13_hw_cp220_Q.ipynb

{
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    "id": "view-in-github"
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   "source": [
    "<a href=\"https://colab.research.google.com/github/AshrithSagar/CP220-M4RAS-2024/blob/main/13_hw_cp220_Q.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ogEoifF21Dh5"
   },
   "source": [
    "---\n",
    "**HW13**\n",
    "\n",
    "Chapter 5: Bertsekas <br>\n",
    "Chapter 3,4: Probabilistic Machine Learning, Kevin Murpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "id": "BCQf0fhp6305"
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "rng = np.random.default_rng(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "j8xILkQg7Fmr"
   },
   "source": [
    "**Concentration Inequalities**\n",
    "\n",
    "These essentially allow one to obtain bounds on tail probabilities\n",
    "\n",
    "---\n",
    "*$\\underline{Markov\\ Inequality}$*\n",
    "\n",
    "If $X$ is a non-negative r.v., then\n",
    "\n",
    "$P(X \\geq a) \\leq \\frac{EX}{a}$ for $a > 0$\n",
    "\n",
    "We can see this by:\n",
    "\n",
    "$aP(X \\geq a) = \\int_a^∞ af_x(x)dx$\n",
    "\n",
    "$EX = \\int_0^∞ xf_X(x)dx = \\int_0^a xf_X(x)dx + \\int_a^∞ xf_X(x)dx$\n",
    "\n",
    "Hence $EX - aP(X\\geq a) = \\int_0^a xf_X(x)dx + \\int_a^∞ (x-a)f_X(x)dx \\geq 0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "7xyQRQTf7Ubo"
   },
   "source": [
    "---\n",
    "Example:\n",
    "\n",
    "Let $X_i$ be 1 if the $i^{th}$ person says they will vote for BJP in the national elections 2024 and 0 otherwise. Let the sequence { $X_i$} be independent and identical bernoulli rvs $\\ \\sim Ber(p)$\n",
    "\n",
    "Let $B_n = \\frac{\\sum_{i=1}^n X_i}{n}$ estimates the chance of BJP winning based on a sampling of $n$ people.\n",
    "\n",
    "We would like to know how many people to ask for their opinion inorder to get some estimate with some accuracy and confidence.\n",
    "\n",
    "\n",
    "$EB_n = EX_i = p$\n",
    "\n",
    "\n",
    "$P(B_n  \\geq p+ϵ) \\leq \\frac{p}{p+ϵ}$\n",
    "\n",
    "Unfortunately - we dont get any insight as to the number of samples needed from this."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-_sbcnx_7Vci"
   },
   "source": [
    "---\n",
    "*$\\underline{Chebychev\\ Inequality}$*\n",
    "\n",
    "Let $X$ be r.v. with mean $\\mu$ and variance $\\sigma^2$.\n",
    "\n",
    "Then, the chance that it is more than $k$ std. deviations from the mean is upper bounded as:\n",
    "\n",
    "\n",
    "$P(|X-\\mu| \\geq k\\sigma) \\leq \\frac{1}{k^2}$\n",
    "\n",
    "This can be seen by considering $Z = (X-\\mu)^2$ and by Markov inequality:\n",
    "\n",
    "$P(Z \\geq (k\\sigma)^2) \\leq \\frac{EZ}{(k\\sigma)^2} = \\frac{E(X-\\mu)^2}{(k\\sigma)^2} = \\frac{1}{k^2}$\n",
    "\n",
    "and $P(Z \\geq (k\\sigma)^2) = P((X-\\mu)^2 \\geq (k\\sigma)^2) = P(|X-\\mu| \\geq k\\sigma)$\n",
    "\n",
    "We can also write this (equivalently) as:\n",
    "\n",
    "$P(|X-\\mu| \\geq c) \\leq \\frac{\\sigma^2}{c^2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "NMiKGhU77bNT"
   },
   "source": [
    "---\n",
    "Example continued:\n",
    "\n",
    "We know that $\\sigma_{X_i}^2 = p(1-p)$\n",
    "\n",
    "and\n",
    "\n",
    "$\\sigma_{B_n}^2 = \\frac{p(1-p)}{n}$\n",
    "\n",
    "\n",
    "From Chebychev's inequality\n",
    "\n",
    "$P(|B_n - p| \\geq ϵ) \\leq \\frac{\\sigma_{B_n}^2}{ϵ^2} = \\frac{p(1-p)}{nϵ^2} \\leq \\frac{1}{4n\\epsilon^2}$\n",
    "\n",
    "Since $p(1-p)$ achieves peak value of $1/4$ for $p=1/2$\n",
    "\n",
    "So if we want to be within $0.01$ of $p$, with $95\\%$ confidence, then\n",
    "\n",
    "$\\epsilon = 0.01$ and $\\frac{1}{4n\\epsilon^2} < 0.05$ which means:\n",
    "\n",
    " $n > 500$\n",
    "\n",
    " In other words, if you ask 500 independent people for their opinion of if they will vote for BJP, then you can say that\n",
    "\n",
    " $B_{500}-0.01 < p < B_{500}+0.01$ with 95% confidence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "exOqCl4J7gp8"
   },
   "source": [
    "*$\\underline{Chernoff\\ Bound}$*\n",
    "\n",
    "Let X be a rv and\n",
    "\n",
    "$M_X(s) = Ee^{sX}$ and let it be finite at $s=0$,\n",
    "\n",
    "\n",
    "For $s \\geq 0$\n",
    "\n",
    "$ X \\geq a ⇔ sX \\geq sa ⇔ e^{sX} \\geq e^{sa}$\n",
    "\n",
    "Hence\n",
    "\n",
    "$P(X \\geq a) = P(e^{sX} \\geq e^{sa}) \\leq e^{-sa}Ee^{sX}$\n",
    "\n",
    "Similarly for $s < 0 $\n",
    "\n",
    "$P(X \\leq a ) \\leq e^{-sa}Ee^{sX}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "owDk6kuR7mCk"
   },
   "source": [
    "---\n",
    "Example contd. (some parts from\n",
    "\n",
    "https://sachdevasushant.github.io/courses/15s-cpsc665/notes/concentration.pdf\n",
    ")\n",
    "\n",
    "for all $s \\geq 0$ by the Chernoff bound\n",
    "\n",
    "$P(B_n - p \\leq ((1+\\delta) p)) \\leq e^{-s(1+\\delta) p)}Ee^{sB_n}$\n",
    "\n",
    "$Ee^{sB_n} = Ee^{\\frac{s}{n}\\sum_1^n X_i} = (Ee^{\\frac{s}{n}X})^n$ where $X\\ \\ \\tilde\\ \\ Ber(p)$\n",
    "\n",
    "$Ee^{sX/n} = pe^{s/n} + 1-p \\leq p(e^{s/n}-1) \\leq e^{p(e^{s/n}-1)}$\n",
    "\n",
    "Therefore. $Ee^{sB_n} \\leq e^{np(e^{s/n}-1)}$\n",
    "\n",
    "and\n",
    "\n",
    "$P(B_n - p \\leq ((1+\\delta) p)) \\leq e^{-s(1+\\delta) p)}e^{np(e^{s/n}-1)}$\n",
    "\n",
    "Above is true for all $s \\geq 0$\n",
    "\n",
    "In particular for $s/n = ln(1+\\delta)$\n",
    "\n",
    "$P((B_n - p) \\leq ((1+\\delta) p)) \\leq (\\frac{e^\\delta}{(1+\\delta)^{(1+\\delta)}})^{np}$\n",
    "\n",
    "We can show that $(1+\\delta)^{1+\\delta)} \\geq e^{\\delta + \\delta^2/3}$\n",
    "\n",
    "Hence\n",
    "\n",
    "$P((B_n - p) \\leq ((1+\\delta) p)) \\leq e^{-\\frac{\\delta^2np}{3}}$\n",
    "\n",
    "Similarly\n",
    "\n",
    "$P((B_n - p) \\geq ((1-\\delta) p)) \\leq e^{-\\frac{\\delta^2np}{2}}$\n",
    "\n",
    "Hence $P(|B_n-p| \\geq \\delta p) \\leq  e^{-\\frac{\\delta^2np}{3}} + e^{-\\frac{\\delta^2np}{2}} < 2e^{-\\frac{\\delta^2np}{3}}$ from Chernoff bound.\n",
    "\n",
    "Recall from Chebychev bound:\n",
    "\n",
    "$P(|B_n-p| \\geq \\delta p) \\leq \\frac{Var(B_n)}{(\\delta p)^2} = \\frac{1-p}{np\\delta^2 }$\n",
    "\n",
    "For example, if we know that $p$ is about 0.5 and we want to estimate it within 1% (i.e. $\\delta = 0.01) and with a confidence exceeding 99.9%, then Chebychev bound says we need at least 10million samples, while Chernoff bounds tells that about 13.6Lakh samples is enough\n",
    "(see calculation below)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "IEg3Zhah7uLM",
    "outputId": "1873a9f7-4f94-42d5-bf9f-5e71d4f9a782"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "From Chebychev's ineuality, n should be atleast 10000000.0\n",
      "From Chernoff bound, n should be at least 1368162.4427175745\n"
     ]
    }
   ],
   "source": [
    "q = 0.001 #(1-confidence)\n",
    "p = 0.5\n",
    "d = 0.01 #delta (accuracy)\n",
    "\n",
    "print(f\"From Chebychev's ineuality, n should be atleast {(1-p)/q/d**2/p}\")\n",
    "\n",
    "print(f\"From Chernoff bound, n should be at least {3*np.log(2/q)*3/p/d**2}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 467
    },
    "id": "0AjU2-iV9fan",
    "outputId": "661297e8-d7e8-4192-9bff-9677f4c33425"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EX = 10.080579401528137, Var(X)=20.23578067643614\n"
     ]
    },
    {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# testing the bounds\n",
    "nSamples = 10000\n",
    "#draw samples from chisquare distribution\n",
    "#https://numpy.org/doc/stable/reference/random/generated/numpy.random.RandomState.chisquare.html\n",
    "df = 10 #Number of degrees of freedom, obtained by adding squares of 'df' std normal variables\n",
    "X_D = rng.chisquare(df,nSamples)\n",
    "\n",
    "nbins = 20\n",
    "amin=0\n",
    "amax=40\n",
    "x_axis = np.arange(amin, amax, (amax-amin)/nbins)\n",
    "#print(x_axis)\n",
    "\n",
    "X_D_Hist,_ = np.histogram(X_D,nbins,(amin,amax))\n",
    "\n",
    "#convert to pdf\n",
    "X_pdf = X_D_Hist/nSamples\n",
    "\n",
    "print(f\"EX = {np.mean(X_D)}, Var(X)={np.var(X_D)}\")\n",
    "\n",
    "plt.plot(x_axis,X_pdf)\n",
    "plt.legend(['Chisquare density'])\n",
    "plt.ylabel(\"pdf f(x)\")\n",
    "plt.xlabel(\"X\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Khii1qqY768Q"
   },
   "source": [
    "---\n",
    "*$\\underline{Weak\\ Law\\ of\\ Large\\ Numbers}$*\n",
    "\n",
    "Let $X_1, X_2, ..., X_n$ be i.i.d with mean $\\mu$ and variance $\\sigma^2$\n",
    "\n",
    "Let $M_n=\\frac{\\sum_{i=1}^n X_i}{n}$\n",
    "\n",
    "Then we know that the average of $M_n$ =  $EM_n = \\mu$ and $Var(M) = \\frac{\\sigma^2}{n}$\n",
    "\n",
    "The weak law states that even the sample mean ($M_n$) also converges to the mean $EX_i$ in probability.\n",
    "\n",
    "i.e.\n",
    "\n",
    "$lim_{n→∞} P(|M_n - \\mu| \\geq \\epsilon) = 0$\n",
    "\n",
    "Since from Chebuchev's inequality:\n",
    "\n",
    "$P(|M_n - \\mu| \\leq \\epsilon ) \\leq \\frac{\\sigma^2}{n\\epsilon^2}$\n",
    "\n",
    "\n",
    "This is also stated $M_n$ converges to $\\mu$ in probability.\n",
    "\n",
    "*$\\underline{Convergence\\ in\\ Probability}$*\n",
    "\n",
    "Let $Y_1, Y_2, ..... Y_n$ be a sequence of r.v. and let $a$ be a real number. Then\n",
    "\n",
    "$Y_n$ converges to $a$ in probability if:\n",
    "\n",
    "$lim_{n→∞} P(|Y_n-a| \\geq \\epsilon) =0$ for all $\\epsilon > 0 $\n",
    "\n",
    "What this means is that the pdf of $Y_n$ gets increasingly concentrated around $a$ with increasing $n$.\n",
    "\n",
    "\n",
    "*$\\underline{Central\\ Limit\\ Theorem}$*\n",
    "\n",
    "Let $X_1, X_2, .., X_n$ be i.i.d with mean $\\mu$ and variance $\\sigma^2$\n",
    "\n",
    "$Z_n = \\frac{\\sum_{i=1}^n X_i - n\\mu}{\\sqrt{n}\\sigma}$.\n",
    "\n",
    "We know that $EZ_n=0$ and $Var(Z_n)=1$\n",
    "\n",
    "Then the CDF $F_{Z_n}(z)$ converges to CDF of standard normal. This is more precisely expressed as:\n",
    "\n",
    "$lim_{n\\rightarrow ∞} P(Z_n \\leq z) = \\Phi(z)$ for all $z$, where $\\Phi(z)$ is CDF of standard normal.\n",
    "\n",
    "Note that this holds regardless of pdf of $X_i$\n",
    "\n",
    "\n",
    "*$\\underline{Almost\\ Sure\\ Convergence}$*\n",
    "\n",
    "If $P(lim_{n→∞} Y_n = a) =1$, it is called a.s. convergence.\n",
    "\n",
    "The way to interpret this is at in the sample space of infinitely long sequences drawn from $Y_i$, all the outcomes are sequences with all $a$ except for a finite sub-sequence in the beginnnig.\n",
    "\n",
    "\n",
    "*Example*\n",
    "\n",
    "As an example of a sequence which converges in probability but not almost surely,\n",
    "\n",
    "Let $Y_i = 0$ with probability $\\frac{n-1}{n}$\n",
    "\n",
    "$\\ \\ = 1$ with probability $\\frac{1}{n}$\n",
    "\n",
    "Then $\\lim_{n→∞} P(Y_n=0)=1$ and converges in probability to the 0.\n",
    "\n",
    "However, it doesnt almost surely converge to 0, i.e. $P(lim_{n→∞} Y_n =0) \\neq 1$. To see this, assume that $lim{n→∞} Y_n =0$.\n",
    "\n",
    "Then for every $\\delta$, there exists $N(\\delta)$ such that\n",
    "\n",
    "$Y_i = 0$ for all $n > N(\\delta)$.\n",
    "\n",
    "However the probability of this happening is $lim_{n→∞}(\\frac{n-1}{n})^{n-N(\\delta)} = e^{-1} < 1$\n",
    "\n",
    "*$\\underline{Strong\\ Law\\ of\\ Large\\ Numbers}$*\n",
    "\n",
    "Let $X_1, X_2, ..., X_n$ be i.i.d with mean $\\mu$ and variance $\\sigma^2$\n",
    "\n",
    "Let $M_n=\\frac{\\sum_{i=1}^n X_i}{n}$\n",
    "\n",
    "$P(lim_{n→∞} M_n = \\mu)= 1$\n",
    "\n",
    "i.e. $M_n$ almost surely converges to $\\mu$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "bVTjeo9Q1Mw5"
   },
   "source": [
    "---\n",
    "**Chapters 3 & 4 for Probabilistic Machine Learning by Kevin Murphy**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "-Cre4hK4793c"
   },
   "source": [
    "---\n",
    "*$\\underline{Multivariate\\ Models}$*\n",
    "\n",
    "Let $X$ be a $n$ dimensional random vector: $X = \\begin{bmatrix} X_1 \\\\ X_2 \\\\ ... \\\\ X_n \\end{bmatrix}$ (every component is a random variable)\n",
    "\n",
    "Then the probability density function associated with this vector is given by the join density function:\n",
    "\n",
    "$f_X(x_1,x_2, ..., x_n)$\n",
    "\n",
    "And the mean of the component 1 is : $\\mu_1 = EX_1 = \\int_{x_1}\\int_{x_2}...\\int_{x_n} x_1 f_X(x_1, ..., x_n)dx_1dx_2...dx_n = \\int_{x_1}x_1(\\int_{x_2}...\\int_{x_n} f_X(x_1, ..., x_n)dx_2...dx_n)dx_1 = \\int_{x_1}x_1f_{X_1}(x_1)dx_1$\n",
    "\n",
    "Here $f_{X_1}(x_1)$ is the marginal distribution of $X_1$\n",
    "\n",
    "Similarly the expections of other components and covariances, higher moments etc can be obtained.\n",
    "\n",
    "\n",
    "Then the covariance matrix is\n",
    "\n",
    "$\\Sigma = Cov(X) = E(X-EX)(X-EX)^T  = \\begin{bmatrix} Var(X_1) & Cov(X_1,X_2) & ... & Cov(X_1,X_n) \\\\ Cov(X_2,X_1) & Var(X_2) & ...& Cov(X_2,X_n) \\\\ ... \\\\ Cov(X_1,X_n) & Cov(X_2,X_n) & ... & Var(X_n)\\end{bmatrix} = \\begin{bmatrix} EX_1^2 - (EX_1)^2 & EX_1X_2 - EX_1EX_2 & ... & EX_1X_n - EX_1EX_n \\\\ EX_2X_1-EX_2EX_1 & EX_2^2-(EX_2)^2 & ... & EX_2X_n - EX_2EX_n \\\\ ... \\\\ EX_nEX_1 - EX_nEX_1 & EX_nX_2 - EX_nEX_2 & ... & EX_n^2-(EX_n)^2 \\end{bmatrix}= EXX^T - \\mu\\mu^T$\n",
    "\n",
    "\n",
    "Where $\\mu = \\begin{bmatrix} \\mu_1 \\\\ \\mu_2 \\\\ ... \\\\ \\mu_n \\end{bmatrix} = EX $ a $D$ dimensional column vector.\n",
    "\n",
    "That is:\n",
    "\n",
    "$EXX^T = \\Sigma + \\mu\\mu^T$\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "l3AJaAnaBJxQ"
   },
   "source": [
    "---\n",
    "**Q1**\n",
    "\n",
    "Show that the covariance of an affine map $Y = AX+b$ is given as:\n",
    "\n",
    "$\\Sigma_Y = A\\Sigma_XA^T$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "bTzSV0bVBtup"
   },
   "source": [
    "Ans:\n",
    "\n",
    "Let $X$ be a random vector with mean $\\mu_X$ and covariance matrix $\\Sigma_X$.\n",
    "\n",
    "For the affine map $Y = AX+b$, the mean of $Y$ is given by\n",
    "\n",
    "$\n",
    "\\displaystyle\n",
    "\\mu_Y\n",
    "= \\operatorname{E}[Y]\n",
    "= \\operatorname{E}[AX+b]\n",
    "= A\\operatorname{E}[X]+b\n",
    "= A\\mu_X+b\n",
    "$\n",
    "\n",
    "and the covariance matrix of $Y$ is given by\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "\\Sigma_Y\n",
    "&\n",
    "= \\operatorname{E}[(Y-\\mu_Y)(Y-\\mu_Y)^T]\n",
    "= \\operatorname{E}[(AX+b-A\\mu_X-b)(AX+b-A\\mu_X-b)^T]\n",
    "\\\\ &\n",
    "= \\operatorname{E}[(AX-A\\mu_X)(AX-A\\mu_X)^T]\n",
    "= \\operatorname{E}[A(X-\\mu_X)(X-\\mu_X)^TA^T]\n",
    "\\\\ &\n",
    "= A \\operatorname{E}[(X-\\mu_X)(X-\\mu_X)^T] A^T\n",
    "= A \\Sigma_X A^T\n",
    "\\end{aligned}\n",
    "$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "H00UaYfmLfUH"
   },
   "source": [
    "---\n",
    "**Q2**\n",
    "\n",
    "Show that the covariance matrix is positive semi-definite"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "o45V2VG2Lk3n"
   },
   "source": [
    "Ans:\n",
    "\n",
    "A square matrix $\\mathbf{A} \\in \\mathbb{R}^{n \\times n}$ is positive semi-definite if $\\mathbf{v}^T\\mathbf{A}\\mathbf{v} \\geq 0 \\quad \\forall \\ \\mathbf{v} \\in \\mathbb{R}^n \\setminus \\{ \\mathbf{0} \\}$.\n",
