Intro_ParticleFilter.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to Particle Filter\n",
    "** AOE 5984 Final Project**<br/>\n",
    "** Man Jun Koh ** <br />\n",
    "** Kevin T. Croft Aerospace and Ocean Engineering, Virginia Tech**\n",
    "\n",
    "This is a simple documentation/tutorial via Jupyter Notebook to show the basics of particle filter and its derivations. Particle filter is an advanced method for estimating non-linear and non-Gaussian systems since filtering method such as Kalman filter does not work in those cases. <br />\n",
    "\n",
    "**Note:** Make sure to run the cells chronically otherwise the code won't work or the plots will look extremely unproportional. The best way I found was to navigate to the cell you want to run, then click `Run All Above` from the `'Cell' tab`. then run your cell `(Ctr + Enter)`. \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Motivation: Why Particle Filter?**\n",
    "\n",
    "It has been over 50 years since Kalman filter was invented. And today it is still known to be one of the most robust and revolutionary filtering methods. But just like everything in life, it has limitations especially when it comes to non-Gaussian/non-linear systems. Kalman filter uses linear Gaussian assumption and single Gaussians for the posterior density. While this makes the process simpler and faster, it is incapable of handling higher dimensional or turly non-Gaussian systems. In many systems today, it is becoming more and more important to include non-linearity and non-Gaussianity in order to accurately model the underlying dynamics of a physical system. Enter particle filter. \n",
    "\n",
    "Particle filter is also part of the Bayesian estimation. People also call it Sequential Monte Carlo method because it is a recursive Bayesian filter that uses Monte Carlo technique. The idea of particle filter is that the posterior probability density function (pdf) is represented as point masses (particles) with corresponding likelihoods (weights). In other words we generate a plethora of particles and send them to where we think the target is and estimate the state of the target using algorithms; then when the particles combined together, they make up the the posterior pdf. Because of such flexivity, particle filter has become on of the most popular techniques for estimating stochoastic systems. I know all this sounds like another language rn, but hopefully by the end you will have much better understanding.\n",
    "\n",
    "Before diving straight into particle filtering, you have to know these few topics as prerequisites: a little bit of estimation basics, Bayesian inference, and some rudimentary knowledge of Kalman filter. The first two are briefly covered, but Kalman filter won't be since it is a lot of material. A lot of good resources online are out there and I suggest looking over before starting this document. "
   ]
  },
  {
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"
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   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Estimation Basics \n",
    "\n",
    "**Definition** \n",
    "\n",
    "Estimation is a process of inferring the value of a quantity of interest from indirect, inaccurate and uncertain observations. \n",
    "\n",
    "Filtering \n",
    "\n",
    "There are two models required to perform the estimation: \n",
    "\n",
    "**I. State Space System/Dynamics Model**\n",
    "\n",
    "This system describes how the state system is evolving with time including the process noise because it's a stochastic system.\n",
    "\n",
    "Let $x_k$ be an $n_x$ dimensional state vector that describes the system:\n",
    "\n",
    "$$x_{k+1}\\ =\\ f_k x_k + v_k\\hspace{55pt} k = 0,1,...$$\n",
    "\n",
    "and if there is no control input: \n",
    "$$ E \\{v_k\\}\\ =\\ 0,\\ E\\{v_k v_j\\}\\ =\\ \\delta _ {jk} Q_k\\ $$\n",
    "\n",
    "$f_k$: A known possibly nonlinear function of $x_k$<br />\n",
    "$v_k$: Process noise. Usually zero mean and white<br />\n",
    "$Q_k$: Process noise coavariance. <br/>\n",
    "$k$: Current time step<br />\n",
    "\n",
    "The state vector has all the information to describe the system. A good example of a state vector is a kinematic characteristics of a missile as a 6 x 1 vector containing position and velocity.\n",
    "\n",
    "**II. Measurement Model**\n",
    "\n",
    "Let $z_k$ be an $n_z$ dimensional known measurement model: \n",
    "\n",
    "$$ z_{k+1}\\ =\\ h_k x_k + w_k \\hspace{55pt} k\\ =\\ 0,1,...$$\n",
    "\n",
    "and\n",
    "\n",
    "$$ E \\{ w_k \\}\\ =\\ 0,\\ E \\{w_k w_j\\}\\ =\\ \\delta_{jk} R_k\\ $$\n",
    "\n",
    "$h_k$: A known possibly nonlinear measurement function for the problem<br />\n",
    "$w_k$: Measurement noise. Also usually zero mean and white<br />\n",
    "$Q_k$: Measurement noise coavariance. <br/>\n",
    "    \n",
    "Example: A radar measurement that tracks the state ($x_k$) of the missile.\n",
    "\n",
    "**Bayesian Estimation**\n",
    "\n",
    "given data $Z^k\\ =\\ \\{z_1,\\dots,z_k\\}$, we want to estimate the $x$ which can be defined as: <br />\n",
    "\n",
    "$$\\hat{x}_k\\ =\\ \\hat{x}(k,Z^k) $$\n",
    "\n",
    "and $\\hat{x}_k$ is usally the approximation of conditional mean given the data which can be described as:\n",
    "\n",
    "$$ \\hat{x}_k\\ \\widetilde{=}\\ E[x_k | Z^k] $$ \n",
    "\n",
    "\n",
    "\n",
    "\n",
    "![image.png](attachment:image.png)\n",
    "   \n",
    "The standard Bayesian approach begins with a prior pdf, a belief about information before incorporating the data. using Bayes' rule, a posterior pdf can be obtained based on all available information, including the set of received measurement. Why is posterior pdf so important? Because the posterior pdf is the best description of the state and can be used to compute the state estimation $\\hat{x}$.\n",
    "\n",
    "Suppose the posterior pdf of the previous time step, $p(x_{k-1}|Z^{k-1})$ is known, then the prediction stage can evolve to the prior pdf at time step k shown below: \n",
    "\n",
    "$$p(x_{k}|Z_{k-1}) = \\int p(x_k|x_{k-1})\\ p(x_{k-1}|Z^{k-1})\\ dx_{k-1}$$ <br />\n",
    "known as the Chapman - Kolmogorov equation.\n",
    "\n",
    "As mentioned previously, the posterior pdf at time step k can be computed using Bayes' rule: \n",
    "\n",
    "$$p(x_k|Z^k) = \\frac{p(z_k|x_k)\\ p(x_k|Z^{k-1})}{p(z_k|Z^{k-1})}\\ =\\ \\frac{likelihood \\times a\\ priori}{marginal\\ likelihood}$$\n"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# General Approach to Particle Filter\n",
    "\n",
    "Recall that particle filter is a recursive Bayesian filter using Monte Carlo technique. The posterior pdf is represented as particles with corresponding weights. That means that the posterior is an approximation. In this case approximation is better than finding the exact solution because the system is too complex.  \n",
    "<img src=\"nongaussdistrib.PNG\">\n",
    "The figure above (K. Hsiao) is a good illustration of a non-Gaussian pdf $p(x)$ represented with particles $\\chi_k ^i$ and weights $w_k ^i$. We can draw $N_s$ independent samples $\\chi_k ^i$'s, which are $i$th particles at time step $k$ where each represents the value of the system state, and $w_k ^i$ which are the weight (likelihood) of each particle. \n",
    "\n",
    "For example, suppose we have no idea about the state of an object we are tracking; then we would have the generated particles to be scattered around the location where the object might be. A location with more particles and bigger weight would mean the object is more likely to be in that place. As mentioned earlier combination of those particles and weights would form the pdf we want to generate or to estimate from.\n",
    "\n",
    "**Monte Carlo Technique** \n",
    "\n",
    "Suppose we want to evaluate the evaluate the expectation of the pdf to find the estimation: \n",
    "\n",
    "$$ E[f(x)]\\ =\\ \\int\\limits_{-\\infty}^{\\infty} f(x) p(x) dx\\ \\widetilde{=}\\ \\frac{1}{N_s} \\sum\\limits_{i = 1}^{N_s} f(\\chi^i) $$\n",
    "\n",
    "This integral is easy to perform analytically if the pdf is Gaussian. However, in most real life cases, pdf's are non-Gaussian and are extremely difficult or impossible to integrate analytically. That is where the Monte Carlo integration technique comes in. Monte Carlo technique can numerically integrate any arbitrary function, including non-integrable and noncontinuous functions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monte Carlo Example: Computing $\\pi$\n",
    "\n",
    "A quick and solid example is estimating the value of $\\pi$. This technique was developed by Stanley Ulam at Los Alamos National Laboratory. He couldn't calculate some nuclear reactions because he could not integrate analytically. \n",
    "\n",
    "As briefly shown in the equation above, the idea is simple. Integration is calculating the area under the curve. So you take the curve you wish to integrate, and put a bounding box around it. Then you generate random points that are positioned inside the bounding box and see how many of those points land under the curve. If the area of the bounding box is 1 and 70% of the points are under the curve then the approximation of that integral would be 0.7. And as you might intuitively think, more points means more accuracy of the integral. \n",
    "\n",
    "Now let's look at how we can compute $\\pi$ using Python. The example below is taken from Kalman and Bayesian Filters in Python by Roger Labbe. The bounding box is a square with a length of two and we draw a circle inside with a radius of 1. \n",
    "\n",
    "The algorithm is simple: it generates 20000 samples random distribution anywhere inbetween -1 and 1  using numpy's `random.uniform` function then compute the distance from the origin using `numpy.linalg.norm` which calculates magnitude of a vector. \n",
    "\n",
    "Next, it creates a bool array to see if the distance of each point falls inside the circle\n",
    "\n",
    "```python\n",
    "in_circle = dist <= 1\n",
    "```\n",
    "Then the only thing left is to compute pi by taking the ratio of points that fell inside the circle and vice versa. The visual representation is also shown by scatter plotting those points.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mean pi(N=20000)= 3.1530\n",
      "error  pi(N=20000)= -0.0114\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from numpy.random import uniform \n",
    "\n",
    "\n",
    "N = 20000  # number of points\n",
    "radius = 1.\n",
    "area = (2*radius)**2\n",
    "\n",
    "pts = uniform(-1, 1, (N, 2))\n",
    "\n",
    "# distance from (0,0) \n",
    "dist = np.linalg.norm(pts, axis=1)\n",
    "in_circle = dist <= 1\n",
    "\n",
    "pts_in_circle = np.count_nonzero(in_circle)\n",
    "pi = 4 * (pts_in_circle / N)\n",
    "\n",
    "# plot results\n",
    "plt.scatter(pts[in_circle,0], pts[in_circle,1], \n",
    "            marker=',', edgecolor='k', s=1)\n",
    "plt.scatter(pts[~in_circle,0], pts[~in_circle,1], \n",
    "            marker=',', edgecolor='r', s=1)\n",
    "plt.axis('equal')\n",
    "\n",
    "print('mean pi(N={})= {:.4f}'.format(N, pi))\n",
