deriv.ipynb

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   "cell_type": "code",
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   "source": [
    "from sympy import *\n",
    "N, kc, kt, sc2, st2, sct2 = symbols('N kc kt sc2 st2 sct2')"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {
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   "source": [
    "# Set the keff estimators and population covariance matrices\n",
    "k = Matrix([[kc], [kt]])\n",
    "Sigma = Matrix([[sc2, sct2], [sct2, st2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "# Set the transformations\n",
    "A = Matrix([[1, 0], [1, -1]])\n",
    "z = A * k\n",
    "d = Matrix(z[1:])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
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   "outputs": [],
   "source": [
    "Sigmaz = A * Sigma * A.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Sz22 = (N - 1) * Sigmaz[1, 1]\n",
    "Sz12 = (N - 1) * Sigmaz[0, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "kc - (kc - kt)*(sc2 - sct2)/(sc2 - 2*sct2 + st2)\n"
     ]
    }
   ],
   "source": [
    "# Get khat\n",
    "khat = kc - Sz12 / Sz22 * d[0]\n",
    "print(khat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix([[(sc2*st2 - sct2**2)/(sc2 - 2*sct2 + st2)]])\n",
      "sc2 - (sc2 - sct2)**2/(sc2 - 2*sct2 + st2)\n"
     ]
    }
   ],
   "source": [
    "# Get the combined estimator first with the asymptotic variance of khat\n",
    "e = Matrix([[1], [1]])\n",
    "eT = e.T\n",
    "sopt2 = (eT * Sigma**-1 * e)\n",
    "sopt2 = sopt2**-1\n",
    "print(sopt2)\n",
    "a = Sigmaz[0,0] - Sigmaz[0,1] * Sigmaz[1,0] / Sigmaz[1,1]\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(N*(kc - kt)**2 + (N - 1)*(sc2 - 2*sct2 + st2))*(sc2*st2 - sct2**2)/(N*(N - 1)*(sc2 - 2*sct2 + st2)**2)\n"
     ]
    }
   ],
   "source": [
    "# Now use this asymptotic variance to get khat\n",
    "sk2 = sopt2 * (1 / N + d[0] * Sz22**-1 * d[0])\n",
    "print(sk2[0,0].simplify())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "# Compare to my hand calculations\n",
    "g = sc2 + st2 - 2 * sct2\n",
    "hand_calcs = (N*(kc - kt)**2 + (N - 1)*g)*(sc2*st2 - sct2**2)/(N*(N - 1)*(g)**2)\n",
    "print(sk2[0,0].simplify() / hand_calcs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Good! we're all done!"
   ]
  }
 ],
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