diff --git a/.ipynb_checkpoints/Duffing-checkpoint.ipynb b/.ipynb_checkpoints/Duffing-checkpoint.ipynb deleted file mode 100644 index 3b7d1ab..0000000 --- a/.ipynb_checkpoints/Duffing-checkpoint.ipynb +++ /dev/null @@ -1,1011 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 1, - "metadata": {}, - "outputs": [], - "source": [ - "%matplotlib notebook\n", - "import numpy as np\n", - "from scipy.integrate import odeint\n", - "import matplotlib.pyplot as plt\n", - "from matplotlib import animation\n", - "#from matplotlib import rc\n", - "#import seaborn as sbs\n", - "#from scipy.integrate import quad\n", - "#from scipy.optimize import brentq\n", - "#rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size': 20})\n", - "#rc('text', usetex=True)\n", - "#rc('animation', html='html5')" - ] - }, - { - "cell_type": "code", - "execution_count": 2, - "metadata": {}, - "outputs": [], - "source": [ - "# The potential and its first derivative, as callables.\n", - "V = lambda x: 0.5 * x**2 * (0.5 * x**2 - 1)\n", - "dVdx = lambda x: x**3 - x" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "metadata": {}, - "outputs": [], - "source": [ - "# The potential energy function on a grid of x-points.\n", - "xgrid = np.linspace(-1.5, 1.5, 100)\n", - "Vgrid = V(xgrid)" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "metadata": { - "scrolled": false - }, - "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" - }, - { - "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": [ - "# Plot freqency scans (mean +- variation)\n", - "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", - "for exp_no in range(5):\n", - " xy_data = read_xy('data/XYPost.mat', exp_no)\n", - " plot_std_freqscan(xy_data)\n", - " plot_xy(rows, xy_data)\n", - " plot_xyt(rows, xy_data)\n", - " break" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "## Model\n", - "\n", - "Model derived from Duffing equations:\n", - "\n", - "\\begin{align}\n", - "m_1 \\ddot{y}_1 &= F_1 - \\dot{y}_1(c_1 + c_3) + \\dot{y}_2c_3 - y_1(k_1 + k_3) + y_2k_3 - \\alpha_1y_1^3 + \\alpha_3(y_2 - y_1)^3 \\\\\n", - "m_2 \\ddot{y}_2 &= F_2 - \\dot{y}_2(c_2 + c_3) + \\dot{y}_1c_3 - x_2(k_2 + k_3) + y_1k_3 - \\alpha_2y_2^3 + \\alpha_3(y_2 - y_1)^3 \\\\\n", - "\\end{align}\n", - "\n", - "where $F = Ce^{i\\omega t}$. Transform to first-order form by variable substitutions $x_3 = \\dot{y}_1, x_1 = y_1$ and $x_4 = \\dot{y}_2, x_2 = y_2$:\n", - "\n", - "\\begin{align}\n", - "\\dot{x}_1 &= x_3 \\\\\n", - "\\dot{x}_2 &= x_4 \\\\\n", - "m_1\\dot{x}_3 &= F_1 - x_3(c_1 + c_3) + x_4c_3 - x_1(k_1 + k_3) + x_2k_3 - \\alpha_1x_1^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", - "m_2\\dot{x}_4 &= F_2 - x_4(c_2 + c_3) + x_3c_3 - x_2(k_2 + k_3) + x_1k_3 - \\alpha_2x_2^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", - "\\end{align}\n", - "\n", - "\n", - "For some reason, this model doesn't work out when working backwards from resonnance frequencies. I may be missing something obvious, otherwise the fact that mass goes into the estimations may mess things up. See plots of simulation below.\n", - "\n", - "#### Notes on solvers\n", - "\n", - " * We use SciPy's `solve_ivp` to simulate the system. Different methods (RK45, LSODA, Radeau) has been tested with no noticable differences.\n", - " * The standard `odeint` from SciPy is super shit. It easily diverges and is unstable. They claim to use the standard LSODA solver (same as `solve_ivp` with method='LSODA') but the results are entirely different.\n", - " * Once we hit the right parameters, the simulation is considerable slower because these are adaptive solvers.\n", - " * There is an ODE implementation from the [PyDSTool package](https://github.com/robclewley/pydstool) which compiles to C and is much much faster.\n", - " \n", - "TODO: Solve ODE with PyDSTool solver. Simulations takes 10-30 seconds per frequency, which results in way too long test cycles. With PyDSTool we can expect a magnitudal speedup." - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "metadata": {}, - "outputs": [], - "source": [ - "@curry\n", - "def model(omega, p, t, X):\n", - " x1, x2, xd1, xd2 = X\n", - " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", - " F1 = C1*np.cos(omega*t)\n", - " F2 = C2*np.cos(omega*t)\n", - " xdd1 = F1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*x1**3 + a3*(x2 - x1)**3\n", - " xdd2 = F2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*x2**3 + a3*(x2 - x1)**3\n", - " return xd1, xd2, m1*xdd1, m2*xdd2\n", - "\n", - "def _simulate(T, t_trans, steps, x0, v0, omega, p): \n", - " X0 = x0 + v0\n", - " dt = (T-t_trans)/steps\n", - " t = np.linspace(0, T, int(T/dt))\n", - " X = odeint(model(omega, p), X0, t, tfirst=True)\n", - " return X[-steps:].T\n", - "\n", - "def simulate(T, t_trans, steps, x0, v0, omega, p):\n", - " X0 = x0 + v0\n", - " dt = (T-t_trans)/steps\n", - " t = np.linspace(0, T, int(T/dt))\n", - " sol = solve_ivp(model(omega, p), [0, T], X0, t_eval=t, method='RK45')\n", - " return sol.y[:, -steps:]\n", - "\n", - "# Rely on the same functional signatures by mimicing the data structure of lab measurement.\n", - "def simulate_xy_data(rows, xy_data, T, t_trans, steps, x0, v0, p, progress=True):\n", - " sim_xy_data = {\n", - " 'x': np.zeros((101, 100)),\n", - " 'y': np.zeros((101, 100)),\n", - " 'f': np.zeros((101, 100)),\n", - " 'XResfFreq': xy_data['XResfFreq'],\n", - " 'YResfFreq': xy_data['YResfFreq']\n", - " }\n", - " \n", - " _rows = tqdm(rows) if progress else rows\n", - " for i in _rows:\n", - " f = xy_data['f'][i,0]\n", - " omega = 2*np.pi*f\n", - " X = simulate(T, t_trans, steps, x0, v0, omega, p)\n", - " x1, x2, xd1, xd2 = X\n", - " sim_xy_data['x'][i,:] = x1\n", - " sim_xy_data['y'][i,:] = x2\n", - " sim_xy_data['f'][i,:] = f\n", - " return sim_xy_data" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "scrolled": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Omega0 1 = 47763.0\n", - "Omega0 2 = 50905.3\n", - "k1 = 3421950.0\n", - "k2 = 3887017.6\n", - "c1 = 2.1\n", - "c2 = 2.3\n", - "Variables (rows x observations): ('x', 'y', 's', 'f', 'v', 't', 'XResAmp', 'XResfFreq', 'YResAmp', 'YResfFreq')\n", - "Resonnance frequencies: (7600, 8100)\n", - "Resonnance amplitudes: (1.03, 1.10)\n", - "T = 55.78\n", - "t in [0.00, 55.78]\n", - "f in [0.0, 10000.0]\n", - "x shape: 101 x 100\n", - "y shape: 101 x 100\n" - ] - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "57d6963c79b24bfc82d284c6fd96e296", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(IntProgress(value=0, max=8), HTML(value='')))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n" - ] - }, - { - "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 = $('
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');\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" - }, - { - "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" - }, - { - "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", - " '
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')\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": [ - "# Work backwards from resonnance frequency of a harmonic oscillator.\n", - "# Assume the system is significantly underdamped\n", - "exp_no = 0\n", - "print(\"Read data, experiment %d\" % exp_no)\n", - "xy_data = read_xy('data/XYPost.mat', exp_no)\n", - "\n", - "# Amplitude at resonnance should be 1\n", - "# Aim for better numerical stability by setting C and m to approx. the same numeric precision\n", - "C1 = 1.5e3\n", - "C2 = 1.5e3\n", - "m1 = 1e-4\n", - "m2 = 1e-4\n", - "\n", - "# Damping is probably small, 0 < zeta << 1/sqrt(2)\n", - "zeta1 = 0.1\n", - "zeta2 = 0.1\n", - "\n", - "# Fetch resonnance frequencies from data\n", - "f_r1 = xy_data['XResfFreq']\n", - "f_r2 = xy_data['YResfFreq']\n", - "omega_r1 = 2*np.pi*f_r1\n", - "omega_r2 = 2*np.pi*f_r2\n", - "omega_01 = omega_r1/(np.sqrt(1-2*zeta1**2))\n", - "omega_02 = omega_r2/(np.sqrt(1-2*zeta2**2))\n", - "\n", - "# Compute parameters from identities\n", - "c1 = 2*zeta1*omega_01\n", - "c2 = 2*zeta2*omega_02\n", - "c3 = 0 #0.05*(c1 + c2)\n", - "k1 = omega_01**2\n", - "k2 = omega_02**2\n", - "k3 = 0 #0.05*(k1 + k2)\n", - "a1 = 0 #1e3*k1\n", - "a2 = 0 #1e3*k2\n", - "a3 = 0 #0.05*(a1 + a2)\n", - "\n", - "p = (\n", - " C1, C2,\n", - " m1, m2,\n", - " c1, c2, c3,\n", - " k1, k2, k3,\n", - " a1, a2, a3\n", - ")\n", - "\n", - "print(\"Parameters:\")\n", - "print(\"Omega_r 1 = %.1f\" % omega_r1)\n", - "print(\"Omega_r 2 = %.1f\" % omega_r2)\n", - "print(\"Omega0 1 = %.1f\" % omega_01)\n", - "print(\"Omega0 2 = %.1f\" % omega_02)\n", - "print(\"c1 = %.1f\" % c1)\n", - "print(\"c2 = %.1f\" % c2)\n", - "print(\"c3 = %.1f\" % c3)\n", - "print(\"k1 = %.1f\" % k1)\n", - "print(\"k2 = %.1f\" % k2)\n", - "print(\"k3 = %.1f\" % k3)\n", - "print(\"a1 = %.1f\" % a1)\n", - "print(\"a2 = %.1f\" % a2)\n", - "print(\"a3 = %.1f\" % a3)\n", - "\n", - "# Start from any state, the sytem stabilize quickly\n", - "x0, v0 = (-0.05, 0.02), (-0.01, 0.02)\n", - "#x0, v0 = (0.0, 0.0), (0.0, 0.0)\n", - "T = 0.6\n", - "t_trans = 0.1\n", - "steps = 100\n", - "\n", - "# Simulate mostly around resonnance\n", - "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", - "\n", - "xyhat_data = simulate_xy_data(rows, xy_data, T, t_trans, steps, x0, v0, p)\n", - "plot_std_freqscan(xyhat_data)\n", - "plot_xy(rows, xyhat_data)\n", - "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### With duffing term\n", - "\n", - "The duffing term has to be pretty large to see any stiffening effect. Set $\\alpha_1 = 1500k_1$ and $\\alpha_2 = 1500k_2$ (still without coupling)." - ] - }, - { - "cell_type": "code", - "execution_count": 22, - "metadata": { - "scrolled": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Read data, experiment 0\n", - "Variables (rows x observations): ('x', 'y', 's', 'f', 'v', 't', 'XResAmp', 'XResfFreq', 'YResAmp', 'YResfFreq')\n", - "Resonnance frequencies: (7600, 8100)\n", - "Resonnance amplitudes: (1.03, 1.10)\n", - "T = 55.78\n", - "t in [0.00, 55.78]\n", - "f in [0.0, 10000.0]\n", - "x shape: 101 x 100\n", - "y shape: 101 x 100\n", - "Parameters:\n", - "Omega_r 1 = 47752.2\n", - "Omega_r 2 = 50893.8\n", - "Omega0 1 = 48237.0\n", - "Omega0 2 = 51410.5\n", - "c1 = 9647.4\n", - "c2 = 10282.1\n", - "c3 = 0.0\n", - "k1 = 2326809592.7\n", - "k2 = 2643039774.5\n", - "k3 = 0.0\n", - "a1 = 3490214389022.0\n", - "a2 = 3964559661768.2\n", - "a3 = 0.0\n" - ] - }, - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "41dadc16b11e4dbc9fcb14cfb9f53858", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "HBox(children=(IntProgress(value=0, max=8), HTML(value='')))" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n" - ] - }, - { - "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 = $('
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');\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" - }, - { - "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" - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "X idx: 32.0000 mean, 0.0000 std\n", - "Y idx: 1.0000 mean, 0.0000 std\n", - "X coeffs: 0.1348 mean, 0.0000 std\n", - "Y coeffs: 0.1414 mean, 0.0000 std\n", - "X MSEs: 1.4426 mean, 0.0000 std\n", - "Y MSEs: 1.7080 mean, 0.0000 std\n", - "Loss: 3.1506\n", - "\n", - "X idx: 5.0000 mean, 0.0000 std\n", - "Y idx: 15.0000 mean, 0.0000 std\n", - "X coeffs: 0.9416 mean, 0.0000 std\n", - "Y coeffs: 0.8535 mean, 0.0000 std\n", - "X MSEs: 0.0022 mean, 0.0000 std\n", - "Y MSEs: 0.0127 mean, 0.0000 std\n", - "Loss: 0.0149\n", - "\n", - "X idx: 18.5000 mean, 13.5000 std\n", - "Y idx: 8.0000 mean, 7.0000 std\n", - "X coeffs: 0.5382 mean, 0.4034 std\n", - "Y coeffs: 0.4975 mean, 0.3560 std\n", - "X MSEs: 0.7224 mean, 0.7202 std\n", - "Y MSEs: 0.8603 mean, 0.8476 std\n", - "Loss: 1.5827\n" - ] - } - ], - "source": [ - "# Test loss function\n", - "t = np.linspace(0, 100, 100)\n", - "s1 = {\n", - " 'f': np.array([\n", - " np.ones((100,)),\n", - " 2*np.ones((100,))\n", - " ]),\n", - " 'x': np.array([\n", - " np.random.randn(100),\n", - " np.array([1*np.sin(i/np.pi) for i in t]),\n", - " ]),\n", - " 'y': np.array([\n", - " np.random.randn(100),\n", - " np.array([1*np.cos(i/np.pi) for i in t]),\n", - " ])\n", - "}\n", - "s2 = {\n", - " 'f': np.array([\n", - " np.ones((100,)),\n", - " 2*np.ones((100,))\n", - " ]),\n", - " 'x': np.array([\n", - " np.random.randn(100),\n", - " np.array([1*np.cos(i/np.pi) for i in t]),\n", - " ]),\n", - " 'y': np.array([\n", - " np.random.randn(100),\n", - " np.array([1*np.sin(i/np.pi) for i in t]),\n", - " ])\n", - "}\n", - "\n", - "plot_xyt([0,1], s1, sim_xy_data=s2)\n", - "\n", - "print(\"Loss: %.4f\\n\" % xy_loss([0], s1, s2, verbose=True))\n", - "print(\"Loss: %.4f\\n\" % xy_loss([1], s1, s2, verbose=True))\n", - "print(\"Loss: %.4f\" % xy_loss([0, 1], s1, s2, verbose=True))" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "#### Loss for model\n", - "\n", - "Plotting normalized signals." - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": { - "scrolled": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "X idx: 25.0000 mean, 17.5357 std\n", - "Y idx: 15.2500 mean, 9.2432 std\n", - "X coeffs: 0.0720 mean, 0.0817 std\n", - "Y coeffs: 0.0795 mean, 0.0590 std\n", - "X MSEs: 1.8373 mean, 0.1778 std\n", - "Y MSEs: 1.8084 mean, 0.1515 std\n", - "Loss: 3.6457\n", - "\n" - ] - }, - { - "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" - }, - { - "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": [ - "# Work backwards from resonnance frequency of a harmonic oscillator.