    "print('error  pi(N={})= {:.4f}'.format(N, np.pi-pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pretty neat right? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Importance Sampling \n",
    "\n",
    "But there is another problem; when the system and its non-Gaussian pdf's are complex, it is also difficult or impossible to sample from $p(x)$ which is a required piece to perform the Monte Carlo technique. For example, what if the object we are tacking moves very differently than we predicted with our state model? we cant draw from the $p(x)$, let alone perform Monte Carlo Integration. Importance sampling allows us to draw a sample from a target pdf that is possibly unknown.\n",
    "\n",
    "This is a very difficult concept to understand intuitively. you might question, how does importance sampling allow you to sample and compute the integral of a  target pdf? Let's get down to business and look how this scheme works mathematically. \n",
    "\n",
    "We are still going to use the same expectation integral equation, but we are going to tweak it a little bit by introducing $q(x)$: \n",
    "\n",
    "$$ E[f(x)] = \\int\\limits_{-\\infty}^{\\infty} f(x)\\frac{p(x)}{q(x)} q(x) dx$$\n",
    "\n",
    "$q(x)$ is called the importance density, or some also call it proposal distribution. It is statistically indepdent from our samples and we know the pdf. $q(x)$ could be drawn from uniform distribution, measurements, or even a prior estimate, but the important thing to note is that $q(x)$ is known. We can manipulate this equation to look like: \n",
    "\n",
    "$$ E[f(x)] = \\int\\limits_{-\\infty}^{\\infty} f(x)q(x)\\ \\  \\frac{p(x)}{q(x)}\\ \\ dx$$\n",
    "\n",
    "$\\frac{p(x)}{q(x)}$ is the weight of the particle. Note that terms first start out as unnormalized form because we don't know the exact value; to avoid the confusion we will define unnormalized forms of $p(x)$ and $q(x)$ as $\\tilde{p}(x)$ and $\\tilde{q}(x)$ respectively. Then we can define both unnormalized and normalized weight of $i$th particle at time step $k$ can be described as:\n",
    "\n",
    "$$ \\tilde{w}_k ^i = \\frac{\\tilde{p}(\\chi_k ^i)}{\\tilde{q}(\\chi_k ^i)},\\quad w_k ^i = \\frac{\\tilde{w}_k ^i}{\\sum\\limits_{j = 1}^{N_s} \\tilde{w}_k ^j} $$\n",
    "\n",
    "Putting all the pieces back together to the original equation, the expectation integral can be approximated as: \n",
    "    $$ E[f(x)] = \\int\\limits_{-\\infty}^{\\infty} f(x) p(x) dx\\ \\widetilde{=}\\ \\sum\\limits_{i = 1}^{N_s}  w_k ^i f(\\chi_k ^i) $$\n",
    "\n",
    "And the Important Sampling approximation of $p(x)$ can be written as: \n",
    "\n",
    "$$ p(x)\\ =\\ \\sum\\limits_{i = 1}^{N_s} w_k ^i \\delta(x_k - \\chi_k ^i) $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Importance Sampling Example: Estimating a Beta Distribution\n",
    "\n",
    "Consider a Beta Distribution as our target pdf $p(x)$ with the x range from 0 and 1, and 2 and 11 as its shape parameters. Thanks to `scipy` library we can easily construct it with just a couple lines:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8d7d540850>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import beta\n",
    "x = np.linspace(0,1,20000)      # set the x range \n",
    "targetPdf = beta.pdf(x,2,11)    # construct the beta pdf\n",
    "plt.plot(x,targetPdf,linewidth = 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We already know what the distribution looks like, but the goal is to estimate $p(x)$ by using Importance Sampling technique with a given  importance density $q(x)$, a distribution that we know and can sample from. $q(x)$ will be drawn from uniform distribution and we will use 20,000 samples. Before applying Importance Sampling, the pdf comparison looks like below: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8d7f029f10>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ns = 20000                      # number of samples\n",
    "qx = uniform(0,1,Ns)            # draw samples of importance density q as \n",
    "                                # uniform distribution \n",
    "\n",
    "# plot pre-Importance Sampling \n",
    "plt.hist(qx,60,color='c',edgecolor='k',label = 'q(x)')\n",
    "plt.plot(x,targetPdf*450,linewidth = 2,color='k',label = 'Target pdf')\n",
    "\n",
    "#plot lables \n",
    "plt.title('Before Importance Sampling',size=20)\n",
    "plt.xlabel('x', fontsize=15)\n",
    "plt.ylabel('Samples', fontsize=15)\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see in the plot above $q(x)$ looks nothing like our target pdf $p(x)$. Now we can apply the Importance Sampling and see how the uniform distribution turns into a close estimation of $p(x)$.\n",
    "\n",
    "We start with generating the initial weight distribution and uniformly distribute it across all samples. with the value of 1/Ns. We do that because when you initialize our assumption is that no particle is biased yet; similar to how we did with $q(x)$. Then we evaluate $f(\\chi^i_k)$, recall that $\\chi^i_k$ is the ith particle at time step k. In this case it would be samples from $q(x)$. Finally, we compute the normalized weights. \n",
    "We can apply the normalized weights to our original $q(x)$ to re-generate the distribution to how the post-Importance sampling estimation compares to the target pdf $p(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8d7d501410>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "w_initial = 1/Ns;               # initial unnormalized weight in scalar \n",
    "                                # same for every distribution value    \n",
    "\n",
    "# compute the weight distribution \n",
    "                                \n",
    "fx = beta.pdf(qx,2,11)            # evaluate the f(x) integral part \n",
    "w_unnormal = fx / w_initial       # generate unnormalize weight distribution\n",
    "w_normal = w_unnormal / np.sum(w_unnormal) # normalize the weight \n",
    "\n",
    "# generate the new pdf with the new weight distribution \n",
    "pdf_impSamp = np.random.choice(qx,Ns,replace=True,p=w_normal)\n",
    "\n",
    "# plot post-Importance Sampling \n",
    "plt.hist(pdf_impSamp,30,color='c',edgecolor='k',label = 'Estimation p(x)')\n",
    "plt.plot(x,targetPdf*450,linewidth = 2,color='k',label = 'Target pdf')\n",
    "\n",
    "\n",
    "plt.title('After Importance Sampling', fontsize=20)\n",
    "plt.xlabel('x', fontsize=15)\n",
    "plt.ylabel('Probability', fontsize=15)\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now our estimation looks much like our target pdf. To summerize, Importance Sampling is a very powerful technique that allows us to sample from pdf that is difficult to sample from or even is unknown. It generates particles from importance density, a known statistically independent pdf $q(x)$ and weigh those particles according to the true/target pdf. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SIS (Sequential Importance Sampling) Particle Filter\n",
    "\n",
    "Now finally, let's apply the Importance Sampling into a particle filter. This is the most elementary particle filter and simple enough to get it started. Importance Sampling is 90% of this algorithm (hence the name). To reiterate, the main goal is to represent the required posterior pdf by a set of random samples with associated weights and to compute estimates based on these samples and weights. Remember all the variables from the Estimation Basics section? We are going to actually estimate a system using all the things we learned. Before that, there is a bit more of derivations using the estimation variables. I know this is a very hefty material, and derivations are all we've been doing, but you will understand the example much better knowing the foundation.\n",
    "\n",
    "Set up the importance density q\n",
    "\n",
    "$$q(x_0,\\dots,x_k | Z^k) = q(x_k | x_0,\\dots,x_k , Z^k)\\ q(x_0,\\dots,x_{k-1} | Z^{k-1})  $$\n",
    "\n",
    "Then,\n",
    "\n",
    "$$ p(x_0,\\dots,x_k | Z^k)\\ =\\ p(z_k | x_k)\\ p(x_k | x_{k-1})\\ p(x_0,\\dots,x_{k-1} | Z^{k-1})/ p(z_k | Z^{k-1}) $$\n",
    "\n",
    "Given this formula and suggested form of q, the particle weight can  be:\n",
    "\n",
    "$$w_k^i\\ =\\ \\frac{p(z_k|\\chi_k^i)p(\\chi_k^i |\\chi_{k-1}^i) p(\\chi_0^i,\\dots,\\chi_{k-1}^i | Z^{k-1})}{q(\\chi_k^i | \\chi_0^i,\\dots,\\chi_{k-1}^i , Z^k) q(\\chi_0^i,\\dots,\\chi_{k-1}^i)| Z^{k-1})} $$\n",
    "\n",
    "$$=\\ w_{k-1}^i\\ \\frac{p(z_k|\\chi_k^i)p(\\chi_k^i |\\chi_{k-1}^i)} {q(\\chi_k^i | \\chi_0^i,\\dots,\\chi_{k-1}^i , Z^k)} $$\n",
    "\n",
    "This allows us to develop a recursive form for particle weights. Also note that Importance Sampling Algorithm was designed that \n",
    "\n",
    "$$q(x_k | x_0,\\dots,x_{k-1}, Z^k)\\ =\\ q(x_k | x_{k-1}, Z^k)$$\n",
    "\n",
    "meaning that importance density becomes only dependent on $x_{k-1}$ and $z_k$, not the data with whole time history. Then the weight propagation can be written as: \n",
    "\n",
    "$$\\bar{w}_k^i\\ =\\ w_{k-1}^i\\ \\frac{p(z_k|\\chi_k^i)p(\\chi_k^i |\\chi_{k-1}^i)} {q(\\chi_k^i | \\chi_{k-1}^i , Z^k)}$$ (Unnormalized)\n",
    "\n",
    "$$ w_k ^i\\ = \\frac{\\bar{w}_k ^i}{\\sum\\limits_{j = 1}^{N_s} \\bar{w}_k ^j} \\qquad $$ (Normalized)\n",
    "\n",
    "Now we can finally Compute the state estimation: \n",
    "\n",
    "$$ \\hat{x}_k\\ =\\ \\sum\\limits_{i = 1}^{N_s} w_k ^i \\chi_k^i $$\n",
    "\n",
    "And the error covariance if needed:\n",
    "\n",
    "$$ P_k\\ =\\ \\sum\\limits_{i = 1}^{N_s} w_k ^i [\\chi_k^i - \\hat{x}_k][\\chi_k^i - \\hat{x}_k]^T $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Example \n",
    "\n",
    "Take a look at this one dimensional, but extremely chaotic non-linear system:\n",
    "\n",
    "$$x_{k+1} = f [ k , x_k , v_k ] = 2 \\operatorname { atan } [ x_k ] + 0.5 * \\cos ( \\pi k / 3 ) + v_k$$\n",
    "\n",
    "$$z_k = h [ k , x_k ] + w_k = x_k + x_k ^ { 2 } + x_k ^ { 3 } + w_k$$\n",
    "\n",
    "$$E \\{ v_k \\} = 0 , E \\{ v_k v_j \\} = \\delta _ { j k } Q_k\\ where\\ Q_k = 1$$\n",
    "\n",
    "$$E \\{ w_k \\} = 0 , E \\{ w_k w_j \\} = \\delta _ { j k } R_k\\ where\\ R_k= 0.25$$\n",
    "\n",
    "$$\\quad \\hat { x }_0 = 4 , P_0 = 2$$\n",
    "\n",
    "This example is from one of the classes I took taught by Dr. Mark Psiaki. This is a great example to apply the particle filter on. As shown above, The system is chaotic with two stable equilibria at $x = \\pm \\ 2.31122$. The sinusoidal behavior and random process noise cause extremely unpredictable jumps between the two equilibria.The measurement function $h[k,x_k]$ is also very non-linear. $\\frac{\\partial h}{\\partial x}$ is small near $x = 0$. which is where the most uncertainty about the jumps occurs.\n",
    "\n",
    "Lets start with the two estimation models state dynamics model $f(x_k,v_k)$ and measurement model $h(x_k)$. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# compute f the state space/dynamics model \n",
    "def dynamics_model(xk,vk,k):\n",
    "\n",
    "    atanxk = np.arctan(xk)\n",
    "    pi = np.pi\n",
    "    f = 2*atanxk + 0.5*np.cos(pi*k/3) + vk\n",
    "    \n",
    "    return f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# compute h the measurement model \n",
    "def measurement_model(xk,Rk,k):\n",
    "\n",
    "    oneoversqrtRk = 1/np.sqrt(Rk)\n",
    "    h = oneoversqrtRk * (xk**3 + xk**2 + xk)\n",
    "\n",
    "    return h"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far pretty simple. Before we spin the wheels with particle filter, we want to generate the truth state first so we can compare each other in the future and measure the filter accuracy. the function below is considered as a truth state simulator because you manually add the process noise variable `vkhist` using the given process noise covariance `Qk`. With the noisy measurement `zk` the variable was normalized by using `1/np.sqrt(Rk)` because this process makes the measurement noise zero mean. \n",
    "\n",
    "**Note:** whenever there is a `hist` added to a variable that means it's a vector that stores all the values through out the time steps k. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "    # initialize the given variables \n",
    "    Qk = 1            # process noise covariance matrix at time step k\n",
    "    P0 = 2            # initial a posteriori state estimation error covariance matrix\n",
    "    Rk = 0.25         # measurement noise covariance matrix at time step k\n",
    "    xhat0 = 4         # initial a posteirori state estimate \n",
    "    K = 100           # number of time steps (history) \n",
    "    Ns = 500          # number of particles \n",
    "    # This function computes truth simulation of both dynamics and measurement model\n",
    "    # through out the given time history\n",
    "    # \n",
    "    \n",
    "    from numpy.random import randn     # random standard normal distribution \n",
    "    \n",
    "    def tru_simulation(xhat0,P0,Q,R,Ns):\n",
    "\n",
    "        # initialize the variables \n",
    "        vkhist = np.sqrt(Q)*randn(Ns,1)\n",
    "        xkhist = np.zeros((Ns+1,1))\n",
    "        zkhist = randn(Ns,1)\n",
    "        \n",
    "        # initialize the truth state vector with xhat0 and error covariance\n",
    "        xkhist[0,0] = xhat0 + np.sqrt(P0)*randn(1,1)\n",
    "\n",
    "        for k in range(0,Ns):\n",
    "            kp1 = k + 1\n",
    "\n",
    "            # compute the truth state for each model \n",
    "            xkhist[kp1,0] = dynamics_model(xkhist[k, :], vkhist[k, :], k)\n",
    "            zkperfect = measurement_model(xkhist[k, :],R,k)\n",
    "\n",
    "            # update the measurement model \n",
    "            zkhist[k,:] = zkperfect + zkhist[k,:]\n",
    "\n",
    "        return xkhist, zkhist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's run this simulation to see what the truth state through 100 times steps looks like."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8d7d2beb50>"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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xoG+MMSliQd8YY1LEgr4xCdXS2sarSz+jpbWt0kUxERL2M3KNMSGY3bSMKffNp6aqivaODq44YS+OqR9W6WKZCLCavjEJ09LaxpT75rO2vYPVbetZ297B5PvmW43fABb0jUmc5k/XUFPV/addU1VF86drKlSi3rE0lb8svWNMwgzfoh/tHR3dprV3dDB8i34VKlH5LE3lP6vpG5Mwg/vXcsUJe9G3pooBtX3oW1PFFSfsxeD+tZUumieWpgqG1fSNSaBj6ocxdpchNH+6huFb9ItdwIeNaaq1bLxq6UxTxXF7osKCvjEJNbh/bayDY6401boNG/h8zTpaWttivW2VZOkdY0wkZaep+lRBh8K5M19h7LRG5jQtq3QRY8mCfhHWcyDZ7PgGrzf7+Jj6YTw75VCuOWlfqquqaN+glt/vpYqkd0SkGpgLLFPVoytRhlJYz4Fks+MbPD/28eD+tQzsV8Mm1VW0rbf8fm9Vqqb/c+D1sFZWTk3Deg4kW9qPbxhXOH7u4yR1Q6200IO+iAwHJgAzwljf7KZljJ3WyMQZL3jKAybtBhfTXZKOr9cAXu5vwis/93FSuqFGQSXSO/8FTAYG5JtBRCYBkwDq6urKXlFmTaOz29fk++YzdpchRb8sVrNItqQcX6/pk978Jrzyex8noRtqFHiq6YvIQBE5TEROEpHTReR4EdnDw+ePBj5R1XmF5lPV6araoKoNQ4cO9VLEbnpT07CaRbIl4fiWkz4J8woniH08uH8te28/KFbHKWqK1vRFZHPgh8DJwBh6nihURD4HZgEzVPXvBRY3FjhGRI4C+gKbi8htqjqxrNIX0duahtUski3ux7ecm5fCvsKJ+z5OooI1fRH5FfA+cB7wJPAdYEec1MwmwFbA/sAvgcFAo4g0isjoXMtT1X9R1eGqOgL4AdAYVMAHf2oaVrNItjgf33ICeCWucOK8j5OoWE3/AOAYVX0mz/sr3b+5wHUiMhA4G6dG/5pvpewFq2mYpOoM4JOzcvrFvuP2m0g3UdVKl6GghoYGnTt3bqWLYUxktbS2WQA33YjIPFVtyPVeyQ25IvL1Iu+f7LVgxmSzO2S9s/SJ8cJLl81HRWS8qs7PfkNEfgJcBdziW8lM6tgdssYEz0uXzduBx0RkVOZEEfklTsA/y8+CmXRJ+x2yxoSl5KCvqj8F5uD00NkZQER+C0wFJqrqDcEU0VRaGCmXJN0ha0yUeb0jdxJOCqdRRBpxul2eoKoP+V4yE4pijYBhpVyC7j9ujZ3GODwFfVVVETkFuBM4AThKVZ8MpGQmcMUCepi37Jfb/bAU1lZgzEYFg76IrABy9ensg3Nz1l0i0jVRVbfytXQmMKUE9LAfVxdE//EwT1zGxEGxmv415A76JuZKCeiVGJTM70f82XNWjemuYNBX1UtDKocJWSkBPciUS1iSMpqmiZ+otiOVnNMXkd1UNe+DT0Rkgqo+7E+xTNBKDehxv2U/CScuEz9RbkcqeRgGEVkDXAxcqRkfEpEBwH8DJ6tqtd8FtGEYghXV2ojf0rKdpvJaWtsYO62Rte0brzD71lTx7JRDQ/vuFRqGwUvvnV8BlwHHisgpqvquiHwD5wlYfYBv976oJmx+59CjKi3baSov6u1IXm7OuhJnPP1aYL6IzAIeBf4GjFbVR4IpojHGxEfU25E8PTnLzelfBFQDxwKvAOeo6qcBlM0YEwIb5M5fUX8qm5eG3Frgt8DPgPuBO3DG3HlNRE5X1ceCKaIxJihRbnCMsyh3gPBS02/CeWziRFX9nqrOAvYEHscZgfPaIApojAmGDXIXrKgOee0l6C8G9lTVOzsnqOpqVT0NJ9VzjN+FM8YExwa5S6eS0zuqmrd3jqo+lO+5uMaYaIp6g6MJRrEHo0uh9zNlNuZ6+ZwxvWUNkeWJeoOjCUaxmv5bIjINuENVvyy2MBEZA/wceB2n0TfS7Iad+LOGyN6JcoOjCUaxoH8RcClwtYj8BXgOeA1YCbQBg4AdcfrvHwFsj3OzVuQfqGLBIv5sBE1/2I1r/olDRbLYgGv3icj9wOHAycB5wLZsHHlTgHXAPOCPwK2q+klwxfWHBYtkiPqdjyZd4lKRLNqQ646z85j7h4hsC2wD9AVWAe+p6rogC+k3CxbJYA2RJiriVJH0dEcugKouV9VXVPXvqvpm3AI+WLAIWhgNq52X0Rcfvbs1RGawRu3KiFP3V6/PyE0EG243OGFc4mav4+IJuzN62MBI51HDEJf0QhLFqSJZ8tDKlRLk0MpxaHSJkzCGlI3CsLVRZPul8uY0LetRkazUSdevoZX9KEhf4GmckTr7APeq6tQwy5DJei34K4y2EmuPyc32S+XFpftr2OmdNuBQVW0VkRrgGRH5k6o+H3I5TADCuMSN02V0mGy/REMcKpKeG3J7Qx2t7ssa9y/a+SVTsjDu8LS7SHOz/WJK5SmnLyJ7Af8KNADDgQNV9WUR+Q3wjKr+qYRlVOP0698FuEZVp+SYZxIwCaCurm7MkiVLSi6jqbww2kqsPSa3tO6XtG53PoVy+l6ekXskMAfnrtxGYCrQ4Ab9S4ADVPUoD4UaBMwCfqqqr+WbrxLPyLUvkDHxYb2WeioU9L2kd34L3KSq/wj8Juu9JqDeS6FU9TPgKZzhGyJjdtMyxk5rZOKMFxg7rZE5TcsqXSRjIqvS9wXYMwG88xL0RwF3uf/Pvjz4Atiy2AJEZKhbw0dE+uEM7/CGhzIEyr5AxpQuChWkON0UFRVegv4nwE553tsD+KCEZWwLPCki84GXgMdU9SEPZQiUfYHSodK1Uz9UehuKVZDCKp8fvZYqvS/D5qXL5p3Av4nIIuDv7jQVka8BU4Driy1AVecD+3guZUis21vyRTX/66UdKQrbUOi+gGcWrwytfL29uz4K+zJsXoL+xcDuwF+Bj9xps3EGX/sLcLm/RQtf2odnSHoDdhQGxcq1j70EnihsA+SvIG22SXXo5Sv3pqio7MuweXlcYhtwtIgcBhwGDMEZZfMJVX0soPKFLi531fktDTWeSt+1mmsfj91liKfAU+lt6JSvgvTlug0VKV85N0VFZV+GreSgLyJ1wHJVfQJ4Iuu9PsB2qlpKXj/y4nBXnZ/SUuOpZPou3z6e/sMxngJPlFKQuSpILa1tkSlfMVHal2Hy0pD7Hvnz8Xu775sYSksDdiXvWs23j0E8BZ6o3Xk7uH8te28/qGv9UStfIXEqq5+85PQLPey8L864OiaG0lTjqVT6Lt8+3mO7zUtuR+psDxi7yxCenXJoZFOQcUqRxqmsfikY9N1hFzJvujpKREZlzdYX+B7wls9lMyFJWwN2JdJ3hfZxduABeHXpZ2U39kZBnFKkcSqrHwoOwyAiU3GGWwDnhqx8tf33gLNU9XF/i1eZYRjSKum9d6Kg2D7O19hrY+UbL3ozDMPlwABgc5yAf6j7OvOvVlV3DiLghyltN2jkkp2fNf4rtI/z3fC08MPPU9HmYsJRML2jqu1Au/sy1GGYwxS3S2eTTPm6EHpt7DWmEM8PURGR4cDXcHL53ajqI34UKkxp6a5oos+Pxl4TfZVOo3rppz8AuBv4Zuck99/MRoFqn8oVmrTeoGGix0tjr303Kx88yxGFrIKXmv5vgTrgH4BngOOBT4GJOLn+E30vXQjS1F3RRF+h4J62XiaFRCF4etHS2sbCD79g8r3zaVtf2ayClzz9UTjj6L/gvv5QVZ9W1Uk4Y/Bc6HfhwpDWGzRMdFWyQT0OHRriNgR65xDUZ986j7b13SuYlWiQ91LT3xpYqqobRORLuo+f/whwn68lC0iuS0K7dDYmPrXnOKVkM09QuVQiq+Al6C/FGWQN4G3gaODP7uv9gbU+lisQhb7UdunsXRxzqia3qHVoKPTdilNKNtcJCmDTTarpUOXiCbt31fTD2s9egv5jOE+6mgVcBdwsImNwhl84BLjS/+L5J2pf6riLS63QlCas2nMpFYVi36043UGe6wRV20e4buK+LF21hsseXhT6b8hL0J8CbAqgqreKSCvwXaAf8BPgj/4Xzz9xuiQsptI1bDuBJk8YtedSKgqlfrfikpLNd4LaY7uBTLp1XkV+Q17G0/8K+Crj9SycWn8shHlJGGRQLvTDCetk4McJtNInrlyiWCY/Fdq+oGvPpQZzL9+tuKRkc52gXl36WcUqoV766W8ADlTVF3O8NwZ4UVUj208/rEvCINMehX44YT6irrcn0CimhqJYJj+Vsn1B1p5LDeZxytd7kX2CquR2eumyWWho5RpgfS/LErhj6ofx7JRDue2M/Xl2yqG+/6iD7kqWb0z2hR9+EWoXtt50c41id7solslPXrYvqO6ipQa5tHShruR2FhtauQ4YkTFpHxHJHn6hL3AKMXmISpCXhEG3G+T74YCGfqlYbq0wim0rUSyTn6LQSOvlStvPK44op+wq1S5RLL3zI5yhldX9uzbPfGuAM3wsVywFfclWqFHIy3pL+SGUMk85J9AoXr5HsUx+ikojrZcg50flrLcpuzBOGJVolyg2nv5QYCuc1M584CT330zrgA/cB6f7Lm7j6c9pWtYjKAeRRsr+Mpa63lJ+CEHnt8PYR0kok5+C3L6W1rbIjfff2zLFvY2n0Hj6BYN+1kJ2wHkw+jo/C1dM3II+VO6Ssth6S/khhPUDjuJldxTL5Kegtu/VpZ8xccYLrG7b2Kw3oLYPt52xP3tvP8i39YRVJq+/gSh+bwoFfS9dNpdkLHBT4HRgFPARcEvm+0mVeXCBgvnLShz8YustJbcbVv43it3tolgmPwW1feWmj4IMlr1JaXn5DcTxiqBYQ+6VwLdV9WsZ0wYALwG74oyyORC4QET2U9XEPic38+CuaV+PiNC3T3VsDjSU9kNIen7b+K+c7tBBB8vedNEu9TcQ15sUi9X0xwO3ZU37Z5yHqJyhqje4ef/HgIuBH/pfxMrLdXBBad/gXDrG4UBDaT+EON3ibqLDSyNtWMGy3N4xpf4G4trrq1jQHwHMy5p2ArBIVW8AUNUV7hXBr4utTES2B24BtgE6gOmqerXXQoct36BJnapFePKNTxg/aqtIH2wo7YdQzo8lqEv1KOZLTW6lpo/CDJblprRK+Q3E9aq4WNDvQ8bomSKyJbAbcE3WfO/jBPJi1gMXqOrLbpponog8pqqLSi9y+HId3ExfrtvApQ8u5FezX4tFqqeUH4KXH0tQl+pxzJea4uISLIv9BuJ6VVzsjty3gHEZr492//1z1nxbAauKrUxVl6vqy+7/VwOvA5H/FWffPdenCmqqhc022TjqRGvbhsTdyVmKoO5mTfpdsmmWpLtug77LPwjFavp/AP5XRAYCHwM/w7nz9i9Z830TeM3LikVkBLAPG5/ElfneJGASQF1dnZfFBib7cg/gyTc+4dIHF9LatqFrvjjk9PxU7FK93PRMXPOlpjRxGSWzFHHr9VUw6KvqTSKyLXAuMAh4GThXVds753Ebco+lhJx+xmf64zxp6zxV/SLHeqcD08Hpp1/qcoOWfXDHj9qKX83ufq6L4mVqkApdqvcmPROXFIApXyWDZRTbisIqU9EB11T1t6o6XFX7q+ohqrog6/0VqrqNquYboqEbEanBCfgzVfX+8oodDVG+TA3rWaf59gHQq/RMlPetibfOZ9ZOnPECY6c1MqdpWaWLFGqZSr4j15eViQhwM7BKVc8r5TNxuCM3aj1XsmvYF0/YndHDBgZag8guq193aXrdB1GsweUTp7ImRRKHjMjFlztyfTIWpy//AhFpcqf9UlUfCbkcvgriMrXc1EiuPtD/+sBr9K+tZn2HBtYDJnsf+JWeiUIvoiDEqaxJEsW2olxlCrIbuJfx9HtNVZ9RVVHVvVS13v2LdcAPQm96ruQacx/C710UdnomTr194lTWMIWRkoxiW1GuMnV2Aw8i1RN2Td+UoDe1kWL3FIRZqwmzh0bYtaXeiGJts9LCuvKJYt/6zDJVi/DlOqc3YGevQL/vVragH6JSc7i9qY3k+wJ5XU4+XvPQYfXQKFRbitpNc1GsbVZS2GPYRLG7aGeZwugGHmp6J828tM73NjXSecPI7WcewG+OH+1biiWKvR46Ze6zqN805+X4htULq5LyPWZ8rUcAAA7+SURBVAa0+dM1ga1zcP/ePRYyiOMyuH8t40dtxfqO7p1r/K4QhNp7pxxx6L1TTLmt83717vBjOVHs9ZBLS2tbztpSpcd3z6XYcUlLY29cvlud4vCQoSj13kmlcnO4fqVG/FhOXPLQnbWlONw0V+i4xHXY3nJEMc+eTxjHJej0kwX9ECQhhxunbYhTEMknLidZv0Qxz55LEh4yZEE/BEkIQnHbhrgEkXzidJL1SxzGsEnCcbGcfoiCuAMz7Ls6k7ANcZH0h7XHVRyOiy8PRq+UJAV9vyWhoS8J2xAkOyFGU9SPizXkJlASGvqSsA1Bi0PKI43ifFysn35MVaJvs9+SsA3GxI0F/ZhKQoNSErbBmLixoB9TSRhvPgnbYJIlDXdAW0NuhUTpbttKS8I2mPhLUqcCa8iNGD+/XHFuUOqUhG0w8ZamTgWW3gmZjaVuTPSkqVOBBf2QpenLlQZpyAGnQZo6FVjQD1h2UEjTlyvpojzUtPEmTZ0KLKcfoHy5+ziNYRMlUWrwTVMOOC3iPl5TqSzoB6RQUEjLl8tPYfesKHaCSdsomGmRhk4FFvQDUiwopOHL5Zewa9WlnGAsTRc/UbpSrCTL6QfEgoJ/wmz8LrV3VZpywElg7S8bWU0/IHEbfz7KwjyBeknbpDVNF7casx9XinHb5kIs6AcorUHBb2GeQL2eYNKWpovjXau9bX+J4zYXYkE/YKUEhSTVIoIS1gnUrtDyi2uPpd5cKcZ1mwuxoF9hSatFBCmsWrVdoeUW1x5LvTmRx3WbC7GgX0FJrEUkRdrSNqWIc+eEck/kcd7mfELtvSMiN4jIJyLyWpjrjSobkiEdkjJUQ9x7LA3uX8ve2w/yVN64b3MuYdf0bwL+ANwS8nojyUstwvL+8ZS09F0aU19J2+ZQg76qPi0iI8JcZ5SVmmtMWuBIi7in7/JVNNKY+krSNkcypy8ik4BJAHV1dRUuTbCK1SLiHjjSLM6NgFbRSK5I3pGrqtNVtUFVG4YOHVrp4gSuUK7R8v7xFddGQHvmQ7JFMuibjeIaOEx8GwHjXtFISsN5UCKZ3jEb2c1C8RbHRsA4VzQsLVVcqA9GF5E7gHHAEOBjYKqqXl/oM0l9MLpX1nsn+pJ0jOY0LetR0Yh68GxpbWPstEbWtm88YfWtqeLZKYfG/nh4FZkHo6vqiWGuL0mS1HsgiZJWw4zjFUqcG87DZOkdY3opqT2s4lbRiHNaKkzWkGuMB7kaCePe8JkUcW04D5vV9I0pojNX/9qyz7ns4UU9UjhWw4yOOKalwmZB35gCOnP11SJ8uW4DQM4UjvWwio64paXCZkHfmDwyc/W5ZDYSWg3TxIUFfWPyyNUbJFN2CsdqmCYOLOgbk0euXD3AZrXVbOhQS+GYWLKgb0weuXL1F0/YndHDBloKx8SWBX0TmjjesWq5epM0FvRNKOJ8x6rl6k2S2M1ZJnA2VK8x0WFB3wTO7lg1Jjos6JvApfGOVRvT3USVBX0TuLSNiTK7aRljpzUyccYLjJ3WyJymZZUuUq/ZSSw5rCHXhCItvWCSOOJmnBvhTU9W0zehKfQs4KRIWvuFNcInjwV9Y3yUtPaLpJ3EjAV9Y3yVtPaLpJ3EjOX0IyeOd62a7pLUfmHDRiePBf0IsQaz5EjSXbxJOokZC/qRkcReHyY5knQSSzvL6UeENZgZY8JgQT8irMHMGBMGC/oRkbReH8aYaLKcfoRYg5kxJmgW9CPGGsyMMUEKPb0jIkeIyJsislhELgp7/cYYk2ahBn0RqQauAY4EdgdOFJHdwyyDMcakWdg1/f2Axar6rqquA+4Ejg25DMYYk1phB/1hwNKM183utG5EZJKIzBWRuStWrAitcMYYk3RhB33JMU17TFCdrqoNqtowdOjQEIpljDHpEHbvnWZg+4zXw4EPC31g3rx5K0VkSZnrGwKsLPOzcWXbnHxp216wbfZqh3xviGqPinZgRKQP8BZwGLAMeAn4J1VdGND65qpqQxDLjirb5uRL2/aCbbOfQq3pq+p6EfkJ8GegGrghqIBvjDGmp9BvzlLVR4BHwl6vMcaY5I+9M73SBagA2+bkS9v2gm2zb0LN6RtjjKmspNf0jTHGZLCgb4wxKZLIoJ+GQd1EZHsReVJEXheRhSLyc3f6liLymIi87f67RaXL6jcRqRaRV0TkIff1jiLygrvNd4nIJpUuo59EZJCI3Csib7jH+8CkH2cROd/9Xr8mIneISN+kHWcRuUFEPhGR1zKm5Tyu4vhvN6bNF5F9y11v4oJ+igZ1Ww9coKq7AQcA57rbeRHwhKruCjzhvk6anwOvZ7yeBlzlbvOnwOkVKVVwrgYeVdVRwN44257Y4ywiw4CfAQ2qOhqne/cPSN5xvgk4ImtavuN6JLCr+zcJuLbclSYu6JOSQd1Udbmqvuz+fzVOIBiGs603u7PdDBxXmRIGQ0SGAxOAGe5rAQ4F7nVnSdQ2i8jmwCHA9QCquk5VPyPhxxmnO3k/94bOTYHlJOw4q+rTwKqsyfmO67HALep4HhgkItuWs94kBv2SBnVLEhEZAewDvABsrarLwTkxAFtVrmSB+C9gMtD5QOHBwGequt59nbTjvROwArjRTWnNEJHNSPBxVtVlwH8AH+AE+8+BeST7OHfKd1x9i2tJDPolDeqWFCLSH7gPOE9Vv6h0eYIkIkcDn6jqvMzJOWZN0vHuA+wLXKuq+wBfkqBUTi5uHvtYYEdgO2AznPRGtiQd52J8+54nMeh7HtQtrkSkBifgz1TV+93JH3de9rn/flKp8gVgLHCMiLyPk7Y7FKfmP8hNA0Dyjncz0KyqL7iv78U5CST5OB8OvKeqK1S1HbgfOIhkH+dO+Y6rb3EtiUH/JWBXt6V/E5wGoDkVLpPv3Fz29cDrqvqfGW/NAU5x/38KMDvssgVFVf9FVYer6gic49qoqicBTwLfdWdL2jZ/BCwVkZHupMOARST4OOOkdQ4QkU3d73nnNif2OGfId1znACe7vXgOAD7vTAN5pqqJ+wOOwhnN8x3gXytdnoC28WCcy7v5QJP7dxROjvsJ4G333y0rXdaAtn8c8JD7/52AF4HFwD1AbaXL5/O21gNz3WP9ALBF0o8z8GvgDeA14FagNmnHGbgDp82iHacmf3q+44qT3rnGjWkLcHo2lbVeG4bBGGNSJInpHWOMMXlY0DfGmBSxoG+MMSliQd8YY1LEgr4xxqSIBX3jGxHREv7Gicip7v/7V7CsecsgIpeKyMqM1+PceUeXuOyt3GWM8K/EvSMiJ7gjNFa7r4seAxG5RkSuD6+UJgyhPyPXJNqBGf/vBzQC/w48nDF9EbDQnfer8IrWKy/jlPedEuffCpgKPAW8H0yRSiciVTj93n+vqhs8fPT3wBsi8ltVXRxM6UzYLOgb36gz+h/QNSYQwDuZ0zOsCKdUvafOmEa5tiEUItJPVdf0YhGHATsDt3v5kKq+LyLPAD8GLujF+k2EWHrHhC47tSAiI9zXPxCRG0XkCxFpFpGJ7vuTReRDEVkhItPcmmvm8kaLyMMistr9u0dEtvGxvD3SOyJyujgP+VgjIitF5K8isoeb0lngzvZkZ1or43M7isgD7jauFpEHRWSXrPWpiPxCRP5LRFYAC0TkXHf+/lnzjnfn36vAJpwC/EWdIbgLbeeFIrJWRI7JmHwfcFL2PjfxZQfSRMk0nNvSTwD+BtwsIlfiPCPhNDYOq/y9zg+4AfNZoC/wQ+BUYA/gQXfclmKqRaRP5h9FfhcicghwHXAbzuiPpwHPAQPd8p/kznouTlroQPdztTi31u8GnOmWdUfgryKyZdZqLgS2dbfpZ8BMnCvz72bNdyrwsqrOL1DkQ93yFdqmi3FSQMeqauZYVc8BWwN7Fvq8iZFKjz9hf8n8A/rjjA10ao73TnXf6+++HuG+vjFjns1xxiR5G6jOmP4icFfG61uBN4FNMqbtCmwAJhQoX2cZ8v2tzJh3nDtttPv6n4F5BZY92p1/XNb0s3GeeLZTxrThwDrgXzKmKfBKjuXeBvw1ax+3Aj8pUJbt3OVNyLP9/YHLgdXZ5XXn6+OW+cxKf6fsz58/q+mbKHmi8z/q5NFX4AS5zMbHxXR/eMThwCygI6Om/h5OA2pDCes8BPh61t//FvlME7CPiFwlIodI6c9q3Q+nVv5u5wRVbca5Ujk4a96H6el64B9EZCf39fdwgnKhXH1nmmtlnvf/EzgH+JaqPpX9pjoPLfksYzkm5izomyj5LOv1ujzT+ma8HgJMwbkqyPzbie7jj+fziqrOzfyjyDjlqvo48COcE8ZTwEoR+R9xnmhVyLbAxzmmfwxkp3dyzfcU8C5OLR23DLNVNfuRe5k691VbnvdPwHkq1YsFltFG931uYsx675i4W4VT05+R4718tdteU9WbcdochgLfAa4CvqDwU62W47Q3ZNuans9K7TH8raqqiNwATBKRW3GuDnI9USpT53IH5Xn/aOAh4BYRmaiqHTnmGZSjfCamrKZv4u4JnBz6vOwau6q+H/TK1Xm60x9xGp53dyevc//Nrh2/AIwRkR07J4jIMJynQj1T4ipvwmkHuAFYBjxWZP733PLsmOf9BTgnjqNxGqe7cU9qm+I8n8IkgAV9E3eX4gT9h0Xku273ypNE5CYRGRfECkXk1yLy/927XP9RRM4D/hHnZjRwnvy0BjhFRA4Ukc62hZvc9/4kIt8TkROAR3GuSP5YyrpV9UP3MwcDt2iRm61UtQ0nfTOmwDwv4gT9iSLyH1lvN+BcdRTs/WPiw4K+iTVVfQs4AOfu3unAn3C6HrbhNPoG4SWcWv11wJ9xbl66FLjaLdNanC6ZY4C/uvN3BuDDcZ4IdT1wM7AEp9eMl/TJA+6/N5Y4//3AEYVmUNWncdJUPxWRqRlvHYHTmN7ioXwmwuzJWcbEjIjcDWyrqv9Q4vxb41xhHKyqL3lYTzXOSekiVb2trMKayLGavjExISJ7isiPcGrkV5f6OVX9GKeh++ceV/n/cNJUd3r8nIkw671jTHw8iNNF9X9U9V6Pn70MOF1Eqou1A2QQ4HS3r75JCEvvGGNMilh6xxhjUsSCvjHGpIgFfWOMSREL+sYYkyIW9I0xJkX+D6oxyb45W28rAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xkhist_tru, zkhist_tru = tru_simulation(xhat0,P0,Qk,Rk,K)\n",
    "\n",
    "# plot the result \n",
    "plt.scatter(np.linspace(0, 100, 100), xkhist_tru[0:-1, 0],s=20, label = 'Truth State (xk)')\n",
    "plt.title('Truth Simulation Plot')\n",
    "plt.xlabel('Time History (k)', fontsize=15)\n",
    "plt.ylabel('State (xk)', fontsize=15)\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you already have predicted by looking at the equations, the plot is very funky. If you haven't tried try to run it multiple times, and you can see that the plots drastically change every time you run it.\n",
    "\n",
    "The truth states are generated, and now we are going to construct our own state estimation via SIS particle filter and compare by plotting them together. First we have to initialize the variables as our first step. Below function shows the initialization and what kind of data we need to run the filter. <br/>\n",
    "I used a dictionary so that way it's easier to see which variables are which and just pass the whole dict as an argument when running functions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def intializeData():\n",
    "    pfData = {}\n",
    "    pfData['current time step'] = 0                 # k\n",
    "    pfData['number of particles'] = 500             # Ns\n",
    "    pfData['state estimation'] = xhat0              # xhat0\n",
    "    pfData['error covariance'] = 2                  # P0\n",
    "    pfData['process noise covariance'] = 1          # Q0\n",
    "    pfData['next measurement'] = zkhist_tru[0,:]\n",
    "    pfData['measurement noise covariance'] = 0.25   # R0\n",
    "    pfData['particle matrix'] = np.nan              # chi0 \n",
    "    pfData['particle weight'] = np.nan              # wk0\n",
    "    pfData['resample threshold'] = 40\n",
    "\n",
    "    return pfData"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "** Steps of SIS **\n",
    "\n",
    "Now we will look at step by step on how SIS filter works. Note that this code uses a bit of Kalman filter knowledge to solve for some variables, but I don't expect all readers to understand all that. The goal is to have a general idea, and understand the each steps of algorithm below without Kalman filter details.\n",
    "\n",
    "Algorithm is for 1 iteration:\n",
    "\n",
    "**FOR** i = 1 to $N_s$: \n",
    "\n",
    "**1)** draw the particles $\\chi^i_{k+1}$ from importance density $q(x)$ <br/>\n",
    "  - In this case $q(x)$ will be our a priori distribution from the previous timestep. \n",
    "\n",
    "**2)** Assign unnormalized particle weight $\\bar{w}_{k+1}^i$: \n",
    "\n",
    "**END FOR**\n",
    "\n",
    "**3)** Compute the normalize the weight $w_{k+1}^i$: \n",
    "\n",
    "**4)** Compute the estimations $\\hat{x}_{k+1}$ and $P_{k+1}$: \n",
    "\n",
    "When you implement the algorithm into a code:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This function calculates one iteration of SIS particle filter\n",
    "def SIS_singleIter(data):\n",
    "\n",
    "    # extract the variables from the data dict \n",
    "    k = data['current time step']               # int \n",
    "    Ns = data['number of particles']            # int\n",
    "    xhatk = data['state estimation']            # nx x 1\n",
    "    Pk = data['error covariance']               # nx x nx \n",
    "    Qk = data['process noise covariance']       # nv x nv\n",
    "    zkp1 = data['next measurement']             # nz x 1\n",
    "    Rk = data['measurement noise covariance']   # nz x nz \n",
    "    chikmat = data['particle matrix']           # Ns x Nx \n",
    "    wk = data['particle weight']                # Ns x 1\n",
    "\n",
    "    # obtain the variable sizes\n",
    "    Nx = 1\n",
    "    Nz = 1\n",
    "    Nv = 1\n",
    "\n",
    "    # initialize the filter if time step (k) is at 0 other wise skip\n",
    "    if k == 0: \n",
    "        # obtain square root of covariance by taking cholesky factorization \n",
    "        if np.ndim(Pk) == 0:\n",
    "            Sxk = np.sqrt(Pk)\n",
    "        else:\n",
    "            Sxk = np.linalg.cholesky(Pk)\n",
    "        \n",
    "        #######################\n",
    "        # STEP 1 of algorithm #\n",
    "        #######################\n",
    "        \n",
    "        # draw particles and initialize the weight \n",
    "        chikmat = xhatk + Sxk*randn(Ns,Nx)\n",
    "        wk = (1/Ns) * np.ones((Ns,1))\n",
    "\n",
    "    # compute square roots of covariance \n",
    "    if np.ndim(Qk) == 0:\n",
    "        Svk = np.sqrt(Qk)\n",
    "        invRkp1 = 1/Rk\n",
    "    else:\n",
    "        Svk = np.linalg.cholesky(Qk)\n",
    "        invRkp1 = np.linalg.inv(Rk)\n",
    "    \n",
    "    # generate particles for process noise covariance \n",
    "    vkmat = Svk * randn(Ns,Nv)\n",
    "\n",
    "    # particle propagation \n",
    "    chikp1mat = np.zeros((Ns,Nx))\n",
    "    Zkp1mat = np.zeros((Ns,Nz))\n",
    "    logwbarkp1 = np.zeros((Ns,1))\n",
    "    logwk = np.log(wk)\n",
    "\n",
    "    for i in range(0,Ns):\n",
    "        # measurement prediction for particles \n",
    "        chikp1mat[i,:] = dynamics_model(chikmat[i,:],vkmat[i,:],k)\n",
    "        zbarkp1i = measurement_model(chikp1mat[i,:],Rk,k+1)\n",
    "        Zkp1mat[i,:] = zbarkp1i\n",
    "\n",
    "        #######################\n",
    "        # STEP 2 of algorithm #\n",
    "        #######################\n",
    "        \n",
    "        # compute natural log of weights \n",
    "        nukp1i = zkp1 - zbarkp1i\n",
    "        logwbarkp1[i,:] = logwk[i,:] - 0.5 * np.transpose(nukp1i)*invRkp1*nukp1i\n",
    "\n",
    "    #######################\n",
    "    # STEP 3 of algorithm #\n",
    "    #######################\n",
    "    \n",
    "    # normalize the weights \n",
    "    logWbarMax = np.amax(logwbarkp1)\n",
    "    wDoubleBarKp1 = np.exp(logwbarkp1-logWbarMax)\n",
    "    sumWDoulbeBar = np.sum(wDoubleBarKp1)\n",
    "    oneOverSumWDoubleBar = 1 / sumWDoulbeBar\n",
    "    wkp1 = oneOverSumWDoubleBar * wDoubleBarKp1\n",
    "\n",
    "    #######################\n",
    "    # STEP 4 of algorithm #\n",
    "    ####################### \n",
    "    \n",
    "    # update state estimation\n",
    "    wkp1TimesChikp1 = np.multiply(wkp1,chikp1mat) \n",
    "    xhatkp1 = np.sum(wkp1TimesChikp1)\n",
    "\n",
    "    # update A Posteriori Error Covariance\n",
    "    chikp1MinusxhatMat = chikp1mat - xhatkp1\n",
    "    Pkp1 = np.zeros((Nx,Nx))\n",
    "\n",
    "    for i in range(0,Ns):\n",
    "        Pkp1 = Pkp1 + wkp1[i,:] * chikp1MinusxhatMat[i,:] * np.transpose(chikp1MinusxhatMat[i,:])\n",
    "    \n",
    "    return xhatkp1, Pkp1, chikp1mat, wkp1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's run this 100 times to see if our estimates are good or bad. We will use 500 particles.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/junniboi/anaconda3/lib/python3.7/site-packages/ipykernel_launcher.py:51: RuntimeWarning: divide by zero encountered in log\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# This function computes all given iterations of particle filter either SIS or Generic\n",
    "def SIS_multiIter(zkhist_tru):\n",
    "\n",
    "    xhathist = np.zeros((K, Nx))\n",
    "    Phist = np.zeros((Ns, Nx, Nx))\n",
    "    pfData = intializeData()\n",
    "\n",
    "    for k in range(0,K):\n",
    "\n",
    "        kp1 = k + 1\n",
    "        # pass in the current measurement \n",
    "        pfData['next measurement'] = zkhist_tru[k, :]\n",
    "        xhatkp1, Pkp1, chikp1mat, wkp1 = SIS_singleIter(pfData)\n",
    "\n",
    "        # store the estimation\n",
    "        xhathist[k,:] = xhatkp1\n",
    "        Phist[k,:,:] = Pkp1\n",
    "\n",
    "        # iterate \n",
    "        pfData['state estimation'] = xhatkp1\n",
    "        pfData['error covariance'] = Pkp1\n",
    "        pfData['particle matrix'] = chikp1mat\n",
    "        pfData['particle weight'] = wkp1\n",
    "        pfData['current time step'] = kp1\n",
    "    \n",
    "    return xhathist, wkp1\n",
    "\n",
    "def plotResult(xhathist,xhist_tru,pf_type,label,figureNum):\n",
    "    # set up the plot layout \n",
    "    #plt.figure(figureNum)\n",
    "    #plt.rcParams.update({'font.size': 35})\n",
    "    #ax = plt.gca()\n",
    "    #ax.tick_params(axis='both', which='major', labelsize=24)\n",
    "\n",
    "    # plot the result \n",
    "    plt.scatter(np.linspace(0, 100, 100), xkhist_tru[0:-1, 0],s=30, label = 'Truth State (xk)')\n",
