dim_reduct/PCA.ipynb

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    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Dimensionality-Reduction\" data-toc-modified-id=\"Dimensionality-Reduction-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Dimensionality Reduction</a></span><ul class=\"toc-item\"><li><span><a href=\"#PCA\" data-toc-modified-id=\"PCA-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>PCA</a></span></li><li><span><a href=\"#PCA-From-Scratch\" data-toc-modified-id=\"PCA-From-Scratch-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>PCA From Scratch</a></span><ul class=\"toc-item\"><li><span><a href=\"#Standardize\" data-toc-modified-id=\"Standardize-1.2.1\"><span class=\"toc-item-num\">1.2.1&nbsp;&nbsp;</span>Standardize</a></span></li><li><span><a href=\"#Eigendecomposition---Computing-Eigenvectors-and-Eigenvalues\" data-toc-modified-id=\"Eigendecomposition---Computing-Eigenvectors-and-Eigenvalues-1.2.2\"><span class=\"toc-item-num\">1.2.2&nbsp;&nbsp;</span>Eigendecomposition - Computing Eigenvectors and Eigenvalues</a></span></li><li><span><a href=\"#Choosing-Top-K-Eigenvectors\" data-toc-modified-id=\"Choosing-Top-K-Eigenvectors-1.2.3\"><span class=\"toc-item-num\">1.2.3&nbsp;&nbsp;</span>Choosing Top K Eigenvectors</a></span></li><li><span><a href=\"#Projection-Onto-the-New-Feature-Space\" data-toc-modified-id=\"Projection-Onto-the-New-Feature-Space-1.2.4\"><span class=\"toc-item-num\">1.2.4&nbsp;&nbsp;</span>Projection Onto the New Feature Space</a></span></li></ul></li><li><span><a href=\"#Takeaway\" data-toc-modified-id=\"Takeaway-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>Takeaway</a></span></li></ul></li><li><span><a href=\"#Reference\" data-toc-modified-id=\"Reference-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Reference</a></span></li></ul></div>"
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   "execution_count": 1,
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   "outputs": [
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       " font-weight: normal;\n",
       " font-style: normal;\n",
       " src: local('Exo_2'), url('fonts/exo-II-regular.ttf') format('truetype');\n",
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       " src: local('Fira Code'), url('fonts/firacode.otf') format('opentype');\n",
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       " font-family: 'Lora';\n",
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       " font-style: normal;\n",
       " src: local('Lora'), url('fonts/Lora-Regular.ttf') format('truetype');\n",
       "}\n",
       "div#notebook {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " line-height: 170%;\n",
       " color: #303030;\n",
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       "div.body {\n",
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       " font-size: 13pt;\n",
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       "}\n",
       "div#cluster_list > div.list_item.row,\n",
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       "div#clusters.tab-pane.active {\n",
       " font-size: 12.0pt;\n",
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       "#running .panel-group .panel .panel-heading {\n",
       " font-size: 14pt;\n",
       " color: #303030;\n",
       " padding: 8px 8px;\n",
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       "#running .panel-group .panel .panel-heading a {\n",
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       "#running .panel-group .panel .panel-body .list_container .list_item:hover {\n",
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       " background-color: #f7f7f7;\n",
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       "#running .panel-group .panel .panel-body {\n",
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       "div.running_list_info.toolbar_info {\n",
       " font-size: 12.0pt;\n",
       " padding: 4px 0 4px 0;\n",
       " height: inherit;\n",
       " line-height: inherit;\n",
       " text-shadow: none;\n",
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       ".list_placeholder {\n",
       " font-weight: normal;\n",
       "}\n",
       "#tree-selector {\n",
       " padding: 0px;\n",
       "}\n",
       "#project_name > ul > li > a > i.fa.fa-home {\n",
       " color: #ff7823;\n",
       " font-size: 17pt;\n",
       " display: inline-block;\n",
       " position: static;\n",
       " padding: 0px 0px;\n",
       " font-weight: normal;\n",
       " text-align: center;\n",
       " vertical-align: text-top;\n",
       "}\n",
       "#project_name {\n",
       " display: inline-flex;\n",
       " padding-left: 7px;\n",
       " margin-left: -2px;\n",
       " margin-bottom: -20px;\n",
       " text-align: -webkit-auto;\n",
       " vertical-align: text-top;\n",
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       "div#notebook_toolbar div.dynamic-instructions {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
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       ".toolbar_info {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " color: #303030;\n",
       " text-shadow: none;\n",
       " border: none;\n",
       " height: inherit;\n",
       " line-height: inherit;\n",
       "}\n",
       ".list_container {\n",
       " font-size: 12.0pt;\n",
       " color: #303030;\n",
       " border: none;\n",
       " text-shadow: none !important;\n",
       "}\n",
       ".list_container > div {\n",
       " border-bottom: 1px solid rgba(180,180,180,.14);\n",
       " font-size: 12.0pt;\n",
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       ".list_item > div {\n",
       " padding-left: 0px;\n",
       "}\n",
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       ".list_item > div input {\n",
       " top: 0px;\n",
       "}\n",
       ".list_header > div .item_link,\n",
       ".list_item > div .item_link {\n",
       " margin-left: -1px;\n",
       " vertical-align: middle;\n",
       " line-height: 22px;\n",
       " font-size: 12.0pt;\n",
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       ".item_icon {\n",
       " font-size: 12.0pt;\n",
       " vertical-align: middle;\n",
       "}\n",
       ".list_item input:not([type=\"checkbox\"]) {\n",
       " padding-right: 0px;\n",
       " height: auto;\n",
       " width: 20%;\n",
       " margin: 6px 0 0;\n",
       " margin-top: 1px;\n",
       "}\n",
       "#button-select-all {\n",
       " height: auto;\n",
       " font-size: 12.0pt;\n",
       " padding: 5px;\n",
       " min-width: 65px;\n",
       " z-index: 0;\n",
       "}\n",
       "button#tree-selector-btn {\n",
       " height: auto;\n",
       " font-size: 12.0pt;\n",
       " padding: 5px;\n",
       "}\n",
       "input#select-all.pull-left.tree-selector {\n",
       " margin-left: 7px;\n",
       " margin-right: 2px;\n",
       " margin-top: 5px;\n",
       "}\n",
       "input[type=\"radio\"],\n",
       "input[type=\"checkbox\"] {\n",
       " margin: 6px 0 0;\n",
       " margin-top: 1px;\n",
       " line-height: normal;\n",
       "}\n",
       ".list_container a {\n",
       " font-size: 17px;\n",
       " color: #303030;\n",
       " border: none;\n",
       " text-shadow: none !important;\n",
       " font-weight: normal;\n",
       " font-style: normal;\n",
       "}\n",
       "div.list_container a:hover {\n",
       " color: #2f2f2f;\n",
       "}\n",
       "div.list_item:hover {\n",
       " background-color: #fafafa;\n",
       "}\n",
       ".breadcrumb > li {\n",
       " font-size: 12.0pt;\n",
       " color: #303030;\n",
       " border: none;\n",
       " text-shadow: none !important;\n",
       "}\n",
       "ul#tabs a {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
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       " font-weight: normal;\n",
       " font-style: normal;\n",
       " border-color: transparent;\n",
       " text-shadow: none !important;\n",
       "}\n",
       ".nav-tabs {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " font-weight: normal;\n",
       " font-style: normal;\n",
       " background: #ffffff;\n",
       " text-shadow: none !important;\n",
       " border-color: transparent;\n",
       " border-bottom-color: rgba(180,180,180,.30);\n",
       "}\n",
       ".nav-tabs > li > a:hover {\n",
       " color: #2f2f2f;\n",
       " background-color: rgba(180,180,180,.14);\n",
       "}\n",
       ".nav-tabs > li > a:active,\n",
       ".nav-tabs > li > a:focus,\n",
       ".nav-tabs > li.active > a,\n",
       ".nav-tabs > li.active > a:focus,\n",
       ".nav-tabs > li.active > a:hover,\n",
       ".nav-tabs > li.active > a,\n",
       ".nav-tabs > li.active > a:hover,\n",
       ".nav-tabs > li.active > a:focus {\n",
       " color: #1c1c1c;\n",
       " background-color: #eeeeee;\n",
       " border: 1px solid transparent;\n",
       " border-bottom-color: transparent;\n",
       " cursor: default;\n",
       "}\n",
       ".nav > li > a:hover,\n",
       ".nav > li > a:focus {\n",
       " text-decoration: none;\n",
       " background-color: rgba(180,180,180,.14);\n",
       "}\n",
       ".nav > li.disabled > a,\n",
       ".nav > li.disabled > a:hover {\n",
       " color: #aaaaaa;\n",
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       "div#notebook {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " padding-top: 4px;\n",
       "}\n",
       ".notebook_app {\n",
       " background-color: #ffffff;\n",
       "}\n",
       "#notebook-container {\n",
       " padding: 13px;\n",
       " background-color: #ffffff;\n",
       " min-height: 0px;\n",
       " box-shadow: none;\n",
       " width: 980px;\n",
       " margin-right: auto;\n",
       " margin-left: auto;\n",
       "}\n",
       "div#ipython-main-app.container {\n",
       " width: 980px;\n",
       " margin-right: auto;\n",
       " margin-left: auto;\n",
       " margin-right: auto;\n",
       " margin-left: auto;\n",
       "}\n",
       ".container {\n",
       " width: 980px;\n",
       " margin-right: auto;\n",
       " margin-left: auto;\n",
       "}\n",
       ".notebook_app #header {\n",
       " box-shadow: none !important;\n",
       " background-color: #ffffff;\n",
       " border-bottom: 2px solid rgba(180,180,180,.14);\n",
       "}\n",
       "#header {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " box-shadow: none;\n",
       " background-color: #ffffff;\n",
       "}\n",
       "#header .header-bar {\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       "}\n",
       "body > #header .header-bar {\n",
       " width: 100%;\n",
       " background: #ffffff;\n",
       "}\n",
       "#menubar {\n",
       " background-color: #ffffff;\n",
       "}\n",
       "#menubar .navbar,\n",
       ".navbar-default {\n",
       " background-color: #ffffff;\n",
       " margin-bottom: 0px;\n",
       "}\n",
       ".navbar {\n",
       " border: none;\n",
       "}\n",
       ".navbar-default {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " background-color: #ffffff;\n",
       " border-color: rgba(180,180,180,.14);\n",
       " line-height: 1.5em;\n",
       " padding-bottom: 0px;\n",
       "}\n",
       ".navbar-default .navbar-nav > li > a {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " color: #303030;\n",
       " display: block;\n",
       " line-height: 1.5em;\n",
       " padding-top: 8px;\n",
       " padding-bottom: 6px;\n",
       "}\n",
       ".navbar-default .navbar-nav > li > a:hover,\n",
       ".navbar-default .navbar-nav > li > a:focus {\n",
       " color: #2f2f2f;\n",
       " background-color: rgba(180,180,180,.14);\n",
       " border-color: rgba(180,180,180,.14);\n",
       " line-height: 1.5em;\n",
       "}\n",
       ".navbar-default .navbar-nav > .open > a,\n",
       ".navbar-default .navbar-nav > .open > a:hover,\n",
       ".navbar-default .navbar-nav > .open > a:focus {\n",
       " color: #1c1c1c;\n",
       " background-color: rgba(180,180,180,.14);\n",
       " border-color: rgba(180,180,180,.14);\n",
       " line-height: 1.5em;\n",
       "}\n",
       ".edit_mode .modal_indicator:before {\n",
