Removed overuse of word 'just'.

I just use it just all the time, and it is just unnecessary.

Author: rlabbe

01_g-h_filter.ipynb

@@ -404,7 +404,7 @@
     "\n",
     "where $\\sigma$ is the notation for the standard deviation and $\\mu$ is the mean.\n",
     "\n",
-    "If this is the first time you have seen this it may not have a lot of meaning for you. But let's just work through that with the data from the three classes to be sure we understand the formula. We just subtract the mean of x from each value of x, square it, take the average of those, and then take the square root of the result.  The mean of $[1.8, 2.0, 1.7, 1.9, 1.6]$ is 1.8, so we compute the standard deviation as\n",
+    "If this is the first time you have seen this it may not have a lot of meaning for you. But let's work through that with the data from the three classes to be sure we understand the formula. We subtract the mean of x from each value of x, square it, take the average of those, and then take the square root of the result.  The mean of $[1.8, 2.0, 1.7, 1.9, 1.6]$ is 1.8, so we compute the standard deviation as\n",
     "\n",
     "$$ \n",
     "\\begin{aligned}\n",
@@ -657,7 +657,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Many texts alternatively use *VAR(x)* to denote the variance of x, and I opted to use that here just to expose you to that convention."
+    "Many texts alternatively use *VAR(x)* to denote the variance of x."
    ]
   },
   {
@@ -666,7 +666,7 @@
    "source": [
     "### Why the Square of the Differences\n",
     "\n",
-    "As an aside, why are we taking the *square* of the difference? I could go into a lot of math, but let's just look at this in a simple way. Here is a chart of the values of x plotted against the mean for $x=[3,-3,3,-3]$\n"
+    "As an aside, why are we taking the *square* of the difference? I could go into a lot of math, but let's look at this in a simple way. Here is a chart of the values of x plotted against the mean for $x=[3,-3,3,-3]$\n"
    ]
   },
   {
@@ -869,7 +869,7 @@
     "\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 somewhat closely 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 choice. Even in this book Gaussians will fail to model reality, forcing us to computationally expensive alternative. \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, so I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and just talk about a *Gaussian* or *normal* - these are both typical shortcut names for the *Gaussian distribution*. "
+    "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, so 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*. "
    ]
   },
   {

