\n", "
\n", "
\n", " Capacity and Overfitting \n", "\n", "First let us examine the accuracy of our results as a function of the order of the polynomial." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "scrolled": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "C:\\Users\\FSU\\Anaconda2\\envs\\ML\\lib\\site-packages\\ipykernel_launcher.py:9: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n", "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n", " if __name__ == '__main__':\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.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n", " 'ui-button-icon-only');\n", " button.attr('role', 'button');\n", " button.attr('aria-disabled', 'false');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", "\n", " var icon_img = $('');\n", " icon_img.addClass('ui-button-icon-primary ui-icon');\n", " icon_img.addClass(image);\n", " icon_img.addClass('ui-corner-all');\n", "\n", " var tooltip_span = $('');\n", " tooltip_span.addClass('ui-button-text');\n", " tooltip_span.html(tooltip);\n", "\n", " button.append(icon_img);\n", " button.append(tooltip_span);\n", "\n", " nav_element.append(button);\n", " }\n", "\n", " var fmt_picker_span = $('');\n", "\n", " var fmt_picker = $('');\n", " fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n", " fmt_picker_span.append(fmt_picker);\n", " nav_element.append(fmt_picker_span);\n", " this.format_dropdown = fmt_picker[0];\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = $(\n", " '', {selected: fmt === mpl.default_extension}).html(fmt);\n", " fmt_picker.append(option)\n", " }\n", "\n", " // Add hover states to the ui-buttons\n", " $( \".ui-button\" ).hover(\n", " function() { $(this).addClass(\"ui-state-hover\");},\n", " function() { $(this).removeClass(\"ui-state-hover\");}\n", " );\n", "\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "}\n", "\n", "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n", "}\n", "\n", "mpl.figure.prototype.send_message = function(type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "}\n", "\n", "mpl.figure.prototype.send_draw_message = function() {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n", " }\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_resize = function(fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1]);\n", " fig.send_message(\"refresh\", {});\n", " };\n", "}\n", "\n", "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n", " var x0 = msg['x0'] / mpl.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n", " var x1 = msg['x1'] / mpl.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0, 0, fig.canvas.width, fig.canvas.height);\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "}\n", "\n", "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "}\n", "\n", "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch(cursor)\n", " {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "}\n", "\n", "mpl.figure.prototype.handle_message = function(fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "}\n", "\n", "mpl.figure.prototype.handle_draw = function(fig, msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "}\n", "\n", "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Called whenever the canvas gets updated.\n", " this.send_message(\"ack\", {});\n", "}\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function(fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = \"image/png\";\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src);\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig[\"handle_\" + msg_type];\n", " } catch (e) {\n", " console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n", " }\n", " }\n", " };\n", "}\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function(e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e)\n", " e = window.event;\n", " if (e.target)\n", " targ = e.target;\n", " else if (e.srcElement)\n", " targ = e.srcElement;\n", " if (targ.nodeType == 3) // defeat Safari bug\n", " targ = targ.parentNode;\n", "\n", " // jQuery normalizes the pageX and pageY\n", " // pageX,Y are the mouse positions relative to the document\n", " // offset() returns the position of the element relative to the document\n", " var x = e.pageX - $(targ).offset().left;\n", " var y = e.pageY - $(targ).offset().top;\n", "\n", " return {\"x\": x, \"y\": y};\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys (original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object')\n", " obj[key] = original[key]\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", " if (name === 'button_press')\n", " {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * mpl.ratio;\n", " var y = canvas_pos.y * mpl.ratio;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " // Handle any extra behaviour associated with a key event\n", "}\n", "\n", "mpl.figure.prototype.key_event = function(event, name) {\n", "\n", " // Prevent repeat events\n", " if (name == 'key_press')\n", " {\n", " if (event.which === this._key)\n", " return;\n", " else\n", " this._key = event.which;\n", " }\n", " if (name == 'key_release')\n", " this._key = null;\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which != 17)\n", " value += \"ctrl+\";\n", " if (event.altKey && event.which != 18)\n", " value += \"alt+\";\n", " if (event.shiftKey && event.which != 16)\n", " value += \"shift+\";\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, {key: value,\n", " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n", " if (name == 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message(\"toolbar_button\", {name: name});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function() {\n", " comm.close()\n", " };\n", " ws.send = function(m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function(msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overridden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data'])\n", " });\n", " return ws;\n", "}\n", "\n", "mpl.mpl_figure_comm = function(comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = $(\"#\" + id);\n", " var ws_proxy = comm_websocket_adapter(comm)\n", "\n", " function ondownload(figure, format) {\n", " window.open(figure.imageObj.src);\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy,\n", " ondownload,\n", " element.get(0));\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element.get(0);\n", " fig.cell_info = mpl.find_output_cell(\"\");\n", " if (!fig.cell_info) {\n", " console.error(\"Failed to find cell for figure\", id, fig);\n", " return;\n", " }\n", "\n", " var output_index = fig.cell_info[2]\n", " var cell = fig.cell_info[0];\n", "\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", " var width = fig.canvas.width/mpl.ratio\n", " fig.root.unbind('remove')\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('
\n", "
\n", "
\n", " Regularization \n", "\n", "If we examine the magnitude of the parameters as a function of polynomial order, we see that they quickly become unwieldly.\n", "\n" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "def J2(w,x,t,order,lamda):\n", " c = 1./(2*x.size)\n", " diff = t-model(x,w,order)\n", " cost = c*(diff**2).sum()\n", " return cost" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[array([ 0.49224809, -1.35925975]),\n", " array([ 0.35260734, -0.41668471, -0.94257504]),\n", " array([ -0.10881211, 7.180257 , -20.96488346, 13.34820561]),\n", " array([ -0.15517354, 8.91881064, -29.74457933, 27.43049008,\n", " -7.04114223]),\n", " array([-8.70368596e-02, 2.71155869e+00, 2.19967154e+01, -1.18824903e+02,\n", " 1.60600653e+02, -6.70567180e+01]),\n", " array([-3.76131911e-02, -1.04771473e+01, 1.84709317e+02, -8.19155197e+02,\n", " 1.51397206e+03, -1.27090423e+03, 4.01282503e+02]),\n", " array([-3.52100231e-02, -1.30759847e+01, 2.27592168e+02, -1.07345541e+03,\n", " 2.23302141e+03, -2.31948776e+03, 1.16224630e+03, -2.17418227e+02]),\n", " array([-3.27317764e-02, -3.08119819e+01, 5.85263957e+02, -3.77122332e+03,\n", " 1.24102018e+04, -2.35073993e+04, 2.58814764e+04, -1.53516407e+04,\n", " 3.78355562e+03]),\n", " array([-3.41762896e-02, 1.19987505e+02, -2.89794148e+03, 2.76032890e+04,\n", " -1.34983074e+05, 3.78046632e+05, -6.31638933e+05, 6.22402317e+05,\n", " -3.33634682e+05, 7.49818306e+04])]" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ws" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[1.8515078333930484,\n", " 1.7118670871393113,\n", " 41.60215818342165,\n", " 73.29019581812514,\n", " 371.2775848815156,\n", " 4200.538059734789,\n", " 7246.332467678018,\n", " 85321.60581656045,\n", " 2206308.7212477215]" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ws_abs = [np.abs(wsi).sum() for wsi in ws]\n", "ws_abs" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '');\n", " var titletext = $(\n", " '');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas = $('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", " $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $('')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button = $('');\n", " button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n", " 'ui-button-icon-only');\n", " button.attr('role', 'button');\n", " button.attr('aria-disabled', 'false');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", "\n", " var icon_img = $('');\n", " icon_img.addClass('ui-button-icon-primary ui-icon');\n", " icon_img.addClass(image);\n", " icon_img.addClass('ui-corner-all');\n", "\n", " var tooltip_span = $('');\n", " tooltip_span.addClass('ui-button-text');\n", " tooltip_span.html(tooltip);\n", "\n", " button.append(icon_img);\n", " button.append(tooltip_span);\n", "\n", " nav_element.append(button);\n", " }\n", "\n", " var fmt_picker_span = $('');\n", "\n", " var fmt_picker = $('');\n", " fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n", " fmt_picker_span.append(fmt_picker);\n", " nav_element.append(fmt_picker_span);\n", " this.format_dropdown = fmt_picker[0];\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = $(\n", " '', {selected: fmt === mpl.default_extension}).html(fmt);\n", " fmt_picker.append(option)\n", " }\n", "\n", " // Add hover states to the ui-buttons\n", " $( \".ui-button\" ).hover(\n", " function() { $(this).addClass(\"ui-state-hover\");},\n", " function() { $(this).removeClass(\"ui-state-hover\");}\n", " );\n", "\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "}\n", "\n", "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n", "}\n", "\n", "mpl.figure.prototype.send_message = function(type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "}\n", "\n", "mpl.figure.prototype.send_draw_message = function() {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n", " }\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_resize = function(fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1]);\n", " fig.send_message(\"refresh\", {});\n", " };\n", "}\n", "\n", "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n", " var x0 = msg['x0'] / mpl.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n", " var x1 = msg['x1'] / mpl.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0, 0, fig.canvas.width, fig.canvas.height);\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "}\n", "\n", "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "}\n", "\n", "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch(cursor)\n", " {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "}\n", "\n", "mpl.figure.prototype.handle_message = function(fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "}\n", "\n", "mpl.figure.prototype.handle_draw = function(fig, msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "}\n", "\n", "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Called whenever the canvas gets updated.\n", " this.send_message(\"ack\", {});\n", "}\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function(fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = \"image/png\";\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src);\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig[\"handle_\" + msg_type];\n", " } catch (e) {\n", " console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n", " }\n", " }\n", " };\n", "}\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function(e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e)\n", " e = window.event;\n", " if (e.target)\n", " targ = e.target;\n", " else if (e.srcElement)\n", " targ = e.srcElement;\n", " if (targ.nodeType == 3) // defeat Safari bug\n", " targ = targ.parentNode;\n", "\n", " // jQuery normalizes the pageX and pageY\n", " // pageX,Y are the mouse positions relative to the document\n", " // offset() returns the position of the element relative to the document\n", " var x = e.pageX - $(targ).offset().left;\n", " var y = e.pageY - $(targ).offset().top;\n", "\n", " return {\"x\": x, \"y\": y};\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys (original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object')\n", " obj[key] = original[key]\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", " if (name === 'button_press')\n", " {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * mpl.ratio;\n", " var y = canvas_pos.y * mpl.ratio;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " // Handle any extra behaviour associated with a key event\n", "}\n", "\n", "mpl.figure.prototype.key_event = function(event, name) {\n", "\n", " // Prevent repeat events\n", " if (name == 'key_press')\n", " {\n", " if (event.which === this._key)\n", " return;\n", " else\n", " this._key = event.which;\n", " }\n", " if (name == 'key_release')\n", " this._key = null;\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which != 17)\n", " value += \"ctrl+\";\n", " if (event.altKey && event.which != 18)\n", " value += \"alt+\";\n", " if (event.shiftKey && event.which != 16)\n", " value += \"shift+\";\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, {key: value,\n", " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n", " if (name == 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message(\"toolbar_button\", {name: name});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function() {\n", " comm.close()\n", " };\n", " ws.send = function(m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function(msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overridden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data'])\n", " });\n", " return ws;\n", "}\n", "\n", "mpl.mpl_figure_comm = function(comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = $(\"#\" + id);\n", " var ws_proxy = comm_websocket_adapter(comm)\n", "\n", " function ondownload(figure, format) {\n", " window.open(figure.imageObj.src);\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy,\n", " ondownload,\n", " element.get(0));\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element.get(0);\n", " fig.cell_info = mpl.find_output_cell(\"\");\n", " if (!fig.cell_info) {\n", " console.error(\"Failed to find cell for figure\", id, fig);\n", " return;\n", " }\n", "\n", " var output_index = fig.cell_info[2]\n", " var cell = fig.cell_info[0];\n", "\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", " var width = fig.canvas.width/mpl.ratio\n", " fig.root.unbind('remove')\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('
\n", "
\n", "
\n", " Dataset Size and Overfitting \n", "\n", "Now we examine how our training and test error changes as the number of observations increase." ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "C:\\Users\\FSU\\Anaconda2\\envs\\ML\\lib\\site-packages\\ipykernel_launcher.py:10: FutureWarning: `rcond` parameter will change to the default of machine precision times ``max(M, N)`` where M and N are the input matrix dimensions.\n", "To use the future default and silence this warning we advise to pass `rcond=None`, to keep using the old, explicitly pass `rcond=-1`.\n", " # Remove the CWD from sys.path while we load stuff.\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.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n", " 'ui-button-icon-only');\n", " button.attr('role', 'button');\n", " button.attr('aria-disabled', 'false');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", "\n", " var icon_img = $('');\n", " icon_img.addClass('ui-button-icon-primary ui-icon');\n", " icon_img.addClass(image);\n", " icon_img.addClass('ui-corner-all');\n", "\n", " var tooltip_span = $('');\n", " tooltip_span.addClass('ui-button-text');\n", " tooltip_span.html(tooltip);\n", "\n", " button.append(icon_img);\n", " button.append(tooltip_span);\n", "\n", " nav_element.append(button);\n", " }\n", "\n", " var fmt_picker_span = $('');\n", "\n", " var fmt_picker = $('');\n", " fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n", " fmt_picker_span.append(fmt_picker);\n", " nav_element.append(fmt_picker_span);\n", " this.format_dropdown = fmt_picker[0];\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = $(\n", " '', {selected: fmt === mpl.default_extension}).html(fmt);\n", " fmt_picker.append(option)\n", " }\n", "\n", " // Add hover states to the ui-buttons\n", " $( \".ui-button\" ).hover(\n", " function() { $(this).addClass(\"ui-state-hover\");},\n", " function() { $(this).removeClass(\"ui-state-hover\");}\n", " );\n", "\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "}\n", "\n", "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n", "}\n", "\n", "mpl.figure.prototype.send_message = function(type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "}\n", "\n", "mpl.figure.prototype.send_draw_message = function() {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n", " }\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_resize = function(fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1]);\n", " fig.send_message(\"refresh\", {});\n", " };\n", "}\n", "\n", "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n", " var x0 = msg['x0'] / mpl.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n", " var x1 = msg['x1'] / mpl.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0, 0, fig.canvas.width, fig.canvas.height);\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "}\n", "\n", "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "}\n", "\n", "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch(cursor)\n", " {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "}\n", "\n", "mpl.figure.prototype.handle_message = function(fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "}\n", "\n", "mpl.figure.prototype.handle_draw = function(fig, msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "}\n", "\n", "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Called whenever the canvas gets updated.\n", " this.send_message(\"ack\", {});\n", "}\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function(fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = \"image/png\";\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src);\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig[\"handle_\" + msg_type];\n", " } catch (e) {\n", " console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n", " }\n", " }\n", " };\n", "}\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function(e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e)\n", " e = window.event;\n", " if (e.target)\n", " targ = e.target;\n", " else if (e.srcElement)\n", " targ = e.srcElement;\n", " if (targ.nodeType == 3) // defeat Safari bug\n", " targ = targ.parentNode;\n", "\n", " // jQuery normalizes the pageX and pageY\n", " // pageX,Y are the mouse positions relative to the document\n", " // offset() returns the position of the element relative to the document\n", " var x = e.pageX - $(targ).offset().left;\n", " var y = e.pageY - $(targ).offset().top;\n", "\n", " return {\"x\": x, \"y\": y};\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys (original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object')\n", " obj[key] = original[key]\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", " if (name === 'button_press')\n", " {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * mpl.ratio;\n", " var y = canvas_pos.y * mpl.ratio;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " // Handle any extra behaviour associated with a key event\n", "}\n", "\n", "mpl.figure.prototype.key_event = function(event, name) {\n", "\n", " // Prevent repeat events\n", " if (name == 'key_press')\n", " {\n", " if (event.which === this._key)\n", " return;\n", " else\n", " this._key = event.which;\n", " }\n", " if (name == 'key_release')\n", " this._key = null;\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which != 17)\n", " value += \"ctrl+\";\n", " if (event.altKey && event.which != 18)\n", " value += \"alt+\";\n", " if (event.shiftKey && event.which != 16)\n", " value += \"shift+\";\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, {key: value,\n", " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n", " if (name == 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message(\"toolbar_button\", {name: name});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function() {\n", " comm.close()\n", " };\n", " ws.send = function(m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function(msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overridden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data'])\n", " });\n", " return ws;\n", "}\n", "\n", "mpl.mpl_figure_comm = function(comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = $(\"#\" + id);\n", " var ws_proxy = comm_websocket_adapter(comm)\n", "\n", " function ondownload(figure, format) {\n", " window.open(figure.imageObj.src);\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy,\n", " ondownload,\n", " element.get(0));\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element.get(0);\n", " fig.cell_info = mpl.find_output_cell(\"\");\n", " if (!fig.cell_info) {\n", " console.error(\"Failed to find cell for figure\", id, fig);\n", " return;\n", " }\n", "\n", " var output_index = fig.cell_info[2]\n", " var cell = fig.cell_info[0];\n", "\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", " var width = fig.canvas.width/mpl.ratio\n", " fig.root.unbind('remove')\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('
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\n", " Probability and Bayesian Primer \n", "\n", "Let our dataset be given by $\\mathcal{D} = \\lbrace \\mathbf{x}, \\mathbf{t} \\rbrace$ and our model by $y(x,\\mathbf{w})$. Our model is parameterized by $\\mathbf{w}$ where $\\dim{\\mathbf{w}}=L$.\n", "\n", "The **joint distribution** tells us everything about the data and is given by\n", "\n", "$$p(w, \\mathcal{D})$$\n", "\n", "The **product rule** for probability theory allows us to express the joint distribution in terms of conditional and marginal distributions and is given by\n", "\n", "$$p(w, \\mathcal{D}) = p(w \\lvert \\mathcal{D})p(\\mathcal{D}) = p(\\mathcal{D} \\lvert w)p(w)$$\n", "\n", "The **sum rule** for probability theory allows us to eliminate certain variables and find what are called *marginal* distributions. It is given by\n", "\n", "$$p(w) = \\int p(w, \\mathcal{D}) d\\mathcal{D}$$\n", "$$p(\\mathcal{D}) = \\int p(w,\\mathcal{D}) dw$$\n", "for continuous random variables, and\n", "$$p(w) = \\sum_{\\mathcal{D}} p(w, \\mathcal{D})$$\n", "$$p(\\mathcal{D}) = \\sum_{w} p(w,\\mathcal{D})$$\n", "for discrete random variables.\n", "\n", "Dividing by $p(\\mathcal{D})$ in the product rule above we arrive at **Bayes Formula**.\n", "\n", "$$p(w \\lvert \\mathcal{D}) = \\frac{p(\\mathcal{D}\\lvert w)p(w)}{p(\\mathcal{D})}$$\n", "\n", "where\n", "\n", "$$p(\\mathcal{D}\\lvert w) \\hspace{5pt} \\text{is called the Likelihood}$$\n", "\n", "$$p(w) \\hspace{5pt} \\text{is called the Prior}$$\n", "\n", "$$p(\\mathcal{D}) \\hspace{5pt} \\text{is called the Marginal}$$\n", "\n", "$$p(w \\lvert \\mathcal{D}) \\hspace{5pt} \\text{is called the Posterior}$$\n", "\n", "Another common way of expressing Bayes Formula involves combining both the product and sum rules\n", "\n", "$$p(w \\lvert \\mathcal{D}) = \\frac{p(\\mathcal{D}\\lvert w)p(w)}{\\sum_{w}p(\\mathcal{D}\\lvert w)p(w)}$$\n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "
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\n", " Probabilistic Perspective " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us now cast the regression problem in a stochastic framework. Again our goal is to predict the target value $t$ given an input $x$ based on a set of traning data $\\mathbf{x} = (x_1,\\dots,x_n)^T$ and and their corresponding target values $\\mathbf{t} = (t_1,\\dots,t_n)^T$. We know there is noise in the data, there always is, and until now we have not taken it into consideration. Now however we will directly consider the uncertainty in our predictions. We will assume that given an input $x$, our target value $t$ is given by a Gaussian distribution with a mean $y(x,\\mathbf{w})$ and a standard deviation $\\sigma$. As per our previous formulation the mean $y(x,\\mathbf{w})$ is our prediction, however unlike our previous formulation here we explicitly consider our uncertainty via the standard deviation $\\sigma$. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$p(t \\lvert x,\\mathbf{w},\\sigma) = \\mathcal{N}(t\\lvert y(x,\\mathbf{w}),\\sigma)$$\n", "\n", "Now if we assume that our training data $\\lbrace \\mathbf{x}, \\mathbf{t} \\rbrace$ are independent and identically distributed (iid), then we can express our likelihood function as\n", "\n", "$$p(\\mathbf{t} \\lvert \\mathbf{x},\\mathbf{w},\\sigma) = \\prod_{n=1}^N \\mathcal{N}(t_n\\lvert y(x_n,\\mathbf{w}),\\sigma)$$\n", "\n", "We notice a few things here. Firstly in casting this problem stochastically we have increased the number of parameters from $\\dim{w}$ to $\\dim{w}+1$ for the extra $\\sigma$ term. By assuming $\\sigma$ to be constant for all values of $x$ we're assuming the noise is what is called, homoscedastic. If instead we believed the noise were to change for different values of $x$, then we could allow the standard deviation to be a function of the input $\\sigma(x)$, in which case we call the noise, heteroscedastic.\n", "\n", "This new formulation changes our task **T**. Before we were fitting an $M^{th}$ order polynomial to the data, here however we wish to fit parameters $\\theta = \\lbrace{\\mathbf{w}, \\sigma}\\rbrace$. Our goal this time will be to find the parameters $\\theta^*$ that maximize the probability $p(\\mathbf{t} \\lvert \\mathbf{x}, \\mathbf{w}, \\sigma)$. This goal, maximizing the likelihood, is the quintessential frequentist example. Where before we explicity chose the performance measure **P** to be the (SSE), here we will analytically optimize the likelihood and use the result as our performance measure." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The log function is monotonic, therefore, the extrema of $\\ln f(x)$ equal the extrema of $f(x)$. Considering this and the fact that the logarithm of exponential distributions are analytically convenient, we will optimize the log of the likelihood.\n", "\n", "$$ \\ln p(\\mathbf{t} \\lvert \\mathbf{x}, \\mathbf{w}, \\sigma) = -\\frac{1}{2\\sigma} \\sum_{n=1}^N \\lbrace y(x_n,\\mathbf{w}) - t_n \\rbrace^2 - \\frac{N}{2} \\ln \\sigma- \\frac{N}{2} \\ln(2\\pi)$$\n", "\n", "Let us maximize the log likelihood with respect to $\\mathbf{w}$. We notice two things, firstly that the second two terms are independent of $\\mathbf{w}$, and secondly that maximizing the likelihood is equivalent to minimizing the negative log likelihood. \n", "\n", "$$ \\mathbf{w}_{ML} = \\max_\\mathbf{w} \\hspace{4pt} \\ln p(\\mathbf{t} \\lvert \\mathbf{x}, \\mathbf{w}, \\sigma) = \\min_\\mathbf{w} \\hspace{4pt} -\\ln p(\\mathbf{t} \\lvert \\mathbf{x}, \\mathbf{w}, \\sigma) = \\min_{\\mathbf{w}} \\frac{1}{2} \\sum_{n=1}^N \\lbrace y(x_n,\\mathbf{w}) - t_n \\rbrace^2 = \\min \\text{SSE}$$\n", "\n", "Thus the SSE has arisen as a consequence of maximizing the likelihood under the assumption of Gaussian noise. The optimal parameters found with such a technique are aptly called the **maximum likelihood** solutions (ML). With the maximum likelihood solution for $\\mathbf{w}$ found we can also find the maximum likelihood solution for the standard deviation.\n", "\n", "$$\\sigma_{ML} = \\frac{1}{N} \\sum_{n=1}^N \\lbrace y(x_n, \\mathbf{w}_{ML})-t_n \\rbrace^2$$\n", "\n", "With the optimal parameters found we can define our *predictive distribution*\n", "\n", "$$p(t \\lvert x, \\mathbf{w}_{ML}, \\sigma_{ML}) = \\mathcal{N}(t \\lvert y(x, \\mathbf{w}_{ML}), \\sigma_{ML})$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "
\n", "
\n", " Let's go Deeper \n", "\n", "What is the expected value of our SSE loss function $L(t,y(\\bf{x}))$? We have dropped the dependence of $y$ on $\\bf{w}$ for notational convenience and here $y$ operates element-wise on the vector $\\bf{x}$.\n", "\n", "$$ \\mathbb{E}[L] = \\iint L(t,y(\\mathbf{x}))p(\\mathbf{x},t)\\text{d}\\mathbf{x}\\text{d}t $$\n", "\n", "$$ \\mathbb{E}[L] = \\iint \\lbrace y(\\mathbf{x})-t \\rbrace^2 p(\\mathbf{x},t)\\text{d}\\mathbf{x}\\text{d}t $$\n", "\n", "We wish to choose a $y(\\mathbf{x})$ that minimizes the expected loss. We use the calculus of variations\n", "\n", "$$ \\frac{\\partial \\mathbb{E}[L]}{\\partial y(\\mathbf{x})} = 2 \\int \\lbrace y(\\mathbf{x})-t \\rbrace p(\\mathbf{x},t)\\text{d}t $$\n", "\n", "We now set it to zero and solve for $y(\\mathbf{x})$\n", "\n", "$$ y(\\mathbf{x})\\int p(\\mathbf{x},t)\\text{d}t - \\int t p(\\mathbf{x},t)\\text{d}t = 0$$\n", "\n", "$$ y(\\mathbf{x}) = \\frac{\\int t p(\\mathbf{x},t)\\text{d}t}{p(\\mathbf{x})} = \\int t p(t \\lvert \\mathbf{x})\\text{d}t = \\mathbb{E}_t[t \\lvert \\mathbf{x}]$$\n", "\n", "**Another way to view this result comes as follows**\n", "\n", "$$ \\lbrace y(\\mathbf{x}) - t \\rbrace^2 = \\lbrace y(\\mathbf{x}) -\\mathbb{E}_t[t \\lvert \\mathbf{x}] + \\mathbb{E}_t[t \\lvert \\mathbf{x}]- t \\rbrace^2$$\n", "\n", "$$ \\lbrace y(\\mathbf{x}) - t \\rbrace^2 = \\lbrace y(\\mathbf{x}) -\\mathbb{E}_t[t \\lvert \\mathbf{x}] \\rbrace^2 + 2\\lbrace y(\\mathbf{x}) -\\mathbb{E}_t[t \\lvert \\mathbf{x}] \\rbrace\\lbrace \\mathbb{E}_t[t \\lvert \\mathbf{x}]- t \\rbrace + \\lbrace \\mathbb{E}_t[t \\lvert \\mathbf{x}]- t \\rbrace^2$$\n", "\n", "We now substitute this expression into the loss function\n", "\n", "$$ \\mathbb{E}[L] = \\iint \\lbrace y(\\mathbf{x}) -\\mathbb{E}_t[t \\lvert \\mathbf{x}] \\rbrace^2 p(\\mathbf{x},t)\\text{d}\\mathbf{x}\\text{d}t + 2\\iint \\lbrace y(\\mathbf{x}) -\\mathbb{E}_t[t \\lvert \\mathbf{x}] \\rbrace\\lbrace \\mathbb{E}_t[t \\lvert \\mathbf{x}]- t \\rbrace p(\\mathbf{x},t)\\text{d}\\mathbf{x}\\text{d}t + \\iint \\lbrace \\mathbb{E}_t[t \\lvert \\mathbf{x}]- t \\rbrace^2 p(\\mathbf{x},t)\\text{d}\\mathbf{x}\\text{d}t $$\n", "\n", "Integrating over $t$ we notice that the cross term equals 0 and are left with\n", "\n", "$$ \\mathbb{E}[L] = \\int \\lbrace y(\\mathbf{x}) -\\mathbb{E}_t[t \\lvert \\mathbf{x}] \\rbrace^2 p(\\mathbf{x})\\text{d}\\mathbf{x} + \\int \\lbrace \\mathbb{E}_t[t \\lvert \\mathbf{x}]- t \\rbrace^2 p(\\mathbf{x})\\text{d}\\mathbf{x}$$\n", "\n", "Where the first term goes to zero exactly when $y(\\mathbf{x}) = \\mathbb{E}_t[t \\lvert \\mathbf{x}]$ and the second term represents the irreducible minimum loss due to noise." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "
\n", "
