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Adding Interact and Lorenz examples.
This commit is contained in:
Brian E. Granger
MinRK
parent
1d6cde4877
commit
0cf8002ebf
475
examples/widgets/Interact.ipynb
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475
examples/widgets/Interact.ipynb
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{
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"metadata": {
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"name": ""
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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"worksheets": [
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{
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"cells": [
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{
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"cell_type": "heading",
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"level": 1,
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"metadata": {},
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"source": [
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"Interact Demos"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This Notebook shows basic demonstrations of IPython `interact` module. This provides a high-level interface for creating user interface controls to use in exploring code and data interactively."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"%pylab inline"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"from IPython.html.widgets.interact import interact, interactive\n",
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"from IPython.html import widgets\n",
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"from IPython.display import clear_output, display, HTML"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"Basic interact"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here is a simple function that displays its arguments as an HTML table:"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def show_args(**kwargs):\n",
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" s = '<h3>Arguments:</h3><table>\\n'\n",
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" for k,v in kwargs.items():\n",
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" s += '<tr><td>{0}</td><td>{1}</td></tr>\\n'.format(k,v)\n",
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" s += '</table>'\n",
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" display(HTML(s))"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"show_args(a=10, b='Hi There', c=True)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let's use this function to explore how `interact` works."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"interact(show_args,\n",
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" Temp=(0,10),\n",
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" Current=(0.,10.,0.01),\n",
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" z=(True,False),\n",
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" Text=u'Type here!',\n",
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" Algorithm=['This','That','Other'],\n",
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" a=widgets.FloatRangeWidget(min=-10.0, max=10.0, step=0.1, value=5.0)\n",
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" )"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The keyword arguments to `interact` can be any `Widget` instance that has a `value` and `description` attribute, or one of the shorthand notations shown above."
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]
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"Factoring polynomials"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Here is an example that uses [SymPy](http://sympy.org/en/index.html) to factor polynomials."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"from sympy import Symbol, Eq, factor, init_printing\n",
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"init_printing(use_latex=True)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"x = Symbol('x')"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def factorit(n):\n",
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" display(Eq(x**n-1, factor(x**n-1)))"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Notice how the output of the `factorit` function is properly formatted LaTeX."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"interact(factorit, n=(2,40))"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"A simple image browser"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This example shows how to browse through a set of images with a slider."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"from sklearn import datasets"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We will use the digits dataset from [scikit-learn](http://scikit-learn.org/stable/)."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"digits = datasets.load_digits()"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def browse_images(digits):\n",
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" n = len(digits.images)\n",
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" def view_image(i):\n",
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" imshow(digits.images[i], cmap=cm.gray_r, interpolation='nearest')\n",
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" title('Training: %s' % digits.target[i])\n",
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" show()\n",
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" interact(view_image, i=(0,n-1))"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"browse_images(digits)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"Explore random graphs"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In this example, we build a simple UI for exploring random graphs with [NetworkX](http://networkx.github.io/)."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"import networkx as nx"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def plot_random_graph(n, p, generator):\n",
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" g = generator(n,p)\n",
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" nx.draw(g)\n",
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" show()"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"interact(plot_random_graph, n=(2,30), p=(0.0, 1.0, 0.001),\n",
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" generator={'gnp': nx.gnp_random_graph,\n",
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" 'erdos_renyi': nx.erdos_renyi_graph,\n",
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" 'binomial': nx.binomial_graph})"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"Image manipulation"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This example builds a simple UI for performing basic image manipulation with [scikit-image](http://scikit-image.org/)."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"import skimage\n",
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"from skimage import data, filter, io"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"i = data.coffee()"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"io.Image(i)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def edit_image(image):\n",
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" def apply_filter(sigma, r, g, b):\n",
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" new_image = filter.gaussian_filter(image, sigma=sigma)\n",
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" new_image[:,:,0] = r*new_image[:,:,0]\n",
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" new_image[:,:,1] = g*new_image[:,:,1]\n",
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" new_image[:,:,2] = b*new_image[:,:,2]\n",
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" new_image = io.Image(new_image)\n",
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" display(new_image)\n",
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" return new_image\n",
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" lims = (0.0,1.0,0.01)\n",
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" return interactive(apply_filter, sigma=(0.1,10.0,0.01), r=lims, g=lims, b=lims)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"w = edit_image(i)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"display(w)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"w.arguments"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"w.result"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"Playing with audio"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"This example uses the `Audio` object and Matplotlib to explore the phenomenon of beat frequencies."
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"from IPython.display import Audio\n",
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"import numpy as np"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"def beat_freq(f1=220.0, f2=224.0):\n",
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" max_time = 3\n",
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" rate = 8000.0\n",
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" times = np.linspace(0,max_time,rate*max_time)\n",
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" signal = np.sin(2*np.pi*f1*times) + np.sin(2*np.pi*f2*times)\n",
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" print f1, f2, abs(f1-f2)\n",
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" display(Audio(data=signal, rate=rate))\n",
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" return signal"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"v = interactive(beat_freq, f1=(200.0,300.0), f2=(200.0,300.0))\n",
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"display(v)"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"plot(v.result[0:6000])"
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],
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"language": "python",
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"metadata": {},
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"outputs": []
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}
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],
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"metadata": {}
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}
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]
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}
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269
examples/widgets/Lorenz.ipynb
Normal file
269
examples/widgets/Lorenz.ipynb
Normal file
@ -0,0 +1,269 @@
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{
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"metadata": {
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||||
"name": ""
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||||
},
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"nbformat": 3,
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"nbformat_minor": 0,
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"worksheets": [
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{
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"cells": [
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{
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"cell_type": "heading",
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"level": 1,
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"metadata": {},
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"source": [
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"Exploring the Lorenz System of Differential Equations"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In this Notebook we explore the Lorenz system of differential equations:\n",
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"\n",
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"$$\n",
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"\\begin{aligned}\n",
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"\\dot{x} & = \\sigma(y-x) \\\\\n",
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"\\dot{y} & = \\rho x - y - xz \\\\\n",
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"\\dot{z} & = -\\beta z + xy\n",
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"\\end{aligned}\n",
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"$$\n",
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"\n",
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"This is one of the classic systems in non-linear differential equations. It exhibits a range of different behaviors as the parameters ($\\sigma$, $\\beta$, $\\rho$) are varied."