       " font-size: 13pt;\n",
       " color: #2c85f7;\n",
       " content: \"\\f040\";\n",
       "}\n",
       ".item_icon {\n",
       " color: #126dce;\n",
       "}\n",
       ".item_buttons .kernel-name {\n",
       " font-size: 13pt;\n",
       " color: #126dce;\n",
       " line-height: 22px;\n",
       "}\n",
       ".running_notebook_icon:before {\n",
       " color: #009e07 !important;\n",
       "}\n",
       ".item_buttons .running-indicator {\n",
       " padding-top: 2px;\n",
       " color: #009e07;\n",
       "}\n",
       "#modal_indicator {\n",
       " float: right !important;\n",
       " color: #126dce;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       "}\n",
       "#kernel_indicator {\n",
       " float: right !important;\n",
       " color: #ff7823;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " font-size: 14.5pt;\n",
       " border-left: 2px solid #ff7823;\n",
       " padding-bottom: 2px;\n",
       "}\n",
       "#kernel_indicator .kernel_indicator_name {\n",
       " color: #ff7823;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " font-size: 14.5pt;\n",
       " padding-left: 5px;\n",
       " padding-right: 5px;\n",
       "}\n",
       "div.notification_widget.info,\n",
       ".notification_widget.info,\n",
       ".notification_widget:active:hover,\n",
       ".notification_widget.active:hover,\n",
       ".open > .dropdown-toggle.notification_widget:hover,\n",
       ".notification_widget:active:focus,\n",
       ".notification_widget.active:focus,\n",
       ".open > .dropdown-toggle.notification_widget:focus,\n",
       ".notification_widget:active.focus,\n",
       ".notification_widget.active.focus,\n",
       ".open > .dropdown-toggle.notification_widget.focus,\n",
       "div#notification_notebook.notification_widget.btn.btn-xs.navbar-btn,\n",
       "div#notification_notebook.notification_widget.btn.btn-xs.navbar-btn:hover,\n",
       "div#notification_notebook.notification_widget.btn.btn-xs.navbar-btn:focus {\n",
       " color: #126dce;\n",
       " background-color: #ffffff;\n",
       " border-color: #ffffff;\n",
       "}\n",
       "#notification_area,\n",
       "div.notification_area {\n",
       " float: right !important;\n",
       " position: static;\n",
       "}\n",
       "#kernel_logo_widget,\n",
       "#kernel_logo_widget .current_kernel_logo {\n",
       " display: none;\n",
       "}\n",
       "div#ipython_notebook {\n",
       " display: none;\n",
       "}\n",
       "i.fa.fa-icon {\n",
       " -webkit-font-smoothing: antialiased;\n",
       " -moz-osx-font-smoothing: grayscale;\n",
       " text-rendering: auto;\n",
       "}\n",
       ".fa {\n",
       " display: inline-block;\n",
       " font: normal normal normal 12pt/1 \"FontAwesome\", \"Exo_2\", sans-serif;\n",
       " text-rendering: auto;\n",
       " -webkit-font-smoothing: antialiased;\n",
       " -moz-osx-font-smoothing: grayscale;\n",
       "}\n",
       ".dropdown-menu {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " box-shadow: none;\n",
       " padding: 0px;\n",
       " text-align: left;\n",
       " border: 2px solid rgba(180,180,180,.30);\n",
       " background-color: #ffffff;\n",
       " background: #ffffff;\n",
       " line-height: 1.3;\n",
       " margin: 0px;\n",
       "}\n",
       ".dropdown-menu:hover {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " border: 2px solid rgba(180,180,180,.30);\n",
       " background-color: #ffffff;\n",
       " box-shadow: none;\n",
       " line-height: 1.3;\n",
       "}\n",
       ".dropdown-header {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " display: block;\n",
       " color: #ff7823;\n",
       " text-decoration: underline;\n",
       " white-space: nowrap;\n",
       " padding: 8px 0px 0px 6px;\n",
       " line-height: 1.3;\n",
       "}\n",
       ".dropdown-menu > li > a {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " line-height: 1.3;\n",
       " display: block;\n",
       " padding: 10px 25px 10px 14px;\n",
       " color: #303030;\n",
       " background-color: #ffffff;\n",
       " background: #ffffff;\n",
       "}\n",
       ".dropdown-menu > li > a:hover {\n",
       " color: #2f2f2f;\n",
       " background-color: rgba(180,180,180,.14);\n",
       " background: rgba(180,180,180,.14);\n",
       " border-color: rgba(180,180,180,.14);\n",
       "}\n",
       ".dropdown-menu .divider {\n",
       " height: 2px;\n",
       " margin: 0px 0px;\n",
       " overflow: hidden;\n",
       " background-color: rgba(180,180,180,.30);\n",
       "}\n",
       ".dropdown-submenu > .dropdown-menu {\n",
       " top: 0;\n",
       " left: 100%;\n",
       " margin-top: -2px;\n",
       " margin-left: 0px;\n",
       " padding-top: 0px;\n",
       "}\n",
       ".dropdown-menu > .disabled > a,\n",
       ".dropdown-menu > .disabled > a:hover,\n",
       ".dropdown-menu > .disabled > a:focus {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " font-weight: normal;\n",
       " color: #aaaaaa;\n",
       " padding: none;\n",
       " display: block;\n",
       " clear: both;\n",
       " line-height: 1.2;\n",
       " white-space: nowrap;\n",
       "}\n",
       ".dropdown-submenu > a:after {\n",
       " color: #303030;\n",
       " margin-right: -16px;\n",
       "}\n",
       ".dropdown-submenu:hover > a:after,\n",
       ".dropdown-submenu:active > a:after,\n",
       ".dropdown-submenu:focus > a:after,\n",
       ".dropdown-submenu:visited > a:after {\n",
       " color: #ff7823;\n",
       " margin-right: -16px;\n",
       "}\n",
       "div.kse-dropdown > .dropdown-menu,\n",
       ".kse-dropdown > .dropdown-menu {\n",
       " min-width: 0;\n",
       " top: 94%;\n",
       "}\n",
       ".btn,\n",
       ".btn-default {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " color: #303030;\n",
       " background: #ebebeb;\n",
       " background-color: #ebebeb;\n",
       " border: 2px solid #e8e8e8;\n",
       " font-weight: normal;\n",
       " box-shadow: none;\n",
       " text-shadow: none;\n",
       " border-radius: 2px;\n",
       " font-size: inherit;\n",
       "}\n",
       ".btn:hover,\n",
       ".btn:active:hover,\n",
       ".btn.active:hover,\n",
       ".btn-default:hover,\n",
       ".open > .dropdown-toggle.btn-default:hover,\n",
       ".open > .dropdown-toggle.btn:hover {\n",
       " color: #2f2f2f;\n",
       " background-color: #e4e4e4;\n",
       " background: #e4e4e4;\n",
       " border-color: #e4e4e4;\n",
       " background-image: none;\n",
       " box-shadow: none !important;\n",
       " border-radius: 2px;\n",
       "}\n",
       ".btn:active,\n",
       ".btn.active,\n",
       ".btn:active:focus,\n",
       ".btn.active:focus,\n",
       ".btn:active.focus,\n",
       ".btn.active.focus,\n",
       ".btn-default:focus,\n",
       ".btn-default.focus,\n",
       ".btn-default:active,\n",
       ".btn-default.active,\n",
       ".btn-default:active:hover,\n",
       ".btn-default.active:hover,\n",
       ".btn-default:active:focus,\n",
       ".btn-default.active:focus,\n",
       ".btn-default:active.focus,\n",
       ".btn-default.active.focus,\n",
       ".open > .dropdown-toggle.btn:focus,\n",
       ".open > .dropdown-toggle.btn.focus,\n",
       ".open > .dropdown-toggle.btn-default {\n",
       " color: #1c1c1c;\n",
       " background-color: #e4e4e4;\n",
       " background: #e4e4e4;\n",
       " border-color: #e4e4e4;\n",
       " background-image: none;\n",
       " box-shadow: none !important;\n",
       " border-radius: 2px;\n",
       "}\n",
       ".item_buttons > .btn,\n",
       ".item_buttons > .btn-group,\n",
       ".item_buttons > .input-group {\n",
       " margin-left: 5px;\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border: 2px solid #eeeeee;\n",
       "}\n",
       ".item_buttons > .btn:hover,\n",
       ".item_buttons > .btn-group:hover,\n",
       ".item_buttons > .input-group:hover {\n",
       " margin-left: 5px;\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border: 2px solid #e9e9e9;\n",
       "}\n",
       ".btn-group > .btn-mini,\n",
       ".btn-sm,\n",
       ".btn-group-sm > .btn,\n",
       ".btn-xs,\n",
       ".btn-group-xs > .btn,\n",
       ".alternate_upload .btn-upload,\n",
       ".btn-group,\n",
       ".btn-group-vertical {\n",
       " font-size: 12.0pt;\n",
       " font-weight: normal;\n",
       "}\n",
       ".btn-xs,\n",
       ".btn-group-xs > .btn {\n",
       " font-size: 12.0pt;\n",
       " background-image: none;\n",
       " font-weight: normal;\n",
       " text-shadow: none;\n",
       " display: inline-table;\n",
       "}\n",
       ".alternate_upload .btn-upload {\n",
       " display: none;\n",
       "}\n",
       ".alternate_upload input.fileinput {\n",
       " display: none;\n",
       "}\n",
       "button.close {\n",
       " border: 0px none;\n",
       " font-family: sans-serif;\n",
       " font-size: 25pt;\n",
       "}\n",
       ".dynamic-buttons {\n",
       " font-size: inherit;\n",
       " padding-top: 0px;\n",
       " display: inline-block;\n",
       "}\n",
       ".close {\n",
       " color: #de143d;\n",
       " opacity: .5;\n",
       " text-shadow: none;\n",
       "}\n",
       ".close:hover {\n",
       " color: #de143d;\n",
       " opacity: 1;\n",
       "}\n",
       "div.btn.btn-default.output_collapsed {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border-color: #eeeeee;\n",
       "}\n",
       "div.btn.btn-default.output_collapsed:hover {\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border-color: #e9e9e9;\n",
       "}\n",
       "div.nbext-enable-btns .btn[disabled],\n",
       "div.nbext-enable-btns .btn[disabled]:hover,\n",
       ".btn-default.disabled,\n",
       ".btn-default[disabled],\n",
       ".btn-default.disabled:hover,\n",
       ".btn-default[disabled]:hover,\n",
       "fieldset[disabled] .btn-default:hover,\n",
       ".btn-default.disabled:focus,\n",
       ".btn-default[disabled]:focus,\n",
       "fieldset[disabled] .btn-default:focus,\n",
       ".btn-default.disabled.focus,\n",
       ".btn-default[disabled].focus,\n",
       "fieldset[disabled] .btn-default.focus {\n",
       " color: #4a4a4a;\n",
       " background: #e8e8e8;\n",
       " background-color: #e8e8e8;\n",
       " border-color: #e8e8e8;\n",
       "}\n",
       ".input-group-addon {\n",
       " padding: 2px 5px;\n",
       " font-size: 12.0pt;\n",
       " font-weight: normal;\n",
       " height: auto;\n",
       " color: #303030;\n",
       " text-align: center;\n",
       " background-color: #ffffff;\n",
       " border: none;\n",
       "}\n",
       ".btn-group > .btn + .dropdown-toggle {\n",
       " padding-left: 8px;\n",
       " padding-right: 8px;\n",
       " height: 100%;\n",
       " border-left: 2px solid #ff7823 !important;\n",
       "}\n",
       ".btn-group > .btn + .dropdown-toggle:hover {\n",
       " border-left: 2px solid #ff7823 !important;\n",
       "}\n",
       ".input-group-btn {\n",
       " position: relative;\n",
       " font-size: inherit;\n",
       " white-space: nowrap;\n",
       "}\n",
       ".input-group-btn:first-child > .btn,\n",
       ".input-group-btn:first-child > .btn-group {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border: 1px solid #e9e9e9;\n",
       " margin: 2px;\n",
       " font-size: inherit;\n",
       "}\n",
       ".input-group-btn:first-child > .btn:hover,\n",
       ".input-group-btn:first-child > .btn-group:hover {\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border: 1px solid #e9e9e9;\n",
       " margin: 2px;\n",
       " font-size: inherit;\n",
       "}\n",
       "div.modal .btn-group > .btn:first-child {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border: 1px solid #e9e9e9;\n",
       " margin-top: 0px !important;\n",
       " margin-left: 0px;\n",
       " margin-bottom: 2px;\n",
       "}\n",
       "div.modal .btn-group > .btn:first-child:hover {\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border: 1px solid #e9e9e9;\n",
       "}\n",
       "div.modal > button,\n",
