Added Pearson correlation notebook

Author: seanlaw

docs/Pearson.ipynb

@@ -0,0 +1,104 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "86920570-6be2-4aee-b24d-7d82cc923b5a",
+   "metadata": {},
+   "source": [
+    "# Pearson Correlation Vs. Z-normalized Euclidean Distance\n",
+    "\n",
+    "It is [well understood](https://arxiv.org/pdf/1601.02213.pdf) that the z-normalized Euclidean distance, $ED_{z-norm}$, and the Pearson correlation, $PC$, between any two subsequences with length $m$ share the following relationship:\n",
+    "\n",
+    "$ED_{z-norm} = \\sqrt {2 * m * (1 - PC)}$\n",
+    "\n",
+    "Naturally, when the two subsequences are perfectly correlated (i.e., $PC = 1$), then we get:\n",
+    "\n",
+    "\\begin{align}\n",
+    "    ED_{z-norm} ={}&\n",
+    "        \\sqrt {2 * m * (1 - PC)}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m * (1 - 1)}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m * 0}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {0}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        0\n",
+    "    \\\\\n",
+    "\\end{align}\n",
+    "\n",
+    "Similarly, when the two subsequences are completely uncorrelated (i.e., $PC = 0$), then we get:\n",
+    "\n",
+    "\\begin{align}\n",
+    "    ED_{z-norm} ={}&\n",
+    "        \\sqrt {2 * m * (1 - PC)}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m * (1 - 0)}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m * 1}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m}\n",
+    "    \\\\\n",
+    "\\end{align}\n",
+    "\n",
+    "In other words, the largest possible z-normalized distance between any pair of subsequences with length $m$ is $\\sqrt{2 * m}$. The maximum distance can never be bigger!\n",
+    "\n",
+    "Finally, when two subsequences are anti-correlated (i.e., $PC = -1$), then we get:\n",
+    "\n",
+    "\\begin{align}\n",
+    "    ED_{z-norm} ={}&\n",
+    "        \\sqrt {2 * m * (1 - PC)}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m * (1 - (-1))}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {2 * m * 2}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        \\sqrt {4 * m}\n",
+    "    \\\\\n",
+    "    ={}&\n",
+    "        2 * \\sqrt {m}\n",
+    "    \\\\\n",
+    "\\end{align}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b5a437dc-8c9c-4075-9548-87e97676630e",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "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": "3.9.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}