A sample proof to show both general and deformed formula of Covariance give same result

covariance deformation cross check.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The general covariance formula:\n",
    "\n",
    "$$\n",
    "\\mathrm{Cov}(X,Y) = \\dfrac{\\sum\\limits_{i=1}^N(x_i - \\overline{x})(y_i - \\overline{y})}{N}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let X=[1, 2, 3, 4, 5] and Y=[2, 1, 4, 3, 0]"
   ]
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     "end_time": "2018-11-03T03:40:31.346008Z",
     "start_time": "2018-11-03T03:40:31.332002Z"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.4\n"
     ]
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   ],
   "source": [
    "X, Y = [1,2,3,4,5], [2,1,4,3,0]\n",
    "N = len(X)  # or len(Y) also should do as both should be equal\n",
    "Xb, Yb = sum(X)/len(X), sum(Y)/len(Y)\n",
    " \n",
    " # first formula\n",
    "num = 0\n",
    "for i in range(N):\n",
    "    num += (X[i] - Xb)*(Y[i] - Yb)\n",
    "\n",
    "cov1 = num/N\n",
    "print(cov1)"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Test B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The deformed formula\n",
    "\n",
    "$$\n",
    "\\mathrm{Cov}(X,Y) = \\dfrac{\\sum\\limits_{i=1}^N\\sum\\limits_{j=i+1}^N(x_i-x_j)(y_i - y_j)}{N^2}\n",
    "$$"
   ]
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     "end_time": "2018-11-03T04:11:30.734958Z",
     "start_time": "2018-11-03T04:11:30.725961Z"
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-0.4\n"
     ]
    }
   ],
   "source": [
    "# second formula\n",
    "num = 0\n",
    "for i in range(0,N):\n",
    "    for j in range(i+1,N):\n",
    "        num += (X[i] - X[j])*(Y[i] - Y[j])\n",
    "\n",
    "cov2 = num/(N**2)\n",
    "print(cov2)"
   ]
  }
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