       "div.modal-footer > button {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border-color: #eeeeee;\n",
       "}\n",
       "div.modal > button:hover,\n",
       "div.modal-footer > button:hover {\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border-color: #e9e9e9;\n",
       "}\n",
       ".modal-content {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " position: relative;\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border: none;\n",
       " border-radius: 1px;\n",
       " background-clip: padding-box;\n",
       " outline: none;\n",
       "}\n",
       ".modal-header {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " color: #303030;\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border-color: rgba(180,180,180,.30);\n",
       " padding: 12px;\n",
       " min-height: 16.4286px;\n",
       "}\n",
       ".modal-content h4 {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 16pt;\n",
       " color: #303030;\n",
       " padding: 5px;\n",
       "}\n",
       ".modal-body {\n",
       " background-color: #ffffff;\n",
       " position: relative;\n",
       " padding: 15px;\n",
       "}\n",
       ".modal-footer {\n",
       " padding: 10px;\n",
       " text-align: right;\n",
       " background-color: #f7f7f7;\n",
       " border-top: 1px solid rgba(180,180,180,.30);\n",
       "}\n",
       ".alert-info {\n",
       " background-color: #fdfdfd;\n",
       " border-color: rgba(180,180,180,.30);\n",
       " color: #303030;\n",
       "}\n",
       ".modal-header .close {\n",
       " margin-top: -5px;\n",
       " font-size: 25pt;\n",
       "}\n",
       ".modal-backdrop,\n",
       ".modal-backdrop.in {\n",
       " opacity: 0.75;\n",
       " background-color: #eeeeee;\n",
       "}\n",
       "div.panel,\n",
       "div.panel-default,\n",
       ".panel,\n",
       ".panel-default {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " background-color: #f7f7f7;\n",
       " color: #303030;\n",
       " margin-bottom: 14px;\n",
       " border: 0;\n",
       " box-shadow: none;\n",
       "}\n",
       "div.panel > .panel-heading,\n",
       "div.panel-default > .panel-heading {\n",
       " font-size: 14pt;\n",
       " color: #303030;\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border: 0;\n",
       "}\n",
       ".modal .modal-dialog {\n",
       " min-width: 950px;\n",
       " margin: 50px auto;\n",
       "}\n",
       "div.container-fluid {\n",
       " margin-right: auto;\n",
       " margin-left: auto;\n",
       " padding-left: 7px;\n",
       " padding-right: 12px;\n",
       "}\n",
       "div.form-control,\n",
       ".form-control {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: inherit;\n",
       " color: #303030;\n",
       " background-color: #ffffff;\n",
       " border: 2px solid #e7e7e7;\n",
       " margin-left: 2px;\n",
       " height: auto;\n",
       " box-shadow: none;\n",
       " padding: 6px 12px;\n",
       " transition: border-color 0.15s ease-in-out 0s, box-shadow 0.15s ease-in-out 0s;\n",
       "}\n",
       ".form-group.list-group-item {\n",
       " color: #303030;\n",
       " background-color: #f7f7f7;\n",
       " border-color: rgba(180,180,180,.30);\n",
       " margin-bottom: 0px;\n",
       "}\n",
       "input,\n",
       "button,\n",
       "select,\n",
       "textarea {\n",
       " background-color: #ffffff;\n",
       " font-weight: normal;\n",
       " border: 2px solid rgba(180,180,180,.30);\n",
       "}\n",
       "select.form-control.select-xs {\n",
       " height: auto;\n",
       "}\n",
       "div.output.output_scroll {\n",
       " box-shadow: none;\n",
       "}\n",
       "::-webkit-scrollbar-track {\n",
       " -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,0.11);\n",
       " background-color: #d0d0d0;\n",
       " border-radius: 6px;\n",
       "}\n",
       "::-webkit-scrollbar {\n",
       " width: 14px;\n",
       " height: 10px;\n",
       " background-color: #d0d0d0;\n",
       " border-radius: 6px;\n",
       "}\n",
       "::-webkit-scrollbar-thumb {\n",
       " background-color: #ffffff;\n",
       " background-image: -webkit-gradient(linear,40% 0%,75% 86%,from(#ff6b0f ),color-stop(0.5,#ff8b42 ),to(#ff6b0f ));\n",
       " min-height: 60px;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.input_area {\n",
       " background-color: #efefef;\n",
       " padding-right: 1.2em;\n",
       " border: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " border-top-right-radius: 2px;\n",
       " border-bottom-left-radius: 0px;\n",
       " border-bottom-right-radius: 0px;\n",
       "}\n",
       "div.cell {\n",
       " padding: 0px;\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       " border: medium solid #ffffff;\n",
       " border-top-right-radius: 2px;\n",
       " border-top-left-radius: 2px;\n",
       "}\n",
       "div.cell.selected {\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       " border: medium solid #ff7823;\n",
       " padding: 0px;\n",
       " border-top-right-radius: 2px;\n",
       " border-top-left-radius: 2px;\n",
       "}\n",
       ".edit_mode div.cell.selected {\n",
       " padding: 0px;\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       " border: medium solid #ffd5bb;\n",
       " border-top-right-radius: 2px;\n",
       " border-top-left-radius: 2px;\n",
       "}\n",
       "div.cell.edit_mode {\n",
       " padding: 0px;\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       " border: medium solid #ffd5bb;\n",
       " border-top-right-radius: 2px;\n",
       " border-top-left-radius: 2px;\n",
       "}\n",
       "div.prompt,\n",
       ".prompt {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 9.5pt;\n",
       " font-weight: normal;\n",
       " color: #aaaaaa;\n",
       " line-height: 170%;\n",
       " padding: 0px;\n",
       " padding-top: 4px;\n",
       " padding-left: .25em;\n",
       " text-align: left !important;\n",
       " min-width: 12ex;\n",
       " width: 12ex;\n",
       "}\n",
       "div.prompt.input_prompt {\n",
       " background-color: #efefef;\n",
       " border-right: 2px solid rgba(240,147,43,.50);\n",
       " border-top-left-radius: 2px;\n",
       " border-top-right-radius: 0px;\n",
       " border-bottom-left-radius: 0px;\n",
       " border-bottom-right-radius: 0px;\n",
       " min-width: 12ex;\n",
       " width: 12ex !important;\n",
       "}\n",
       "div.output_wrapper {\n",
       " background-color: #ffffff;\n",
       " border: 0px;\n",
       " margin-bottom: 0em;\n",
       " margin-top: 0em;\n",
       " border-top-right-radius: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " border-bottom-left-radius: 2px;\n",
       " border-bottom-right-radius: 2px;\n",
       "}\n",
       "div.output_subarea.output_text.output_stream.output_stdout,\n",
       "div.output_subarea.output_text {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 10.0pt;\n",
       " line-height: 150% !important;\n",
       " background-color: #ffffff;\n",
       " color: #303030;\n",
       " border-top-right-radius: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " border-bottom-left-radius: 2px;\n",
       " border-bottom-right-radius: 2px;\n",
       "}\n",
       "div.output_area pre {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 10.0pt;\n",
       " line-height: 150% !important;\n",
       " color: #303030;\n",
       " border-top-right-radius: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " border-bottom-left-radius: 2px;\n",
       " border-bottom-right-radius: 2px;\n",
       "}\n",
       "div.output_area {\n",
       " display: -webkit-box;\n",
       "}\n",
       "div.output_html {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 10.0pt;\n",
       " color: #353535;\n",
       " background-color: #ffffff;\n",
       " background: #ffffff;\n",
       "}\n",
       "div.output_subarea {\n",
       " overflow-x: auto;\n",
       " padding: .8em;\n",
       " -webkit-box-flex: 1;\n",
       " -moz-box-flex: 1;\n",
       " box-flex: 1;\n",
       " flex: 1;\n",
       " max-width: 90%;\n",
       "}\n",
       "div.prompt.output_prompt {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 9.5pt;\n",
       " background-color: #ffffff;\n",
       " color: #ffffff;\n",
       " border-bottom-left-radius: 2px;\n",
       " border-top-right-radius: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " border-bottom-right-radius: 0px;\n",
       " min-width: 12ex;\n",
       " width: 12ex;\n",
       "}\n",
       "div.out_prompt_overlay.prompt {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 9.5pt;\n",
       " background-color: #ffffff;\n",
       " border-bottom-left-radius: 2px;\n",
       " border-top-right-radius: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " border-bottom-right-radius: 0px;\n",
       " min-width: 12ex;\n",
       " width: 12ex;\n",
       "}\n",
       "div.out_prompt_overlay.prompt:hover {\n",
       " background-color: #ffffff;\n",
       " box-shadow: #e8e8e8 2px 1px 2px 2.5px inset;\n",
       " border-bottom-left-radius: 2px;\n",
       " -webkit-border-: 2px;\n",
       " -moz-border-radius: 2px;\n",
       " border-top-right-radius: 0px;\n",
       " border-top-left-radius: 0px;\n",
       " min-width: 12ex;\n",
       " width: 12ex !important;\n",
       "}\n",
       "div.text_cell,\n",
       "div.text_cell_render pre,\n",
       "div.text_cell_render {\n",
       " font-family: \"Lora\", serif;\n",
       " font-size: 13pt;\n",
       " line-height: 170% !important;\n",
       " color: #353535;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.cell.text_cell.rendered.selected {\n",
       " font-family: \"Lora\", serif;\n",
       " border: medium solid #126dce;\n",
       " line-height: 170% !important;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.cell.text_cell.unrendered.selected {\n",
       " font-family: \"Lora\", serif;\n",
       " line-height: 170% !important;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " border: medium solid #126dce;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.cell.text_cell.selected {\n",
       " font-family: \"Lora\", serif;\n",
       " line-height: 170% !important;\n",
       " border: medium solid #126dce;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " border-radius: 2px;\n",
       "}\n",
       ".edit_mode div.cell.text_cell.selected {\n",
       " font-family: \"Lora\", serif;\n",
       " line-height: 170% !important;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " border: medium solid #87b0db;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.text_cell.unrendered,\n",
       "div.text_cell.unrendered.selected,\n",
       "div.edit_mode div.text_cell.unrendered {\n",
       " font-family: \"Lora\", serif;\n",
       " line-height: 170% !important;\n",
       " background: #ffffff;\n",
       " background-color: #ffffff;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.cell.text_cell.rendered .input_prompt {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 9.5pt;\n",
       " font-weight: normal;\n",
       " color: #aaaaaa;\n",
       " text-align: left !important;\n",
       " min-width: 0ex;\n",
       " width: 0ex !important;\n",
       " background-color: #ffffff;\n",
       " border-right: 2px solid transparent;\n",
       "}\n",
       "div.cell.text_cell.unrendered .input_prompt {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 9.5pt;\n",
       " font-weight: normal;\n",
       " color: #aaaaaa;\n",
       " text-align: left !important;\n",
       " min-width: 0ex;\n",
       " width: 0ex !important;\n",
       " border-right: 2px solid transparent;\n",
       "}\n",
       "div.rendered_html code {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt;\n",
       " padding-top: 3px;\n",
       " color: #303030;\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       "}\n",
       "pre,\n",
       "code,\n",
       "kbd,\n",
       "samp {\n",
       " white-space: pre-wrap;\n",
       "}\n",
       "code {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt !important;\n",
       " line-height: 170% !important;\n",
       " color: #353535;\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       "}\n",
       "kbd {\n",
       " padding: 4px;\n",
       " font-size: 11pt;\n",
       " color: #303030;\n",
       " background-color: #efefef;\n",
       " border: 0;\n",
       " box-shadow: none;\n",
       "}\n",
       "pre {\n",
       " display: block;\n",
       " padding: 8.5px;\n",
       " margin: 0 0 9px;\n",
       " font-size: 12.0pt;\n",
       " line-height: 1.42857143;\n",
       " color: #303030;\n",
       " background-color: #efefef;\n",
       " border: 1px solid #e7e7e7;\n",
       " border-radius: 2px;\n",
       "}\n",
       "div.rendered_html {\n",
       " color: #353535;\n",
       "}\n",
       "div.rendered_html pre,\n",
       "div.text_cell_render pre {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt !important;\n",
       " line-height: 170% !important;\n",
       " color: #353535;\n",
       " background: #efefef;\n",
       " background-color: #efefef;\n",
       " border: 2px #e7e7e7  solid;\n",
       " max-width: 86%;\n",
       " border-radius: 2px;\n",
       " padding: 5px;\n",
       "}\n",
       "div.text_cell_render h1,\n",
       "div.rendered_html h1,\n",
       "div.text_cell_render h2,\n",
       "div.rendered_html h2,\n",
       "div.text_cell_render h3,\n",
       "div.rendered_html h3,\n",
       "div.text_cell_render h4,\n",
       "div.rendered_html h4,\n",
       "div.text_cell_render h5,\n",
       "div.rendered_html h5 {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       "}\n",
       ".rendered_html h1:first-child,\n",
       ".rendered_html h2:first-child,\n",
       ".rendered_html h3:first-child,\n",
       ".rendered_html h4:first-child,\n",
       ".rendered_html h5:first-child,\n",
       ".rendered_html h6:first-child {\n",
       " margin-top: 0.2em;\n",
       "}\n",
       ".rendered_html h1,\n",
       ".text_cell_render h1 {\n",
       " color: #126dce;\n",
       " font-size: 220%;\n",
       " text-align: center;\n",
       " font-weight: lighter;\n",
       "}\n",
       ".rendered_html h2,\n",
       ".text_cell_render h2 {\n",
       " text-align: left;\n",
       " font-size: 170%;\n",
       " color: #126dce;\n",
       " font-style: normal;\n",
       " font-weight: lighter;\n",
       "}\n",
       ".rendered_html h3,\n",
       ".text_cell_render h3 {\n",
       " font-size: 150%;\n",
       " color: #126dce;\n",
       " font-weight: lighter;\n",
       " text-decoration: italic;\n",
       " font-style: normal;\n",
       "}\n",
       ".rendered_html h4,\n",
       ".text_cell_render h4 {\n",
       " font-size: 120%;\n",
       " color: #126dce;\n",
       " font-weight: underline;\n",
       " font-style: normal;\n",
       "}\n",
       ".rendered_html h5,\n",
       ".text_cell_render h5 {\n",
       " font-size: 100%;\n",
       " color: #2f2f2f;\n",
       " font-weight: lighter;\n",
       " text-decoration: underline;\n",
       "}\n",
       ".rendered_html table,\n",
       ".rendered_html tr,\n",
       ".rendered_html td {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 10.0pt !important;\n",
       " line-height: 150% !important;\n",
       " border: 1px solid #d6d6d6;\n",
       " color: #353535;\n",
       " background-color: #ffffff;\n",
       " background: #ffffff;\n",
       "}\n",
       "table.dataframe,\n",
       ".rendered_html tr,\n",
       ".dataframe * {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 10.0pt !important;\n",
       " border: 1px solid #d6d6d6;\n",
       "}\n",
       ".dataframe th,\n",
       ".rendered_html th {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 11pt !important;\n",
       " font-weight: bold;\n",
       " border: 1px solid #c4c4c4;\n",
       " background: #eeeeee;\n",
       "}\n",
       ".dataframe td,\n",
       ".rendered_html td {\n",
       " font-family: \"Fira Code\", monospace;\n",
       " font-size: 10.0pt !important;\n",
       " color: #353535;\n",
       " background: #ffffff;\n",
       " border: 1px solid #d6d6d6;\n",
       " text-align: left;\n",
       " min-width: 4em;\n",
       "}\n",
       ".dataframe-summary-row tr:last-child,\n",
       ".dataframe-summary-col td:last-child {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 11pt !important;\n",
       " font-weight: bold;\n",
       " color: #353535;\n",
       " border: 1px solid #d6d6d6;\n",
       " background: #eeeeee;\n",
       "}\n",
       "div.widget-area {\n",
       " background-color: #ffffff;\n",
       " background: #ffffff;\n",
       " color: #303030;\n",
       "}\n",
       "div.widget-area a {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " font-weight: normal;\n",
       " font-style: normal;\n",
       " color: #303030;\n",
       " text-shadow: none !important;\n",
       "}\n",
       "div.widget-area a:hover,\n",
       "div.widget-area a:focus {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12.0pt;\n",
       " font-weight: normal;\n",
       " font-style: normal;\n",
       " color: #2f2f2f;\n",
       " background: rgba(180,180,180,.14);\n",
       " background-color: rgba(180,180,180,.14);\n",
       " border-color: transparent;\n",
       " background-image: none;\n",
       " text-shadow: none !important;\n",
       "}\n",
       "div.widget_item.btn-group > button.btn.btn-default.widget-combo-btn,\n",
       "div.widget_item.btn-group > button.btn.btn-default.widget-combo-btn:hover {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border: 2px solid #eeeeee !important;\n",
       " font-size: inherit;\n",
       " z-index: 0;\n",
       "}\n",
       "div.jupyter-widgets.widget-hprogress.widget-hbox,\n",
       "div.widget-hbox,\n",
       ".widget-hbox {\n",
       " display: inline-table;\n",
       "}\n",
       "div.jupyter-widgets.widget-hprogress.widget-hbox .widget-label,\n",
       "div.widget-hbox .widget-label,\n",
       ".widget-hbox .widget-label {\n",
       " font-size: 11pt;\n",
       " min-width: 100%;\n",
       " padding-top: 5px;\n",
       " padding-right: 10px;\n",
       " text-align: left;\n",
       " vertical-align: text-top;\n",
       "}\n",
       ".progress {\n",
       " overflow: hidden;\n",
       " height: 20px;\n",
       " margin-bottom: 10px;\n",
       " padding-left: 10px;\n",
       " background-color: #c6c6c6;\n",
       " border-radius: 4px;\n",
       " -webkit-box-shadow: none;\n",
       " box-shadow: none;\n",
       "}\n",
       ".rendered_html :link {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 100%;\n",
       " color: #2c85f7;\n",
       " text-decoration: underline;\n",
       "}\n",
       ".rendered_html :visited,\n",
       ".rendered_html :visited:active,\n",
       ".rendered_html :visited:focus {\n",
       " color: #2e6eb2;\n",
       "}\n",
       ".rendered_html :visited:hover,\n",
       ".rendered_html :link:hover {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 100%;\n",
       " color: #eb6a18;\n",
       "}\n",
       "a.anchor-link:link:hover {\n",
       " font-size: inherit;\n",
       " color: #eb6a18;\n",
       "}\n",
       "a.anchor-link:link {\n",
       " font-size: inherit;\n",
       " text-decoration: none;\n",
       " padding: 0px 20px;\n",
       " visibility: none;\n",
       " color: #126dce;\n",
       "}\n",
       "div#nbextensions-configurator-container.container {\n",
       " width: 980px;\n",
       " margin-right: 0;\n",
       " margin-left: 0;\n",
       "}\n",
       "div.nbext-selector > nav > .nav > li > a {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12pt;\n",
       "}\n",
       "div.nbext-readme > .nbext-readme-contents > .rendered_html {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 12pt;\n",
       " line-height: 145%;\n",
       " padding: 1em 1em;\n",
       " color: #353535;\n",
       " background-color: #ffffff;\n",
       " -webkit-box-shadow: none;\n",
       " -moz-box-shadow: none;\n",
       " box-shadow: none;\n",
       "}\n",
       ".nbext-icon,\n",
       ".nbext-desc,\n",
       ".nbext-compat-div,\n",
       ".nbext-enable-btns,\n",
       ".nbext-params {\n",
       " margin-bottom: 8px;\n",
       " font-size: 12pt;\n",
       "}\n",
       "div.nbext-readme > .nbext-readme-contents {\n",
       " padding: 0;\n",
       " overflow-y: hidden;\n",
       "}\n",
       "div.nbext-readme > .nbext-readme-contents:not(:empty) {\n",
       " margin-top: 0.5em;\n",
       " margin-bottom: 2em;\n",
       " border: none;\n",
       " border-top-color: rgba(180,180,180,.30);\n",
       "}\n",
       ".nbext-showhide-incompat {\n",
       " padding-bottom: 0.5em;\n",
       " color: #4a4a4a;\n",
       " font-size: 12.0pt;\n",
       "}\n",
       ".shortcut_key,\n",
       "span.shortcut_key {\n",
       " display: inline-block;\n",
       " width: 16ex;\n",
       " text-align: right;\n",
       " font-family: monospace;\n",
       "}\n",
       "mark,\n",
       ".mark {\n",
       " background-color: #ffffff;\n",
       " color: #353535;\n",
       " padding: .15em;\n",
       "}\n",
       "a.text-warning,\n",
       "a.text-warning:hover {\n",
       " color: #aaaaaa;\n",
       "}\n",
       "a.text-warning.bg-warning {\n",
       " background-color: #ffffff;\n",
       "}\n",
       "span.bg-success.text-success {\n",
       " background-color: transparent;\n",
       " color: #009e07;\n",
       "}\n",
       "span.bg-danger.text-danger {\n",
       " background-color: #ffffff;\n",
       " color: #de143d;\n",
       "}\n",
       ".has-success .input-group-addon {\n",
       " color: #009e07;\n",
       " border-color: transparent;\n",
       " background: inherit;\n",
       " background-color: rgba(83,180,115,.10);\n",
       "}\n",
       ".has-success .form-control {\n",
       " border-color: #009e07;\n",
       " -webkit-box-shadow: inset 0 1px 1px rgba(0,0,0,0.025);\n",
       " box-shadow: inset 0 1px 1px rgba(0,0,0,0.025);\n",
       "}\n",
       ".has-error .input-group-addon {\n",
       " color: #de143d;\n",
       " border-color: transparent;\n",
       " background: inherit;\n",
       " background-color: rgba(192,57,67,.10);\n",
       "}\n",
       ".has-error .form-control {\n",
       " border-color: #de143d;\n",
       " -webkit-box-shadow: inset 0 1px 1px rgba(0,0,0,0.025);\n",
       " box-shadow: inset 0 1px 1px rgba(0,0,0,0.025);\n",
       "}\n",
       ".kse-input-group-pretty > kbd {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " color: #303030;\n",
       " font-weight: normal;\n",
       " background: transparent;\n",
       "}\n",
       ".kse-input-group-pretty > kbd {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " color: #303030;\n",
       " font-weight: normal;\n",
       " background: transparent;\n",
       "}\n",
       "div.nbext-enable-btns .btn[disabled],\n",
       "div.nbext-enable-btns .btn[disabled]:hover,\n",
       ".btn-default.disabled,\n",
       ".btn-default[disabled] {\n",
       " background: #e8e8e8;\n",
       " background-color: #e8e8e8;\n",
       " color: #282828;\n",
       "}\n",
       "label#Keyword-Filter {\n",
       " display: none;\n",
       "}\n",
       ".nav-pills > li.active > a,\n",
       ".nav-pills > li.active > a:hover,\n",
       ".nav-pills > li.active > a:focus {\n",
       " color: #ffffff;\n",
       " background-color: #126dce;\n",
       "}\n",
       ".input-group .nbext-list-btn-add,\n",
       ".input-group-btn:last-child > .btn-group > .btn {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border-color: #eeeeee;\n",
       "}\n",
       ".input-group .nbext-list-btn-add:hover,\n",
       ".input-group-btn:last-child > .btn-group > .btn:hover {\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border-color: #e9e9e9;\n",
       "}\n",
       "#notebook-container > div.cell.code_cell.rendered.selected > div.widget-area > div.widget-subarea > div > div.widget_item.btn-group > button.btn.btn-default.dropdown-toggle.widget-combo-carrot-btn {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border-color: #eeeeee;\n",
       "}\n",
       "#notebook-container > div.cell.code_cell.rendered.selected > div.widget-area > div.widget-subarea > div > div.widget_item.btn-group > button.btn.btn-default.dropdown-toggle.widget-combo-carrot-btn:hover {\n",
       " background: #e9e9e9;\n",
       " background-color: #e9e9e9;\n",
       " border-color: #e9e9e9;\n",
       "}\n",
       "input.raw_input {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt !important;\n",
       " color: #303030;\n",
       " background-color: #efefef;\n",
       " border-color: #ececec;\n",
       " background: #ececec;\n",
       " width: auto;\n",
       " vertical-align: baseline;\n",
       " padding: 0em 0.25em;\n",
       " margin: 0em 0.25em;\n",
       " -webkit-box-shadow: none;\n",
       " box-shadow: none;\n",
       "}\n",
       "audio,\n",
       "video {\n",
       " display: inline;\n",
       " vertical-align: middle;\n",
       " align-content: center;\n",
       " margin-left: 20%;\n",
       "}\n",
       ".cmd-palette .modal-body {\n",
       " padding: 0px;\n",
       " margin: 0px;\n",
       "}\n",
       ".cmd-palette form {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       "}\n",
       ".typeahead-field input:last-child,\n",
       ".typeahead-hint {\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " z-index: 1;\n",
       "}\n",
       ".typeahead-field input {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " color: #303030;\n",
       " border: none;\n",
       " font-size: 28pt;\n",
       " display: inline-block;\n",
       " line-height: inherit;\n",
       " padding: 3px 10px;\n",
       " height: 70px;\n",
       "}\n",
       ".typeahead-select {\n",
       " background-color: #eeeeee;\n",
       "}\n",
       "body > div.modal.cmd-palette.typeahead-field {\n",
       " display: table;\n",
       " border-collapse: separate;\n",
       " background-color: #f7f7f7;\n",
       "}\n",
       ".typeahead-container button {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 28pt;\n",
       " background-color: #d0d0d0;\n",
       " border: none;\n",
       " display: inline-block;\n",
       " line-height: inherit;\n",
       " padding: 3px 10px;\n",
       " height: 70px;\n",
       "}\n",
       ".typeahead-search-icon {\n",
       " min-width: 40px;\n",
       " min-height: 55px;\n",
       " display: block;\n",
       " vertical-align: middle;\n",
       " text-align: center;\n",
       "}\n",
       ".typeahead-container button:focus,\n",
       ".typeahead-container button:hover {\n",
       " color: #2f2f2f;\n",
       " background-color: #ff7823;\n",
       " border-color: #ff7823;\n",
       "}\n",
       ".typeahead-list > li.typeahead-group.active > a,\n",
       ".typeahead-list > li.typeahead-group > a,\n",
       ".typeahead-list > li.typeahead-group > a:focus,\n",
       ".typeahead-list > li.typeahead-group > a:hover {\n",
       " display: none;\n",
       "}\n",
       ".typeahead-dropdown > li > a,\n",
       ".typeahead-list > li > a {\n",
       " color: #303030;\n",
       " text-decoration: none;\n",
       "}\n",
       ".typeahead-dropdown,\n",
       ".typeahead-list {\n",
       " font-family: \"Exo_2\", sans-serif;\n",
       " font-size: 13pt;\n",
       " color: #303030;\n",
       " background-color: #ffffff;\n",
       " border: none;\n",
       " background-clip: padding-box;\n",
       " margin-top: 0px;\n",
       " padding: 3px 2px 3px 0px;\n",
       " line-height: 1.7;\n",
       "}\n",
       ".typeahead-dropdown > li.active > a,\n",
       ".typeahead-dropdown > li > a:focus,\n",
       ".typeahead-dropdown > li > a:hover,\n",
       ".typeahead-list > li.active > a,\n",
       ".typeahead-list > li > a:focus,\n",
       ".typeahead-list > li > a:hover {\n",
       " color: #2f2f2f;\n",
       " background-color: #f7f7f7;\n",
       " border-color: #f7f7f7;\n",
       "}\n",
       ".command-shortcut:before {\n",
       " content: \"(command)\";\n",
       " padding-right: 3px;\n",
       " color: #aaaaaa;\n",
       "}\n",
       ".edit-shortcut:before {\n",
       " content: \"(edit)\";\n",
       " padding-right: 3px;\n",
       " color: #aaaaaa;\n",
       "}\n",
       "ul.typeahead-list i {\n",
       " margin-left: 1px;\n",
       " width: 18px;\n",
       " margin-right: 10px;\n",
       "}\n",
       "ul.typeahead-list {\n",
       " max-height: 50vh;\n",
       " overflow: auto;\n",
       "}\n",
       ".typeahead-list > li {\n",
       " position: relative;\n",
       " border: none;\n",
       "}\n",
       "div.input.typeahead-hint,\n",
       "input.typeahead-hint,\n",
       "body > div.modal.cmd-palette.in > div > div > div > form > div > div.typeahead-field > span.typeahead-query > input.typeahead-hint {\n",
       " color: #aaaaaa !important;\n",
       " background-color: transparent;\n",
       " padding: 3px 10px;\n",
       "}\n",
       ".typeahead-dropdown > li > a,\n",
       ".typeahead-list > li > a {\n",
       " display: block;\n",
       " padding: 5px;\n",
       " clear: both;\n",
       " font-weight: 400;\n",
       " line-height: 1.7;\n",
       " border: 1px solid #ffffff;\n",
       " border-bottom-color: rgba(180,180,180,.30);\n",
       "}\n",
       "body > div.modal.cmd-palette.in > div {\n",
       " min-width: 750px;\n",
       " margin: 150px auto;\n",
       "}\n",
       ".typeahead-container strong {\n",
       " font-weight: bolder;\n",
       " color: #ff7823;\n",
       "}\n",
       "#find-and-replace #replace-preview .match,\n",
       "#find-and-replace #replace-preview .insert {\n",
       " color: #ffffff;\n",
       " background-color: #ff7823;\n",
       " border-color: #ff7823;\n",
       " border-style: solid;\n",
       " border-width: 1px;\n",
       " border-radius: 0px;\n",
       "}\n",
       "#find-and-replace #replace-preview .replace .match {\n",
       " background-color: #de143d;\n",
       " border-color: #de143d;\n",
       " border-radius: 0px;\n",
       "}\n",
       "#find-and-replace #replace-preview .replace .insert {\n",
       " background-color: #009e07;\n",
       " border-color: #009e07;\n",
       " border-radius: 0px;\n",
       "}\n",
       "div.CodeMirror,\n",
       "div.CodeMirror pre {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt;\n",
       " line-height: 170%;\n",
       " color: #303030;\n",
       "}\n",
       "div.CodeMirror-lines {\n",
       " padding-bottom: .6em;\n",
       " padding-left: .5em;\n",
       " padding-right: 1.5em;\n",
       " padding-top: 4px;\n",
       "}\n",
       "span.ansiblack {\n",
       " color: #dc4384;\n",
       "}\n",
       "span.ansiblue {\n",
       " color: #009e07;\n",
       "}\n",
       "span.ansigray {\n",
       " color: #ff7823;\n",
       "}\n",
       "span.ansigreen {\n",
       " color: #333333;\n",
       "}\n",
       "span.ansipurple {\n",
       " color: #653bc5;\n",
       "}\n",
       "span.ansicyan {\n",
       " color: #055be0;\n",
       "}\n",
       "span.ansiyellow {\n",
       " color: #ff7823;\n",
       "}\n",
       "span.ansired {\n",
       " color: #de143d;\n",
       "}\n",
       "div.output-stderr {\n",
       " background-color: #ebb5b7;\n",
       "}\n",
       "div.output-stderr pre {\n",
       " color: #000000;\n",
       "}\n",
       "div.js-error {\n",
       " color: #de143d;\n",
       "}\n",
       ".ipython_tooltip {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt;\n",
       " line-height: 170%;\n",
       " border: 2px solid #dadada;\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " border-radius: 2px;\n",
       " overflow-x: visible;\n",
       " overflow-y: visible;\n",
       " box-shadow: none;\n",
       " position: absolute;\n",
       " z-index: 1000;\n",
       "}\n",
       ".ipython_tooltip .tooltiptext pre {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt;\n",
       " line-height: 170%;\n",
       " background: #eeeeee;\n",
       " background-color: #eeeeee;\n",
       " color: #303030;\n",
       " overflow-x: visible;\n",
       " overflow-y: visible;\n",
       " max-width: 900px;\n",
       "}\n",
       "div#tooltip.ipython_tooltip {\n",
       " overflow-x: wrap;\n",
       " overflow-y: visible;\n",
       " max-width: 800px;\n",
       "}\n",
       "div.tooltiptext.bigtooltip {\n",
       " overflow-x: visible;\n",
       " overflow-y: scroll;\n",
       " height: 400px;\n",
       " max-width: 800px;\n",
       "}\n",
       ".cm-s-ipython.CodeMirror {\n",
       " font-family: \"Droid Sans Mono\", monospace;\n",
       " font-size: 11pt;\n",
       " background: #efefef;\n",
       " color: #303030;\n",
       " border-radius: 2px;\n",
       " font-style: normal;\n",
       " font-weight: normal;\n",
       "}\n",
       ".cm-s-ipython div.CodeMirror-selected {\n",
       " background: #e0e1e3;\n",
       "}\n",
       ".cm-s-ipython .CodeMirror-gutters {\n",
       " background: #e0e1e3;\n",
       " border: none;\n",
       " border-radius: 0px;\n",
       "}\n",
       ".cm-s-ipython .CodeMirror-linenumber {\n",
       " color: #aaaaaa;\n",
       "}\n",
       ".cm-s-ipython .CodeMirror-cursor {\n",
       " border-left: 2px solid #ff711a;\n",
       "}\n",
       ".cm-s-ipython span.cm-comment {\n",
       " color: #8d8d8d;\n",
       " font-style: italic;\n",
       "}\n",
       ".cm-s-ipython span.cm-atom {\n",
       " color: #055be0;\n",
       "}\n",
       ".cm-s-ipython span.cm-number {\n",
       " color: #ff8132;\n",
       "}\n",
       ".cm-s-ipython span.cm-property {\n",
       " color: #e22978;\n",
       "}\n",
       ".cm-s-ipython span.cm-attribute {\n",
       " color: #de143d;\n",
       "}\n",
       ".cm-s-ipython span.cm-keyword {\n",
       " color: #713bc5;\n",
       " font-weight: normal;\n",
       "}\n",
       ".cm-s-ipython span.cm-string {\n",
       " color: #009e07;\n",
       "}\n",
       ".cm-s-ipython span.cm-meta {\n",
       " color: #aa22ff;\n",
       "}\n",
       ".cm-s-ipython span.cm-operator {\n",
       " color: #055be0;\n",
       "}\n",
       ".cm-s-ipython span.cm-builtin {\n",
       " color: #e22978;\n",
       "}\n",
       ".cm-s-ipython span.cm-variable {\n",
       " color: #303030;\n",
       "}\n",
       ".cm-s-ipython span.cm-variable-2 {\n",
       " color: #de143d;\n",
       "}\n",
       ".cm-s-ipython span.cm-variable-3 {\n",
       " color: #aa22ff;\n",
       "}\n",
       ".cm-s-ipython span.cm-def {\n",
       " color: #e22978;\n",
       " font-weight: normal;\n",
       "}\n",
       ".cm-s-ipython span.cm-error {\n",
       " background: rgba(191,97,106,.40);\n",
       "}\n",
       ".cm-s-ipython span.cm-tag {\n",
       " color: #e22978;\n",
       "}\n",
       ".cm-s-ipython span.cm-link {\n",
       " color: #ff7823;\n",
       "}\n",
       ".cm-s-ipython span.cm-storage {\n",
       " color: #055be0;\n",
       "}\n",
       ".cm-s-ipython span.cm-entity {\n",
       " color: #e22978;\n",
       "}\n",
       ".cm-s-ipython span.cm-quote {\n",
       " color: #009e07;\n",
       "}\n",
       "div.CodeMirror span.CodeMirror-matchingbracket {\n",
       " color: #1c1c1c;\n",
       " background-color: rgba(30,112,199,.30);\n",
       "}\n",
       "div.CodeMirror span.CodeMirror-nonmatchingbracket {\n",
       " color: #1c1c1c;\n",
       " background: rgba(191,97,106,.40) !important;\n",
       "}\n",
       "div.cell.text_cell .cm-s-default .cm-header {\n",
       " color: #126dce;\n",
       "}\n",
       "div.cell.text_cell .cm-s-default span.cm-variable-2 {\n",
       " color: #353535;\n",
       "}\n",
       "div.cell.text_cell .cm-s-default span.cm-variable-3 {\n",
       " color: #aa22ff;\n",
       "}\n",
       ".cm-s-default span.cm-comment {\n",
       " color: #8d8d8d;\n",
       "}\n",
       ".cm-s-default .cm-tag {\n",
       " color: #009fb7;\n",
       "}\n",
       ".cm-s-default .cm-builtin {\n",
       " color: #e22978;\n",
       "}\n",
       ".cm-s-default .cm-string {\n",
       " color: #009e07;\n",
       "}\n",
       ".cm-s-default .cm-keyword {\n",
       " color: #713bc5;\n",
       "}\n",
       ".cm-s-default .cm-number {\n",
       " color: #ff8132;\n",
       "}\n",
       ".cm-s-default .cm-error {\n",
       " color: #055be0;\n",
