lecture for today

lectures/lecture-07/2 - optimization.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Optimization\n",
+    "\n",
+    "Let's consider the following code:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "# position of flower in 2D\n",
+    "flower = np.array([29.1, 19.1])\n",
+    "\n",
+    "def make_bee_track(t):\n",
+    "    \"\"\"Find bee position at time t\"\"\"\n",
+    "    pos0 = (13.21, 4.56)\n",
+    "    velocity = (3.1, 0.25)\n",
+    "    pos_x = pos0[0] + t*velocity[0]\n",
+    "    pos_y = pos0[1] + t*velocity[1]\n",
+    "    return np.array([pos_x,pos_y])\n",
+    "\n",
+    "t = np.linspace(0,15,10)\n",
+    "bee_track = make_bee_track(t)\n",
+    "\n",
+    "fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(4,3))\n",
+    "ax.plot( [flower[0]], [flower[1]], 'or', label='flower' )\n",
+    "ax.plot( bee_track[0], bee_track[1], '.-k', label='bee')\n",
+    "ax.axis('equal')\n",
+    "ax.legend()\n",
+    "ax.set_xlabel('x')\n",
+    "ax.set_ylabel('y')\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "In the above code, we *parameterized* the bee trajectory by the variable `t` in the function `make_bee_track()`. This means we could get a new point on the track by choosing a new value of `t`. For example:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(make_bee_track(0.0))\n",
+    "print(make_bee_track(0.1))\n",
+    "print(make_bee_track(0.2))\n",
+    "print(make_bee_track(1.0))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now let's measure the distance between the bee and the flower."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def compute_distance(a,b):\n",
+    "    a = np.array(a)\n",
+    "    b = np.array(b)\n",
+    "    return np.sqrt(np.sum((a-b)**2))\n",
+    "\n",
+    "n_time_steps = bee_track.shape[1]\n",
+    "distance = np.zeros(n_time_steps)\n",
+    "for i in range(n_time_steps):\n",
+    "    bee_pos = bee_track[:,i]\n",
+    "    distance[i] = compute_distance(bee_pos, flower)\n",
+    "display(distance)\n",
+    "\n",
+    "fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(4,3))\n",
+    "ax.plot( t, distance )\n",
+    "ax.set_xlabel('t')\n",
+    "ax.set_ylabel('distance')"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Given the plot of distance versus t above, we can see the distance is minimized when t is near 5. What is the bee's position when t is 5?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(make_bee_track(5))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We can check back to the xy plot to see, indeed, this point is pretty close to the flower.\n",
+    "\n",
+    "What if we want to know, however, exactly the closest point? There are several ways to find this. Here we are going to use a \"brute force\" approach which will work on many different problems. Specifically, we will use [scipy.optimize.minimize_scalar](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize_scalar.html). The overall idea of this kind of *numerical optimization* is that we find the best fitting parameter to minimize our \"error\".\n",
+    "\n",
+    "In this example, we are not so much concerned with the exact algorithm being used, but with the way we call this algorithm.\n",
+    "\n",
+    "Let's create a function which uses the `flower` location to compute distance:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def calc_distance_func(t):\n",
+    "    # assume variable 'flower' in our global scope\n",
+    "    x1, y1 = flower\n",
+    "    # calculate bee position (also assuming 'make_bee_track' in scope)\n",
+    "    x2, y2 = make_bee_track(t)\n",
+    "    dist = compute_distance((x1,y1), (x2,y2))\n",
+    "    print(f't: {t} -> dist: {dist}')\n",
+    "    return dist"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "calc_distance_func(0)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "calc_distance_func(5)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import scipy.optimize\n",
+    "result = scipy.optimize.minimize_scalar(calc_distance_func)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "scipy.optimize.minimize_scalar?"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "result"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "type(result)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "result.x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(calc_distance_func(5))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "calc_distance_func(5.468493134264144)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# Where is the bee for this value of `t`?\n",
+    "make_bee_track(5.468493134264144)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(4,3))\n",
+    "ax.plot( [flower[0]], [flower[1]], 'or', label='flower' )\n",
+    "ax.plot( bee_track[0], bee_track[1], '.-k', label='bee')\n",
+    "\n",
+    "x,y = make_bee_track(5.468493134264144)\n",
+    "ax.plot( [x], [y] ,'x', label='closest')\n",
+    "ax.axis('equal')\n",
+    "ax.legend()\n",
+    "ax.set_xlabel('x')\n",
+    "ax.set_ylabel('y')\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.10"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}