    "\n",
    "Let $\\mathbf{X}$ be a random vector with mean $\\mu_X$ and covariance matrix $\\Sigma_X$, and let $\\mathbf{Y} = \\mathbf{X} - \\mu_X$.\n",
    "Then, we have\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "\\mathbf{v}^T\\Sigma_X\\mathbf{v}\n",
    "&\n",
    "= \\mathbf{v}^T\\operatorname{E}[(\\mathbf{X}-\\mu_X)(\\mathbf{X}-\\mu_X)^T]\\mathbf{v}\n",
    "= \\mathbf{v}^T\\operatorname{E}[\\mathbf{Y}\\mathbf{Y}^T]\\mathbf{v}\n",
    "\\\\ &\n",
    "= \\operatorname{E}[\\mathbf{v}^T\\mathbf{Y}\\mathbf{Y}^T\\mathbf{v}]\n",
    "= \\operatorname{E}[(\\mathbf{Y}^T\\mathbf{v})^T(\\mathbf{Y}^T\\mathbf{v})]\n",
    "= \\operatorname{E}[\\|\\mathbf{Y}^T\\mathbf{v}\\|^2]\n",
    "\\geq 0\n",
    "\\end{aligned}\n",
    "$\n",
    "\n",
    "Thus, the covariance matrix is positive semi-definite.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "nzRijcXtjvrr"
   },
   "source": [
    "*$\\underline{Positive\\ definiteness\\ of\\ Covariance matrix}$*\n",
    "\n",
    "From above $\\Sigma ≽ 0 $ (positive semi-definite).\n",
    "\n",
    "Also it can be diagonalized by an orthogonal matrix S such that\n",
    "\n",
    "$\\Sigma = SDS^T$ and $SS^T=S^TS=I$\n",
    "\n",
    "Therefore:\n",
    "\n",
    "$EXX^T = \\Sigma + \\mu\\mu^T = E(S^TX)(S^TX)^T = S^T\\Sigma S + (S^T\\mu)(S^T\\mu)^T = EYY^T = D + \\mu_Y\\mu_Y^T$\n",
    "\n",
    "Here $Y = S^TX$\n",
    "\n",
    "Note that the components of $Y$ are uncorrelated as $\\Sigma_Y = D$ a diagonal matrix.\n",
    "\n",
    "Hence we can go from a correlated set of rv to uncorrelated by a suitable linear transform of the random vector.\n",
    "\n",
    "If the random vector is Gaussian, then the uncorrelated vector's component sare also independent.\n",
    "\n",
    "For eigen values of 0, the corresponding components of $Y$ are also 0 and we can drop them and work in a lower dimensional space where the reduced covariance matrix has only non-zero eigenvalues and hence invertible.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "6kUNoCSIIyaN"
   },
   "source": [
    "*$\\underline{Multivariate\\ Gaussian}$*\n",
    "\n",
    "$X$ is a $n$ dimensional random vector $X\\ \\ \\tilde\\ \\ N(x|\\mu,\\Sigma)$, with $x^T=[x_1\\ x_2\\ ...\\ x_n]\\in ℝ^n$\n",
    "\n",
    "then\n",
    "\n",
    "$f_X(x) = \\frac{1}{\\sqrt{(2\\pi)^n|\\Sigma|)}}e^{-\\frac{1}{2}(x-\\mu)^T\\Sigma^{-1}(x-\\mu)}$\n",
    "\n",
    "is the pdf of $X$\n",
    "\n",
    "$\\Lambda = \\Sigma^{-1}$ is also called the \"precision\" matrix (we are assuming that $\\Sigma$ is invertible (as discussed in above cell)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "ummRciY1eL76"
   },
   "source": [
    "*$\\underline{Mahalanobis\\ distance}$*\n",
    "\n",
    "Consider the level sets of (log) of constant probability of multivariate gaussian\n",
    "\n",
    "$-log(f_X(x)) = \\frac{1}{2}(x-u)^T\\Lambda(x-u) = C$ a constant.\n",
    "\n",
    "Geometrically it looks like an ellipse.\n",
    "\n",
    "If eigenvalues of $\\Sigma$ are:\n",
    "\n",
    "$\\lambda(\\Sigma) = \\{\\lambda_1, \\lambda_2, ..., \\lambda_n\\}$, with $\\lambda_i > 0$,\n",
    "\n",
    "then the eigen values of the precision matrix are\n",
    "\n",
    "$\\lambda(\\Lambda)=\\{1/\\lambda_i:i = 1..n\\}$\n",
    "\n",
    "The precision matrix is also positive definite.\n",
    "\n",
    "Let $\\Lambda = S^TDS$ for orthogonal $S$ and Diagonal $D$\n",
    "\n",
    "then\n",
    "\n",
    "$\\Lambda = \\sum_{i=1}^n \\frac{s_is_i^T}{\\lambda_i}$\n",
    "\n",
    "$(x-u)^T\\Lambda(x-u) = y^TDy = \\sum_{i=1}^n \\frac{y_iy_i^T}{\\lambda_d} > 0 $ with $y_i=(x-\\mu)^Ts_i$. the locus for all $x$ is the probability level set and  forms an ellipsoids in n-dimensional space.\n",
    "\n",
    "Note that $\\lambda_i$ corresponds to eigen value of covariance matrix. If it is high - it means the uncertainty in that direction (eigen vector) is high and hence is given a lower weight in the distance computation. The standard deviation along that direction is: $\\sqrt{\\lambda_i}$\n",
    "\n",
    "\n",
    "Also $D_\\Sigma(x,\\mu)=(x-u)^T\\Sigma^{-1}(x-u)$ is called the Mahalanobis distance - analogous to the Euclidean distance - but scaled by the covariance matrix.\n",
    "\n",
    "\n",
    "P C Mahalanobis: https://en.wikipedia.org/wiki/Prasanta_Chandra_Mahalanobis\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "BeJCWEs44b3u"
   },
   "source": [
    "*$\\underline{Linear\\ transform\\ of\\ Multivariate\\ Normal}$*\n",
    "\n",
    "Let $X$ be $n\\times 1$ random vector and\n",
    "$X\\ \\sim N(x|\\mu,\\Sigma)$\n",
    "\n",
    "Let $Y=AX+b$ with $A$ an $n\\times n$ invertible matrix and $b$ an $n\\times 1$ vector.\n",
    "\n",
    "Consider a volume element $dV_Y = dy_1dy_2..dy_n$. It is related to the volume element $dV_X=dx_1dx_2..dx_n$ as\n",
    "\n",
    "$dV_Y = det(A)dV_X$\n",
    "\n",
    "(you can recall that det(A) is the determinant and represents the volume of the tranform of a unit hypercube)\n",
    "\n",
    "We will also write $|A|$ to represent $det(A)$\n",
    "\n",
    "We also know that:\n",
    "\n",
    "$\\mu_Y = A\\mu_X + b$\n",
    "\n",
    "$\\Sigma_Y = A\\Sigma_X A^T$\n",
    "\n",
    "Then $P(Y=y\\in dV_Y) \\approx f_Y(y)dV_Y = P(X=A^{-1}(y-b) \\in dV_Y/det(A))=\\frac{1}{\\sqrt{(2\\pi)^\\frac{n}{2}|\\Sigma_X|}}\\frac{dV_Y}{|A|}e^{-\\frac{1}{2}(A^{-1}(y-b)-\\mu_X)^T\\Sigma_X^{-1}(A^{-1}(y-b)-\\mu_X)}= \\frac{1}{\\sqrt{(2\\pi)^\\frac{n}{2}|\\Sigma_X|}}e^{-\\frac{1}{2}((y-\\mu_Y)^TA^{-T}\\Sigma_X^{-1}A^{-1}(y-\\mu_Y)}\\frac{dV_Y}{\\sqrt{|A||A^T|}}= N(y|\\mu_Y, \\Sigma_Y)dV_Y$\n",
    "\n",
    "Hence $Y$ is also a MVN"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "0SL1PeRiHdX2"
   },
   "source": [
    "*$\\underline{Inverse\\ of\\ Block\\ Matrices}$*\n",
    "\n",
    "Let $\\Sigma = \\begin{bmatrix} \\Sigma_{11} & \\Sigma_{12} \\\\ \\Sigma_{21} & \\Sigma_{22} \\end{bmatrix}$\n",
    "\n",
    "Applying Gaussian elimination at block level, we can determine,\n",
    "\n",
    "$L = \\begin{bmatrix} I & 0 \\\\ -\\Sigma_{21}\\Sigma_{11}^{-1} & I \\end{bmatrix}$\n",
    "\n",
    "$U = \\begin{bmatrix} I & -\\Sigma_{11}^{-1}\\Sigma_{12} \\\\ 0 & I \\end{bmatrix}$\n",
    "\n",
    "and\n",
    "$L\\Sigma U = \\begin{bmatrix} \\Sigma_{11} & 0 \\\\ 0 & \\Sigma/\\Sigma_{11} \\end{bmatrix}$\n",
    "\n",
    "where $\\Sigma/\\Sigma_{11} = \\Sigma_{22} - \\Sigma_{21}\\Sigma_{11}^{-1}\\Sigma_{12}$ is the Schur complement\n",
    "\n",
    "Therefore $\\Sigma^{-1} = \\begin{bmatrix} I & -\\Sigma_{11}^{-1}\\Sigma_{12} \\\\ 0 & I \\end{bmatrix}\\begin{bmatrix} \\Sigma_{11}^{-1} & 0 \\\\ 0 & (\\Sigma/\\Sigma_{11})^{-1} \\end{bmatrix}\\begin{bmatrix} I & 0 \\\\ -\\Sigma_{21}\\Sigma_{11}^{-1} & I \\end{bmatrix}$\n",
    "\n",
    "\n",
    "Note also that $|L\\Sigma U| = |\\Sigma| = |\\Sigma_{11}||\\Sigma/\\Sigma_{11}|$ as $|L|=|U|=1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "DN7jqatrpaTL"
   },
   "source": [
    "*$\\underline{Marginals\\ and\\ Conditionals\\ of\\ Multivariate Normal (MVN)}$*\n",
    "\n",
    "\n",
    "Let an  n-dimensional random vector, $X^T = [X_1^T\\ X_2^T]$ with $X_1$ and $X_2$ having dimensions $n_1$ and $n_2$ ($n_1+n_2=n$)\n",
    "\n",
    "Let $EX_1 = \\mu_1$, $EX_2=\\mu_2$\n",
    "\n",
    "and\n",
    "\n",
    "$\\Sigma = \\begin{bmatrix} \\Sigma_{11} & \\Sigma_{12} \\\\ \\Sigma_{21} & \\Sigma_{22} \\end{bmatrix}$\n",
    "\n",
    "with $\\Sigma_{ij} = Cov(X_i,X_j)$,\n",
    "\n",
    "(Note cross covariance $\\Sigma_{ij} = E(X_i-EX_i)(Y_j-EY_j)^T$\n",
    "\n",
    "\n",
    "We can reorganize the exponent of the gaussian using the schur complement decomposition from previous cell as:\n",
    "\n",
    "$[(X_1-\\mu_1)^T\\ (X_2-\\mu_2)^T]\\ \\Sigma^{-1}\\begin{bmatrix} X_1 - \\mu_1 \\\\ X_2 - \\mu_2\\end{bmatrix}= [(X_1-\\mu_1)^T (X_2-\\mu_2)^T ]\\begin{bmatrix} I & -\\Sigma_{11}^{-1}\\Sigma_{12} \\\\ 0 & I \\end{bmatrix}\\begin{bmatrix} \\Sigma_{11}^{-1} & 0 \\\\ 0 & (\\Sigma/\\Sigma_{11})^{-1} \\end{bmatrix}\\begin{bmatrix} I & 0 \\\\ -\\Sigma_{21}\\Sigma_{11}^{-1} & I \\end{bmatrix} \\begin{bmatrix} X_1 - \\mu_1 \\\\ X_2 - \\mu_2\\end{bmatrix} = (X_1-\\mu_1)^T\\Sigma_{11}^{-1}(X_1-\\mu_1) + (X_2-\\mu_2 - \\Sigma_{21}\\Sigma_{11}^{-1}(X_1-\\mu_1))^T(\\Sigma/\\Sigma_{11})^{-1}(X_2-\\mu_2 - \\Sigma_{21}\\Sigma_{11}^{-1}(X_1-\\mu_1))$\n",
    "\n",
    "Hence we can factor $N(x|\\mu,\\Sigma) = N(x_1|\\mu_1,\\Sigma_{11})N(x_2|\\mu_2+\\Sigma_{21}\\Sigma_{11}^{-1}(x_1-\\mu_1),\\Sigma/\\Sigma_{11})$\n",
    "\n",
    "\n",
    "Marginal of $X_1$:\n",
    "\n",
    "$f_{X_1}(x_1) = \\int_{X_2} f_X(x_1,x_2)dx_2 = \\int_{X_2}N(x_1|\\mu_1,\\Sigma_{11})N(x_2|\\mu_2\n",
    "+\\Sigma_{21}\\Sigma_{11}^{-1}(x_1-\\mu_1),\\Sigma/\\Sigma_{11}) = N(x_1|\\mu_1, \\Sigma_{11}) $\n",
    "\n",
    "\n",
    "is also a multi-variate normal\n",
    "\n",
    "and from the above the conditional $X_2|X_1$ is ,\n",
    "\n",
    "$f_{X_2|X_1} = N(x_2|\\mu_2+\\Sigma_{21}\\Sigma_{11}^{-1}(x_1-\\mu_1),\\Sigma/\\Sigma_{11})$ is also a MVN.\n",
    "\n",
    "Similarly the marginal $f_{X_2}(x_2) = N(x_2|\\mu_2,\\Sigma_{22})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "mo_Z7v3STsm4"
   },
   "source": [
    "*$\\underline{Example}$*\n",
    "\n",
    "Let $X,Y$ be two univariate gaussian rv, which are jointly Gaussian and with correlation $\\rho$,  mean, $\\mu_x, \\mu_y$ and variance $\\sigma_x, \\sigma_y$ respectively.\n",
    "\n",
    "Then $\\Sigma = \\begin{bmatrix} \\sigma_x^2 & \\rho \\sigma_x\\sigma_y \\\\ \\rho\\sigma_x\\sigma_y & \\sigma_y^2\\end{bmatrix}$\n",
    "\n",
    "$f_{Y|X}(y) = N(y|\\mu_y-\\frac{\\rho\\sigma_y}{\\sigma_x}(x-\\mu_x),\\sigma_y^2(1-\\rho^2))$\n",
    "\n",
    "We can interpret it by looking at the level set for 2D correlated Gaussian and seeing that if $X=x$, then we are taking a slice of the ellipses at $X=x$ and hence get a modified gaussian for the $Y|X=x$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "vgjrmUyRP4_M"
   },
   "source": [
    "*$\\underline{Uncorrelated\\ implies\\ Independence\\ for\\ Gaussians}$*\n",
    "\n",
    "Let $X_1, X_2$ be jointly Gaussian and uncorrelated. This means $\\Sigma_{12}=0$ and $\\Sigma_{21}=\\Sigma_{12}^T = 0$.\n",
    "\n",
    "Hence $\\Sigma/\\Sigma_{11} = \\Sigma_{22}$\n",
    "\n",
    "and $f_{X_2|X_1}(x_2) = f_{X_2}(x_2)$\n",
    "\n",
    "therefore $f_X(x_1, x_2) = f_{X_1}(x_1)f_{X_2}(x_2)$ and they are independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "n5h_bs_bOsdx"
   },
   "source": [
    "---\n",
    "**Q3**\n",
    "\n",
    "Let $X\\ \\sim \\ N(x|0,1)$ and $Y=WX$ where $P(W=1)=P(W=-1)=0.5$\n",
    "\n",
    "a) Show $Y \\sim N(y|0,1)$\n",
    "\n",
    "b) Show $Cov(X,Y)=0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "fTYmr2wHPEtI"
   },
   "source": [
    "Ans:\n",
    "\n",
    "(a)\n",
    "Since $X \\sim \\mathcal{N}(0,1)$, we have $f_X(x) = \\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}$.\n",
    "\n",
    "The random variable $W$ is a discrete random variable with $P(W=1)=P(W=-1)=0.5$, thereby we have that $f_W(w) = \\frac{1}{2}\\mathbf{1}_{\\{w=1\\}} + \\frac{1}{2}\\mathbf{1}_{\\{w=-1\\}}$.\n",
    "\n",
    "Now, we have\n",
    "\n",
    "$\n",
    "\\displaystyle\n",
    "f_Y(y)\n",
    "= \\sum_{w \\in \\{-1,1\\}} f_{Y|W}(y|w)f_W(w)\n",
    "= \\frac{1}{2}f_{X}(y) + \\frac{1}{2}f_{-X}(y)\n",
    "= \\frac{1}{2}\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{y^2}{2}} + \\frac{1}{2}\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{y^2}{2}}\n",
    "= \\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{y^2}{2}}\n",
    "\\implies \n",
    "Y \\sim \\mathcal{N}(0,1)\n",
    "$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "J8Uy3ze3V3Yf"
   },
   "source": [
    "---\n",
    "**Inference and Estimation**\n",
    "\n",
    "*$\\underline{Statistics}$*\n",
    "\n",
    "*Inference* refers to figuring out, quantitatively, aspects of the population given some samples. It involves building estimators (or statistics - which are formulae for combining data - e.g. average), for the various parameters of the population distribution and quantify their uncertainties. Here estimation is part of the inferencing process.\n",
    "\n",
    "$\\underline{Machine\\ Learning}$\n",
    "\n",
    "Usually develops a parametrized model of the phenomenon, where the model parameters are *estimated* (aka **learning**) from the data.\n",
    "\n",
    "Inference is the process of using the model  to make predictions. It has a narrower connotation than that in Statistics.\n",
    "\n",
    "*$\\underline{Example:\\ Bayesian\\ Inference}$*\n",
    "\n",
    "A patient is tested positive for a disease and the test is 99% accurate (i.e given the disease the test shows positive 99% of the time and given no disease it indicates negative 99% of the time). However the disease is prevalent in only about 1 in 10000 people. What is the chance that the patient has the disease?\n",
    "\n",
    "Ans:\n",
    "\n",
    "Let $D=1$ if someone has the diease and 0 otherwise.\n",
    "\n",
    "$P(D=1)=0.0001$\n",
    "\n",
    "Let $T=1$ if test returns positive and 0 otherwise.\n",
    "\n",
    "Then $P(T=1|D=1) = 0.99$ (true positive) and $P(T=0|D=1)=0.01$ (false negative)\n",
    "\n",
    "Similarly,\n",
    "\n",
    "$P(T=0|D=0)=0.99$ (true negative), $P(T=1|D=0)=0.01$ (false positive)\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "We need to find $P(D=1|T=1)$\n",
    "\n",
    "$P(D=1|T=1)=\\frac{P(T=1|D=1)P(D=1)}{P(T=1)}= \\frac{P(T=1|D=1)P(D=1)}{P(T=1|D=1)P(D=1)+P(T=1|D=0)P(D=0)} = \\frac{0.99*0.0001}{0.99*0.0001+0.01*0.9999} = 0.0098$\n",
    "\n",
    "So even though test seems very accurate, since the disease is very rare, the chance of having the disease is pretty small. Hence one needs to drive down the chance of false positive even more for the test.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "uvxm2JAnY3xl"
   },
   "source": [
    "**Maximum Likelihood Estimation**\n",
    "\n",
    "Given data samples $x_D=\\{x_1,x_2,...\\}$ and a parametrized probability model for generating the sample, $X\\ \\ \\tilde\\ \\ p(x;\\theta)$, then the MLE estimate of the parameter is the one that gives the maximum probability of observing that data $x_D$.\n",
    "\n",
    "$\\theta_{mle} = argmax_\\theta p(x_D|\\theta)$\n",
    "\n",
    "Usually the data samples are assumed to be independently sampled hence\n",
    "\n",
    "$p(x_D;\\theta) = ∏ p(x_i;\\theta)$ and is also called the likelihood $L(\\theta)$.\n",
    "\n",
    "Note that the likelihood is a function of the parameter to be estimated.\n",
    "\n",
    "Log likelihood is:\n",
    "\n",
    " $LL(\\theta) = log L(\\theta) = \\sum log(p(x_i;\\theta))$\n",
    "\n",
    "and is more convenient to work with (due to sum rather than a product).\n",
    "\n",
    "Hence MLE estimate of the parameter $\\theta$ is:\n",
    "\n",
    "$\\theta_{mle} = argmax_\\theta LL(\\theta) = argmin_\\theta -LL(\\theta)$\n",
    "\n",
    "where $-LL(\\theta)=NLL(\\theta)$ is sometimes more convenient and is called negative log likelihood."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "WqBW9u_Qr5Fk"
   },
   "source": [
    "*$\\underline{Example:Estimating\\ probability\\ of\\ head}$*\n",
    "\n",
    "Let a coin toss give heads with rv $H\\ \\ \\tilde\\ \\ Ber(\\theta)$\n",
    "\n",
