from this presentation. Following chapters will fill in the gaps. Still, let's get a feel for it by designing and implementing a Kalman filter for a thermometer. The sensor for the thermometer outputs a voltage that corresponds to the temperature that is being measured. We have read the manufacturer's specifications for the sensor, and it tells us that the sensor exhibits white noise with a standard deviation of 0.13 volts.\n",
     "\n",
     "We can simulate the temperature sensor measurement with this function:"
    ]
@@ -12381,7 +12381,7 @@
    "source": [
     "## Example: Extreme Amounts of Noise\n",
     "\n",
-    "With the dog filter I didn't put a lot of noise in the signal, and I 'guessed' that the dog was at position 0. How does the filter perform in real world conditions? I will start by injecting more noise in the RFID sensor while leaving the process variance at 2 m$^2$.  I will inject an extreme amount of noise - noise that apparently swamps the actual measurement. What does your intuition say about the filter's performance if the sensor has a standard deviation of 300 meters?. In other words, an actual position of 1.0 m might be reported as 287.9 m, or -589.6 m, or any other number in roughly that range. Think about it before you scroll down."
+    "With the dog filter I didn't put a lot of noise in the signal, and I 'guessed' that the dog was at position 0. How does the filter perform in real world conditions? I will start by injecting more noise in the RFID sensor while leaving the process variance at 2 m$^2$.  I will inject an extreme amount of noise - noise that apparently swamps the actual measurement. What does your intuition say about the filter's performance if the sensor has a standard deviation of 300 meters? In other words, an actual position of 1.0 m might be reported as 287.9 m, or -589.6 m, or any other number in roughly that range. Think about it before you scroll down."
    ]
   },
   {

05-Multivariate-Gaussians.ipynb

@@ -4608,9 +4608,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "This is a plot of multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}10&0\\\\0&4\\end{smallmatrix}]$. The three dimensional shape shows the probability density of for any value of ($X, Y$) in the z-axis. I have projected the variance for x and y onto the walls of the chart - you can see that they take on the Gaussian bell curve shape. The curve for $X$ is wider than the curve for $Y$, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. The highest point of the 3D surface is at the the means for $X$ and $Y$. \n",
+    "This is a plot of multivariate Gaussian with a mean of $\\mu=[\\begin{smallmatrix}2\\\\17\\end{smallmatrix}]$ and a covariance of $\\Sigma=[\\begin{smallmatrix}10&0\\\\0&4\\end{smallmatrix}]$. The three dimensional shape shows the probability density for any value of ($X, Y$) in the z-axis. I have projected the variance for x and y onto the walls of the chart - you can see that they take on the Gaussian bell curve shape. The curve for $X$ is wider than the curve for $Y$, which is explained by $\\sigma_x^2=10$ and $\\sigma_y^2=4$. The highest point of the 3D surface is at the the means for $X$ and $Y$. \n",
     "\n",
-    "All multivariate Gaussians have this shape. If we think of this as a the Gaussian for the position of a dog, the z-value at each point of ($X, Y$) is the probability density of the dog being at that position. Strictly speaking this is the *joint probability density function*, which I will define soon. So, the dog has the highest probability of being near (2, 17), a modest probability of being near (5, 14), and a very low probability of being near (10, 10). As with the univariate case this is a *probability density*, not a *probability*. Continuous distributions have an infinite range, and so the probability of being exactly at (2, 17), or any other point, is 0%. We can compute the probability of being within a given range by computing the area under the curve with an integral."
+    "All multivariate Gaussians have this shape. If we think of this as the Gaussian for the position of a dog, the z-value at each point of ($X, Y$) is the probability density of the dog being at that position. Strictly speaking this is the *joint probability density function*, which I will define soon. So, the dog has the highest probability of being near (2, 17), a modest probability of being near (5, 14), and a very low probability of being near (10, 10). As with the univariate case this is a *probability density*, not a *probability*. Continuous distributions have an infinite range, and so the probability of being exactly at (2, 17), or any other point, is 0%. We can compute the probability of being within a given range by computing the area under the curve with an integral."
    ]
   },
   {