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]
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},
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{
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"cell_type": "heading",
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"level": 2,
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"metadata": {},
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"source": [
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"Imports"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"First, we import the needed things from IPython, NumPy, Matplotlib and SciPy."
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]
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},
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{
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||||
"cell_type": "code",
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||||
"collapsed": false,
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"input": [
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"%pylab inline"
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||||
],
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||||
"language": "python",
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"metadata": {},
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||||
"outputs": []
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||||
},
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{
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"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"from IPython.html.widgets.interact import interact, interactive\n",
|
||||
"from IPython.display import clear_output, display, HTML"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"import numpy as np\n",
|
||||
"from scipy import integrate\n",
|
||||
"\n",
|
||||
"from matplotlib import pyplot as plt\n",
|
||||
"from mpl_toolkits.mplot3d import Axes3D\n",
|
||||
"from matplotlib.colors import cnames\n",
|
||||
"from matplotlib import animation"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "heading",
|
||||
"level": 2,
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Computing the trajectories and plotting the result"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We define a function that can integrate the differential equations numerically and then plot the solutions. This function has arguments that control the parameters of the differential equation ($\\sigma$, $\\beta$, $\\rho$), the numerical integration (`N`, `max_time`) and the visualization (`angle`)."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"def solve_lorenz(N=10, angle=0.0, max_time=4.0, sigma=10.0, beta=8./3, rho=28.0):\n",
|
||||
"\n",
|
||||
" fig = plt.figure()\n",
|
||||
" ax = fig.add_axes([0, 0, 1, 1], projection='3d')\n",
|
||||
" ax.axis('off')\n",
|
||||
"\n",
|
||||
" # prepare the axes limits\n",
|
||||
" ax.set_xlim((-25, 25))\n",
|
||||
" ax.set_ylim((-35, 35))\n",
|
||||
" ax.set_zlim((5, 55))\n",
|
||||
" \n",
|
||||
" def lorenz_deriv((x, y, z), t0, sigma=sigma, beta=beta, rho=rho):\n",
|
||||
" \"\"\"Compute the time-derivative of a Lorentz system.\"\"\"\n",
|
||||
" return [sigma * (y - x), x * (rho - z) - y, x * y - beta * z]\n",
|
||||
"\n",
|
||||
" # Choose random starting points, uniformly distributed from -15 to 15\n",
|
||||
" np.random.seed(1)\n",
|
||||
" x0 = -15 + 30 * np.random.random((N, 3))\n",
|
||||
"\n",
|
||||
" # Solve for the trajectories\n",
|
||||
" t = np.linspace(0, max_time, int(250*max_time))\n",
|
||||
" x_t = np.asarray([integrate.odeint(lorenz_deriv, x0i, t)\n",
|
||||
" for x0i in x0])\n",
|
||||
" \n",
|
||||
" # choose a different color for each trajectory\n",
|
||||
" colors = plt.cm.jet(np.linspace(0, 1, N))\n",
|
||||
"\n",
|
||||
" for i in range(N):\n",
|
||||
" x, y, z = x_t[i,:,:].T\n",
|
||||
" lines = ax.plot(x, y, z, '-', c=colors[i])\n",
|
||||
" setp(lines, linewidth=2)\n",
|
||||
"\n",
|
||||
" ax.view_init(30, angle)\n",
|
||||
" show()\n",
|
||||
"\n",
|
||||
" return t, x_t"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Let's call the function once to view the solutions. For this set of parameters, we see the trajectories swirling around two points, called attractors. "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"t, x_t = solve_lorenz(angle=0, N=10)"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Using IPython's `interactive` function, we can explore how the trajectories behave as we change the various parameters."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"w = interactive(solve_lorenz, angle=(0.,360.), N=(0,50), sigma=(0.0,50.0), rho=(0.0,50.0))\n",
|
||||
"display(w)"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The object returned by `interactive` is a `Widget` object and it has attributes that contain the current result and arguments:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"t, x_t = w.result"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"w.arguments"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"After interacting with the system, we can take the result and perform further computations. In this case, we compute the average positions in $x$, $y$ and $z$."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"xyz_avg = x_t.mean(axis=1)"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"xyz_avg.shape"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Creating histograms of the average positions (across different trajectories) show that on average the trajectories swirl about the attractors."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"hist(xyz_avg[:,0])\n",
|
||||
"title('Average $x(t)$')"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"collapsed": false,
|
||||
"input": [
|
||||
"hist(xyz_avg[:,1])\n",
|
||||
"title('Average $y(t)$')"
|
||||
],
|
||||
"language": "python",
|
||||
"metadata": {},
|
||||
"outputs": []
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
}
|
||||
]
|
||||
}
|
||||
Loading…
Reference in New Issue
Block a user