       "}\n",
       ".CodeMirror-cursor {\n",
       " border-left: 2px solid #ff711a;\n",
       " border-right: none;\n",
       " width: 0;\n",
       "}\n",
       ".cm-s-default div.CodeMirror-selected {\n",
       " background: #e0e1e3;\n",
       "}\n",
       ".cm-s-default .cm-selected {\n",
       " background: #e0e1e3;\n",
       "}\n",
       "div#maintoolbar {\n",
       " display: none !important;\n",
       "}\n",
       "#header-container {\n",
       " display: none !important;\n",
       "}\n",
       "\n",
       "/**********************************\n",
       " MathJax Settings and Style Script\n",
       "**********************************/\n",
       ".MathJax_Display,\n",
       ".MathJax nobr>span.math>span {\n",
       "    border: 0 !important;\n",
       "    font-size: 110% !important;\n",
       "    text-align: center !important;\n",
       "    margin: 0em !important;\n",
       "}\n",
       "/* Prevents MathJax from jittering */\n",
       "/* cell position when cell is selected */\n",
       ".MathJax:focus, body :focus .MathJax {\n",
       "    display: inline-block !important;\n",
       "}\n",
       "\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "        \"HTML-CSS\": {\n",
       "            preferredFont: \"TeX\",\n",
       "            availableFonts: [\"STIX\",\"TeX\"],\n",
       "            styles: {\n",
       "                scale: 110,\n",
       "                \".MathJax_Display\": {\n",
       "                    \"font-size\": \"110%\",\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    });\n",
       "</script>\n",
       "\n",
       " </style>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from jupyterthemes import get_themes\n",
    "from jupyterthemes.stylefx import set_nb_theme\n",
    "themes = get_themes()\n",
    "set_nb_theme(themes[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ethen 2017-10-07 14:49:39 \n",
      "\n",
      "CPython 3.5.2\n",
      "IPython 6.1.0\n",
      "\n",
      "numpy 1.13.3\n",
      "pandas 0.20.3\n",
      "matplotlib 2.0.0\n",
      "sklearn 0.19.0\n"
     ]
    }
   ],
   "source": [
    "# 1. magic for inline plot\n",
    "# 2. magic to print version\n",
    "# 3. magic so that the notebook will reload external python modules\n",
    "# 4. magic to enable retina (high resolution) plots\n",
    "# https://gist.github.com/minrk/3301035\n",
    "%matplotlib inline\n",
    "%load_ext watermark\n",
    "%load_ext autoreload\n",
    "%autoreload 2\n",
    "%config InlineBackend.figure_format = 'retina'\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from sklearn.decomposition import PCA\n",
    "from sklearn.pipeline import Pipeline\n",
    "from sklearn.datasets import load_iris\n",
    "from sklearn.metrics import accuracy_score\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "from sklearn.linear_model import LogisticRegression\n",
    "from sklearn.model_selection import train_test_split\n",
    "\n",
    "%watermark -a 'Ethen' -d -t -v -p numpy,pandas,matplotlib,sklearn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Dimensionality Reduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The starting point most data analysis problems is to perform some sort of exploratory data analysis on our raw data. This step is important because until we have basic understanding of the structure of our raw data, it might be hard to know whether the data is suitable for the task at hand or even derive insights that we can later share. During this exploratory process, unsupervised methods such as dimensionality reduction can help us identify simpler and more compact representations of the original raw data to either aid our understanding or provide useful input to other stages of analysis. Here, we'll be focusing on a specific dimensionality reduction technique called **Principal Component Analysis (PCA)**.\n",
    "\n",
    "Imagine that we're trying to understand some underlying phenomenon, in order to do so we measure various quantities potentially related to it. If we knew exactly what to measure in advance, we might be able to find some simple relationships in our data. But we typically don't, so we often measure anything that might be relevant and end up having irrelevant or redundant signals in our measurement. To make this a bit more concrete, we'll generate a toy 2-dimensional dataset to work with using the equation below.\n",
    "\n",
    "$$x_2 = 500 + 20 \\times x_1 + \\epsilon$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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68NBb/tZ7yi7lG3V3NGx7CdxW73OuqfRj771Z/9bVqi/9nxb9W1erHnvvzTq2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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114955e48>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 397,
       "width": 567
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(123)\n",
    "x1 = np.arange(12, 56, 0.5)\n",
    "e  = np.random.normal(0, 100, x1.shape[0])\n",
    "x2 = 500 + 20 * x1 + e\n",
    "X = np.c_[x1, x2]\n",
    "\n",
    "# change default style figure and font size\n",
    "plt.style.use('fivethirtyeight')\n",
    "plt.rcParams['figure.figsize'] = 8, 6\n",
    "plt.rcParams['font.size'] = 12\n",
    "\n",
    "def plot2var(m):\n",
    "    plt.scatter(m[:, 0], m[:, 1], s = 40, alpha = 0.8)\n",
    "    plt.xlabel('x1')\n",
    "    plt.ylabel('x2')\n",
    "\n",
    "plot2var(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can clearly see from the plot that there's linear relationship between the two variables, thus we probably don't need to include both of them as $x_1$ can be easily explained by $x_2$ and vice versa."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PCA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If it is the case that one of the features is considered redundant, we should be able to summarize the data with less characteristics (features). So, the way PCA tackles this problem is: Instead of simply picking out the useful features and discarding the others, it uses a linear combination of the existing features and constructs some new features that are good alternative representation of the original data. In our 2D toy dataset, PCA will try to pick the best single direction, or often referred to as **first principal components** in 2D, and project our points onto that single direction. So the next question becomes, out of the many possible lines in 2D, what line should we pick?\n",
    "\n",
    "It turns out, there are two different answers to this question. First answer is that we are looking for some features that strongly differ across data points, thus, PCA looks for features that captures as much variation across data points as possible. The second answer is that we are looking for the features that would allow us to \"reconstruct\" the original features. Imagine that we come up with a feature that has no relation to the original ones; If we were to use this new feature, there is no way we can relate this to the original ones. So PCA looks for features that minimizes the reconstruction error. These two notions can be depicted in the graph below, where the black dots represents the original data point, the black line represents the projected line, the red dot on the left shows the points on the projected line and the red line on the right shows the reconstruction error.\n",
    "\n",
    "<img src='img/objective1.png' width='80%'>\n",
    "\n",
    "Surprisingly, it turns out that these two aims are equivalent and PCA can kill two birds with one stone. To see why minimizing squared residuals is equivalent to maximizing variance consider the 2 dimensions visualization below.\n",
    "\n",
    "<img src='img/objective2.png' width='80%'>\n",
    "\n",
    "Consider a datapoint $a_i$. The contribution of this specific data point to the total variance is $a_i^Ta_i$, or equivalently the squared Euclidean length $\\lVert \\mathbf{a}_i \\lVert^2$. Applying the Pythagorean theorem shows that this total variance equals the sum of variance lost (the squared residual) and variance remaining. Thus, it is equivalent to either maximize remaining variance or minimize lost variance to find the principal components.\n",
    "\n",
    "Before we go deeper, let's build some intuition using the scikit-learn library. The following section standardizes the data, fits the PCA model and prints out some of the important informations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Components:\n",
      "  [[-0.70710678 -0.70710678]\n",
      " [ 0.70710678 -0.70710678]]\n",
      "Explained Variance Ratio:\n",
      "  [ 0.95588995  0.04411005]\n"
     ]
    }
   ],
   "source": [
    "# we start by standardizing our dataset\n",
    "X_std = StandardScaler().fit_transform(X)\n",
    "\n",
    "# call PCA specifying we only want the\n",
    "# first two principal components (since\n",
    "# we only have a 2d datset)\n",
    "pca = PCA(n_components = 2)\n",
    "pca.fit(X_std)\n",
    "\n",
    "# important information\n",
    "print('Components:\\n ', pca.components_)\n",
    "print('Explained Variance Ratio:\\n ', pca.explained_variance_ratio_)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After fitting the PCA on the dataset, the fit learns some quantities from the data, most importantly the \"components\", which is the principal components (the new direction that our data points will be projected upon) and \"explained variance ratio\", which corresponds to the percentage of variance explained by each of the principal components. To get a better sense of what these numbers mean, let's visualize them over our standardized input data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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g8W/4hDxevfL4DX6tesVRKbS6uLCi5VK1TSc9nu0qm42gvnIMP7SaHOuLVbXM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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114955cc0>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 379,
       "width": 536
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "def draw_vector(v0, v1):\n",
    "    \"\"\"draw principal components as vectors\"\"\"\n",
    "    ax = plt.gca()\n",
    "    arrowprops = dict(arrowstyle = '->',\n",
    "                      linewidth = 2, edgecolor = 'black',\n",
    "                      shrinkA = 0, shrinkB = 0)\n",
    "    plt.annotate('', v1, v0, arrowprops = arrowprops)\n",
    "\n",
    "# plot data\n",
    "plt.scatter(X_std[:, 0], X_std[:, 1], s = 40, alpha = 0.8)\n",
    "for length, vector in zip(pca.explained_variance_, pca.components_):\n",
    "    # the larger the explained variance, the longer the\n",
    "    # annotated line will be\n",
    "    v = vector * 2 * np.sqrt(length)\n",
    "    draw_vector(pca.mean_, pca.mean_ + v)\n",
    "\n",
    "plt.axis('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the plot, we see that our PCA model gives us two principal components (the black line) and the length of the lines indicate of how important that principal component is in describing the variation of the original data. So if we were to choose 1 principal component to summarize our 2d dataset, it will be line that has the longest length (largest explained variance ratio). There're two things worth noting:\n",
    "\n",
    "- The number of principal components matches the total number of features\n",
    "- The first and second principal components are orthogonal to each other\n",
    "\n",
    "The reason for this is that principal components transform the data into a new set of dimensions, and these new dimensions have to be equal to the original amount of dimensions. And just like the original $x$, $y$ axis that we're used to, they have to be orthogonal to each other. Let's now reduce the dimension of our 2d dataset into 1d by transforming our data onto the most important principal component and plot it along with the original data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "original shape:    (88, 2)\n",
      "transformed shape: (88, 1)\n"
     ]
    },
    {
     "data": {
      "image/png": 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sIgqMTkxFOj8ba7aVaTKya77WEuc1F+cBxompiM0/N4AQAwAAANjnKHcfLu7I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jxg/vcujOSioiZ/5OeVDg7t9rIBAACAvtaRSoxnnnlGn/70p7f12E984hM6f/58J5axJXeS/Cyk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\nkTU6PlTo6SAGAAAAA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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1151eb0f0>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 379,
       "width": 536
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# dimensionality reduction, keeping only\n",
    "# the first principal component\n",
    "pca = PCA(n_components = 1)\n",
    "X_pca = pca.fit_transform(X_std)\n",
    "print(\"original shape:   \", X.shape)\n",
    "print(\"transformed shape:\", X_pca.shape)\n",
    "\n",
    "# inverse transform to obtain the projected data\n",
    "# and compare with the original\n",
    "X_new = pca.inverse_transform(X_pca)\n",
    "plt.scatter(X_std[:, 0], X_std[:, 1], s = 40, alpha = 0.2)\n",
    "plt.scatter(X_new[:, 0], X_new[:, 1], s = 40, alpha = 0.8)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the plot above, the lighter points are the original data and the darker points are the projected version. Looking at the projections we see that the points projected onto the first principal component all seem close to their initial representations and at the same time, it also captures most of the variations in our original data points.\n",
    "\n",
    "Hopefully, this elucidate what a PCA dimensionality reduction is doing. In a nutshell, PCA aims to find the directions or so called the principal components of maximum variance in high-dimensional data and project it onto a smaller dimensional subspace while retaining most of the information. In our 2d example, the linear relationship between $x_1$ and $x_2$ is mostly preserved using only 1 feature instead of 2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## PCA From Scratch"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Now that we have a high level understanding of what PCA is accomplishing, let's now formalize this with some notations. Recall that PCA is trying the find the direction such that the projection of the data on it has the highest variance. So given $X$, the centered data matrix, the projection, $Xw$ (dot product between the data point, $X$ and the projection weight, $w$), its variance can be computed as follows:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\frac{1}{n-1}(\\mathbf{Xw})^\\top \\mathbf{Xw} = \\mathbf w^\\top (\\frac{1}{n-1}\\mathbf X^\\top\\mathbf X) \\mathbf w = \\mathbf w^\\top \\mathbf{Cw}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Where $\\mathbf{C}$ is the covariance matrix of the data $\\mathbf{X}$. In case this looks unfamiliar, the covariance matrix is a $d \\times d$ matrix ($d$ is the total number of dimensions, features) where each element represents the covariance between two features. The covariance between two features $x_j$ and $x_k$ is calculated as follows:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\sigma_{jk} = \\frac{1}{n-1} \\sum_{i=1}^{n} \\left( x_{ij} - \\bar{x}_j \\right) \\left( x_{ik} - \\bar{x}_k \\right)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Where $\\bar{x}_j$ is simply the mean of vector (feature) $\\bar{x}_j = \\sum \\limits_{i=1}^n x_{ij}$.\n",
    "\n",
    "We can also express the calculation of the covariance matrix via the following matrix equation: \n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbf{C} = \\frac{1}{n-1} \\big( (X - \\bar{x})^\\top( X - \\bar{x}) \\big)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Because we already assume the data point $X$ has been centered, thus the $X - \\bar{x}$ part can be simplified into just $X$. And the formula above now becomes:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\mathbf{C} = \\frac{1}{n-1} X^\\top X\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Next, apart from the original objective function $\\mathbf w^\\top \\mathbf{Cw}$, we also introduce an additional constraint $\\|\\mathbf w\\|=\\mathbf w^\\top \\mathbf w=1$, saying our weight vector should have unit length. The intuition behind this is that: If we were to simply maximize the formula $\\mathbf w^\\top \\mathbf{Cw}$, we can multiply $w$ by any number and the objective function will increase by the square of this number. So the problem becomes ill-defined since the maximum of this expression is infinite. Given these two pieces of information, our objective function for PCA becomes:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "&  \\underset{\\mathbf{w}}{\\text{maximize}}\n",
    "&& \\mathbf w^\\top \\mathbf{Cw} \\nonumber \\\\\n",
    "&  \\text{subject to}\n",
    "&& \\mathbf w^\\top \\mathbf w=1\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "This objective function can be solved by the Lagrange multiplier, minimizing the loss function:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "L &= \\mathbf w^\\top \\mathbf{Cw}-\\lambda(\\mathbf w^\\top \\mathbf w-1)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "If this looks unfamiliar, check out the following tutorial on Lagrange multiplier. [Khan Academy: Lagrange multipliers, examples](https://www.khanacademy.org/math/multivariable-calculus/applications-of-multivariable-derivatives/constrained-optimization/a/lagrange-multipliers-examples). Next, to solve for $w$, we set the partial derivative of $L$ with respect to $w$ to 0.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\frac{\\partial L}{\\partial \\mathbf w} \n",
    "& = \\mathbf{Cw} - \\lambda \\mathbf w = 0 \\\\\n",
    "& = \\mathbf{Cw} = \\lambda \\mathbf w\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "Hopefully the formula above looks familar, since it's essentially an eigendecomposition problem. The notion of eigendecomposition is basically trying to solve the equation:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "Ax &= \\lambda x\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "In our case, $A$ is our covariance matrix correponding to $\\mathbf C$; $x$ is our eigenvector, correponding to $\\mathbf w$ and $\\lambda$ is our eigenvalue correponding to explained variance. After solving the equation above, we'll obtain eigenvector and eigenvalue pairs, where every eigenvector has a corresponding eigenvalue. An eigenvector is essentially the direction of each principal component and the eigenvalue is a number telling us how much variance there is in the data in that direction, in other words, how spread out the data is on the line. For those interested, the following blog walks through the calculation of eigenvalue and eigenvectors from scratch. [Blog: What are eigenvectors and eigenvalues?](http://www.visiondummy.com/2014/03/eigenvalues-eigenvectors/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given all of that, let's see how this works in code. Just to summarize, the general framework for computing PCA is as follows:\n",
    "\n",
    "- Standardize the data.\n",
    "- Obtain the Eigenvectors and Eigenvalues from the covariance matrix (technically the correlation matrix after performing the standardization).\n",
    "- Sort eigenvalues in descending order and choose the $k$ eigenvectors that correspond to the $k$ largest eigenvalues where $k$ is the number of dimensions of the new feature subspace.\n",
    "- Projection onto the new feature space. During this step we will take the top $k$ eigenvectors and use it to transform the original dataset $X$ to obtain a k-dimensional feature subspace $X'$.\n",
    "\n",
    "For the section below, we will be working with the famous \"Iris\" dataset. The iris dataset is a 150×4 matrix where the columns are the different features (sepal length in cm, sepal width in cm, petal length in cm, petal width in cm) and every every row represents a separate flower sample. The three classes in the Iris dataset are: Iris-setosa (n=50), Iris-versicolor (n=50), Iris-virginica (n=50). We'll use histograms to get a feeling of how the 3 different flower classes are distributes along the 4 different features."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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55ZRMmjQpQ4cOzebNm7Nw4cLMmzcvK1asyCWXXJLx48fn4x//eJdqvvPOO523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nfvDjMzM3Tp0gUeHh746quvkJubW2kf2WrLCxcuFJ7z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\nAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115569d30>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 419,
       "width": 708
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "iris = load_iris()\n",
    "X = iris['data']\n",
    "y = iris['target']\n",
    "\n",
    "label_dict = {\n",
    "    0: 'Iris-Setosa',\n",
    "    1: 'Iris-Versicolor',\n",
    "    2: 'Iris-Virgnica'}\n",
    "\n",
    "feature_dict = {\n",
    "    0: 'sepal length [cm]',\n",
    "    1: 'sepal width [cm]',\n",
    "    2: 'petal length [cm]',\n",
    "    3: 'petal width [cm]'}\n",
    "\n",
    "fig = plt.figure(figsize = (10, 6))\n",
    "for feature in range(len(feature_dict)):\n",
    "    plt.subplot(2, 2, feature + 1)\n",
    "    for index, label in label_dict.items():\n",
    "        plt.hist(X[y == index, feature], label = label,\n",
    "                 bins = 10, alpha = 0.3)\n",
    "        plt.xlabel(feature_dict[feature])\n",
    "        \n",
    "plt.legend(loc = 'upper right', fancybox = True, fontsize = 12)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Standardize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In general, it's important to standardize the data prior to a PCA on the covariance matrix depends on the measurement scales of the original features. Since PCA yields a feature subspace that maximizes the variance along the axes, it makes sense to standardize the data, especially if it was measured on different scales so different features will have equal contribution in terms of their scales."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# subtract the mean and divide the\n",
    "# standard deviation of each features\n",
    "mean = np.mean(X, axis = 0)\n",
    "scale = np.std(X, axis = 0)\n",
    "X_std = (X - mean) / scale"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Eigendecomposition - Computing Eigenvectors and Eigenvalues"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenvectors and eigenvalues of a covariance matrix is the core of PCA: The eigenvectors (principal components) determine the directions of the new feature space, and the eigenvalues determine their magnitude (variance explained along the principal components)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Covariance matrix \n",
      " [[ 1.00671141 -0.11010327  0.87760486  0.82344326]\n",
      " [-0.11010327  1.00671141 -0.42333835 -0.358937  ]\n",
      " [ 0.87760486 -0.42333835  1.00671141  0.96921855]\n",
      " [ 0.82344326 -0.358937    0.96921855  1.00671141]]\n",
      "Covariance matrix \n",
      " [[ 1.00671141 -0.11010327  0.87760486  0.82344326]\n",
      " [-0.11010327  1.00671141 -0.42333835 -0.358937  ]\n",
      " [ 0.87760486 -0.42333835  1.00671141  0.96921855]\n",
      " [ 0.82344326 -0.358937    0.96921855  1.00671141]]\n",
      "NumPy covariance matrix: \n",
      " [[ 1.00671141 -0.11010327  0.87760486  0.82344326]\n",
      " [-0.11010327  1.00671141 -0.42333835 -0.358937  ]\n",
      " [ 0.87760486 -0.42333835  1.00671141  0.96921855]\n",
      " [ 0.82344326 -0.358937    0.96921855  1.00671141]]\n"
     ]
    }
   ],
   "source": [
    "# for those unfamiliar with the @ syntax it\n",
    "# is equivalent of .dot, the dot product\n",
    "vec_mean = np.mean(X_std, axis = 0)\n",
    "vec_diff = X_std - vec_mean\n",
    "cov_mat  = vec_diff.T @ vec_diff / (X_std.shape[0] - 1)\n",
    "print('Covariance matrix \\n {}'.format(cov_mat))\n",
    "\n",
    "# note that since we already standardize the data,\n",
    "# meaning the mean vector for each features, vec_mean \n",
    "# will be 0 (really small numbers if you were to print them out),\n",
