Appendix-E-Ensemble-Kalman-Filters.ipynb

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    "[Table of Contents](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/table_of_contents.ipynb)"
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   "source": [
    "# Ensemble Kalman Filters"
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   "source": [
    "#format the book\n",
    "%matplotlib inline\n",
    "%load_ext autoreload\n",
    "%autoreload 2  \n",
    "from __future__ import division, print_function\n",
    "import sys\n",
    "sys.path.insert(0,'./code')\n",
    "from book_format import load_style\n",
    "load_style()"
   ]
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   "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 also will, 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 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",
    "$$\\begin{bmatrix}\n",
    "32 & 15 \\\\ 15 & 40\n",
    "\\end{bmatrix}$$\n",
    "\n",
    "I've drawn the covariance ellipse representing two standard deviations to illustrate how the points are distributed."
   ]
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+x+Kr+fNPXcfy5S7V1e8AsHLlcpYsCUZNpPR9TW8v3H67P1jfrYhGDb4/hY4O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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x6adad68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from numpy.random import multivariate_normal\n",
    "from filterpy.stats import (covariance_ellipse, \n",
    "                            plot_covariance_ellipse)\n",
    "\n",
    "mean = (5, 3)\n",
    "P = np.array([[32, 15],\n",
    "              [15., 40.]])\n",
    "\n",
    "x,y = multivariate_normal(mean=mean, cov=P, size=2500).T\n",
    "plt.scatter(x, y, alpha=0.3, marker='.')\n",
    "plt.axis('equal')\n",
    "\n",
    "plot_covariance_ellipse(mean=mean, cov=P,\n",
    "                        variance=2.**2,\n",
    "                        facecolor='none')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Algorithm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As I already stated, when the filter is initialized a large number of sigma points are drawn from the initial state ($\\mathbf{x}$) and covariance ($\\mathbf{P}$). From there the algorithm proceeds very similarly to the UKF. During the prediction step the sigma points are passed through the state transition function, and then perturbed by adding a bit of noise to account for the process noise. During the update step the sigma points are translated into measurement space by passing them through the measurement function, they are perturbed by a small amount to account for the measurement noise. The Kalman gain is computed from the \n",
    "\n",
    "We already mentioned the main difference between the UKF and EnKF - the UKF choses the sigma points deterministically. There is another difference, implied by the algorithm above. With the UKF we generate new sigma points during each predict step, and after passing the points through the nonlinear function we reconstitute them into a mean and covariance by using the *unscented transform*. The EnKF keeps propagating the originally created sigma points; we only need to compute a mean and covariance as outputs for the filter! \n",
    "\n",
    "Let's look at the equations for the filter. As usual, I will leave out the typical subscripts and superscripts; I am expressing an algorithm, not mathematical functions. Here $N$ is the number of sigma points, $\\chi$ is the set of sigma points."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Initialize Step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\require{autobold}\\boldsymbol\\chi \\sim \\mathcal{N}(\\mathbf{x}_0, \\mathbf{P}_0)$$\n",
    "\n",
    "This says to select the sigma points from the filter's initial mean and covariance. In code this might look like\n",
    "\n",
    "```python\n",
    "N = 1000\n",
    "sigmas = multivariate_normal(mean=x, cov=P, size=N)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Predict Step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "\\boldsymbol\\chi &= f(\\boldsymbol\\chi, \\mathbf{u}) + v_Q \\\\\n",
    "\\mathbf{x} &= \\frac{1}{N} \\sum_1^N \\boldsymbol\\chi\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "That is short and sweet, but perhaps not entirely clear. The first line passes all of the sigma points through a use supplied state transition function and then adds some noise distributed according to the $\\mathbf{Q}$ matrix. In Python we might write\n",
    "\n",
    "```python\n",
    "for i, s in enumerate(sigmas):\n",
    "    sigmas[i] = fx(x=s, dt=0.1, u=0.)\n",
    "\n",
    "sigmas += multivariate_normal(x, Q, N)\n",
    "```\n",
    "\n",
    "The second line computes the mean from the sigmas. In Python we will take advantage of `numpy.mean` to do this very concisely and quickly.\n",
    "\n",
    "```python\n",
    "x = np.mean(sigmas, axis=0)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now optionally compute the covariance of the mean. The algorithm does not need to compute this value, but it is often useful for analysis. The equation is\n",
    "\n",
    "$$\\mathbf{P} = \\frac{1}{N-1}\\sum_1^N[\\boldsymbol\\chi-\\mathbf{x}^-][\\boldsymbol\\chi-\\mathbf{x}^-]^\\mathsf{T}$$\n",
    "\n",
    "$\\boldsymbol\\chi-\\mathbf{x}^-$ is a one dimensional vector, so we will use `numpy.outer` to compute the $[\\boldsymbol\\chi-\\mathbf{x}^-][\\boldsymbol\\chi-\\mathbf{x}^-]^\\mathsf{T}$ term. In Python we might write\n",
    "\n",
    "```python\n",
    "    P = 0\n",
    "    for s in sigmas:\n",
    "        P += outer(s-x, s-x)\n",
    "    P = P / (N-1)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Update Step"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the update step we pass the sigma points through the measurement function, compute the mean and covariance of the sigma points, compute the Kalman gain from the covariance, and then update the Kalman state by scaling the residual by the Kalman gain. The equations are\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\boldsymbol\\chi_h &= h(\\boldsymbol\\chi, u)\\\\\n",
    "\\mathbf{z}_{mean} &= \\frac{1}{N}\\sum_1^N \\boldsymbol\\chi_h \\\\ \\\\\n",
    "\\mathbf{P}_{zz} &= \\frac{1}{N-1}\\sum_1^N [\\boldsymbol\\chi_h - \\mathbf{z}_{mean}][\\boldsymbol\\chi_h - \\mathbf{z}_{mean}]^\\mathsf{T} + \\mathbf{R} \\\\\n",
    "\\mathbf{P}_{xz} &= \\frac{1}{N-1}\\sum_1^N [\\boldsymbol\\chi - \\mathbf{x}^-][\\boldsymbol\\chi_h - \\mathbf{z}_{mean}]^\\mathsf{T} \\\\\n",
    "\\\\\n",
    "\\mathbf{K} &= \\mathbf{P}_{xz} \\mathbf{P}_{zz}^{-1}\\\\ \n",
    "\\boldsymbol\\chi & = \\boldsymbol\\chi + \\mathbf{K}[\\mathbf{z} -\\boldsymbol\\chi_h + \\mathbf{v}_R] \\\\ \\\\\n",
    "\\mathbf{x} &= \\frac{1}{N} \\sum_1^N \\boldsymbol\\chi \\\\\n",
    "\\mathbf{P} &= \\mathbf{P} - \\mathbf{KP}_{zz}\\mathbf{K}^\\mathsf{T}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is very similar to the linear KF and the UKF. Let's just go line by line.\n",
    "\n",
    "The first line,\n",
    "\n",
    "$$\\boldsymbol\\chi_h = h(\\boldsymbol\\chi, u),$$\n",
    "\n",
    "just passes the sigma points through the measurement function $h$. We name the resulting points $\\chi_h$ to distinguish them from the sigma points. In Python we could write this as\n",
    "\n",
    "```python\n",
    "sigmas_h = h(sigmas, u)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next line computes the mean of the measurement sigmas.\n",
    "\n",
    "$$\\mathbf{z}_{mean} = \\frac{1}{N}\\sum_1^N \\boldsymbol\\chi_h$$\n",
    "\n",
    "In Python we write\n",
    "\n",
    "```python\n",
    "z_mean = np.mean(sigmas_h, axis=0)\n",
    "```\n",
    "    \n",
    "Now that we have the mean of the measurement sigmas we can compute the covariance for every measurement sigma point, and the *cross variance* for the measurement sigma points vs the sigma points. That is expressed by these two equations\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathbf{P}_{zz} &= \\frac{1}{N-1}\\sum_1^N [\\boldsymbol\\chi_h - \\mathbf{z}_{mean}][\\boldsymbol\\chi_h - \\mathbf{z}_{mean}]^\\mathsf{T} + \\mathbf{R} \\\\\n",
    "\\mathbf{P}_{xz} &= \\frac{1}{N-1}\\sum_1^N [\\boldsymbol\\chi - \\mathbf{x}^-][\\boldsymbol\\chi_h - \\mathbf{z}_{mean}]^\\mathsf{T}\n",
    "\\end{aligned}$$\n",
    "\n",
    "We can express this in Python with\n",
    "\n",
    "```python\n",
    "P_zz = 0\n",
    "for sigma in sigmas_h:\n",
    "    s = sigma - z_mean\n",
    "    P_zz += outer(s, s)\n",
    "P_zz = P_zz / (N-1) + R\n",
    "\n",
    "P_xz = 0\n",
    "for i in range(N):\n",
    "    P_xz += outer(self.sigmas[i] - self.x, sigmas_h[i] - z_mean)\n",
    "P_xz /= N-1\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computation of the Kalman gain is straightforward $\\mathbf{K} = \\mathbf{P}_{xz} \\mathbf{P}_{zz}^{-1}$.\n",
    "\n",
    "In Python this is\n",
    "\n",
    "```python\n",