\n", " A Step Towards Bayes " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Frequentist \n", "\n", "$$p(\\bf{t} \\lvert \\bf{x},\\bf{w},\\sigma)$$\n", "\n", "This is the likelihood function and is a key to understanding the frequentist perspective, or the perspective of frequencies of random repeatable events. It poses the problem as *\"what is the probability of the data, given a certain set of parameters\"*. In the previous example we found the set of parameters that maximized this distribution using a popular estimator called maximum likelihood. In other words, we estimated the parameters that maximize the probability of our data, which is maximizing the probability of the true data **ON AVERAGE**. This is the underlying distinction between Bayes and frequentist perspectives.\n", "\n", " In the frequentist perspective, the data is considered the random variable and it aims at finding solutions considering our uncertainty of the observed data. The parameters are considered fixed and their value is determined by some form of 'estimator'. Error bars or uncertainty in the parameters are found by considering the distribution of possible datasets $\\mathcal{D}$. \n", "\n", "**In a frequentist perspective the data is the random variable**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Bayesian \n", "\n", "$$p(\\bf{w} \\lvert \\bf{x},\\bf{t},\\sigma) \\propto p(\\bf{t} \\lvert \\bf{x},\\bf{w},\\sigma)p(\\bf{w})$$\n", "\n", "\n", "On the right we have the Bayesian viewpoint. From this perspective there is only one single data set $\\mathcal{D}$, the one that is actually observed. The uncertainty in the parameters is expressed via the probability distribution describing them. \n", "\n", "**In a Bayesian perspective the parameters are the random variable.**\n", "\n", "Let us examine the curve fitting problem from a more Bayesian perspective by introducing a prior over the parameters. For ease of exposition we will assume a simple Gaussian form\n", "\n", "$$p(\\bf{w} \\lvert \\alpha) = \\mathcal{N}(\\bf{w} \\lvert \\bf{0}, \\alpha \\bf{I}) = \\bigg ( \\frac{1}{2\\pi \\alpha} \\bigg)^{(M+1)/2}\\exp\\bigg\\{-\\frac{1}{2\\alpha}\\bf{w}^T\\bf{w}\\bigg\\}$$\n", "\n", "Using Bayes theorem above we can express the posterior in terms of the likelihood and the prior. We interpret their product as a distribution describing the probability of $\\bf{w}$ given the observed data $\\mathcal{D} = \\{\\bf{x},\\bf{t}\\}$. If we find the $\\bf{w}$ that maximize this posterior distribution we are finding the so-called **maximum posterior** solutions (MAP). Using techniques similar to ML we take the negative logarithm of the posterior and find that its maximum is given by the minimum of \n", "\n", "$$\\bf{w}_{MAP} = \\min_{\\bf{w}} \\frac{1}{2\\sigma} \\sum_{n=1}^N \\lbrace y(x_n,\\mathbf{w}) - t_n \\rbrace^2 + \\frac{1}{2\\alpha}\\bf{w}^T\\bf{w}$$\n", "\n", "We see that maximizing the posterior distribution is equivalent to minimizing the SSE with a shrinkage regularization term where $\\lambda = \\sigma/\\alpha$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "
\n", "
\n", " Full Bayes \n", "\n", "Even though we've included a prior we're still making a point estimate of $\\bf{w}$. A true fully Bayesian approach involves consistently applying the sum and product rules... " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " Coin Flipping with Bayes\n", "\n", "To understand the difficult and complexity in using Bayes, let us explore a very simple problem. Coin flipping." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "flip = lambda: np.random.binomial(n=1,p=0.5) # Bernoulli random variable" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let us pose the problem of learning the probability distribution of a new unseen coin with probability of heads $q$ and tails $1-q$. We begin by assuming no information about the coin. Therefore our prior will be a uniform distribution. The likelihood function will be a Bernoulli and the posterior will be the normalized product of the two.\n", "\n", "$$ p(q \\lvert x) = \\frac{p(x \\lvert q)p(q)}{p(x)}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us derive the update rule. First we must define all the terms on the left hand side. We start with the Likelihood...\n", "\n", "$$p(x_1 \\lvert q) = q^{x_1} (1-q)^{1-x_1}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the prior, we begin with the assumption that every coin is equally likely. We will allow the data to teach us the properties of the coin\n", "\n", "$$p_0(q) = 1$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For the marginal we must integrate over all possible values of $q$.\n", "\n", "$$p(x_1) = \\int_0^1 p(x_1 \\lvert q)p_0(q)\\text{d}q = \\int_0^1 q^{x_1} (1-q)^{1-x_1}\\text{d}q$$\n", "\n", "Using repeated integration by parts this integral can be shown to equal\n", "\n", "$$\\int_0^1 q^{x_1} (1-q)^{1-x_1}\\text{d}q = \\frac{x_1!(1-x_1)!}{2}$$\n", "\n", "However, we observe that $x \\in \\{0,1\\}$ and under either circumstance, the numerator is always equal to 1, therefore \n", "\n", "$$\\int_0^1 q^{x_1} (1-q)^{1-x_1}\\text{d}q = \\frac{1}{2}$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We combine the above equations to see what the updated prior would look like after observing one coin toss...\n", "\n", "$$p_1(q \\lvert x_1) = \\frac{p(x_1 \\lvert q)p_0(q)}{p(x_1)} = \\frac{q^{x_1} (1-q)^{1-x_1}(1)}{\\frac{1}{2}} = 2q^{x_1} (1-q)^{1-x_1}$$\n", "\n", "Notice how the posterior of the previous observation becomes the prior of the next observation.\n", "\n", "$$p_2(q \\lvert x_1, x_2) = \\frac{p(x_2 \\lvert q)p_1(q\\lvert x_1)}{\\int_0^1 p(x_2 \\lvert q)p_1(q \\lvert x_1)\\text{d}q }$$" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", " $(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " '');\n", " var titletext = $(\n", " '');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $('');\n", "\n", " canvas_div.attr('style', 'position: relative; 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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.