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   機械学習は化学研究の  
「経験と勘」を合理理化できるか?
北北海道⼤大学  ⼤大学院情報科学研究科

情報理理⼯工学専攻  ⼤大規模知識識処理理研究室

瀧川  ⼀一学
本⽇日の話題
① 私と「化学」「材料料」との関わり  
② 化学データとデータ科学(情報科学)  
③ 機械学習(or  AI?)の仕組み  (何ができる?)
⼀一⾔言で⾔言うと:  
多くの化学データから規則や例例外を⾒見見出し新たな

物質...
⾃自⼰己紹介:データ/情報を視るための道具職⼈人(?)
北北⼤大で学ぶ  (10年年)  内訳:  ⼤大学4,  ⼤大学院2+3,  研究員1
😊多変量量解析・統計的信号処理理  (⾳音響信号の分離離技術)
北北⼤大で働く  (5年年)  内訳...
私と「化学  (or  材料料)」との関わり
その1:京都⼤大学化学研究所  (2005)
その2:京都⼤大学薬学研究科  (2007)
Bioinformatics  (分⼦子⽣生物学〜~⽣生化学)
代謝反応,  代謝制御,  糖鎖構造,  ...
私と「化学  (or  材料料)」との関わり
その4:北北海道⼤大学創成研究機構  (2012)
その5:JSTさきがけ  (2015)
ゲノミクス/遺伝学〜~分⼦子⽣生物学
コピー数/⼀一塩基多型解析,  ChIP-‐‑‒Seq解析
マテリア...
物質や化学反応の理理解・予測はとても難しい…  
• 原理理上は構造・エネルギー・多くの物性あるいは化学反
応はシュレディンガー⽅方程式で記述できるが、現実的な
系では基本的に解けない  (巨⼤大な計算をしても近似…)  
• 溶媒・触媒・温度...
化学研究の「経験と勘」:熟練者と初⼼心者の違い
「経験と勘」≠「偶然」
• データおよび実験から(多⾓角的に?)学習する。  
• ⼀一連の実験を⾏行行い、結果を解析し、共通の特徴や例例外を探す。  
• 既知事実を総合し、それらを説明できるモ...
帰納  (経験と勘)  と  演繹  (科学法則)
The grand aim of science is to cover the greatest
number of experimental facts by logical deduct...
帰納  (経験と勘)  と  演繹  (科学法則)
The grand aim of science is to cover the greatest
number of experimental facts by logical deduct...
帰納  (経験と勘)  と  演繹  (科学法則)
The grand aim of science is to cover the greatest
number of experimental facts by logical deduct...
帰納  (経験と勘)  と  演繹  (科学法則)
The grand aim of science is to cover the greatest
number of experimental facts by logical deduct...
データ駆動科学による科学研究の効率率率化・合理理化
データ駆動のもたらす⼆二⾯面的な恩恵
• 効率率率化:そもそも⼈人⼿手でさばける規模を超えている  
• 合理理化:“マイサイド・バイアス  (確証バイアス)”

