{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Problem sheet 3\n",
"The previous exercises gave an introduction to Python, Numpy and Pandas. Beginning with this exercise, we shift our focus to statistical learning itself. To this end, we will employ the module scikit-learn which offers many functions we will cover over the remaining semester.\n",
"\n",
"If not already done, please download the file [Advertising.csv](https://www.tu-chemnitz.de/mathematik/numa/lehre/ds-2018/exercises/Advertising.csv) and move it into a subfolder called `datasets`."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Exercise 1:\n",
"We start this exercise with the Advertising dataset known from the lecture."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We read the dataset using Pandas:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" TV radio newspaper sales\n",
"1 230.1 37.8 69.2 22.1\n",
"2 44.5 39.3 45.1 10.4\n",
"3 17.2 45.9 69.3 9.3\n"
]
}
],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"%matplotlib inline\n",
"\n",
"adv = pd.read_csv('./datasets/Advertising.csv', index_col=0)\n",
"\n",
"# Print first entries of adv\n",
"print(adv.head(3))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For convenience, we extract the values from this pandas-DataFrame"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"X = adv.values[:,0:3]\n",
"tv, radio, newspaper = np.hsplit(X,3)\n",
"Y = adv.values[:,3]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (a)\n",
"Compute for each of the 3 predictor variables **TV**, **radio** and **newspaper** simple (1-dimensional) linear regressions, e.g.\n",
"\n",
"$$ y^{TV}_i \\approx \\beta_0^{TV} + \\beta_1^{TV} \\, x_i^{TV}$$\n",
"\n",
"Use the following function:\n",
"\n",
" from sklearn.linear_model import LinearRegression\n",
" \n",
"You can use a command similar to\n",
"\n",
" print('y = %5.4f + %5.4f x TV' % (intercept, lincoef))\n",
" \n",
"to print your results in a nice fashion."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y = 7.0326 + 0.0475 x TV\n"
]
}
],
"source": [
"from sklearn.linear_model import LinearRegression\n",
"\n",
"reg_tv = LinearRegression().fit(tv, Y)\n",
"print('y = %5.4f + %5.4f x TV' % (reg_tv.intercept_, reg_tv.coef_[0]))"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y = 9.3116 + 0.2025 x radio\n"
]
}
],
"source": [
"reg_radio = LinearRegression().fit(radio, Y)\n",
"print('y = %5.4f + %5.4f x radio' % (reg_radio.intercept_, reg_radio.coef_[0]))"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y = 12.3514 + 0.0547 x newspaper\n"
]
}
],
"source": [
"reg_newspaper = LinearRegression().fit(newspaper, Y)\n",
"print('y = %5.4f + %5.4f x newspaper' % (reg_newspaper.intercept_, reg_newspaper.coef_[0]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You should observe, that the regression coefficients for **TV** and **newspaper** are very similar.\n",
"As you already know from the lecture, it is not satisfying from a mathematical point of view to restrict our investigation to the absolute values of the coefficients.\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (b)\n",
"\n",
"In the lecture you learned about different measures for assessing the quality of a linear fit.\n",
"In the last exercise, we already implemented a function to compute the mean squared error (MSE).\n",
"\n",
"This time, we want to compare the $R^2$ scores. You can use the method `score()` of a `LinearRegression` to get the $R^2$ values.\n",
"Remember that this value is the proportion of variability in $Y$ explained using **TV**, **radio** or **newspaper** as predictor in a 1-dimensional linear regression fit."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"R^2 for TV: 0.611875050850071\n",
"R^2 for radio: 0.33203245544529525\n",
"R^2 for newspaper: 0.05212044544430516\n"
]
}
],
"source": [
"print(\"R^2 for TV: \", reg_tv.score(tv,Y))\n",
"print(\"R^2 for radio: \", reg_radio.score(radio,Y))\n",
"print(\"R^2 for newspaper: \", reg_newspaper.score(newspaper,Y))\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (c)\n",
"Now we want to compute the predicted value of sales if we restrict our prediction to one input, i.e. **TV**, **radio** or **newspaper**, resp.\n",
"Predict the values $\\hat{y}^{TV}$ $\\hat{y}^{radio}$ and $\\hat{y}^{newspaper}$ using the method `predict()`"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": [
"y_tv = reg_tv.predict(tv)\n",
"y_radio = reg_radio.predict(radio)\n",
"y_newspaper = reg_newspaper.predict(newspaper)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (d)\n",
"\n",
"Plot the datapoints as well as the corresponding regression line for each of the inputs **TV**, **radio** or **newspaper**.\n",
"\n",
"You can use the functions `subplots` or `fig.add_subplot` to arrange the plots in one figure."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
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" }\n",
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"\tthis.context.mozBackingStorePixelRatio ||\n",
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"\tthis.context.oBackingStorePixelRatio ||\n",
"\tthis.context.backingStorePixelRatio || 1;\n",
"\n",
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" function toolbar_event(event) {\n",
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" 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",
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"\n",
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" icon_img.addClass(image);\n",
" icon_img.addClass('ui-corner-all');\n",
"\n",
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"\n",
"mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
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"}\n",
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"\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",
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"}\n",
"\n",
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" var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
" var x1 = msg['x1'] / mpl.ratio;\n",
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" x0 = Math.floor(x0) + 0.5;\n",
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"\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",
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" // Updates the figure title.\n",
" fig.header.textContent = msg['label'];\n",
"}\n",
"\n",
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" 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",
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" 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",
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" // Request the server to send over a new figure.\n",
" fig.send_draw_message();\n",
"}\n",
"\n",
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" fig.image_mode = msg['mode'];\n",
"}\n",
"\n",
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" // Called whenever the canvas gets updated.\n",
" this.send_message(\"ack\", {});\n",
"}\n",
"\n",
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" 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",
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"\n",
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" (window.URL || window.webkitURL).revokeObjectURL(\n",
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" }\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",
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"\n",
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" var callback = fig[\"handle_\" + msg_type];\n",
" } catch (e) {\n",
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" }\n",
"\n",
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" // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
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" }\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",
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" 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",
" fig.close_ws(fig, msg);\n",
"}\n",
"\n",
"mpl.figure.prototype.close_ws = function(fig, msg){\n",
" fig.send_message('closing', msg);\n",
" // fig.ws.close()\n",
"}\n",
"\n",
"mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
" // Turn the data on the canvas into data in the output cell.\n",
" var width = this.canvas.width/mpl.ratio\n",
" var dataURL = this.canvas.toDataURL();\n",
" this.cell_info[1]['text/html'] = '';\n",
"}\n",
"\n",
"mpl.figure.prototype.updated_canvas_event = function() {\n",
" // Tell IPython that the notebook contents must change.\n",
" IPython.notebook.set_dirty(true);\n",
" this.send_message(\"ack\", {});\n",
" var fig = this;\n",
" // Wait a second, then push the new image to the DOM so\n",
" // that it is saved nicely (might be nice to debounce this).\n",
" setTimeout(function () { fig.push_to_output() }, 1000);\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) { continue; };\n",
"\n",
" var button = $('');\n",
" button.click(method_name, toolbar_event);\n",
" button.mouseover(tooltip, toolbar_mouse_event);\n",
" nav_element.append(button);\n",
" }\n",
"\n",
" // Add the status bar.\n",
" var status_bar = $('');\n",
" nav_element.append(status_bar);\n",
" this.message = status_bar[0];\n",
"\n",
" // Add the close button to the window.\n",
" var buttongrp = $('');\n",
" var button = $('');\n",
" button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
" button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
" buttongrp.append(button);\n",
" var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
" titlebar.prepend(buttongrp);\n",
"}\n",
"\n",
"mpl.figure.prototype._root_extra_style = function(el){\n",
" var fig = this\n",
" el.on(\"remove\", function(){\n",
"\tfig.close_ws(fig, {});\n",
" });\n",
"}\n",
"\n",
"mpl.figure.prototype._canvas_extra_style = function(el){\n",
" // this is important to make the div 'focusable\n",
" el.attr('tabindex', 0)\n",
" // reach out to IPython and tell the keyboard manager to turn it's self\n",
" // off when our div gets focus\n",
"\n",
" // location in version 3\n",
" if (IPython.notebook.keyboard_manager) {\n",
" IPython.notebook.keyboard_manager.register_events(el);\n",
" }\n",
" else {\n",
" // location in version 2\n",
" IPython.keyboard_manager.register_events(el);\n",
" }\n",
"\n",
"}\n",
"\n",
"mpl.figure.prototype._key_event_extra = function(event, name) {\n",
" var manager = IPython.notebook.keyboard_manager;\n",
" if (!manager)\n",
" manager = IPython.keyboard_manager;\n",
"\n",
" // Check for shift+enter\n",
" if (event.shiftKey && event.which == 13) {\n",
" this.canvas_div.blur();\n",
" event.shiftKey = false;\n",
" // Send a \"J\" for go to next cell\n",
" event.which = 74;\n",
" event.keyCode = 74;\n",
" manager.command_mode();\n",
" manager.handle_keydown(event);\n",
" }\n",
"}\n",
"\n",
"mpl.figure.prototype.handle_save = function(fig, msg) {\n",
" fig.ondownload(fig, null);\n",
"}\n",
"\n",
"\n",
"mpl.find_output_cell = function(html_output) {\n",
" // Return the cell and output element which can be found *uniquely* in the notebook.\n",
" // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
" // IPython event is triggered only after the cells have been serialised, which for\n",
" // our purposes (turning an active figure into a static one), is too late.\n",
" var cells = IPython.notebook.get_cells();\n",
" var ncells = cells.length;\n",
" for (var i=0; i= 3 moved mimebundle to data attribute of output\n",
" data = data.data;\n",
" }\n",
" if (data['text/html'] == html_output) {\n",
" return [cell, data, j];\n",
" }\n",
" }\n",
" }\n",
" }\n",
"}\n",
"\n",
"// Register the function which deals with the matplotlib target/channel.\n",
"// The kernel may be null if the page has been refreshed.\n",
"if (IPython.notebook.kernel != null) {\n",
" IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
"}\n"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
""
],
"text/plain": [
""
]
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"metadata": {},
"output_type": "display_data"
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{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# We plot our findings using subplots\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib notebook\n",
"fig = plt.figure()\n",
"fig.add_subplot(1,3,1)\n",
"plt.plot(X[:,0],Y,'ro', alpha=0.1)\n",
"plt.plot(X[:,0],y_tv,'k-')\n",
"\n",
"fig.add_subplot(1,3,2)\n",
"plt.plot(X[:,1],Y,'b+')\n",
"plt.plot(X[:,1],y_radio,'k-')\n",
"\n",
"fig.add_subplot(1,3,3)\n",
"plt.plot(X[:,2],Y,'g.')\n",
"plt.plot(X[:,2],y_newspaper,'k-')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (e)\n",
"Take a closer look at the correlation matrix.\n",
"You can use the method `corr()` that is implemented for pandas `DataFrames`.\n",
"Which features are correlated most strongly?"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
\n",
"\n",
"
\n",
" \n",
"
\n",
"
\n",
"
TV
\n",
"
radio
\n",
"
newspaper
\n",
"
sales
\n",
"
\n",
" \n",
" \n",
"
\n",
"
TV
\n",
"
1.000000
\n",
"
0.054809
\n",
"
0.056648
\n",
"
0.782224
\n",
"
\n",
"
\n",
"
radio
\n",
"
0.054809
\n",
"
1.000000
\n",
"
0.354104
\n",
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0.576223
\n",
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\n",
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\n",
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newspaper
\n",
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0.056648
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0.354104
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1.000000
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0.228299
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\n",
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\n",
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sales
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0.782224
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1.000000