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n", " 'ui-button-icon-only');\n", " button.attr('role', 'button');\n", " button.attr('aria-disabled', 'false');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", "\n", " var icon_img = $('');\n", " icon_img.addClass('ui-button-icon-primary ui-icon');\n", " icon_img.addClass(image);\n", " icon_img.addClass('ui-corner-all');\n", "\n", " var tooltip_span = $('');\n", " tooltip_span.addClass('ui-button-text');\n", " tooltip_span.html(tooltip);\n", "\n", " button.append(icon_img);\n", " button.append(tooltip_span);\n", "\n", " nav_element.append(button);\n", " }\n", "\n", " var fmt_picker_span = $('');\n", "\n", " var fmt_picker = $('');\n", " fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n", " fmt_picker_span.append(fmt_picker);\n", " nav_element.append(fmt_picker_span);\n", " this.format_dropdown = fmt_picker[0];\n", "\n", " for (var ind in mpl.extensions) {\n", " var fmt = mpl.extensions[ind];\n", " var option = $(\n", " '', {selected: fmt === mpl.default_extension}).html(fmt);\n", " fmt_picker.append(option)\n", " }\n", "\n", " // Add hover states to the ui-buttons\n", " $( \".ui-button\" ).hover(\n", " function() { $(this).addClass(\"ui-state-hover\");},\n", " function() { $(this).removeClass(\"ui-state-hover\");}\n", " );\n", "\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "}\n", "\n", "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n", " // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n", " // which will in turn request a refresh of the image.\n", " this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n", "}\n", "\n", "mpl.figure.prototype.send_message = function(type, properties) {\n", " properties['type'] = type;\n", " properties['figure_id'] = this.id;\n", " this.ws.send(JSON.stringify(properties));\n", "}\n", "\n", "mpl.figure.prototype.send_draw_message = function() {\n", " if (!this.waiting) {\n", " this.waiting = true;\n", " this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n", " }\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " var format_dropdown = fig.format_dropdown;\n", " var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n", " fig.ondownload(fig, format);\n", "}\n", "\n", "\n", "mpl.figure.prototype.handle_resize = function(fig, msg) {\n", " var size = msg['size'];\n", " if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n", " fig._resize_canvas(size[0], size[1]);\n", " fig.send_message(\"refresh\", {});\n", " };\n", "}\n", "\n", "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n", " var x0 = msg['x0'] / mpl.ratio;\n", " var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n", " var x1 = msg['x1'] / mpl.ratio;\n", " var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n", " x0 = Math.floor(x0) + 0.5;\n", " y0 = Math.floor(y0) + 0.5;\n", " x1 = Math.floor(x1) + 0.5;\n", " y1 = Math.floor(y1) + 0.5;\n", " var min_x = Math.min(x0, x1);\n", " var min_y = Math.min(y0, y1);\n", " var width = Math.abs(x1 - x0);\n", " var height = Math.abs(y1 - y0);\n", "\n", " fig.rubberband_context.clearRect(\n", " 0, 0, fig.canvas.width, fig.canvas.height);\n", "\n", " fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n", "}\n", "\n", "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n", " // Updates the figure title.\n", " fig.header.textContent = msg['label'];\n", "}\n", "\n", "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n", " var cursor = msg['cursor'];\n", " switch(cursor)\n", " {\n", " case 0:\n", " cursor = 'pointer';\n", " break;\n", " case 1:\n", " cursor = 'default';\n", " break;\n", " case 2:\n", " cursor = 'crosshair';\n", " break;\n", " case 3:\n", " cursor = 'move';\n", " break;\n", " }\n", " fig.rubberband_canvas.style.cursor = cursor;\n", "}\n", "\n", "mpl.figure.prototype.handle_message = function(fig, msg) {\n", " fig.message.textContent = msg['message'];\n", "}\n", "\n", "mpl.figure.prototype.handle_draw = function(fig, msg) {\n", " // Request the server to send over a new figure.\n", " fig.send_draw_message();\n", "}\n", "\n", "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n", " fig.image_mode = msg['mode'];\n", "}\n", "\n", "mpl.figure.prototype.updated_canvas_event = function() {\n", " // Called whenever the canvas gets updated.\n", " this.send_message(\"ack\", {});\n", "}\n", "\n", "// A function to construct a web socket function for onmessage handling.\n", "// Called in the figure constructor.\n", "mpl.figure.prototype._make_on_message_function = function(fig) {\n", " return function socket_on_message(evt) {\n", " if (evt.data instanceof Blob) {\n", " /* FIXME: We get \"Resource interpreted as Image but\n", " * transferred with MIME type text/plain:\" errors on\n", " * Chrome. But how to set the MIME type? It doesn't seem\n", " * to be part of the websocket stream */\n", " evt.data.type = \"image/png\";\n", "\n", " /* Free the memory for the previous frames */\n", " if (fig.imageObj.src) {\n", " (window.URL || window.webkitURL).revokeObjectURL(\n", " fig.imageObj.src);\n", " }\n", "\n", " fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n", " evt.data);\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", " else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n", " fig.imageObj.src = evt.data;\n", " fig.updated_canvas_event();\n", " fig.waiting = false;\n", " return;\n", " }\n", "\n", " var msg = JSON.parse(evt.data);\n", " var msg_type = msg['type'];\n", "\n", " // Call the \"handle_{type}\" callback, which takes\n", " // the figure and JSON message as its only arguments.\n", " try {\n", " var callback = fig[\"handle_\" + msg_type];\n", " } catch (e) {\n", " console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n", " return;\n", " }\n", "\n", " if (callback) {\n", " try {\n", " // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n", " callback(fig, msg);\n", " } catch (e) {\n", " console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n", " }\n", " }\n", " };\n", "}\n", "\n", "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n", "mpl.findpos = function(e) {\n", " //this section is from http://www.quirksmode.org/js/events_properties.html\n", " var targ;\n", " if (!e)\n", " e = window.event;\n", " if (e.target)\n", " targ = e.target;\n", " else if (e.srcElement)\n", " targ = e.srcElement;\n", " if (targ.nodeType == 3) // defeat Safari bug\n", " targ = targ.parentNode;\n", "\n", " // jQuery normalizes the pageX and pageY\n", " // pageX,Y are the mouse positions relative to the document\n", " // offset() returns the position of the element relative to the document\n", " var x = e.pageX - $(targ).offset().left;\n", " var y = e.pageY - $(targ).offset().top;\n", "\n", " return {\"x\": x, \"y\": y};\n", "};\n", "\n", "/*\n", " * return a copy of an object with only non-object keys\n", " * we need this to avoid circular references\n", " * http://stackoverflow.com/a/24161582/3208463\n", " */\n", "function simpleKeys (original) {\n", " return Object.keys(original).reduce(function (obj, key) {\n", " if (typeof original[key] !