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线性回归实验 html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; } .mac-os-11 { --title-bar-height:28px; } html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; } body { margin: 0px; padding: 0px; height: auto; inset: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; } iframe { margin: auto; } a.url { word-break: break-all; } a:active, a:hover { outline: 0px; } .in-text-selection, ::selection { text-shadow: none; background: var(--select-text-bg-color); color: var(--select-text-font-color); } #write { margin: 0px auto; height: auto; width: inherit; word-break: normal; overflow-wrap: break-word; position: relative; white-space: normal; overflow-x: visible; padding-top: 36px; } #write.first-line-indent p { text-indent: 2em; } #write.first-line-indent li p, #write.first-line-indent p * { text-indent: 0px; } #write.first-line-indent li { margin-left: 2em; } .for-image #write { padding-left: 8px; padding-right: 8px; } body.typora-export { padding-left: 30px; padding-right: 30px; } .typora-export .footnote-line, .typora-export li, .typora-export p { white-space: pre-wrap; } .typora-export .task-list-item input { pointer-events: none; } @media screen and (max-width: 500px) { body.typora-export { padding-left: 0px; padding-right: 0px; } #write { padding-left: 20px; padding-right: 20px; } } #write li > figure:last-child { margin-bottom: 0.5rem; } #write ol, #write ul { position: relative; } img { max-width: 100%; vertical-align: middle; image-orientation: from-image; } button, input, select, textarea { color: inherit; font: inherit; } input[type="checkbox"], input[type="radio"] { line-height: normal; padding: 0px; } *, ::after, ::before { box-sizing: border-box; } #write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p, #write pre { width: inherit; } #write h1, #write h2, #write h3, #write h4, #write h5, #write h6, #write p { position: relative; } p { line-height: inherit; } h1, h2, h3, h4, h5, h6 { break-after: avoid-page; break-inside: avoid; orphans: 4; } p { orphans: 4; } h1 { font-size: 2rem; } h2 { font-size: 1.8rem; } h3 { font-size: 1.6rem; } h4 { font-size: 1.4rem; } h5 { font-size: 1.2rem; } h6 { font-size: 1rem; } .md-math-block, .md-rawblock, h1, h2, h3, h4, h5, h6, p { margin-top: 1rem; margin-bottom: 1rem; } .hidden { display: none; } .md-blockmeta { color: rgb(204, 204, 204); font-weight: 700; font-style: italic; } a { cursor: pointer; } sup.md-footnote { padding: 2px 4px; background-color: rgba(238, 238, 238, 0.7); color: rgb(85, 85, 85); border-radius: 4px; cursor: pointer; } sup.md-footnote a, sup.md-footnote a:hover { color: inherit; text-transform: inherit; text-decoration: inherit; } #write input[type="checkbox"] { cursor: pointer; width: inherit; height: inherit; } figure { overflow-x: auto; margin: 1.2em 0px; max-width: calc(100% + 16px); padding: 0px; } figure > table { margin: 0px; } thead, tr { break-inside: avoid; break-after: auto; } thead { display: table-header-group; } table { border-collapse: collapse; border-spacing: 0px; width: 100%; overflow: auto; break-inside: auto; text-align: left; } table.md-table td { min-width: 32px; } .CodeMirror-gutters { border-right: 0px; background-color: inherit; } .CodeMirror-linenumber { user-select: none; } .CodeMirror { text-align: left; } .CodeMirror-placeholder { opacity: 0.3; } .CodeMirror pre { padding: 0px 4px; } .CodeMirror-lines { padding: 0px; } div.hr:focus { cursor: none; } #write pre { white-space: pre-wrap; } #write.fences-no-line-wrapping pre { white-space: pre; } #write pre.ty-contain-cm { white-space: normal; } .CodeMirror-gutters { margin-right: 4px; } .md-fences { font-size: 0.9rem; display: block; break-inside: avoid; text-align: left; overflow: visible; white-space: pre; background: inherit; position: relative !important; } .md-fences-adv-panel { width: 100%; margin-top: 10px; text-align: center; padding-top: 0px; padding-bottom: 8px; overflow-x: auto; } #write .md-fences.mock-cm { white-space: pre-wrap; } .md-fences.md-fences-with-lineno { padding-left: 0px; } #write.fences-no-line-wrapping .md-fences.mock-cm { white-space: pre; overflow-x: auto; } .md-fences.mock-cm.md-fences-with-lineno { padding-left: 8px; } .CodeMirror-line, twitterwidget { break-inside: avoid; } svg { break-inside: avoid; } .footnotes { opacity: 0.8; font-size: 0.9rem; margin-top: 1em; margin-bottom: 1em; } .footnotes + .footnotes { margin-top: 0px; } .md-reset { margin: 0px; padding: 0px; border: 0px; outline: 0px; vertical-align: top; background: 0px 0px; text-decoration: none; text-shadow: none; float: none; position: static; width: auto; height: auto; white-space: nowrap; cursor: inherit; -webkit-tap-highlight-color: transparent; line-height: normal; font-weight: 400; text-align: left; box-sizing: content-box; direction: ltr; } li div { padding-top: 0px; } blockquote { margin: 1rem 0px; } li .mathjax-block, li p { margin: 0.5rem 0px; } li blockquote { margin: 1rem 0px; } li { margin: 0px; position: relative; } blockquote > :last-child { margin-bottom: 0px; } blockquote > :first-child, li > :first-child { margin-top: 0px; } .footnotes-area { color: rgb(136, 136, 136); margin-top: 0.714rem; padding-bottom: 0.143rem; white-space: normal; } #write .footnote-line { white-space: pre-wrap; } @media print { body, html { border: 1px solid transparent; height: 99%; break-after: avoid; break-before: avoid; font-variant-ligatures: no-common-ligatures; } #write { margin-top: 0px; padding-top: 0px; border-color: transparent !important; padding-bottom: 0px !important; } .typora-export * { -webkit-print-color-adjust: exact; } .typora-export #write { break-after: avoid; } .typora-export #write::after { height: 0px; } .is-mac table { break-inside: avoid; } .typora-export-show-outline .typora-export-sidebar { display: none; } } .footnote-line { margin-top: 0.714em; font-size: 0.7em; } a img, img a { cursor: pointer; } pre.md-meta-block { font-size: 0.8rem; min-height: 0.8rem; white-space: pre-wrap; background: rgb(204, 204, 204); display: block; overflow-x: hidden; } p > .md-image:only-child:not(.md-img-error) img, p > img:only-child { display: block; margin: auto; } #write.first-line-indent p > .md-image:only-child:not(.md-img-error) img { left: -2em; position: relative; } p > .md-image:only-child { display: inline-block; width: 100%; } #write .MathJax_Display { margin: 0.8em 0px 0px; } .md-math-block { width: 100%; } .md-math-block:not(:empty)::after { display: none; } .MathJax_ref { fill: currentcolor; } [contenteditable="true"]:active, [contenteditable="true"]:focus, [contenteditable="false"]:active, [contenteditable="false"]:focus { outline: 0px; box-shadow: none; } .md-task-list-item { position: relative; list-style-type: none; } .task-list-item.md-task-list-item { padding-left: 0px; } .md-task-list-item > input { position: absolute; top: 0px; left: 0px; margin-left: -1.2em; margin-top: calc(1em - 10px); border: none; } .math { font-size: 1rem; } .md-toc { min-height: 3.58rem; position: relative; font-size: 0.9rem; border-radius: 10px; } .md-toc-content { position: relative; margin-left: 0px; } .md-toc-content::after, .md-toc::after { display: none; } .md-toc-item { display: block; color: rgb(65, 131, 196); } .md-toc-item a { text-decoration: none; } .md-toc-inner:hover { text-decoration: underline; } .md-toc-inner { display: inline-block; cursor: pointer; } .md-toc-h1 .md-toc-inner { margin-left: 0px; font-weight: 700; } .md-toc-h2 .md-toc-inner { margin-left: 2em; } .md-toc-h3 .md-toc-inner { margin-left: 4em; } .md-toc-h4 .md-toc-inner { margin-left: 6em; } .md-toc-h5 .md-toc-inner { margin-left: 8em; } .md-toc-h6 .md-toc-inner { margin-left: 10em; } @media screen and (max-width: 48em) { .md-toc-h3 .md-toc-inner { margin-left: 3.5em; } .md-toc-h4 .md-toc-inner { margin-left: 5em; } .md-toc-h5 .md-toc-inner { margin-left: 6.5em; } .md-toc-h6 .md-toc-inner { margin-left: 8em; } } a.md-toc-inner { font-size: inherit; font-style: inherit; font-weight: inherit; line-height: inherit; } .footnote-line a:not(.reversefootnote) { color: inherit; } .reversefootnote { font-family: ui-monospace, sans-serif; } .md-attr { display: none; } .md-fn-count::after { content: "."; } code, pre, samp, tt { font-family: var(--monospace); } kbd { margin: 0px 0.1em; padding: 0.1em 0.6em; font-size: 0.8em; color: rgb(36, 39, 41); background: rgb(255, 255, 255); border: 1px solid rgb(173, 179, 185); border-radius: 3px; box-shadow: rgba(12, 13, 14, 0.2) 0px 1px 0px, rgb(255, 255, 255) 0px 0px 0px 2px inset; white-space: nowrap; vertical-align: middle; } .md-comment { color: rgb(162, 127, 3); opacity: 0.6; font-family: var(--monospace); } code { text-align: left; vertical-align: initial; } a.md-print-anchor { white-space: pre !important; border-width: initial !important; border-style: none !important; border-color: initial !important; display: inline-block !important; position: absolute !important; width: 1px !important; right: 0px !important; outline: 0px !important; background: 0px 0px !important; text-decoration: initial !important; text-shadow: initial !important; } .os-windows.monocolor-emoji .md-emoji { font-family: "Segoe UI Symbol", sans-serif; } .md-diagram-panel > svg { max-width: 100%; } [lang="flow"] svg, [lang="mermaid"] svg { max-width: 100%; height: auto; } [lang="mermaid"] .node text { font-size: 1rem; } table tr th { border-bottom: 0px; } video { max-width: 100%; display: block; margin: 0px auto; } iframe { max-width: 100%; width: 100%; border: none; } .highlight td, .highlight tr { border: 0px; } mark { background: rgb(255, 255, 0); color: rgb(0, 0, 0); } .md-html-inline .md-plain, .md-html-inline strong, mark .md-inline-math, mark strong { color: inherit; } .md-expand mark .md-meta { opacity: 0.3 !important; } mark .md-meta { color: rgb(0, 0, 0); } @media print { .typora-export h1, .typora-export h2, .typora-export h3, .typora-export h4, .typora-export h5, .typora-export h6 { break-inside: avoid; } } .md-diagram-panel .messageText { stroke: none !important; } .md-diagram-panel .start-state { fill: var(--node-fill); } .md-diagram-panel .edgeLabel rect { opacity: 1 !important; } .md-fences.md-fences-math { font-size: 1em; } .md-fences-advanced:not(.md-focus) { padding: 0px; white-space: nowrap; border: 0px; } .md-fences-advanced:not(.md-focus) { background: inherit; } .typora-export-show-outline .typora-export-content { max-width: 1440px; margin: auto; display: flex; flex-direction: row; } .typora-export-sidebar { width: 300px; font-size: 0.8rem; margin-top: 80px; margin-right: 18px; } .typora-export-show-outline #write { --webkit-flex:2; flex: 2 1 0%; } .typora-export-sidebar .outline-content { position: fixed; top: 0px; max-height: 100%; overflow: hidden auto; padding-bottom: 30px; padding-top: 60px; width: 300px; } @media screen and (max-width: 1024px) { .typora-export-sidebar, .typora-export-sidebar .outline-content { width: 240px; } } @media screen and (max-width: 800px) { .typora-export-sidebar { display: none; } } .outline-content li, .outline-content ul { margin-left: 0px; margin-right: 0px; padding-left: 0px; padding-right: 0px; list-style: none; } .outline-content ul { margin-top: 0px; margin-bottom: 