    "    plt.scatter(np.linspace(0, 100, 100), xhathist[:, 0], marker = 'x', linewidth = 4,s=30, label = label)#'PF (Ns = 500, Nt = 40)')\n",
    "    plt.title('Truth State vs ' + pf_type)\n",
    "    plt.xlabel('Time History (k)', fontsize=15)\n",
    "    plt.ylabel('System State (x)', fontsize=15)\n",
    "    plt.legend()\n",
    "    plt.grid(True)\n",
    "\n",
    "    return plt\n",
    "\n",
    "\n",
    "if __name__ == '__main__':\n",
    "    \n",
    "    # obtain the variable sizes\n",
    "    Nx = 1\n",
    "    Nz = 1\n",
    "    Nv = 1\n",
    "    \n",
    "    # Run the particle filter for given iteration K \n",
    "    xhathistSIS, wkp1SIS = SIS_multiIter(zkhist_tru)\n",
    "    \n",
    "    plt1 = plotResult(xhathistSIS, xkhist_tru,'SIS Particle Filter', 'PF (Ns = 500)',1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Issues with SIS** \n",
    "\n",
    "**1. Degeneracy**\n",
    "\n",
    "As you could clearly see, that is one bad estimation. The estimated particles missed almost all truth states. It was accurate for only for the first 2-3 iterations of given time history. This problem is called degeneracy. After a few iterations all particles weights collapse except one, meaning all but one particles weight become very close to 0. The variance of importance weights only increase over time, so that means it is impossible to avoid this phenomenon. Below shows the weight distribution \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "500\n"
     ]
    }
   ],
   "source": [
    "print(Ns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8d7d1e0f90>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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XvE51Om9ZfkJJE0laFwwaNIiqqqpmbbC2trbZ626tOmqd16z7mAs/VfgZL4N6sHHZs3Nmsahrp1IWrV209ufcp08fampqGk03+Y/PNth6m/zHZ/nl1wo9Er04NTU1rF+/nunTp3PEEUdQU1PD9ddfT//+/amtreXwww/nyCOPZMCAAbz//vt88pOf5KGHHgLg1ltvZfr06QwYMICamhoigtraWhYvXswPf/hDHnzwQQYMGMA777xDTU0NH374IV26dKGmpoazzjqLa6+9lt133505c+Zw7rnncv/9mz/q+4ADDuCqq66ipqaGWbNm8e1vf5trrrmGmpoaHnnkEUaOHElNTQ3f+MY3OPfccxkzZgxvvvkmJ5xwAnPnzmXt2rWsW7eOmpoazj//fCZOnMiJJ57IrbfeurHua9as4amnnmL27NkMHjyYz33uczz44IOceeaZXH311ZvVL9fatWubtD9sacGh0JB6wQdORMQU0qcmVVRURHNvPOUbdXUc1avW8I2rHi647MJPrefq55Ld/R8XjWFov56lLFq7aO3P+YUXXmj0GoLqVWt48s33Gkzz5JvvsXp9p2Z9Bh988AGHHXYYkLQczj//fLp27crVV1/NH/7wBzZs2MCSJUv45z//ybBhw+jUqRNf/epX6dQpORiQRK9evTbWo256+vTpnHTSSdTd3bluebdu3ejWrRuSmD17NmeeuenR1h9++GHm/aisrOTUU09lm222YcOGDQwePJjdd9+dt956izlz5nDxxRfTu3dvHnnkEV5++eWN69XW1gLJxWpdu3ald+/ezJkzh/vvv5/OnTvz9a9/nR/84Af07t2bnj17MnLkSPbee28ARowYwfLly+ndu3emfrm6d+/OgQceWPR7vaUFh2pg55zpoUChB8ibZQzt15NDhvVrsGtp5LD+ZREY2stb7xV3Lv1b761t1udQN+aQq6qqioceeohZs2bx8ccfc+yxx248p7979+4bA0NDIqLB0z03bNhA3759M9vO17NnT3bffXduu+02DjooOZ9m9OjRzJgxg+XLl7PXXnttzG/WrFn06NGj0bIV0q1bt42vt9lmG9avb/2nom5pF8FNA76WnrU0GlgdEZkuJbP6XHLsvvSsp8uoZ9dOTD52eIlLVF6KvaakNa89Wb16Nf369aNnz5689NJLPP744/Wm7d27d8GuscMPP5y7775745lE77yz+dDodtttx6677so999wDJMHkmWeeKbiNsWPHct111zFmzBgAxowZw/XXX8/o0aM3BqAjjzySG2+8ceM6hYLO6NGjuffeewGYOnVqvXUqpn7NUepTWe8CZgF7SaqWdJak8ySdlyaZASwGFgE/B75RyvLZ1m+/IX24+9wxjBzWf7P523btzN3njvF1Dm2srvXWkNZuvY0fP57169czYsQIrrjiCkaPHl1v2okTJ3LUUUdlBmz33Xdfvv/97/OZz3yG/fffnwsuuCCz7p133smtt97K/vvvz7777st9991XcBtjx45l8eLFG4PDQQcdRHV19cbBaIAbbriBuXPnMmLECIYPH17wrKvrrruOa665hpEjR7Js2TL69Gl8362vfs3RIZ4hXVFREX7YT/HKpc65V0gveuaJsqhzrrYYc9hnn30aTVd3tlKhQemeXTu1aZDuSPdWWrNmDT169EASU6dO5a677ioYkIqtc6HPT9K8iKgolH5LG3MwazVD+/XceIS6qJ3LUk7qWm+XT3+eJ17b1D0zclh/Jh873K23Is2bN49JkyYREfTt25fbbrutpNt3cDCzVrffkD7cfd6YzVpvPhGgaQ477LB6xzVKwcHBzIrW2Fk9+XJbb9Z+mjN8sKWdrWRmW6ju3buzcuXKZv3QWPupe55D9+5NO0PMLQczK8rQoUOprq5mxYoV7V2Ueq1du7bJP4Jbu2LqXPckuKZwcDCzonTp0qVJTxJrD1VVVU26CrgjaKs6u1vJzMwyHBzMzCzDwcHMzDIcHMzMLMPBwczMMhwczMwsw8HBzMwyHBzMzCzDwcHMzDIcHMzMLMPBwczMMhwczMwsw8HBzMwyHBzMzCzDwcHMzDIcHMzMLMPBwczMMhwczMwsw8HBzMwyHBzMzCzDwcHMzDIcHMzMLMPBwczMMkoeHCSNl7RQ0iJJFxdY/glJD0t6StKzko4udRnNzMpdSYODpE7ATcBRwHDgFEnD85L9ALg7Ig4ETgZ+WsoymplZ6VsOI4FFEbE4ItYBU4Hj89IEsF36ug+wtITlMzMzQBFRuo1JE4DxEXF2On0aMCoiJuWkGQz8BegHbAscERHzCuQ1EZgIMGjQoIOnTp3arDLV1tbSq1evZq27tXKdy4PrXB5aUudx48bNi4iKQss6t6hUTacC8/Kj0ynAHRFxtaQxwK8l7RcRGzZbKWIKMAWgoqIiKisrm1Wgqqoqmrvu1sp1Lg+uc3loqzqXulupGtg5Z3oo2W6js4C7ASJiFtAdGFiS0pmZGVD64DAH2EPSrpK6kgw4T8tL8wZwOICkfUiCw4qSltLMrMyVNDhExHpgEvAA8ALJWUkLJF0u6bg02YXAOZKeAe4CzohSDoyYmVnJxxyIiBnAjLx5k3NePw+MLXW5zMxsE18hbWZmGQ4OZmaW4eBgZmYZDg5mZpbh4GBmZhkODmZmluHgYGZmGQ4OZmaW4eBgZmYZDg5mZpbh4GBmZhkODmZmluHgYGZmGQ4OZmaW4eBgZmYZDg5mZpbh4GBmZhkODmZmluHgYGZmGQ4OZmaW4eBgZmYZDg5mZpbh4GBmZhkODmZmluHgYGZmGQ4OZmaW4eBgZmYZDg5mZpZRdHCQ9AlJXepZ1lnSJ1qvWGZm1p6a0nJ4FTiwnmX7p8vNzKwDaEpwUAPLugMfFpWJNF7SQkmLJF1cT5qTJD0vaYGk/2lCGc3MrBV0bmihpBHAATmzjpa0d16y7sBJwEuNbUxSJ+Am4HNANTBH0rSIeD4nzR7A94CxEbFK0g5F1cTMzFpNg8EBOAG4JH0dwOR60r0KnFvE9kYCiyJiMYCkqcDxwPM5ac4BboqIVQARsbyIfM3MrBUpIupfmAxAdyXpUnoP+CwwJy/Zuoj4qKiNSROA8RFxdjp9GjAqIiblpPkjSStkLNAJuDQi/rdAXhOBiQCDBg06eOrUqcUUIaO2tpZevXo1a92tletcHlzn8tCSOo8bN25eRFQUWtZgyyH90a/74W+N014LjVvkR6fOwB5AJTAU+Luk/SLi3byyTQGmAFRUVERlZWWzClRVVUVz191auc7lwXUuD21V58a6lTIk7Unyo909f1lEzGhk9Wpg55zpocDSAmkeTwPTq5IWkgSL/BaLmZm1kaKDg6ThwG+B4dTfAujUSDZzgD0k7QosAU4GvpKX5o/AKcAdkgYCewKLiy2nmZm1XFNaDreQjD/8H5IB5HVN3VhErJc0CXiAJJDcFhELJF0OzI2IaemyIyU9D3wMfCciVjZ1W2Zm1nxNCQ4HAidHxP0t2WDa9TQjb97knNcBXJD+mZlZO2jKIPMrFBhnMDOzjqcpweFC4D8k7dZWhTEzsy1DY1dIz2HzU02HAC9Keg14Nz99RIxs1dKZmVm7aGzMYQGbB4cFbVgWMzPbQjR2EdwZJSqHmZltQfywHzMzy2jKRXC3NbB4A8m9l54Gfh8RtS0tmJmZtZ+mXOfwKZJbX+wAvAWsALYHBgHLgdXAJOBKSYdHRKO38DYzsy1TU7qVJpOcoTQqIgZHxIiIGAyMJgkM3wH2AmqA/2r1kpqZWck0JTj8GLgkIja7AV5EPAFcClwVEa8CPwL+pdVKaGZmJdeU4LA78EE9y9YAw9LXrwPdWlAmMzNrZ00JDk8Bl0jaMXempMEkT4ubl87ahextuM3MbCvSlAHp80jumPqapHlsGpCuAFYCn0/T7QT8vDULaWZmpVV0cIiIZ9P7Kn2dJCDsSPI4zzuB2yPigzTdj9qioGZmVjpNehJcGgBuaqOymJnZFsJXSJuZWUZjd2VdDnw+Ip6StILNb8KXERE7tGbhzMysfTTWrXQTydXQda8bDA5mZtYxNHZX1styXl/a5qUxM7MtQpMGpAEk9QP2I7nP0p8jYpWk7sC6iNjQ2gU0M7PSK3pAWlJnST8GqoFHgF8Du6aL7yW5EM7MzDqAppytdCVwDsmdV3cDlLPsPuDYViyXmZm1o6Z0K30NuDgibpfUKW/ZKyQBw8zMOoCmtBz6kgSBQroC+QHDzMy2Uk0JDvOB4+tZdhTwZMuLY2ZmW4KmdCtdAdwrqQdwD8k1DwdIOgE4FziuDcpnZmbtoOiWQ0TcB3wFOAL4M8mA9C+AM4DTIuKBtiigmZmVXmO3zxgNzIuIjwAi4m7gbkl7AQOAd4CFEeErp83MOpDGupVmAmslzU1fPwY8FhEL27xkZmbWbhrrVvo8yTOh15CMK9wHrJD0oqRbJX1d0t5N2aCk8ZIWSlok6eIG0k2QFJIqmpK/mZm1XGP3VnoQeBBAkoB9gbHAGOAw4EwgJK0CZkZEg4PS6fURNwGfI7nSeo6kaRHxfF663sA3gdnNqZSZmbVMUwakIyLmR8QtEXEGsBfwWZLB6X7AF4rIZiSwKCIWR8Q6YCqFT4/9IfBjYG2x5TMzs9ZT9KmskrYFRgGHpn+jgd7A8yTPjJ5VRDZDgDdzpqvTPHO3cyCwc0TcL+nbDZRnIjARYNCgQVRVVRVblc3U1tY2e92tletcHlzn8tBWdW7sbKWvsCkYjADeBR4nGZy+GpgdEbVN2J4KzNt4ppOkbYBrSU6PbVBETAGmAFRUVERlZWUTirFJVVUVzV13a+U6lwfXuTy0VZ0bazn8Bngf+BXJtQwLWri9apJbfdcZCizNme5NcjvwqmSIgx2BaZKOi4i5Ldy2mZkVqbExh/8CniI5kn9C0qOSrpL0RUmDmrG9OcAeknaV1BU4GZhWtzAiVkfEwIgYFhHDSFopDgxmZiXW2NlKF0HyLAfgQJKzlA4FTgGGSnqVZKxhFjArIhq8v1JErJc0CXiA5EZ9t0XEAkmXA3MjYlpD65uZWWkUNSAdEetJjvrnADcASBpCEihOB64vNr+ImAHMyJs3uZ60lcWUz8zMWleTHhMqqRtwCJsGqccA26eL32jdopmZWXtp7GylndgUCA4FDiB5dsN64GngLjbdUmNpffmYmdnWpbGWQzXJqaarSMYVLiM5jfWJiPigjctmZmbtpLHgcDbJbTFeLEVhzMxsy9DY2Uq3laogZma25WjKY0LNzKxMODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZlllDw4SBovaaGkRZIuLrD8AknPS3pW0l8l7VLqMpqZlbuSBgdJnYCbgKOA4cApkobnJXsKqIiIEcDvgB+XsoxmZlb6lsNIYFFELI6IdcBU4PjcBBHxcESsSScfB4aWuIxmZmVPEVG6jUkTgPERcXY6fRowKiIm1ZP+RuCfEXFFgWUTgYkAgwYNOnjq1KnNKlNtbS29evVq1rpbK9e5PLjO5aEldR43bty8iKgotKxzi0rVdCowr2B0kvRVoAL4TKHlETEFmAJQUVERlZWVzSpQVVUVzV13a+U6lwfXuTy0VZ1LHRyqgZ1zpocCS/MTSToC+D7wmYj4sERlMzOzVKnHHOYAe0jaVVJX4GRgWm4CSQcCtwDHRcTyEpfPzMwocXCIiPXAJOAB4AXg7ohYIOlyScelyf4L6AXcI+lpSdPqyc7MzNpIqbuViIgZwIy8eZNzXh9R6jKZmdnmfIW0mZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWU4OJiZWYaDg5mZZTg4mJlZhoODmZllODiYmVmGg4OZmWUoItq7DC1WUVERczi3RdkAAAn3SURBVOfObfJ61avW8OTsx3i9yy4sefcDhOjVvTM1az9q9HWd9kzb3PU+uf4NXun8iTbdRinq0VhagO16dIEItq9dXBZ1zn097KPXmf/xkK1+f93S6rwl/Q7s1K8Hu3z4GgeNHsvQfj1pKknzIqKi0LLOhWZ2dPOXrOa7v3uG55fVcOGn1nP1cy+3d5FK6sJPrefnz73a3sUoqXKt813PvdnexSipcq3zN696mOGDt+PHE0aw35A+rZJvybuVJI2XtFDSIkkXF1jeTdJv0+WzJQ1rze3PX7KaCT+byfPLalozWzOzdvX8sveYcPNM5i9Z3Sr5lTQ4SOoE3AQcBQwHTpE0PC/ZWcCqiNgduBa4qjXLcNn0Baxdv6E1szQz2yKs/WgDl09/vlXyKnXLYSSwKCIWR8Q6YCpwfF6a44Ffpq9/BxwuSa2x8epVa5jz2qrWyMrMbIv0xGvvUL1qTYvzKemAtKQJwPiIODudPg0YFRGTctLMT9NUp9OvpGnezstrIjARYNCgQQdPnTq10e2vWfcxr6yo3WzeoB7w1gctqtZWx3UuD65zeShU509u34ueXTs1uu64ceO2mAHpQi2A/OhUTBoiYgowBZKzlSorKxvdePWqNXzjqoc3m5cMSJfXuLzrXB5c5/JQqM7/uGhMs85eylXqbqVqYOec6aHA0vrSSOoM9AHeaY2ND+3Xk0OG9WuNrMzMtkgjh/VvcWCA0geHOcAeknaV1BU4GZiWl2YacHr6egLwt2jFvq9Ljt2X7p197Z+ZdTzdu2zD5GPzz/FpnpL+SkbEemAS8ADwAnB3RCyQdLmk49JktwIDJC0CLgAyp7u2xH5D+vC7fz2UfQdv15rZmpm1q3132o7fnXdoq13n4CukfYV0u9fZV0i3zTZ8hfTWW49i0/oK6TYytF9PFnXvwr9V7tneRSmpqqrlnFzZOk3PrUVV1dIyrPNyTq0c0d7FKKnyrPOSVhljyOfOdzMzy3BwMDOzDAcHMzPLcHAwM7MMBwczM8twcDAzs4wOcZ2DpBXA681cfSDwdqOpOhbXuTy4zuWhJXXeJSK2L7SgQwSHlpA0t76LQDoq17k8uM7loa3q7G4lMzPLcHAwM7MMB4f0mRBlxnUuD65zeWiTOpf9mIOZmWW55WBmZhkODmZmllHWwUHSeEkLJS2S1KoPFWpPkm6TtFzS/Jx5/SU9KOnl9H+/dL4k3ZC+B89KOqj9St48knaW9LCkFyQtkPStdH5HrnN3SU9Ieiat82Xp/F0lzU7r/Nv0iYtI6pZOL0qXD2vP8reEpE6SnpJ0fzrdoess6TVJz0l6WtLcdF6b79tlGxwkdQJuAo4ChgOnSOooN/y/AxifN+9i4K8RsQfwVzY9Ye8oYI/0byLwsxKVsTWtBy6MiH2A0cD56WfZkev8IfDZiNgfOAAYL2k0cBVwbVrnVcBZafqzgFURsTtwbZpua/UtkidJ1imHOo+LiANyrmdo+307IsryDxgDPJAz/T3ge+1drlas3zBgfs70QmBw+nowsDB9fQtwSqF0W+sfcB/wuXKpM9ATeBIYRXKlbOd0/sZ9nOTRvGPS153TdGrvsjejrkPTH8PPAvcDKoM6vwYMzJvX5vt22bYcgCHAmznT1em8jmpQRCwDSP/vkM7vUO9D2nVwIDCbDl7ntHvlaWA58CDwCvBuJM9qh83rtbHO6fLVwIDSlrhVXAd8F9iQTg+g49c5gL9ImidpYjqvzfftcn5MqArMK8fzejvM+yCpF3Av8O8R8Z5UqGpJ0gLztro6R8THwAGS+gJ/APYplCz9v9XXWdIxwPKImCepsm52gaQdps6psRGxVNIOwIOSXmwgbavVuZxbDtXAzjnTQ4Gl7VSWUnhL0mCA9P/ydH6HeB8kdSEJDHdGxO/T2R26znUi4l2gimS8pa+kuoO+3HptrHO6vA/wTmlL2mJjgeMkvQZMJelauo6OXWciYmn6fznJQcBISrBvl3NwmAPskZ7p0BU4GZjWzmVqS9OA09PXp5P0y9fN/1p6lsNoYHVdc3VroaSJcCvwQkRck7OoI9d5+7TFgKQewBEkg7QPAxPSZPl1rnsvJgB/i7RTemsREd+LiKERMYzk+/q3iDiVDlxnSdtK6l33GjgSmE8p9u32Hmxp54Geo4GXSPpqv9/e5WnFet0FLAM+IjmSOIukr/WvwMvp//5pWpGctfUK8BxQ0d7lb0Z9P03SdH4WeDr9O7qD13kE8FRa5/nA5HT+bsATwCLgHqBbOr97Or0oXb5be9ehhfWvBO7v6HVO6/ZM+reg7neqFPu2b59hZmYZ5dytZGZm9XBwMDOzDAcHMzPLcHAwM7MMBwczM8twcLCyJOlSSZHzt1TSvZI+2Qp575DmPyxvfmW6rf2akFeVpN+1tExmTeXgYOVsNcmN2sYA3ya5u+lf04uNWmIH4BKSmx/mejLd1istzN+szZXzvZXM1kfE4+nrxyW9Afyd5AK6e5qToaTu9S2LiPeAx+tbbrYlccvBbJN56f9hksZImpZ2N72fPmjl1NzEks5Iu4lGpt0/HwDfIbkyFeDhum6rNH2mWym9s+r3JL0k6UNJ1ZLuaKiQkvaT9CdJNenfPZJ2bLV3wQy3HMxyDUv//xPYBXgMuBlYS3LTt9slbYiIu/LWu4vkoSqXAWtIuo3uBM4n6UpqyC3A14AfA48A/dl0n6AMSbun5ZoLnAZ0An4ITJc0MnzLA2slDg5W1nLu5rkb8FOgBngocm5Wlt7Y71GSO1yeQxIMct0QEdfnpH8/ffl8TrdVoW3vTXLfq29FxA05i37bQJEvIQleR0XEujSfZ4EXSbrD/tTAumZFc7eSlbMBJDcn/IjkiVm7AV+OiGWS+qXP4n09J81EYM8C+TT3B3lc+v+OJqxzBMltmzdI6pwGt1dJnhZW0dCKZk3hloOVs9UkP7ZBcjS+NKdb5g6S5yP8EHgeeA/4V+D4Avm81cztDwDeTweqizUQuCj9y7dzgXlmzeLgYOVsfUTMzZ+ZnnH0BWBSRNycM7++lnZz+/lXAttK2q4JAeIdkpbDLwose7uZ5TDLcHAwy+pGMtD7Yd2M9IErx1FcIFiX/q/3tNbU39L/XwNuLLJsfwX2A+Z58NnakoODWZ6IWC1pDjBZ0nskD7O/mKQbarsisngD+AA4XdJq4KNCLZSIWChpCnB1+nzgR4G+wISIOLmevC8leXDNnyTdRtJaGAJ8DrgjIqqKr6lZ/RwczAr7CjAF+BVJ98+NQE9gUmMrRsRaSeeQnFn0CNCFwg9+B/gG8DpwNkkAWg482EDeL6WPf7wiLV8PYAlJi2JRMRUzK4afBGdmZhk+ldXMzDIcHMzMLMPBwczMMhwczMwsw8HBzMwyHBzMzCzDwcHMzDIcHMzMLOP/A8cjmUqRwLi/AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the weight distribution along the particles for degeneracy example \n",
    "\n",
    "plt.scatter(np.linspace(0,500,500),wkp1SIS,marker = 'o', linewidth = 4,s=30, label = 'Particle Weight')\n",