\n", - "# Assume the system is significantly underdamped\n", - "\n", - "C1 = 1\n", - "C2 = 1\n", - "m1 = 1.5e-3\n", - "m2 = 1.5e-3\n", - "zeta1 = 0.015\n", - "zeta2 = 0.015\n", - "omega_r1 = 2*np.pi*7.6e3\n", - "omega_r2 = 2*np.pi*8.1e3\n", - "\n", - "omega_01 = omega_r1/(np.sqrt(1-2*zeta1**2))\n", - "omega_02 = omega_r2/(np.sqrt(1-2*zeta2**2))\n", - "k1 = m1*omega_01**2\n", - "k2 = m2*omega_02**2\n", - "c1 = 2*zeta1*np.sqrt(m1*k1)\n", - "c2 = 2*zeta2*np.sqrt(m2*k2)\n", - "k3 = 0.0\n", - "c3 = 0.0\n", - "\n", - "print(\"Omega0 1 = %.1f\" % omega_01)\n", - "print(\"Omega0 2 = %.1f\" % omega_02)\n", - "print(\"k1 = %.1f\" % k1)\n", - "print(\"k2 = %.1f\" % k2)\n", - "print(\"c1 = %.1f\" % c1)\n", - "print(\"c2 = %.1f\" % c2)\n", - "\n", - "T = 0.6\n", - "t_trans = 0.1\n", - "steps = 100\n", - "x0, v0 = (-0.05, 0.02), (-0.01, 0.02)\n", - "\n", - "p = (\n", - " C1, C2,\n", - " m1, m2,\n", - " c1, c2, c3,\n", - " k1, k2, k3,\n", - " 0, 0, 0\n", - ")\n", - "\n", - "xy_data = read_xy('data/XYPost.mat', 0)\n", - "\n", - "# Simulate mostly around resonnance\n", - "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", - "\n", - "xyhat_data = simulate_xy_data(rows, xy_data, T, t_trans, steps, x0, v0, p)\n", - "plot_std_freqscan(xyhat_data)\n", - "plot_xy(rows, xyhat_data)\n", - "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "\n", - "## Rewritten model\n", + "### Transformed model\n", "\n", "Eliminate mass, + easier to reason about physical constants:\n", "\n", @@ -5218,7 +2677,16 @@ "\\dot{x}_4 &= \\frac{1}{m_2}F_2 - x_4(c_2 + c_3) + x_3c_3 - x_2(k_2 + k_3) + x_1k_3 - \\alpha_2x_2^3 + \\alpha_3(x_2 - x_1)^3 \\\\\n", "\\end{align}\n", "\n", - "With harmonic oscillator identities\n", + "With the Jacobian $\\mathbf{J} = \\frac{\\partial \\mathbf{f}}{\\partial \\mathbf{X}}$\n", + "\n", + "\\begin{bmatrix}\n", + "0 & 0 & 1 & 0 \\\\\n", + "0 & 0 & 0 & 1 \\\\\n", + "-k_1 - k_3 - 3\\alpha_1x_1^2 - 3\\alpha_3(x_2 - x_1)^2 & k_3 + 3\\alpha_3(x_2 - x_1)^2 & -c_1 - c_3 & c_3 \\\\\n", + "k_3 - 3\\alpha_3(x_2 - x_1)^2 & -k_2 - k_3 - 3\\alpha_2x_2^2 + 3\\alpha_3(x_2 - x_1)^2 & c_3 & -c_2 - c_3\n", + "\\end{bmatrix}\n", + "\n", + "Use the harmonic oscillator identities\n", "\n", " * Undamped angular frequency:\n", "\n", @@ -5238,7 +2706,7 @@ "\\omega_r &= \\omega_0\\sqrt{1-2\\zeta^2}, \\zeta < \\frac{1}{\\sqrt{2}}\n", "\\end{align}\n", "\n", - "express the constants subject to estimation as\n", + "and express the constants subject to estimation as\n", "\n", "\\begin{align}\n", "c_1 &= 2 \\zeta_1 \\omega_{0,1} \\\\\n", @@ -5249,45 +2717,72 @@ "k_3 &= g_k(k_1, k_2) \\\\\n", "\\alpha_1 &= f_1(\\mathbf{X} ; \\theta) \\\\\n", "\\alpha_2 &= f_2(\\mathbf{X} ; \\theta) \\\\\n", - "\\alpha_3 &= g_c(\\alpha_1, \\alpha_2)\n", + "\\alpha_3 &= g_{\\alpha}(\\alpha_1, \\alpha_2)\n", "\\end{align}\n", "\n", "\n", - "With the model expressed this way, things make sense and we get resonnance where it should be." + "With the model expressed this way, things make sense and we get resonnance where it should be.\n", + "\n", + "\n", + "### Notes on solvers\n", + "\n", + " * We use SciPy's `solve_ivp` to simulate the system. Different methods (RK45, LSODA, Radeau) has been tested with no noticable differences.\n", + " * The standard `odeint` from SciPy is super shit. It easily diverges and is unstable. They claim to use the standard LSODA solver (same as `solve_ivp` with method='LSODA') but the results are entirely different.\n", + " * Once we hit the right parameters, the simulation is considerable slower because these are adaptive solvers.\n", + " * There is an ODE implementation from the [PyDSTool package](https://github.com/robclewley/pydstool) which compiles to C and is much much faster.\n", + " \n", + "#### PyDSTool solver\n", + "\n", + "The model is implemented with this solver. It's slightly faster, but notoriously more complicated to use. Besides, it requires SciPy version `<1.0` which is not compatible with the rest of this code. Let's stick to scipy's modern solvers." ] }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 36, "metadata": {}, "outputs": [], "source": [ "@curry\n", - "def model(omega, p, t, X):\n", + "def model1(omega, p, t, X):\n", " x1, x2, xd1, xd2 = X\n", " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", " F1 = C1*np.cos(omega*t)\n", " F2 = C2*np.cos(omega*t)\n", - " xdd1 = F1/m1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*x1**3 + a3*(x2 - x1)**3\n", - " xdd2 = F2/m2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*x2**3 + a3*(x2 - x1)**3\n", + " xdd1 = F1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*x1**3 + a3*(x2 - x1)**3\n", + " xdd2 = F2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*x2**3 + a3*(x2 - x1)**3\n", + " return xd1, xd2, m1*xdd1, m2*xdd2\n", + "\n", + "@curry\n", + "def model2(omega, p, t, X):\n", + " x1, x2, xd1, xd2 = X\n", + " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", + " xdd1 = C1*np.cos(omega*t)/m1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*x1**3 + a3*(x2 - x1)**3\n", + " xdd2 = C2*np.cos(omega*t)/m2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*x2**3 + a3*(x2 - x1)**3\n", " return xd1, xd2, xdd1, xdd2\n", "\n", - "def _simulate(T, t_trans, steps, x0, v0, omega, p): \n", - " X0 = x0 + v0\n", - " dt = (T-t_trans)/steps\n", - " t = np.linspace(0, T, int(T/dt))\n", - " X = odeint(model(omega, p), X0, t, tfirst=True)\n", - " return X[-steps:].T\n", + "@curry\n", + "def jacobian2(omega, p, t, X):\n", + " x1, x2, xd1, xd2 = X\n", + " C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p\n", + " return np.array([\n", + " [ 0 , 0 , 1 , 0 ],\n", + " [ 0 , 0 , 0 , 1 ],\n", + " [-k1-k3-3*a1*x1**2-3*a3*(x2-x1)**2, k3+3*a3*(x2-x1)**2 , -c1-c3 , c3 ],\n", + " [ k3-3*a3*(x2-x1)**2 , -k2-k3-3*a2*x2**2+3*a3*(x2-x1)**2 , c3 , -c2-c3]\n", + " ])\n", "\n", - "def simulate(T, t_trans, steps, x0, v0, omega, p):\n", - " X0 = x0 + v0\n", - " dt = (T-t_trans)/steps\n", + "def odeint_integrate(model, jac, dt, T, X0, rtol, atol):\n", " t = np.linspace(0, T, int(T/dt))\n", - " sol = solve_ivp(model(omega, p), [0, T], X0, t_eval=t, method='RK45')\n", - " return sol.y[:, -steps:]\n", + " X = odeint(model, X0, t, Dfun=jac, tfirst=True, rtol=rtol, atol=atol)\n", + " return X.T\n", + "\n", + "def solve_ivp_integrate(model, jac, dt, T, X0, rtol, atol):\n", + " t = np.linspace(0, T, int(T/dt))\n", + " sol = solve_ivp(model, [0, T], X0, t_eval=t, jac=jac, method='Radau', first_step=dt, rtol=rtol, atol=atol)\n", + " return sol.y\n", "\n", "# Rely on the same functional signatures by mimicing the data structure of lab measurement.\n", - "def simulate_xy_data(rows, xy_data, T, t_trans, steps, x0, v0, p, progress=True):\n", + "def simulate_xy_data(integrator, rows, xy_data, T, t_trans, t_scale, steps, x0, v0, p, progress=True):\n", " sim_xy_data = {\n", " 'x': np.zeros((101, 100)),\n", " 'y': np.zeros((101, 100)),\n", @@ -5295,17 +2790,88 @@ " 'XResfFreq': xy_data['XResfFreq'],\n", " 'YResfFreq': xy_data['YResfFreq']\n", " }\n", - " \n", " _rows = tqdm(rows) if progress else rows\n", " for i in _rows:\n", " f = xy_data['f'][i,0]\n", - " omega = 2*np.pi*f\n", - " X = simulate(T, t_trans, steps, x0, v0, omega, p)\n", + " omega = 2*np.pi*f/t_scale\n", + " X = integrator(\n", + " model2(omega, p),\n", + " jacobian2(omega, p),\n", + " (T-t_trans)/steps,\n", + " T,\n", + " x0 + v0,\n", + " 1e-3,\n", + " [1e-4, 1e-4, 1e-2, 1e-2]\n", + " )\n", " x1, x2, xd1, xd2 = X\n", - " sim_xy_data['x'][i,:] = x1\n", - " sim_xy_data['y'][i,:] = x2\n", + " sim_xy_data['x'][i,:] = x1[-steps:]\n", + " sim_xy_data['y'][i,:] = x2[-steps:]\n", " sim_xy_data['f'][i,:] = f\n", - " return sim_xy_data" + " return sim_xy_data\n", + "\n", + "def simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True):\n", + " # Amplitude at resonnance should be 1\n", + " # Aim for better numerical stability by setting C and m to approx. the same numeric precision\n", + " t_scale = 1\n", + " C1 = 1.5e7/t_scale\n", + " C2 = 1.5e7/t_scale\n", + " m1 = 1\n", + " m2 = 1\n", + "\n", + " # Fetch resonnance frequencies from data\n", + " f_r1 = xy_data['XResfFreq']/t_scale\n", + " f_r2 = xy_data['YResfFreq']/t_scale\n", + " omega_r1 = 2*np.pi*f_r1\n", + " omega_r2 = 2*np.pi*f_r2\n", + " omega_01 = omega_r1/(np.sqrt(1-2*zeta1**2))\n", + " omega_02 = omega_r2/(np.sqrt(1-2*zeta2**2))\n", + "\n", + " # Compute parameters from identities\n", + " c1 = 2*zeta1*omega_01\n", + " c2 = 2*zeta2*omega_02\n", + " c3 = gc(c1, c2)\n", + " k1 = omega_01**2\n", + " k2 = omega_02**2\n", + " k3 = gk(k1, k2)\n", + " a1 = f1(k1, k2)\n", + " a2 = f2(k1, k2)\n", + " a3 = ga(k1, k2)\n", + "\n", + " p = (\n", + " C1, C2,\n", + " m1, m2,\n", + " c1, c2, c3,\n", + " k1, k2, k3,\n", + " a1, a2, a3\n", + " )\n", + "\n", + " if verbose:\n", + " print(\"Parameters:\")\n", + " print(\"Omega_r 1 = %.3f\" % omega_r1)\n", + " print(\"Omega_r 2 = %.3f\" % omega_r2)\n", + " print(\"Omega0 1 = %.3f\" % omega_01)\n", + " print(\"Omega0 2 = %.3f\" % omega_02)\n", + " print(\"c1 = %.3f\" % c1)\n", + " print(\"c2 = %.3f\" % c2)\n", + " print(\"c3 = %.3f\" % c3)\n", + " print(\"k1 = %.3f\" % k1)\n", + " print(\"k2 = %.3f\" % k2)\n", + " print(\"k3 = %.3f\" % k3)\n", + " print(\"a1 = %.3f\" % a1)\n", + " print(\"a2 = %.3f\" % a2)\n", + " print(\"a3 = %.3f\" % a3)\n", + "\n", + " # Start from any state, the system stabilize quickly\n", + " x0, v0 = (-0.002, 0.01), (-0.004, 0.03)\n", + " #x0, v0 = (0.0, 0.0), (0.0, 0.0)\n", + " \n", + " # Set transient period to 0.1 seconds and simulate 100 steps over 0.5 seconds\n", + " t_trans = 0.1*t_scale\n", + " T = t_trans + 0.5*t_scale\n", + " steps = 100\n", + "\n", + " # Simulate mostly around resonnance\n", + " return simulate_xy_data(solve_ivp_integrate, rows, xy_data, T, t_trans, t_scale, steps, x0, v0, p, progress=verbose)" ] }, { @@ -5319,7 +2885,7 @@ }, { "cell_type": "code", - "execution_count": 21, + "execution_count": 28, "metadata": { "scrolled": false }, @@ -5338,25 +2904,25 @@ "x shape: 101 x 100\n", "y shape: 101 x 100\n", "Parameters:\n", - "Omega_r 1 = 47752.2\n", - "Omega_r 2 = 50893.8\n", - "Omega0 1 = 48237.0\n", - "Omega0 2 = 51410.5\n", - "c1 = 9647.4\n", - "c2 = 10282.1\n", - "c3 = 0.0\n", - "k1 = 2326809592.7\n", - "k2 = 2643039774.5\n", - "k3 = 0.0\n", - "a1 = 0.0\n", - "a2 = 0.0\n", - "a3 = 0.0\n" + "Omega_r 1 = 47752.208\n", + "Omega_r 2 = 50893.801\n", + "Omega0 1 = 48237.015\n", + "Omega0 2 = 51410.503\n", + "c1 = 9647.403\n", + "c2 = 10282.101\n", + "c3 = 0.000\n", + "k1 = 2326809592.681\n", + "k2 = 2643039774.512\n", + "k3 = 0.000\n", + "a1 = 0.000\n", + "a2 = 0.000\n", + "a3 = 0.000\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { - "model_id": "87be653ba05448fdbf322890cf15de91", + "model_id": "4b9f94d45aa9415ca7ca60b84d497a17", "version_major": 2, "version_minor": 0 }, @@ -5367,13 +2933,6 @@ "metadata": {}, "output_type": "display_data" }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n" - ] - }, { "data": { "application/javascript": [ @@ -6157,7 +3716,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -6949,7 +4508,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -7741,7 +5300,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -7752,76 +5311,30 @@ } ], "source": [ - "# Work backwards from resonnance frequency of a harmonic oscillator.\n", - "# Assume the system is significantly underdamped\n", "exp_no = 0\n", "print(\"Read data, experiment %d\" % exp_no)\n", "xy_data = read_xy('data/XYPost.mat', exp_no)\n", "\n", - "# Amplitude at resonnance should be 1\n", - "# Aim for better numerical stability by setting C and m to approx. the same numeric precision\n", - "C1 = 1.5e3\n", - "C2 = 1.5e3\n", - "m1 = 1e-4\n", - "m2 = 1e-4\n", + "def gc(c1, c2):\n", + " return 0.0\n", "\n", - "# Damping is probably small, 0 < zeta << 1/sqrt(2)\n", + "def gk(k1, k2):\n", + " return 0.0\n", + "\n", + "def f1(k1, k2):\n", + " return 0.0\n", + "\n", + "def f2(k1, k2):\n", + " return 0.0\n", + "\n", + "def ga(k1, k2):\n", + " return 0.0\n", + "\n", + "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", "zeta1 = 0.1\n", "zeta2 = 0.1\n", + "xyhat_data = simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True)\n", "\n", - "# Fetch resonnance frequencies from data\n", - "f_r1 = xy_data['XResfFreq']\n", - "f_r2 = xy_data['YResfFreq']\n", - "omega_r1 = 2*np.pi*f_r1\n", - "omega_r2 = 2*np.pi*f_r2\n", - "omega_01 = omega_r1/(np.sqrt(1-2*zeta1**2))\n", - "omega_02 = omega_r2/(np.sqrt(1-2*zeta2**2))\n", - "\n", - "# Compute parameters from identities\n", - "c1 = 2*zeta1*omega_01\n", - "c2 = 2*zeta2*omega_02\n", - "c3 = 0 #0.05*(c1 + c2)\n", - "k1 = omega_01**2\n", - "k2 = omega_02**2\n", - "k3 = 0 #0.05*(k1 + k2)\n", - "a1 = 0 #1e3*k1\n", - "a2 = 0 #1e3*k2\n", - "a3 = 0 #0.05*(a1 + a2)\n", - "\n", - "p = (\n", - " C1, C2,\n", - " m1, m2,\n", - " c1, c2, c3,\n", - " k1, k2, k3,\n", - " a1, a2, a3\n", - ")\n", - "\n", - "print(\"Parameters:\")\n", - "print(\"Omega_r 1 = %.1f\" % omega_r1)\n", - "print(\"Omega_r 2 = %.1f\" % omega_r2)\n", - "print(\"Omega0 1 = %.1f\" % omega_01)\n", - "print(\"Omega0 2 = %.1f\" % omega_02)\n", - "print(\"c1 = %.1f\" % c1)\n", - "print(\"c2 = %.1f\" % c2)\n", - "print(\"c3 = %.1f\" % c3)\n", - "print(\"k1 = %.1f\" % k1)\n", - "print(\"k2 = %.1f\" % k2)\n", - "print(\"k3 = %.1f\" % k3)\n", - "print(\"a1 = %.1f\" % a1)\n", - "print(\"a2 = %.1f\" % a2)\n", - "print(\"a3 = %.1f\" % a3)\n", - "\n", - "# Start from any state, the sytem stabilize quickly\n", - "x0, v0 = (-0.05, 0.02), (-0.01, 0.02)\n", - "#x0, v0 = (0.0, 0.0), (0.0, 0.0)\n", - "T = 0.6\n", - "t_trans = 0.1\n", - "steps = 100\n", - "\n", - "# Simulate mostly around resonnance\n", - "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", - "\n", - "xyhat_data = simulate_xy_data(rows, xy_data, T, t_trans, steps, x0, v0, p)\n", "plot_std_freqscan(xyhat_data)\n", "plot_xy(rows, xyhat_data)\n", "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" @@ -7838,7 +5351,7 @@ }, { "cell_type": "code", - "execution_count": 22, + "execution_count": 37, "metadata": { "scrolled": false }, @@ -7857,25 +5370,25 @@ "x shape: 101 x 100\n", "y shape: 101 x 100\n", "Parameters:\n", - "Omega_r 1 = 47752.2\n", - "Omega_r 2 = 50893.8\n", - "Omega0 