Added link to SIAM News article on EnKF.

11_Ensemble_Kalman_Filter/Ensemble_Kalman_Filters.ipynb

@@ -273,9 +273,10 @@
      "source": [
       "> I am not well versed with Ensemble filters. I have implemented one for this book, and made it work, but I have not used one in real life. Different sources use slightly different forms of these equations. If I implement the equations given in the sources the filter does not work. It is possible that I am doing something wrong. However, in various places on the web I have seen comments by people stating that they do the kinds of things I have done in my filter to make it work. In short, I do not understand this topic well, but choose to present my lack of knowledge rather than to gloss over the subject. I hope to master this topic in the future and to author a more definitive chapter. At the end of the chapter I document my current confusion and questions. In any case if I got confused by the sources perhaps you will to, so documenting my confusion can help you avoid the same.\n",
       "\n",
+      "\n",
       "The ensemble Kalman filter (EnKF) is very similar to the unscented Kalman filter (UKF) of the last chapter. If you recall, the UKF uses a set of deterministically chosen weighted sigma points passed through nonlinear state and measurement functions. After the sigma points are passed through the function, we find the mean and covariance of the points and use this as the filter's new mean and covariance. It is only an approximation of the true value, and thus suboptimal, but in practice the filter is highly accurate. It has the advantage of often producing more accurate estimates than the EKF does, and also does not require you to analytically derive the linearization of the state and measurement equations. \n",
       "\n",
-      "The ensemble Kalman filter works in a similar way, except it uses a *Monte Carlo* method to choose a large numbers of sigma points. It came about from the geophysical sciences as an answer for the very large states and systems needed to model things such as the ocean. The filter starts by randomly generating a large number of points distributed about the filter's initial state. This distribution is proportional to the filter's covariance $\\mathbf{P}$. In other words 68% of the points will be within one standard deviation of the mean, 95% percent within two standard deviations, and so on. Let's look at this in two dimensions. We will use `numpy.random.multivariate_normal()` function to randomly create points from a multivariate normal distribution drawn from the mean (5, 3) with the covariance\n",
+      "The ensemble Kalman filter works in a similar way, except it uses a *Monte Carlo* method to choose a large numbers of sigma points. It came about from the geophysical sciences as an answer for the very large states and systems needed to model things such as the ocean and atmosphere. There is an interesting article on it's development in weather modeling in *SIAM News* [1]. The filter starts by randomly generating a large number of points distributed about the filter's initial state. This distribution is proportional to the filter's covariance $\\mathbf{P}$. In other words 68% of the points will be within one standard deviation of the mean, 95% percent within two standard deviations, and so on. Let's look at this in two dimensions. We will use `numpy.random.multivariate_normal()` function to randomly create points from a multivariate normal distribution drawn from the mean (5, 3) with the covariance\n",
       "\n",
       "$$\\begin{bmatrix}\n",
       "32 & 15 \\\\ 15 & 40\n",
@@ -620,11 +621,11 @@
       "\n",
       "All of this should be considered as *my* questions, not lingering questions in the literature. However, I am copying equations directly from well known sources in the field, and they do not address the discrepencies.\n",
       "\n",
-      "First, in Brown [1] we have all sums multipied by $\\frac{1}{N}$, as in \n",
+      "First, in Brown [2] we have all sums multipied by $\\frac{1}{N}$, as in \n",
       "\n",
       "$$ \\hat{x} = \\frac{1}{N}\\sum_{i=1}^N\\chi_k^{(i)}$$\n",
       "\n",
-      "The same equation in Crassidis [2] reads (I'll use the same notation as in Brown, although Crassidis' is different)\n",
+      "The same equation in Crassidis [3] reads (I'll use the same notation as in Brown, although Crassidis' is different)\n",
       "\n",
       "$$ \\hat{x} = \\frac{1}{N-1}\\sum_{i=1}^N\\chi_k^{(i)}$$\n",
       "\n",
@@ -656,9 +657,11 @@
       "I am not comfortable saying either book is wrong; it is quite possible that I missed some point that makes each set of equations work. I can say that when I implemented them as written I did not get a filter that worked. I define \"work\" as performs essentially the same as the linear KF for linear problems. Between reading implementation notes on the web and reasoning about various issues I have chosen the implementation in this chapter, which does in fact seem to work correctly. I have yet to explore the significant amount of original literature that will likely definitively explain the discrepencies. I would like to leave this here in some form even if I do find an explanation that reconciles the various differences, as if I got confused by these books than probably others will as well.\n",
       "\n",
       "## References\n",
-      "- [1] Crassidis, John L., and John L. Junkins. *Optimal estimation of dynamic systems*. CRC press, 2011.\n",
+      "- [1] Mackenzie, Dana. *Ensemble Kalman Filters Bring Weather Models Up to Date* Siam News,  Volume 36, Number 8, October 2003. http://www.siam.org/pdf/news/362.pdf\n",
       "\n",
-      "- [2]  Brown, Robert Grover, and Patrick Y.C. Hwang. *Introduction to Random Signals and Applied Kalman Filtering, With MATLAB\u00ae excercises and solutions.* Wiley, 2012."
+      "- [2]  Brown, Robert Grover, and Patrick Y.C. Hwang. *Introduction to Random Signals and Applied Kalman Filtering, With MATLAB\u00ae excercises and solutions.* Wiley, 2012.\n",
+      "\n",
+      "- [3] Crassidis, John L., and John L. Junkins. *Optimal estimation of dynamic systems*. CRC press, 2011."
      ]
     }
    ],