    "plt.title('Particle Weight Distribution During Degeneracy')\n",
    "plt.xlabel('Particle', fontsize=15)\n",
    "plt.ylabel('Weight', fontsize=15)\n",
    "plt.grid(True)\n",
    "plt.legend() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see from the plot above, one particle has all the weight and the other 499 particles weights went to 0. Once degeneracy occurs, the filter is no longer reliable which is most cases for SIS .\n",
    "\n",
    "# Resampling \n",
    "\n",
    "One of the solutions is to resample. The idea is to resample from the distrbution $p[x_k | Z^k]$ every time degeneracy occurs. Particles with more weights (likelihood) gets sampled more than once. The bottom tier particles are eliminated. Finally generates new set of $\\chi_k^i$ by resampling $N_s$ times fromapproximate discrete representation of $p(x_k | z_1,\\dots,z_k)$ \n",
    "\n",
    "The function below shows how to resample. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def resample(chikp1mat,wkp1,Ns,Nx):\n",
    "    \n",
    "    # construct the CDF, c0 is zero\n",
    "    cVec = np.zeros((Ns,1))\n",
    "    for i in range(1,Ns):\n",
    "        cVec[i,0] = cVec[i-1,0] + wkp1[i,:]\n",
    "    \n",
    "    # draw starting point - sample eta from uniform distribution \n",
    "    oneOverNs = 1/Ns\n",
    "    eta1 = oneOverNs * uniform(0,1)\n",
    "    \n",
    "    # For each eta(j), find i such that c(i) <= eta(j) < c(i+1)\n",
    "    chikp1matNew = np.zeros((Ns,Nx))\n",
    "    jVec = np.zeros((Ns,1))\n",
    "    \n",
    "    # move along the cdf \n",
    "    for j in range(0,Ns):\n",
    "        etaj = eta1 + j * oneOverNs\n",
    "        i = 0\n",
    "        while etaj > cVec[i,0] and i < Ns-1:\n",
    "            i = i+1\n",
    "        \n",
    "        jVec[j,0] = i\n",
    "        chikp1matNew[j,:] = chikp1mat[i,:].copy() \n",
    "    \n",
    "    # normalize the weights again\n",
    "    chikp1mat = chikp1matNew.copy()\n",
    "    wkp1 = oneOverNs * np.ones((Ns,1))\n",
    "\n",
    "    return chikp1mat, wkp1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The key features of this algorithm is that high weight particles get sampled multiple times and low weight particles get eliminated. he algorithm not only prevents degeneracy, but also  decreases loss of diversity of particles by always sampling $\\chi_k^i$ with $w_k^i \\geq \\frac{1}{N_s}$. The particles remain diverse because the resampling occurs after the weight propagation.\n",
    "\n",
    "# Generic Particle Filter \n",
    "\n",
    "Now we are at the final stage. There were a lot of materials in which most are hard to consume at once, but this is the complete form of a basic particle filter. We are going to combine all the improvements we made to create a more robust particle filter with more details. \n",
    "\n",
    "** Effective Number of Particles**\n",
    "\n",
    "Let $N_{eff}$ be the effective number of particles:\n",
    "    $$N_{eff}\\ =\\ \\frac{1}{\\sum\\limits_{i=1}^{N_s} w_k ^i},\\quad where\\quad 1 < N_{eff} < N_s$$\n",
    "    \n",
    "Small $N_{eff}$ means degeneracy, so we want to set a certain threshold $N_T$ and resample every time $N_{eff} < N_T$\n",
    "\n",
    "Below is the code for single iteration of generic particle filter. It is similar to SIS, but now with resampling and particle threshold `Nt`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def generic_pf(data,resampleFlag):\n",
    "\n",
    "    # extract the variables from the data dict \n",
    "    k = data['current time step']               # int \n",
    "    Ns = data['number of particles']            # int\n",
    "    xhatk = data['state estimation']            # nx x 1\n",
    "    Pk = data['error covariance']               # nx x nx \n",
    "    Qk = data['process noise covariance']       # nv x nv\n",
    "    zkp1 = data['next measurement']             # nz x 1\n",
    "    Rk = data['measurement noise covariance']   # nz x nz \n",
    "    chikmat = data['particle matrix']           # Ns x Nx \n",
    "    wk = data['particle weight']                # Ns x 1\n",
    "    Nt = data['resample threshold']             # int\n",
    "    \n",
    "    # obtain the variable sizes\n",
    "    Nx = 1#len(xhatk)\n",
    "    Nz = 1#len(zkp1)\n",
    "    Nv = 1#len(Qk)\n",
    "\n",
    "    # initialize the filter if time step (k) is at 0 other wise skip\n",
    "    if k == 0: \n",
    "        # obtain square root of covariance by taking cholesky factorization \n",
    "        if np.ndim(Pk) == 0:\n",
    "            Sxk = np.sqrt(Pk)\n",
    "        else:\n",
    "            Sxk = np.linalg.cholesky(Pk)\n",
    "        \n",
    "        chikmat = xhatk + Sxk*randn(Ns,Nx)\n",
    "        wk = (1/Ns) * np.ones((Ns,1))\n",
    "\n",
    "    # compute square roots of covariance \n",
    "    if np.ndim(Qk) == 0:\n",
    "        Svk = np.sqrt(Qk)\n",
    "        invRkp1 = 1/Rk\n",
    "    else:\n",
    "        Svk = np.linalg.cholesky(Qk)\n",
    "        invRkp1 = np.linalg.inv(Rk)\n",
    "\n",
    "    # generate particles for process noise covariance \n",
    "    vkmat = Svk * randn(Ns,Nv)\n",
    "\n",
    "    # particle propagation \n",
    "    chikp1mat = np.zeros((Ns,Nx))\n",
    "    Zkp1mat = np.zeros((Ns,Nz))\n",
    "    logwbarkp1 = np.zeros((Ns,1))\n",
    "    logwk = np.log(wk)\n",
    "\n",
    "    for i in range(0,Ns):\n",
    "        # measurement prediction for particles \n",
    "        chikp1mat[i,:] = dynamics_model(chikmat[i,:],vkmat[i,:],k)\n",
    "        zbarkp1i = measurement_model(chikp1mat[i,:],Rk,k+1)\n",
    "        Zkp1mat[i,:] = zbarkp1i\n",
    "\n",
    "        # compute natural log of weights \n",
    "        nukp1i = zkp1 - zbarkp1i\n",
    "        logwbarkp1[i,:] = logwk[i,:] - 0.5 * np.transpose(nukp1i)*invRkp1*nukp1i\n",
    "\n",
    "    # normalize the weights \n",
    "    logWbarMax = np.amax(logwbarkp1)\n",
    "    wDoubleBarKp1 = np.exp(logwbarkp1-logWbarMax)\n",
    "    sumWDoulbeBar = np.sum(wDoubleBarKp1)\n",
    "    oneOverSumWDoubleBar = 1 / sumWDoulbeBar\n",
    "    wkp1 = oneOverSumWDoubleBar * wDoubleBarKp1\n",
    "\n",
    "    # update state estimation\n",
    "    wkp1TimesChikp1 = np.multiply(wkp1,chikp1mat) \n",
    "    xhatkp1 = np.sum(wkp1TimesChikp1)\n",
    "\n",
    "    # update A Posteriori Error Covariance\n",
    "    chikp1MinusxhatMat = chikp1mat - xhatkp1\n",
    "    Pkp1 = np.zeros((Nx,Nx))\n",
    "\n",
    "    for i in range(0,Ns):\n",
    "        Pkp1 = Pkp1 + wkp1[i,:] * chikp1MinusxhatMat[i,:] * np.transpose(chikp1MinusxhatMat[i,:])\n",
    "\n",
    "    # Resample if degeneracy (N_eff < Nt occurs)\n",
    "    wkp1Squared = np.square(wkp1)\n",
    "    Neff = np.sum(wkp1Squared)\n",
    "    if resampleFlag:\n",
    "        if Neff < Nt: \n",
    "            chikp1mat,wkp1 = resample(chikp1mat,wkp1,Ns,Nx)\n",
    "\n",
    "    \n",
    "    return xhatkp1, Pkp1, chikp1mat, wkp1\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot this to see how much improvement we have made. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WrVoBkJGRQV1dnc9tcnJyOOecczjnnHPo1KkTixcvbub03333XT744AM+/fRTcnNzOeWUUzh8+HCzY6kql19+OX/+858D2pWbm9vI9urqasrLy6mvr6e6urqhzfn5+XTv3p2PP/64kdM/dOgQubmpVduTmuEdOOL4L5oLHftYf43DNyQRs047hjatMsl0WQU4WS6hdatMLv9J9O9hl8tFhw4d2LRpE263m0WLFjWsa9euHfv27QvpeKtXr6a8vBywBmXXrVvH0UcfTWam1e/0/FBUVVXRsWNHcnNz+eqrr1i5ciVAs+3OOOMMXnrpJXbu3AlYWUnFxcXNzjto0CA2b97c8P7GG2/k0ksvZfbs2fzqV79qWN6qVSsWL17M008/zUsvvdSwfOPGjQwePDiktiY6qev0wXLwnkHbQZOMwzckFd3zc3nz2jFMHdWLoT3zuGRUL968dgxd2+fE5Pz3338/Z511Fqeffjo9e/ZsWH7JJZdwzz33NBrIDcYPP/zAxIkTG1I2c3Nz+fWvfw3AzJkzGTJkCNOnT2fixIkcOHCAoUOHcueddzJq1KiGY3hvd/zxx/OnP/2JM844gyFDhjBhwgQqKiqanXfixIksX74cgKVLl7J27Vquv/56ZsyYgdvtZu7cuQ3btm3bliVLlnD//ffz+uuvA/Dee+8xceLEUD+6xMaf/GaivIy0cuiY9iYuX3/9ddjH2Lt3bwQsSR7Cae/+/ft19OjRWldXF/K+Bw4c0FGjRrVo35bSkrb6uqcIIK2c2j19g8GQ1rRu3ZrZs2c3hJZCobi4mAceeICMjIwoWBY/UnMg12AwGGzOPvvsFu03cOBABg4cGGFr4o/p6RsMBkMaYZy+wWAwpBHG6RsMBkMaYZy+wWAwpBHG6RsMiUxlyRGtqKIlEZEFz8jIYNiwYQwePJgLL7yQAwcONFruefnKwS8vL2fSJKv2Zfny5YgI//rXvxrWT5o0qSEvPtKMHTuWgQMHNti3fbulpHv48GGmTJlCv379GDVqVCO77733Xvr168fAgQP597//7egcHmkHgFWrVjF27FjAklp+4403Itqm3/72t7Rte6TC2l9b1q1bx6WXXhqRcxqnbzAkKpUllpTIS9PgxanW32cnIXvDk/rNzc1lzZo1rF+/nuzsbJ588slGyz0vj5iaNw8//DBXXnllw/uePXty9913h2VPKLzwwgsN9nXp0gWAp59+mg4dOrB582auu+46/vjHPwKWgNz8+fP56quveOutt7j66qupr68Peo7t27fz5ptvNlseaae/atUqKisrGy177rnnfLbl+OOPp7S01GfVcagYp28wJCIeh79nG6gbvnnd+rtnG61fuihiEwGNGTOmkUxBMF599VXOOuushvdDhw4lLy+Pd955p9m2N998M8cddxxDhgzhhhtuiIi9vli8eDEzZswA4Oc//zlLly5FVVm8eDEXX3wxrVq1ok+fPvTr148VK1YEPd6NN97IXXfd1WhZTU0Ns2fPZsGCBQwbNixsueX6+npuvPFGHnjggUbLX3/9dZ9tAUty2peSaKiYPH2DIREpX2uJBfpA9pZY68OUFamrq+PNN99scOIHDx5smEGqT58+jfR2ALZu3UqHDh0axNU83Hbbbdx2222ceeaZDct2797NokWL2LBhAyLSrEcLlsTBdddd12x569atfWryA1x22WVkZGRwwQUXcNtttyEifP/99w0Km5mZmeTl5bFr1y6+//57Ro8e3bBvz549HU2I8uMf/5hFixbx3nvv0a5dOwCys7O58847WbVqFY8//nizfb755humTJni83jLly9vNDcBwOOPP87kyZPp1q1bo+Xl5eU+29KpUydGjBjBfffdx0033RS0DYEwTt9gSEQGTYIBZ1s9/CbU9T2TrDAmAvJ27mPGjGHmzJnAkfCOP5rKFHsYM2YMQCPd+fbt25OTk8MVV1zBxIkTG8YBvBk3blzQaRa9eeGFF+jRowf79u3jggsuYO7cuUyfPr2RGqgHEfG73Am33XYbd911F/fff7+j7QcOHOi4LWVlZbz88ss+xz4C2RwpmWfj9A2GRKRoCWxsHlcGyNzyjrW+hY4/mHMPtJ8/ieVbb72Vu+++u0ENMzMzkxUrVrB06VLmz5/P448/3myWqlB7+p6JXdq1a8fUqVNZsWIF06dPp2fPnpSUlNCzZ0/q6uoalDo9yz2UlpY20+X3x/jx47n99tv59NNPHW0fSk//iy++YPPmzfTr1w+AAwcO0K9fPzZv3kz37t19tgUiJ/NsnL7BkIgEmAhI2xcicZgIaMCAAX5VNSdMmMDtt9/e0BOtrq7mwIEDnHPOOYwePbrBwXkTSk+/rq6OyspKOnXqRG1tLUuWLOGMM84AYPLkyTz77LP8+Mc/5pVXXmH8+PGICJMnT2bq1Kn84Q9/oKysjE2bNjFy5EgATj/9dJ577jm/M4SB9UM2a9Ys+vbtCwSWlA6lpz9x4kR++OGHhvdt27ZtGFc555xzfLYFIifzbAZyDYZExDMfRIfe1gRAAydafzv05sBFL8VFJrxNmzYcc8wxfgd+b731VkpLSwHYt28fkyZNYsiQIZx22mk88sgjYZ378OHD/PSnP2XIkCEMGzaMHj16NGQRzZw5k127dtGvXz8efvjhhgnUf/SjH3HRRRdx3HHHcdZZZ/HEE0+QkZGB2+1m8+bNDT1of5xzzjmNwlnjxo3j66+/jshArj+mT5/usy0QQZlnf/KbifIy0sqhY9qbuIQsrbynWPXrf9k7/0t1T3FcpZUXLlyot956a0zPGen2rlu3Tq+77rqIHjNS+GvroUOHdNSoUVpbW9tsXajSyia8YzAkMvmFR3r1nhh+iLNWRZLzzjuPXbt2xe38kWDw4ME8/PDD8TYjJIqLi7nvvvsaxkzCIeZOX0S2AfuAeqBOVUcE3sNgSC1U1XEWSSJyxRVXxNuEtKN///7079+/2XL1ke0TjHj19Mep6s44ndtgiBs5OTns2rWLo446KqkdvyH+qCq7du0iJye06TNNeMdgiCE9e/aktLSUHTt2tPgYhw4dCvmLnsykU3tDbWtOTk6j+YudIC15PAgHEdkK7AEU+D9VfcrHNlcBVwEUFBQMD7f0uLq6upGoUapj2pvamPamLpFq67hx41b7DZ37G+GN1gvobv/tAqwFTg20vcneCR3T3tTGtDd1iVRbSaSJ0VW1zP67HVgEjIy1DQaDwZCuxNTpi0gbEWnn+R+YAKyPpQ0Gg8GQzsR6ILcAWGRnLWQC81T1rRjbYDAYDGlLTJ2+qm4BYi8aYjAYDAbAaO8YDAZDWmGcvsFgMKQRxukbDAZDGmGcvsFgMKQRjgdyRSQPGAF0BXKA3cBGVf0qSrYZDAaDIcIEdPoi0h6YBkwHhtP8yUBFpAqryOrvqvqfqFhpMBgMhojgN7wjIrcB24DfA+8B5wN9gHZANpaMwijgFuAoYJmILBOR8OfzMhgMBkNUCNTTHw1MVtWP/Kzfab9WAU/a4Z9ZwMmYKluDwWBISPw6fVWdFMqBVLUKuD9siwwGg8EQNRxl74hIQIFnEekeGXMMBoPBEE2cpmyuEZFRvlaIyAxMOMdgMBiSAqdOfyPwkYjcIyJZACLSRUT+CTwN/D1aBhoMBoMhcjhy+qo6GWsmq18Dq0TkWuAr4DhgjKreFD0TDQaDwRApHBdnqeocEfkC+A/wMPAFcIqqHoqWcQaDwWCILI5lGERkMvAmUAY8DgwBFohIlyjZZjAYDIYI4zR751ngn8BiYIiqXgv8BOgPfC0iU6JnosFgCJnKEihaYv1ftMR6bzDgPLwzHjhLVd/2LFDVVSJyAnA38DywIAr2GQyGUKksoW7ORFxVJazIHsnImhW48wrJvOx1yC+Mt3WGOOM0vDPY2+F7UNXDqnoDcFpkzTIYDC3CdviZVd/hws3omk9x4Saz6jvq5kw0PX6D4+ydqiDrP4mMOQaDISzK1+Kq8u3YXVUlUL42xga1EO/w1KEq82MVQQIJrj0tIv2dHkhEskTkchGZFhnTDAZDyAyaxIrskT5XrcgeCYNCUleJD/bTinvBND6956ewZ6t5SokggXr6B4C1IvKhiFwrIieJSK73BiJytIicLyJ/w8rq+R3wTRTtNRgMgShawsiaFT5XjaxZcaT3nKj4CE8BJjwVQfw6fVX9LTAQ+AC4HvgMqBaR/SKyW0TqgC3AfKyJVa5U1WGq6vuOMxgM0afbUNx5vgdr3XmF0G1ojA0KkVQJTyUwAWP6qlqiqreqai/gWOCXwG3Ag8A1wOlAvqr+TFX/GXVrDQZDYPKtLJ26vKNx4+LT7NG4cVGXd3RyZO8EC0+B6e2HSSgVuRuxNHgMBkMiYzt+ytcyetAkKFqCq9vQxHf4EDA8NarmU9wLPsOd1ys5fsASFDMxuiF9SeUCpvzCI4O2gybFx0G25PMNEJ4SwIWa+H6YOO7ppwSVdkzQ7v0QSu8nnH0NiYcpYIouLf18PeGpORNxVRUjqM/NGuL75lqFTFx6+iKSISJfiEjsUgkqS+DZSfDSNHhxqvX32UnOegtNUsjcC6aZnkYyk84FTLF4ugn387Udv2vK83yWPdrnJkmTfpqAxCu8cy1QFLOz1ddaDn7PNlA3fPO69XfPtuCOP50dRKqSShkioTjxcDo+oRCJz9fuwSd1+mmCEnOnLyI9gYnEcuKV2gNoZbHPVVpZHPgmTCUHYbBIhQImCO0J1OPwW9LxCZVIfb7Jnn6aoIiq75hZsw1FWgGXAyOAQuAaVd1kK2x+qaqOeu4i8gpwL9AOuMHXBOwichXWpC0UFBQMnz9/viMb/VFdXU3mgR/Iqa9utu5QRltyuvQLuP/+HzbRRvc3Xy5taNPVcdFyzKiurqZt27bxNiNmhNzeQ1WwZ6v/9R36QE5e+IZFierqatrmtoJdm6C+pvkGGdlwVH/IyDqyLJZtjuS56mup3ltJ20PfW9833e+7fSlCpL6748aNW62qI3ytczSQKyIDgHeAPGA1MBbLaQOMweq5T3dwnEnAdlVdLSJj/W2nqk8BTwGMGDFCx471u6kjlr+1mDFf30gG7mbr6nGRcfxc/72PoiW4l9+Ey8e+bly4hgbYN04sX76ccD+zZCLk9laWUDfndjKrvmu2qi7vaDLPTuzB3OXLlzO2oBr3h7f6vy+nzIVBZzZaXvTIJAZVfdhs+6K8MQy6JIKhkgh/vsuXvsPYY3PTIokiFt9dp+Gd/wWKgd7AT7Gypzy8D5zi8DgnA5NFZBtWJe94EXne4b4tJ6s1Vdldfa6qyu4a+DHRPGKmHslewAShh1CKljCg6mOf2w+o+jiy8fFIf74ZWfFPP00hnDr9McC9qloJzXKoKoBuTg6iqv+tqj1VtTdwMbBMVX/p1NgWk5FF7S9fo4QC6lV4u3449SqUUEDtL19zlkKWzA7C0JyGDJG5jL7l37imzE2u6xmqxk63oS3v+LSEZP98UxinefqHgFw/63oAlZExJ3oU9OpPxeWvs+C9d1iwbwg7233J6ePOpKCXg5h8Mlc4GvyTX3jkGiZYiC4o9hOoy0cIxZ1XaN2f3uQXWh2fZybSXbez1H0ip7s+p0y6kB2s49NSkvnzTWGcOv13gFtE5F3AMxqq9uDub4E3Qj2xqi4Hloe6XzgU9OrP1Bn9mQo4j0jZmBvYkEg0KmJyVgAVVsfHkDI4dfo3Ah8Dm7F+ABSYDfwIyAbOj4p1hvTDVD47pwVPoGF1fAwpgSOnr6olIjIU+AOWsua3WHH8l4GHVXVX9Ew0pA1GGiF0zBOoIUQcF2ep6h5VvV1Vf6KqA1R1tC27bBx+qpMMpftBjp2ywmoGQ4g4zdPfApynqs3KT0VkMPCaqvaNtHGGBMBTyVlZDAPOho1vQn4vmLEksr1vJ5XPLTlfrOw3GJIEpz393kArP+taAz0jYo0h9gTqBSdj6b43sbTfYEgS/Pb0RaQ9kO+1qKuI9GqyWQ5Wvv33UbDNEG2CxdDL16KVxY0q8TxoZTESSWlbJ3nnoTr+WNpvMCQJgXr61wHbgK1Y2TqL7P+9X0XA77Eqdg3JhJMY+qBJbGh/ss/dN7Q/ObIDh9GofI6l/QZDkhDI6c8DfigAPFoAACAASURBVAaciyW7cCMwucnrLKC3qj4SZTsNkcZJDD2ZS/chtvYbDEmC3/COqm4CNgGIyDjgc1XdFyvDDFHGjqGPrvm02aoV2SOtvO/KEqqyu9KxpqzZNlXZXekYpdL9iFU+29IDMbPfkNZUFG9iqV34NiWBC98cDeSq6vsehy8iLhFp3fQVXTMNEcdJDN1Tut8SzaKWEsm5XeNhvyEtqSjeRM0zE5my5RauqZjNlC23UPPMRCqKN8XbtGY4TdkU4CbgSqCPn80yImWUIQY41G5J9tL9ZLffkARUlpD1/GQKqACBCRmrASikgt3PT4ar306oDoZTGYbfATcDDwB3A3cB9ViZO9nAPVGxzhA9QtBuSfbS/WS335DglK8lr+YHn6vyan5IuAncnebpXwn8CcvpA/xTVf8HS3tnA2C6TcmIkb9Nb0ylcmQYNImNeb6zxDbmJV6WmNOefh9gjarWi0gtdv6+qrpF5C9Y893eHCUbDdHEaLekJ0bnKHIULeHYvb6zxI7d+3HLakyiiNOe/i7AM3FjMXCC17oO+NfaNxgMiUY0dY7SkW5DkfymdasWkt8r4WbXc+r0PwZOsv+fB9whIneLyJ+Ah4Gl0TDOYDBEASc1Ggbn5BdaWk4deoO4YOBE62+H3gmp8eQ0vHMH1gxZYA3a5gOXYvXw38GaSMVgMCQDTmo0DKHhcfxJMBeEUz39b4Bv7P8PA9faL4MhvlSWwKEq6/8E/qIlFNHQOTIkzfiYo/COiCwTkWP9rBsgIssia1aMMVkMyYkdm2bPVj6956e4F0wzMWknREPnyJA0OI3pjwXa+1nXHjg1ItbEA4/87kvT4MWp1l8ju5v4eA1GAmYwMhSioXNkSBqcxvTBUtpshIhkA+MB35UJiY633jpYeutwRG89AQdhDDbRmnQlXYi0zlGakix6O94E0tP/E9bk52A5/E8tNQafPBhhu2KD0VtPXsxgZPgkSQw6UWnQ29HtdHKfyOk7Pqds66NUXP56Qjv+QD39N4CdWLLK/ws8hKWv700NsEFVP4yKddHG1lsfVNXc/A3tT2aQ+SK0jMqS6GcxmMFIQzxJMr0dbwJJK68EVgKIyD5gScpNgu5Eb904jtCIQaVnRfEm/vPJl4xydaGbu3lk0VswLq2IxY+twSLJ9Ha8cSqt/Ky3wxeR40XkQhE5VUScDgYnHrbeui+qsruaLIZQiUGlp+eR+mfFD7CptjOqgMLqzGHpPRhpf/buBdNMJlMsSDK9HW/8OmwRuVxEXvax/AVgDbAAeA9YISL5TbdLCozeemSJdqWn/UhdSAUZopyasQ4RQKCPazuuSY+ktcM3sgoxxIneToISqJc+nSZZOSJyBXAJ8A9gCHAh0AtLaz8pKejVn+zLX2dB33t4ouBOFvS9h+wEH4hJWOzBVV+syB4Zfu8n2CN1m07p5/DByCrEgyTT2/EmkNM/FvigybJpWD8EV6nqelVdiCXL8F9OTiYiOSKyQkTWishXIvI/LbI6wlh661ez+DenMHXG1cbhtxQng6vhkMSP1FEl2j+2iUo8iyqTTG/Hm0DZO+2B7Z43ItIKGA28qKr1Xtt9gdXbd8JhYLyqVotIFvCRiLypqs3z7gzJh8PZuFpMkknYxox0zGTy1NhUFsOAs2Hjm5DfK7YON4n0drwJ1NMvxpokxcOpQBZWHN+b1sBBJydTi2r7bZb9alb0ZUhSol3pmcSP1FEl3WQVvIsq1W0VVar7SFFlrHv8kZrTOUaIqm+fKyJ/Bq4ArgEqgMeAAcDRqrqnyXZnqOqPHZ1QJANYDfQDnlDVP/rY5irgKoCCgoLh8+fPD6VNzaiurqZt27bBN2xCXc1h9u7byx53Lh1cB2nfrj2Z2a3CsiUWtLS9EaO+FmoPQE6eJYaW1RoysiJ37F2boL4GWuXB4Sqqc3rQtn1+5M6R4Pi8vl6fy35pQxvdDxnZcFT/pP9cmrX3UBXs2ep/hw59rHsvCYnUd3fcuHGrVXWEr3WBnH5rYCEwwV60H7hSVed7bZMDfAv8TVXvCMUoO+NnEfBbVV3vb7sRI0boqlWrQjl0M5YvX87YsWND2seTGthdt7PUfSKnuz6nTLokxSBvS9qbVDTJR19e1oqxp5/Zon2T4XG8KX6vbwzbFkv5AV/tLXpkks+iyqK8MQy6LnEzZ4IRqe+uiPh1+oGKsw4AZ4lIPyz9/G9UdZ+P/SdjOf6QUNVKEVkOnAX4dfpxIYmr7dKCpvIBFcud7ZcIceBoEiNZhYDyA+1zYlKNHXZRZQr8+LeUoIJrqro5wLpqrFCNI0SkM1BrO/xc4Azgfqf7x4wkrrYz+CGFxPXqag4z79m/xEfkK0CHqHLuOdA6G/aWRfdH1S6q7FhT1mxVVXZXOgYbw0j1H/8gxLqathvwnoh8iSXx8I6qJt6zmEkNTD1scT1faGVxfHPZQ0g9rCjehO7axJQtt3BNxWymbLmFmmcmUlG8KTa2BuoQ1W6HqtLoD66GU1SZSIPAcSIUaeWwUdUvaTypemKSrqmBqfzImwDiej7j4O1znGsV2b3srD6/IkM0PmFHu0Pk63P0RzQUawt69afi8tdZYH+eO50+8Rhl3dg6/aTBkxroCQV4kbKpgTEQSosrcRbX8xUHr9j6EDVtssneb4UpPDLRLls+odlnnwhhxwAdIn9E60fVKqrsz1QATnG2UwL8+Meb5BVLiyZJXG3XItJBuyWe4npNNIMmZKwmQ5Tu7CRrf/O4NPiRT0iEsGOAWgl/NPyoJgJOfvxTHOP0/eFx/BfNhUvmWX9T0eFDemi3xFNcL0AP3V9pYjP5hMoSeOeO+It8+esQ5fWkKqvA5y4JpVhrlHWdh3dEZCRwHtADyGmyWlV1SiQNSwjSZWahNJmFqsVx4HAJFAf3MxldI/mEhtBbMTVk+9w+pmFHP/IDh/ceosRXbUsiKdZ6fvwT3c4o4sjpi8h1WDNnVQBbsGbMMqQKaaTd0qI4cLi0IA7uzivEtX8nFK+g7tUrGiaAz+HwkW0yc3HVH45PuqGPDlFBPvH5UQ2RuP34JwhOe/rXY8kw/EH9lfCmCqmcweKPaAulpTuBEgPyelLnVlz7ylmTNZQTar8AYN2Bjgxdch3a+ihcB/xMWFd3GE6+Fk6amTD3aFx+VFtAstgZDZzG9FsBr6eFw392Erw0DV6cav1Nh9zdaAulpTuBEgMue4vMmf/GNekRhrTejT0nDCfUfoELNxkHdlCDb+2cFdkj4cw7zPUxhITTnv4/gPOBd6NnSpxJoYrNFmE7fsrXWjH8oiVWDz+V2xxLgsnwtunkdzC9lVdIx5tUC70ZYoNTp/9H4HEReRdYBlQ2Wa+q+teIWhZrTNFG+gxcx4tAn2+AwfTDtGoUy/eQlqG3JAy/xlKczglOnf544BdAO/v/piiQ3E7fFG0Y4kmAwfRsaqhv3Rk5sKthhqy0DL0loWZOQHG6ODl+pzH9vwCfYU2q0kpVXU1eGdEzMUaYog1DPAk4EUovMi6eh2vKXEbf8m/o0CfyDj+eUw86Ick0cyqKN/HPp++h9pmJzYryCqkg6/nJcbPZaU+/O3C1qhZF05iY0vQxsU2X8JT7DIZw8AymO5HCyMmLuMNPeAmOJAq/enr3P9MKq1ftw+h4qvU67em/C6SO1/OVpbPoSur/66n4VGwmK4neO0w2bMfv6dG7psyNvuNNJAmOQPeTHX71xYb2CaR820hyA8RP8V081Xqd9vT/F3jS1sD3NZCLqn4dScOiRoAsnc7v/IYdFz3NgpXr0rJoIySSoXeYjMR6MN2JBEcsrmeg+wniLpjnmECSG17EU603lJ5+P+BO4ENgnddrvf03OQiiq95Zqpg642oW/+YUps64uuUOP5V7wYnUOzSEh5015Itm+j9hUFG8iXnP/oVzH/+Iec/+pbH+f7D7qb42eTRzAojiefODdGHxJ2tjNw+CF06d/jiv1/gmL8+y5CAWj4kxKPLy+yWqLLEmjobo/dikokBbiv9I+22bEwmOMGnIYPE38Uuw+6n2QHwF80IhgOSGel4Km2o7M6n4wdhOgGPjKLyjqu9H25CYEe3HxBgUeflLA9tx0dN0fuc30OUy68cmWiltkRBoS6R861QOVQVrW7QlOJzMNx3kfiInD0gSzZwAkhtuMsigHgROzbCCI/GYdzskaWUROVtEbheRp0Skl73sVBHpHh3zokC0HxOjPS2fH232Qiro+NK5jX9sopXSFm7v0HZE7gXT+PSen+JeMC1+YaFUDlU5aVskJDgCPUk4mfgl2P3keXLFo5kTgfBrtPAnudGmM4Lb5y4Nn0OMcOT0RaRARD4D/gXMAGYCnezVlwG3R8e8KBDtx8Roh48CfIlc1PtcHvE5YAPmlBcG/uFMNCebiqEqD07bFk7WULBQppOJX4LdT1mtg9vR1KaWhuoiEebzNRfHle/xTZ5vYbdYZ/I47en/f0Bb4Fj75Z2I9C5weoTtiioFvfqTffnrLOh7D08U3MmCvveQHakKuWgXeQX4Eu2WDj6XN/uxCffGDqd3mGhONkYDmXEhlLblFx55P2hSaA4/UMGUk/mmg91PGb4F5/zZ1OKnyEiOxTX9PMvXxn8CHBunKZtnATNUdbOINK2+LcWaWCWpiJq0qh0+ilqRV4AvUUfd43N5o7EKh/HrssqDPPn+t6wtqWRoYT6zTjuG7vm5Rw7aUoG2RJuwJZXnEoh228rXWpIIvvA8XTqdbzrg/fStM3u8niLBwZzDTfaN6lhcAs27HUpM33fswArzHIyALalBtMNHAeYodeNbDaNhrMJhaKWs8iCXPbqQ7SteZW1pFdtXvMpljy6krLLJZW5J7zAG2SIhEU6oKtGJdtsGTeJgnwk+Vx3sM+HIPeF0vumW3E/ehPMUGe2xuASad9up0/8Q+G2TXr5HW/9yrIItg01Uw0cBbp7dFy2mhAJQfP/YOPxSvPjOx/xN7+CJzEd4Kushnsh8hL/pHbz4TmizP/kk0ZxsKs8lEO22FS2h1Za3fa5qteXtIz/gsZpvOpxQXQhjcQFrDgKRIPNuhyKt/BFWIdYiLId/pYgMBgYDo6NjXmLiRCo1qjPz+NFm75xfSMXlr7N75RqeLbizeUqbk9BKZQlTi66hm2wHjqTY9WI7U4uugcp3w7tJQ9GYiRWpPJdANNvWbSgVri50czdPLKhwdaGb9w+4V6VxWbfT7dDhd75Dhy0lnHCWw1TusFUzE0C+3Gme/noRGQH8CbgUK9RzPrAUmKmqsS8rixPNL/pqdm+5j+n5D3N++42MHdCZ/DFXRj/33M/NU9CrPx23fM/iC06h2Y+Nw9BKgXu7z20K3NsjU5afiE42Ab6MUSNabcsvZN6gJ7hw/a/pwY6GSca/pzMvD3qC631cT0/osE/tZtbWj6Bb+VIu+6Ifc35/fviOP5yaAydjcU5qDpKgo+C0p4+qbgamRdGWxMfPRe9MFf9XOYucyloogUOb/kVOyYeJp/Xt5EuRX8jhvhPI3fJWs20O951AbqScRio72TTikjNP5rKv7qBP7Wbeqh/B2Rmr2JLVjzln+g6VeEKHPTJ3sNRl/0hoZ158p4DrLzwjPGPCeYr0jMU9M5Huur3hB6xMupDtCY8WLQlec5Ao3/UAOM3TXyYix/pZN0BEHMX0RaRQRN4TkSIR+UpErg3F2LgTIEc+V2oRsXJZc4rfT0ytbycx3qIl5G71HafN3fq234HWssqDzF68nnMf/4jZi9c3H/RtAdE4piGydM/PZc7vz6fLyAsY2jOPziMv8N9rt0OHvWR7o8LCXuIJHUbgexJGzUHQsTgnNQdJgNOe/ligvZ917YFTHR6nDrheVT8XkXbAahF5J2kUOu2L7mt2LX8kmtZ30NBKt6HWE4qP1DL8pJZF45E9qmEAQ0Tpnp/LnecODr5h+drohw7B/1OkA+mPgGNxTmoOksDxOw7vcCRbpwERycYSWwuuJQqoajlQbv+/T0SKsHL8k8PpB7jo/kjIqRYDhVY8g8QhTEsXjUf2qIYBDPFh0KTYhA59EYmpFhMo1z4cRLWZL7dWiPwJmO3wOA+q6s0hnVikN/ABMFhV9zZZdxVwFUBBQcHw+fPnh3LoZlRXV9O2bduwjgFYEq+7NkF9TWj7dejTIBoVCyLS3vpaS90wJ8/SPslq7bsysr6W2u0byaK22apaspCj+rH9oHKwpo7c7Ew6t21FVoafmSUcHjOry4BGtkTs+iYJSdveQ1WwZ6v/9X6+J2G3N9D3NiMbjurvvOrX+1it8uBwVejH8D5Wk+9Y9cHDEbm248aNW62qI3ytC+T0TwJGYoWp/xd4CNjWZLMaYIOqOo93WMduC7wP3K2qCwNtO2LECF21alUoh2/G8uXLGTt2bFjHaKChcu87yMq1LloAdmd3p2OMR/Uj2t5gFC3BvWAaLh9iUm5c/IHrWVJzInVuJdMltGmVyZvXjgkcoglyTNeUuY2eUGLa3gQgadvbtOrVmwBFSmG3t2iJJamgPgTPxGXly4fylBEJhVg/lfEfDXmIsaefGdqxfCAifp2+34FcVV2pqk+o6uNYomr32++9X39rgcPPAl4FXgjm8BMST/hjyvNwzQpo0wXExaFepzVoZX9Qf3zkqnAjJR616h/WqyXHCYT9yO6Lr9v9hNU1vRiPlSY6nhXkHf6BJ98PUlYf4JiH+07w/wVNJk38ZLI1UsSrKtVJ5XAohFs5HKAynl2bon4vOI3pvwCNa/xFZAJwHPCBqn7u5CAiIsDTQJGqPhyKoQmFd0z8ymVQvpacQZOo/PBvvL9xO88cPI3SSGh9h6Pz7rXvmqyhnFD7BQDy9WLYujxy6aQBsn0G7fuYhVnr6MjehhS4Uu3Mndvux6rpC/2YDRlETb+oyaSJn0y2Rho/hYVRbbeTyuFYjrsFqIynvgZWPg0nzYzaZ+JUhmEB8FfPGxH5HfAWcC/wqYg4/cROxsr1Hy8ia+zXOaEYnHB4/ernj7mSc2feGhmt73AkiOtrG+17Yu0XCLY06pZlkU0n9WT7+MRFZ6lqlJ53tGs7D+6/NfB5Ax3T14BZk/bGXa45EIkmLR0Pwu0ph4pdOeyLCleX2A/ABpCLqNMM3B8/FtV7wanTHw284fX+RuAhVc0F/g7c6uQgqvqRqoqqDlHVYfbrjeB7piHhiEfVHvDfk/AiqkJSbTrjEt/jRR1qg0waEWoYIEB7XVUlVs8pUZxpoklLx5mY1GLYlcPF2qWRCGKxdmHeoCdi/3QVoDI+U+qj3glw6vSPwk7LFJHjge7Ak/a6l7HCPIZIEo54VE6e3329idicwH4mjZABZ/vcXAacHfy8oYhTBWjvTm0f9Z5TSKSyfn+IOFZzjQCXnHkyV8odXFN3HVfVXs9v6q7jSrmDS/xUDkeVAKKD3kSrE+DU6VcAve3/zwK+U1XPaFwu+JkHzNAMxz2bcCSID1X53deblkzq4ldh0MekEWx80/dBNr7p7LxOwwAB2ttFKhMrfBLKdU3xwd6oqrk2IaTK4WjTpDJ+u+b73CxanQCnA7kvA/eLyFCsTJ7HvdadAKSN4Fo4hFRlGo54VFZrv/t6E+qkLiEpDLagsrfFOGxvQ88pnoOlTq9roMHeVCDaaq4+cFw5HAs8lfErn6bTx4/53CRak/g47enfDPwf1lSJfwXu8Vo3HGug1xCEkHo24WihZ2Q12ndVxjBUw0wnDTAhe9bzk5v3QmOZntekvbHuOYWEk+sabLC3vnnRWtLhRJIhgWixhn4g8gvhpJkxn1/CqbRyHXCnn3XnR9SiVKUlPZtwJIi99h0RiXTSAGJzfhUGY5meF8eeU8gEu67BBnuDFAQmBfGUZAiRsDX0A9FEGXS/tMGNK6opvI6cvoisAp4B5qlqZcStSAdaKjblRIK4aYVgfatm++aPuZJzx8C5QIsmdQkgNrcxL4C+UCwllD09p/ULWxYWiyWBPpcgk93EUtIjarSkFiMexEJD36sT0KaiLa6hc6M6v4TT8M5XwP1AuYgsEJEJdqGVwSktrTINhqe0/aVp8OJU6280qvqcKAwmAqkw/WGwwd5DVTE2KAqEWosRL5w84UaCGNYuOHL6qjoD6ApcY/99CygWkbtFJAITv6YBLdSpD4i3lom64ZvXrb/1NZHX8Q8wIXu4CoMRz9UOQ1M9IQg2j3BW6xgbFAXiJckQKimioe+N054+qrpfVZ9R1dOA/sAc4BfABhH5QEQuFZGcaBma9ESjZ1O+1iqw8kFECq+8idKXNGq52rGu+owkwZ5WQlVzTFRCqcWIF8nyhBsCoejpe+PmiL5+PVaF/1+A+0Rkmqq+EwnjUooW6NQHZdAkS6/fR5w9Kjr+URiYNbr5fgg42BtEsC6ZSPRpM1NEQ98bx05fRFoDF2JNjD4G2Izl6J9V1QoR6YiVv/9/QN/Im5oCRNppFi2xCqx80FB4FQ3HH6kvaRxytZOKRHeI6UA0Omtxxmn2ztNYDt8FvALMbiqprKq7ReQx4OKIW5lKRPKL3G2oVWBVU9ZsVaiFV3EhVtPnGQzhEA9l0CjiNKZ/PHAD0E1VLw2gof8VMC4ilhmCk19I7S9fo4SCRkJStWSGr+MfC6KV0RQLUlwiIZ1wlEiQzGNETXBanBVcvcvarhprRiyDAyqKN7H0vXdYsG8IU1qov1/Qqz8Vl7/OAvs4O9t9SZejjgm/aCQWJEuudlMiMd+qISEoqzzI2Y99yP7DddS5la/K9rJ4TVnw2d2SmJAHcu3Y/kwsSYYfgOdUNbDoiaEZkazyK+jVn6kz+jMVgFNYvnx5FCyOArHU54kUTaf8+8bWwvHMT2Acf1Lx5Pvf0v7wD4xiC29zEuNZwYbDfXny/W8TR6cnwvgN74jIQyKyscmydsDnwKPAFKyJ09eKyICoWplqhKpjk6okS662N+VrrR6+LyKdJmuIOt9v28jzmX/mr1mP8lTWQ/w161HmZv6Z77dtDL5zkhIopj8OeL7JshuAAcCVqtoJS1d/G3B7VKxLVWJV5ZcMJEOutjeDJlkhHV84mSfAkDhUlvDg/ls52rU99NndkphATr83sLrJsguAr1X1GQBV3QE8hDUNosEPzRT62gxMuSq/sEi0QbJAg7RFS1A/8wSo03kCDIlB+VprFjcfBJ3dLYkJFNPPBA553th5+IOAJ5pstw1LmsHgA1+x+z1b7+NY2edz+4Yqv3Rz/EEoqzzIk+9/y9qSSoYW5jPrtGOiM9AWbJC221D2ZPlOk92TlQRpsoYjDJpkzeL2TfM5ChzN7pakBOrpbwTGer33fAL/brJdF2B3BG1KHfzE7jtRhVt969Ula5VfNInZtHr+tIy8J5HPL+TGNnfznbvxfKvfubtwY5u74/+UEk1SLU21aEn4s7slIYF6+o8DfxORPKzpEn8HbAWa5thNANZHx7