    "Given a sequence of $n$ tosses with observations $h_D=\\{h_1, h_2, ..... h_n\\}$, where $h_i$ is either 0 or 1 based on tail of head,\n",
    "\n",
    "we need to find an estimator for the parameter $\\theta$, the probability of getting a head.\n",
    "\n",
    "$L(\\theta) = p(h_D;\\theta) = \\prod_1^n p(h_i;\\theta)= \\prod_1^n\\theta^{h_i}(1-\\theta)^{(1-h_i)}$\n",
    "\n",
    "Therefore\n",
    "\n",
    "$LL(\\theta) = \\sum_1^n h_i log(\\theta)+(1-h_i)log(1-\\theta) = log(\\theta)(\\sum h_i) + log(1-\\theta)\\sum (1-h_i)$\n",
    "\n",
    "Since $\\theta_{mle} = argmax_\\theta LL(\\theta)$,\n",
    "\n",
    "\n",
    "$\\frac{dLL(\\theta)}{d\\theta} = \\frac{\\sum h_i}{\\theta} - \\frac{\\sum 1-h_i}{1-\\theta} =0 $ and we get $\\theta_{mle} = \\frac{\\sum_1^n h_i}{n}$ which is just to sum up binary sample values!  (you can check that this is the maximum by checking the sign of second derivative)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "iQt5PBHAxY3B"
   },
   "source": [
    "*$\\underline{Example:MLE\\ for\\ Univariate\\ Gaussian\\ Parameters}$*\n",
    "\n",
    "Let $X\\ \\ \\tilde\\ \\ N(x|\\mu,\\sigma^2)$\n",
    "\n",
    "and $x_D = \\{x_1, x_2, ...., x_n\\}$ be the observed data samples, each being sampled independently and we would like to estimate the parameters $\\mu, \\sigma$.\n",
    "\n",
    "Then the likelihood\n",
    "\n",
    "$L(\\mu,\\sigma) = \\prod_1^n \\frac{1}{\\sqrt{2\\pi}\\sigma} e^{-\\frac{(x_i-\\mu)^2}{2\\sigma^2}}$\n",
    "\n",
    "and negative log likelihood is:\n",
    "\n",
    "$NLL(\\mu,\\sigma) = \\frac{1}{2}\\sum_i^n \\frac{(x_i-\\mu)^2}{\\sigma^2} + nlog\\sigma + \\frac{n}{2}log2\\pi$\n",
    "\n",
    "$\\frac{dNLL}{d\\mu}= \\frac{1}{\\sigma^2}\\sum_1^n (x_i-\\mu) = 0$ leads to $\\mu_{mle} = \\frac{\\sum_1^nx_i}{n}$\n",
    "\n",
    "and\n",
    "\n",
    "$\\frac{dNLL}{d\\sigma} = \\frac{-\\sum (x_i-\\mu)^2}{\\sigma^3} + \\frac{n}{\\sigma}=0$ and leads to $\\sigma_{mle}^2 = \\frac{\\sum_1^n (x_i-\\mu_{mle})^2}{n}$\n",
    "\n",
    "The above is a biased estimator (i.e. $E\\sigma_{mle} \\neq \\sigma$)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "gwMZwXrD1_9j"
   },
   "source": [
    "*$\\underline{MLE\\ for\\ MVN}$*\n",
    "\n",
    "Let $X\\ \\ \\tilde\\ \\ N(x|\\mu,\\Sigma)$ be a n-dimensional random vector which is multivariate normally distributed.\n",
    "\n",
    "Let $x_D = \\{x_1,x_2,...,x_k\\}$ be $k$ observations each of dimension $n$\n",
    "\n",
    "We will write it as a $n\\times k$ data matrix: $x_D = \\begin{bmatrix}x_1 & x_2 & ... & x_k\\end{bmatrix}$\n",
    "\n",
    "Then the Maximum Likelihood Estimate of $\\mu, \\Sigma$ are:\n",
    "\n",
    "$\\mu_{mle} = \\frac{\\sum_{i=1}^k x_i}{k}$\n",
    "\n",
    "$\\Sigma_{mle} = \\frac{x_Dx_D^T}{D} - \\mu_{mle}\\mu_{mle}^T$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "vr3siZXo946S"
   },
   "source": [
    "**Computing uncertainties in the estimates via Bayesian Inference**\n",
    "\n",
    "\n",
    "Given an estimate of parameters $\\hat{\\theta}$ computed from data $D$, we would also like to estimate the uncertainty of the estimate.\n",
    "\n",
    "We have seen how inequality bounds can give you one approach to get confidence intervals, i.e. bounds on $P(|\\hat{\\theta}-\\theta_{actual}|\\leq 1-\\epsilon)$\n",
    "\n",
    "\n",
    "Another approach is to estimate the entire posterior distribution $f_{\\hat{\\Theta}}(\\hat{\\theta})$ of the random vector $\\hat{\\Theta}$\n",
    "\n",
    "In order to simplify typing, we will drop the hat notation (for the estimate).\n",
    "\n",
    "Treating the parameters  as a random vector (note this is a modeling choice - to represent unknown but fixed parameters as a random vector with a distribution):\n",
    "\n",
    "Let $f_\\Theta(\\theta)$ be the density we believe the parameter has.\n",
    "\n",
    "Then by Bayes rule, once we get new data $D$,\n",
    "\n",
    "we can do\n",
    "\n",
    "$f_{\\Theta|D}(\\theta) = \\frac{P(D|\\theta)f_\\Theta(\\theta)}{P(D)}$\n",
    "\n",
    "Here $P(D|\\theta)$ is the likelihood of seeing the data for a particular value of the parameter.\n",
    "\n",
    "$f_\\Theta(\\theta)$ is called the prior and\n",
    "\n",
    "$f_{\\Theta|D}(\\theta)$ is called the posterior distribution\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "DUjDFfGtNB2L"
   },
   "source": [
    "*$\\underline{Conjugate\\ Prior}$*\n",
    "\n",
    "As a modeling choice, certain analytical forms of the prior density, combines with speicific forms of the likelihood function to results in  a posterior density which can be analytically computed.\n",
    "\n",
    "In such cases, it is easy to do the computations and the prior is a conjugate prior to that likelihood function.\n",
    "\n",
    "\n",
    "*$\\underline{Example: Ber(\\theta)}$\n",
    "\n",
    "Let $D=\\{x_1,x_2,...,x_N\\}$ data from N independent samples of a Bernoulli random variable\n",
    "\n",
    "As we have seen, the likelihood is:\n",
    "\n",
    "$L(\\theta;D) = \\prod_{i=1}^n \\theta^{x_i}(1-\\theta)^{(1-x_i)} = \\theta^{N_1}(1-\\theta)^{N_0}$\n",
    "\n",
    "Where $N_1$ is number of 1's and $N_0$ is number of 0s.\n",
    "\n",
    "A conjugate prior is the Beta distribution:\n",
    "\n",
    "$f_\\Theta(\\theta) = Beta(\\theta|a,b) =\n",
    "\\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\theta^{a-1}(1-\\theta)^{b-1}$ with $0 \\leq \\theta \\leq 1$\n",
    "\n",
    "with $\\Gamma(a) = \\int_0^∞ x^{a-1}e^{-x}dx$\n",
    "\n",
    "for integer $n$, $\\Gamma(n) = n-1!$ (using integration by parts)\n",
    "\n",
    "$\\Gamma(a+1)=a\\Gamma(a)$\n",
    "\n",
    "\n",
    "Hence the posterior is also a Beta distribution with\n",
    "\n",
    "$f_\\Theta(\\theta) = Beta(\\theta|a+N_1, b+N_0)$\n",
    "\n",
    "<img src='https://drive.google.com/uc?export=view&id=17qEUod78-VGPa4xVmxbB1clguqnhidk8' width=800px>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "qK1JReZrRnaL"
   },
   "source": [
    "---\n",
    "**Q4**\n",
    "\n",
    "If $\\theta\\ \\ \\tilde\\ \\ Beta(\\theta|a,b)$, then show that\n",
    "\n",
    "a) $E\\theta = \\frac{a}{a+b}$\n",
    "\n",
    "b) $Var(\\theta) = \\frac{ab}{(a+b)^2(a+b+1)}$\n",
    "\n",
    "c) $mode(\\theta) = \\frac{a-1}{a+b-2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "X7uI0bQjR2S5"
   },
   "source": [
    "Ans:\n",
    "\n",
    "With $\\Theta \\sim \\text{Beta}(a,b)$, we have\n",
    "\n",
    "(a)\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "\\operatorname{E}[\\Theta]\n",
    "&\n",
    "= \\int_0^1 \\theta \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\theta^{a-1}(1-\\theta)^{b-1} d\\theta\n",
    "= \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^1 \\theta^{a}(1-\\theta)^{b-1} d\\theta\n",
    "\\\\ &\n",
    "= \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\frac{\\Gamma(a+1)\\Gamma(b)}{\\Gamma(a+b+1)}\n",
    "= \\frac{a}{a+b}\n",
    "\\end{aligned}\n",
    "$\n",
    "\n",
    "(b)\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "\\text{Var}[\\Theta]\n",
    "&\n",
    "= \\operatorname{E}[\\Theta^2] - (\\operatorname{E}[\\Theta])^2\n",
    "= \\int_0^1 \\theta^2 \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\theta^{a-1}(1-\\theta)^{b-1} d\\theta - \\left( \\frac{a}{a+b} \\right)^2\n",
    "\\\\ &\n",
    "= \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\int_0^1 \\theta^{a+1}(1-\\theta)^{b-1} d\\theta - \\left( \\frac{a}{a+b} \\right)^2\n",
    "\\\\ &\n",
    "= \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} \\frac{\\Gamma(a+2)\\Gamma(b)}{\\Gamma(a+b+2)} - \\left( \\frac{a}{a+b} \\right)^2\n",
    "\\\\ &\n",
    "= \\frac{a(a+1)}{(a+b)(a+b+1)} - \\left( \\frac{a}{a+b} \\right)^2\n",
    "= \\frac{ab}{(a+b)^2(a+b+1)}\n",
    "\\end{aligned}\n",
    "$\n",
    "\n",
    "(c)\n",
    "\n",
    "$\n",
    "\\begin{aligned}\n",
    "\\text{mode}[\\Theta]\n",
    "&\n",
    "= \\frac{a-1}{a+b-2}\n",
    "\\end{aligned}\n",
    "$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "y8gW5oelOjwW"
   },
   "source": [
    "*$\\underline{Gaussian\\ Conjugate\\ Prior\\ and\\ Kalman\\ gain}$*\n",
    "\n",
    "Let $X\\ \\ \\tilde\\ \\ N(x|\\mu_X,\\Sigma_X)$ be a random vector which we would like to estimate (i.e. estimate its distribution or its moments).\n",
    "\n",
    "And let $Z$ be a measurement of $X$ with an additive  Gaussian noise $Y\\ \\ \\tilde\\ \\ N(y|0,\\Sigma_y)$\n",
    "\n",
    "i.e. $Z = WX + Y$ where $W is a  matrix.\n",
    "\n",
    "Then we know that,\n",
    "$\\mu_Z = W\\mu_X$ and\n",
    "\n",
    "$\\Sigma_Z = W\\Sigma_X W^T + \\Sigma_Y$ and that\n",
    "\n",
    "$Z$ is gaussian with $Z=N(z|\\mu_Z,\\Sigma_Z)$\n",
    "\n",
    "\n",
    "We would like to get $f_{X|Z}(x)$, i,e, the posterior after the measurement.\n",
    "\n",
    "We will use the following to get this:\n",
    "\n",
    "$f_{X|Z}(x)f_Z(x) =f_{XZ}(x,z)= f_{Z|X}(z)f_X(x)$\n",
    "\n",
    "We will guess that the joint distribution $f_{XZ}(x,z)$ is also Gaussian. To determine its moments, lets find the cross co-variance $\\Sigma_{XZ} = EXZ^T-(EX)(EZ)^T = XX^TW^T+XY^T - EX(EX)^T\\mu^T = \\Sigma_X W^T$ as $X,Y$ are independent and $EXY^T = EXEY^T = 0$\n",
    "\n",
    "\n",
    "Therefore, the we can write down the joint distribution of $X,Z$ is\n",
    "\n",
    "$f_{XZ}(x,z) = N(x,z|(\\mu_X,\\mu_Z),\\Sigma)$\n",
    "\n",
    "with\n",
    "$\\Sigma = \\begin{bmatrix} \\Sigma_X & \\Sigma_XW^T \\\\ W\\Sigma_X & W\\Sigma_XW^T+\\Sigma_Y \\end{bmatrix}$\n",
    "\n",
    "You can cross check that this gives the right marginals for $f_{Z|X}(z) = N(z|Wx,\\Sigma_y)$\n",
    "\n",
    "Now we can calculate the posterior distribution of $X$, given a measurement $Z=z$ as $\\frac{f_{X,Z}(x,z)}{f_Z(z)}$, which we can write down be refactoring the joint to result in:\n",
    "\n",
    "$f_{X|Z}(x) = N(x|\\mu_{X|Z},\\Sigma_{X|Z})$\n",
    "\n",
    "with\n",
    "\n",
    "$\\mu_{X|Z} = \\mu_X + \\Sigma_XW^T\\Sigma_Z^{-1}(z-W\\mu_X)$\n",
    "\n",
    "$\\Sigma_{X|Z} = \\Sigma/\\Sigma_Z = \\Sigma_X - \\Sigma_XW^T\\Sigma_Z^{-1}W\\Sigma_X$\n",
    "\n",
    "and $\\Sigma_Z = W\\Sigma_X W^T + \\Sigma_Y$\n",
    "\n",
    "In a more compact form, this is written as:\n",
    "\n",
    "$K = \\Sigma_XW^T\\Sigma_Z^{-1}$ also called the Kalman gain\n",
    "\n",
    "then,\n",
    "$\\mu_{X|Z} = \\mu_X + K(z-W\\mu_X) = (I-KW)\\mu_X + Kz$\n",
    "\n",
    "$\\Sigma_{X|Z} = \\Sigma/\\Sigma_Z = (I - KW)\\Sigma_X$\n",
    "\n",
    "Note that in the formula: $f_{X|Z}(x) \\propto f_{Z|X}(z)f_X(x)$, the likelihood, prior and posterior are all Gaussian. Hence Gaussian is also the conjugate prior for a Gaussian likelihood."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "lhwKKT4aDsB_"
   },
   "source": [
    "---\n",
    "*Example*\n",
    "\n",
    "Consider a univariate case, with $\\Sigma_X = \\sigma_x^2$ and $W = w$ as the measurement gain and $\\Sigma_Y = \\sigma_y^2$ as the measurement noise.\n",
    "\n",
    "\n",
    "Then $\\Sigma_Z = \\sigma_z^2 = w^2\\sigma_x^2 + \\sigma_y^2$\n",
    "\n",
    "and the Kalman gain is $K = \\frac{w\\sigma_x^2}{w^2\\sigma_x^2 + \\sigma_y^2}$.\n",
    "\n",
    "\n",
    "\n",
    "Therefore $\\mu_{X|Z} = \\mu_X + K(z-w\\mu_X) = \\frac{\\sigma_y^2}{w^2\\sigma_x^2+\\sigma_y^2}\\mu_X + \\frac{w^2\\sigma_x^2}{w^2\\sigma_x^2+\\sigma_y^2}\\frac{z}{w}$ and $\\sigma_{X|Z}^2 = (1-Kw)\\sigma_x^2 = \\frac{\\sigma_y^2\\sigma_x^2}{w^2\\sigma_x^2+\\sigma_y^2}$\n",
    "\n",
    "\n",
    "Note if $\\sigma_y → 0$ , it means the measurements are more accurate, then $Kw → 1$ and. more weightage is given to measurements $(1-Kw)$\n",
    "\n",
    "On the other hand, with poor measurements,  $\\sigma_y^2 >> w^2\\sigma_x^2$, then $Kw << 1$ and tends to 0 and less weightage is given to measurements.\n",
    "\n",
    "Thus with good measurements $Kw→1$ and posterior is more influenced by data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "jzX9IL54F3mA"
   },
   "source": [
    "---\n",
    "*$\\underline{Multisensor\\ Fusion}$*\n",
    "\n",
    "Assume there are two independent sensors, giving a measurement of $X$ as:\n",
    "\n",
    "$Z_1 = W_1 X + Y_1$ with $Y_1\\ \\ \\tilde\\ \\ N(y_1|0,\\Sigma_{Y_1})$\n",
    "\n",
    "and\n",
    "\n",
    "$Z_2 = W_2 X + Y_2$ with $Y_2\\ \\ \\tilde\\ \\ N(y_2|0,\\Sigma_{Y_2})$\n",
    "\n",
    "and the conditional measurement independence is described formally as :\n",
    "\n",
    "$f_{Z_1,Z_2|X}(z_1,z_2) = f_{Z_1|X}(z_1)f_{Z_2|X}(z_2)$\n",
    "\n",
    "\n",
    "We would like to figure out how to best estimate the vector $X$ given the measurements $Z_1, Z_2$, i.e. find the posterior $f_{X|Z_1,Z_2}(x)$.\n",
    "\n",
    "We can concatenate the measurements as:\n",
    "\n",
    "$Z = \\begin{bmatrix} Z_1 \\\\ Z_2 \\end{bmatrix} = \\begin{bmatrix} W_1 \\\\ W_2 \\end{bmatrix} X + \\begin{bmatrix}Y_1 \\\\ Y_2 \\end{bmatrix}$\n",
    "\n",
    "\n",
    "$\\Sigma_Z = W\\Sigma_XW^T+\\begin{bmatrix}\\Sigma_{Y_1} & 0 \\\\ 0 & \\Sigma_{Y_2} \\end{bmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "$K = \\Sigma_X W^T \\Sigma_Z^{-1}$\n",
    "\n",
    "and apply the formulae  from above cell for deriving $\\mu_{X|Z}$ and $\\Sigma_{X|Z}$.\n",
    "\n",
    "However we can get a bit more insight by looking at this as follows:\n",
    "\n",
    "\n",
    "$f_{X|Z_1,Z_2}(x) = \\frac{f_{Z_1,Z_2|X}(z_1, z_2)f_X(x)}{f_{Z_1,Z_2}(z_1,z_2)} = \\gamma f_{Z_1,Z_2|X}(x)f_X(x)$ where $\\gamma$ is independent of $x$ and is whatever needed to normalize the pdf.\n",
    "\n",
    "Now $Z_1, Z_2$ are independent measurements given a particular value of $X$ (or conditioned on X).\n",
    "\n",
    "Hence $f_{Z_1, Z_2|X}(z_1, z_2) = f_{Z_1|X}(z_1)f_{Z_2|X}(z_2) = N(z_1|W_1x, \\Sigma_{Y_1})N(z_2|W_2x, \\Sigma_{Y_2})$\n",
    "\n",
    "Hence $f_{X|Z_1,Z_2}(x) = \\gamma N(z_1|W_1x, \\Sigma_{Y_1})N(z_2|W_2x, \\Sigma_{Y_2}) N(x|\\mu_X, \\Sigma_X)$\n",
    "\n",
    "Lets look at the argument of the exponent on both sides:\n",
    "\n",
    "Left hand side:\n",
    "\n",
    "$x^T\\Sigma_{X|Z}^{-1}x - 2x^T\\Sigma_{X|Z}^{-1}\\mu_{X|Z} + \\mu_{X|Z}^T\\Sigma_{X|Z}^{-1}\\mu_{X|Z}$\n",
    "\n",
    "Right hand side:\n",
    "\n",
    "$(z_1-W_1x)^T\\Sigma_{Y_1}^{-1}(z_1-W_1x) + (z_2-W_2x)^T\\Sigma_{Y_2}^{-1}(z_2-W_2x) + (x-\\mu_X)\\Sigma_X^{-1}(x-\\mu_X)$\n",
    "\n",
    "Equating the terms corresponding the $x$, we get\n",
    "\n",
    "$\\Sigma_{X|Z}^{-1} = W_1^T\\Sigma_{Y_1}^{-1}W_1 + W_2^T\\Sigma_{Y_2}^{-1}W_2 + \\Sigma_X^{-1}$\n",
    "\n",
    "and\n",
    "\n",
    "$\\Sigma_{X|Z}^{-1}\\mu_{X|Z} = W_1^T\\Sigma_{Y_1}^{-1}z_1 + W_2^T\\Sigma_{Y_2}^{-1}z_2 +  \\Sigma_X^{-1} \\mu_X$\n",
    "\n",
    "\n",
    "(note the terms not dependent on $x$ get aborbed into the normalization constant).\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "hXoWbPG6O0cu"
   },
   "source": [
    "*Example*\n",
    "\n",
    "Let two sensors have variances $\\sigma_1^2$ and $\\sigma_2^2$ respectively. Then the optimal way to combine their measurements $z_1$ and $z_2$ is:\n",
    "\n",
    "$\\sigma_{Z}^{-2} = \\sigma_1^{-2}+\\sigma_2^{-2}$\n",
    "\n",
    "and $Z = \\frac{\\sigma_Z^2}{\\sigma_1^2}z_1+\\frac{\\sigma_Z^2}{\\sigma_2^2}z_2 = \\frac{\\sigma_2^2}{\\sigma_1^2+\\sigma_2^2}z_1  + \\frac{\\sigma_1^2}{\\sigma_1^2+\\sigma_2^2}z_2$\n",
    "\n",
    "Thus the sensor with low variance is given high weightage and vice-versa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "aa3sPoyoQn-I"
   },
   "source": [
    "---\n",
    "**Q5**\n",
    "\n",
    "\n",
    "Consider two temperature sensors with the data on their errors during calibration as below:\n",
    "\n",
    "error data sensor 1: [-0.11575496  0.02897558  0.07808541  0.05439736 -0.09613826  0.10710087  0.07014557  0.07049735  0.07450626  0.11043472]\n",
    "\n",
    "error data for sensor 2:\n",
    "[ 0.44859448 -0.12229862  0.00944224  0.35084694 -0.26759597  0.06511489\n",
    " -0.13782354 -0.00396436  0.09495065 -0.38622028]\n",
    "\n",
    "\n",
    "Find the optimal temperature estimate if sensor 1 report 21.1 and sensor 2 reports 23.2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "p42dhBgSR8Ji"
   },
   "source": [
    "Ans:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/"
    },
    "id": "C__tztpXSHAR",
    "outputId": "66d2724e-ac26-4dc5-c69a-3cf321b63eb4"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sensor1 bias: bias=0.03822499, variance=0.006499477687241411\n",
      "Sensor2 bias=0.005104642999999992, variance=0.06579805574772601\n",
      "The optimal temperature estimate is = 21.253540447087815\n"
     ]
    }
   ],
   "source": [
    "nSamp = 10\n",
    "\n",
    "\n",
    "#sensor 1\n",
    "data_1 = np.array([-0.11575496,0.02897558,0.07808541,0.05439736,-0.09613826,0.10710087,\n",
    "  0.07014557,0.07049735, 0.07450626, 0.11043472])\n",
    "var1=np.sum(data_1*data_1)/(nSamp-1)\n",
    "#print(f\"Variance of sensor 1 is {var1}\")\n",
    "m1 = np.sum(data_1)/nSamp\n",
    "var_act1= var1-m1*m1\n",
    "print(f\"Sensor1 bias={m1}, variance={var_act1}\")\n",
    "\n",
    "\n",
    "data_2 = np.array([0.44859448,-0.12229862, 0.00944224,  0.35084694, -0.26759597,  0.06511489,\n",
    " -0.13782354, -0.00396436,  0.09495065, -0.38622028])\n",
    "var2 = np.sum(data_2*data_2)/(nSamp-1)\n",
    "m2 = np.sum(data_2)/nSamp\n",
    "#print(f\"Variance of sensor 2 is {var2}\")\n",
    "var_act2 = var2-m2*m2\n",
    "print(f\"Sensor2 bias={m2}, variance={var_act2}\")\n",
    "\n",
    "T1 = 21.1\n",
    "T2 = 23.2\n",
    "\n",
    "#print(f\"The optimal temperature estimate without considering bias is = {var2*T1/(var1+var2) + var1*T2/(var1+var2)}\")\n",
    "\n",
    "print(f\"The optimal temperature estimate is = {<yourcode>}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Ufa1IcxPZw0F"
   },
   "source": [
    "---\n",
    "**Q6**\n",
    "\n",
    "Exercise 3.7 from PML text:\n",
    "\n",
    "Suppose we have two sensors with known (and different) variances $\\sigma_1^2$ and $\\sigma_2^2$, but unknown (and the same)\n",
    "mean μ.\n",
    "\n",
    "Suppose we observe $n_1$ observations $z_1 ∼ N(μ,\\sigma_1)$ from the first sensor and $n_2$ observations\n",
    "$z_2 ∼ N(μ,\\sigma_2)$ from the second sensor. (For example, suppose μ is the true temperature outside, and sensor 1\n",
    "is a precise (low variance) digital thermosensing device, and sensor 2 is an imprecise (high variance) mercury thermometer.)\n",
    "\n",
    "Let $D$ represent all the data from both sensors.\n",
    "\n",
    "What is the posterior $p(μ|D)$, assuming a non-informative prior for μ (which we can simulate using a Gaussian with a precision of 0)?\n",
    "\n",
    "Give an explicit expression for the posterior mean and variance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "OF78F6OIb69n"
   },
   "source": [
    "Ans:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "Rad85MqQvYKK"
   },
   "source": [
    "---\n",
    "\n",
    "**Kalman filter for a linear dynamical system**\n",
    "\n",
    "\n",
    "We will now build upon the previous results to derive the optimal state estimate for a linear dynamical system, by combining state evolution based on the state transition equation, with measurements.\n",
    "\n",
    "Let at time step $k$\n",
    "\n",
    "$X_k$ be the state of the system. We will model this as gaussian with mean $\\mu_{X_k}$ and covariance $\\Sigma_{X_k}$\n",
    "\n",
    "\n",
    "$u_k$ be the input to the system\n",
    "\n",
    "$A_k$ state transition matrix\n",
    "\n",
    "$B_k$ be the input to state matrix\n",
    "\n",
    "$P_k$ be the process noise which we will model as 0 mean gaussian with covariance $\\Sigma_{P_k}$\n",
    "\n",
    "The dynamical evolution of the system is then given as\n",
    "\n",
    "$X_{k+1} = A_k X_k + B_k u_k + P_k$\n",
    "\n",
    "\n",
    "$W_k$ be the measurment matrix\n",
    "\n",
    "$M_k$ be the measurement noise which is also assumed to be Gassian with zero mean and  covariance $\\Sigma_{M_k}$\n",
    "\n",
    "$Z_k$ is the measurement and is given as\n",
    "\n",
    "$Z_k = W_k X_k + M_k$\n",
    "\n",
    "Kalman filtering is done in two steps:\n",
    "\n",
    "*$\\underline{Predict\\ step}$*\n",
    "\n",
    "Here we use the state value at time k to predict its value at k+1 using the dynamical equation.\n",
    "\n",
    "Let $X_{k+1}^p$ be the predicted state at $k+1$\n",
    "\n",
    "Note since all input variables are assumed to be distributed as gaussian, the outputs will also be gaussian. Hence we only need to keep track of the means and the co-variances\n",
    "\n",
    "Then\n",
    "\n",
    "$\n",
    "\\mu_{X_{k+1}}^p = A_k \\mu_{X_k} + B_k u_k\n",
    "$\n",
    "\n",
    "$\n",
    "\\Sigma_{X_{k+1}}^p = A_k \\Sigma_{X_k} A_k^T + \\Sigma_{P_k}\n",
    "$\n",
    "\n",
    "We can use these as prior to the measurement $Z_{k+1}$ as it measures $X_{k+1}^p$\n",
    "\n",
    "Therefore, using the formula derived above the posterior moments is can be derived as\n",
    "\n",
    "\n",
    "Kalman gain is:\n",
    "\n",
    "$\n",
    "K_{k+1} = \\Sigma_{X_{k+1}}^p W_{k+1}^T(W_{k+1}\\Sigma_{X_{k+1}}^pW_{k+1}^T + \\Sigma_{M_k})^{-1}\n",
    "$\n",
    "\n",
    "*$\\underline{Update\\ step}$*\n",
    "\n",
    "Use the measurement $y_{k+1}$ to updat ethe  Covariance and mean of posterior at time $k+1$:\n",
    "\n",
    "$\n",
    "\\Sigma_{X_{k+1}} = (I - K_{k+1}W_{k+1})\\Sigma_{X_{k+1}}^p\n",
    "$\n",
    "\n",
    "$\n",
    "\\mu_{X_{k+1}} = \\mu_{X_{k+1}}^p + K_{k+1}(z_{k+1}-W_{k+1}\\mu_{X_{k+1}}^p)\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "0yJpNRYAuC5K"
   },
   "source": [
    "---\n",
    "**Q7**\n",
    "\n",
    "*$\\underline{Simple\\ pendulum\\ Model}$*\n",
    "\n",
    "𝑙  : Length of pendulum\n",
    "\n",
    "𝑚  : Mass of pendulum\n",
    "\n",
    "$g$  : Acceleration due to gravity\n",
    "\n",
    "𝑑𝑡  : Time step\n",
    "\n",
    "$𝜃_𝑛$  : Angular deviation from vertical at timestep  𝑛 (counter clockwise is positive)\n",
    "\n",
    "$\\omega_n=\\dot{𝜃}_𝑛$ : Angular velocity at timestep  𝑛\n",
    "\n",
    "$𝜃_{𝑛+1}=𝜃_𝑛+\\dot{𝜃}_n 𝑑𝑡 $\n",
    "\n",
    "$\\omega_{n+1}=\\dot{𝜃}_{𝑛+1}=\\dot{𝜃}_𝑛 − \\frac{g sin(𝜃_𝑛)}{l}𝑑𝑡 $\n",
    "\n",
    "For small $\\theta_n$, $sin(\\theta_n) ≈ \\theta_n$\n",
    "and the above equation can be linearized as:\n",
    "\n",
    "$\\dot{𝜃}_{𝑛+1}=\\dot{𝜃}_𝑛 − \\frac{g𝜃_𝑛}{l}𝑑𝑡 $\n",
    "\n",
    "Therefore:\n",
    "\n",
    "$\\begin{bmatrix} \\theta_{n+1} \\\\ \\omega_{n+1} \\end{bmatrix} = \\begin{bmatrix} 1 & dt \\\\-\\frac{gdt}{l} & 1\\end{bmatrix} \\begin{bmatrix} \\theta_n \\\\ \\omega_n \\end{bmatrix} + P_n$\n",
    "\n",
    "with the process noise $P_n\\ \\ \\tilde\\ \\ N(r_n|0,\\Sigma_P)$ where $\\Sigma_P$ is a 2x2 matrix\n",
    "\n",
    "Measurement of angle:\n",
    "\n",
    "$𝑧_𝑛 = 𝜃_𝑛+M$\n",
    "\n",
    "where  $M\\ \\tilde\\ 𝑁(m|0,\\sigma_M^2)$   is normal.\n",
    "\n",
    "\n",
    "a) Write the Kalman filter based state estimation equations and\n",
    "\n",
    "b) Complete the simulation code below"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "SMG28Lh8x05I"
   },
   "source": [
    "Ans:\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "colab": {
     "base_uri": "https://localhost:8080/",
     "height": 562
    },
    "id": "QNcnjLHFrA55",
    "outputId": "bfd37b3d-14fa-497c-b75f-61c7624d744b"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0: ang=[0.01],vel=[0.], predang=[0.01], corr_ang=[0.01], K=[[0.]\n",
      " [0.]]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x78045889b160>"
      ]
     },
     "execution_count": 180,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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FqK2tRV1dXZP15QkaMrJA1dTUAIAisn379o1ar6CgIO4OIEBvrnj44IMP8Lvf/Q5WqxUFBQUoLi7GM888E9X2GTNmYMKECbjssstQWlqKCy+8EMuXL09Lx1F+eMXKPyW7seR1bDZbXOtpSePfMzc3F1arVXFtqJfLvzEA/PLLL/jxxx9RXFwc9erfvz8A4MiRIwBSO/e//PILAGDWrFlNjvPCCy8gEAg0uUb79esX9b5Pnz7geV5J0ZFIe+RrUN2ZYGQ3mdSzESNGYODAgVEu7GXLlqGoqEgxoo4ePYra2lo899xzTa7p2bNnA4jcO4kSj2aKoojHH38c/fr1izo/3333XdT5ueWWW5CTk4OxY8eiX79+uOqqq/R3L0q0pplqHUy3Znbt2rWJHuTm5qJbt25NlgGR6yiR3z2Vc//LL7+AEIJ+/fo1Oc727dubXFudO3eGw+GIWiZruKyZibQnHZrJYhYzxNVXX40lS5bguuuuw/jx45GbmwuO43DhhRfGfIA2Nxs33s5daxQWFgJAVMekOb744gtMnToVJ554Ip5++mmUlZXBZDJhyZIlUbPgbDYbPv/8c6xZswYffvghVqxYgWXLluGUU07Bxx9/nPQM43goKysDABw+fLjJZ4cPH0ZBQYFiGZeVlWHNmjUghETdbPK2cjyRlsT67vH8xqIoYtiwYXjsscdiriuLZyrnXr7+Hn74YYwcOTLmOq3llGssWom0R74GG3ecGdlLpvVsxowZuO+++1BZWQmn04n33nsPF110kTKjV27DH//4xyaxjTLDhw9P6JhqzezatWuL695///2488478ec//xn33HMPCgoKwPM8rrvuuqjzM2jQIOzYsQMffPABVqxYgTfffBNPP/005s+fj7vuuiuh9iWKWjMbd8IOHz4cFacsZ6VojF6a2dz10tp1lMjvnsq5F0URHMfhf//7X8w2JZODM5H21NTUwG636zKwIcM6ixnijTfewKxZs/Doo48qy/x+f9LJPnv06AEA2LVrF3r16qUsr6qqiqsD2L17d9hsNuzevbvVdd98801YrVasXLkyyhWxZMmSJuvyPI9TTz0Vp556Kh577DHcf//9uP3227FmzRpMmjRJN0uoS5cuKC4uxjfffNPks40bN0Z1gkaOHIkXXngB27dvx+DBg5XlGzZsUD5viXSOgPXp0wfbtm3Dqaee2upxkz33ffr0AUBTZkyaNCmudv3yyy9R192uXbsgimJU5Z/W2iOze/du8DyvWNqM7CfTejZjxgzcddddePPNN1FaWgq32x3ltiwuLobT6YQgCHFf060hp5HZvXt3s5M5ZN544w2cfPLJ+Ne//hW1vLa2tolR5HA4MGPGDMyYMQPBYBDnnnsu7rvvPtx2221KUm09kHXum2++ieoYHjp0CAcOHFBSjcnrfvHFFxBFMWqSy4YNG2C321u9d9OlmYn+7sme+z59+oAQgl69esWlW4cOHYLH44kaXdy5cycARGlma+2R2b17d5MJmlrD3NAZwmAwNLGin3zySQiCkNT+Tj31VBiNRjzzzDNRy+W8ga1hMpkwZsyYmJ2rxhgMBnAcF9XWPXv2KDNxZaqrq5tsKwuS7MKQbxY9MuKfd955+OCDD7B//35l2erVq7Fz505ccMEFyrKzzz4bJpMJTz/9tLKMEILFixejS5curc7O1vM7NGb69Ok4ePAgnn/++Saf+Xw+eDweAKmd+9GjR6NPnz545JFHokqYycSqgLNo0aKo908++SQA4Pe//33c7ZHZvHkzhgwZoriUGNlPpvVs0KBBGDZsGJYtW4Zly5ahrKxMySAgt++8887Dm2++GbOcZKxrujVGjx4Ns9kct2Y2Pj+vv/46Dh48GLVMjoOUMZvNGDx4MAghCIVCAPTTmyFDhmDgwIF47rnnon63Z555BhzHKfkUAeD8889HRUVFVJxoZWUlXn/9dZx11lkx4xnVOByOtOhlIr97Kuf+3HPPhcFgwF133dXkdyaENNl3OByOqroiV2EpLi7G6NGj426PzJYtWzTJItISbGQxQ/zhD3/AK6+8gtzcXAwePBjr16/HqlWrFNdGopSWluLaa6/Fo48+iqlTp2LKlCnYtm0b/ve//6GoqCguS+7ss8/G7bff3mrKkjPPPBOPPfYYpkyZgosvvhhHjhzBokWL0LdvX3z33XfKenfffTc+//xznHnmmejRoweOHDmCp59+Gl27dlVS0fTp0wd5eXlYvHgxnE4nHA4Hxo0bFzWa0JinnnoKtbW1OHToEADg/fffx4EDBwBQd5jcyfi///s/vP766zj55JNx7bXXoqGhAQ8//DCGDRumxKsANB7muuuuw8MPP4xQKIRjjz0W77zzDr744gu8+uqrrbrL5Zv7mmuuweTJk2EwGKJGNbTkT3/6E5YvX46//vWvWLNmDSZMmABBEPDzzz9j+fLlWLlyJcaMGZPyuX/hhRfw+9//HkOGDMHs2bPRpUsXHDx4EGvWrIHL5cL7778f1a7du3cr19369evxn//8BxdffDFGjBgBIL5rAaCTIz777LMmExsY2U026NmMGTMwf/58WK1WzJkzp0lalwcffBBr1qzBuHHjcPnll2Pw4MGorq7Gli1bsGrVqpgGTUtYrVacfvrpWLVqVasJ5P/whz/g7rvvxuzZs3Hcccfh+++/x6uvvorevXtHrXf66aejU6dOmDBhAkpLS7F9+3Y89dRTOPPMM5VUMLLe3H777bjwwgthMplw1llnNYmBk6mrq1OMNznm7amnnkJeXh7y8vIwb948Zd2HH34YU6dOxemnn44LL7wQP/zwA5566ilcdtllUSNX559/Pn73u99h9uzZ+Omnn1BUVISnn34agiDE5S4fPXo0nnnmGdx7773o27cvSkpKdEsoHe/vnsq579OnD+69917cdttt2LNnD6ZNmwan04ndu3fj7bffxty5c3HTTTcpbercuTP+/ve/Y8+ePejfvz+WLVuGrVu34rnnnoPJZIq7PQA1rqurq3H22Wfrcv4UdJ1rzWiWmpoaMnv2bFJUVERycnLI5MmTyc8//9wk5YKcamLTpk1R28dKMRAOh8mdd95JOnXqRGw2GznllFPI9u3bSWFhYZNyTbGoqKggRqORvPLKK1HLY6XO+de//kX69etHLBYLGThwIFmyZImSRkVm9erV5OyzzyadO3cmZrOZdO7cmVx00UVNykG9++67ZPDgwcRoNMaVsiJWUlf51TilwQ8//EBOP/10YrfbSV5eHpk5cyYpLy9vsk9BEMj9999PevToQcxmMxkyZAj5z3/+0+o5I4Se96uvvpoUFxcTjuPiSsrdOIXMrFmziMPhaLJvOYGrmmAwSP7+97+TIUOGEIvFQvLz88no0aPJXXfdRerq6ggh2pz7b7/9lpx77rmksLCQWCwW0qNHDzJ9+nSyevXqJt/np59+Iueffz5xOp0kPz+fzJs3Lyopd7zt+d///kcAkF9++aWVs87IJrJBz3755RdFBxonj5epqKggV111FenWrRsxmUykU6dO5NRTTyXPPfecsk4i5f7eeustwnFck3QzsVLn3HjjjUqC8QkTJpD169eTiRMnRqWQefbZZ8mJJ56o3HN9+vQhN998s3Jfy9xzzz2kS5cuhOf5VtPoyN8n1itWGde3336bjBw5klgsFtK1a1dyxx13KOlc1FRXV5M5c+aQwsJCYrfbycSJE5v8rs1RXl5OzjzzTOJ0OuNOyt2YHj16kDPPPLPJcgDkqquuiloWz++uxbl/8803yfHHH08cDgdxOBxk4MCB5KqrriI7duxo8n3USbl79OhBnnrqqajjxNueW265hXTv3l33cn8cIRrNkGBkJbW1tcjPz8e9996rJEBtiTlz5mDnzp344osv0tA6Rltn4cKFuOuuu3D06FFNJqRMmzYNHMfh7bff1qB1jPZGonqmN4IgYPDgwZg+fTruueeeTDeH0QY46aSTUFlZGdMtniiBQAA9e/bErbfeimuvvVaD1jUPi1lsR8TKbfXEE08AoBdoPCxYsACbNm1KW7oGBkNm+/bt+OCDD9hDlwFAGz3TG4PBgLvvvhuLFi2KGd/LYOjJkiVLYDKZ8Ne//lX3Y7GYxXbEsmXL8NJLL+GMM85ATk4OvvzyS/z3v//F6aefjgkTJsS1j+7du8dVHpDB0JpBgwYhHA5nuhmMLEELPUsH8mxVBiPd/PWvf01LRxFgncV2xfDhw2E0GvHQQw/B7XYrQeL33ntvppvGYDAYCcH0jMHIHljMIoPBYDAYDAajWVjMIoPBYDAYDAajWVhnkcFgMBgMBoPRLCxmsRVEUcShQ4fgdDrTWtaNwWBoDyEE9fX16Ny5c5OkzYymMP1jMNoPqegf6yy2wqFDh5oUVWcwGG2b/fv3o2vXrpluRtbD9I/BaH8ko3+ss9gKclmd/fv3t1gCj8FgZD9utxvdunWLKpfFaB6mfwxG+yEV/WOdxVaQXS8ul4uJJYPRTmAu1fhg+sdgtD+S0b82F7SzaNEi9OzZE1arFePGjcPGjRtbXL+2thZXXXUVysrKYLFY0L9/f3z00Udpai2DwWBoB9M/BoORCdrUyOKyZctwww03YPHixRg3bhyeeOIJTJ48GTt27EBJSUmT9YPBIE477TSUlJTgjTfeQJcuXbB3717k5eWlv/EMBoORAkz/GAxGpmhTSbnHjRuHY489Fk899RQAOlOvW7duuPrqq3Hrrbc2WX/x4sV4+OGH8fPPP8NkMiV1TLfbjdzcXNTV1TE3DIPRxmnL9zPTPwaDkQqp3M9tZmQxGAxi8+bNuO2225RlPM9j0qRJWL9+fcxt3nvvPYwfPx5XXXUV3n33XRQXF+Piiy/GLbfcAoPBEHObQCCAQCCgvHe73dp+EUZcEEIQDochCEKmm8JoY5hMpmbv77YK07/shWkVI5vQS//aTGexsrISgiCgtLQ0anlpaSl+/vnnmNv89ttv+PTTTzFz5kx89NFH2LVrF6688kqEQiEsWLAg5jYPPPAA7rrrLs3bz4ifYDCIw4cPw+v1ZropjDYIx3Ho2rUrcnJyMt0UzWD6l50wrWJkG3rpX5vpLCaDKIooKSnBc889B4PBgNGjR+PgwYN4+OGHmxXL2267DTfccIPyXp5qzkgPoihi9+7dMBgM6Ny5M8xmM5u5yogbQgiOHj2KAwcOoF+/fu1uhDERmP7pC9MqRrahp/61mc5iUVERDAYDKioqopZXVFSgU6dOMbcpKytrMiQ7aNAglJeXIxgMwmw2N9nGYrHAYrFo23hG3ASDQSUWy263Z7o5jDZIcXEx9uzZg1Ao1G46i0z/sg+mVYxsRC/9azOpc8xmM0aPHo3Vq1cry0RRxOrVqzF+/PiY20yYMAG7du2CKIrKsp07d6KsrCymUDKyB1aKjZEs7XF0h+lf9sK0ipFN6KV/beoqv+GGG/D888/j3//+N7Zv344rrrgCHo8Hs2fPBgBccsklUQHgV1xxBaqrq3Httddi586d+PDDD3H//ffjqquuytRXYDAYjKRg+sdgMDJFm3FDA8CMGTNw9OhRzJ8/H+Xl5Rg5ciRWrFihBH3v27cvysrr1q0bVq5cieuvvx7Dhw9Hly5dcO211+KWW27J1FdgMBiMpGD6x2AwMkWbyrOYCViesfTi9/uxe/du9OrVC1arNdPNYbRBWrqG2P2cGOx8NQ/TKkamWLhwId555x1s3bq1yWd66V+bckMzGO2RPXv2gOO4mDc+g8FgZBNMrzomrLPIYDAYDAaDwWgW1llkZDWEEHiD4Yy8EonQEEURDzzwAHr16gWbzYYRI0bgjTfeUD6vqanBzJkzUVxcDJvNhn79+mHJkiUAgF69egEAjjnmGHAch5NOOinmMdauXQuO47By5Uocc8wxsNlsOOWUU3DkyBH873//w6BBg+ByuXDxxRdHJQlurW2CIGDOnDnK5wMGDMA//vGPqGNfeumlmDZtGh555BGUlZWhsLAQV111FUKhULPn5Ndff8XZZ5+N0tJS5OTk4Nhjj8WqVaui1unZsyfuv/9+/PnPf4bT6UT37t3x3HPPRa3z1VdfYeTIkbBarRgzZgzeeecdNrLByEqYXkVgetVUr0RRRNeuXfHMM89EbfPtt9+C53ns3bsXAFBbW4vLLrsMxcXFcLlcOOWUU7Bt27Zm254O2tQEF4Y2uP0hPLJyB6Yd0wWjuufThavuAvZ9DfzpbcCUPfE3vpCAwfNXZuTYP909GXZzfLfIAw88gP/85z9YvHgx+vXrh88//xx//OMfUVxcjIkTJ+LOO+/ETz/9hP/9738oKirCrl274PP5AAAbN27E2LFjsWrVKgwZMqTVtCYLFy7EU089BbvdjunTp2P69OmwWCxYunQpGhoacM455+DJJ59UJjK01jZZwF5//XUUFhbiq6++wty5c1FWVobp06crx12zZg3KysqwZs0a7Nq1CzNmzMDIkSNx+eWXx2xnQ0MDzjjjDNx3332wWCx4+eWXcdZZZ2HHjh3o3r27st6jjz6Ke+65B//3f/+HN954A1dccQUmTpyIAQMGwO1246yzzsIZZ5yBpUuXYu/evbjuuuvi+k0YjFjoqX9Mr5rC9Oo6ZVue53HRRRdh6dKluOKKK5Tlr776KiZMmIAePXoAAC644ALYbDb873//Q25uLp599lmceuqp2LlzJwoKCuL6jbWGdRY7IO9tPYSX1+/FniovXv7zWLpw07+AQB1Q/h3QbWxmG9jGCAQCuP/++7Fq1Sol513v3r3x5Zdf4tlnn8XEiROxb98+HHPMMRgzZgwAaqHKFBcXAwAKCwubTbCs5t5778WECRMAAHPmzMFtt92GX3/9Fb179wYAnH/++VizZg1uueWWuNpmMpmiSrz16tUL69evx/Lly6PENz8/H0899RQMBgMGDhyIM888E6tXr25WfEeMGIERI0Yo7++55x68/fbbeO+99zBv3jxl+RlnnIErr7wSAHDLLbfg8ccfx5o1azBgwAAsXboUHMfh+eefh9VqxeDBg3Hw4MFmj8lgtEZH1z+mV5nVq5kzZ+LRRx/Fvn370L17d4iiiNdeew133HEHAODLL7/Exo0bceTIESVB/iOPPIJ33nkHb7zxBubOndvqOdcD1lnsgOw60gAA2F8tDf0HvVQoAcBXk6FWxcZmMuCnuydn7NjxsGvXLni9Xpx22mlRy4PBII455hgANOfdeeedhy1btuD000/HtGnTcNxxxyXVruHDhyv/l5aWwm63K8IrL9u4cWPcbQOARYsW4cUXX8S+ffvg8/kQDAYxcuTIqG2GDBkSVRGgrKwM33//fbPtbGhowMKFC/Hhhx/i8OHDCIfD8Pl82LdvX7Pfh+M4dOrUCUeOHAEA7NixA8OHD4+a1Td2bPt+mDP0RU/9Y3rVFKZX0Xo1cuRIDBo0CEuXLsWtt96Kzz77DEeOHMEFF1wAANi2bRsaGhpQWFgYtZ3P58Ovv/7abPv1hnUWOyC/VXoAAAdrfBBFAr6hPPKhtzpDrYoNx3Fxu1YyRUMDffh8+OGH6NKlS9RnsmX4+9//Hnv37sVHH32ETz75BKeeeiquuuoqPPLIIwkfz2QyKf9zHBf1Xl4mV+2Ip22vvfYabrrpJjz66KMYP348nE4nHn74YWzYsKHZ4zY+TixuuukmfPLJJ3jkkUfQt29f2Gw2nH/++QgGgyntl8FIBT31j+lVU5heNWXmzJlKZ3Hp0qWYMmWK0jlsaGhAWVkZ1q5d22S7vLy8hI6jJdl9VTN04bej9IYMCiKONgRQWq+qN5tlI4ttgcGDB8NisWDfvn2YOHFis+sVFxdj1qxZmDVrFk444QTcfPPNeOSRR5SYH0EQMtK2devW4bjjjlNcKwA0sWDXrVuHSy+9FOeccw4AKoJ79uxJaB8DBgzAf/7zHwQCAeVhsWnTppTbxui4dHT9Y3oVm3Tq1cUXX4w77rgDmzdvxhtvvIHFixcrn40aNQrl5eUwGo1R7v9MwzqLHQx/SMDBWp/y/kCNF6X1hyMr+LJrZLEt4HQ6cdNNN+H666+HKIo4/vjjUVdXh3Xr1sHlcmHWrFmYP38+Ro8ejSFDhiAQCOCDDz7AoEGDAAAlJSWw2WxYsWIFunbtCqvVitzc3LS1rV+/fnj55ZexcuVK9OrVC6+88go2bdqkzHpMln79+uGtt97CWWedBY7jcOeddyZsgV988cW4/fbbMXfuXNx6663Yt2+fMrrRHmtAM/SF6R/Tq+ZIp1717NkTxx13HObMmQNBEDB16lTls0mTJmH8+PGYNm0aHnroIfTv3x+HDh3Chx9+iHPOOUeJI003LHVOB2NPlQfqDAsHanxAQ8eyrPXgnnvuwZ133okHHngAgwYNwpQpU/Dhhx8qAmY2m3Hbbbdh+PDhOPHEE2EwGPDaa68BAIxGI/75z3/i2WefRefOnXH22WentW1/+ctfcO6552LGjBkYN24cqqqqoqz2ZHnssceQn5+P4447DmeddRYmT56MUaNGJbQPl8uF999/H1u3bsXIkSNx++23Y/78+QDAqmYwEobpH4XpVVPSrVczZ87Etm3bcM4558BmsynLOY7DRx99hBNPPBGzZ89G//79ceGFF2Lv3r1Kac9MwMr9tUJ7K3f10feHceWrW5T3N08egKvCLwPrpDxVQ84FLliSodaxElqM1nn11Vcxe/Zs1NXVRYmsDCv3px3t7XxpqX9Mqxjx0JpeaY1e+sfc0B0MOV5H5kCNF0DHs6wZbYeXX34ZvXv3RpcuXbBt2zbccsstmD59elqEl9G+YPrH0Jv2qless9jB+O0onQnYq8iB3ZUe6oYxdqyYHUbbory8HPPnz0d5eTnKyspwwQUX4L777st0sxhtEKZ/DL1pr3rFOosdjF+ltBEn9iuKiKWNWdaM7OVvf/sb/va3v2W6GYx2ANM/ht60V71iE1w6EIQQ7JbcMCf2p1n4D9b4QNSzAb1MLBkMRvuD6R+DkTyss9iBqPIE4faHwXHA73oXwsBz4AQ/OH9dZKVgPSA0X2ydwWAw2iJM/xiM5GGdxQ6EHK/TJc8Gh8WIslwrSjjJkjZaAUh5oJgrhsFgtDOY/jEYycM6ix0IeSZg7+IcAEDXfBtKUEs/dHYCrFJiVSaWDAajncH0j8FIHtZZ7EDINVF7FzkAAF3z7SiVLWtnGWDLp/9nWX1oBoPBSBWmfwxG8rDOYgdib1UkbQQgWdZcLf0wpxSwF9D/mWXNYDDaGUz/GIzkYZ3FDsSBGloTtVsBTQ7arGXNxLJdcemll2LatGmZbgaDkVGY/rUN2pNevfTSS8jLy8t0MzSBdRY7EAdrqVh2ybMDaGRZO0sBm2xZMzdMulm4cCFGjhyZ6WYwGO0Wpn/awfSq48GScncQ6v0h1HppSogu+dSy7lFoRwjUig7aS2G2SfnGmGXNYDDaEUz/GIzUYCOLHQTZqs63m5BjoTZCJ5cVnQ00x9huf04kZiebArwJAYKezLwIibuZK1aswPHHH4+8vDwUFhbiD3/4A3799deodQ4cOICLLroIBQUFcDgcGDNmDDZs2ICXXnoJd911F7Zt2waO48BxHF566SXs2bMHHMdh69atyj5qa2vBcRzWrl0LABAEAXPmzEGvXr1gs9kwYMAA/OMf/0joFFdVVeGiiy5Cly5dYLfbMWzYMPz3v/+NWuekk07CNddcg7/97W8oKChAp06dsHDhwqh1fv75Zxx//PGwWq0YPHgwVq1aBY7j8M477yTUHgZDa9Kmf0yv2o1eHXfccbjllluitjl69ChMJhM+//xzAEAgEMBNN92ELl26wOFwYNy4ccp3bW+wkcUOwoFqyQWTHylmznEcOvG1gAhsrbVhQGEWxuyEvMD9nTNz7P87BJgdca3q8Xhwww03YPjw4WhoaMD8+fNxzjnnYOvWreB5Hg0NDZg4cSK6dOmC9957D506dcKWLVsgiiJmzJiBH374AStWrMCqVasAALm5uaioqGjlqIAoiujatStef/11FBYW4quvvsLcuXNRVlaG6dOnx9V2v9+P0aNH45ZbboHL5cKHH36IP/3pT+jTpw/Gjh2rrPfvf/8bN9xwAzZs2ID169fj0ksvxYQJE3DaaadBEARMmzYN3bt3x4YNG1BfX48bb7wxruMzGHqTNv1jetUibUmvZs6ciYceeggPPvggOI7m4Fy2bBk6d+6ME044AQAwb948/PTTT3jttdfQuXNnvP3225gyZQq+//579OvXL67v01ZgncUOgmxZd5XidQAAnio4xHoAwNdHjJjRVRbLLBpZbCOcd955Ue9ffPFFFBcX46effsLQoUOxdOlSHD16FJs2bUJBAR3B6Nu3r7J+Tk4OjEYjOnXqlNBxTSYT7rrrLuV9r169sH79eixfvjxu8e3SpQtuuukm5f3VV1+NlStXYvny5VHiO3z4cCxYsAAA0K9fPzz11FNYvXo1TjvtNHzyySf49ddfsXbtWuU73HfffTjttNMS+j4Mhh4w/YuG6VXrejV9+nRcd911+PLLL5XO4dKlS3HRRReB4zjs27cPS5Yswb59+9C5MzUQbrrpJqxYsQJLlizB/fffn9C5yXZYZ7GDcKDGC4AGdaO+AnjrcmDPlwAAL7Hgi/1hkAn5tIZBNo0smuzUYs7UsePkl19+wfz587FhwwZUVlZCFEUAwL59+zB06FBs3boVxxxzjCK8WrJo0SK8+OKL2LdvH3w+H4LBYELB54Ig4P7778fy5ctx8OBBBINBBAIB2O3R33/48OFR78vKynDkyBEAwI4dO9CtW7eoh4dauBmMTJI2/WN61SptRa+Ki4tx+umn49VXX8UJJ5yA3bt3Y/369Xj22WcBAN9//z0EQUD//v2jtgsEAigsLIz7+7QVWGexgyCnjeiabwN+/gDY/RkAQCzqj/vKT0RlIIgjoTyUAoA3izqLHBe3ayWTnHXWWejRoweef/55dO7cGaIoYujQoQgGgwAAm83Wyh6awvM0pJioYpFCoei6ta+99hpuuukmPProoxg/fjycTicefvhhbNiwIe7jPPzww/jHP/6BJ554AsOGDYPD4cB1112ntF3GZDJFvec4TnnIMBjZTNr0j+lVu9KrmTNn4pprrsGTTz6JpUuXYtiwYRg2bBgAoKGhAQaDAZs3b4bBYIjaLicnJ6HjtAXYBJcOgpI2It8O1JfThaNmgZ+3CT90OgcA8F2NdDlk08hiG6Cqqgo7duzAHXfcgVNPPRWDBg1CTU30ORw+fDi2bt2K6urYLi6z2QxBEKKWFRcXAwAOHz6sLFMHjwPAunXrcNxxx+HKK6/EMcccg759+zYJVG+NdevW4eyzz8Yf//hHjBgxAr1798bOnTsT2seAAQOwf//+qLilTZs2JbQPBkMvmP5FYHoVv16dffbZ8Pv9WLFiBZYuXYqZM2cqnx1zzDEQBAFHjhxB3759o16JuufbAqyz2EGIsqwbJLHM7QoAOKY7jdX5plyyCEMeIBxIexvbKvn5+SgsLMRzzz2HXbt24dNPP8UNN9wQtc5FF12ETp06Ydq0aVi3bh1+++03vPnmm1i/fj0AoGfPnti9eze2bt2KyspKBAIB2Gw2/O53v8ODDz6I7du347PPPsMdd9wRtd9+/frhm2++wcqVK7Fz507ceeedCXfS+vXrh08++QRfffUVtm/fjr/85S9xBaurOe2009CnTx/MmjUL3333HdatW6e0VQ4OZzAyBdO/CEyv4tcrh8OBadOm4c4778T27dtx0UUXKZ/1798fM2fOxCWXXIK33noLu3fvxsaNG/HAAw/gww8/TKg9bQHWWewAeAJhVHvoEH0XOWYHoCWuAIzqQcXyy/0hEOmSqKlK7ObryPA8j9deew2bN2/G0KFDcf311+Phhx+OWsdsNuPjjz9GSUkJzjjjDAwbNgwPPvig4r4477zzMGXKFJx88skoLi5WUkG8+OKLCIfDGD16NK677jrce++9Ufv9y1/+gnPPPRczZszAuHHjUFVVhSuvvDKh9t9xxx0YNWoUJk+ejJNOOkl5SCSCwWDAO++8g4aGBhx77LG47LLLcPvttwMArFZrQvtiMLSE6V80TK8S06uZM2di27ZtOOGEE9C9e/eoz5YsWYJLLrkEN954IwYMGIBp06Zh06ZNTdZrD3CEJJCcqQPidruRm5uLuro6uFyuTDcnKXZW1OP0xz9Hrs2EbQtOB549ETi8Dbh4OdB/Mg7UeHH839cAALZY5qKAa8B7x72Bqaenfyar3+/H7t270atXL9bJaOOsW7cOxx9/PHbt2oU+ffqk7bgtXUPt4X5OJ+3hfOmlf0yr2heZ0iut0Uv/2ASXDsDBGrnMlRS0LMfsSJZ1lzwbehbasafKixriRAHXgLCnfcftMLTn7bffRk5ODvr164ddu3bh2muvxYQJE9q08DLaPkz/GLFgepUYrLPYAYhKGyEKgOco/cBJg3A5jsOS2WPx65EGmD4oALyHwXeAXGMMbamvr8ctt9yCffv2oaioCJMmTcKjjz6a6WYxOjhM/xixYHqVGKyzqCNufwgrfijH5CGdkGsztb6BTkSCu+1UKIkIcDzgKFbW6VXkQK8iB3atzAMAGAO1GWgpoy1zySWX4JJLLsl0MxhZAtM/RjbD9Cox2AQXHXlp3R787Y3v8K8vd2e0HQdqVaWuZBeMoxjgDU3WDZpzAQDGYG26msdgMNohTP8YjPYD6yzqyOE6KlK/HmnIaDui00ZEzwRsTFgSS1PQnZa2NQebd8VIFnbtZAcdRf/Y9cbIJvS6HllnUUfcvjCASELYTFFZT3OGlbqsEcvaGTtpqGChYmkOZ6azKGfd93q9GTk+o+0jV3JoXFWBkV7au/4xrWJkI3rpH4tZ1JE6Hy11lGmxDIRpiSOriW/VshateXTdcF06mtYEg8GAvLw8pYan3W5nSZ0ZcSOKIo4ePQq73Q6jkclbJmnv+se0ipFt6Kl/TE11xO2nYnm0PoBAWIDFmJmRjkCYlmUyG3iVZV0Wc12iiGV9OpoWE7lUkizCDEYi8DyP7t27swd3hukI+se0ipFt6KV/rLOoByE/4K2C2xcpon641o+eRZkpMB+ULGuLyRCxrJ2xLWvYaDUDu5C5ziLHcSgrK0NJSUmTQvQMRmuYzWbwPIuwyRgdSP+YVjGyDb30j3UW9eD9a4Ef3kR3shB70AsAcKjWlxGxJIQobpgoyzondswOb6di6RAz11mUMRgMLO6MwWhrdED9Y1rFaO8w81sPKn4ExBDOCa9QFh3IUNxOSIjMjDIb+VYDvHl7AQAgR8zsDEYGg9FGYfrHYLQ72lxncdGiRejZsyesVivGjRuHjRs3xrXda6+9Bo7jEi44nhQhDwBgMr8RDlCRPJQhsQwKIrrgKEZxO2ExcK0GeBsd1LLOgYdWO2AwGFkD07/EYPrHYGhDm+osLlu2DDfccAMWLFiALVu2YMSIEZg8eXKrwcV79uzBTTfdhBNOOCE9DQ1RYbRzAZxh2AAgUp803QRCAp40P4k3zHfB/OtKQJTiapoRS5OjIPLGn5kZ0QwGoylM/xKH6R+DoQ1tqrP42GOP4fLLL8fs2bMxePBgLF68GHa7HS+++GKz2wiCgJkzZ+Kuu+5C796909PQUCTv1vmGzwEAh+oyZ1l3446A5wj4tffThbYCwGiOub7VakEDsdI3vpo0tZLBYLQG07/EYfrHYGhDm+ksBoNBbN68GZMmTVKW8TyPSZMmYf369c1ud/fdd6OkpARz5syJ6ziBQAButzvqlXhjI2I5jv8Z3biKDFrWIhygSWlR8QP920y8DgDYTAbUIgcAQJhYMhhZAdO/5GD6x2BoQ5vpLFZWVkIQBJSWRrsPSktLUV5eHnObL7/8Ev/617/w/PPPx32cBx54ALm5ucqrW7duiTVUCCmuju9EOhPwLP5rHKrzQxTTXxYqGArBzgWiFzbjggFoegk3obMWwx4mlgxGNsD0LzmY/jEY2tBmOouJUl9fjz/96U94/vnnUVRUFPd2t912G+rq6pTX/v37EzuwygXzmTgCANCdP4JgWESlJ9DcVroR9nuaLmzBsraaeNRKYhlqqNKrWQwGQ0eY/lGY/jEY2tBm8iwWFRXBYDCgoqIianlFRYWSRV/Nr7/+ij179uCss85SlokizbdlNBqxY8cO9OnTp8l2FosFFosl+YZKwd0ieBwihQCALsZ6IAQcqvWjxGlNft/JNMffILWHA29xAgF3i2JpNvBwg4qlwCxrBiMrYPqXZHOY/jEYmtBmRhbNZjNGjx6N1atXK8tEUcTq1asxfvz4JusPHDgQ33//PbZu3aq8pk6dipNPPhlbt25N3L0SL5JlHTZYUUloUfpSA03wmom4HcFPj+2DFRj3F7qw27hm1+c4DvW8k27rrda9fYz2ybtbD+Li579GZUP6R5PaI0z/koPpHyMTtEf9azMjiwBwww03YNasWRgzZgzGjh2LJ554Ah6PB7NnzwYAXHLJJejSpQseeOABWK1WDB06NGr7vLw8AGiyXFOk4O4gZ0UVcQEACjmagiETucaEALWs/ZwNjpNvB0bPBlydW9zGyzsBAoje2jS0kNEeWbJuD7bur8Wan4/ggjE6dUw6GEz/EofpHyMTtEf9a1OdxRkzZuDo0aOYP38+ysvLMXLkSKxYsUIJ+t63b1/ma8JKbpgAZ8FRUMvaJdQAIDiYAbEUJbEM8FaA44DcLq1u4zU4gTBY6ghG0hxx+wEAFdJfRuow/Uscpn+MTNAe9a9NdRYBYN68eZg3b17Mz9auXdviti+99JL2DWqM5IbxI+KGMYkBOODPiFiSgGzp2+LexieLpb9Wn0Yx2jWEEByV3C8V7vbjhskGmP4lBtM/Rrppr/rXZmIW2wySWHphhg9WhA1UpAo5N8rr0m9lENmyNsQvln4jFXnezyxrRuLUeENKTd72ZFkz4oDpH6OD0171j3UWtUYWS0IrBISsNG1FEerg9ofS354gTR0R4uMXy5CJxhoZAkkk5GV0eI7URwSyor79WNaMOGD6x+jgtFf9Y51FrZFidhpEKpaiXRJLrg51vgyIZYha1kGDPe5NgmYqlsZgrR4tYrRz1K6XigyMJjEyCNM/Rgenveof6yxqjTQbsEEw0feOYgBAEeeG2xcCIemtYsAF5VQW8VvWYZMUaxRkljUjcY6oXC9HGwIQMlC5g5EhmP4xOjjtVf9YZ1FrJDdMvWRZG1x0pmIR6iASwBMU0tocXmqPYIzfsg5b8wAARtGvjBQwGPFyROV6EUSCqgxU7mBkCKZ/jA5Oe9U/1lnUGklcvIRWQTBJYlnCUys13a4YPkxjdhIRS87shEA4+sZXq0OrGO2ZI42Cuo+0oxmBjFZg+sfo4LRX/WOdRa0JUXHywQybyQCDswQA0MlIKwm40yyWhnDilrXFbESdVPKKpY/IQsIBYPEJwNtXZLolMTnSKKg7E7NgGRmC6R9Db5j+ZQTWWdQaybL2wwKXzajE7JTwtIpBui1rWSxFU/xiaTMZUEcksWSJabOPoz8D5d8BP72b6ZbERBZLA09HZyrq24dYMuKA6R9Db5j+ZQTWWdQalRsm12ZSxLIA1A2TbsvaKND2EKMj7m2sJgPqkEPfMDdM9uE5Sv+GvECaJwzEg5w6ol8JvYbaU2JaRisw/WPoDdO/jMA6i1oTjLhhXFYTkEPdMPliLQDA7Q+ntTlGybIm5kTEkmeWdTbjqZL+IdQlk0UQQhRxHNaFziptHMPDaMcw/WPoDdO/jMA6i1ojWdY+WOBSWdYO0gATwml3w5jFZMTSgFrZsmYxO9mHtzLyvzTbM1tw+8IIhkUAwLCuVCzLtRbLQH1WjigwwPSPoT9M/zKif6yzqDXSxesjFrisRsCaB/C0BHch6tLuhjFLbhg+QbFklnUWI7thgKwTS9kFM8G6ByceWAwLgtq6Yap+BR7qDbx3tXb7ZGgH0z+G3jD9y4j+sc6i1shiCTON2eF5VWLa9Je8MomSVZOwZS2LZa32jWKkhkdlWQezSyxlYbzZ8F/0/PEZnMxv1dYNc+hbQAgCe7/Sbp8M7WD6x9Abpn8Z0T/WWdSaxm4YIKqKQbrdMBZC28NZcuLehs0GzHI82euGkS3rrqgAAORxDajyBBXXTMrI12P9YeaKzkaY/jH0hulfRvSPdRa1Rg7wJhYa4A1EW9a+9AZ4W0UqlgZr/GJpNfGoJU76xletR7MYLXFoK1D9W/OfR8XsZFeFiSP1AfAQkS/QIHQX75eWa2Rde6XrMeQF/HXa7JOhHUz/GKnC9K95Mqh/rLOoNVGWNY3VkWcEFsKd3pgdIQQz6PH4BCxrq8mAKrjoG3V8CEN/vNXAC5OAf09tfp0oy9qjf5sSoMLtRxHqYAAt61ZiCUvLNYrbUT+868u12SdDO5j+MVKB6V/LZFD/WGdRaxSxNMe2rNMZsxOM3EjGhCxrAyqJLJZVLa/M0JbavYAYAur2N285erLLsg4JIm5+fRuWbtiHI/UBdOYi10yxOQhAw/QRXrVYHtJmnwztYPrHSAWmfy2TQf0zpvVo7R1CFEvHSyywmQ10eZQbJo1iKcVzhIgBJrM17s2sJh7VRGVZEwJwnB4tZEgEwyLe2nIAp5oOoVheWHcAsOZGrxgOAMH6yPssEMtv99Xi9c0H8PrmA3BajTheJZYFRiqWFVqJJRtZzF6Y/jGShOlfnLCRxXaCEAQIDWT1wwKzUTq9KjdMWgO8JcvaCwssJkPcm0W5YcRQRmPDdpTXY/Fnv2oXIJylfPJTBW5963u8v/77yMK6A01XVFvVQNToSaZQdwDq/eEoyzrXQMWyXDM3jGrCgZuNLGYVTP80h+lfI5j+qQ7MRhbbLqqZWT6YYTFKApXXAwAwlN+NQCCAsCDCaEhDPz3YAADwwAqLMf7j2UwGBGBGA7Eih/MD3irAlqdTI1vmvo+24/OdR1HitODcUV0z0oZ0sL+GXjtBtypGqm5/0xUbx1BlgWXtCUZPWihTiWUOR9tX6w1qczAvG1nMWpj+aQ7Tv0Yw/Yv8z0YW2zBSzqcQjAjDGBGorseCOIpRwDVgPP9T+kpeqWYmmhMQS6tkhVeRzAd576ui32HLvvadwqKqgVqepoBKDGpjiKW3kWWdBakjGgL0ej6xfzFOH1yK4a6ItW+TUpfI66RMlBvmsDb7ZGgD0z/NYfrXCKZ/kf/TrH+ss6glkpXjhxkAIgJlMIIbRGd3ncl/nb64HUm8qWWdiBuGtrtamRFY2cLa+kEIweE6GuuxdX9tRtqQLqoaqOVpC6keCjHdMI0C7rNALD2SEBY5zHjukjEYWxCx9i1SuTWPFmIphKNdgqyzmF0w/dMUpn9M/6LIsP6xzqKWqEpdAYh2fQw5BwAwxbAJ9Z70xFmIAeqG8cKamGUtCWtlhi3rGm8IASlW5+fD9fCHhIy0Ix1UeqhYFnANkYUxxTL73DB+rwcvmx7A6XXL6YK6g8pncrk1T1CD365xnV436yxmFUz/NIXpH9O/6APVRr9Ps/6xzqKWSGLpBRXLKIHqcRyquTzkcR5wuz9PS3MEP5015iGJiSXPczAbeZUbJjOW9aHaiBCERYIfD7kz0o50ILthCjjVd4wlllnohsmv/hYnGr7HyeVLgJAfaIjE0pgEDS3rxtU0GioAsf0+QNscTP80hekf078oMqx/rLOoJYplLblh1EHcvAEbbScAAHJ/+yAtzQn7qZXmgyWhAG+ABnkrbpjGN2iaKK+LTjewrR27YmQ3TCFUYll/iLoe1MiWtS2f/s2C2qi8JGIW0Qvs+kSZEQsAxjAdRdJELOXg7txuAMcDRGBJk7MJpn+awvSP6V8UGdY/1lnUElVCWgBN0jVsc50CACg9tCotFoEQiMwGNPKJ5QmzmviMB3gfrot2MWw7UJuRdugNIQRVHmpZ53OqHGJEbBqXIsfs5HWnf7PAsuYDtZE33y2jf402+lnIA4Bo44aRg7sdRYCDpmNhcYtZBNM/TWH6x/QvigzrH+ssaolk5XilmB1zo/QQR/OHQyAczOH6tLg25JidAGcFl2BSWavJkHE3jBzc3avIAaD9WtZufxghgYCHiDxI8VwW6dzX7aeJaA9tpcmB5VEORSzTF7NztD6Ag7VNj2cOqtwjO1fSv0X9AAAcCGwIaGtZ2woAVxn9n8UtZg9M/zSF6R+Y/qnJsP6xzqKWyG4YKWbHZIgWqBybLeLaaKjQvTligN54Ad6W8La2qPqomRXLyUM6AQD2VHm1y1cVg817q/HUp78gJKQ3Aa4cr5OHBvAcoQtLh9K/dQcQXHEn8NxECFteiYxy5KbXsvaHBExbtA5Tnvi8Sck2S1A1Q0+Qfp+ifgDo9Z8DP7xBAaJIUmuEbFnbCwCnJJZsZDF7YPqnKUz/wPRPTYb1j3UWtUSVOsJi5JtYs7k2EyqJVL6o4Yj+7ZEta96e8KYWkyFS8ipDMTuyG2ZQmTNiXR/Qr5rCfR9uxyMf78Sbm2MEVutIZYM8E5DG69RzTiC/JwCg6sAO+L5ZCgCo++LZjLlhVv5YjoO1PtT7w9h9NHo2qyUcI/A+tytgpvV4HVJiWm+qsznlAG8b6yxmJUz/NIXpH9O/KDKsf6yzqCVKXdTYs+9cNhOOKmKpv2VNpKS0IT7+uqgyViOvSh1RCYjpLzclW9adXFaM6ErP29Z9tbodr8ZLLca3vj3YypraIlvWRVK8Tg1cVGwAuDcsRS7o8oLaHyJ1UdPshlm6YZ/yf+NYKns4xgPM1RWwULF0cvR3TNkV42Uji1kN0z9NYfrH9C+KDOsf6yxqiSrAO9bsO5fViKPIo2/SIJacZHWFDIlb1jazajYgEZrmeNIZIorKTdk5z4YBnWhb9lTpl6NNzrK/cXc19lenL3BazjE2OI+KdaWYAyKJZS8uRv1P3gQ4S+n/aZgN+OvRBmzYHakccKg2epZmjkgta1/xiMhCV2fFsi400++VchUD2Q1jy2cxi9kI0z/NYPrH9K8JGdY/1lnUEsUNY4lZMSDXZsJRkkffpGOGnVQbNWRIPGbHajQgBCOCRiddkOa4ndDL5+Fj/joM535FicuCYieNg6ry6Bezo7b83tuWviLtsmU90En/HhWd2CcURK3zpnBC5I2jCDBRt1Q63DCvbdwX9b6xZe0k9DoL9jsjsjC3i2JZFxrpb6aZZW0rAJw0jovVh84imP5pBtM/pn9NyLD+sc6ilgRlN0zsWqTpdsPw0o0kGBO3rOWSVz6TlM8qnekjAg0w7/kUPfkKvG65G5Yf30BhDk3HIQuL1ogigVeV3uCtLQdASIoByXEi5xjrYqa/VzVxYtUhi/J5yOjAXaFL4OWk39FeBJikB6DOYhkIC3hDimE6rk8hAOCQKv9bSBAVNxHf91TaNnMOkN9LsazzjfQ38wQ0itmx5wPOzvT/+vQ91BitwPRPG5j+Mf2LRYb1j3UWtUTlhmmcNgJoZFmnIcCbl2KIBGMSswHNdGTAY8qjC9IZ5O2OxMxYEALenouu3u0AIsKiNergY7OBx69HPfj+oH7B5Goa5xirhhNv/BIR6qrOJ8MNB74wHw8AEO2FmPXqj/TDsF/XeKqfD9ejxhtCvt2Ei8fROKHDqvQRHn8QeaCWtbWgM/DnlcCcjwFbniKWeVpZ1pJYvrjFjQ1V1siyev07How4YPqnDUz/mP7FIsP6xzqLWiKJkw8WWEzNWNagljVJh2Udphe1YHQkvK3sRvIYMmBZ1+0HAOwUu2CHlcaBFNVLYukJ6GLxeqUbmeeA04bQeJhV29MwYxOR2YC5IhXnauLC9qqwEmDv7UPdGy9hKlA6DAd7noONB1SukLB+Qd7yaENhjgVd86llr47Z8dTXwCCluzA5CoGivkDpEPqh5IbJ4yXLOqiNG2bJt7W45cO9QJcxdPn3r6e2X4Y2MP3TBqZ/TP9iNiaz+sc6i1oiW9bE0oJlnT43jDFMh+hFU+JiaZWqL9Qb8ugCOWVBOqilYrmflKDe0QMAkBOuBQCEBAK3X4MEp42Qg48dFiN6FdLzVadjTjM1smvJIchiSeOk7gvNhPeYOcAAKpbf+0uAK77Er53OgF+qkgFA1yDvQJiKpcXIo3MutWaP1PsRlnKxBeroiIsPFsDUaNapZFk7DRq4YUI+5aFQS5zYU+VF3YDz6X43vYJv9lSnnseMkRpM/7SB6R8Apn9RZIH+sc6ilqhnA8awrB1mA+oMNHCX89fRzPR6QQiMgiyWScwGlMSyjpfEPa2WNY0ROUQKwTtonIgpUIMcixGAPnE78o3sMBsVF5RXixJNcSAHrVtD1M0gz8LcnDcZ9rMfQ76TindDIIxgWERVQxAEvFKDV8+4nUCYiqLFyKOoegumGDdDJEBFPf0NAvX0IermnE03NtN2a5I6QrKqBRhQD+pW/Nx8AmAww1HzM+Y/+1/8d9O+lvbA0Bumf9rA9A8A078oskD/WGdRS4IRN0wsy5rjOBjt+QgSaaagnnE7QhA8kS7OJMRSDvCu5TInlgdJEUwuqQampzIS5K3DjEDZReCwGGCXxTLVJKpxEBJE1Er5zcx+KgiyZT2mB3WBuWwmyPmN63whVEoPC7kGr565xvzSObAYePCvXYRFxsdRiDolbifUQC3rBt7VdGML/R45klimlDpCShtRzzshV0b44oAAb8/TAADnGb7AaYNKk98/I3WY/mkD0z8ATP+iyAL9Y51FLVEs69izAQGg0GlR5RrTUSyDkXxcJAU3TA0nVzFIoxtGsayLYM+TxNJbiUKHPCNQB7GUbuQci1ERS18aLOtqSfgNPMD5osVydM986TMOLqsJAFDrDSoPC7msmhwrpgeyZZ1n9AP+WhggooSrVWYEih7aZq8hlmUtVTCAVMEglZgdKbi7WsxRFq3/rQprbJMAAOeb16PE0TRdCyONMP3TBqZ/AJj+RZEF+sc6i1qixOyYY+YZA4AChyU9cTuSWAaIESazuZWVm5JnpzfngaB0cabRsiZSgPdBUghngZRPyluFwhwLTAij//qbgG3LND2mbPXZzUbYzdTdk9LNHSeyldzVLoIT6P/VkC3rSK6xfOn3qPGGUCm5QHxEFkv9LGtZLAulmYoAkMt5FMtalNwjPmNu042lAG8rkS3rFB4+0nGqRDpKZOA57K/24f4dXVBJXMgVa4Fdq5PfPyN1mP5pAtM/pn9NyAL9Y51FLVHNBmzOsi5ymFXpI3QUS1WdVrMhcYujSx6Ni/jNKwXtpisprSgAbpo76hApgqtQGlr3VKEox4xx/Hb0Pvg+8PnDLe5m15GGhGJE5PgcR5ota3mUoJeN/l7EaEPYYEOpy4J+JRErMtdOH3i13iCOptENE5DcMHLdVgDIRYNSikweDQiYYoilZFnbCG1fSjE70nFqiRNFORYMl8qfHawPY7l4CoKDz49UNWBkBqZ/qcP0j+lfLLJA/4y67r2jIV203hbEssBhRiVJQxxMOCKWsYLNW6NMEsu99TxgQtoKtqOhApwYQpjwqDMWwCLH7HirUGg3I8xJ7iCpOkNj9ld7cc8HP+Hjnypw8oBiLJk9Nq7DegKRmJ10BnjLOca623xAPcA5ivDWpRPgshnB85yynmxZ13pDisB6IT3Igvq7YfKJSiw5Dw5JljUvlUELmfOabiyJpUWk105KIxWSZV1DctCryI4xPQvwrVQnd0vfeTBPPzb5fTO0gelf6jD9Y/oXiyzQPzayqBWEKILiJ5aYtVEBmq9JzjWmr2Xtl9oSO0Fua5Q6LeA5oEEwSfvz0u+oN1K8TjkKkOuwgXMU0eVCAKW2MEohZbGPkS5h/a9VmPTYZ/j4J3pe91bFL/DKbECL2g2TPsu6i1kSPHsBhnXNRY/C6DirfMmyrvEGFdeNX5oNSPScDShZ1nlqsYRHsayNgVoAQNiS33RjyQ1jlmalphbgTX/3GuSgZ6ED43sXKh+dNaJz8vtlaAPTP21g+sf0LxZZoH9trrO4aNEi9OzZE1arFePGjcPGjRubXff555/HCSecgPz8fOTn52PSpEktrp8SqqHwlizrwnS5YVSWdXNtaQmjgUcnlzUy1A/om+pCRonXKUKe3UzTD0gVGMpMXnTipPqYwYYm4v35L0cRCIsoyqGxLL4EZvPJswHVAd7pidmhYllqlMWyMOZ6uTb60Kr2BpWgcDnA2++NPcqgBbJlnUsi1RzyuAalPqo5WAsAEK15TTeWLGuTJJYp5RmT3IDVxIWeRQ6M6ZmPQocZRTkWTOpAs6CZ/sUJ0z+mfxrA9C9Cm+osLlu2DDfccAMWLFiALVu2YMSIEZg8eTKOHIk9q27t2rW46KKLsGbNGqxfvx7dunXD6aefjoMHD8ZcPyV4AzD1SXzQ5Trq+mjGmi3MMaen5JVsWcPcrJXfGmV5tujkp+lwxahyjBU4JKteEpASvh6lXC1dRgRAiJ4VKKc5GNU9D0BiYhkJ8DYoOdbSYVlXS26YAoNUFcAaI/YFEct6X5UXYSn5ashA3TB1bv3Kcsli6RQix8iFB5UNQQTCAswhabk9lmVNA9WNYfogSClmR3JZVsOJXkUO2M1GfHjNCfjwmuPhsHSMaBqmfwnA9I/pnwYw/YvQpjqLjz32GC6//HLMnj0bgwcPxuLFi2G32/Hiiy/GXP/VV1/FlVdeiZEjR2LgwIF44YUXIIoiVq9uftZQIBCA2+2OesWF0QKMugRf5J8HETwspuZmA5rTMxtQHbOTpFh2zrMhDCNETvouYX/LG2iBSixlgYCUmLaAq0epbFkDTWJVZLGUt0skQNsbI3VEICxC0DkrvizIOZAeRJYY+boA5EsPjl1HqBWdazPBaKGuGk99fcxttECuYJAj1CrLCnh63svr/LCFqVjy9oIm28pJaQ1hLwCSWrkrqTZvJclFT8lF1SnXilKXtaWt2hVM/xKA6R/TPw1g+hehzXQWg8EgNm/ejEmTJinLeJ7HpEmTsH79+rj24fV6EQqFUFAQ44eVeOCBB5Cbm6u8unXrllg7pTJAzcXJFOVE8oyRBh0DvCXLOkBMzaaxaA25vFGI0z9FgYIqx1iBlFdMtqxzSR1KuZrIuo0sfVkc86XtAmEx7hJIcloDu9kYZaklYp0ngz9Erxe7KAm/NbZYym6YPVV0vcIcM0xWKhper45iKbXPEY6c9xITvbYO1frhEOixDY4Y7iPJDcMRETYEUnLDiA2yG8aJnkWJJ1lu6zD9SxCmf0z/NIDpX4Q201msrKyEIAgoLY32z5eWlqK8vDyufdxyyy3o3LlzlOA25rbbbkNdXZ3y2r9/f0LtlC2ReGYDciEPENAp3iLFmB2AWtYAEEirWEZyjOXJlrWdBnk7glUogsrl0CjIWxYeeeYcAPjD8d2g6tmAFiOvVAzQO25HHg2wymJpadkNExKo+BflWGC2UTEK6CiW8vmzS7VpAaCAp+f9YJUbDkKvX6MciK/G7IBcbSAHfniCYZBkJgkQAnhpx4LLKVYC8DsSTP8ShOkfXcb0L7X2Mf1T6DCq++CDD+K1117D2rVrYbU2P3RrsVhgsViSPk5QVUsyFnazAaLJDg+xwMEFqCvGkhNz3ZRQxewkK5ZlkmXtI2ZarTMdYllLxfIQKcJJ9uiYHf7odoBT3WyN0kf4GrlhAGptx3NzeVUB3hzHwW4ywBMUdM81JrfZIkjfpRnLOk/1AACAohwzbCKNiQn6dEwdIT2AbKGIZZ3H0bYeKD+sLDM7Y4xWcRy1roP1sHN+EEK/b8JiF2wAL8VnuQpYLsVkYPqXOEz/mP4x/YvQZkYWi4qKYDAYUFERHedSUVGBTp06tbjtI488ggcffBAff/wxhg8frmczlYDY5gSK4zgUOiz6B3nLljVJLWYHADyiKWqfuhGoB6S8VYdIoeJOkWN2UPFD9Pqhxpa1ZAVaDMr590u/R2uoKxgAgE36m9IMtjiQxdgSlsSyuZgd1QMAoJZ1joOKpRDQP8+YJRgRS9maPlJBR7TcxI4cWzMdEKkjkFJ9VCm420fM6FwSe7Zke4fpX4Iw/aPLmP6lBNO/CG2ms2g2mzF69Oio4Gw5WHv8+PHNbvfQQw/hnnvuwYoVKzBmzBjd29maWAI03qJSzjWmV2JaDSxrWSwbZLHU27Kuo7M06zkHGmBXxexIQ/yVO6PXb+KGkVwaxsiMvngtYyXQWorXUaoYhNLjhjGHE7WsLXC66DVEYuRc04pAWIANfhiFSHC/OVQPDiKqj1KxrCUO5FibsZalIO8iE7WMk3r4eGgi4iq4UOxMftSrLcP0L0GY/gFg+pcqTP8itJnOIgDccMMNeP755/Hvf/8b27dvxxVXXAGPx4PZs2cDAC655BLcdtttyvp///vfceedd+LFF19Ez549UV5ejvLycjQ06JeXKeKGaT6ousBhRo1ULF2e5aQ5UbMBkwvwzrebYDXxkfQReoulNDuykuRLx48O8IbYSLgaFZCXXRo2VfoHf5wB2oplbaHb2dNUxUBun1GxrGMUpAcVcaOqokFhjhm5LiqsvOBTrjutCYTFSF1UngoiBwInvAhJQdc1cMJhaeYak4K8CxWxTOLhI90jVcTVIeMVZZj+JQDTPwBM/1KF6V+ENqW8M2bMwNGjRzF//nyUl5dj5MiRWLFihRL0vW/fPvB8pP/7zDPPIBgM4vzzz4/az4IFC7Bw4UJd2hiMx7J2WFAli6VkNWiOYlmbkrasOY5D51wb/HVpEkspS32VSK2xiBsmRvAw0GyAt9UUKVkVz2w+QohyE8uWdbpKXsntM4YkQWrGDcNxHPLsJiWJbVGOBTkcvYZsCOBwna9J1QMtCIREFEBKn+IoAfx1QMiDXo4g8qRYoVqSg37N5fqSxD/PkIJYSqNPVcTVvCh3AJj+JQDTP7pLpn8pwfQvQpvqLALAvHnzMG/evJifrV27Nur9nj179G9QI5TZgC2UmCrMMaMa0k2ht2WdZLkrmc55KrHUO2ZHLmlEqDVW0Niybkzj1BGyG8bEw5qAGyYQFiFnmJDTRjgkCy5dAd58UBLLZtwwAJBnN6vE0gwuRMXRhiBqvSH00CGcxR8WUMbJYllIg7ZDHgzJF2Hz0zbXIUcZyWiCZFnnGyWxTGZ2pVy9AC7ld+moMP2LE6Z/dBnTv5Rg+hehTbmh2wJynrGWitcXOswqy1onsZTzjMHcYltaoyzXqpRVSpdlXUNyYDHyinWrxOxIBIm0vNFsQNmlYTMZYJO+czyWtTro2C7d9OmwrAkh8IdEGCCAl4W/mdQRAJBni8TtFOVYABPNt2VFQLd8aIGQiEJIQm4vAmzURTYgV0CZlCC4ls8Hx3GxdyDF7ORKFRqSi9mRE9J27JHFtgDTvxRg+sf0LxZZon+ss6gxihumBWu2wGFGNdHbslYFeKdqWZN0uWGkmw85keBuALDlQc5XBQD7iJRrrrkAb5UbJp6YHY+q1BUvxcWkoz6qPBkgB6rz2oplLUPFkgbg27mAbiMAgbCAAsWyLgakGqi9ckLowdGZrBXGFgrZS7MBXTwt65VKzE41cXbomMW2ANO/FGD6x/QvFlmif6yzqDGBVvKMAfRCV9wwOsXskJAqdUQKlnXnPCt86QrwlizrWuKITpXAGwBVOaXdRMo1pXLDhAVRSdhKLev43TAeVfUCmXQEeMttc3LS9zDaAIOp2fXlZLtWE0/bZ5bdMAHd2hkV4O0okh5cQDdrAN05GpBfZW5BLM1y6ggqlsmljlC5YdjIYlbD9C8FmP4x/YtFlugf6yxqTLyzAat0ng0oi6UPZlgMyV9gnfNskdmAetdG9dUCoDEgUZY1EBW386sslqraqOp8YjazIRKzE49lrSSkjZwnm4kKp65iKbWtkJfOawtWNRBJH1GUY6FuD8mytiKonxsmrArwthcqYllq8qK7ZFnXWrs2vwMpYF1+ICTnhqEB3pUdfDZ0W4DpXwow/WtxfaZ/mdU/1lnUmLhmA+ZE3DDEU0nL+WiMnHsqlTxjANDJZVWljtAvnxWAqJidxnm15LgdYs5RyoWpk7GqLWiLkY9Y1gm4YQrMYeAozWWm5BnT0Q0jt61AqjXa3ExAGdkNU5gjxVBJMTsWLgy/X/sHWUgQIYhE5YYpUtwwdvdvsHFBCISD195CVQEpxsdJqHWelFvLS0efqgmb4JLtMP1LAaZ/La7P9I9NcGk3CCJBWJpW1mrqCMkNw4khIODWvC0knHpSWgCwW4yqmB2dLWtvMzE7QMQN4+oMv1SrNeSL1AT1q2YCchynWNZyOomWkK2927yPAIuOBQ5uSUuAtyzwBUZZLGPnGJORy491yZOqBZgiBeWDfu2rGMguRcUNowrwxsEtAGhZMqvF1vxOpPVzRLqPhN0whNAOBTKfOoLRMkz/UoTpX4vrM/1jqXPaDerEoC3F7NjMBhjMtkh9VE8lYG1+FlhSSG6YEGeBgW9mplYcWI28MhuQhLxIfk9xIFnWdSSnSXknOdcY5+wEvtoBiEDYH5kNqA7uBpBUgHcPYQ9d8MsncJinAwC8Ork3gEiakQKDHwijVTfM74eWoaohiEmDpQB3owUiOPAgCOkhlrLlD5Vl3ZBH/6/6BQCwl5QoudliYqdi6RDoPhIO8A7UgxNovE8VXEpqD0b2wfQvRZj+tbg+07/M6h8bWdQQtVi2Zs1GzwjUIchbsoKJsZmalXFiMxsUN4wY1DHAm5CoAO+mlrWUPsLZGQYrDRoWVW4Y2YKW3S+J5BmTY3ZyhVq6YO+XSmyInnnGfEHa5jw+PjeMzWzA5Sf2Rq8iKfksxyHM099X1KE+aouWtUS5oRMmDihuodF0RMQargMAeBI9n1JMm5dYEOQsSdf5ZegP078UYPrH9C8WWaR/THk1RLaUOA5RpYliUegwowr65RrjwtqIpdVogI/IYqljzE7IC0gWVA2cTWN2hp4H9DgeGHUJTJJYElUMkU+VY0z9t7mYHUIIPv25AodqffAEwrDBDwuRRGv/JtiNdLukUh3Eidy2XE56CLViWcciZKAuEEEnsbQgiBy5fY5CJWZH5rxTT8DZI7s0vxPJfWYJSWKZ6PlUu2DMxubzmTEyDtO/FGD6x/QvFlmkf8ynoyHqtBGt/agum0nXXGOcIF3gKYolz3MQeOqGEfVMHSFZ1SEY4YWlqWVdOhiY/SEAwGLfBVQDnGo2oCw8FkUsW05Ku3lvDf780jcY27MAo3rkR6xHAAj7UNrwc4vba4G8bxffekLa5ggb7ECoGiSgfb3fQFhAgZyQljdSoZRmA8pwBb1a3olkiRsFPywIJi+WcCp1axnZCdO/FGD6x/QvFlmkf2xkUUPk6gXxJIHNtZlUucY0FktRBC/Q8kK8qYXg2zgRZMHV07KWxNKNHABc05gdFRY7HZHgw5H2RKoX0HOvxOw0Hvbf8yXw5BjU/bgKALBlXw2qPQEUoi5qtZLqbwDoG+Att9mJ5C1rwSgFeQe1F0t/SESB4oKRSl01csOgNbG0uKjQAshDQ9JumCqSy2ZCZzlM/1KA6R/Tv1hkkf6xzqKGBEJy2ojWLYBcmwnVSq4xjWN2VPnAeHPqYkkM1ib71RwlbQSNR8lvbFmrsOdQC9QoRCz9xgHezeYZ2/4+UPULivZ+AAAIiwSb9tSgkIuekZl/dBPdPi1iKVvWSYiliZ4v9SiDVgRCAhyykMtta+SGQX4rYqkS2HyuIfHzKeUYqyaZt6wZLcP0LwWY/jH9i0UW6R/rLGqIUhc1jiDUdIklp0ovkCxE2genpxtGShtRTWg8TkELlnWOk964ZtGv5GjzxxuzIx3H7DmsLNpd6UERJ1nWTpozy1HxDXiIupa7koXDAUnokrCsiSyWIX1iduxS5QE5AS6dtSq5GO2F8bVZEcv6xN1aUoWPKrAci9kO078UYPrH9C8WWaR/rLOoIcE4Sl3J5NpMSq4xzd0wctoIYoDZ0rzoxI3khuEE/S3rOpIDq4lX3CixyHFSy5qHCITpzSwLj9UcnTqiiSUn1V91BI5ELS6S0yP0mghYXDCEGjCY25OWCgZ2MXnLmkjlpIxhfcTSCurOU3Ka8YaIQOb3jG9H0ozAXHiStqxpjjHWWcxmmP6lANM/pn+xyCL9Y51FDZFnA8aTBDbastZYLFUJaW0p1EWVIWZ6o/Bh/QO8m9RFjUGuSxUILc0IlMtdWY3RlnWTPGOSZZ0fjhZLxQ3j7AR0/x0AYBz/MwJhmsVfD5QZjEQSulaS0sZEKlRvCGsfTxUIC7BBsqzNqhEaOW6nNReMjDQjULasxUTOp3RvVBOXUlWCkZ0w/UsBpn9M/2KRRfrHOosakqhlrcwG9GjshpEsaz9MSuxKKnCSZW0Q/LqU5gIQEUvktGpBFThtCBAptUSwAfBWw1X1HQDAZqbnvtkKBpLLywlvJB4FQKHshnEUA6VDAQBdOWrV6eWKkWO8rELybhheEkuTDpa1P6R2w6jEUo7baS24W0Z2w4AGoQfCYktrRyOXuoIz424YRssw/UsBpn9M/2KRRfrHOosaEk9dVJkoN4xOlnUAZsXCTAXeorpR9AryltwjNZIbpiUKHWZ4pKoKYsADfHQTLv7uUpzEb1Us62YDvCVRBoBuhhr0L6ViU6hk6S9Wbu48jgqQXkHe8n4tgjSTL4nUEbxkjZtE7Uc9AmEBVjSK2QHo6AMAFPWPb0fK+aTfM6GHj58+xOqII+MB3oyWYfqXAkz/mP7FIov0j3UWNSSSZ6z1H9WldsOEvNqmZZAta2LWxLLm1VaVXkHevloAQB1yWhX4PLsZXlBrv6HeDRzeBgA4x/ClEqujxOyoxVKIrkM7KKceo7rTG7lIXSxeyqVVwFOx1CtuxxcSwEGEWZB++yQsa7mag0XQwQ0TEmFrHLMDAKcuAE6dDww+O74dSW6YwmTOp5/+LvWwZ9yyZrQM078UYPrH9C8WWaR/rLOoIYla1g2wIUikC0BL61oVs6OFWJrNZoSItB/dxFKO2clptc1mI48ARy3r+vo6wH0IAHAK/y3shuhZgUF1zI3KqgaAATY3jumeB0AVs+MoVtwMeRwVID3FMgd+cJDal0SAt9FGt7EQX2KxMHEQPRtQJZalg4ETbgSMlvh2ZJPFklrW8dSrBUBdftLDrZ7YMh7gzWgZpn8pwPSP6V9jskz/WGdRQwKJJKW1mwBw+swIVGJ2tHHDqOuj6ueGkWN2HHGNTIR46hbwVx9QgrydnA993DQ/mPp7KzenFNwt09NUi2O654ODqCoWH8MNE9InZscfEiI5xgxmwJR4tQmTnVrWOfDDH9ZW1JsN8E4UVZ4xIIGHT8gLiPTcu+GAg7mhsxqmfynA9I/pX2OyTP9YZ1FDAkrJpdZPq9NihIHn9Mk1JlvWxNRq/Es8WIwqsQxpP9wPQBGyWuKMq81hqSYoObozannfSlqZQB1kr7hifNFi2YWrRt/iHJzawwQjJwUd2wsVN4wLCd7cCeIPCXByyaeNAACTZFnbOb/m7fSH1KkjUkhuLLlh5JiduHONSS4YATy8sMDO3NBZDdO/FGD6l9Q+mP6lD9ZZ1JBEyl1xHAeX1YgqorNlrcF0e5vZAD+RxVIHy5qQqNQR8biO5DJPptrfaLNAZwd2PbIW2P4B+H9NwlVmWqVACdBuZFkXkkrwPIcXzutJF1jzAKNZsQRzSAMAoqsbRrGsk4jXASKzAXPg1zwQPRAWYrthEkVyw7iIJJbxtlNywfg4OwAODpY6J6th+pckTP+Y/sVsQHbpH+ssakgiMTuAXB9Vh1xjKrHUImbHajTAp7hhdIjZCfkAgd6UtXEEeANQbl5H/W4AwI/mYThKXDCH3MCymcDBbzCPfxNWBFRuGDp6EZC+S26QpoaQE58ip4T+lWJ2TAjDhoBuqSN8QQE5nFxOKokcYwBgptvZ4U+8OkArBMJixA2TkljSh4+T1AMgCVjWdCZgPUerNNhZzGJWw/QvSZj+Mf2LRZbpH+ssakgiswGBxrnGtAzwpjdgQCOxtJl5lRtGB7GU3CMCZ4AH1vhcRxZ6A+V69wIAyrlirBSOpZ9xPGDOgQ0BTOS3NXHD7CTdAABWXzldLoulo5j+NTtUxd89OrphRLga1x5NFDM9Dw7Op3k7A6EYFQySQXLDGCHAiQTaKblhGiA9GNnIYlbD9C9JmP4x/YtFlukf6yxqSKKWtctmQq1UCxT+Wu0aEpJjdrQJ8LaaDPBJeb30EUvqgvEaaN3NeATeIImliVDLrwIFeCQ8HQeGXgXMWQWMvhQAMMWwqYkb5gehO91H0A0E6iMPKkcR/asq/p7LJVGiKU6iYnasiecYA6BUMHDoMAIQ7YZJIWbHZAOMdPtcriF+yzpALWu3KIklG1nMapj+JQnTP6Z/MRuQXfrHOosakowbphb0pm8cT5ISYW1nA1pNhkjFAF3F0qkcrzWMUn4tmUNCAWrhROW4vwFdRwODpgIATuW3IBCQ2ixZ1gdIMeolaw3uQ00ta0CVPqIhPTE7SVvW9DzYuQD8gaBGLaNEuWFSmQ0IRFUx8MUr6pIbppZQoc10njFGyzD9SxKmf0z/YpFl+sc6ixoi10aNp9wVIImlPBuwUQ6slJAsax/MmswGjLKs9YjZkb67h6fnIp7zZ7ZFx7gcEvMAIPJ9ux6Lar4ALs4H+4F1dJk3UlKr2iAJo/tgbLGULWud3DCE0NiViGWdmlgCQNDXoEHLIvhDgjZuGCC6PmowznJXkhumVqRimekKBoyWYfqXJEz/mP7FbEB26R/rLGpIIrVRgUaWtZTBXxNky1qjCgZWo84xO9KogpujghHPDEaLPVos94Xz6Lby9+V5bLFPAAAU7V9Jl6lKatWbS+myOrVYFkV2KKeP4DzxW4IJEAiLIARwphqzY7RAkG7jkNfdysqJEZ2UNgU3DBDJ3YYGeOPN2yZZ1m4ix+ywkcVshulfkjD9Y/oXiyzTP9ZZ1BAldURClrVkGelgWWs1G9BmNsBHdBRLKV6pQZr1ZY0jQN7miBbLvSEa86J2O23NmQgAKDu8GhAFRZRr4ETALolla24Y6OOGCYToteKSEt8mHbPDcQjwVEzC/notmqYQPRvQkdrOVLnG/AmmjnDDDo6DJqNEDP1g+pckTP+Y/sVsQHbpH1NfDZFvgMQsa1ksdYrZ0WAGldVk0NeyDtOhfp8UFxSPwNsdEUuUmOyoJfRmtqi23e8aCQ+xwBKqBap2KakjaogTQUcZXcl9QCWWJZEDqAK89RBLOci5UErUCnth0vsKStUcRL+2bphASFDVRtXGss5P5OEj10UltC4qx3GptYGhK0z/koTpH9O/WGSZ/rHOooakZFmHvJolfCXSfgIwaVPuymSAX4nZ0SEprRgCAARE2tZ4LCiLXRWrYisFQG8k9fe1mM34mdCZfzj8nTJ6UUNyEJLFcvfnSm3V6JidPACyZa29GyYilpI1rHYBJUjQQB8UosaWdTgUgoWjv03KMTu2iGWdcJ4x2DNe6orROkz/koTpH9O/WGSZ/rHOoobIecbMhvjzjNXDpsRcaJU+QgyqY3a0CPDm4Yc8G1CHclcCtd78ili2fv44VWCz20xFzsBzMBki1pfNZMBPYg/6Zu86gNCbtBY5CBYMpMtr9kQeAFExOzpb1tI+C2SxlNwUyRCWqjmQoLZiGTWKkupsQJUbJtEKBm7JsmZkN0z/koTpH9O/WGSZ/mW+Be2IQIIB3i6bCQQ86uFAHuqp5efslHI7xJAPBkgxO3EmyG0Jq8kAH5HzjOlgWQvUevMlIJZyMlYAOALqwrAa+aiheqvZgO1EEsvdnwMA/LAgADNCpSOBS94D9n0NHPkR6HyMYk3Tjen/ufBEKiBoiFz0Pg9SUHYKbhhZLBHU1g3DSe48Ag6c0ZraztSpIxKsjVoPe8ZnAjJah+lfkjD9Y/oXiyzTP9ZZ1JBk8owBQA1xII+r1yzIm0gWkWCwgOdTj3NQx+yQkBeaR040sazjOH8qt8B+eSZgo/gkm8mADbJlXf0rAMDN0cDwHJsJ6D2RvmIhCWcu50FIIPF8i4TwBwVYEYikZkhBLEU5+Dro0aBlEQyCD+AAYrKlHi+juGHqE4jZicwGtGeBZc1oGaZ/ScL0j+lfLLJM/5gbWkPkPGNxi6WdiqUSt6NVYlpJLEVDikG5EjZTpDaq7OLRFEksfSI9b/FZ1hGx/MVPg70blxmzmQzYQbpCVMl7rVSL1mk1tbx/VZ4xORZLS3whAQWQ3CYGc1S+sESRxZLXWCx52bI2anAdSXnUnPDFP1IRiFjWmS51xWgdpn9JwvSP6V8sskz/WGdRQxLNM+a0GGHgOdRonD6Ck2NQUh06l7CaDPATWSz1iNmhbhivkIAbRpXKYHsDPX9NLGuzAT5YccTUVVlWJZ3rnNZKJ6kqGIR0Est8JV6nkJbYShIiCa0hrN1vI4gEJlG6jlIN7gaU2YQWLhSfZS2KtBQZ6GxAOyv1l/Uw/UsSpn9M/xqThfrHOosakqgbhuM4uKxGVfoIjXKNSWLJpTrdX8LAcwjxNGaH6JE6QrKsA7JlHc/5U8Xs7BeoFdx45qMsuntNfZRllQLdzmVt5eaTLGsXvAjHm0Q1AXxBQRXcnbwLBoBilRvD2lnWQVVCWi7V4G5AqY1qQyC+mJ2AGwB1f9XDlhWWNaNlmP4lCdO/1HbG9C8tsM6ihiQa4A3QuJ06Ilcx0EYseUksiUaWNQCI0r6ILpY1FcuQFEIbV240kx3EQAX8EJECvBvF+sjiudvQS1lWI5UXy2lVLPMAADxHYBK0/87+sIgCJbg7+ZmAAMBbZLHUrp3qUlecJpY1vX6sCMY3G1BywYQ5MwIww5EFljWjZZj+JQnTv5T2xfQvPbDOooZE3DDxWwHRVQw0iNkhBLxAxZI3a2NZA6r4Hz3yjElumKAklnHNYOR5CGc/jVtDl6MKNPt/Y/eNLJY7uZ7KshrkwMBzredfM1qUB4RNqIvnWySEX0PLmpPE0iRoZ1mrqxdoaVlbEYQvFAYhrQTNSzMB/VIOtWxIHcFoGaZ/ScL0L6V9Mf1LD6yzqCGJJqUFaPoITd0w4QA4afhaKzcMAOVi53R0w4SIEWYDH/cMRuPw87HW8XvlfWOxlN//jB7KslqSgxxLfNnwRQt1xdgFbVMyAI1jdpJPSAsABil42iJq99sEwoKqLqp2lrWBI+DFcOtB89JMQC9P741sSB3BaBmmf0nC9C+lfTH9Sw+ss6gRYUGEIFKRMhsSc8NoGuAdjtwwBot2Ykmki50T9LOsQzDCkmAS3c55EVdTY2vZZqb7OhjOVQSphjjhbM0FI0GkeqU5orYF6oFGswFTtKyNNnr9WETt3DCBsBhJa6HFQ1cluFYE4Q+2IpaSG8bL0e2ywbJmNA/TvxRg+pfSvpj+pQfWWdQItaWQyA2fazOhTkvLWkoaKxIOJpN2MTuylc6HdbSsYYxvJqCKznmRG7m5AG9fSAT6nQbCGfAj6dl62ghlh9SyziEeiKK2ucb8jWcDpoDRRi1rK9HuQRYIRdwwmljWBjPkkmRWBOBtLWhesqwbOOqGsWdBgDejeZj+pQDTv5T2xfQvPWS+u9pOsBgN+PCa46lFkkDMTt+SHPygBHjXpt4QScz8MMOi5QUm3TAGwQ8QklKqgyZIYhmEMeFarl3yI2LZXIC3LyQAZy/CJ12vwS9v7sbYeIOFVYlpg4IIK6/d+fSHBBRAcu+kGOBtttGgdTvxQRAJDBokIvaHBdhlsdQiZofjqIUe8sLKBVtPHyFXLyCSZZ0FAd6M5mH6lwJM/1LaF9O/9NDmRhYXLVqEnj17wmq1Yty4cdi4cWOL67/++usYOHAgrFYrhg0bho8++kiXdhl4DkM652JU9/yEqgaM71OoxOwQLZLSSpa1H6aEhaclooLFtQ7yVrlhEq3l2kVlWVtj5BkDqDCBN6BacnfF64bhJbHMg/a5xnxB7Sxri4O6i+ycX7PSXIGQCBsnu2E0EEtAyXtnRaj1GYEBalkfDtD8dp1ytRslassw/WsFpn8KTP+Sh+lfU9pUZ3HZsmW44YYbsGDBAmzZsgUjRozA5MmTceTIkZjrf/XVV7joooswZ84cfPvtt5g2bRqmTZuGH374Ic0tb57+JU6lFBAX8gDhQGo7VFnWuoml1kHeqgDvhN0wuSqxjFHBAABCAkFYENEQoEP/raaNkOAli9fFeZSZnlrhC2k3G9Bil9JhwB9/KalWCIRpOS4A2sTsqPZDZwS2ZllTsawKW+G0GjG8S642bWjDMP2LA6Z/Ckz/kofpX1PaVGfxsccew+WXX47Zs2dj8ODBWLx4Mex2O1588cWY6//jH//AlClTcPPNN2PQoEG45557MGrUKDz11FPNHiMQCMDtdke99ITnOQzt3RUikazxVF0xsmVNzAkLT0uYzRaEiLQ/vcQShoRcWEC0G6ZxfjL19/eGBLj9VCzjtaw5KWaHWtbaxuz4ggLyNQrw5iySGwZ++ALaJNANhMWIG0Yry1otlnG6YdzEjuP7FsGYwKSJ9grTvzhg+qfA9C95mP41JfMtiJNgMIjNmzdj0qRJyjKe5zFp0iSsX78+5jbr16+PWh8AJk+e3Oz6APDAAw8gNzdXeXXr1k2bL9ACv+tbgjpolJhWsqwDMMeX3DVO1PVRtXfD0Bs8mNRswOYDvC1GXgkMrm4IokESyxxLvAHeeQBozI7Wbhg+6IaJkwQjxZgduZqDkRPh82uT5iIQFmDTMnUEEKliwAVaHwFQ1UU9oV+xNsdvwzD9ixOmfwpM/5KH6V9T2kxnsbKyEoIgoLS0NGp5aWkpysvLY25TXl6e0PoAcNttt6Gurk557d+/P/XGt8L43oWolYK8A/WVqe1MidkxJ1RJoTWsJgO8kOImAhqPNqhmAybqOnJZjUqd08bxPhzHodRF21zh9qPeT2OD4rWsoVjWHqU6hVZYgvShGDbaU3dzqEp/BT3a/Db+kCp1hBYB3kBUFYPWYovC3loANMD7xP6p5WFrDzD9ixOmfwpM/5KH6V9T2kxnMV1YLBa4XK6ol970KXbAY6DH2b3/QGo7C0fcMFpa1laTQSkVBS0C0dWkkDqC4zgl11isbUuctCRWRX1AidmJWyylmqM2zq+5ZW0JSGIpJb5NCd4AP+j3DHrrlcXBsNh6pYBmCIQE7d0wqgDv1izrhjp6jdlcBeiar9HxGa3C9C82TP+Y/qVMG9e/NtNZLCoqgsFgQEVFRdTyiooKdOrUKeY2nTp1Smj9TMFxHDhpKH7vgVTFkl7gAY1nA1pNPKp1E8tIuatEZwMCwCkDS+EwGzA0RhCwbFkfcftRn2DMDgzUXWOGoLlYWsO1AADBllq8joyPo9Z5yCe5L/whTPj7p7js398ktT9/WFS5YbQN8LZwrQd4Bz21AIC+3Tprc+w2DtO/OGH6FwXTP6Z/WtFmOotmsxmjR4/G6tWrlWWiKGL16tUYP358zG3Gjx8ftT4AfPLJJ82un0kcuXSoufJo8y6iuFDl7NIywNtmMqAGslhWabZfACnNBgSAW38/EFsXnI4+xTlNPit1SZa124962bKON2bHSLc1I6S5WNrDdLYbsaUYryMR4KkQhaWYnZ/L63G0PoD1vyX3W3mDAmzQOHVEVIB3y4HovOTqG9y7qzbHbuMw/YsTpn9RMP1j+qcVmc/0mAA33HADZs2ahTFjxmDs2LF44okn4PF4MHv2bADAJZdcgi5duuCBBx4AAFx77bWYOHEiHn30UZx55pl47bXX8M033+C5557L5NeIiSmHWlgGf21qO1JZqQ5NLWuDyrLWUCwJAcRInrFkRwNMzcwWi8TsBJSYnXhTR9Cs+4AJYc1jdnJEjcXSYAcEQPRRN0yFm7rjvEE6KtDc+WkOTyCsbQUDIBLgjUCLlrU3GIaDeAAOGNhT/wkWbQWmf3HA9C8Kpn9M/7SiTXUWZ8yYgaNHj2L+/PkoLy/HyJEjsWLFCiWIe9++feD5yEVx3HHHYenSpbjjjjvwf//3f+jXrx/eeecdDB06NFNfoVl4O43dsEoWV9KkEP/SElYTj3I9LGtJ3AG5Nqq2ZY1KVAHeDUm6YUxcWPPUES7BDRgAzqFN8HLIQAUtKLlhKtyRfHVuXwiFOZaE9ucJhHVww0RidupbiNnx+XwolBLi5ri0eZi0B5j+xQHTvyiY/jH904o21VkEgHnz5mHevHkxP1u7dm2TZRdccAEuuOACnVuVOkbJsrYJ9a2s2QpRLg1tZwPqYllL7QWkPGMathkASqUA7yP1gUjMTrxuGIPshgkjpKFlLYgEuYSKGu/QJmYHphzAB/ga6MP2iDuS3sPtDycslg2BcMQNo5ptmBKSZW3lgjjagmUd8EQ6DLxV/wkWbQmmf63A9C8Kpn9M/7SizcQstnfMTnrTOEV30jO4AESVjtJvNqBeYmlMOClta8humEO1PmXoP37LmrphzAgjqGHMTiAcqV5gyNHGsuZtNLg93EBTj1SoxdIXirlNS9CYHf0qGLQ0GzDopQ8SP0yAoc3Zs4wkYPrH9C8VmP7pD+ssZgkWF3UlFcDdeimglkihKH1L2EwGVCtuGA1nA0riLoKDAF5T1xEAlEgB3uqYm7hjdoyRmB0tA7z9IVGpi2p0aiOW4aIBAICC+h0Aot0wdUmIpc/vjyTN1aPcVQtiGZJypXmh0XEZWQ/TP6Z/qcD0T39YZzFLsBTQKfKlXI0SW5IMRLeYHX0t6zCMADjYzNpeknazEU5LRBytJj7+YGfFsg5pWhvVH4qUujKkWOpKxtT1GABAj8BOAEBFvdoNk7hYhv0e1c61csNIMTutpI4ISaWufFx25BeLxf79+3FAleZl48aNuO6667Jy8khbgOkf079UYPqnP6yzmCVwzjIAQD7XgPqG5EsWiWFVnVGtLWu1WKbiKlIjiyUnVSHQ2A0DRKxrIIFSV4AilgaOIBROXHCaIxAWkcNJ9WU1iklx9j4WANCdHEbQU4eKuohYJmNZkyAVS8IZlED3lInTsg5LQep+PrvEUs3FF1+MNWvWAKCVUk477TRs3LgRt99+O+6+++4Mt67twfSP6V8qMP3TH9ZZzBZs+QiAXpSB2sNJ70YI0eH3UJIJXpvDauJRAymPlxgCAikGostIbpiw9N21dsMAkbgdgJbHihtJLAFACAZaWDEx/CF1PIw2VmtBcRccJgXgOYLDOzbAoxIjty/xkRoh6AUAiEYbwHGatDFSwaDlmB3RT68tP59dbhg1P/zwA8aOHQsAWL58OYYOHYqvvvoKr776Kl566aXMNq4twvRPOg7Tv2Rg+qc/rLOYLXAcaniaPiJYeyjp3chiGYYR5gRzS7WE1URLKnmJZKVq5YpRxRgBgEXj2YBAtFjGHdwNRImlGAq2sGJi+IPhSCkpjeqO8jyHXca+AICqXzZGfZaMG4ZIYkm0yjEGRFnWLdVGJdKDOGjQyP2jA6FQCBYLvRdWrVqFqVOnAgAGDhyIw4eT7+x0WJj+AWD6lyxM//SHdRaziFoDDfYldamIJb2pCW8Gp5VFhIjFq3mQtzJ70RB1HC2JcsMkJJYR94MQ1s6yDgT8MHJSDJCGYnTYPpD+c2hr1PJE3TCiSMCHqFhyWgV3AxGx5Fq2rIk/O8VSzZAhQ7B48WJ88cUX+OSTTzBlyhQAwKFDh1BYqFE6kA4G0z+mf6nA9E9fWGcxi6g3SQ+Z+uRLXskxO5rFWUjILp0aIrlitLasCRUxLWcwypQ6VZZ1IjE7HIcwR9cXQv5WVo6fkF8Vk6VVDi8AdXlDAABF9dujlieaOsIbEmCVksJyGrZPdsPY0Ept1CA9P2FjdsXsqPn73/+OZ599FieddBIuuugijBgxAgDw3nvvKe5pRmIw/WP6lwpM//Qle5L4MOAxlwBegPdUJL0PUbYAVS4ELZAtXs1nBEqlrmSx1DtmJyHLGoDAmWAkIZCwdm4YQRLLEIwwafhQC5cOB/YDXYWDcMAHP2+HIJKELWt1qStOIzcRAMWytrQS4M0pYpldljUAeL1e2O12nHTSSaisrITb7UZ+fr7y+dy5c2G3Z5fItxWY/