1 = 48237.0\n", - "Omega0 2 = 51410.5\n", - "c1 = 9647.4\n", - "c2 = 10282.1\n", - "c3 = 0.0\n", - "k1 = 2326809592.7\n", - "k2 = 2643039774.5\n", - "k3 = 0.0\n", - "a1 = 3490214389022.0\n", - "a2 = 3964559661768.2\n", - "a3 = 0.0\n" + "Omega_r 1 = 47752.208\n", + "Omega_r 2 = 50893.801\n", + "Omega0 1 = 48237.015\n", + "Omega0 2 = 51410.503\n", + "c1 = 9647.403\n", + "c2 = 10282.101\n", + "c3 = 0.000\n", + "k1 = 2326809592.681\n", + "k2 = 2643039774.512\n", + "k3 = 0.000\n", + "a1 = 3490214389021.967\n", + "a2 = 3964559661768.200\n", + "a3 = 0.000\n" ] }, { "data": { "application/vnd.jupyter.widget-view+json": { - "model_id": "41dadc16b11e4dbc9fcb14cfb9f53858", + "model_id": "663cb0ef7baa4e26b8279ffba570751a", "version_major": 2, "version_minor": 0 }, @@ -7886,13 +5399,6 @@ "metadata": {}, "output_type": "display_data" }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "\n" - ] - }, { "data": { "application/javascript": [ @@ -8676,7 +6182,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -9468,7 +6974,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -10260,7 +7766,7 @@ { "data": { "text/html": [ - "" + "" ], "text/plain": [ "" @@ -10271,76 +7777,74 @@ } ], "source": [ - "# Work backwards from resonnance frequency of a harmonic oscillator.\n", - "# Assume the system is significantly underdamped\n", "exp_no = 0\n", "print(\"Read data, experiment %d\" % exp_no)\n", "xy_data = read_xy('data/XYPost.mat', exp_no)\n", "\n", - "# Amplitude at resonnance should be 1\n", - "# Aim for better numerical stability by setting C and m to approx. the same numeric precision\n", - "C1 = 1e3\n", - "C2 = 1e3\n", - "m1 = 1e-4\n", - "m2 = 1e-4\n", + "def gc(c1, c2):\n", + " return 0.0\n", "\n", - "# Damping is probably small, 0 < zeta << 1/sqrt(2)\n", + "def gk(k1, k2):\n", + " return 0.0\n", + "\n", + "def f1(k1, k2):\n", + " return 1.5e3*k1\n", + "\n", + "def f2(k1, k2):\n", + " return 1.5e3*k2\n", + "\n", + "def ga(k1, k2):\n", + " return 0.0\n", + "\n", + "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", "zeta1 = 0.1\n", "zeta2 = 0.1\n", + "xyhat_data = simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True)\n", "\n", - "# Fetch resonnance frequencies from data\n", - "f_r1 = xy_data['XResfFreq']\n", - "f_r2 = xy_data['YResfFreq']\n", - "omega_r1 = 2*np.pi*f_r1\n", - "omega_r2 = 2*np.pi*f_r2\n", - "omega_01 = omega_r1/(np.sqrt(1-2*zeta1**2))\n", - "omega_02 = omega_r2/(np.sqrt(1-2*zeta2**2))\n", + "plot_std_freqscan(xyhat_data)\n", + "plot_xy(rows, xyhat_data)\n", + "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "#### With coupling" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "scrolled": false + }, + "outputs": [], + "source": [ + "exp_no = 0\n", + "print(\"Read data, experiment %d\" % exp_no)\n", + "xy_data = read_xy('data/XYPost.mat', exp_no)\n", "\n", - "# Compute parameters from identities\n", - "c1 = 2*zeta1*omega_01\n", - "c2 = 2*zeta2*omega_02\n", - "c3 = 0 #0.05*(c1 + c2)\n", - "k1 = omega_01**2\n", - "k2 = omega_02**2\n", - "k3 = 0 #0.05*(k1 + k2)\n", - "a1 = 1.5e3*k1\n", - "a2 = 1.5e3*k2\n", - "a3 = 0 #0.05*(a1 + a2)\n", + "def gc(c1, c2):\n", + " return 0.05*(c1 + c2)\n", "\n", - "p = (\n", - " C1, C2,\n", - " m1, m2,\n", - " c1, c2, c3,\n", - " k1, k2, k3,\n", - " a1, a2, a3\n", - ")\n", + "def gk(k1, k2):\n", + " return 0.05*(k1 + k2)\n", "\n", - "print(\"Parameters:\")\n", - "print(\"Omega_r 1 = %.1f\" % omega_r1)\n", - "print(\"Omega_r 2 = %.1f\" % omega_r2)\n", - "print(\"Omega0 1 = %.1f\" % omega_01)\n", - "print(\"Omega0 2 = %.1f\" % omega_02)\n", - "print(\"c1 = %.1f\" % c1)\n", - "print(\"c2 = %.1f\" % c2)\n", - "print(\"c3 = %.1f\" % c3)\n", - "print(\"k1 = %.1f\" % k1)\n", - "print(\"k2 = %.1f\" % k2)\n", - "print(\"k3 = %.1f\" % k3)\n", - "print(\"a1 = %.1f\" % a1)\n", - "print(\"a2 = %.1f\" % a2)\n", - "print(\"a3 = %.1f\" % a3)\n", + "def f1(k1, k2):\n", + " return 1.5e3*k1\n", "\n", - "# Start from any state, the sytem stabilize quickly\n", - "x0, v0 = (-0.05, 0.02), (-0.01, 0.02)\n", - "#x0, v0 = (0.0, 0.0), (0.0, 0.0)\n", - "T = 0.6\n", - "t_trans = 0.1\n", - "steps = 100\n", + "def f2(k1, k2):\n", + " return 1.5e3*k2\n", + "\n", + "def ga(k1, k2):\n", + " return 0.05*(a1 + a2)\n", "\n", - "# Simulate mostly around resonnance\n", "rows = [50, 65, 70, 75, 80, 85, 90, 100]\n", + "zeta1 = 0.1\n", + "zeta2 = 0.1\n", + "xyhat_data = simulate_experiment(xy_data, rows, zeta1, zeta2, gc, gk, ga, f1, f2, verbose=True)\n", "\n", - "xyhat_data = simulate_xy_data(rows, xy_data, T, t_trans, steps, x0, v0, p)\n", "plot_std_freqscan(xyhat_data)\n", "plot_xy(rows, xyhat_data)\n", "plot_xyt(rows, xy_data, sim_xy_data=xyhat_data)" @@ -10362,7 +7866,7 @@ }, { "cell_type": "code", - "execution_count": 23, + "execution_count": null, "metadata": {}, "outputs": [], "source": [ @@ -10419,833 +7923,11 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": null, "metadata": { "scrolled": false }, - "outputs": [ - { - "data": { - "application/javascript": [ - "/* Put everything inside the global mpl namespace */\n", - "window.mpl = {};\n", - "\n", - 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' +\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" - } - ], + "outputs": [], "source": [ "print(\"Loss: %.4f\\n\" % xy_loss(rows, xy_data, xyhat_data, verbose=True))\n", "plot_xyt(rows, xy_data, normalizer=normalize_signal, sim_xy_data=xyhat_data)" diff --git a/pydstool_integrator.py b/pydstool_integrator.py new file mode 100644 index 0000000..cf3ae1e --- /dev/null +++ b/pydstool_integrator.py @@ -0,0 +1,144 @@ +"""Simulate duffing model with PyDSTool integrator. + +See + https://github.com/robclewley/pydstool/blob/master/examples/forced_spring.py + +for forced-spring example. +""" + +import numpy as np +import PyDSTool + + +def simulate(T, t_trans, steps, x0, v0, omega, p): + x1, x2 = x0 + xd1, xd2 = v0 + C1, C2, m1, m2, c1, c2, c3, k1, k2, k3, a1, a2, a3 = p + + args = PyDSTool.args(name='duffing-stuffing') + args.varspecs = { + 'x1': 'xd1', + 'x2': 'xd2', + 'xd1': 'C1*F(t)/m1 - xd1*(c1 + c3) + xd2*c3 - x1*(k1 + k3) + x2*k3 - a1*pow(x1, 3) + a3*pow(x2 - x1, 3)', + 'xd2': 'C2*F(t)/m2 - xd2*(c2 + c3) + xd1*c3 - x2*(k2 + k3) + x1*k3 - a2*pow(x2, 3) + a3*pow(x2 - x1, 3)' + } + args.fnspecs = { + 'F': (['t'], 'cos(omega*t)') + } + args.pars = { + 'omega': omega, + 'C1': C1, 'C2': C2, + 'm1': m1, 'm2': m2, + 'c1': c1, 'c2': c2, 'c3': c3, + 'k1': k1, 'k2': k2, 'k3': k3, + 'a1': a1, 'a2': a2, 'a3': a3 + } + args.ics = { + 'x1': x0[0], 'x2': x0[1], + 'xd1': v0[0], 'xd2': v0[1] + } + args.xdomain = { + 'x1': [-10, 10], 'x2': [-10, 10], + 'xd1': [-1e6, 1e6], 'xd2': [-1e6, 1e6] + } + #args.tdomain = [0, T] + args.tdata = [0, T] + + # Integrator + ds = PyDSTool.Generator.Vode_ODEsystem(args) + traj = ds.compute('experiment') + pts = traj.sample(dt=(T-t_trans)/steps, precise=True) + return np.stack([ + pts['x1'][-steps:], + pts['x2'][-steps:], + pts['xd1'][-steps:], + pts['xd2'][-steps:] + ]) + + +def test(): + C1 = 1.5e5 + C2 = 1.5e5 + m1 = 1.0 #1e-3 + m2 = 1.0 #1e-3 + + c1 = 9647.403 + c2 = 10282.101 + c3 = 0.000 + k1 = 2326809592.681 + k2 = 2643039774.512 + k3 = 0.000 + a1 = 0.000 + a2 = 0.000 + a3 = 0.000 + + p = ( + C1, C2, + m1, m2, + c1, c2, c3, + k1, k2, k3, + a1, a2, a3 + ) + + omega = 2*np.pi*5e3 + + x0, v0 = (0.00001, 0.00002), (0.000001, 0.0002) + T = 0.6 + t_trans = 0.1 + steps = 100 + + return simulate(T, t_trans, steps, x0, v0, omega, p) + + +if __name__ == '__main__': + X = test() + print(X.shape) + print(X.mean(axis=1)) + print(X.std(axis=1)) + print(X) + + +#pars = {'D1': 1.0 , 'D2': 10.0, 'A': 2.1, 'B': 3.5, 'lambda': 5} +# +#tdomain = [0,50] +# +#icdict = {'X1': 1.087734, 'X2': 1.087734, 'X3': 4.124552, \ +# 'Y1': 1.8488063, 'Y2': 1.8488063, 'Y3': 1.0389936} +# +## Set up model +#X1str = 'D1*(X2 - 2*X1 + X3) + lambda*(A - (B+1)*X1 + pow(X1,2)*Y1)' +#X2str = 'D1*(X3 - 2*X2 + X1) + lambda*(A - (B+1)*X2 + pow(X2,2)*Y2)' +#X3str = 'D1*(X1 - 2*X3 + X2) + lambda*(A - (B+1)*X3 + pow(X3,2)*Y3)' +#Y1str = 'D2*(Y2 - 2*Y1 + Y3) + lambda*(B*X1 - pow(X1,2)*Y1)' +#Y2str = 'D2*(Y3 - 2*Y2 + Y1) + lambda*(B*X2 - pow(X2,2)*Y2)' +#Y3str = 'D2*(Y1 - 2*Y3 + Y2) + lambda*(B*X3 - pow(X3,2)*Y3)' +# +#DSargs = args(name='Brusselator') +#DSargs.tdomain = tdomain +#DSargs.pars = pars +#DSargs.varspecs = {'X1': X1str, 'X2': X2str, 'X3': X3str, 'Y1': Y1str, 'Y2': Y2str, 'Y3': Y3str} +#DSargs.xdomain = {'X1': [0, 50], 'X2': [0, 50], 'X3': [0, 50], 'Y1': [0, 50], 'Y2': [0, 50], 'Y3': [0, 50]} +#DSargs.ics = icdict +# +#testDS = Generator.Vode_ODEsystem(DSargs) +# +## Set up continuation class +#PyCont = ContClass(testDS) +# +#PCargs = args(name='EQ1', type='EP-C') +#PCargs.freepars = ['lambda'] +#PCargs.StepSize = 5e-3 +#PCargs.LocBifPoints = ['BP','LP'] +#PCargs.verbosity = 2 +#PCargs.SaveEigen = True +#PyCont.newCurve(PCargs) +# +#print('Computing curve...') +#start = clock() +#PyCont['EQ1'].forward() +#print('done in %.3f seconds!' % (clock()-start)) +# +## Plot +#PyCont['EQ1'].display(('lambda','X1'), stability=True) +#PyCont.plot.toggleAll('off', byname='P1') +#show() diff --git a/requirements.txt b/requirements.txt new file mode 100644 index 0000000..fe3b8e3 --- /dev/null +++ b/requirements.txt @@ -0,0 +1,2 @@ +scipy>=0.5.1,<1.0 +PyDSTool