{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Curve-Fitting comparison: Python and Mathematica\n", "\n", "+ Implementation of curve-fitting in Python.
\n", "+ Starting point: version of mlr.py written for PHYS 310 Jack Gallimore.
\n", "+ Comments based on treatment in Numerical Recipes (Chapter 14.3 in my copy).
\n", "+ Compare with results of Mathematica for same data sets: see pythonTest.nb.\n", "\n", "Define function that is a linear combination of \"Basis Functions,\" $X_k(x)$, evaluated at all values of independent variable $x_j$ -- see Eq. (14.3.2) in Numerical Recipes. These are combined into the single matrix that gets weighted for the illustrated matrix in Fig. 14.3.1 of Numerical Recipes.\n", "\n", "Covariance Matrix; See Eq. (14.3.8), and following discussion:\n", "\n", "$$\\alpha_{kj} = \\sum_{i=1}^N\\frac{X_j(x_i)X_k(x_i)}{\\sigma_i^2} \\hspace{0.2in}\\mbox{and}\n", " \\hspace{0.2in}C_{jk} = [\\alpha]_{jk}^{-1}$$\n", "\n", "This result gives covariance matrix if uncertainties are measurement uncertainties with well-known values. To get covariance matrix from uncertainties estimated from observed spread of values, multiply $C_{jk}$ by $\\chi_R^2$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Linear fitting " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from scipy import linalg\n", "from scipy import stats\n", "\n", "import matplotlib as mpl # As of July 2017 Bucknell computers use v. 2.x \n", "import matplotlib.pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Following is an Ipython magic command that puts figures in the notebook.\n", "# For figures in separate windows, comment out following line and uncomment\n", "# the next line\n", "# Must come before defaults are changed.\n", "%matplotlib notebook\n", "#%matplotlib\n", "\n", "# As of Aug. 2017 reverting to 1.x defaults.\n", "# In 2.x text.ustex requires dvipng, texlive-latex-extra, and texlive-fonts-recommended, \n", "# which don't seem to be universal\n", "# See https://stackoverflow.com/questions/38906356/error-running-matplotlib-in-latex-type1cm?\n", "mpl.style.use('classic')\n", " \n", "# M.L. modifications of matplotlib defaults using syntax of v.2.0 \n", "# More info at http://matplotlib.org/2.0.0/users/deflt_style_changes.html\n", "# Changes can also be put in matplotlibrc file, or effected using mpl.rcParams[]\n", "plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures\n", "plt.rc('axes', labelsize=16, titlesize=14)\n", "plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### THE FOLLOWING CELL DEFINING BASIS FUNCTIONS IS ONLY FUNCTION CELL THAT SHOULD REQUIRE MODIFICATION FOR NEW MODEL" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# Basis functions for linear model: func = a0*X0 + a1*X1 + a2*X2 + ...\n", "def basis(x):\n", " ''' Returns basis functions for linear model\n", " \n", " The function to be fit is assumed to be of the form\n", " f(x) = a0*X0 + a1*X1 + a2*X2 + a3*X3 + ...\n", " where a0, a1, a2, ... are constants, and X0, X1, X2, ... are defined below.\n", " '''\n", " X2 = x**2 \n", " X1 = x\n", " X0 = 0.*X1 + 1. # Need array of len(x), thus the 0.*X1\n", " return np.array([X0,X1,X2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Evaluate function that is a linear combination of basis functions, with cofficients contained in array `a`" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "def func(x, a):\n", " '''Given basis functions and coefficients, returns value of linear function'''\n", " return np.dot(basis(x).T, a)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Function that does fitting, and formats output:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "def LinearModelFit(x, y, u):\n", " '''Performs linear least squares fit to a set of 2-d data with uncertainties\n", " \n", " x = array of x values [x0, x1, x2, ...]\n", " y = array of values of dependent variable [y0, y1, y2, ...]\n", " u = array of uncertainties for dependent variable [u0, u1, u2, ...]\n", " '''\n", " X = basis(x).T # Basis functions evaluated at all x (the X_j(x_i)) of N.R.)\n", " W = np.diag(1/u) # Matrix with uncertainties on diagonal\n", " Xw = np.dot(W,X) # A_ij of Eq. (14.3.4)\n", " Yw = np.dot(y,W) # b_i of Eq. (14.3.5)\n", " fit = np.linalg.lstsq(Xw,Yw,rcond=1) # lstq returns: best values, chi2, ....\n", " covariance = np.linalg.inv(np.dot(Xw.T,Xw))\n", " uncertainty = np.sqrt(np.diag(covariance))\n", " return(fit[0], uncertainty,fit[1], covariance)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Generate some noisy data (x,y) with uncertainties u, or import sample data file." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# Cell for generating data; overwritten by following cell if data is coming from file.