wkJ0DsHtzQpgsc2GnS/oIQM6kGJ4O0+YX06D2AX1bcziC28Lb7JCa4VrKBvoztncL5DEmaphrwCTEZs8cigN+evqr+Ays753zgv4FvgPNUtWHaHhHpjCXRvji6ZiYpARX6ToErlyXPAGa8sKUaeknjwbZe4pFqiGBvc9AkDvbxXSx2sM+RYrFZpx3D3lZdWYZVvvIeI6lq1ZVZpx0TOVsSCSdPQAlI0CfEYNljkFpPNjYB8/RV9V7g3gDrd2Di+f4JptDnKesGq1fhkR1I8jLviOJoWr22kTlX0RJabfFdLNZqy5Fise75ubx57RhnYwxNJ7hJxuvq8Ako0XD0hOhPYgGS8snGCS1V2TQ4walCX5I+OscEJ9PqVSyPzLm6DaXC1YVu7uYhuQpXF7p5Pe47mmQ7Va6r/QTk6xoc7JNYUxs2EIqYX1M9rBQvwHOsp29oAU6Kj0J5dE61gTQnRGPyGX/kFzJv0BMUa+NB2mLtwrxBT4T2RU/SkIhPnDwBJRrhTLwey3kq4oDp6UebYAp99g3mK5dHK4sRz6NzqvQaQ8XJYFtF5PTlLznzZC776g761G7mrfoRnJ2xii1Z/ZhzZoilKEkaEvFJCE9ACUM4E6/Hep6KGGN6+rEgUPGRfYP5YkP7k5s/biZ7rzFUYizV0D0/lzm/P58uIy9gaM88Oo+8gDm/Pz/0moBUqtyN5BNQrAjnCdHJPBVJTEx7+iLyDFa+/3ZVTU01o1BxeoOlSq+xJcRYz9xRvD4YTnLAk8jxR+wJKFaEk46Z7PNUBCHWPf1/AGfF+JyJjX2D+aIqu6t1c6ZSr7GlJJpUQzCiMSdyrPAxdhSxJ6BYEc4Top95KkooSI55KoIQ056+qn4gIr1jec6Ex3ODPTOR7rqdpW4rtaxMupDtucFsvRefcf+NbyJJ1mtMC/ILqTjvFWp8XdfzXqEgUR1HZQl1cybiqiphRfZIRtaswJ1nzdfbPb8wueSGw3hC9DVPRUvmu0hERFWDbxXJE1pOf0mg8I6IXAVcBVBQUDB8/vz5YZ2zurqatm0jlMsdJepqDrN33172uHPp4DpI+3btycxuZa2sr6V+x0YyjtTFNVAvWWR0HgAZWQ3LkqG9kSRm7a2vhdoDkJMHh6ogq3Wjz70pZVWH2Fe9nxxq2Etr2nOAQ2TTrm0buufltNiMqLW3vhZ2bYL6mubrMrLhqP4B2xst0ul+jlRbx40bt1pVR/hal5DZO6r6FPAUwIgRI3Ts2LFhHW/58uWEe4x4M/Oxfcze9Ud6yo6GXmOpdubOo+7n6YvPbLRtKrQ3FGLS3hZkT537+EesLa0HPA6+PQBDe7Zl8bmntNiUqLW3aAnuD2/FhbvZKjcuXFPmwqAzfewYXdLpfo5FWxPS6Ruak5Z6L4lCC4t1hhbm81XZXurcR56ms1zC0ML86NvcEgZNYkX2SEbXfNps1YrskYxO4BBiJKYeTRdMymaSkHZ6L4lEC2fLmnXaMbRplUmmyxqNyXIJrVtlJu41K1rCyJoVPleNZ8JwDAAADrBJREFUrFmRsKmKDfLlW27hmorZTNlyCzXPTKSieFO8TUtIYur0ReRF4D/AQBEpFZGZsTx/MuPRe5k6qhdDe+ZxyaheKT15c0LhIHuqrPIgsxev59zHP2L24vWUVR5MvmvWbSjuPN+hKndeYWJmHJmpR0Mm1tk7l8TyfKlGRPLHDaETJOd+96qFnP16W/YfrqPOrXxVtpfFa8oaHHzSXLN8K0vHX/ZOQqYqOpl6NBHtjiMmvGMwBKKyBPbvDJhzP2dr+waHD1DnVg4cruPJ9yMnDxEzbMfvmjKX0bf8G9eUuYnr8CGIfHkaTj3qAOP0DQZ/eAZwX78OOvRFAQW+yDoBNy7q8o6GGUv4oCKHLu4dTHCtBGCCayWd3TtYW1IZV/NbTDIVwgWTL0/QcYh4YrJ3DAZfNM3Y2bKsoTiu4+Hvua3+clZXncgcOnFqwTdcuOMuenAknfZ7OvNywV/jZX364FS+3NCA6ekbDL4IIK/bU3awU9uxpaYDL77zMdeW/sHnzF7Xlv4h+QcSE13OO8aCfKmA6ekbDL4IIK+71H0ib7tPApTDxV+QWe3bEWbuLUnugcRkkfOOsSBfsmN6+gaDLwKon57u+pwJrpVkuYRD/c5OTTG8ZJPzTqZxiDhjnL7B4IsA6qel2pkN9KV1q0x+32NjcAnlZKSFBWmGxMc4fUPsSPT4sDd+5HWLtQu3d7iXsaNO5M1rx9Cx30nJK6EcCCPnnbKYmL4hNgSQ7E3UR3F/8rrPNdJ0sePJyRD7DoUUmwTGcATj9A3Rx3b4mVXfATQIermqvrOWJ7jjnzqjP1MB8KOMmYoDieHMPGVIaEx4xxB9ytfiqvIdynFVlaRGfDjVBhJNKmTKYnr6huiTxJK9aU0qPsEYjNM3xAAnkr0p5PhTSts9v/CIk0+ha5TOGKefiFSWpFbvypbsddkxfW/ceYW4Uig+3KDtrtvp5D6R03d8TtnWR6m4/PWkdPxllQd58v1vWVtSydDCfGaddkziSkMbHGFi+omGpyjmpWnw4lTrbyIWw4SCR7I372jcuPg0e3SDYFkiD+KGTIppu5dVHuTsxz5k3mfFrC2tYt5nxZz92IeUVR6Mt2mGMDBOP5FItirIUEg2yd6W4ETbPYl48v1vU0cy2tCAcfqJRKpXQaZahktTUkzbfW1JZaP5fQFq3Zq8ktEGwDj9xMJUQSY3KabtPrQwv2F+Xw8JPbG7wRHG6ScSRUtQP1WQmsw6LumCR9vdB8mo7Z50E7sbHGGcfiLRbSh7snyLfO3J6pp0TiPtSLGCpqSb2N3gCJOymUjkF3Jjm7uZfeiP9JQjszCVamfubHM3TyeZ00hLUqygKakmdjc4wjj9BKNH7wH8suJ2BrGFt90nMcG1kg30ZWzvAfE2zeAUU9BkSGBMeCfBmHXaMext1ZVljATgPUZS1aqriaMaDIaIYHr6CYYnjmqqIA0GQzQwTj8BMXFUg8EQLUx4x2AwGNKImDt9ETlLRL4Rkc0icnOsz28wGAzpTEydvohkAE8AZwPHAZeIyHGxtMFgMBjSmVj39EcCm1V1i6rWAPOBc2Nsg8FgMKQtoqrBt4rUyUR+DpylqlfY76cBo1T1N022uwq4CqCgoGD4/PnzwzpvdXU1bdu2DesYyYRpb2pj2pu6RKqt48aNW62qI3yti3X2jvhY1uxXR1WfAp4CGDFihI4dOzasky5fvpxwj5FMmPamNqa9qUss2hprp18KeNej9wTKAu2wevXqnSLSfMql0OgE7AzzGMmEaW9qY9qbukSqrUf7WxHr8E4msBE4HfgeWAlMVdWvonzeVf4edVIR097UxrQ3dYlFW2Pa01fVOhH5DfBvIAN4JtoO32AwGAxHiHlFrqq+AbwR6/MaDAaDIX0qcp+KtwExxrQ3tTHtTV2i3taYxvQNBoPBEF/SpadvMBgMBozTNxgMhrQi5Z1+qgu8iUihiLwnIkUi8pWIXGsv7ygi74jIJvtvh3jbGilEJENEvhCRJfb7PiLymd3WBSKSHW8bI4WI5IvIKyKywb7GP07xa3udfR+vF5EXRSQnla6viDwjIttFZL3XMp/XUyz+1/ZdX4rIiZGwIaWdfpoIvNUB16vqIGA0cI3dxpuBparaH1hqv08VrgWKvN7fDzxit3UPMDMuVkWHx4C3VPVYYChWu1Py2opID+B3wAhVHYyV1n0xqXV9/wGc1WSZv+t5NtDffl0F/DUSBqS00ycNBN5UtVxVP7f/34flFHpgtfNZe7Nngf+Kj4WRRUR6AhOBv9vvBRgPvGJvkkptbQ+cCjwNoKo1qlpJil5bm0wg1y7kbA2Uk0LXV1U/AHY3Wezvep4LPKcWnwL5ItItXBtS3en3AEq83pfay1ISEekNnAB8BhSoajlYPwxAl/hZFlEeBW4C3Pb7o4BKVa2z36fSNe4L7ADm2OGsv4tIG1L02qrq98D/A4qxnH0VsJrUvb4e/F3PqPivVHf6jgTeUgERaQu8CvxeVffG255oICKTgO2qutp7sY9NU+UaZwInAn9V1ROA/aRIKMcXdiz7XKAP0B1ogxXiaEqqXN9gROXeTnWnH7LAWzIiIllYDv8FVV1oL67wPAraf7fHy74IcjIwWUS2YYXqxmP1/PPtcACk1jUuBUpV9TP7/StYPwKpeG0BzgC2quoOVa0FFgI/IXWvrwd/1zMq/ivVnf5KoL89+p+NNSj0Wpxtiih2TPtpoEhVH/Za9Roww/5/BrA41rZFGlX9b1Xtqaq9sa7lMlX9BfAe8HN7s5RoK4Cq/gCUiMhAe9HpwNek4LW1KQZGi0hr+772tDclr68X/q7na8B0O4tnNFDlCQOFhaqm9As4B0vZ81vg1njbE4X2nYL1yPclsMZ+nYMV614KbLL/doy3rRFu91hgif1/X2AFsBl4GWgVb/si2M5hwCr7+v4T6JDK1xb4H2ADsB6YC7RKpesLvIg1XlGL1ZOf6e96YoV3nrB91zqsrKawbTAyDAaDwZBGpHp4x2AwGAxeGKdvMBgMaYRx+gaDwZBGGKdvMBgMaYRx+gaDwZBGGKdviBgiog5eY0XkUvv/tnG01a8NInKHiOz0ej/W3naww2N3sY/RO3IWh4eIXGCrNWbY74NeAxF5QkSejp2VhlgQ8zlyDSnNj73+zwWWAXcBr3st/xr4yt72QOxMC4vPsez91uH2XYA/AcuBbdExyTki4sLKf39QVetD2PVBYIOI3Kuqm6NjnSHWGKdviBhqKQECDVpAAN96L/diR2ysCh+1tIx8tSEmiEiuqh4M4xCnA8cA80LZSVW3ichHwK+B68M4vyGBMOEdQ8xpGloQkd72+4tFZI6I7BWRUhH5pb3+JhEpE5EdInK/3XP1Pt5gEXldRPbZr5dFpGsE7W0W3hGRmWJN9nFQRHaKyPsi8iM7pLPO3uw9T1jLa78+IvJPu437RORfItKvyflURP4gIo+KyA5gnYhcY2/ftsm24+zthwRowgzgbbWktwO180YROSQik70Wvwr8oulnbkhezIU0JBL3Y5WoXwB8CDwrIg9hzYtwOUdklS/y7GA7zI+BHGAacCnwI+Bftn5LMDJEJNP7RZDvhYicCjwJPI+lAnk58AmQZ9v/C3vTa7DCQj+292uFVWY/CLjStrUP8L6IdGxymhuBbnabfge8gPVk/vMm210KfK6qXwYwebxtX6A23Y4VAjpXVb31qT4BCoDjA+1vSCLirUVhXqn5AtpiaQJd6mPdpfa6tvb73vb7OV7btMfSJ9kEZHgtXwEs8Ho/F/gGyPZa1h+oByYGsM9jg7/XTq9tx9rLBtvvbwD+//bOJrSKK4rjvz/pokgXLpQQ241Zil1IUmkhGhdCA3Xl10ahSGgRsbQ725WWQldSkYI0WjWxLqQUP7BiSxtIxI1fWHAnoi20llBbpBZssvB0ce7AZPLevPfU0Dd55weX9+beM/feNzzOnHvOnTk3SvpemeTXFep34pnOenN1rwAzwEe5OgNu1uj3JDBZuMb/ALtL5rIs9fdWnd//EvAp8Kg43yT3QprzO//3fyrK8ylh6QftxHj2xdyP/geu5PLBxzvMTiSxHjgDPMlZ6vfwAGp/E2OuBV4rlCMNzvkJWCXpgKS1aj5n62rcKr+bVZjZr/hKZaAge4G5HAXWSOpNx1txpVzmq8/cXA/qtH8G7ALeNLOJYqN58pKHuX6CihNKP2gnHhaOZ+rUvZg7XgLswVcF+dLL7HeR1+OmmV3PFxq8s9zMfgR24DeMCeCBpEPyrFZl9ABTNeqngKJ7p5bcBHAXt9JJczhnZsX0e3myazVdp30Tnp3qakkf08y+5kGFid07QdX5C7f0v6zRVs+6fWbMbAyPOSwFNgIHgL8pz2z1Ox5vKNLN3Lypc15/a2Ym6RjwrqSv8NVBrcxSebJ+F9dp3wB8C5yQtN3MntSQWVxjfkFFCUs/qDrjuA/9RtFiN7Of53tw8yxPI3jgeUWqnkmfRev4CtAnaXlWIellPDvU5SaHHMXjAMeA34AfGsjfS/NZXqf9Fn7j2IAHp2eRbmqL8JwUwQIglH5QdfbhSv+CpM1pe+U2SaOS1s3HgJI+lvR5esp1UNIHwCD+MBp4BqjHwNuS3pCUxRZGU9tFSVslbQK+w1ckI82MbWb30zkDwAlr8LCVmU3j7pu+EpmruNLfLml/obkfX3WU7v4JqkMo/aDSmNlt4HX86d7DwEV86+E0HvSdD67hVv0XwPf4w0v7gINpTv/iWzL7gMkknyng9XhmqKPAGPALvmumFffJ2fR5vEn508BQmYCZXcLdVO9J2ptrGsKD6X+2ML+gjYnMWUFQMSR9DfSY2Zom5bvxFcaAmV1rYZwu/Kb0oZmdfKrJBm1HWPpBUBEkvSppB26RH2z2PDObwgPd77c45BbcTXWqxfOCNiZ27wRBdTiPb1E9ZGbftHjuJ8CwpK5GcYAcAobTXv1ggRDunSAIgg4i3DtBEAQdRCj9IAiCDiKUfhAEQQcRSj8IgqCDCKUfBEHQQfwHVXfdYHFRR1AAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# This function computes all given iterations of particle filter either SIS or Generic\n",
    "def pf_MultiStep(zkhist_tru,resampleFlag):\n",
    "\n",
    "    xhathist = np.zeros((K, Nx))\n",
    "    Phist = np.zeros((Ns, Nx, Nx))\n",
    "    pfData = intializeData()\n",
    "\n",
    "    for k in range(0,K):\n",
    "\n",
    "        kp1 = k + 1\n",
    "        # pass in the current measurement \n",
    "        pfData['next measurement'] = zkhist_tru[k, :]\n",
    "        xhatkp1, Pkp1, chikp1mat, wkp1 = generic_pf(pfData,resampleFlag)\n",
    "\n",
    "        # store the estimation\n",
    "        xhathist[k,:] = xhatkp1\n",
    "        Phist[k,:,:] = Pkp1\n",
    "\n",
    "        # iterate \n",
    "        pfData['state estimation'] = xhatkp1\n",
    "        pfData['error covariance'] = Pkp1\n",
    "        pfData['particle matrix'] = chikp1mat\n",
    "        pfData['particle weight'] = wkp1\n",
    "        pfData['current time step'] = kp1\n",
    "    \n",
    "    return xhathist, wkp1\n",
    "\n",
    "# Run the generic particle filter, True means you are resmapling \n",
    "xhathistGeneric, wkp1Generic = pf_MultiStep(zkhist_tru,True)\n",
    "plt2 = plotResult(xhathistGeneric, xkhist_tru,'Generic Particle Filter', 'PF (Ns = 500, Nt = 40)',2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just by preventing degeneracy via resampling, the result is exceptionally better, and the filter predicted most of them accurately. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f8d7d1d3fd0>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the error comparison for each time iteration \n",
    "xkhist_tru = xkhist_tru[0:-1, 0] \n",
    "xtildkhist_generic = xkhist_tru - xhathistGeneric[:,0] \n",
    "xtildkhist_SIS = xkhist_tru - xhathistSIS[:,0] \n",
    "plt.title('State Estimation Error in Each Time Step')\n",
    "plt.scatter(np.linspace(0, 100, 100), xtildkhist_generic, marker = '*', linewidth = 2,s=5, label = 'Generic PF (Ns = 500, Nt = 40)')\n",
    "plt.scatter(np.linspace(0, 100, 100), xtildkhist_SIS, marker = 'o', linewidth = 2,s=5, label = 'SIS PF (Ns = 500)')\n",
    "plt.xlabel('Time History (k)', fontsize=15)\n",
    "plt.ylabel('Error in State Estimation', fontsize=15)\n",
    "plt.grid(True)\n",
    "plt.legend()  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This plot shows a better visual representation of how accurate the generic particle filter is. \n",
    "\n",
    "** Diversity Is Still an Issue**\n",
    "\n",
    "Even with Resampling diversity issues don't fully go away. The reason is that in the resampling stage, samples are drawn from a discrete distribution rather than a continuous one. If this problem is not addressed properly, it may lead to particle collapse. There is a solution method called regularized particle filter. regularized PF deals with lack of particle diversity and too large of weight on a single particle. It won't be covered for now, but I may or may not add on to this documentation in the future. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Conclusions\n",
    "\n",
    "If assumptions of the Kalman filter hold, then no other algorithm can out-perform them. However, those assumptions in many other real life situations do not hold. Particle filter approximates the posterior density as a finite number of samples. If the system is non-Gaussian and multi-modal PF is the optimal choice. It is critical to choose a good important density and implement the resampling scheme whenever being close to degeneracy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# References \n",
    "\n",
    "[1] M. Sanjeev Arulampalam, S. Maskell, N. Gordon, and T. Clapp.A Tutorialon Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking.IEEE, Feb 2002.\n",
    "\n",
    "[2] R. Labbe. Kalman and Bayesian Filters in Python.https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python,2015\n",
    "\n",
    "[3] M. Psiaki.AOE 6984:  Model Based Estimation Lecture Notes. VirginiaTech. Dec 2018\n",
    "\n",
    "[4] Y. Bar-Shalom, X. Rong Li, and T. Kirubarajan.Estimation withApplications to Tracking and Navigation. John Wiley and Sons, Inc., 2001.\n",
    "\n",
    "[5] K. Hsiao, H. Plinval-Salgues, J. Miller.Particle Filters and TheirApplications. MIT, Apr 11 2005.\n",
    "\n",
    "[6] B. Ristic, M. Sanjeev Arulampalam, and N. Gordon.Beyond the KalmanFilter. Particle Filters for Tracking Applications.. Artech House, Dec 2003."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "display_name": "Python 3",
   "language": "python",
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    "name": "ipython",
    "version": 3
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