jH9SwWmf/qSdGextrYWGzduxJEjRyCK0dPqL7nkkpQb1hHx24qBWsDkO5L0PuSbmjNqK5YmAw8jz6ncMFrH7MhuGD1idiJumIRidgCIvBEQVSMWGiCnZQhyVmg5/pFX0hWHSQHKuGoM4vaipnA0fj3qgTvBVCQNgUhMkaZuGFXqiKAgIiyIMMaIK+MlsRRNOdodWyOKiopwyimnYOrUqTj77LOVUnsyPXv2zEzD2gFM/5j+pQLTP31JqrP4/vvvY+bMmWhoaIDL5YqKDeE4jnUWkyRkow8eaypiKVcE0FgsAeoiqRG0FstGlrUOqSOiArwtiVrW9DwSDd0woiQGAd4KLW3HslwrfhB7ocxQjWH8bpSXnoRfj3pQn6gbJhBxw2jpJpLjk6xSGS1fSIAzhlgawvRhImiV30xDfv75Z7z77rtYvnw5rrnmGowYMQJTp07F1KlTMWzYsEw3r03D9I/pXyow/dOXpMyYG2+8EX/+85/R0NCA2tpa1NTUKK/qag2z23cwBEcJAMARPJr0PmTLmtfYDQMAdosO9VEbzQbUww1T7FRb1onZsiJP1xcF7fKMiX4aPB3SODVClzwbvhd7AQCG8nvQr4Rapom6YRpUbhjNqhcAgCkSswPQSg6xMIbow4SYndodWyO6d++Oq6++GqtWrUJFRQWuu+46fP/99zjhhBPQu3dvXHfddfj0008hCClUIemgMP1j+pcKTP/0JanO4sGDB3HNNdew2BytkRLTOkOVye9Duql5HSzrfLsZNZq7YaTZgISKpMWovRvGajIgz05FL9GYHVkstYzZEaWEryGDtZU1E6Msz4ZfSBcAQE+uHH1L6W/l9ocSqrfrCYTh0DgPGgDASIXXxgUBkGbrzRrD9GGCLBPLxuTm5uKiiy7Ca6+9hqNHj2Lx4sUQBAGzZ89GcXExXn311Uw3sW3B9I/pXwow/dOXpK7MyZMn45tvvtG6LR0e3kXF0i42ACFfcjsR5JgdSysrJk6e3aSyrLVKHREpz2Ux8uB57dJdqCmRrOvEY3akh46GqSPk6gBhg7aWdY7FiGozvYa6c0fQv5Ra1iGBJFRv1xMMw8VJ5a6sudo10BR5OFhaKCFmEuixOUt2uWFawmQy4fTTT8eTTz6JvXv3YvXq1ejfv3+mm9WmYPrH9C8VmP7pS1Ixi2eeeSZuvvlm/PTTTxg2bBhMpuihbTlBLSMxrDn58BEztTzqy4GCXgnvgxP1s6wLHHpY1vrUcm3MtGO6YPmm/Ti2Z0FC2xF5tp6gnWUNKYeX1mIJAGFXD8ANFHN1cDoBA89BEAncvjDs5vhud09AgAuSdaulWBoj39eCYLNiaRHosXlrdlnWjWlukh/HcfjTn/6UwZa1TZj+Mf1LFaZ/+pFUZ/Hyyy8HgJg1UDmOY/E6SZJjM6GC5KMnV5GVYplnN+NXzWN2pABvjctzNebKk/riypP6JrwdkS1rLcVSsqwFo/ZimVtQjNo6B/I4D6wNB+CyGlHjDcHtD6FTbnxuH08gjDLZsrbladc4gwngDAARYG3BsraI2SmWalqb5Mc6i4nD9I/pX6ow/dOPpK5OURSbfbGOYvK4rCZUQMrZVp9cyTBeuqkNJj1idlRuGF81IMa+2BNCam8YBl0t62QhUqA80VAs+TB1sYkG7WN+O+dZsY/QiQKo2QOXjY4MJBLk7QmG4YIObhiOUwLGbVwAwWa0wkqoWBpsGh5bY9gkP+1h+sf0L1WY/ukHq+CSReRYjDhK8uibJKsY8ESyrE3aBg8DNMC7VnbDEBHw16a+U9kNQ4y6VC9IGckNw2kplpIbRtSy7qhEWa4tIpa1e5EriWUiVQw8gTBcnA5uGCCSawxBBGJZ1qIIG6FpOow2l7bH1hA2yU97mP4x/UsVpn/6kXRSbo/Hg88++wz79u1DMBh9IV1zzTUpN6wjkmM1ooJQy1p0H06qJ89LbhijLpa1GSEY4eXssBMvDfK2JxYD0wSlNqoRliwUS2KggeGye0sLDGG5SL32AcyTBpVi81ddgBCoZW0dBYDOCIyXhoAApx4xO0CkPmpzMTvBSCkwkz27xFKNPMmvd+/emW5Ku4HpH9O/VGH6px9JdRa//fZbnHHGGfB6vfB4PCgoKEBlZSXsdjtKSkpYZzFJnFYjKiTLOuw+jGTkjhdptnqDSfvZgPkOaUifc0liWQkg/jgYQSQwNJ7tp8ozZtUhbUTK6GBZGwQ5NYL2lvWATk4MmHwi8MFbkhuG3uJ13vjF0qvXbEBAZVk3E7Mjl7oiPKzW7B21Y5P8tIfpH9O/VGH6px9JdRavv/56nHXWWVi8eDFyc3Px9ddfw2Qy4Y9//COuvfZardvYYbAYDajiCgEAYl1yMTsGIlvWOoilncp3NXGhDOUJBXnX+UKY9NhnGNurAIsuHhX5IE2zAZOFk2J2tLSsjYKUFkQHNwwAIL8n/VuzB7llkhvGH0YgLOBQrR+9ilq26D3+EHKVmJ08bdsmW9ZSyasmBKhYNsAGa5yzFzMBm+SnPUz/mP5pAtM/XUjKlNm6dStuvPFG8DwPg8GAQCCAbt264aGHHsL//d//ad3GDkWDuQgAwDUkF7MTEUt93DAAcFSUalYmIJa7jtTjaH0AH31/GNUelZWqcsPoORswaaR8bbyonWVtEmhMim55tFRi6ZLKe9X5Qnjgo59x8iNr8f62Qy1uHvR7YOakzo5ubphAbMs6UA+AiqXNnH0PTxk2yU8fmP5lGUz/tG1bG9a/pK5Ok8kEnqeblpSUYN++fQBoRYP9+/dr17oOiMdSDAAweCoS31gUYAC9AE0WHQK8HVQsq0TJIvTVxtEmEdi3AT6PVMKIAF/8oirnpbKsszHAW7aseQ0ta5NILWtey7qjanK70hQNYT86GeoAULFc+SN9AD/56S8tVjTgAm4AgMgZtK2NCkQFeMcSS9FPxdJDrFl5PTD0helfdsH0j+mfTFLjnMcccww2bdqEfv36YeLEiZg/fz4qKyvxyiuvYOjQoVq3sUMRshYCPsAYqqdZ8xOpRKCq32kya++GcVmNMPAc3ES6gfx1rW/041vAm3PQtd8sAJMBAJ/tPIqzR9KyTErMDslSN4yUr43XMGbHLFLLmtfLsjaYqGDW7kUZqQBgwk+H3DhcR4+7s6IBn+08ipMGlMTcnA/Q31Uwu8BzGleUUNwwoZhumJCvDhYAHljRLRtHWlSwSX7aw/Qvu2D6x/RPJqnO4v3334/6etoDvu+++3DJJZfgiiuuQL9+/fDiiy9q2sCOhkGdiDPoSVAsIze02ay9Zc1xHPJsJrgDkmUdj1jW0lFnZ+VWyGL5+c5KiCKhpa0EGpAeggEOS3bFaAAAJ8U+ye4tLZDF0qhnOaf8HkDtXhSHDgPojp8Ou6M+fuGL3c2KpTEoWdYWHWbjtTIbMOh1wwKggdhgNWbfw1OGTfLTB6Z/2QXTP41pw/qXVNd1zJgxOPnkkwFQN/SKFSvgdruxefNmjBgxQtMGdjQcNiv8RJpZKcUvxI3KsjbrYFkD1BXjJgmIpVRT1Fn/KwA69F/ZEIjcvCo3TDaKJS+5YQwaumGsUh4tgzVHs302QYrbKQhGx+dMHdEZBp7Dl7sq8eOh2L+fKSRdd1oHdwNKyavm8oyFffTYXs6mW51cLZAn+dXU1MBms+Hrr7/G3r17MXr0aDzyyCOZbl6bhelfdsH0T2PasP5l1zgnA06LEQ2QyiBJZZHiRq4GQHhYzKZWVk6OfLsJbiTghglTYTCHG1CKGmXxZzuluB21WGZZQC8A8JJlzZOwZvuUxdJk07GckySWLv+BqMXnjOqCM4aVAQD+u3Ffk80IITCH6YOM0zq4GwCkZMk2LnaAtyBdUwE+u9JGNIZN8tMHpn/ZBdM/jWnD+hd3Z3HUqFGoqaEX+zHHHINRo0Y1+2Ikj9NqhJdIVrEqQWdcqIRHrwSvefZELWu/8m9f/qAStBsRy0htVHs2WtbSrEojCbYYFJ0IVtDRBpNVTzdMTwCA3RMRS54DxvTIxykD6SSC3ZVNH8a+UCQhLW/P075dimUdO2ZH9NFrPqBDKTAtYZP89IHpX3bB9E9j2rD+xX11nn322bBY6E08bdo0vdrT4cmxGuFRLOtExVJVDUCnBK8FdjP2JWFZA0A/7iDyBp6O3J/+g24HjqLeNxpOVbmrbLSsDVLMlBFhCCKB0ZCaa0AQCWyQRhvSYFlban8DdX9xGNI5F06rCaUuat2W1/mbbNYQCMMFuTZpnvbtkixrSzMxOyRI3TDBLLSs1bBJfvrA9C+7YPqnMW1Y/+LuLC5YsCDm/wxtcVpN8EAKzg4kZ1kHdUzwmucw4YckYnYAoC93EKWFwOWmJTBCxJ6Dv0TEMktjduRKEGaEERREGA2pPYQCAT/sUg4vs13HmJ2SIYDRCt5XiX7cQfxCumJsL1qarCyXPowP1/lBCAGnmvHnDQjIlaoXcHoUsm8lwJtI13zQqOO50QA2yU8fmP5lF0z/NKYN61/2XZ0dnByLER4iiWUqbhgdLWs3knPD9OMPwu3ZAqOUC83vro4aDXBkWcZ6IFosQ2GCpGqQqQh4G+SzB6uelrXJCvSYAPy6GqeYvscvwYhYdpIsa29QQH0gDJc1Et9FLWudqhcAETcMF0QgHCN5tSSW4Sx0w6gZM2aM8r88yY+ROkz/sgumfxrThvUv7jsqPz8fBQUFcb0YyeO0GtEgW9YJBngTxaVh0M2yzrebI3nGQp6oGYgxaWRZ96z9Wnkf9NZFjQbYLdnnhuGlPGMmybJOlYA02y1MeCV4XDf60IwFZ+XsQLcCG47rQ0up2cwG5NmpQDZ2xXgCYbg4qXarngHezVjWvOSGEUw6xjMxshamf9kF0z+NacP6F7cp88QTTyj/V1VV4d5778XkyZMxfvx4AMD69euxcuVK3HnnnZo3siPhtBpRJVvWCaaOCAUDMINaqXk6JfTMs5sisxUBwO8GHIXNb6CyrAu5euRUrFLehzx1UZZ1Tha6YTgp475WYhmSxNLHWeDUOuFrY/qcAgAYEvwen/3tOPCqGaLDctxY57XgcJ0f/UsjFr4nmCbLGrFro/IhalkLpuxzw4waNQqrV69Gfn4+jjnmmCj3VWO2bNmSxpa1H5j+ZRdM/zSmDetf3FfnrFmzlP/PO+883H333Zg3b56y7JprrsFTTz2FVatW4frrr9e2lR0Ip9WEvUmmjghKYhmEUbeEngUOMwQY4IENDvgAf20rYhmIemsJRtJHhHzuSLoLGGDPwgBvGKjAmLkwQrFqeSZIyEd/Uz+s0NEJQykZDOSUgmuoAHdgA9B7Il2+4Tm84r4Z/zCei/K66LyonoCAYk4WS/0saysX27I2hOixRb1KgaUAm+SnP0z/sgymf9rShvUvKVNm5cqV+Pvf/95k+ZQpU3Drrbem3KiOTI7FCC+SSx0RDlArNgQjTCnOWmuOPDt1S7hhj4hli41StQnRuboEfz2IEAQHWu4qG2N2ICWlNSOMkBaWtZ/+pn5O+woTTeA4oPfJwHevAb+toWJ5dAfw8R0AgGHcb/i+Lvph5lHNBtRHLGksTnNJaY1hKpbErPujJGHYJD/9YfqXZTD905Y2rH9JjdUXFhbi3XffbbL83XffRWFhC1YWo1WcViM8JLnUEeEQvfAFztSiiywVChxUPGrFOIO8Q1Qsf+O6N/mI+N0g4UhQejbG7EDjmB0hQMUgkA6xBJS4HexaDYSDwNt/AQR6nRRybpS7fVGrN+gds2NsOXWESaDH5izZJ5YM/WH6l2Uw/dOWNqx/SZkyd911Fy677DKsXbsW48aNAwBs2LABK1aswPPPP69pAzsa6tQRgr8eichHKCiLpX4Waq7NBI5D/FUMJMt6O3piAH4DABzIGYauDd+DC9YrQemEN8KSZbUwAags6xA8GrhhBGm2W4C3tbKmRvQ+if4t/w64twRyzjGAoJBz43CjAG+vejagLnnG6Pe2xYrZIQRmybKGJftidvLz8+PuhFRXV+vcmvYJ078sg+mftrRh/UtqZPHSSy/FunXr4HK58NZbb+Gtt96Cy+XCl19+iUsvvVTjJnYsXFYjfBy9oEI+dytrR6O2rPXCwHNwWU3xVzGQYnY2hfvSttmLUZ43GgDABRqUAG+DKU2WZqIYqEvMhDBCQuoVDIhkWYf4NH1fZydgyDny0emfU6gbphDuJrMBQz43DJy0no6WdcyYnbAfPGg6CT4LLesnnngCjz/+OB5//HHccQc9h5MnT8bChQuxcOFCTJ48GQDYJL8UYPqXZTD905Y2rH9Jm2Djxo3Dq6++qmVbGAAdubDkAAIg+JNzw4i8fmIJUFeMuy5esaQ348ZwX1wrXIm7Z04FNnwEQJr5JVnWJnOKCbz0QgrwNnGCJjE7oiKWabKsAeCCl4BpzwC+WhrHY3YAn94DGxdEbW1t1KqClwbghzkTjEYdBL2lpLSqJMwGa/aJJZvkpz9M/7IMpn/a0ob1L+X8An6/H263O+rFSA2jZFWQBCsYiCEqPHpa1gBNH6HkGovTsg7AhHfF42HpORac1QUAMIYbwInUsjaZs9WypiJuQShmjEmikCCNSQkb0iiWABUpVxm1tM05IJIQGgNV8AUFiCJBhdsPr5u6T4NGJxVWPdqBZlJHSDnGGogVtmwM9lexcuVKTJkypcnyKVOmYNWqVTG2YMQL078sgumf9u1A29S/pDqLXq8X8+bNQ0lJCRwOB/Lz86NejNQw2qlYcgmmjkibZZ1IFQPJsg4QEww8B4uRh9FGxdIcbgAvi6UlS8XSGHHDaBHgLacDSbtYquE4wF4EgLpiDtf5cOHzX2Pc/avx02/7afv0mo2nVDAIIRSKnh0qW9YeWGHLxjQiKtgkP/1g+pdFMP3Tljasf0l1X2+++WasWbMGzzzzDP70pz9h0aJFOHjwIJ599lk8+OCDWrexw2GRxFJO0BkvomTFEl5fl0Zhjjm+mB1RACQxDMAEu9kAjuNgtFOxtAuRUWizWeds/ski5xnTKHUEF6aWtWDMoFgC4BxFgPsACrh6rN5+BJ32vo8rDZU4ZKKzNi1OnTo8qtgsrlEOOjkJcwOx6VaBQyvYJD/9YPqXRTD905Y2rH9JjSy+//77ePrpp3HeeefBaDTihBNOwB133IH7779ftzjG6upqzJw5Ey6XC3l5eZgzZw4aGpoXk+rqalx99dUYMGAAbDYbunfvjmuuuQZ1dXHU88wwFjsNrDVKN1a8yG4Y0aCvZV3itMY3G1B1M8hiCQBmSSxzxEiFBnOWu2FMCGvihuFC9DcVTRmu/emglnURV4cXv/gVD5pewN9My/DE0N0AAItDJw+B6iHBCdHB5XKqlAZkp1iqSfckP6Z/rcP0TweY/mlLG9a/pEYWq6ur0bt3bwCAy+VS0kQcf/zxuOKKK7RrnYqZM2fi8OHD+OSTTxAKhTB79mzMnTsXS5cujbn+oUOHcOjQITzyyCMYPHgw9u7di7/+9a84dOgQ3njjDV3aqBV2JxVLk+in1ikf34UjhuU0DDqLpcuCX+KxrFWlrgIwK0lnrY48AEAuiVjWVmu2WtaSG4YTEAqHW1m5dXjpASgaMyyWkhumAPUI11fAbpUebD++Rf/qMRMQAAxGEIMZnBAEL0TnOJOvpXpigy0LxbIx6Zzkx/SvdZj+6QDTP21pw/qXVGexd+/e2L17N7p3746BAwdi+fLlGDt2LN5//33k5eVp3ERg+/btWLFiBTZt2oQxY8YAAJ588kmcccYZeOSRR9C5c+cm2wwdOhRvvvmm8r5Pnz6477778Mc//hHhcBhGY/YFkMo4cvIib4INcV+4shtGtgb1osRpidOypmIp8iaI4JWks9YcarXZIMUYEQ5WS7aKZeTBE5ZGLlKBD1OBIFliWRdybnTjjkaWE2n0QC+xBECMNnBCEEbBD0JIJHehLJawoygLY3aaw+/3IxiMvjZcLpdm+2f6x/QvYzD905y2qn9JuaFnz56Nbdu2AQBuvfVWLFq0CFarFddffz1uvvlmTRsIAOvXr0deXp4ilAAwadIk8DyPDRs2xL2furo6uFyuFoUyEAhkfHZ3rjMHYSL9NAkEeRMpZxf0dsO4rPHF7EhiKUgxRHbJsra78qJWC8EIhyVLH16qB48QCrSwYpy7k8QSWSSWXbjKpp/rKJbqxLRRudsC9F5zE0dWWtZq0jnJj+lffDD90wGmf9rTRvUv4c5iKBTCBx98gN///vcAqGj9/PPPWLp0Kb799ltce+21mjeyvLwcJSUlUcuMRiMKCgpQXl4e1z4qKytxzz33YO7cuS2u98ADDyA3N1d5devWLel2J0t+jkWpYoAE0kfIpaO4tFjW9GYnccTsCDy1mh2StWRzRN+IwTYilqIGYmmUXA+cOdNiWQyAzgbsKoulXRXUrUf1AhmlPmogeoalYllnZ8yOmptvvhmffvopnnnmGVgsFrzwwgu466670LlzZ7z88suaHovpX3ww/dMBpn/a00b1L+HOoslkwnfffRe1rEePHjj33HMxfPjwhPZ16623guO4Fl8///xzok1sgtvtxplnnonBgwdj4cKFLa572223oa6uTnnt378/5eMnSoHdjAYkUR9Vqnkp1/PUi2KnBXVSnjEu5KU1N2MhWdbhRpY1b7ZHRg4AhGBQ4nmyDp6HIBUd00IsTXKcitmR8r5SQo7Z4dzoZ6GJaHHMHwGH1CnR0bKWHxR2LhAdNO+nlnU9sWdl6gg1WkzyY/oXG6Z/WQTTP81pq/qX1BX6xz/+Ef/6179STpNz4403tjpzsHfv3ujUqROOHDkStTwcDqO6uhqdOnVqcfv6+npMmTIFTqcTb7/9Nkymll0UFosFlgzHj+Q7TPASK8ABJFCPuFODCumxrC1GA80VJo+gB9yAsajpipJlHeZksZRuAI6Dl7PDBfogCMGoxPNkIwJvgkEUIDROdZAEJpGKJW/JsFhKlvWAHD/6FgWBfQAK+gBnPARseBYYcKZuh+Yky9qGaLEkfjc4AG7Ys9INo0aLSX5M/2LD9C+7YPqnLW1V/5LqLIbDYbz44otYtWoVRo8eDYcj+od/7LHH4tpPcXExiouLW11v/PjxqK2txebNmzF6NK2r+emnn0IURSXHWSzcbjcmT54Mi8WC9957D1ZrlqYnaESBw4ydkhsm4K1HvK3mpJgdXmfLGgCKXHa4a21wcT46fO6IJZbUsg5JYql2tXg5O1wkIpZZa1lDrgjhV2ZbpoJZEktDxsWSulwswRrAe5Auy+sO9DlZVUtVJ5opeSX662AADfC2mlIuLqUrWkzyY/oXG6Z/2QXTP41po/qXVIt++OEHjBo1Ck6nEzt37sS3336rvLZu3apxE4FBgwZhypQpuPzyy7Fx40asW7cO8+bNw4UXXqjMBDx48CAGDhyIjRs3AqBCefrpp8Pj8eBf//oX3G43ysvLUV5eDkEQNG+jlthMBvgkifTU18a/oWRZp0Mso3ON1cZeSbJEg6CjGXbV0HpAVRs0RAzZG7MDalkDANFgNqBFpA8Q3pKT8r5SQrKsEfYD1b/R//O6p+fYsmXNBRFU3YuirxYAdcNYjdlnWatJ5yQ/pn9xwvRPF5j+aUwb1b+krtA1a9Zo3Y5WefXVVzFv3jyceuqp4Hke5513Hv75z38qn4dCIezYsQNeL83jtGXLFmWmYN++faP2tXv3bvTs2TNtbU8UjuMQNDoAEfB54p+NKJeO4k36u5FKXBbUEzvAofkZgXKpKzS1rAMGByDdJ9Syzr6bQ0aUYo6IkLobxkLoOTFaM2xZmx00QWzYB4hS/rTcruk5tmRZ2+FHIEbMjs+QA57XoS6rRsiT/BYvXgwgMslv8+bN6Nu3b8Kx2/HA9K91mP7pA9M/jWmj+pe95kwjCgoKmk1ACwA9e/YEIZFp6CeddFLU+7aGYLQDQcDfEH/FBbkoffos61bSR4RksaSWqToOI2TMASRDlcbsZO+lKNeaJam6YUQBZki1YK0ZtqwBal3X7aP/53RS6sDqjlmO2Yl2w3BS6oigMQvOTQs0N8mvR48euh2T6V/rMP3TB6Z/GtNG9S/7HOMMAIBoohdMyJe4ZW1Ih2XttMBNWklMK1vWhIqNQxXEHTZGLMsQjMjJ4gBvWSyRaoC3KmecyaZTofpEcKhSRaTLBQNEUkc0mg0oi2XYlAXnphXkSX4MfWD6lz0w/dOYNqp/2WvOdHTMDsADhH3xp46IiGUaLGtXJNcYpFiLJkji4iNyzE7kchNMEespCGPUZ9kGUdwwKVrWUl1UgXAwW2ytrJwGHKrJFXlpzKenSkqr5BkTwjBIpcCyVSzVaDXJj9EMTP+yBqZ/GtNG9S97r9AOjhwALPrrW1lTtQ2hYmlMi2VtxY+tVTGQLGtfDMuaWCI3RIhk92xAopUbJkB/Sw9ssGbD97WrZnBmwLK2q1NHBCIjSNkqlmrkSX4AsHPnzqjPlPJdjKRh+pc9MP3TmDaqf1nwizFiYbDR2rIkgaS0BjGdYmnBemk2IPHXxc6FJlnWXpFeZjaT6nIzq8Qyy/OMESlvG5eqZS0FMLuzJTWCOt1Hbjot6xhuGOmB6yUWuBxZMOrQCpmY5NeRYPqXPTD905g2qn9Z8IsxYmGWYjr4RMSS0FldBnN6ZgPKVQyEyl9jryRZ1h5JLNWWNWeNiGWYM8JkyN5LURZLSA+jZBGUDP227EiNoBbLPP0mZzRB5YYJNLKs3bCjwKG/G5GR3TD9yx6Y/mlMG9W/7L1COzhmB7WseSmOIR6MkhvGlAbL2m42YpNxNATCwbhnLbB/Y9OVJLH0CkZlGxmDSixlN0fWIotlim6YkIeWlaJJV7NBLDMVsxOjgoGq1FVhloolI30w/csimP5pSxvVP9ZZzFLkYvMmwdPKmhEMoJa1yZyeSg0eVy+8Lkykbz5ZADRO1SG5YRoEKgxqy9poj9TelAOosxYDFXNOTE0swz6pUDyxw2LMglvPnik3jGRZc0EEhGg3TD1sKHBkttwcI/Mw/csimP5pSxvVvyz4xRixsOdQMbEIiVjWslim52IrcVrwRPg8CLwF2PcV8MvH0SuEaWknn9h0NqBJJZayGGUtBno++VhuGEKAvV8B3upWdxPy1AIAvJwtO5Kuusro35xOSu6vtBDLsg5ELOuCnCx/eDJ0h+lfFsH0T1vaqP6xzmKWkuPKA0Az3seTXJcQApNkWZst6bGsS5xWlKMQP3S9iC747O/RK0iWdSBGuSuLQ2VZG7Lz5lAwSpZ1rADvvV8BS34PfHBdq7sJNtQCAELGLJntVjoUOHUBMPWfra+rJebm3TBuZK8bhpE+mP5lEUz/tKWN6h/rLGYpTkksHfDB7Q+3un4gLCpiabKkz7IGgK9cU+iCyl+iVwhHKhiYDXxUELdVJZZcloslp1jWMcSySvrONXtb3U/IWwsgi1IjcBxwwg1A/8npPa5iWQebzAasJ3bk27P7emDoD9O/7IHpn8a0Uf1jncUsxWKnAd4O+FHT0HrmfLVYWtIUs1PqosfZ65XEOeAGxEhhdNmy9sPcJDWEzZmv/M8Zs9sNw0nlw3gxxkPLR4O2EcesTUGK2VHnWOuQSDE7Vi6AoCBdL6rZgIVZ6oZhpA+mf9kD0z+NaaP6xzqL2YqUlNbAEVS7Wy95FQiGYeHSG7MzoBO96TeVq0RElVxUXe6qcdJZuzNP+Z9PV03OJJHF0kBiWNZyrE6g9eTBREowTKwuzdrWJpHEUp2UVpQeJG6SvakjGGmE6V/WwPRPY9qo/rHOYrZiipQPq6yqanX1QDByI8s3t96M6JYHAPi1OgRilBKJ+tViGYnZsZmjLWujLSIY6WpvsvBGOoIQM8DbJ4tl65a1XPvTYM1tZc12jnRt2xBEMEQtazmtRgOy1w3DSCNM/7IGpn8a00b1j3UWsxWeR4CjAlRRWdnq6oGAP/ImTTEwuTYT+pbQEYCgHLSsLn2lxOyY4WgkljCY4Idksaahlmsq8IplHUMsvZIbJuQBRLHp5yoMIWpZR82E7IhIljXPEYjSNRL20utGMDthyIaZkozMwvQva2D6pzFtVP9YZzGLCRtpIGxVVetpCYLB9IslABwjWdf/3969R0dV3nsD/+65ZnKdBHIh3AKogDdUKHlRtJ5D3gqe49JKW9S89dBSsC1plaKruFat1T9K355jvZVVV1dPsT2HlrbnVG1pS19aRKonDZjiqSKkglQ4QEgxTjJh7rOf94+955bMBEIm8+xn5vtZKythLuHZDPnm95vn2fvxa1n2SU3rrMuz7AUaMH8Z2K3eWTtHhmXXu+/j9i2vYWjgb6kHnmfdjjNq3O+s8OZ9jEpxprazElHj8iJ64v9Nqb/rQEnMP2tg/uWZovnHYtHChMt4u/oD3wWEZXpnbSvclt/XzTQWan8QM38A0tfsmD8IYeHMuCBtQkgzjs9egB0XxiOxpsghYtB14zIeL/33Kfz3CR+CYwhLd9y4P31xe0myOxHXzP+jEeM6elrYCEu7xytpUGQ1zD9rYP7lmaL5x2LRwmzmllCDA77zPjYaMbrYCBzGJQEK5NoZXgBAb8QMvBydtSdLZx22G924o0AL0i+Wwzy70okYQjFjjcmQeTkPdyTteM+zbsejG7tRlJd6WAKI2Y1frpr5C9Ue4RQVZWL+WQPzL/9UzD8WixbmNBdBR4ODCEXjoz42EZZxrbCXYbi0oQoVLjs+0BMLvLOt2XGisWpkIHoqvQCAlgbvBI9yfFxmmLsQxVDYCEnjs0B5PO14I6OcERiLwA1jEX6Vd9JEDVUZ8URYxo2wTExRufiLhEzMP2tg/uWfivnHYtHCEmfMlSOI//lg9G2vEmEZQ+GmYADAbtOwYLoXfpFYs5PtbEAXptV6Rjy36dKFAID6lisnfJzjoZnTME7E4Dc76qFQDBUIwYG0X2KjddZp01PVNXUTMk6V6IkzLKNBIBqCw7wsR3kV/23IwPyzBuZf/qmYfywWLUwzrzVWiRD+enb0sIyZC7wL3VkDwHUzajGIYQu8hUh21iHhwrTaLHtvLt8MbDgIzLy+QCO9SOaCeRdiyekXfziGWm1YOI6yZidkbnV1TrhRWzXyF0ep0c1LjdhjgeQvEl1oyW3eiJh/FsH8yzsV84/FopW5jLCsQBDv9Z+vszY6k+TC2QK6doYXg8K8LloiLONRAMZi6DCcmFaXJSBsdqBmWmEGOR7m2YouLa2zDkfhxbBpl8SFaV/+OvCdGzKmpPwDxrXi/CgfeRmNEpS4Lp0tHkq+GzMED+oqC7P7BimA+WcNzL+8UzH/WCxambktUoUWwvH3z4360LjZxcZthe+sWyZXwI/Emh2f8TmWOjsxDCemehXuJs3O2okYhsLG5SOGQlk660RYvvFj4MxbwIn9ybuGBowzOgNaBbQCLsC3KmHuj2qPBwHzTMBBlGNShbUX+1MBMf+sgfmXdyrmH4tFK3MlpmHO31nHzM5alxCWVW5HsrMWibUpsdR+ruUeD6rKrL3/6ajSwtIfikEIgaFwDF7kmIZJ7JcaTF3yI+g3bgvZK0CpztoRDyXfgfALj2W3uiIJmH/WwPzLOxXzj8WilZlrdozOevSw1M1w0iWs2alwO5JrdoS5x2VqvY4T0+qyrNdRSdqaHX8ohnBMRzQu4B3RWQ8Z00+JswID/Wl3+QAAEUdlIUZsfcnOOgQ9aPyCHUQFJlVaNyypwJh/1sD8yz8F84/FopW5Ugu8T3wQQNy8IGo28ajRWQt74cOy3GVPC0ufcWPaZSOmeYsjLI1pmNS6ndpsnXWiqwaAQGpP26i5LVbcybAEkNzFwKWHEPQbv1T8wmPZfVFJAuafNTD/8k/B/GOxaGXmmp1KLYRoXOCUL5jzocnOWsI0jKZpiDuNy1wkL5GQti9qtstGKMUMS7cWhT+UutZYrWZ00LpmLtgODwvLtGmY1N6f1QUYsPVpbmM6ypkWliF7BVwORhKZmH/WwPzLOxXzz7ojo2RnXecwguf4KOt2dLOzhk1OZ6K7jRCwhQfNy0aY1xgTTvXD0pHZWScuH1GjGYvuh9xNxuMi/pydtQgZwaqZvwBLnc1lvNvi1EMIDxn/ZlEH/20oDfPPGph/eadi/rFYtDJzzU6NzQie90ZZt6PH5E3DAIDuNrYp0kQciJzLmIaZmu0aYyoZtsDbb54RWGteOuIDZ6PxuPBQxjqd9K818x0HzWPd7ZwKKRGWLhGGGDwFAAi4J8scElkN888amH95p2L+sVi0MrOzLocx/dI7GMr5UBE3O2u7nM7a6S5HVJjTEaGB1ALvopiGMX4BJRZ4JzrrxKUj+uyJsPTnnIaxmYu+HRbeKL6QEmFZhhDsgycBAEFPs8whkdUw/6yB+Zd3KuYfi0UrM9+yd+tGWEZies6HCrOz1iR11pVlztQuBuFBBIPGuwBGZ616WKa2uxoKx5JrdhKXjjiFeuNxIxZ4p8LSGTPC0lXJzhoA7G7j/4oHEbgDRmcdq5oqc0hkNcw/a2D+5Z2K+cdi0cqSYRmABh3R+ChhmeisHXI660q3A4PJ/VEH8MGAEQ5xmwvVKl9jDEhb4B2DPxhJhaXZWb8XN6cPwkMZ3XR6WLpjxmPLKrwTP14F2MsSu3OEUBnqBQCUTZ4pc0hkNcw/a2D+5Z2K+cdi0cpcqcsMlCM8ameNeKKzlnMF+Aq3A/60/VE/GDTXqDitu33RBUs7hmgoAH8oBht0VGvGuwdHoubm78MXeMeCQMR4jEc3FoOXV1t3o/hCspvTMFO1s3AhCl1omHvZXMmjIkth/lkD8y/vVMw/FotW5vQAmvESVSA0amedDEunnM66qiy9sx7EoN9co+JSfAoGAFyV0O1GYLoj72MoHEMNhmAz937tCXqNxw2/dAQABPsRjsVRASM0KxiWAADNvCjtTO0MAKAPtbhmZoPMIZHVMP+sgfmXdyrmH4tFK9M0wJW41lgQkVHD0jhDzSaps650OzAIcyunkA/+IWPawekugrDUNIhK4we5PHIWg8FocnH3oPDgVNjsvPUoYgOnM58beB++QBSV5iJ9hqXJvCitU4sDMM6o9LjsMkdEVsP8swbmX/4pmH8sFq3OnVrbMNo0jE03w1JSZ13hdsCftmYnEDCmHVxuxS8bkVBpnPFXDx/6/OHk4m4fqnAOadM077+X8bSj751Avz+ISs04O9LGS0cYXJl7xOrV0yQNhCyN+WcNzL/8UjD/WCxaXWLLKy046jSMZnbWdlkLvMtS+6MiNIBo2OgkXWVF0FkDsFUbF56t1wZweiCYXNw9ZKuCDht0c2N4d9CYVggK43XYuf8t+D5IW/TNi9IanJn/L8rrrb24myRh/lkC8y/PFMw/FotWd8GdtbFmx+6UNQ1jT63ZCQ8iHjE6yWLprDWzs27QPkDvQCi5L2rAbnTKMYfRKdpgvEah6hYAQG/vaTy07Q8AgAicgEPO62M5w8KyftolkgZClsb