02_Discrete_Bayes.ipynb

@@ -868,7 +868,7 @@
     "\n",
     "This has two benefits. First, there is no process model. You do not need to predict your system state in the next time step. This is of huge value if you do not have a good model for the behavior of the system. For example, it is hard to predict the behavior of a person in a crowd for very long. You don't know where they are going, and they are endlessly being diverted and jostled by the crowd. Second, the algorithm can track arbitrarily many objects at once - some particles will match the behavior on one object, and other particles will match other objects. So this technique is often used to track automobile traffic, people in crowds, and so on. \n",
     "\n",
-    "The costs should be clear. It is computationally expensive to test tens of thousands of points for every step in the filter. But modern CPUs are very fast, and this is a perfect problem for GPUs because the algorithm is highly parallelizable. If we do have the ability to use a process model that is important information that is just being ignored by the filter - you rarely get better results by ignoring information. Of course, there are hybrid filters that do use a process model. Then, the answer is not mathematical. With a Kalman filter my covariance matrix gives me important information about the amount of error in the estimate. The particle filter does not give me a rigorous way to compute this. Finally, the output of the filter is a cloud of points; I then have to figure out how to interpret it. Usually you will be doing something like taking the mean of the points, but this is a difficult problem. There are still many points that do not 'belong' to a tracked object, so you first have to run some sort of clustering algorithm to first find the points that seem to be tracking an object, and then you need another algorithm to produce an state estimate from those points. None of this is intractable, but it is all quite computationally expensive. \n",
+    "The costs should be clear. It is computationally expensive to test tens of thousands of points for every step in the filter. But modern CPUs are very fast, and this is a perfect problem for GPUs because the algorithm is highly parallelizable. If we do have the ability to use a process model that is important information that is being ignored by the filter - you rarely get better results by ignoring information. Of course, there are hybrid filters that do use a process model. Then, the answer is not mathematical. With a Kalman filter my covariance matrix gives me important information about the amount of error in the estimate. The particle filter does not give me a rigorous way to compute this. Finally, the output of the filter is a cloud of points; I then have to figure out how to interpret it. Usually you will be doing something like taking the mean of the points, but this is a difficult problem. There are still many points that do not 'belong' to a tracked object, so you first have to run some sort of clustering algorithm to first find the points that seem to be tracking an object, and then you need another algorithm to produce an state estimate from those points. None of this is intractable, but it is all quite computationally expensive. \n",
     "\n",
     "\n",
     "Finally, we have a new algorithm called the *unscented Kalman filter* (UKF) that does not require you to find analytic solutions to nonlinear equations, and yet almost always performs better than the EKF. It does especially well with highly nonlinear problems - problems where the EKF has significant difficulties. Designing the filter is extremely easy. Some will say the jury is still out, but to my mind the UKF is superior in almost every way to the EKF, and should be the starting point for any implementation, especially if you are not a Kalman filter professional with a graduate degree in the relevant mathematical techniques. The main downside is that the UKF can be a few times slower than the EKF, but this really depends on whether the EKF solves the Jacobian analytically or numerically. If numerically the UKF is almost certainly faster. Finally, it has not been proven (and probably it cannot be proven) that the UKF always yields more accurate results than the EKF. In practice it almost always does, often significantly so. It is very easy to understand and implement, and I strongly suggest this technique as your starting point. "
@@ -889,16 +889,21 @@
     "\n",
     "Until recently the linearized Kalman filter and EKF have been the standard way to solve these problems. They are very difficult to understand and use, and they are also potentially very unstable. \n",
     "\n",
-    "Recent developments have offered what are to my mind superior approaches. The UKF dispenses with the need to find solutions to partial differential equations, but it is usually more accurate than the EKF. It is easy to use and understand. I can get a basic UKF going in just a few minutes by using FilterPy. The particle filter dispenses with mathimatical modeling completely in favor of a Monte Carlo technique of generating a random cloud of thousands of points. It runs slowly, but it can solve otherwise intractable problems with relative ease.\n",
+    "Recent developments have offered what are to my mind superior approaches. The UKF dispenses with the need to find solutions to partial differential equations, but it is usually more accurate than the EKF. It is easy to use and understand. I can get a basic UKF going in a few minutes by using FilterPy. The particle filter dispenses with mathimatical modeling completely in favor of a Monte Carlo technique of generating a random cloud of thousands of points. It runs slowly, but it can solve otherwise intractable problems with relative ease.\n",
     "\n",
-    "I get more email about the EKF than anything else; I suspect that this is because most treatments in books, papers, and on the internet use the EKF. If your interest is in mastering the field of course you will want to learn about the EKF. But if you are just trying to get good results I point you to the UKF and particle filter first. They are so much easier to implement, understand, and use, and they are typically far more stable than the EKF. \n",
+    "I get more email about the EKF than anything else; I suspect that this is because most treatments in books, papers, and on the internet use the EKF. If your interest is in mastering the field of course you will want to learn about the EKF. But if you are just trying to get good results I point you to the UKF and particle filter first. They are much easier to implement, understand, and use, and they are typically far more stable than the EKF. \n",
     "\n",
     "Some will quibble with that advice. A lot of recent publications are devoted to a comparison of the EKF, UKF, and perhaps a few other choices for a given problem. Do you not need to perform a similar comparison for your problem? If you are sending a rocket to Mars, then of course you do. You will be balancing issues such as accuracy, round off errors, divergence, mathematical proof of correctness, and the computational effort required. I can't imagine not knowing the EKF intimately. \n",
     "\n",
     "On the other hand the UKF works spectacularly! I use it at work for real world applications. I mostly haven't even tried to implement an EKF for these applications because I can verify that the UKF is working fine. Is it possible that I might eke out another 0.2% of performance from the EKF in certain situations? Sure! Do I care? No! I completely understand the UKF implementation, it is easy to test and verify, I can pass the code to others and be confident that they can understand and modify it, and I am not a masochist that wants to battle difficult equations when I already have a working solution. If the UKF or particle filters start to perform poorly for some problem then I will turn other techniques, but not before then. And realistically, the UKF usually provides substantially better performance than the EKF over a wide range of problems and conditions. If \"really good\" is good enough I'm going to spend my time working on other problems. \n",
     "\n",
-    "I'm belaboring this point because in most textbooks the EKF is given center stage, and the UKF is either not mentioned at all or just given a 2 page gloss that leaves you completely unprepared to use the filter. This is not due to ignorance on the writer's part. The UKF is still relatively new, and it takes time to write new editions of books. At the time many books were written the UKF was either not discovered yet, or it was just an unproven but promising curiosity. But I am writing this now, the UKF has had enormous success, and it needs to be in your toolkit. That is what I will spend most of my effort trying to teach you. \n",
-    "\n",
+    "I'm belaboring this point because in most textbooks the EKF is given center stage, and the UKF is either not mentioned at all or just given a 2 page gloss that leaves you completely unprepared to use the filter. This is not due to ignorance on the writer's part. The UKF is still relatively new, and it takes time to write new editions of books. At the time many books were written the UKF was either not discovered yet, or it was just an unproven but promising curiosity. But I am writing this now, the UKF has had enormous success, and it needs to be in your toolkit. That is what I will spend most of my effort trying to teach you. "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
     "## References\n",
     "\n",
     "[1] https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Supporting_Notebooks/Computing_and_plotting_PDFs.ipynb"