{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "#### New to Plotly?\n", "Plotly's Python library is free and open source! [Get started](https://plotly.com/python/getting-started/) by dowloading the client and [reading the primer](https://plotly.com/python/getting-started/).\n", "
You can set up Plotly to work in [online](https://plotly.com/python/getting-started/#initialization-for-online-plotting) or [offline](https://plotly.com/python/getting-started/#initialization-for-offline-plotting) mode, or in [jupyter notebooks](https://plotly.com/python/getting-started/#start-plotting-online).\n", "
We also have a quick-reference [cheatsheet](https://images.plot.ly/plotly-documentation/images/python_cheat_sheet.pdf) (new!) to help you get started!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Imports\n", "The tutorial below imports [NumPy](http://www.numpy.org/), [Pandas](https://plotly.com/pandas/intro-to-pandas-tutorial/), [SciPy](https://www.scipy.org/), and [Random](https://docs.python.org/2/library/random.html)." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import plotly.plotly as py\n", "import plotly.graph_objs as go\n", "from plotly.tools import FigureFactory as FF\n", "\n", "import numpy as np\n", "import pandas as pd\n", "import scipy\n", "import random" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "####Tips\n", "A `random walk` can be thought of as a random process in which a tolken or a marker is randomly moved around some space, that is, a space with a metric used to compute distance. It is more commonly conceptualized in one dimension ($\\mathbb{Z}$), two dimensions ($\\mathbb{Z}^2$) or three dimensions ($\\mathbb{Z}^3$) in Cartesian space, where $\\mathbb{Z}$ represents the set of integers. In the visualizations below, we will be using [scatter plots](https://plotly.com/python/line-and-scatter/) as well as a colorscale to denote the time sequence of the walk." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Random Walk in 1D" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The jitter in the data points along the x and y axes are meant to illuminate where the points are being drawn and what the tendancy of the random walk is." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = [0]\n", "\n", "for j in range(100):\n", " step_x = random.randint(0,1)\n", " if step_x == 1:\n", " x.append(x[j] + 1 + 0.05*np.random.normal())\n", " else:\n", " x.append(x[j] - 1 + 0.05*np.random.normal())\n", " \n", "y = [0.05*np.random.normal() for j in range(len(x))]\n", " \n", "trace1 = go.Scatter(\n", " x=x,\n", " y=y,\n", " mode='markers',\n", " name='Random Walk in 1D',\n", " marker=dict(\n", " color=[i for i in range(len(x))],\n", " size=7,\n", " colorscale=[[0, 'rgb(178,10,28)'], [0.50, 'rgb(245,160,105)'],\n", " [0.66, 'rgb(245,195,157)'], [1, 'rgb(220,220,220)']],\n", " showscale=True,\n", " )\n", ")\n", "\n", "layout = go.Layout(\n", " yaxis=dict(\n", " range=[-1, 1]\n", " )\n", ")\n", "\n", "data = [trace1]\n", "fig= go.Figure(data=data, layout=layout)\n", "py.iplot(fig, filename='random-walk-1d')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Random Walk in 2D" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = [0]\n", "y = [0]\n", "\n", "for j in range(1000):\n", " step_x = random.randint(0,1)\n", " if step_x == 1:\n", " x.append(x[j] + 1 + np.random.normal())\n", " else:\n", " x.append(x[j] - 1 + np.random.normal())\n", " \n", " step_y = random.randint(0,1)\n", " if step_y == 1:\n", " y.append(y[j] + 1 + np.random.normal())\n", " else:\n", " y.append(y[j] - 1 + np.random.normal())\n", " \n", "trace1 = go.Scatter(\n", " x=x,\n", " y=y,\n", " mode='markers',\n", " name='Random Walk',\n", " marker=dict(\n", " color=[i for i in range(len(x))],\n", " size=8,\n", " colorscale='Greens',\n", " showscale=True\n", " )\n", ")\n", "\n", "data = [trace1]\n", "py.iplot(data, filename='random-walk-2d')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Advanced Tip\n", "We can formally think of a 1D random walk as a point jumping along the integer number line. Let $Z_i$ be a random variable that takes on the values +1 and -1. Let this random variable represent the steps we take in the random walk in 1D (where +1 means right and -1 means left). Also, as with the above visualizations, let us assume that the probability of moving left and right is just $\\frac{1}{2}$. Then, consider the sum\n", "\n", "$$\n", "\\begin{align*}\n", "S_n = \\sum_{i=0}^{n}{Z_i}\n", "\\end{align*}\n", "$$\n", "\n", "where S_n represents the point that the random walk ends up on after n steps have been taken.\n", "\n", "To find the `expected value` of $S_n$, we can compute it directly. Since each $Z_i$ is independent, we have\n", "\n", "$$\n", "\\begin{align*}\n", "\\mathbb{E}(S_n) = \\sum_{i=0}^{n}{\\mathbb{E}(Z_i)}\n", "\\end{align*}\n", "$$\n", "\n", "but since $Z_i$ takes on the values +1 and -1 then\n", "\n", "$$\n", "\\begin{align*}\n", "\\mathbb{E}(Z_i) = 1 \\cdot P(Z_i=1) + -1 \\cdot P(Z_i=-1) = \\frac{1}{2} - \\frac{1}{2} = 0\n", "\\end{align*}\n", "$$\n", "\n", "Therefore, we expect our random walk to hover around $0$ regardless of how many steps we take in our walk." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "Collecting git+https://github.com/plotly/publisher.git\n", " Cloning https://github.com/plotly/publisher.git to /var/folders/ld/6cl3s_l50wd40tdjq2b03jxh0000gp/T/pip-RrCIlf-build\n", "Installing collected packages: publisher\n", " Found existing installation: publisher 0.10\n", " Uninstalling publisher-0.10:\n", " Successfully uninstalled publisher-0.10\n", " Running setup.py install for publisher ... \u001b[?25l-\b \b\\\b \bdone\n", "\u001b[?25hSuccessfully installed publisher-0.10\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "/Users/brandendunbar/Desktop/test/venv/lib/python2.7/site-packages/IPython/nbconvert.py:13: ShimWarning: The `IPython.nbconvert` package has been deprecated. You should import from nbconvert instead.\n", " \"You should import from nbconvert instead.\", ShimWarning)\n", "/Users/brandendunbar/Desktop/test/venv/lib/python2.7/site-packages/publisher/publisher.py:53: UserWarning: Did you \"Save\" this notebook before running this command? Remember to save, always save.\n", " warnings.warn('Did you \"Save\" this notebook before running this command? '\n" ] } ], "source": [ "from IPython.display import display, HTML\n", "\n", "display(HTML(''))\n", "display(HTML(''))\n", "\n", "! pip install git+https://github.com/plotly/publisher.git --upgrade\n", "import publisher\n", "publisher.publish(\n", " 'python-Random-Walk.ipynb', 'python/random-walk/', 'Random Walk | plotly',\n", " 'Learn how to use Python to make a Random Walk',\n", " title='Random Walk in Python. | plotly',\n", " name='Random Walk',\n", " language='python',\n", " page_type='example_index', has_thumbnail='false', display_as='statistics', order=10,\n", " ipynb= '~notebook_demo/114')" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }