{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Intermittent Cointegration Inference\n", "\n", "## Python notebook implementation\n", "\n", "The code in this notebook is adapted from the original MATLAB implementation by Chris Bracegirdle for the paper [*Bayesian Conditional Cointegration*](http://icml.cc/2012/papers/570.pdf) presented at [*ICML 2012*](http://icml.cc/2012/).\n", "\n", "Contact me" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "' Bayesian Intermittent Cointegration \\nImplementation of inference for intermittent cointegration\\n\\nWritten by Chris Bracegirdle\\n(c) Chris Bracegirdle 2015. All rights reserved.'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from numpy import sum, polyfit, inf, array, isreal, zeros, ones, insert, append, tile, concatenate, atleast_2d\n", "from scipy import log, sqrt, exp, std\n", "from scipy.misc import logsumexp\n", "from scipy.stats import norm \n", "\n", "\"\"\" Bayesian Intermittent Cointegration \n", "Implementation of inference for intermittent cointegration\n", "\n", "Written by Chris Bracegirdle\n", "(c) Chris Bracegirdle 2015. All rights reserved.\"\"\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Some notebook-specific requirements here: we're going to show some plots, so provide a helper function for formatting a live updating the charts in the notebook." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "import matplotlib.dates as mdates\n", "%matplotlib notebook\n", "xtickfmt = mdates.DateFormatter('%b %Y')\n", "def pltsin(ax, x, y, line_number=0, update_x=False):\n", " if ax.lines and len(ax.lines)>line_number:\n", " line = ax.lines[line_number]\n", " if update_x: line.set_xdata(x)\n", " line.set_ydata(y)\n", " else:\n", " ax.plot(x,y)\n", " ax.relim()\n", " ax.autoscale_view()\n", " ax.get_figure().canvas.draw()\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following functions are copied almost exactly from the standard Bayesian Cointegration code except are updated to support vector notation for speed purposes. Numpy works much more quickly with vector functions than looping over array elements!" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def LinearRegression(x,y):\n", " slope, intercept = polyfit(x, y, 1)\n", " std_eta = std( y - intercept - slope * x , ddof=1 )\n", " return slope, intercept, std_eta" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def ParCointInference(epsilon,std_eta,pitgitm1,pi2):\n", " logalpha,loglik,fstats = Filtering(epsilon,std_eta,pitgitm1,pi2)\n", " loggamma,moment1,moment2 = Smoothing(pitgitm1,logalpha,fstats)\n", " return logalpha,loglik,loggamma,moment1,moment2" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def vCalcLogAreaLog(logf,logF):\n", " logf = array(logf); logF = array(logF)\n", " my_ones = tile([1,-1], logf.shape+(1,)).T\n", " lncdf = norm.logcdf(my_ones, loc=exp(logf).real, scale=exp(0.5*array(logF)).real)\n", " logarea = logsumexp(lncdf,b=my_ones,axis=0)-log(2)\n", " return logarea\n", "\n", "def vCalcLogMomentsLog(logf,logF,logarea):\n", " logf = array(logf); logF = array(logF)\n", " lnpdfplogFmlogarea = norm.logpdf(tile([1,-1], logf.shape+(1,)).T, loc=exp(logf).real, scale=exp(0.5*logF).real) + logF - logarea\n", " logmoment1 = logsumexp(concatenate([lnpdfplogFmlogarea, [logf]]),b=tile([-0.5, 0.5, 1], logf.shape+(1,)).T,axis=0)\n", " logmoment2 = logsumexp(concatenate([logf+lnpdfplogFmlogarea, lnpdfplogFmlogarea, [2*logf], [logF]]),\n", " b=tile([-0.5, 0.5, -0.5, -0.5, 1, 1], logf.shape+(1,)).T,axis=0).real\n", " return logmoment1,logmoment2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A function that imports the relevant source data of gas prices. For reference, see the paper." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\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", " 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", " fig.waiting = false;\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " this.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 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);\n", " canvas.attr('height', height);\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", "$(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $(' ')\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button =$('');\n", " button.click(method_name, toolbar_event);\n", " button.mouseover(tooltip, toolbar_mouse_event);\n", " nav_element.append(button);\n", " }\n", "\n", " // Add the status bar.\n", " var status_bar = $('');\n", " nav_element.append(status_bar);\n", " this.message = status_bar[0];\n", "\n", " // Add the close button to the window.\n", " var buttongrp =$('
');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Close figure', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " event.shiftKey = false;\n", " // Send a \"J\" for go to next cell\n", " event.which = 74;\n", " event.keyCode = 74;\n", " manager.command_mode();\n", " manager.handle_keydown(event);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "slope,intercept,std_eta\n", "0.952118070187 0.0632855162392 0.0730737446967\n" ] } ], "source": [ "x,y,dates = LoadGasData()\n", "\n", "pi2 = zeros([2,1])\n", "pi2[0]=0.95; pi2[1]=1-pi2[0]\n", "\n", "pitgitm1 = zeros([2,2])\n", "pitgitm1[0,0]=1-1.0/230\n", "pitgitm1[0,1]=1.0/15\n", "pitgitm1[1,:]=1-pitgitm1[0,:]\n", "\n", "slope,interc,std_eta = LinearRegression(x,y)\n", "\n", "slopes=[]; intercs=[]; std_etas=[]; logliks = [-inf]\n", "\n", "fig, ax = plt.subplots(2,3,figsize=(12, 6), num=\"Intermittent Cointegration Inference: Gas\")\n", "map(lambda myax: myax.xaxis.set_major_formatter(xtickfmt), ax[0:1,0:1].flatten()); \n", "ax[0,0].plot(dates,x,dates,y); ax[0,0].set_xticks(xticks)\n", "plt.show()\n", "for i in range(1000):\n", " slopes.append(slope)\n", " intercs.append(interc)\n", " std_etas.append(std_eta)\n", "\n", " pltsin(ax[1,2],range(i+1),slopes,update_x=True); pltsin(ax[1,2],range(i+1),intercs,1,update_x=True)\n", "\n", " # calculate epsilon\n", " epsilon=y-interc-slope*x;\n", " pltsin(ax[1,0],dates,epsilon); ax[1,0].set_xticks(xticks)\n", "\n", " logalpha,loglik,loggamma,moment1,moment2 = ParCointInference(epsilon,std_eta,pitgitm1,pi2)\n", "\n", " alpha = exp(logalpha).real\n", " logliks.append(loglik)\n", " pltsin(ax[0,2],range(i+1),logliks[1:],update_x=True)\n", "\n", " # plot cointegration filtered posterior\n", " pltsin(ax[0,1],dates[1:],alpha[1:,1]); ax[0,1].set_xticks(xticks)\n", "\n", " gamma = exp(loggamma).real\n", "\n", " pltsin(ax[0,1],dates[1:],gamma[1:,1],1)\n", "\n", " # plot moment1 with error\n", " ax[1,1].clear(); pltsin(ax[1,1],dates[1:],moment1[1:]); ax[1,1].set_xticks(xticks)\n", " ax[1,1].fill_between(ax[1,1].lines[0].get_xdata(), moment1[1:]-sqrt(moment2[1:]-moment1[1:]**2), \n", " moment1[1:]+sqrt(moment2[1:]-moment1[1:]**2), \n", " facecolor='blue', alpha=0.3, linewidth=0)\n", "\n", " slope,interc,std_eta = EMUpdate(x,y,moment1,moment2)\n", "\n", " if logliks[-1]-logliks[-2]<0.0001: break\n", "\n", "print \"slope,intercept,std_eta\"\n", "print slope,interc,std_eta" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "And there we have it. Clockwise, the plots show:\n", "- source data\n", "- filtered (blue) and smoothed (green) posterior of class\n", "- log likelihood of the model fit\n", "- slope (blue) and intercept (green) estimate iterations\n", "- first moment of $\\phi$ with error\n", "- residuals of latest fit\n", "\n", "" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }