2d:null 2e:["$","title",null,{"children":"짧은 머신러닝 안내서: Shapley value | Humble Data Miner | Observable"}] 33:T7b4,--- ## Shapley value? * 협동게임에서 전체 산출에 대한 참여자별 기여도를 측정하는 방식 및 결과 기여도 값 * 협동게임: 여러명의 플레이어가 상호작용을 통해 결과를 산출하는 게임 * ${tex`n`} 명의 플레이어들의 집합 ${tex`N`}, e.g. ${tex`N=\{p_1,p_2,p_3\}`}, ${tex`n=3`} * 특성함수 ${tex`v(S)\rightarrow\mathbb{R}`}, e.g. ${tex`v(\{p_1,p_2\})=10, v(\emptyset)=0`} * ${tex`S \subseteq N`}, ${tex`S`}는 ${tex`N`}의 부분집합 * ${tex`\mathbb{R}`} 은 실수값으로 부분 집합인 ${tex`S`} 로부터 발생한 산출값 * 플레이어 ${tex`i`} 의 기여도 = ${tex`\varphi_i`} * ${tex`\varphi_i = \sum\limits_{S\subseteq N \setminus i}\colorbox{cyan}{$\frac{|S|!\,(n-|S|-1)!}{n!}$}\colorbox{yellow}{$(v(S\cup\{i\})-v(S))$}`} * ${tex`N=\{A,B,C\}`}, ${tex`n=3`} 일때 * ${tex`\varphi_{A} = \colorbox{cyan}{$\frac{|\emptyset|!\,(3-|\emptyset|-1)!}{3!}$}\colorbox{yellow}{$(v({\emptyset}\cup\{A\})-v(\emptyset))$} \\\qquad + \colorbox{cyan}{$\frac{|\{B\}|!\,(3-|\{B\}|-1)!}{3!}$}\colorbox{yellow}{$(v(\{B\}\cup\{A\})-v(\{B\}))$} \\\qquad + \colorbox{cyan}{$\frac{|\{C\}|!\,(3-|\{C\}|-1)!}{3!}$}\colorbox{yellow}{$(v(\{C\}\cup\{A\})-v(\{C\}))$} \\\qquad + \colorbox{cyan}{$\frac{|\{B,C\}|!\,(3-|\{B,C\}|-1)!}{3!}$}\colorbox{yellow}{$(v(\{B,C\}\cup\{A\})-v(\{B,C\}))$}`} * ${tex`\colorbox{cyan}{$\frac{|S|!\,(n-|S|-1)!}{n!}$}=\frac{(\{i\}의\,순열\,개수)\times(S의\,순열\,개수)\times((N-S-\{i\})의\,순열\,개수)}{(N의\,순열\,개수)}`} * ${tex`\colorbox{yellow}{$(v(S\cup\{i\})-v(S))$}=(부분집합S에\,대한\,i의\,추가\,산출)`} * 요컨대 Shapley value ${tex`\varphi_i`} 는 ${tex`i`} 의 참여로 발생한 추가 산출값들의 가중치 평균34:T4d3,{ function _set_range_bar_color(input) { const slider = d3.select(input); slider.style("accent-color",d3.interpolateYlOrRd(input.value/7)); } const table = Inputs.table(pset,{ width:{sub:120,eq:15,rating:200}, maxWidth:500, columns:["sub","eq","rating"], header:{sub:"특성함수", eq:"", rating:"만족도"}, align:{sub:"center",rating:"center"}, format:{ sub:x=>html`${to_vfunc(x)}`, eq:d=>html`${d}`, rating:(x,i)=> { const key = pset[i].str; const slider = Inputs.range([0,5], { value:ratings.hasOwnProperty(key)?ratings[key]:0, step:1 } ); slider.addEventListener("load",e=>{_set_range_bar_color(e.target);}); slider.addEventListener("input", (e) => { ratings[pset[i].str] = e.target.value; _set_range_bar_color(e.target); (viewof update_contributions).dispatchEvent(new Event("input", {bubbles: true})) }); return slider; } }, multiple:false, rows:13, }); d3.select(table).selectAll("input[type=range]").nodes().forEach(slider=>_set_range_bar_color(slider)); return table; }35:T75f,{ const [cw,ch]=[150,30]; const acc_list = contributions.reduce((acc,cur)=>{ const prev_acc = acc.length>0 ? acc[acc.length-1][1] : 0; acc.push([prev_acc,prev_acc+cur.cont]); return acc; },[]); const ex = [Math.min(0,d3.min(acc_list,d=>d[1])), Math.max(d3.max(acc_list,d=>d[1]),5)]; const x = d3.scaleLinear(ex,[0,cw]); function _make_svg_cdraw(acc){ const dir = (acc[1]-acc[0])>=0 ? 1: -1; return svg` `; } return Inputs.table(contributions,{ columns:["p","eq1","stats","eq2","cont","cdraw"], header:{p:"개별기여도",eq1:"",eq2:"",stats:"계산식",cont:"결과",cdraw:"누적 기여도"}, format:{ p:d=>html`${to_phi(d)}`, eq1:d=>html`${d}`, eq2:d=>html`${d}`, stats:d=>html`${d.map((stat,i)=>html`${stat.w.toFixed(3)}*(${to_vfunc(stat.p+stat.sk)}-${to_vfunc(stat.sk)})${i==d.length-1?"":" + "}`)}`, cont:(d,i)=>html`${d>0?"+":""}${numformat.format(d)}`, cdraw:(d,i)=>_make_svg_cdraw(acc_list[i]), }, width:{p:60,eq1:15,eq2:15,cont:45,cdraw:cw}, align:{stats:"center"}, }) }36:T872,/** * Base Class of `js-combinatorics` */ class _CBase { /** * does `new` * @param args */ static of(...args) { return new (Function.prototype.bind.apply(this, [null].concat(args))); } /** * Same as `of` but takes a single array `arg` * * cf. https://stackoverflow.com/questions/1606797/use-of-apply-with-new-operator-is-this-possible */ static from(arg) { return new (Function.prototype.bind.apply(this, [null].concat(arg))); } /** * Common iterator */ [Symbol.iterator]() { return function* (it, len) { for (let i = 0n; i < len; i++) yield it.at(i); }(this, this.length); } /** * returns `[...this]`. */ toArray() { return [...this]; } /** * @deprecated * tells wether you need `BigInt` to access all elements. */ get isBig() { return Number.MAX_SAFE_INTEGER < this.length; } /** * @deprecated * tells wether it is safe to work on this instance. * * * always `true` unless your platform does not support `BigInt`. * * if not, `true` iff `.isBig` is `false`. */ get isSafe() { return typeof BigInt !== 'undefined' || !this.isBig; } /** * check n for nth */ _check(n) { if (n < 0) { if (this.length < -n) throw new RangeError(`${n} is out of range`); return BigInt(this.length) + BigInt(n); 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Shapley value","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":308,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":518,"value":"