    "K = np.dot(P_xz, inv(P_zz))```\n",
    "\n",
    "Next, we update the sigma points with\n",
    "\n",
    "$$\\boldsymbol\\chi  = \\boldsymbol\\chi + \\mathbf{K}[\\mathbf{z} -\\boldsymbol\\chi_h + \\mathbf{v}_R]$$ \n",
    "\n",
    "Here $\\mathbf{v}_R$ is the perturbation that we add to the sigmas. In Python we can implement this with\n",
    "\n",
    "```python\n",
    "v_r = multivariate_normal([0]*dim_z, R, N)\n",
    "for i in range(N):\n",
    "    sigmas[i] += dot(K, z + v_r[i] - sigmas_h[i])\n",
    "```\n",
    "\n",
    "\n",
    "Our final step is recompute the filter's mean and covariance.\n",
    "\n",
    "```python\n",
    "    x = np.mean(sigmas, axis=0)\n",
    "    P = self.P - dot3(K, P_zz, K.T)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementation and Example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I have implemented an EnKF in the `FilterPy` library. It is in many ways a toy. Filtering with a large number of sigma points gives us very slow performance. Furthermore, there are many minor variations on the algorithm in the literature. I wrote this mostly because I was interested in learning a bit about the filter. I have not used it for a real world problem, and I can give no advice on using the filter for the large problems for which it is suited. Therefore I will refine my comments to implementing a very simple filter. I will use it to track an object in one dimension, and compare the output to a linear Kalman filter. This is a filter we have designed many times already in this book, so I will design it with little comment. Our state vector will be\n",
    "\n",
    "$$\\mathbf{x} = \\begin{bmatrix}x\\\\ \\dot{x}\\end{bmatrix}$$\n",
    "\n",
    "The state transition function is\n",
    "\n",
    "$$\\mathbf{F} = \\begin{bmatrix}1&1\\\\0&1\\end{bmatrix}$$\n",
    "\n",
    "and the measurement function is\n",
    "\n",
    "$$\\mathbf{H} = \\begin{bmatrix}1&0\\end{bmatrix}$$\n",
    "\n",
    "The EnKF is designed for nonlinear problems, so instead of using matrices to implement the state transition and measurement functions you will need to supply Python functions. For this problem they can be written as:\n",
    "\n",
    "```python\n",
    "def hx(x):\n",
    "    return np.array([x[0]])\n",
    "\n",
    "def fx(x, dt):\n",
    "    return np.dot(F, x)\n",
    "```\n",
    "\n",
    "One final thing: the EnKF code, like the UKF code, uses a single dimension for $\\mathbf{x}$, not a two dimensional column matrix as used by the linear kalman filter code.\n",
    "\n",
    "Without further ado, here is the code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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V2w434WBaS85kJ+J0eWa3VSqJhGab6dhqBZEhqTQLi6BDXG9aRzfHbLryTZPX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SQH68efw6wnMAbGorpdKFV2lICGk2hVVfGu+YHq3K24RzTvpjHtz/vxv+15dJm+lRRViwnXwG5U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\nbwAAqMAPmgLIcozeIMoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x78dcc18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy.random import randn\n",
    "from filterpy.kalman import EnsembleKalmanFilter as EnKF\n",
    "from filterpy.kalman import KalmanFilter\n",
    "from filterpy.common import Q_discrete_white_noise\n",
    "import book_plots as bp\n",
    "\n",
    "np.random.seed(1234)\n",
    "\n",
    "def hx(x):\n",
    "    return np.array([x[0]])\n",
    "\n",
    "def fx(x, dt):\n",
    "    return np.dot(F, x)\n",
    "    \n",
    "F = np.array([[1., 1.],[0., 1.]])\n",
    "\n",
    "x = np.array([0., 1.])\n",
    "P = np.eye(2) * 100.\n",
    "enf = EnKF(x=x, P=P, dim_z=1, dt=1., N=20, hx=hx, fx=fx)\n",
    "\n",
    "std_noise = 10.\n",
    "enf.R *= std_noise**2\n",
    "enf.Q = Q_discrete_white_noise(2, 1., .001)\n",
    "\n",
    "kf = KalmanFilter(dim_x=2, dim_z=1)\n",
    "kf.x = np.array([x]).T\n",
    "kf.F = F.copy()\n",
    "kf.P = P.copy()\n",
    "kf.R = enf.R.copy()\n",
    "kf.Q = enf.Q.copy()\n",
    "kf.H = np.array([[1., 0.]])\n",
    "\n",
    "measurements = []\n",
    "results = []\n",
    "ps = []\n",
    "kf_results = []\n",
    "\n",
    "zs = []\n",
    "for t in range (0,100):\n",
    "    # create measurement = t plus white noise\n",
    "    z = t + randn()*std_noise\n",
    "    zs.append(z)\n",
    "\n",
    "    enf.predict()\n",
    "    enf.update(np.asarray([z]))\n",
    "    \n",
    "    kf.predict()\n",
    "    kf.update(np.asarray([[z]]))\n",
    "\n",
    "    # save data\n",
    "    results.append (enf.x[0])\n",
    "    kf_results.append (kf.x[0,0])\n",
    "    measurements.append(z)\n",
    "    ps.append(3*(enf.P[0,0]**.5))\n",
    "\n",
    "results = np.asarray(results)\n",
    "ps = np.asarray(ps)\n",
    "\n",
    "plt.plot(results, label='EnKF')\n",
    "plt.plot(kf_results, label='KF', c='b', lw=2)\n",
    "bp.plot_measurements(measurements)\n",
    "plt.plot (results - ps, c='k',linestyle=':', lw=1, label='1$\\sigma$')\n",
    "plt.plot(results + ps, c='k', linestyle=':', lw=1)\n",
    "plt.fill_between(range(100), results - ps, results + ps, facecolor='y', alpha=.3)\n",
    "plt.legend(loc='best');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It can be a bit difficult to see, but the KF and EnKF start off slightly different, but soon converge to producing nearly the same values. The EnKF is a suboptimal filter, so it will not produce the optimal solution that the KF produces. However, I deliberately chose $N$ to be quite small (20) to guarantee that the EnKF output is quite suboptimal. If I chose a more reasonable number such as 2000 you would be unable to see the difference between the two filter outputs on this graph."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Outstanding Questions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "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 discrepancies.\n",
    "\n",
    "First, in Brown [2] we have all sums multiplied 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 [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",
    "The same is true in both sources for the sums in the computation for the covariances. Crassidis, in the context of talking about the filter's covariance, states that $N-1$ is used to ensure an unbiased estimate. Given the following standard equations for the mean and standard deviation (p.2 of Crassidis), this makes sense for the covariance.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mu &= \\frac{1}{N}\\sum_{i=1}^N[\\tilde{z}(t_i) - \\hat{z}(t_i)] \\\\\n",
    " \\sigma^2 &= \\frac{1}{N-1}\\sum_{i=1}^N\\{[\\tilde{z}(t_i) - \\hat{z}(t_i)] - \\mu\\}^2\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "However, I see no justification or reason to use $N-1$ to compute the mean. If I use $N-1$ in the filter for the mean the filter does not converge and the state essentially follows the measurements without any filtering. However, I do see a reason to use it for the covariance as in Crassidis, in contrast to Brown. Again, I support my decision empirically - $N-1$ works in the implementation of the filter, $N$ does not.\n",
    "\n",
    "My second question relates to the use of the $\\mathbf{R}$ matrix. In Brown $\\mathbf{R}$ is added to $\\mathbf{P}_{zz}$ whereas it isn't in Crassidis and other sources. I have read on the web notes by other implementers that adding R helps the filter, and it certainly seems reasonable and necessary to me, so this is what I do. \n",
    "\n",
    "My third question relates to the computation of the covariance $\\mathbf{P}$. Again, we have different equations in Crassidis and Brown. I have chosen the implementation given in Brown as it seems to give me the  behavior that I expect (convergence of $\\mathbf{P}$ over time) and it closely compares to the form in the linear KF. In contrast I find the equations in Crassidis do not seem to converge much.\n",
    "\n",
    "My fourth question relates to the state estimate update. In Brown we have \n",
    "\n",
    "$$\\boldsymbol\\chi  = \\boldsymbol\\chi + \\mathbf{K}[\\mathbf{z} -\\mathbf{z}_{mean} + \\mathbf{v}_R]$$ \n",
    "\n",
    "whereas in Crassidis we have\n",
    "\n",
    "$$\\boldsymbol\\chi  = \\boldsymbol\\chi + \\mathbf{K}[\\mathbf{z} -\\boldsymbol\\chi_h + \\mathbf{v}_R]$$ \n",
    "\n",
    "To me the Crassidis equation seems logical, and it produces a filter that performs like the linear KF for linear problems, so that is the formulation that I have chosen. \n",
    "\n",
    "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 discrepancies. 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."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- [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® excercises and solutions.* Wiley, 2012.\n",
    "\n",
    "- [3] Crassidis, John L., and John L. Junkins. *Optimal estimation of dynamic systems*. CRC press, 2011."
   ]
  }
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