0px; } .outline-content strong { font-weight: 400; } .outline-expander { width: 1rem; height: 1.42857rem; position: relative; display: table-cell; vertical-align: middle; cursor: pointer; padding-left: 4px; } .outline-expander::before { content: ""; position: relative; font-family: Ionicons; display: inline-block; font-size: 8px; vertical-align: middle; } .outline-item { padding-top: 3px; padding-bottom: 3px; cursor: pointer; } .outline-expander:hover::before { content: ""; } .outline-h1 > .outline-item { padding-left: 0px; } .outline-h2 > .outline-item { padding-left: 1em; } .outline-h3 > .outline-item { padding-left: 2em; } .outline-h4 > .outline-item { padding-left: 3em; } .outline-h5 > .outline-item { padding-left: 4em; } .outline-h6 > .outline-item { padding-left: 5em; } .outline-label { cursor: pointer; display: table-cell; vertical-align: middle; text-decoration: none; color: inherit; } .outline-label:hover { text-decoration: underline; } .outline-item:hover { border-color: rgb(245, 245, 245); background-color: var(--item-hover-bg-color); } .outline-item:hover { margin-left: -28px; margin-right: -28px; border-left: 28px solid transparent; border-right: 28px solid transparent; } .outline-item-single .outline-expander::before, .outline-item-single .outline-expander:hover::before { display: none; } .outline-item-open > .outline-item > .outline-expander::before { content: ""; } .outline-children { display: none; } .info-panel-tab-wrapper { display: none; } .outline-item-open > .outline-children { display: block; } .typora-export .outline-item { padding-top: 1px; padding-bottom: 1px; } .typora-export .outline-item:hover { margin-right: -8px; border-right: 8px solid transparent; } .typora-export .outline-expander::before { content: "+"; font-family: inherit; top: -1px; } .typora-export .outline-expander:hover::before, .typora-export .outline-item-open > .outline-item > .outline-expander::before { content: "−"; } .typora-export-collapse-outline .outline-children { display: none; } .typora-export-collapse-outline .outline-item-open > .outline-children, .typora-export-no-collapse-outline .outline-children { display: block; } .typora-export-no-collapse-outline .outline-expander::before { content: "" !important; } .typora-export-show-outline .outline-item-active > .outline-item .outline-label { font-weight: 700; } .md-inline-math-container mjx-container { zoom: 0.95; } .CodeMirror { height: auto; } .CodeMirror.cm-s-inner { background: inherit; } .CodeMirror-scroll { overflow: auto hidden; z-index: 3; } .CodeMirror-gutter-filler, .CodeMirror-scrollbar-filler { background-color: rgb(255, 255, 255); } .CodeMirror-gutters { border-right: 1px solid rgb(221, 221, 221); background: inherit; white-space: nowrap; } .CodeMirror-linenumber { padding: 0px 3px 0px 5px; text-align: right; color: rgb(153, 153, 153); } .cm-s-inner .cm-keyword { color: rgb(119, 0, 136); } .cm-s-inner .cm-atom, .cm-s-inner.cm-atom { color: rgb(34, 17, 153); } .cm-s-inner .cm-number { color: rgb(17, 102, 68); } .cm-s-inner .cm-def { color: rgb(0, 0, 255); } .cm-s-inner .cm-variable { color: rgb(0, 0, 0); } .cm-s-inner .cm-variable-2 { color: rgb(0, 85, 170); } .cm-s-inner .cm-variable-3 { color: rgb(0, 136, 85); } .cm-s-inner .cm-string { color: rgb(170, 17, 17); } .cm-s-inner .cm-property { color: rgb(0, 0, 0); } .cm-s-inner .cm-operator { color: rgb(152, 26, 26); } .cm-s-inner .cm-comment, .cm-s-inner.cm-comment { color: rgb(170, 85, 0); } .cm-s-inner .cm-string-2 { color: rgb(255, 85, 0); } .cm-s-inner .cm-meta { color: rgb(85, 85, 85); } .cm-s-inner .cm-qualifier { color: rgb(85, 85, 85); } .cm-s-inner .cm-builtin { color: rgb(51, 0, 170); } .cm-s-inner .cm-bracket { color: rgb(153, 153, 119); } .cm-s-inner .cm-tag { color: rgb(17, 119, 0); } .cm-s-inner .cm-attribute { color: rgb(0, 0, 204); } .cm-s-inner .cm-header, .cm-s-inner.cm-header { color: rgb(0, 0, 255); } .cm-s-inner .cm-quote, .cm-s-inner.cm-quote { color: rgb(0, 153, 0); } .cm-s-inner .cm-hr, .cm-s-inner.cm-hr { color: rgb(153, 153, 153); } .cm-s-inner .cm-link, .cm-s-inner.cm-link { color: rgb(0, 0, 204); } .cm-negative { color: rgb(221, 68, 68); } .cm-positive { color: rgb(34, 153, 34); } .cm-header, .cm-strong { font-weight: 700; } .cm-del { text-decoration: line-through; } .cm-em { font-style: italic; } .cm-link { text-decoration: underline; } .cm-error { color: red; } .cm-invalidchar { color: red; } .cm-constant { color: rgb(38, 139, 210); } .cm-defined { color: rgb(181, 137, 0); } div.CodeMirror span.CodeMirror-matchingbracket { color: rgb(0, 255, 0); } div.CodeMirror span.CodeMirror-nonmatchingbracket { color: rgb(255, 34, 34); } .cm-s-inner .CodeMirror-activeline-background { background: inherit; } .CodeMirror { position: relative; overflow: hidden; } .CodeMirror-scroll { height: 100%; outline: 0px; position: relative; box-sizing: content-box; background: inherit; } .CodeMirror-sizer { position: relative; } .CodeMirror-gutter-filler, .CodeMirror-hscrollbar, .CodeMirror-scrollbar-filler, .CodeMirror-vscrollbar { position: absolute; z-index: 6; display: none; outline: 0px; } .CodeMirror-vscrollbar { right: 0px; top: 0px; overflow: hidden; } .CodeMirror-hscrollbar { bottom: 0px; left: 0px; overflow: auto hidden; } .CodeMirror-scrollbar-filler { right: 0px; bottom: 0px; } .CodeMirror-gutter-filler { left: 0px; bottom: 0px; } .CodeMirror-gutters { position: absolute; left: 0px; top: 0px; padding-bottom: 10px; z-index: 3; overflow-y: hidden; } .CodeMirror-gutter { white-space: normal; height: 100%; box-sizing: content-box; padding-bottom: 30px; margin-bottom: -32px; display: inline-block; } .CodeMirror-gutter-wrapper { position: absolute; z-index: 4; background: 0px 0px !important; border: none !important; } .CodeMirror-gutter-background { position: absolute; top: 0px; bottom: 0px; z-index: 4; } .CodeMirror-gutter-elt { position: absolute; cursor: default; z-index: 4; } .CodeMirror-lines { cursor: text; } .CodeMirror pre { border-radius: 0px; border-width: 0px; background: 0px 0px; font-family: inherit; font-size: inherit; margin: 0px; white-space: pre; overflow-wrap: normal; color: inherit; z-index: 2; position: relative; overflow: visible; } .CodeMirror-wrap pre { overflow-wrap: break-word; white-space: pre-wrap; word-break: normal; } .CodeMirror-code pre { border-right: 30px solid transparent; width: fit-content; } .CodeMirror-wrap .CodeMirror-code pre { border-right: none; width: auto; } .CodeMirror-linebackground { position: absolute; inset: 0px; z-index: 0; } .CodeMirror-linewidget { position: relative; z-index: 2; overflow: auto; } .CodeMirror-wrap .CodeMirror-scroll { overflow-x: hidden; } .CodeMirror-measure { position: absolute; width: 100%; height: 0px; overflow: hidden; visibility: hidden; } .CodeMirror-measure pre { position: static; } .CodeMirror div.CodeMirror-cursor { position: absolute; visibility: hidden; border-right: none; width: 0px; } .CodeMirror div.CodeMirror-cursor { visibility: hidden; } .CodeMirror-focused div.CodeMirror-cursor { visibility: inherit; } .cm-searching { background: rgba(255, 255, 0, 0.4); } span.cm-underlined { text-decoration: underline; } span.cm-strikethrough { text-decoration: line-through; } .cm-tw-syntaxerror { color: rgb(255, 255, 255); background-color: rgb(153, 0, 0); } .cm-tw-deleted { text-decoration: line-through; } .cm-tw-header5 { font-weight: 700; } .cm-tw-listitem:first-child { padding-left: 10px; } .cm-tw-box { border-style: solid; border-right-width: 1px; border-bottom-width: 1px; border-left-width: 1px; border-color: inherit; border-top-width: 0px !important; } .cm-tw-underline { text-decoration: underline; } @media print { .CodeMirror div.CodeMirror-cursor { visibility: hidden; } } :root { --side-bar-bg-color: #fafafa; --control-text-color: #777; } @include-when-export url(https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext); /* open-sans-regular - latin-ext_latin */ /* open-sans-italic - latin-ext_latin */ /* open-sans-700 - latin-ext_latin */ /* open-sans-700italic - latin-ext_latin */ html { font-size: 16px; -webkit-font-smoothing: antialiased; } body { font-family: "Open Sans","Clear Sans", "Helvetica Neue", Helvetica, Arial, 'Segoe UI Emoji', sans-serif; color: rgb(51, 51, 51); line-height: 1.6; } #write { max-width: 860px; margin: 0 auto; padding: 30px; padding-bottom: 100px; } @media only screen and (min-width: 1400px) { #write { max-width: 1024px; } } @media only screen and (min-width: 1800px) { #write { max-width: 1200px; } } #write > ul:first-child, #write > ol:first-child{ margin-top: 30px; } a { color: #4183C4; } h1, h2, h3, h4, h5, h6 { position: relative; margin-top: 1rem; margin-bottom: 1rem; font-weight: bold; line-height: 1.4; cursor: text; } h1:hover a.anchor, h2:hover a.anchor, h3:hover a.anchor, h4:hover a.anchor, h5:hover a.anchor, h6:hover a.anchor { text-decoration: none; } h1 tt, h1 code { font-size: inherit; } h2 tt, h2 code { font-size: inherit; } h3 tt, h3 code { font-size: inherit; } h4 tt, h4 code { font-size: inherit; } h5 tt, h5 code { font-size: inherit; } h6 tt, h6 code { font-size: inherit; } h1 { font-size: 2.25em; line-height: 1.2; border-bottom: 1px solid #eee; } h2 { font-size: 1.75em; line-height: 1.225; border-bottom: 1px solid #eee; } /*@media print { .typora-export h1, .typora-export h2 { border-bottom: none; padding-bottom: initial; } .typora-export h1::after, .typora-export h2::after { content: ""; display: block; height: 100px; margin-top: -96px; border-top: 1px solid #eee; } }*/ h3 { font-size: 1.5em; line-height: 1.43; } h4 { font-size: 1.25em; } h5 { font-size: 1em; } h6 { font-size: 1em; color: #777; } p, blockquote, ul, ol, dl, table{ margin: 0.8em 0; } li>ol, li>ul { margin: 0 0; } hr { height: 2px; padding: 0; margin: 16px 0; background-color: #e7e7e7; border: 0 none; overflow: hidden; box-sizing: content-box; } li p.first { display: inline-block; } ul, ol { padding-left: 30px; } ul:first-child, ol:first-child { margin-top: 0; } ul:last-child, ol:last-child { margin-bottom: 0; } blockquote { border-left: 4px solid #dfe2e5; padding: 0 15px; color: #777777; } blockquote blockquote { padding-right: 0; } table { padding: 0; word-break: initial; } table tr { border: 1px solid #dfe2e5; margin: 0; padding: 0; } table tr:nth-child(2n), thead { background-color: #f8f8f8; } table th { 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line[data-line] { stroke-width: 70px; fill: none; } g[data-mml-node="mtable"] > rect[data-frame], svg[data-table] > g > rect[data-frame] { stroke-width: 70px; fill: none; } g[data-mml-node="mtable"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed { stroke-dasharray: 140; } g[data-mml-node="mtable"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted { stroke-linecap: round; stroke-dasharray: 0,140; } g[data-mml-node="mtable"] > g > svg { overflow: visible; } [jax="SVG"] mjx-tool { display: inline-block; position: relative; width: 0; height: 0; } [jax="SVG"] mjx-tool > mjx-tip { position: absolute; top: 0; left: 0; } mjx-tool > mjx-tip { display: inline-block; padding: .2em; border: 1px solid #888; font-size: 70%; background-color: #F8F8F8; color: black; box-shadow: 2px 2px 5px #AAAAAA; } g[data-mml-node="maction"][data-toggle] { cursor: pointer; } mjx-status { display: block; position: fixed; left: 1em; bottom: 1em; min-width: 25%; padding: .2em .4em; border: 1px solid #888; font-size: 90%; background-color: #F8F8F8; color: black; } foreignObject[data-mjx-xml] { font-family: initial; line-height: normal; overflow: visible; } mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] { stroke-width: 3; } g[data-mml-node="xypic"] path { stroke-width: inherit; } .MathJax g[data-mml-node="xypic"] path { stroke-width: inherit; } mjx-container[jax="SVG"] path[data-c], mjx-container[jax="SVG"] use[data-c] { stroke-width: 0; } 线性回归实验 线性回归回归问题是非常常见的一类问题，目的是寻找变量之间的关系。比如要从数据中寻找房屋面积与价格的关系，年龄和身高的关系，气体压力和体积的关系等等。而机器学习要做的正是要让机器自己来学习这些关系，并为对未知的情况做出预测。对于线性回归，假设变量之间的关系是线性的，即： hθ(x)=θ0+θ1xh_{\theta}(x)= \theta_{0} + \theta_{1} x 其中 θθ\pmb{\theta} 就是学习算法需要学习的参数，在线性回归的问题上，就是θ1\theta_{1}和θ0\theta_{0}，而 xx 是我们对于问题所选取的特征，也即输入。hh表示算法得到的映射。 代价函数的表示为了找到这个算法中合适的参数，我们需要制定一个标准。一般而言算法拟合出来的结果与真实的结果误差越小越好，试想一下如果算法拟合出来的结果与真实值的误差为零，那么就是说算法完美地拟合了数据。所以可以根据“真实值与算法拟合值的误差”来表示算法的“合适程度”。在线性回归中，我们经常使用最小二乘的思路构建代价函数： J(θθ)=12n∑i=1n(hθ(x(i))−y(i))2J(\pmb{\theta}) = \frac{1}{2n}\sum_{i=1}^{n} \Big( h_{\theta}(x^{(i)}) - y^{(i)} \Big)^2 这里 hθ(x(i))h_{\theta}(x^{(i)}) 由假设模型得出。对线性回归任务，代价函数可以展开为： J(θθ)=12n∑i=1n(θ0+θ1x(i)−y(i))2J(\pmb{\theta}) = \frac{1}{2n} \sum_{i=1}^{n} \Big( \theta_0 + \theta_1 x^{(i)} - y^{(i)} \Big)^2 误差函数的值越小，则代表算法拟合结果与真实结果越接近。 梯度下降梯度下降算法沿着误差函数的反向更新θ\theta的值，知道代价函数收敛到最小值。梯度下降算法更新θi\theta_i的方法为： θi=θi−α∂∂θiJ(θθ)\theta_i = \theta_i - \alpha\frac{\partial }{\partial \theta_i}J(\pmb{\theta})其中 α\alpha表示学习率。对于线性回归的的参数，可以根据代价函数求出其参数更新公式： ∂J∂θ0=1n∑i=1n(hθ(x(i))−y(i))⋅1,\frac{\partial J}{\partial \theta_{0} } = \frac{1}{n} \sum_{i=1}^{n} \Big( h_{\theta}(x^{(i)}) - y^{(i)} \Big) \cdot 1, ∂J∂θ1=1n∑i=1n(hθ(x(i))−y(i))⋅x(i). \frac{\partial J}{\partial \theta_{1} } = \frac{1}{n} \sum_{i=1}^{n} \Big( h_{\theta}(x^{(i)}) - y^{(i)} \Big) \cdot x^{(i)}. 代码实现现在让我们开始动手实现，首先让我们回顾一下numpy和matplotlib：xxxxxxxxxximport numpy as npimport matplotlib.pyplot as plt def warm_up_exercise(): A = None A = np.eye(5,5) return A # 当你的实现正确时，下面会输出一个单位矩阵：print(warm_up_exercise())运行时长: 403毫秒结束时间: 2022-09-21 09:37:05[[1. 0. 0. 0. 0.] [0. 1. 0. 0. 0.] [0. 0. 1. 0. 0.] [0. 0. 0. 1. 0.] [0. 0. 0. 0. 1.]] 你需要实现绘制数据集中图像的函数，当你的实现|正确时，你应该会得到如下的图像： def plot_data(x, y): """绘制给定数据x与y的图像""" plt.figure() # ====================== 你的代码 ========================== # 绘制x与y的图像 # 使用 matplotlib.pyplt 的命令 plot, xlabel, ylabel 等。 # 提示：可以使用 'rx' 选项使数据点显示为红色的 "x"， # 使用 "markersize=8, markeredgewidth=2" 使标记更大 # 给制数据 # 设置y轴标题为 'Profit in $10,000s' # 设置x轴标题为 'Population of City in 10,000s' # ========================================================= plt.xlabel("Population of City in 10,000s") plt.ylabel("Profit in $10,000s") # 数据载入，由","分割，且按列解包 # numpy.loadtxt参见https://numpy.org/doc/stable/reference/generated/numpy.loadtxt.html x, y = np.loadtxt("data/data5984/PRML_LR_data.txt", delimiter=',', unpack=True) # 横纵坐标标签 plt.xlabel("Population of City in 10,000s") plt.ylabel("Profit in $10,000s") # matplotlib.pyplot.plot 参见 https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html # 加入 "markersize=8, markeredgewidth=2" 使标记更大 plt.plot(x, y, "rx", markersize = 8, markeredgewidth = 2) # 让我们测试一下你的实现是否正确# 从txt中加载数据print('Plotting Data ...\n')data = np.loadtxt('./data/data5984/PRML_LR_data.txt', delimiter=',')x, y = data[:, 0], data[:, 1] # 绘图plot_data(x, y)plt.show()运行时长: 190毫秒结束时间: 2022-09-21 09:49:45 现在运用所学的知识，对上述数据利用线性回归进行拟合。首先我们对要学习的参数和数据做一个准备:# Add a column of ones to xm = len(y)X = np.ones((m, 2))X[:, 1] = data[:, 0] # initialize fitting parameterstheta = np.zeros((2, 1)) # Some gradient descent settingsiterations = 1500alpha = 0.01计算初始误差函数的值，你需要实现误差函数的计算：def compute_cost(X, y, theta): """计算线性回归的代价。""" J = 0.0 # ====================== 你的代码 ========================== # 计算给定 theta 参数下线性回归的代价 # 请将正确的代价赋值给 J # predictions = np.dot(X, theta) # sqrErrors = np.multiply((predictions - y), (predictions - y)) # J = np.sum(sqrErrors) / (2 * m) # J /= m J = 1/(2*m) * sum((y - (theta[1] * X[:,1] + theta[0])) **2) # ========================================================= return J # compute and display initial cost# Expected value 32.07J0 = compute_cost(X, y, theta)print(J0) 现在你验证了代价计算的正确性，接下来就需要实现最核心的部分：梯度下降。在实现这一部分之前，确定你理解了上述各种变量及其表示。你需要完成梯度下降的核心代码部分：def gradient_descent(X, y, theta, alpha, num_iters): """执行梯度下降算法来学习参数 theta。""" m = len(y) J_history = np.zeros((num_iters,)) theta_history = np.zeros((2, iterations)) for iter in range(num_iters): # ====================== 你的代码 ========================== # 计算给定 theta 参数下线性回归的梯度，实现梯度下降算法 # ========================================================= # 将各次迭代后的代价进行记录 # print(X.shape) # print(theta.shape) # print(np.dot(X, theta).shape) # print(y.shape) # print(np.dot((np.dot(X, theta).T - y), X).T.shape) theta -= (alpha / m) * np.dot((np.dot(X, theta).T - y), X).T J_history[iter] = compute_cost(X, y, theta) theta_history[0, iter] = theta[0] theta_history[1, iter] = theta[1] return theta, J_history, theta_history # run gradient descent# Expected value: theta = [-3.630291, 1.166362]theta, J_history, theta_history = gradient_descent(X, y, theta, alpha, iterations)print(theta)print(J_history.shape)print(theta_history.shape)plt.figure()plt.plot(range(iterations),J_history)plt.xlabel("iterations")plt.ylabel(r'$J(\theta)$')plt.show()为了验证梯度下降方法实现的正确性，你需要把学习的到的直线绘制出来，确定你的实现是否正确。前面你已经绘制了数据集中的点，现在你需要在点的基础上绘制一条直线，如果你的实现正确，那么得到的图像应该是如下这样： 现在你已经正确实现了线性回归，你可能会对误差函数的优化过程比较好奇。为了更好地理解这个过程，你可以将损失函数的图像绘制出来。为此你需要将需要优化的参数的各个取值时误差函数的取值在图像上绘制出来，以下代码需要你进行填写。def plot_visualize_cost(X, y, theta_best): """可视化代价函数""" # 生成参数网格 theta0_vals = np.linspace(-10, 10, 101) theta1_vals = np.linspace(-1, 4, 101) t = np.zeros((2, 1)) J_vals = np.zeros((101, 101)) for i in range(101): for j in range(101): # =============== 你的代码 =================== # 加入代码，计算 J_vals 的值 J_vals[i][j] = compute_cost(X,y,[theta0_vals[i],theta1_vals[j]]) # =========================================== plt.figure() plt.contour(theta0_vals, theta1_vals, J_vals, levels=np.logspace(-2, 3, 21)) plt.plot(theta_best[0], theta_best[1], 'rx', markersize=8, markeredgewidth=2) plt.xlabel(r'$\theta_0$') plt.ylabel(r'$\theta_1$') plt.title(r'$J(\theta)$') plot_visualize_cost(X, y, theta)plt.show() 在梯度更新时，我们保留了代价的历史信息。在参数的学习过程中，代价函数的变化过程你也可以作一个图来查看。观察最后得到的J(θ)J(\theta)的图像以及代价的变化过程，可以加深你的理解。在梯度下降的迭代中，我们设置终止条件为完成了固定的迭代次数，但是在迭代次数完成时，由于学习率等参数的设置，可能得到的参数并不是使得代价最低的值。你可以通过观察代价函数的变化过程，想办法调整学习率等参数或者改进程序，使得参数的取值为搜索到的最优结果。请利用历史信息，改造plot_visual_cost函数，在等高线图上绘制线性回归模型的优化迭代过程。def plot_visual_history(X, y, theta_histroy): """可视化线性回归模型迭代优化过程""" plt.figure() # =============== 你的代码 =================== # 生成参数网格 theta0_vals = np.linspace(-10, 10, 101) theta1_vals = np.linspace(-1, 4, 101) t = np.zeros((2, 1)) J_vals = np.zeros((101, 101)) for i in range(101): for j in range(101): J_vals[i][j] = compute_cost(X,y,[theta0_vals[i],theta1_vals[j]]) plt.figure() plt.contour(theta0_vals, theta1_vals, J_vals, levels=np.logspace(-2, 3, 21)) for i in range(0, iterations, 100): plt.plot(theta_history[0, i], theta_history[1, i], 'rx', markersize=8, markeredgewidth=1) # =========================================== plt.xlabel(r'$\theta_0$') plt.ylabel(r'$\theta_1$') plt.title(r'$J(\theta)$') plot_visual_history(X, y, theta_history)plt.show()进阶在实现中，你可能采取了像上面公式中给出的结果一样逐个样本计算代价函数，或者在梯度下降的更新时也采用了逐个样本计算的方式。但事实上，你可以采用numpy的矩阵函数一次性计算所有样本的代价函数。可以采用矩阵乘法(np.matmul())求和等方式（np.sum（））。利用你学到的线性代数知识，将其实现更改一下吧。 -
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