== 'object')\n", " obj[key] = original[key]\n", " return obj;\n", " }, {});\n", "}\n", "\n", "mpl.figure.prototype.mouse_event = function(event, name) {\n", " var canvas_pos = mpl.findpos(event)\n", "\n", " if (name === 'button_press')\n", " {\n", " this.canvas.focus();\n", " this.canvas_div.focus();\n", " }\n", "\n", " var x = canvas_pos.x * mpl.ratio;\n", " var y = canvas_pos.y * mpl.ratio;\n", "\n", " this.send_message(name, {x: x, y: y, button: event.button,\n", " step: event.step,\n", " guiEvent: simpleKeys(event)});\n", "\n", " /* This prevents the web browser from automatically changing to\n", " * the text insertion cursor when the button is pressed. We want\n", " * to control all of the cursor setting manually through the\n", " * 'cursor' event from matplotlib */\n", " event.preventDefault();\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " // Handle any extra behaviour associated with a key event\n", "}\n", "\n", "mpl.figure.prototype.key_event = function(event, name) {\n", "\n", " // Prevent repeat events\n", " if (name == 'key_press')\n", " {\n", " if (event.which === this._key)\n", " return;\n", " else\n", " this._key = event.which;\n", " }\n", " if (name == 'key_release')\n", " this._key = null;\n", "\n", " var value = '';\n", " if (event.ctrlKey && event.which != 17)\n", " value += \"ctrl+\";\n", " if (event.altKey && event.which != 18)\n", " value += \"alt+\";\n", " if (event.shiftKey && event.which != 16)\n", " value += \"shift+\";\n", "\n", " value += 'k';\n", " value += event.which.toString();\n", "\n", " this._key_event_extra(event, name);\n", "\n", " this.send_message(name, {key: value,\n", " guiEvent: simpleKeys(event)});\n", " return false;\n", "}\n", "\n", "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n", " if (name == 'download') {\n", " this.handle_save(this, null);\n", " } else {\n", " this.send_message(\"toolbar_button\", {name: name});\n", " }\n", "};\n", "\n", "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n", " this.message.textContent = tooltip;\n", "};\n", "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n", "\n", "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n", "\n", "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n", " // Create a \"websocket\"-like object which calls the given IPython comm\n", " // object with the appropriate methods. Currently this is a non binary\n", " // socket, so there is still some room for performance tuning.\n", " var ws = {};\n", "\n", " ws.close = function() {\n", " comm.close()\n", " };\n", " ws.send = function(m) {\n", " //console.log('sending', m);\n", " comm.send(m);\n", " };\n", " // Register the callback with on_msg.\n", " comm.on_msg(function(msg) {\n", " //console.log('receiving', msg['content']['data'], msg);\n", " // Pass the mpl event to the overridden (by mpl) onmessage function.\n", " ws.onmessage(msg['content']['data'])\n", " });\n", " return ws;\n", "}\n", "\n", "mpl.mpl_figure_comm = function(comm, msg) {\n", " // This is the function which gets called when the mpl process\n", " // starts-up an IPython Comm through the \"matplotlib\" channel.\n", "\n", " var id = msg.content.data.id;\n", " // Get hold of the div created by the display call when the Comm\n", " // socket was opened in Python.\n", " var element = $(\"#\" + id);\n", " var ws_proxy = comm_websocket_adapter(comm)\n", "\n", " function ondownload(figure, format) {\n", " window.open(figure.imageObj.src);\n", " }\n", "\n", " var fig = new mpl.figure(id, ws_proxy,\n", " ondownload,\n", " element.get(0));\n", "\n", " // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n", " // web socket which is closed, not our websocket->open comm proxy.\n", " ws_proxy.onopen();\n", "\n", " fig.parent_element = element.get(0);\n", " fig.cell_info = mpl.find_output_cell(\"\");\n", " if (!fig.cell_info) {\n", " console.error(\"Failed to find cell for figure\", id, fig);\n", " return;\n", " }\n", "\n", " var output_index = fig.cell_info[2]\n", " var cell = fig.cell_info[0];\n", "\n", "};\n", "\n", "mpl.figure.prototype.handle_close = function(fig, msg) {\n", " var width = fig.canvas.width/mpl.ratio\n", " fig.root.unbind('remove')\n", "\n", " // Update the output cell to use the data from the current canvas.\n", " fig.push_to_output();\n", " var dataURL = fig.canvas.toDataURL();\n", " // Re-enable the keyboard manager in IPython - without this line, in FF,\n", " // the notebook keyboard shortcuts fail.\n", " IPython.keyboard_manager.enable()\n", " $(fig.parent_element).html('
\n", "
\n", "
\n", " Strengths/Weaknesses " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "
\n", "
\n", " Summary " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", " Frequentist\n", "\n", " 1. Data is the random variable\n", " 2. Maximizing the likelihood (ML) is estimating which parameters make the data most probable on average\n", " 3. Susceptible to bias and over fitting\n", " 4. Easily interpretable\n", "\n", "
\n", " Bayesian\n", "\n", " 1. Parameters are the random variable\n", " 2. Maximizing the posterior (MAP) is finding which parameters are most probable given the data\n", " 3. Priors are often chosen for mathematical convenience rather than a reflection of prior beliefs, difficult to interpret\n", " 4. Protection against over fitting\n", "\n", "
\n", " Full Bayes\n", "\n", " 1. Parameters and data are random variables\n", " 2. Alternate between considering the data and parameters as being observed and learn the full posterior distribution\n", " 3. Learns full distribution, everything about the data\n", " 4. With a well chosen prior, bias and over fitting is often of little concern\n", " 5. In practice it is computationally intractable. Must use approximation methods such as variational inference and Monte Carlo techniques" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "
\n", "
\n", " Example " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "
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\n", "
\n", " References " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.5" } }, "nbformat": 4, "nbformat_minor": 2 }