        ⼈人はデータ...
これまでに、⼤大変に多くの実験データが蓄積され、それは
いまも増加している。しかしながら、個々のデータは本質
を垣間⾒見見せる⾃自然からの⼀一枚の⼿手紙に過ぎない。  
⼿手紙が蓄積・整理理されその送り⼿手である⾃自然の本質に迫っ
たとき、予測...
変わった例例:Nature  533,  73–76  (05  May  2016)
化学合成:失敗実験を⽤用いた機械学習による材料料の発⾒見見
• 研究室に保管されていた実験ノートから収集した「ダークな(⽇日の⽬目を⾒見見
ていない)」反応...
情報と⽣生命:⽣生命現象は化学の⾔言葉葉で説明可能か?
医薬品化合物  
(⽣生体異異物  =  xenobiotic)
⽣生体細胞  =  分⼦子機械
多数の分⼦子と化学反応
から成る巨⼤大な系
多数の因⼦子が絡む現象の解明を⽬目指す⽣生命科...
事例例:⽣生体細胞内の化学反応系  (代謝反応系)
活性あり(active):  1,737化合物 活性なし(inactive):  26,895化合物
予測モデル 活性(あり  or  なし)
事例例:前⽴立立腺癌細胞株に対する成⻑⾧長阻害アッセイ
?? ?? ??
現在までに調べられた物質 可能な新物質の候補
物性
…
1  2        n
物質1
…
1  2        n
物質2
…
1  2        n
物質N
…
1  2        n
候補1
…
1  2 ...
B
その利利活⽤用(予測)
法則性A
既知データに潜む  
法則性の⾃自動的獲得
⽬目指すゴール:  データ駆動型の帰納的探索索技術の確⽴立立
Q:  どのようなやり⽅方が筋がよく、⾼高速で⾼高精度度な予測を与えるのか?
“演繹的”
“帰納的”...
機械学習:データ科学・⼈人⼯工知能の基幹技術の⼀一つ
「経験と勘」≠「偶然」
• データおよび実験から(多⾓角的に?)学習する。
仮説・モデル 実験的な事実(データ)
演繹
機械学習
多様・多量量化
…
既存の知⾒見見・知識識もデータデータに潜...
分かりやすい事例例:⼿手書き⽂文字認識識
⼊入⼒力力した⼿手書き⽂文字の画像を⼊入⼒力力すると、その⽂文字を出⼒力力
してくれるようなコンピュータプログラムをつくりたい
コンピュータ  
プログラム
…しかし、どういうコードを書けばいいのだろう...
教師つき学習:お⼿手本からの学習
コンピュータプログラムを作るには⼊入⼒力力から出⼒力力を
得る⼿手順がカンペキにわかってなければいけない。
コンピュータ  
プログラム
⼊入⼒力力 出⼒力力
でも、実際には肝⼼心の⼿手順がよく分からないことが...
教師つき学習  =  新しいプログラミングの形
教師つき学習  =  ⼊入出⼒力力の多量量の⾒見見本から計算部
のロジックを⾃自動的に構築する技術
教師つき学習のしくみ:統計的法則性の活⽤用
⼀一つの事例例だけ⾒見見てもわからないが多くの事例例に共通して  
浮かび上がってくる統計的な法則性を使う
「論論述的」理理解から  
「統計的」理理解へ
「あ」とは◯◯ですと明⽰示的規則  
として...
統計的に浮かび上がる法則性
重さ(g)と⾼高さ(cm)を測る
統計的に浮かび上がる法則性
重さ(g)と⾼高さ(cm)を測る
5 6.25 7.5 8.75 10
90112.5135157.5180
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⾼高さ(cm)
●  りんご  
●  みかん
統計的に浮かび上がる法則性
重さ(g)と⾼高さ(cm)を測る
5 6.25 7.5 8.75 10
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統計的に浮かび上がる法則性
重さ(g)と⾼高さ(cm)を測る
5 6.25 7.5 8.75 10
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統計的に浮かび上がる法則性
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5 6.25 7.5 8.75 10
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統計的に浮かび上がる法則性
重さ(g)と⾼高さ(cm)を測る
5 6.25 7.5 8.75 10
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統計的に浮かび上がる法則性
重さ(g)と⾼高さ(cm)を測る
5 6.25 7.5 8.75 10
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機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
機械学習は化学研究の"経験と勘"を合理化できるか?
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機械学習は化学研究の"経験と勘"を合理化できるか?

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電気化学会 第33回ライラックセミナー・第23回若手研究者交流会
日時:平成29年6月10日(土)、11日(日)
場所:おたる自然の村おこばち山荘
主催:電気化学会 北海道支部

http://www.electrochem.jp/event/2017_061011.html

Published in: Science
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機械学習は化学研究の"経験と勘"を合理化できるか?