\n## 머신러닝과의 관계\n* 아래와 같이 문제를 치환\n * ${tex`n`} 명의 플레이어 → ${tex`n`} 개의 특성 (플레이어 ${tex`i`} → 특성 ${tex`i`})\n * 플레이어의 참여/불참 → 특성값이 알려짐/알려지지않음\n * 특성함수 ${tex`v(S)`} → 모델 ${tex`E[M(X)]`}\n * 알려지지 않은 특성값은 기존 학습데이터의 해당 특성값 분포로 대체\n* 하나의 데이터에 대해서\n * 각 특성 ${tex`i`} 의 출력 기여도(shapley value) ${tex`\\varphi_i`} 를 계산\n * ${tex`\\sum{\\varphi_i}=(모델의\\,출력값)`}\n* 다음에 **Tree SHAP**을 설명하면서 상세하게 다룰 예정\n * Tree SHAP: decision tree 모델에 대한 shapley value 측정 알고리즘","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":101,"value":"---\n## (협동게임 예제) 가족구성원 만족감 기여도\n* 협동게임\n * 플레이어 집합: 전체 가족 구성원\n * 특성함수: **(입력)**가족 구성원의 부분집합 → **(출력)**만족도\n * Shapley value: 개별 가족 구성원의 만족 기여도","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":966,"value":"

\n### 1. 같이 살고있는 가족구성원을 선택","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":104,"value":"viewof res = Inputs.checkbox(\"FMSBDC\",{label:\"전체 세대원 집합\",value:\"FMC\",format:x=>fonts[x].cloneNode(false)})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":968,"value":"

\n### 2. 일부 가족구성원과 같이 살때의 만족감 할당","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":241,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":970,"value":"

\n### 3. 구성원별 기여도(shapley value) 계산 결과","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":609,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":993,"value":"

\n## 놀랍게도!\n* 인간의 사고모델도 입력을 바꿔가며 역분석하면\n* 누구를 (혹은 무엇을) 더 중요하게 생각하는지 계량화 가능하다","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":59,"value":"---\n## 참고 자료\n* Shapley value의 이론과 응용: [Wikipedia - Shapley value](https://en.wikipedia.org/wiki/Shapley_value)\n* javascript로 만들어진 조합,멱집합 라이브러리: [Github - dankogai/jscombinatorics](https://github.com/dankogai/js-combinatorics/)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1066,"value":"

\n## 부연설명 동영상\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":9,"value":"---\n## Attribution\n\n${sfonts.F} Icon by Freepik ${sfonts.M} Icon by Freepik
${sfonts.S} Icon by Freepik ${sfonts.B} Icon by Freepik
\n${sfonts.D} Icon by monkik ${sfonts.C} Icon by Freepik\n\n

\n## Appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":305,"value":"html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"fonts = ({\n 'F': await FileAttachment(\"father_2173728.png\").image({width:font_size}),\n 'M': await FileAttachment(\"mother_2173768.png\").image({width:font_size}),\n 'S': await FileAttachment(\"girl_2173862.png\").image({width:font_size}),\n 'B': await FileAttachment(\"boy_2173820.png\").image({width:font_size}),\n 'D': await FileAttachment(\"shiba_1623681.png\").image({width:font_size}),\n 'C': await FileAttachment(\"kitty_763704.png\").image({width:font_size}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":112,"value":"font_size = 48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"sfonts = ({\n 'F': await FileAttachment(\"father_2173728.png\").image({width:sfont_size}),\n 'M': await FileAttachment(\"mother_2173768.png\").image({width:sfont_size}),\n 'S': await FileAttachment(\"girl_2173862.png\").image({width:sfont_size}),\n 'B': await FileAttachment(\"boy_2173820.png\").image({width:sfont_size}),\n 'D': await FileAttachment(\"shiba_1623681.png\").image({width:sfont_size}),\n 'C': await FileAttachment(\"kitty_763704.png\").image({width:sfont_size}),\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":53,"value":"sfont_size = 24","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":94,"value":"function to_imgs(seq) {\n return html`${[...seq].map(k=>sfonts[k].cloneNode(false))}`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":690,"value":"function to_vfunc(seq) {\n return html`${tex`v`}({${[...seq].map((k,i)=>html`${sfonts[k].cloneNode(false)}${i==(seq.length-1)?\"\":\",\"}`)}})`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":706,"value":"function to_phi(seq) {\n return html`${tex`\\varphi`}${[...seq].map((k,i)=>html`${i==1?\"|\":\"\"}${sfonts[k].cloneNode(false)}`)}`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":96,"value":"to_imgs(\"FMSBDC\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"to_imgs(res)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":695,"value":"to_vfunc(res)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":710,"value":"to_phi(res)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":592,"value":"contributions = {\n update_contributions;\n const _R = (k)=>{ return ratings.hasOwnProperty(k)?ratings[k]:0; }\n const n = res.length;\n const phi = Object.fromEntries(res.map((p,i)=>{\n const stats = [...new PowerSet(res.filter(d=>d!=p))].reduce((acc,cur)=>{\n const [sk,sn] = [cur.join(''),cur.length];\n const si = res.filter(x => (x==p || cur.includes(x))).join('')\n const item = {w:1/(n*Number(combination(n-1,sn))),m:_R(si)-_R(sk),sk:sk,p:p};\n item.c = item.w*item.m;\n acc.push(item);\n return acc;\n },[]);\n return [p,{stats:stats,csum:d3.sum(stats,d=>d.c)}];\n }));\n return Object.keys(phi).map(k=>({p:k,stats:phi[k].stats,cont:phi[k].csum,cdraw:0,eq1:\"=\",eq2:\"=\"}));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":208,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":212,"value":"/**\n * Power Set\n */\nclass PowerSet extends _CBase {\n constructor(seed) {\n super();\n this.seed = [...seed];\n const length = 1n << BigInt(this.seed.length);\n this.length = length;\n Object.freeze(this);\n }\n at(n) {\n n = this._check(n);\n if (n === undefined)\n return undefined;\n let bn = BigInt(n);\n let result = [];\n for (let bi = 0n; bn; bn >>= 1n, bi++)\n if (bn & 1n)\n result.push(this.seed[Number(bi)]);\n return result;\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":598,"value":"function permutation(n, k) {\n if (n < 0)\n throw new RangeError(`${n} is out of range`);\n if (k < 0)\n throw new RangeError(`${k} is out of range`);\n if (0 == k)\n return 1n;\n if (n < k)\n return 0n;\n let [bn, bk, bp] = [BigInt(n), BigInt(k), 1n];\n while (bk--)\n bp *= bn--;\n return bp;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":596,"value":"function combination(n, k) {\n if (0 == k)\n return 1n;\n if (n == k)\n return 1n;\n if (n < k)\n return 0n;\n return permutation(n, k) / permutation(k, k);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":215,"value":"pset = [...new PowerSet(res)].filter(x=>x.length>0).map(x=>({sub:x,str:x.join(''),eq:\"=\",rating:0}))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":271,"value":"ratings = ({})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":959,"value":"numformat = new Intl.NumberFormat();","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":777,"value":"viewof update_contributions = Inputs.button(\"Update Contributions\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"\n","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"}],"files":"$W37","stdlib":"1","titling":"auto"}} 32:"$31:body"