09. Interpolation.ipynb

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    "## 9. Interpolation\n",
    "\n",
    "[Data Science Playlist on YouTube](https://www.youtube.com/watch?v=XstK9M4jzBY&list=PLLBUgWXdTBDg1Qgmwt4jKtVn9BWh5-zgy)\n",
    "[![Python Data Science](https://apmonitor.com/che263/uploads/Begin_Python/DataScience09.png)](https://www.youtube.com/watch?v=XstK9M4jzBY&list=PLLBUgWXdTBDg1Qgmwt4jKtVn9BWh5-zgy \"Python Data Science\")\n",
    "\n",
    "Interpolation constructs new prediction points from a discrete set of known data points. There are many types of interpolation such as nearest neighbor (piecewise constant), linear, polynomial, [cubic spline](https://apmonitor.com/wiki/index.php/Main/ObjectCspline), and [basis spline](https://apmonitor.com/wiki/index.php/Main/ObjectBspline). In interpolation, the data provide the shape of the approximate function, with piecewise or higher-order polynomial equations to exactly match the data points at those given discrete locations.\n",
    "\n",
    "![charge](https://apmonitor.com/che263/uploads/Begin_Python/charge.png)\n",
    "\n",
    "### 1D Interpolation\n",
    "\n",
    "The `scipy.interpolate` package has a function for one (`interp1d`) dimensional interpolation where `kind` is `nearest`, `previous`, `next`, `zero`, `linear`, `quadratic`, `cubic`, or a number as `0`-`3`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "x = np.array([0,1,2,3,4,5])\n",
    "y = np.array([0.1,0.25,0.3,0.5,1.0,0.9])\n",
    "\n",
    "# 1D interpolation\n",
    "from scipy.interpolate import interp1d\n",
    "f = interp1d(x,y,kind='cubic')\n",
    "print(f(4.5)) # predict at 4.5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![analyze](https://apmonitor.com/che263/uploads/Begin_Python/analyze.png)\n",
    "\n",
    "### Plot 1D Interpolation\n",
    "\n",
    "Several of the 1d interpolations are displayed graphically. Don't forget to add extra points such as `xp=np.linspace(0,5,100)` when plotting the interpolation or everything will look like a linear interpolation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xp = np.linspace(0,5,100)\n",
    "y1 = interp1d(x,y,kind='nearest')\n",
    "y2 = interp1d(x,y,kind=1)\n",
    "y3 = interp1d(x,y,kind=3)\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.plot(xp,y1(xp),'g-',label='Nearest')\n",
    "plt.plot(xp,y2(xp),'r--',label='Linear Interp')\n",
    "plt.plot(xp,y3(xp),'k:',label='Cubic Spline')\n",
    "plt.plot(x,y,'bo',label='Data')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One caution with retrieving values from an interpolation is that there will be an error such as `ValueError: A value in x_new is above the interpolation range.` if you request a value outside the interpolation region such as `f(5.5)`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![expert](https://apmonitor.com/che263/uploads/Begin_Python/expert.png)\n",
    "\n",
    "### Interpolation Activity\n",
    "\n",
    "Create a linear interpolation with the following data.\n",
    "\n",
    "```python\n",
    "xr = [0.0,1.0,2.0,5.0]\n",
    "yr = [0.2,0.4,1.05,1.7]\n",
    "```\n",
    "\n",
    "Use the interpolation to predict the value at `xr=3.5`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![gekko](https://apmonitor.com/che263/uploads/Begin_Python/gekko.png)\n",
    "\n",
    "The type of spline is important for certain applications such as optimization. Gradient-based solvers perform better with continuous functions that also have continuous gradients. For this reason, a quadratic or cubic spline may be preferable to a linear interpolation. The following is an example of optimization with a cubic spline in Gekko. The original function is $\\frac{1}{1+25 \\, x^2}$ with the objective to find the maximum using only function values at `xr=[-1.0 -0.8 -0.5 -0.25 -0.1 0.1 0.2 0.5]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from gekko import GEKKO\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# generate data\n",
    "def f(x):\n",
    "    return 1.0/(1.0+25.0*x**2)\n",
    "xr = np.array([-1.0,-0.8,-0.5,-0.25,-0.1,0.1,0.2,0.5])\n",
    "yr = f(xr)\n",
