Calculate the integral of a 2d Gaussian using Latin Hypercube sampling

dev_latin_hypercube_integration.ipynb

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   "cell_type": "code",
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   "source": [
    "%matplotlib inline"
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "481b8e94",
   "metadata": {},
   "outputs": [],
   "source": [
    "from jax.scipy.stats import multivariate_normal\n",
    "mu = np.zeros(2)\n",
    "cov = np.eye(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "410bab62",
   "metadata": {},
   "source": [
    "### Create a mesh grid spanning -5, 5 in each dimension"
   ]
  },
  {
   "cell_type": "code",
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(10000, 2)\n"
     ]
    }
   ],
   "source": [
    "npts_per_dim = 100\n",
    "xh, yh = 5, 5\n",
    "\n",
    "xlo, xhi, nx = -xh, xh, npts_per_dim\n",
    "ylo, yhi, ny = -yh, yh, npts_per_dim\n",
    "dx = (xhi-xlo)/nx\n",
    "dy = (yhi-ylo)/ny\n",
    "\n",
    "x = np.linspace(xlo, xhi, npts_per_dim+1)\n",
    "y = np.linspace(ylo, yhi, npts_per_dim+1)\n",
    "x, y = np.array(np.meshgrid(x[:-1], y[:-1]))\n",
    "x, y = x.flatten(), y.flatten()\n",
    "X_grid = np.vstack((x, y)).T\n",
    "print(X_grid.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd4b515b",
   "metadata": {},
   "source": [
    "### Compute the grid sampling weight and do the Riemann integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "24c8587c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.99999875\n"
     ]
    }
   ],
   "source": [
    "npts_grid_sampling = X_grid.shape[0]\n",
    "AREA = (xhi-xlo)*(yhi-ylo)\n",
    "norm_factor_grid = AREA / npts_grid_sampling\n",
    "\n",
    "gaussian_integral_grid = multivariate_normal.pdf(X_grid, mu, cov).sum()*norm_factor_grid\n",
    "print(gaussian_integral_grid)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "290c61e1",
   "metadata": {},
   "source": [
    "### Create a latin hypercube"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "fab05caa",
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.stats.qmc import LatinHypercube\n",
    "LH = LatinHypercube(X_grid.shape[1])\n",
    "npts_lh_sampling = npts_grid_sampling\n",
    "U_LH = LH.random(npts_lh_sampling)\n",
    "X_LH = np.zeros_like(U_LH)\n",
    "X_LH[:, 0] = (U_LH[:, 0] - 0.5)*(xhi-xlo)\n",
    "X_LH[:, 1] = (U_LH[:, 1] - 0.5)*(yhi-ylo)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0ae32af",
   "metadata": {},
   "source": [
    "### Compute the LH sampling weight and do the Riemann integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "ebfca950",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0012085\n"
     ]
    }
   ],
   "source": [
    "norm_factor_lhs = AREA / npts_lh_sampling\n",
    "\n",
    "gaussian_integral_lhs = multivariate_normal.pdf(X_LH, mu, cov).sum()*norm_factor_lhs\n",
    "print(gaussian_integral_lhs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "030b8d8d",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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