{
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
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   "source": [
    "import numpy as np\n",
    "\n",
    "import plotly.graph_objects as go\n",
    "import plotly.io as pio\n",
    "\n",
    "# Our 2-dimensional distribution will be over variables X and Y\n",
    "N = 60\n",
    "x = np.linspace(-3.5, 3.5, N)\n",
    "y = np.linspace(-3.5, 3.5, N)\n",
    "X, Y = np.meshgrid(x, y)\n",
    "\n",
    "\n",
    "# Mean vector and covariance matrix\n",
    "mu = np.array([0., 0.])\n",
    "Sigma = np.array([[ 2.5 , 1], [1,  2.5]])\n",
    "\n",
    "# Pack X and Y into a single 3-dimensional array\n",
    "pos = np.empty(X.shape + (2,))\n",
    "pos[:, :, 0] = X\n",
    "pos[:, :, 1] = Y\n",
    "\n",
    "def multivariate_gaussian(pos, mu, Sigma):\n",
    "    \"\"\"Return the multivariate Gaussian distribution on array pos.\n",
    "\n",
    "    pos is an array constructed by packing the meshed arrays of variables\n",
    "    x_1, x_2, x_3, ..., x_k into its _last_ dimension.\n",
    "\n",
    "    \"\"\"\n",
    "\n",
    "    n = mu.shape[0]\n",
    "    Sigma_det = np.linalg.det(Sigma)\n",
    "    Sigma_inv = np.linalg.inv(Sigma)\n",
    "    N = np.sqrt((2*np.pi)**n * Sigma_det)\n",
    "    # This einsum call calculates (x-mu)T.Sigma-1.(x-mu) in a vectorized\n",
    "    # way across all the input variables.\n",
    "    fac = np.einsum('...k,kl,...l->...', pos-mu, Sigma_inv, pos-mu)\n",
    "\n",
    "    return np.exp(-fac / 2) / N\n",
    "\n",
    "# The distribution on the variables X, Y packed into pos.\n",
    "Z = multivariate_gaussian(pos, mu, Sigma)\n",
    "\n",
    "fig = go.Figure(data=go.Surface(x=X, y=Y, z=Z))\n",
    "fig.update_traces(contours_z=dict(show=True, usecolormap=True, highlightcolor=\"limegreen\", project_z=True),\n",
    "                  showscale=False, opacity=0.6)\n",
    "fig.update_layout(scene=dict(\n",
    "                    xaxis=dict(visible=False),\n",
    "                    yaxis=dict(visible=False),\n",
    "                    zaxis=dict(visible=False, range=[0, 0.07]),\n",
    "                    domain=dict(y=[0.1, 1])),\n",
    "                  height=700,\n",
    "                  margin=dict(r=0, l=0, b=0, t=0, pad=0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "pio.write_html(fig,\n",
    "               file='../_includes/figures/figure.html',\n",
    "               full_html=False,\n",
    "               # include_mathjax='cdn',\n",
    "               include_plotlyjs='cdn')"
   ]
  }
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