fix some minor typing errors

09-Nonlinear-Filtering.ipynb

@@ -310,7 +310,7 @@
     "* homogeneity: $f(ax) = af(x)$\n",
     "\n",
     "\n",
-    "This leads us to say that a linear system is defined as a system whose output is linearly proportional to the sum of all its inputs. A consequence of this is that to be linear if the input is zero than the output must also be zero. Consider an audio amp - if I sing into a microphone, and you start talking, the output should be the sum of our voices (input) scaled by the amplifier gain. But if amplifier outputs a nonzero signal such as a hum for a zero input the additive relationship no longer holds. This is because you linearity requires that $amp(voice) = amp(voice + 0)$ This clearly should give the same output, but if amp(0) is nonzero, then\n",
+    "This leads us to say that a linear system is defined as a system whose output is linearly proportional to the sum of all its inputs. A consequence of this is that to be linear if the input is zero than the output must also be zero. Consider an audio amp - if I sing into a microphone, and you start talking, the output should be the sum of our voices (input) scaled by the amplifier gain. But if the amplifier outputs a nonzero signal such as a hum for a zero input the additive relationship no longer holds. This is because linearity requires that $amp(voice) = amp(voice + 0)$. This clearly should give the same output, but if amp(0) is nonzero, then\n",
     "\n",
     "$$\n",
     "\\begin{aligned}\n",
@@ -633,7 +633,7 @@
    "source": [
     "Unfortunately the nonlinear version is not stable. It drifted significantly from the mean of 0, and the variance is half an order of magnitude larger.\n",
     "\n",
-    "I minimized the issue by using a function that is quite close to a straight line. What happens if the function is $y(x)=x^2$?"
+    "I minimized the issue by using a function that is quite close to a straight line. What happens if the function is $y(x)=-x^2$?"
    ]
   },
   {
@@ -792,7 +792,7 @@
    "source": [
     "That sort of error often leads to disastrous results. The error in this estimate is large. But in the next innovation of the filter that very bad estimate will be used to linearize the next radar measurement, so the next estimate is likely to be markedly worse than this one. After only a few iterations the Kalman filter will diverge, and start producing results that have no correspondence to reality.\n",
     "\n",
-    "This covariance ellipse spans miles. I exaggerated the size to illustrate the difficulties of highly nonlinear systems. In real radar tracking problems the nonlinearity is usually not that bad, but the errors will still accumulate. Other systems you may be work could have this amount of nonlinearity - this was not an exaggeration only to make a point. You will always be battling divergence when working with nonlinear systems."
+    "This covariance ellipse spans miles. I exaggerated the size to illustrate the difficulties of highly nonlinear systems. In real radar tracking problems the nonlinearity is usually not that bad, but the errors will still accumulate. Other systems you might work with could have this amount of nonlinearity - this was not an exaggeration only to make a point. You will always be battling divergence when working with nonlinear systems."
    ]
   },
   {
@@ -814,9 +814,9 @@
     "\n",
     "Recently the field has been changing in exciting ways. First, computing power has grown to the point that we can use techniques that were once beyond the ability of a supercomputer. These use *Monte Carlo* techniques - the computer generates thousands to tens of thousands of random points and tests all of them against the measurements. It then probabilistically kills or duplicates points based on how well they match the measurements. A point far away from the measurement is unlikely to be retained, whereas a point very close is quite likely to be retained. After a few iterations there is a clump of particles closely tracking your object, and a sparse cloud of points where there is no object.\n",
     "\n",
-    "This has two benefits. First, the algorithm is robust even for extremely nonlinear problems. 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",
+    "This has two benefits. First, the algorithm is robust even for extremely nonlinear problems. Second, the algorithm can track arbitrarily many objects at once - some particles will match the behavior of 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 good problem for GPUs because the part of the algorithm is parallelizable. Another cost is that 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 and standard deviations 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 good problem for GPUs because the part of the algorithm is parallelizable. Another cost is that 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 and standard deviations 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 a 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). It does not require you to find analytic solutions to nonlinear equations, and yet almost always performs better than the EKF. It does well with nonlinear problems - problems where the EKF has significant difficulties. Designing the filter is extremely easy. Some will say the jury is still out on the UKF, but to my mind the UKF is superior in almost every way to the EKF. I suggest that the UKF should be the starting point for any implementation, especially if you are not a Kalman filter professional with a graduate degree in control theory. 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. 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 filter as your starting point. "
@@ -837,15 +837,15 @@
     "\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, yet it is also 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",
+    "Recent developments have offered what are to my mind superior approaches. The UKF dispenses with the need to find solutions to partial differential equations, yet it is also 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 mathematical 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 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 to 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",
+    "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 to 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. 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. "
+    "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. 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 as 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. "
    ]
   },
   {