    "# hence we don't have to substract the mean to standardize them\n",
    "cov_mat = X_std.T @ X_std / (X_std.shape[0] - 1)\n",
    "print('Covariance matrix \\n {}'.format(cov_mat))\n",
    "\n",
    "# equivalently, we could have used the np.cov function:\n",
    "# since each row represents a variable for np.cov,\n",
    "# we'll need to transpose the matrix\n",
    "print('NumPy covariance matrix: \\n {}'.format(np.cov(X_std.T)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After obtaining the covariance matrix, we perform an eigendecomposition on it to obtain the eigenvalues and eigenvectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvectors \n",
      "[[ 0.52237162 -0.37231836 -0.72101681  0.26199559]\n",
      " [-0.26335492 -0.92555649  0.24203288 -0.12413481]\n",
      " [ 0.58125401 -0.02109478  0.14089226 -0.80115427]\n",
      " [ 0.56561105 -0.06541577  0.6338014   0.52354627]]\n",
      "\n",
      "Eigenvalues \n",
      "[ 2.93035378  0.92740362  0.14834223  0.02074601]\n"
     ]
    }
   ],
   "source": [
    "# since computing this by hand can be quite tedious\n",
    "# we'll simply use the pre-implemented function\n",
    "eig_vals, eig_vecs = np.linalg.eig(cov_mat)\n",
    "print('Eigenvectors \\n%s' % eig_vecs)\n",
    "print('\\nEigenvalues \\n%s' % eig_vals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Choosing Top K Eigenvectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that the typical goal of a PCA is to reduce the dimensionality of the original feature space by projecting it onto a smaller subspace. In order to decide which eigenvector(s) can dropped without losing too much information for the construction of lower-dimensional subspace, we need to inspect the corresponding eigenvalues. The idea is: eigenvectors with the lowest eigenvalues bear the least information about the distribution of the data and those are the ones can be dropped. In order to do so, the common approach is to rank the eigenvalues from highest to lowest in order choose the top $k$ eigenvectors.\n",
    "\n",
    "After sorting the eigenpairs, the next question is \"how many principal components are we going to choose for our new feature subspace?\" A useful measure is the so-called explained variance\", which can be calculated from the eigenvalues. The explained variance tells us how much extra information (variance) does each of the principal components contribute."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Variance Explained:  [ 0.72770452  0.23030523  0.03683832  0.00515193]\n",
      "Cumulative Variance Explained:  [ 0.72770452  0.95800975  0.99484807  1.        ]\n"
     ]
    },
    {
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uDESGxCZEzBqlqNDapZ/ao0aVlqp+atZsKR8+WnuUqtGq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\nKFWqFJUrV6Zhw4Y0atQow+J+2iwhISFPboGhPBGnT58GoGTJkhl2jnH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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1155955f8>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 419,
       "width": 564
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "eig_vals_total = np.sum(eig_vals)\n",
    "var_exp = eig_vals / eig_vals_total\n",
    "cum_var_exp = np.cumsum(var_exp)\n",
    "print('Variance Explained: ', var_exp)\n",
    "print('Cumulative Variance Explained: ', cum_var_exp)\n",
    "\n",
    "plt.bar(range(var_exp.shape[0]), var_exp, alpha = 0.5, \n",
    "        align = 'center', label = 'individual explained variance')\n",
    "plt.step(range(var_exp.shape[0]), cum_var_exp, \n",
    "         where = 'mid', label = 'cumulative explained variance')\n",
    "plt.ylabel('Explained variance ratio')\n",
    "plt.xlabel('Principal components')\n",
    "plt.ylim(0, 1.1)\n",
    "plt.legend(loc = 'best')\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot above clearly shows that most of the variance (72.77% of the variance to be precise) can be explained by the first principal component alone. The second principal component still bears some information (23.03%) while the third and fourth principal components can be safely dropped without losing to much information. Together, the first two principal components explains 95.8% of the variance. In other words, it contain 95.8% of the information (95% or 99% are common threshold that people use)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Projection Onto the New Feature Space"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the last step, we will use the $4 \\times 2$-dimensional eigenvectors to transform our data onto the new subspace via the equation $Y=XW$, where Y will be our $150 \\times 2$ transformed data (we reduce the dimension from the original 4 down to 2)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_iris_pca(X_pca, y):\n",
    "    \"\"\"a scatter plot of the 2-dimensional iris data\"\"\"\n",
    "    markers = 's', 'x', 'o'\n",
    "    colors = list(plt.rcParams['axes.prop_cycle'])\n",
    "    target = np.unique(y)\n",
    "    for idx, (t, m) in enumerate(zip(target, markers)):\n",
    "        subset = X_pca[y == t]\n",
    "        plt.scatter(subset[:, 0], subset[:, 1], s = 50,\n",
    "                    c = colors[idx]['color'], label = t, marker = m)\n",
    "\n",
    "    plt.xlabel('PC 1')\n",
    "    plt.ylabel('PC 2')\n",
    "    plt.legend(loc = 'lower left')\n",
    "    plt.tight_layout()\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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huhMAAAAQMvbtKS29ckTKvfNeQLPuobSUe0dLrxyVfXuquQcEAASCkAYAAMAv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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1155862b0>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 419,
       "width": 564
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "X_std_pca = X_std.dot(eig_vecs[:, 0:2])\n",
    "plot_iris_pca(X_std_pca, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll put all of this into a single class, train it and confirm the result with scikit-learn's PCA model by printing out the explained variance ratio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class PCAModel:\n",
    "    \"\"\"\n",
    "    Principal component analysis (PCA)\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    n_components : int\n",
    "        top number of principal components to keep\n",
    "    \"\"\"\n",
    "\n",
    "    def __init__(self, n_components):\n",
    "        self.n_components = n_components\n",
    "\n",
    "    def fit(self, X):\n",
    "        # standardize\n",
    "        X = X.copy()\n",
    "        self.mean = np.mean(X, axis = 0)\n",
    "        self.scale = np.std(X, axis = 0)\n",
    "        X = (X - self.mean) / self.scale\n",
    "        \n",
    "        # eigendecomposition\n",
    "        eig_vals, eig_vecs = np.linalg.eig(np.cov(X.T))\n",
    "        self.components = eig_vecs[:, :self.n_components]\n",
    "        var_exp = eig_vals / np.sum(eig_vals)\n",
    "        self.explained_variance_ratio = var_exp[:self.n_components]\n",
    "        return self\n",
    "\n",
    "    def transform(self, X):\n",
    "        X = X.copy()\n",
    "        X = (X - self.mean) / self.scale\n",
    "        X_pca = X @ self.components\n",
    "        return X_pca"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "library:  [ 0.72770452  0.23030523]\n",
      "from scratch:  [ 0.72770452  0.23030523]\n"
     ]
    }
   ],
   "source": [
    "# implementation from scratch\n",
    "pca_model = PCAModel(n_components = 2).fit(X)\n",
    "\n",
    "# using library to confirm results\n",
    "scaler = StandardScaler()\n",
    "X_std = scaler.fit_transform(X)\n",
    "pca = PCA(n_components = 2).fit(X_std)\n",
    "\n",
    "# print explained ratio to see if the matches\n",
    "print('library: ', pca.explained_variance_ratio_)\n",
    "print('from scratch: ', pca_model.explained_variance_ratio)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Takeaway"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's wrap this up with a summary of PCA and some of its applications. In order to identify patterns in our data, we often look for variation across observations to distinguish them from one another. Hence it seems reasonable to be able to find a succinct representation that best captures the variation in our initial data. PCA, in particular, look to explain our data via its maximum directions of variance. By compressing a higher dimensional dataset into lower one, while still retaining most of the variance this allows us to:\n",
    "\n",
    "- Perform Visualization: PCA summarizes our data along the principal components (or eigenvectors), which explains most the variance. Thus we can reduce the dataset to 2 or 3 dimensions and visualize our data's distribution. This can be helpful when we're performing clustering algorithm that needs to choose the cluster number beforehand.\n",
    "- Speed up machine learning algorithms: When dealing with Big Data, we might want to decompose the features into a lower dimension, without a significant loss in variance. By performing dimensionality reduction methods, we can reduce the amount of features we have to speed up the training algorithm and save memory. For this part, we'll use the Iris dataset again and show that by keeping the data's top 3 principal components we can obtain the same level of accuracy as keeping all 4 features."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "PCA Accuracy 0.822\n",
      "Accuracy 0.822\n"
     ]
    }
   ],
   "source": [
    "# split 30% of the iris data into a test set for evaluation\n",
    "X_train, X_test, y_train, y_test = train_test_split(\n",
    "    X, y, test_size = 0.3, random_state = 1)\n",
    "\n",
    "# create the pipeline, where we'll\n",
    "# standardize the data, perform PCA and\n",
    "# fit the logistic regression\n",
    "pipeline1 = Pipeline([\n",
    "    ('standardize', StandardScaler()),\n",
    "    ('pca', PCA(n_components = 3)),\n",
    "    ('logistic', LogisticRegression(random_state = 1))\n",
    "])\n",
    "pipeline1.fit(X_train, y_train)\n",
    "y_pred1 = pipeline1.predict(X_test)\n",
    "\n",
    "# pipeline without PCA\n",
    "pipeline2 = Pipeline([\n",
    "    ('standardize', StandardScaler()),\n",
    "    ('logistic', LogisticRegression(random_state = 1))\n",
    "])\n",
    "pipeline2.fit(X_train, y_train)\n",
    "y_pred2 = pipeline2.predict(X_test)\n",
    "\n",
    "# access the prediction accuracy\n",
    "print('PCA Accuracy %.3f' % accuracy_score(y_test, y_pred1))\n",
    "print('Accuracy %.3f' % accuracy_score(y_test, y_pred2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reference"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- [Youtube: Dimensionality Reduction](https://www.youtube.com/playlist?list=PLnnr1O8OWc6aVexn2BY0qjklobY6TUEIy)\n",
    "- [Blog: What are eigenvectors and eigenvalues?](http://www.visiondummy.com/2014/03/eigenvalues-eigenvectors/)\n",
    "- [Blog: Principal Component Analysis 4 Dummies](https://georgemdallas.wordpress.com/2013/10/30/principal-component-analysis-4-dummies-eigenvectors-eigenvalues-and-dimension-reduction/)\n",
    "- [Blog: Everything you did and didn't know about PCA](http://alexhwilliams.info/itsneuronalblog/2016/03/27/pca/)\n",
    "- [Blog: Principal Component Analysis in 3 Simple Steps](http://sebastianraschka.com/Articles/2015_pca_in_3_steps.html)\n",
    "- [Notebook: In Depth: Principal Component Analysis](http://nbviewer.jupyter.org/github/jakevdp/PythonDataScienceHandbook/blob/95664b978439df948b2cd2f5f5b4e31f28b30394/notebooks/05.09-Principal-Component-Analysis.ipynb)\n",
    "- [StackExchange: Making sense of principal component analysis, eigenvectors & eigenvalues](http://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues)\n",
    "- [StackExchange: What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)?](http://stats.stackexchange.com/questions/217995/what-is-an-intuitive-explanation-for-how-pca-turns-from-a-geometric-problem-wit)"
   ]
  }
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