    "\n",
    "# create cubic spline and maximize\n",
    "c = GEKKO(remote=False)\n",
    "xg = c.Var(); yg = c.Var()\n",
    "c.cspline(xg,yg,xr,yr,True)\n",
    "c.Maximize(yg)\n",
    "c.solve(disp=False)\n",
    "\n",
    "xp = np.linspace(-1,0.5,100)\n",
    "plt.plot(xp,f(xp),'b-',label=r'$f(x) = \\frac{1}{1+25x^2}$')\n",
    "plt.plot(xr,yr,'ro',label='Data')\n",
    "plt.plot(xg.value[0],yg.value[0],'kx',label='C-spline Maximum')\n",
    "plt.legend(loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![expert](https://apmonitor.com/che263/uploads/Begin_Python/expert.png)\n",
    "\n",
    "### Cubic Spline Activity\n",
    "\n",
    "Create a cubic spline with the following data.\n",
    "\n",
    "```python\n",
    "xr = [0.0,1.0,2.0,3.0,4.0,5.0]\n",
    "yr = [46.0,6.6,0.13,0.026,3.84,33.0]\n",
    "```\n",
    "\n",
    "Construct a cubic spline and find the minimum of the function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![idea](https://apmonitor.com/che263/uploads/Begin_Python/idea.png)\n",
    "\n",
    "### 2D Interpolation\n",
    "\n",
    "There is also a function for two dimensional (`interp2d`) interpolation where `kind` is `linear` or `cubic`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 2D interpolation\n",
    "from scipy.interpolate import interp2d\n",
    "x2 = np.arange(-1.5, 1.01, 0.5)\n",
    "y2 = np.arange(-1.5, 1.01, 0.5)\n",
    "xx, yy = np.meshgrid(x2, y2)\n",
    "zz = np.sin(xx**2+yy**2)\n",
    "f2 = interp2d(x2, y2, zz, kind='cubic')\n",
    "print(f2(4.5,1)) # predict z at (x=4.5, y=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![analyze](https://apmonitor.com/che263/uploads/Begin_Python/analyze.png)\n",
    "\n",
    "### Plot 2D Interpolation\n",
    "\n",
    "A 3D plot shows the data points in `red` and the cubic interpolation in `blue`. The plot includes a 3D scatter plot for the data and a surface plot for the cubic spline."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "fig = plt.figure(figsize=(8,5))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "ax.scatter(xx,yy,zz,color='red',label='Data')\n",
    "xe = np.arange(-1.5, 1.01, 0.1)\n",
    "ye = np.arange(-1.5, 1.01, 0.1)\n",
    "xxe, yye = np.meshgrid(xe, ye)\n",
    "fe = np.empty_like(xxe)\n",
    "for i in range(np.size(fe,0)):\n",
    "    for j in range(np.size(fe,1)):\n",
    "        fe[i,j] = f2(xxe[i,j],yye[i,j])\n",
    "ax.plot_surface(xxe,yye,fe,color='blue',alpha=0.7)\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![expert](https://apmonitor.com/che263/uploads/Begin_Python/expert.png)\n",
    "\n",
    "### TCLab Activity\n",
    "\n",
    "![temperature](https://apmonitor.com/che263/uploads/Begin_Python/temperature.png)\n",
    "\n",
    "### Record Temperatures\n",
    "\n",
    "Turn on heater 1 to 100% and record $T_1$ and $T_2$ every 10 seconds for 3 minutes. The data should include a total of 19 data points for each temperature sensor and the recording time, starting at zero.\n",
    "\n",
    "![connections](https://apmonitor.com/che263/uploads/Begin_Python/connections.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![expert](https://apmonitor.com/che263/uploads/Begin_Python/expert.png)\n",
    "\n",
    "### Interpolation\n",
    "\n",
    "Create an interpolation function between the measured points for $T_1$ using a linear interpolation. Show the recorded $T_1$ as red dots and the linear interpolation as a black line. Add x-label and y-labels as well as a plot legend."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![expert](https://apmonitor.com/che263/uploads/Begin_Python/expert.png)\n",
    "\n",
    "### Interpolation versus Regression\n",
    "\n",
    "Create an interpolation function between the measured points for $T_2$ using a cubic spline. Compare the piecewise cubic spline to a 3rd order polynomial regression. Show the recorded $T_2$ as blue dots, the cubic spline interpolation as a black dashed line, and the polynomial regression as a red dotted line. Add appropriate labels to the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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