8swTmX54pmH8sFq3OlQjLIKJxkfNhNmF21i5ZYelMW7MzAD1qhqWnOMISlWZnjQGcHYrAa+6LGjb3hI3YM4+zeuo8AEAd/KgyF3fr3Bc1xZn571XZMEvSQMjSmH/WwPzLLwXzj8Wi1bkTC7xzd9ZxXcAljC2xEqfkF1qF2w6/MLolEUztYFDmqRjtaeowF3g3aD4AgNfcFzXmrgUAhGypf3cdGuyTZgMA7r2mCj++93IAQFmlt0CDVcCwzho11l+zQxIw/6yB+ZdfCuafQ/YA6DzSOutcZwOGonF4YEzDOCVNe1Slddbx4ABc5tdlnuJYs4OqxJodn/EZxud4mXF2XxCp44w6quCuMHY1mKQNATZj/RLK2FknDeus4Z0uZxxkbcw/a2D+5ZeC+cd3Fq0usWZnlM46HNNRZoalQ1JYVqSt2dGDvtR4JHX6eTess55lM666H6k2pg9OBlOXPbBVTALKzUtEBPuBIWPRNzy1hRmrCuxO6DZjZwvhrgLKeJYkZcH8swbmX34pmH8sFq0ucTbgKBelDUXjKNOMcLJJCqfKstQOBlrwA1ymnTDuSNuFQWnmAu/JGIAGHS2aEZa6Od1yJpw6C9NZWQeUTzL+EOgHTnYbXzddXbjxKsBmdtdajfW7apKE+WcNzL+8Uy3/WCxanftCp2GMNTsj3t4ukPS9UZ3hflxmOwm/VgnMXSFlPHlXUQ8dGhyajqnaWTSaHXZ1s7FFU1BLm27y1AIes7MO9AP/87rx9fTFBRywAhLrdhQJS5KA+WcNzL/8Uyz/lCkW+/v70d7ejurqani9XqxZswZD5lXyz0cIgRUrVkDTNLz44osTO9B8c6UWeEdzTMOEoqlpGEjai9TjtGNIy1zM/UvvJ1PTEaqzOxF0eAEAi7XDxm3lk7Dkijl4+q5rcMu1aZc+8NSljnuoF+g7ZHw97UOFG68KEu8CKbC4WzbmH/NPKuZf/imWf8oUi+3t7Th48CB27dqFHTt2YO/evVi3bt0FPfepp56CpmkTPMIJciGddSy1wFtWZ61pGoS7Erow/p2P6Y040LRSylgmStA9GQDwIVuPcUPdHNhtGm6/Zipqa9N+KXhqU2EZjwAQgHdmct0PmZxqhaVMzD/mn2zMvzxTLP+UKBYPHTqEnTt34nvf+x5aW1uxdOlSPPvss9i+fTtOnTo16nPfeOMNPPHEE/j+979foNHmmev8C7yNNTvmNIxDTmcNAJVuF/rgBQBsjt2DmooiuWyEKVJmhOVim9lZT5qTujN9bZKnFnDXAFraXp/sqkdK7ObgnSF3HBbH/GP+WQHzL88Uyz8lisXOzk54vV4sWrQoeVtbWxtsNhu6urpyPi8QCOCee+7Bli1b0NTUdEF/VzgcxuDgYMaHVO7UAu9cnbVxNqBxUdoR128qoAq3A/dHOvCY8wH8P30RaivkbL01UaLlRmc8x3bauKFudupOd1pYltcBNlvm2X9crzPSjRuBa/5P8azrmiDMP+afFTD/8kyx/FPiOou9vb1oaMh8C9vhcKCurg69vb05n7dhwwZcf/31uP322y/479q8eTMee+yxix5r3plrdkbbwSBsgQXegHFGYJeYj/3mUqq6IgtLvWLYNEp6WA7vrAHjjMDAWePraalf9GS69H8bHzQq5h/zzwqYf3mmWP5JfWdx06ZN0DRt1I/Dhw9f1Pf+xS9+gd27d+Opp54a0/MefvhhDAwMJD9OnDhxUX9/3qRdZyyuC8T1kYEZDofh0MyuW9ICb8A4IxAAEkOsLS+usExseZWUPg2TmFIA0sLSXLfjKAMar5rYsZFymH8XgPlnHcy/kib1ncWNGzdi9erVoz5m9uzZaGpqQl9fX8btsVgM/f39OadXdu/ejaNHj8Lr9WbcvnLlStx4443Ys2dP1ue53W643Rba7Nzs2KpgXAU/Gtdht9kzHhILnUv9QWZn7c7871RsnbW9ujHzhrpcYWmGZOJaY1OuARzF9W9B48f8uwDMP8tg/pU2qcVifX096uvrz/u4JUuWwOfzobu7GwsXLgRghKGu62htbc36nE2bNuEzn/lMxm1XXXUVnnzySdx2223jH3yhmD+Ebi0KB2KIxHWUOYeFZdgISx022OzyfihHhqVT0kgmhss7Jfn1kKMWlenbV2VMw3iNz+aWV1yvQ9kw/y4A888ymH+lTYk1i/Pnz8fy5cuxdu1aPPfcc4hGo+jo6MBdd92F5uZmAMDJkyexbNky/PCHP8TixYvR1NSUteueMWMGZs2aVehDuHhpHVsFsp8RGIuYXbfmhlviJTIqhoVlsU3DuL3Nya99nunI2JvBnWXNzv/6HKBpxmeii8T8MzD/5GL+lTYlzoYGgG3btmHevHlYtmwZbr31VixduhTf/e53k/dHo1H09PQgEAhIHOUEsDsBuzEtVIlg1i2vRNg45qhd7vRRVVkqLDUNqPEUV2ddPimts66YOfxOwOYwFuSXeY3b6ucC//gkUN0MovFg/jH/ZGP+lTYl3lkEgLq6OvzoRz/KeX9LSwuEyH62XML57rcsdyUQCOe81lg8anTWMZu8xd1AZmdd43HCYVemF7kg7nIvAsKNci2MYFVL5p1lNcDd2401U7biOm6Sj/nH/JON+VfalCkWS5qrEgi8n7uzjhiddVxyWKav2akrsikYAICmoV/zohxnEK3JMpWn0GUQiJTB/LMG5l9JYwugAnPdToUWQjhLZ50MS4m7FwCZYVlsF6RN+I2zDQf1mQhMXSp7KESlgflnGcy/0sViUQWJLa8Qyn5h2mgIABC3Wygsi7GzBvD2Jetwp/5/cUnLdNlDISoNzD/LYP6VLk5DqyCx5ZUWzLpmR4sZa3aE5M46fc1OsV02IuGJjy/A47dfgaqy4jw+Isth/lkG86908Z1FFWR01tnC0piGkR2W6WcDFus0jM2mMSiJCon5ZxnMv9LFYlEFiS2vkKuzNqZhhEPe7gXAsM66SKdhiKjAmH9E0rFYVIG5hVWZFkEkS2dti4fMx3kKOaoRMs4GLNLOmogKjPlHJB2LRRWYW1i5EM/aWdvNzlpjWBJRsWH+EUnHYlEFdmONiAOxrGt27MnOWu40TJnTBrvN2G6rWNfsEFGBMf+IpGOxqAKzs3YilrWzdupGWNpccjtrTdMwrdYDu03DNK/csRBRkWD+EUnHS+eowOysnbk6az1sfHbL7awBYOvqD6H/XAQN1XLPTCSiIsH8I5KOxaIKbEZYurQ4fFk6a5ceAjTA7pIflrPrKzG7XvYoiKhoMP+IpOM0tArSpmGy7WDgEtbprImI8or5RyQdi0UVjDINE9dFMiwd7oqCD42IaEIx/4ikY7GoglEWeIeicXi0iHF/GTtrIioyzD8i6VgsqiAZlvERnXU4pqMMZliysyaiYsP8I5KOxaIK0qZhwlk660RY2rhmh4iKDfOPSDoWiypIdNbayDU7xjSMsWYHDl6ugYiKDPOPSDoWiyowO2tX1jU7qWkY2TsYEBHlHfOPSDoWiyoY5WzAUCwOTzIsuWsAERUZ5h+RdCwWVZB+NuDwsIzEkmcDMiyJqOgw/4ikY7GogoxLR2RelDYaDqb+wLAkomLD/COSjsWiChLTMFp8RGcdCw2l/uBgWBJRkWH+EUnHYlEF6dtdDVvgneisY3AAdm71TURFhvlHJB2LRRXYUgu8h3fW8dA5AEDE5i74sIiIJhzzj0g6FosqSJ4NOHIHg3jECMuoxrAkoiLE/COSjsWiCsxpmGzXGdMjxjRM1MYL0hJREWL+EUnHYlEFGWcDxjPuSoRlzM6wJKIixPwjko7FogrMaRibJhCPxTLu0qMBAECca3aIqBgx/4ikY7GoArOzBgARj2TcJczOOm7nZSOIqAgx/4ikY7GogrSwRDyaeV/UCEvdwWkYIipCzD8i6VgsqsCchgEAMSwstRjDkoiKGPOPSDoWiyrQNAibecHZYdMwtlgIACC4ewERFSPmH5F0LBZVYTOmYkQ8AiFS+6MmOmuwsyaiYsX8I5KKxaIqzKkYF2KI6amwtMXNsHSWyxgVEdHEY/4RScViURUZ1xpLXZjWHjemYRiWRFS0mH9EUrFYVIXDCEvHsC2vHGZY2lychiGiIsX8I5KKxaIitLRpmMzOOmzc76qQMi4ioonG/COSi8WiKmxGWDoRQySts7aZC7w9HoYlERUp5h+RVCwWVZFYs6OlOutQNA6bOQ1TVVUlbWhERBOK+UckFYtFVdhTnXU0bpwN+Dd/GB7NuO6Yp4JhSURFivlHJBWLRVWYnXX6mp3ewRDKYK7ZcfKitERUpJh/RFKxWFSFPXU2YGLNTu9ACGUwt79iWBJRsWL+EUmlTLHY39+P9vZ2VFdXw+v1Ys2aNRgaGjrv8zo7O/H3f//3qKioQHV1NW666SYEg8ECjDjP0qZhEp31mcEQKrXERWm5wJuoWDH/mH9EMilTLLa3t+PgwYPYtWsXduzYgb1792LdunWjPqezsxPLly/HRz7yEezbtw/79+9HR0cHbDZlDjslMQ2jxZLXGTszGEINzhn3l9fJGhkRTTDmH/OPSCaH7AFciEOHDmHnzp3Yv38/Fi1aBAB49tlnceutt+Jf/uVf0NzcnPV5GzZswBe/+EVs2rQpedvcuXMLMua8S3bW8WRn/beBc6hKdNaeWlkjI6IJxPwD849IMiVazM7OTni93mRQAkBbWxtsNhu6urqyPqevrw9dXV1oaGjA9ddfj8bGRnz4wx/Gq6++OurfFQ6HMTg4mPFhCRlnAxphOeQ7m7q/rEbGqIhogjH/wPwjkkyJYrG3txcNDQ0ZtzkcDtTV1aG3tzfrc959910AwNe+9jWsXbsWO3fuxHXXXYdly5bhnXfeyfl3bd68GTU1NcmP6dOn5+9AxiN9b1QzLCP+vwEAYq5qwGaXNjQimjjMPzD/iCSTWixu2rQJmqaN+nH48OGL+t66bgTKfffdh0996lO49tpr8eSTT2Lu3Ln4/ve/n/N5Dz/8MAYGBpIfJ06cuKi/P++GbXclhEB0qB8AIDxcr0OkGubfGDD/iKSSumZx48aNWL169aiPmT17NpqamtDX15dxeywWQ39/P5qamrI+b8qUKQCAyy+/POP2+fPn4/jx4zn/PrfbDbfbfQGjL7C0S0dE4wKDwRjK44OAHbBzcTeRcph/Y8D8I5JKarFYX1+P+vr68z5uyZIl8Pl86O7uxsKFCwEAu3fvhq7raG1tzfqclpYWNDc3o6enJ+P2v/zlL1ixYsX4B19oGdtdxXHGH4LXPBPQVs7F3USqYf6NAfOPSCol1izOnz8fy5cvx9q1a7Fv3z689tpr6OjowF133ZU8E/DkyZOYN28e9u3bBwDQNA0PPfQQnnnmGfzHf/wHjhw5gkceeQSHDx/GmjVrZB7OxbEZdX1iu6vegRC8mt+4j2cCEhUt5h+Yf0SSKXHpHADYtm0bOjo6sGzZMthsNqxcuRLPPPNM8v5oNIqenh4EAoHkbQ888ABCoRA2bNiA/v5+LFiwALt27cKcOXNkHML4pG13FYzrODMYglfjNcaISgHzj/lHJJMyxWJdXR1+9KMf5by/paUFQogRt2/atCnjOmPKSjsbcCBmhiXMHRzYWRMVNeYf849IJiWmoQmZ213FdfQOhuDVGJZEVAKYf0RSsVhURdoC72hMx5nBcGqrK4YlERUz5h+RVCwWVZGchokjYq7ZqeUCbyIqBcw/IqlYLKpi2HZXxpqdRGfNBd5EVMSYf0RSsVhURVpYBiNx/M0fRg3X7BBRKWD+EUnFYlEVaZeO6Hz3fdhEDNVa0LiPYUlExYz5RyQVi0VVJDvrOM4MhlGN1PXUUFYjaVBERAXA/COSisWiKtLOBgSAuVUR4/ayGsCuzOUyiYjGjvlHJBWLRVWkXZQWAB5Yau4pyykYIip2zD8iqVgsqiJtGub2a5rR2mS+dAxLIip2zD8iqVgsqsLsrOfWu/HUqmuA4AfG7QxLIip2zD8iqbjYQxU2o7O261FA0xiWRFQ6mH9EUvGdRVWY0zCIR43PwX7jMy9IS0TFjvlHJBWLRVWY0zCIm2cBsrMmolLB/COSisWiKhiWRFSqmH9EUrFYVMWIaRiGJRGVCOYfkVQsFlWR6Kx1MywD5pqdcq7ZIaIix/wjkorFoirSp2GEYGdNRKWD+UckFYtFVaRvaaXHgKDP+JphSUTFjvlHJBWLRVUkOmsAiAaA8IDxNcOSiIod849IKhaLqkgPy6G/pb4u8xZ8KEREBcX8I5KKxaIqbGnTMEO9xmd3Teb0DBFRMWL+EUnFYlEVmpbqrv1mWPJMQCIqBcw/IqlYLKokEZZDZ4zPDEsiKhXMPyJpWCyqJDEVk+ysJ8kbCxFRITH/iKRhsaiS4dMwHnbWRFQimH9E0rBYVElyGoadNRGVGOYfkTQsFlWS2B/Vn1izw2uMEVGJYP4RScNiUSUjzgZkZ01EJYL5RyQNi0WVJMIyuXsB1+wQUYlg/hFJw2JRJYlpmAR21kRUKph/RNKwWFTJiLBkZ01EJYL5RyQNi0WVpO+PCrCzJqLSwfwjkobFokqGd9Zcs0NEpYL5RyQNi0WVpHfWrirA4cr9WCKiYsL8I5KGxaJK0jtrXmOMiEoJ849IGhaLKknvrLleh4hKCfOPSBoWiypJD0uu1yGiUsL8I5KGxaJKbI7U1+ysiaiUMP+IpGGxqJKMaRh21kRUQph/RNKwWFQJp2GIqFQx/4ikYbGokoyzARmWRFRCmH9E0rBYVAmnYYioVDH/iKRRpljs7+9He3s7qqur4fV6sWbNGgwNDY36nN7eXnzyk59EU1MTKioqcN111+E///M/CzTiCcBLRxCVJOYfmH9EEilTLLa3t+PgwYPYtWsXduzYgb1792LdunWjPufee+9FT08PfvGLX+DNN9/EnXfeiU984hM4cOBAgUadZ+nTMFyzQ1QymH9g/hFJpESxeOjQIezcuRPf+9730NraiqVLl+LZZ5/F9u3bcerUqZzP+6//+i984QtfwOLFizF79mx85StfgdfrRXd3dwFHn0cZa3bYWROVAuafiflHJI0SxWJnZye8Xi8WLVqUvK2trQ02mw1dXV05n3f99dfjJz/5Cfr7+6HrOrZv345QKISbb74553PC4TAGBwczPiyDa3aISg7zz8T8I5JGiWKxt7cXDQ0NGbc5HA7U1dWht7c35/N++tOfIhqNYtKkSXC73bjvvvvwwgsv4JJLLsn5nM2bN6Ompib5MX369Lwdx7glOmtnOeD0yB0LERUE88/E/COSRmqxuGnTJmiaNurH4cOHL/r7P/LII/D5fPjd736H119/HV/60pfwiU98Am+++WbO5zz88MMYGBhIfpw4ceKi//68S3TWXK9DpDzm3xgx/4ikcZz/IRNn48aNWL169aiPmT17NpqamtDX15dxeywWQ39/P5qamrI+7+jRo/j2t7+Nt956C1dccQUAYMGCBfjDH/6ALVu24Lnnnsv6PLfbDbfbPfaDKYREWHIKhkh5zL8xYv4RSSO1WKyvr0d9ff15H7dkyRL4fD50d3dj4cKFAIDdu3dD13W0trZmfU4gEAAA2GyZb57a7Xbouj7OkUvinWF8rp8ndxxENG7MvzFi/hFJo8Saxfnz52P58uVYu3Yt9u3bh9deew0dHR2466670NzcDAA4efIk5s2bh3379gEA5s2bh0suuQT33Xcf9u3bh6NHj+KJJ57Arl27cMcdd0g8mnGY3grc9wfgH5+UPRIiKhDmn4n5RySNEsUiAGzbtg3z5s3DsmXLcOutt2Lp0qX47ne/m7w/Go2ip6cn2VE7nU78+te/Rn19PW677TZcffXV+OEPf4gf/OAHuPXWW2UdxvhoGjDlasBdKXskRFRAzD8w/4gk0oQQQvYgrGxwcBA1NTUYGBhAdXW17OEQ0Tjw53ls+O9FVDzG8/OszDuLRERERFR4LBaJiIiIKCcWi0RERESUE4tFIiIiIsqJxSIRERER5cRikYiIiIhyYrFIRERERDmxWCQiIiKinFgsEhEREVFOLBaJiIiIKCcWi0RERESUE4tFIiIiIsqJxSIRERER5cRikYiIiIhycsgegNUJIQAAg4ODkkdCROOV+DlO/FzT6Jh/RMVjPPnHYvE8/H4/AGD69OmSR0JE+eL3+1FTUyN7GJbH/CMqPheTf5pgiz0qXddx6tQpVFVVQdO0UR87ODiI6dOn48SJE6iuri7QCAuHx6c2Hp/RUfv9fjQ3N8Nm4yqc82H+pfD41MbjG1/+8Z3F87DZbJg2bdqYnlNdXV2U/xkTeHxqK/Xj4zuKF475NxKPT22lfnwXm39srYmIiIgoJxaLRERERJQTi8U8crvdePTRR+F2u2UPZULw+NTG46OJVOz//jw+tfH4xocnuBARERFRTnxnkYiIiIhyYrFIRERERDmxWCQiIiKinFgsEhEREVFOLBbzaMuWLWhpaUFZWRlaW1uxb98+2UMas82bN+NDH/oQqqqq0NDQgDvuuAM9PT0Zj7n55puhaVrGx2c/+1lJIx67r33tayPGP2/evOT9oVAI69evx6RJk1BZWYmVK1fizJkzEkc8Ni0tLSOOT9M0rF+/HoB6r9/evXtx2223obm5GZqm4cUXX8y4XwiBr371q5gyZQo8Hg/a2trwzjvvZDymv78f7e3tqK6uhtfrxZo1azA0NFTAoyh+zD81MP/Uev2skn8sFvPkJz/5Cb70pS/h0UcfxZ/+9CcsWLAAt9xyC/r6+mQPbUxeeeUVrF+/Hn/84x+xa9cuRKNRfOQjH8G5c+cyHrd27VqcPn06+fHNb35T0ogvzhVXXJEx/ldffTV534YNG/DLX/4SP/vZz/DKK6/g1KlTuPPOOyWOdmz279+fcWy7du0CAHz84x9PPkal1+/cuXNYsGABtmzZkvX+b37zm3jmmWfw3HPPoaurCxUVFbjlllsQCoWSj2lvb8fBgwexa9cu7NixA3v37sW6desKdQhFj/ln3Z+fbJh/6rx+lsk/QXmxePFisX79+uSf4/G4aG5uFps3b5Y4qvHr6+sTAMQrr7ySvO3DH/6wuP/+++UNapweffRRsWDBgqz3+Xw+4XQ6xc9+9rPkbYcOHRIARGdnZ4FGmF/333+/mDNnjtB1XQih9usHQLzwwgvJP+u6LpqamsQ///M/J2/z+XzC7XaLH//4x0IIId5++20BQOzfvz/5mN/85jdC0zRx8uTJgo29mDH/1MH8U/f1k5l/fGcxDyKRCLq7u9HW1pa8zWazoa2tDZ2dnRJHNn4DAwMAgLq6uozbt23bhsmTJ+PKK6/Eww8/jEAgIGN4F+2dd95Bc3MzZs+ejfb2dhw/fhwA0N3djWg0mvFazps3DzNmzFDytYxEIvj3f/93fPrTn4amacnbVX/9Eo4dO4be3t6M16umpgatra3J16uzsxNerxeLFi1KPqatrQ02mw1dXV0FH3OxYf6p9/PD/FP79UsoZP458jfs0nX27FnE43E0NjZm3N7Y2IjDhw9LGtX46bqOBx54ADfccAOuvPLK5O333HMPZs6ciebmZvz5z3/Gl7/8ZfT09ODnP/+5xNFeuNbWVjz//POYO3cuTp8+jcceeww33ngj3nrrLfT29sLlcsHr9WY8p7GxEb29vXIGPA4vvvgifD4fVq9enbxN9dcvXeI1yfazl7ivt7cXDQ0NGfc7HA7U1dUp+ZpaDfNPrZ8f5p/ar1+6QuYfi0XKaf369Xjrrbcy1rMAyFjrcNVVV2HKlClYtmwZjh49ijlz5hR6mGO2YsWK5NdXX301WltbMXPmTPz0pz+Fx+OROLL8+9d//VesWLECzc3NydtUf/2ICoH5pz7mX/5wGjoPJk+eDLvdPuKMsTNnzqCpqUnSqMano6MDO3bswMsvv4xp06aN+tjW1lYAwJEjRwoxtLzzer247LLLcOTIETQ1NSESicDn82U8RsXX8r333sPvfvc7fOYznxn1cSq/fonXZLSfvaamphEnWsRiMfT39yv3mloR80/dnx+A+afy61fI/GOxmAculwsLFy7E73//++Rtuq7j97//PZYsWSJxZGMnhEBHRwdeeOEF7N69G7NmzTrvc9544w0AwJQpUyZ4dBNjaGgIR48exZQpU7Bw4UI4nc6M17KnpwfHjx9X7rXcunUrGhoa8A//8A+jPk7l12/WrFloamrKeL0GBwfR1dWVfL2WLFkCn8+H7u7u5GN2794NXdeTvyjo4jH/3gCg5s8PwPxT+fUraP6N9+wcMmzfvl243W7x/PPPi7ffflusW7dOeL1e0dvbK3toY/K5z31O1NTUiD179ojTp08nPwKBgBBCiCNHjojHH39cvP766+LYsWPipZdeErNnzxY33XST5JFfuI0bN4o9e/aIY8eOiddee020tbWJyZMni76+PiGEEJ/97GfFjBkzxO7du8Xrr78ulixZIpYsWSJ51GMTj8fFjBkzxJe//OWM21V8/fx+vzhw4IA4cOCAACC+9a1viQMHDoj33ntPCCHEN77xDeH1esVLL70k/vznP4vbb79dzJo1SwSDweT3WL58ubj22mtFV1eXePXVV8Wll14q7r77blmHVHSYf9b9+RmO+afW62eV/GOxmEfPPvusmDFjhnC5XGLx4sXij3/8o+whjRmArB9bt24VQghx/PhxcdNNN4m6ujrhdrvFJZdcIh566CExMDAgd+BjsGrVKjFlyhThcrnE1KlTxapVq8SRI0eS9weDQfH5z39e1NbWivLycvHRj35UnD59WuKIx+63v/2tACB6enoyblfx9Xv55Zez/p/8p3/6JyGEcfmIRx55RDQ2Ngq32y2WLVs24rjff/99cffdd4vKykpRXV0tPvWpTwm/3y/haIoX808NzD+1Xj+r5J8mhBBjfOeTiIiIiEoE1ywSERERUU4sFomIiIgoJxaLRERERJQTi0UiIiIiyonFIhERERHlxGKRiIiIiHJisUhEREREObFYJCIiIqKcWCxS0duzZw80TYPP55M9FCKigmL+UT5wBxcqOjfffDOuueYaPPXUUwCASCSC/v5+NDY2QtM0uYMjIppAzD+aCA7ZAyCaaC6XC01NTbKHQURUcMw/ygdOQ1NRWb16NV555RU8/fTT0DQNmqbh+eefz5iGef755+H1erFjxw7MnTsX5eXl+NjHPoZAIIAf/OAHaGlpQW1tLb74xS8iHo8nv3c4HMaDDz6IqVOnoqKiAq2trdizZ4+cAyUiGob5RxOF7yxSUXn66afxl7/8BVdeeSUef/xxAMDBgwdHPC4QCOCZZ57B9u3b4ff7ceedd+KjH/0ovF4vfv3rX+Pdd9/FypUrccMNN2DVqlUAgI6ODrz99tvYvn07mpub8cILL2D58uV48803cemllxb0OImIhmP+0URhsUhFpaamBi6XC+Xl5cmpl8OHD494XDQaxXe+8x3MmTMHAPCxj30M//Zv/4YzZ86gsrISl19+Of7u7/4OL7/8MlatWoXjx49j69atOH78OJqbmwEADz74IHbu3ImtW7fi61//euEOkogoC+YfTRQWi1SSysvLk0EJAI2NjWhpaUFlZWXGbX19fQCAN998E/F4HJdddlnG9wmHw5g0aVJhBk1ElAfMPxorFotUkpxOZ8afNU3Lepuu6wCAoaEh2O12dHd3w263ZzwuPWCJiKyO+UdjxWKRio7L5cpYmJ0P1157LeLxOPr6+nDjjTfm9XsTEeUL848mAs+GpqLT0tKCrq4u/PWvf8XZs2eT3fF4XHbZZWhvb8e9996Ln//85zh27Bj27duHzZs341e/+lUeRk1ENH7MP5oILBap6Dz44IOw2+24/PLLUV9fj+PHj+fl+27duhX33nsvNm7ciLlz5+KOO+7A/v37MWPGjLx8fyKi8WL+0UTgDi5ERERElBPfWSQiIiKinFgsEhEREVFOLBaJiIiIKCcWi0RERESUE4tFIiIiIsqJxSIRERER5cRikYiIiIhyYrFIRERERDmxWCQiIiKinFgsEhEREVFOLBaJiIiIKKf/D8EvtqGUnedEAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 640x480 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "#system parameters\n",
    "g = 9.8 #m/s^2\n",
    "l = 1 #m\n",
    "dt=0.1 #second\n",
    "b = 0.2 #damping factor\n",
    "m = 1 #kg mass of bob\n",
    "\n",
    "#number of time steps for simulation\n",
    "nstep = 1000\n",
    "\n",
    "#process noise\n",
    "Cov_P =np.array([[0.01,0],\n",
    "                 [0,0.01]])\n",
    "P = rng.multivariate_normal([0,0],Cov_P,nstep)\n",
    "\n",
    "#measurement noise\n",
    "Cov_M = np.array([[0.01]])\n",
    "M = rng.multivariate_normal([0],Cov_M,nstep)\n",
    "\n",
    "#process matrix\n",
    "A = np.array([[1,dt],[-g*dt/l, -b/m+1]])\n",
    "#measurement matrix\n",
    "C = np.array([1,0]).reshape((1,2))\n",
    "\n",
    "#state matrices\n",
    "X = np.zeros((nstep,2,1))\n",
    "#mean\n",
    "mu = np.zeros((nstep,2,1))\n",
    "#covariance\n",
    "cov = np.zeros((nstep,2,2))\n",
    "\n",
    "#predicted mean and covariance\n",
    "mu_p = np.zeros((nstep,2,1))\n",
    "cov_p = np.zeros((nstep,2,2))\n",
    "\n",
    "#initial mean and covariance\n",
    "mu[0] = [[0.01],[0]]\n",
    "cov[0] = np.array([[0.05,0],[0,0.05]])\n",
    "cov_p[0] = cov[0]\n",
    "\n",
    "X[0] = mu[0]\n",
    "mu_p[0]=mu[0]\n",
    "\n",
    "#other arrays\n",
    "z = np.zeros(nstep)  #measurement\n",
    "K = np.zeros((nstep,2,1)) #kalman gain\n",
    "\n",
    "print(f\"{0}: ang={X[0,0]},vel={X[0,1]}, predang={mu_p[0,0]}, corr_ang={mu[0,0]}, K={K[0]}\")\n",
    "\n",
    "for i in range(1,nstep):\n",
    "  #simulation of pendulum\n",
    "  #dynamics - use exact -non-linear equation\n",
    "  X[i,0] = X[i-1,0] + X[i-1,1]*dt + P[i,0] #angle update\n",
    "  X[i,1] = -g/l*np.sin(X[i-1,0])*dt + (-b/m+1)* X[i-1,1] + P[i,1] #angular velocity update\n",
    "\n",
    "  #measurement\n",
    "  z[i] = C @ X[i] + M[i]\n",
    "\n",
    "  #predict step\n",
    "  cov_p[i] = #<your code here>\n",
    "  mu_p[i] = #<your code here>\n",
    "\n",
    "  #print(f\"i={i}: {cov_p[i,0,0]}, {cov_p[i] @ C.T}, {C @ cov_p[i] @ C.T },{cov_p[i] @ C.T@np.linalg.inv(C @ cov_p[i] @ C.T + Cov_M)} \")\n",
    "\n",
    "  #update step\n",
    "  K[i] =  #your code here\n",
    "  mu[i] =  #your code here\n",
    "  cov[i] = #your code here\n",
    "\n",
    "  #print(K[i],K[i]@C)\n",
    "\n",
    "  #print(f\"{i}: X={X[i]}, pred={mu_p[i]}, corr={mu[i]}, z={z[i]}, K={K[i]}\")\n",
    "\n",
    "\n",
    "fig,ax = plt.subplots(1,2, layout='constrained')\n",
    "ax[0].plot(mu[-100:,0],label=\"est mean ang\")\n",
    "ax[0].plot(X[-100:,0],label=\"actual ang\")\n",
    "ax[0].set_xlabel(\"time\")\n",
    "ax[0].set_ylabel(\"radian\")\n",
    "ax[0].set_title(\"ang (last 100 timesteps)\")\n",
    "ax[0].legend()\n",
    "\n",
    "ax[1].plot(mu[-100:,0],label=\"est mean angvel\")\n",
    "ax[1].plot(X[-100:,0],label=\"actual angvel\")\n",
    "ax[1].set_xlabel(\"time\")\n",
    "ax[1].set_ylabel(\"radian/s\")\n",
    "ax[1].set_title(\"angvel (last 100 timesteps)\")\n",
    "ax[1].legend()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "id": "MUskeSbxzBcW"
   },
   "source": [
    "---\n",
    "**Online Reference for Kalman Filters**\n",
    "\n",
    "https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python\n",
    "\n"
   ]
  }
 ],
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