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"text/plain": [
" TV radio newspaper sales\n",
"TV 1.000000 0.054809 0.056648 0.782224\n",
"radio 0.054809 1.000000 0.354104 0.576223\n",
"newspaper 0.056648 0.354104 1.000000 0.228299\n",
"sales 0.782224 0.576223 0.228299 1.000000"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# We first take a look at the correlation matrix\n",
"adv.corr()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Answer**: The correlation matrix indicates, that the features **newspaper** and **radio** are correlated with a coefficient of about $0.35$."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (f)\n",
"Investigate the statistical significance of the medium **newspaper** in a linear regression involving only this feature. Use a **t-test** for this purpose as described on slide 80 in the lecture notes.\n",
"\n",
"You should observe the following values:\n",
"\n",
"|Coefficient | Estimate | SE | t-statistic | p-value|\n",
"|:-----------|----------|----|-------------|--------|\n",
"| $\\beta_0$ | 12.351 | 0.621 | 19.88 | < 0.0001 |\n",
"| $\\beta_{newspaper}$ | 0.055 | 0.017 | 3.30 | 0.00115\n",
"\n",
"\n",
"You should use `scipy` to get the $t$-distribution using\n",
"\n",
" from scipy.stats import t\n",
" \n",
"The cumulative distribution function at a point `x` for `n` degrees of freedom can than be called by\n",
"\n",
" t.cdf(x, n)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Intercept: 12.3514\n",
"Lin. coef: 0.0547\n",
"\n",
"std-error Intercept: 0.6214\n",
"std-error Lin. coef: 0.0166\n",
"\n",
"t-statistic Intercept: 19.88\n",
"t-statistic Lin. coef: 3.30\n",
"\n",
"P-value Intercept: 0.00000\n",
"P-value Lin. coef: 0.00115\n"
]
}
],
"source": [
"# We already computed the coefficients\n",
"reg_inv = reg_newspaper\n",
"y = y_newspaper\n",
"x = newspaper\n",
"\n",
"beta_0 = reg_inv.intercept_\n",
"beta_1 = reg_inv.coef_[0]\n",
"\n",
"# Get number of samples\n",
"n,p = X.shape\n",
"\n",
"# Estimate the variance of data X_i\n",
"RSS = np.power(y-Y,2).sum()\n",
"RSE = np.sqrt(RSS/(n-2))\n",
"\n",
"# Compute the standard error for the regression coefficients\n",
"x_mean = np.mean(x)\n",
"\n",
"SE0 = np.sqrt( RSE**2 * (1./n + x_mean**2 / np.power(x - x_mean,2).sum() ) )\n",
"SE1 = np.sqrt( RSE**2 / np.power(x - x_mean,2).sum() )\n",
"\n",
"tstat_0 = (beta_0 - 0) / SE0\n",
"tstat_1 = (beta_1 - 0) / SE1\n",
"\n",
"# Import the t-distribution from scipy\n",
"\n",
"from scipy.stats import t\n",
"pval_0 = 2*(1-t.cdf(tstat_0,n-2))\n",
"pval_1 = 2*(1-t.cdf(tstat_1,n-2))\n",
"# Print information\n",
"print(\"Intercept: %7.4f\" % beta_0)\n",
"print(\"Lin. coef: %7.4f\" % beta_1)\n",
"print(\"\")\n",
"print(\"std-error Intercept: %7.4f\" % SE0)\n",
"print(\"std-error Lin. coef: %7.4f\" % SE1)\n",
"print(\"\")\n",
"print(\"t-statistic Intercept: %6.2f\" % tstat_0)\n",
"print(\"t-statistic Lin. coef: %6.2f\" % tstat_1)\n",
"print(\"\")\n",
"print(\"P-value Intercept: %7.5f\" % pval_0)\n",
"print(\"P-value Lin. coef: %7.5f\" % pval_1)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"5.092480366520192\n",
"5134.804544111939\n"
]
}
],
"source": [
"print(RSE)\n",
"print(RSS)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (g)\n",
"Now construct a linear regression on all three predictor variables, i.e.\n",
"\n",
"$$y_i ≈ \\beta_0 + \\beta_{TV} x^{TV}_i + \\beta_{radio} x^{radio}_i + \\beta_{newspaper} x^{newspaper}_i$$ \n",
"\n",
"What do you observe? Compare your results with your findings from above."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"y = 2.9389 + 0.0458 x TV + 0.1885 x radio + -0.0010 x newspaper\n"
]
}
],
"source": [
"reg_all = LinearRegression().fit(X, Y)\n",
"print('y = %5.4f + %5.4f x TV + %5.4f x radio + %5.4f x newspaper' % ((reg_all.intercept_,)+ tuple(reg_all.coef_)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Answer**: The coefficient for the feature **newspaper** becomes negative, which doas not appear very sensible. \n",
"This question can be answered by the $R^2$-score:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (h)\n",
"What portion of the variance is explained by this linear regression fit?"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.8972106381789521"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# This question can be answered using by the R^2 score:\n",
"reg_all.score(X,Y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Part (i)\n",
"Now perform a linear regression that incorporates only the predictors **TV** and\n",
"**radio**.\n",
"Compute also the $R^2$-value and compare it to the full multiple linear regression.\n",
"\n",
"**Extra task**: Present the datapoints and the regression plane in a 3-dimensional plot.\n"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"reg_2 = LinearRegression().fit(X[:,[0,1]],Y)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.8971942610828956"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"reg_2.score(X[:,[0,1]], Y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Answer**: The $R^2$-score for the linear regression fit incorporating **newspaper** is only marginally larger than the score using only **TV** and **radio** for the prediction.\n",
"Thus, it might be sufficient to exclude the **feature** newspaper from our prediction.\n",
"The procedure we did today is called *feature selection* is should be one of the first steps in every statistical learning problem."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Homework:\n",
"\n",
"We have already seen, that the **t-test** comes in handy when one has to decide whether a coefficient for a single feature is significant or not.\n",
"As has been outlined in the lecture, one can also use the t-test in a multiple linear regression fit\n",
"\n",
"$$ Y = X \\beta + \\varepsilon $$\n",
"\n",
"while the intercept is incorporated into $X$, i.e. a column containing only ones is stacked in front of the original matrix $X$.\n",
"\n",
"The formula to compute the test statistic in this generalized setting is\n",
"\n",
"$$ t_j = \\frac{\\hat{\\beta}_j}{\\hat{\\sigma} \\sqrt{v_j}} $$\n",
"\n",
"while $\\hat \\beta_j$ is the $j$-th entry of the coefficient vector\n",
"\n",
"$$ \\hat \\beta = (X^\\top X)^{-1} X^\\top y, $$\n",
"\n",
"$\\hat{\\sigma}$ is the unbiased estimate of $\\sigma$, which is determined by\n",
"\n",
"$$ \\hat{\\sigma} = \\sqrt{\\frac{1}{n-p-1} \\, \\sum_{i=1}^n (y_i - \\hat{y}_i)^2} $$\n",
"\n",
"and $v_j$ is the $j$-th diagonal element of the matrix $(X^\\top X)^{-1}$.\n",
"\n",
"Then $t_j$ is distributed according to a $t$-distribution with $n-p-1$ degrees of freedom (dofs). \n",
"\n",
"**Task**: Compute the values in the following statistic and try to print it in a similar way. \n",
"\n",
"| Coefficient | Estimate | SE | t-statistic | p-value |\n",
"|:-----------------|-----------|-------|-------------|---------|\n",
"| $\\beta_0$ | 2.939 |0.3119 | 9.42 | < 0.0001|\n",
"| $\\beta_{TV}$ | 0.046 |0.0014 | 32.81 | < 0.0001|\n",
"| $\\beta_{radio}$ | 0.189 |0.0086 | 21.89 | < 0.0001|\n",
"| $\\beta_{news}$ | −0.001 |0.0059 | −0.18 | 0.8599 |\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"| Coefficient | Estimate | SE | t-statistic | p-value |\n",
"---------------------------------------------------------------\n",
"| beta_0 | 2.939 | 0.3119 | 9.42 | < 0.0001 |\n",
"| beta_1 | 0.046 | 0.0014 | 32.81 | < 0.0001 |\n",
"| beta_2 | 0.189 | 0.0086 | 21.89 | < 0.0001 |\n",
"| beta_3 | -0.001 | 0.0059 | -0.18 | 0.8599 |\n"
]
}
],
"source": [
"# Stack one column containing only ones in front of X\n",
"myX = np.hstack((np.ones((n,1)),X))\n",
"\n",
"# Invert X^T * X\n",
"V = np.linalg.inv((myX.T).dot(myX))\n",
"\n",
"# Compute regression coefficients beta\n",
"beta = V.dot( myX.T.dot(Y) )\n",
"\n",
"# Extract diagonal out of matrix (X^T * X)^-1\n",
"v = V.diagonal()\n",
"\n",
"# Predict y using beta\n",
"y_pred = myX.dot(beta)\n",
"\n",
"# Compute estimate of sigma\n",
"sigma_hat = np.sqrt( 1./(n-p-1) * np.power(Y - y_pred,2).sum() )\n",
"\n",
"# Compute the standard errors\n",
"SE = np.sqrt(v) * sigma_hat\n",
"\n",
"# Compute the values of the t-statistic\n",
"t_vals = beta / SE\n",
"\n",
"# Compute the corresponding p values\n",
"p_vals = 2*t.cdf(-np.absolute(t_vals), n-p-1)\n",
"\n",
"# Print header\n",
"print('| Coefficient | Estimate | SE | t-statistic | p-value |')\n",
"print('---------------------------------------------------------------')\n",
"\n",
"pval_str = []\n",
"# Print\n",
"for i in range(p+1):\n",
" pval = p_vals[i]\n",
" if pval < 0.0001:\n",
" pval_str.append('< 0.0001')\n",
" else:\n",
" pval_str.append(' %5.4f' % pval)\n",
" print('| beta_%i | %6.3f | %6.4f | %5.2f | %s |' % (i, beta[i], SE[i], t_vals[i], pval_str[i]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Another way to give present this in a nice way is by using pandas data frames:"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [],
"source": [
"import pandas as pd\n",
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [],
"source": [
"df = pd.DataFrame(index=['Intercept', 'TV', 'radio', 'newspaper'])\n",
"df['beta'] = beta\n",
"df['SE'] = SE\n",
"df['t_vals'] = t_vals\n",
"df['p_vals'] = pval_str"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
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