\n", "x = np.linspace(0, 10, 11)\n", "sigma = 5.0\n", "a = np.array([1.2, 3.4, -0.9])\n", "y = func(x,a) + stats.norm.rvs(0, sigma, size=len(x))\n", "u = sigma*np.ones(len(y))\n", "#sp.savetxt(\"sample.dat\",sp.array([x,y,u]).T)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "data = np.loadtxt(\"sample.dat\") #Each line in file corresponds single data point: x,y,u" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "x = data.T[0]\n", "y = data.T[1]\n", "u = data.T[2]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Two previous cells would be more \"pythonic\" as\n", "\n", " x, y, u = sp.loadtxt(\"sample.dat\", unpack=True)\n", "\n", "The \"unpack = True\" reads columns." ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "x, y, u = np.loadtxt(\"sample.dat\", unpack=True)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '
');\n", " var titletext = $(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('
');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('
')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp = $('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "xfine = np.linspace(np.min(x), np.max(x), 201) # \"quasi-continuous\" set of x's for plotting function\n", "plt.figure(2)\n", "plt.title(\"data with best fit quadratic\",fontsize=14)\n", "plt.xlabel('$x$')\n", "plt.ylabel('$y$')\n", "plt.axhline(0, color='magenta')\n", "plt.xlim(min(x)-0.05*(np.max(x)-np.min(x)), np.max(x)+ 0.05*(np.max(x)-np.min(x))) \\\n", " # Pad x-range on plot\n", "plt.errorbar(x, y, yerr=u, fmt='o')\n", "plt.plot(xfine, func(xfine, a));" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Residuals:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Check $\\chi_R^2$:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Explicit calculation of reduced chi-square parameter:\n", "\n", "\\begin{equation}\n", "\\chi_R^2= \\frac{1}{N-c}\\times\\sum_{i=1}^N \\frac{(y_i-f(x_i))^2}{\\sigma_i^2}, \n", "\\end{equation}\n", "\n", "and compare with return value from `lstsq`." ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.5046871217125679" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.sum((y-func(x,a))**2/u**2)/(len(x)-3)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Version details\n", "\n", "`version_information` is from J.R. Johansson (jrjohansson at gmail.com)
\n", "See Introduction to scientific computing with Python:
\n", "http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb
\n", "for more information and instructions for package installation.
\n", "\n", "If `version_information` has been installed system wide (as it has been on Bucknell linux computers with shared file systems), continue with next cell as written. If not, comment out top line in next cell and uncomment the second line." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Loading extensions from ~/.ipython/extensions is deprecated. We recommend managing extensions like any other Python packages, in site-packages.\n" ] } ], "source": [ "%load_ext version_information\n", "\n", "#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "application/json": { "Software versions": [ { "module": "Python", "version": "3.7.3 64bit [GCC 7.3.0]" }, { "module": "IPython", "version": "7.5.0" }, { "module": "OS", "version": "Linux 3.10.0 862.14.4.el7.x86_64 x86_64 with redhat 7.5 Maipo" }, { "module": "numpy", "version": "1.16.3" }, { "module": "scipy", "version": "1.2.1" }, { "module": "matplotlib", "version": "3.0.3" } ] }, "text/html": [ "
SoftwareVersion
Python3.7.3 64bit [GCC 7.3.0]
IPython7.5.0
OSLinux 3.10.0 862.14.4.el7.x86_64 x86_64 with redhat 7.5 Maipo
numpy1.16.3
scipy1.2.1
matplotlib3.0.3
Mon Sep 09 13:15:54 2019 EDT
" ], "text/latex": [ "\\begin{tabular}{|l|l|}\\hline\n", "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n", "Python & 3.7.3 64bit [GCC 7.3.0] \\\\ \\hline\n", "IPython & 7.5.0 \\\\ \\hline\n", "OS & Linux 3.10.0 862.14.4.el7.x86\\_64 x86\\_64 with redhat 7.5 Maipo \\\\ \\hline\n", "numpy & 1.16.3 \\\\ \\hline\n", "scipy & 1.2.1 \\\\ \\hline\n", "matplotlib & 3.0.3 \\\\ \\hline\n", "\\hline \\multicolumn{2}{|l|}{Mon Sep 09 13:15:54 2019 EDT} \\\\ \\hline\n", "\\end{tabular}\n" ], "text/plain": [ "Software versions\n", "Python 3.7.3 64bit [GCC 7.3.0]\n", "IPython 7.5.0\n", "OS Linux 3.10.0 862.14.4.el7.x86_64 x86_64 with redhat 7.5 Maipo\n", "numpy 1.16.3\n", "scipy 1.2.1\n", "matplotlib 3.0.3\n", "Mon Sep 09 13:15:54 2019 EDT" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version_information numpy, scipy, matplotlib" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python 3", "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": "3.7.3" } }, "nbformat": 4, "nbformat_minor": 1 }