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   "source": [
    "<h2>Multivariate least squares</b>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's do a simple multivariate fit: a fit of a plane to some data. \n",
    "Write a function for the model:\n",
    "$$ z = a + bx + cy$$\n",
    "and for the derivatives with respect to the parameters.\n",
    "Then simulate a data set\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def model(x,y,par) :\n",
    "    \"\"\" Function of two variables, linear in parameters\n",
    "    \"\"\"\n",
    "    return     # model here\n",
    "\n",
    "def deriv(x,y) :\n",
    "    \"\"\" Derivative with respect to each parameter\n",
    "    \"\"\"\n",
    "    return      # return derivatives with respect to each parameter\n",
    "    \n",
    "# simulate some data    \n",
    "x=         # x independent variable\n",
    "y=         # y independent variable\n",
    "truepar=   # parameter values\n",
    "sig=       # uncertainties on data values\n",
    "z=model(x,y,truepar)+np.random.normal(0.,sig,size=len(x))\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the simple form for least squares:\n",
    "$${A^T\\over \\sigma} {A\\over \\sigma} par = {A^T\\over \\sigma} {y\\over \\sigma}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# design matrix\n",
    "design=\n",
    "\n",
    "# solve for best fitting parameters in least squares sense\n",
    "data=\n",
    "ATA=\n",
    "par=\n",
    "print('par: ',par)\n",
    "#use inverse matrix to get uncertainties\n",
    "inv=np.linalg.inv(ATA)\n",
    "print(np.dot(inv,np.dot(design.T,data)))\n",
    "print(inv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are some possibilities for visualizing the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# for plotting in 3D\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "plt3d = plt.figure().gca(projection='3d')\n",
    "ax = plt.gca()\n",
    "ax.scatter(x, y, z)\n",
    "\n",
    "# plot the solution\n",
    "xx, yy = np.meshgrid(np.arange(0,1,0.1),np.arange(0,1,0.1))\n",
    "zfit= model(xx,yy,par)\n",
    "plt3d.plot_surface(xx, yy, zfit, alpha=0.8)\n",
    "plt3d.plot_surface(xx,yy,model(xx,yy,truepar),color='g',alpha=0.8)\n",
    "\n",
    "# 2D representation using color as dependent variable\n",
    "plt.figure()\n",
    "plt.scatter(x,y,c=z)\n",
    "plt.figure()\n",
    "xx,yy=np.mgrid[0:1:0.1,0:1:0.1]\n",
    "plt.imshow(model(xx,yy,par),origin='bottom')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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