  1. 1.    機械学習は化学研究の   「経験と勘」を合理理化できるか? 北北海道⼤大学  ⼤大学院情報科学研究科
 情報理理⼯工学専攻  ⼤大規模知識識処理理研究室
 瀧川  ⼀一学
  2. 2. 本⽇日の話題 ① 私と「化学」「材料料」との関わり   ② 化学データとデータ科学(情報科学)   ③ 機械学習(or  AI?)の仕組み  (何ができる?) ⼀一⾔言で⾔言うと:   多くの化学データから規則や例例外を⾒見見出し新たな
 物質の設計や現象の予測に役⽴立立てようという話
  3. 3. ⾃自⼰己紹介:データ/情報を視るための道具職⼈人(?) 北北⼤大で学ぶ  (10年年)  内訳:  ⼤大学4,  ⼤大学院2+3,  研究員1 😊多変量量解析・統計的信号処理理  (⾳音響信号の分離離技術) 北北⼤大で働く  (5年年)  内訳:  特任助教2.5,  准教授2.5   😆離離散構造を伴う機械学習、科学への機械学習の適⽤用 化学研究所(バイオインフォマティクスセンター)   薬学研究科(医薬創成情報科学専攻) 京⼤大で働く  (7年年)内訳:  助教7 😊⽣生命科学データからの知識識発⾒見見・計算⽣生物学
  4. 4. 私と「化学  (or  材料料)」との関わり その1:京都⼤大学化学研究所  (2005) その2:京都⼤大学薬学研究科  (2007) Bioinformatics  (分⼦子⽣生物学〜~⽣生化学) 代謝反応,  代謝制御,  糖鎖構造,  遺伝⼦子発現 構造活性相関,  低分⼦子-‐‑‒標的相互作⽤用 Chemoinfomatics/Chemometrics〜~創薬化学 その3:ボストン⼤大学  Bioinfo  Program  (2010) 代謝⼯工学〜~計算⽣生物学 代謝フラックス計算,  ネットワーク⽣生物学
  5. 5. 私と「化学  (or  材料料)」との関わり その4:北北海道⼤大学創成研究機構  (2012) その5:JSTさきがけ  (2015) ゲノミクス/遺伝学〜~分⼦子⽣生物学 コピー数/⼀一塩基多型解析,  ChIP-‐‑‒Seq解析 マテリアルズ・インフォマティクス 理理論論・実験・計算科学とデータ科学が連携・融合 した先進的マテリアルズインフォマティクスのた めの基盤技術の構築 材料料科学,  計算化学,  構造物性相関,  触媒科学 関わり始めたばかりでまだまだこれから!
  6. 6. 物質や化学反応の理理解・予測はとても難しい…   • 原理理上は構造・エネルギー・多くの物性あるいは化学反 応はシュレディンガー⽅方程式で記述できるが、現実的な 系では基本的に解けない  (巨⼤大な計算をしても近似…)   • 溶媒・触媒・温度度なども考慮に⼊入れるならさらに近似   • 反応環境が多様  (気相反応,  酵素で触媒される⽣生合成,   ウェルプレート上での合成,  ⼯工業的⼤大量量合成,  etc) 化学研究の「経験と勘」 にも関わらず、なぜ化学者たちは社会が望む素晴らしい 物性を持った化合物を作り出してきたのか?
  7. 7. 化学研究の「経験と勘」:熟練者と初⼼心者の違い 「経験と勘」≠「偶然」 • データおよび実験から(多⾓角的に?)学習する。   • ⼀一連の実験を⾏行行い、結果を解析し、共通の特徴や例例外を探す。   • 既知事実を総合し、それらを説明できるモデルを考える。   • 新しい実験で検証する。   • モデルを修正する。 仮説・モデル 実験的な事実(データ) 演繹 帰納・仮説形成 「経験」 「勘」
  8. 8. 帰納  (経験と勘)  と  演繹  (科学法則) The grand aim of science is to cover the greatest number of experimental facts by logical deduction from the smallest number of hypotheses or axioms. ─── Albert Einstein 仮説・モデル 実験的な事実(データ) 演繹 帰納・仮説形成 「経験」 「勘」(Serendipity?)
  9. 9. 帰納  (経験と勘)  と  演繹  (科学法則) The grand aim of science is to cover the greatest number of experimental facts by logical deduction from the smallest number of hypotheses or axioms. ─── Albert Einstein 仮説・モデル 実験的な事実(データ) 演繹 帰納・仮説形成 「経験」 「勘」(Serendipity?) ここは確かに
 logicalだけど… …
  10. 10. 帰納  (経験と勘)  と  演繹  (科学法則) The grand aim of science is to cover the greatest number of experimental facts by logical deduction from the smallest number of hypotheses or axioms. ─── Albert Einstein 仮説・モデル 実験的な事実(データ) 演繹 帰納・仮説形成 「経験」 「勘」(Serendipity?) ここが全然
 logicalじゃない… ここは確かに
 logicalだけど… …
  11. 11. 帰納  (経験と勘)  と  演繹  (科学法則) The grand aim of science is to cover the greatest number of experimental facts by logical deduction from the smallest number of hypotheses or axioms. ─── Albert Einstein 仮説・モデル 実験的な事実(データ) 演繹 帰納・仮説形成 「経験」 「勘」(Serendipity?) ここが全然
 logicalじゃない… ここは確かに
 logicalだけど… … 今⽇日の話:(⼤大量量)データに基づく推論論により合理理化可能?
  12. 12. データ駆動科学による科学研究の効率率率化・合理理化 データ駆動のもたらす⼆二⾯面的な恩恵 • 効率率率化:そもそも⼈人⼿手でさばける規模を超えている   • 合理理化:“マイサイド・バイアス  (確証バイアス)”
         ⼈人はデータに⾃自分の⾒見見たいものを⾒見見てしまう • 沢⼭山のデータや⾼高度度計測や情報システムの充実?
 (CAS,  ICSD,  SciFinder,  Reaxys,  ⽇日化辞,  …)   • 未だ未解明の現象や未発⾒見見の現象はより複雑な条件?   • 科学法則が⼈人間が把握できるほどsimpleとは限らない?   • 複雑な対象、多量量のデータ、多くの因⼦子の関与…
  13. 13. これまでに、⼤大変に多くの実験データが蓄積され、それは いまも増加している。しかしながら、個々のデータは本質 を垣間⾒見見せる⾃自然からの⼀一枚の⼿手紙に過ぎない。   ⼿手紙が蓄積・整理理されその送り⼿手である⾃自然の本質に迫っ たとき、予測と設計のために美しい知識識の形が出来上がっ てくる。   データから知識識へ。   「J.Gasteiger・T.Engel,  ケモインフォマティックス,  2003」 化学データとデータ科学(情報科学)の接点
  14. 14. 変わった例例:Nature  533,  73–76  (05  May  2016) 化学合成:失敗実験を⽤用いた機械学習による材料料の発⾒見見 • 研究室に保管されていた実験ノートから収集した「ダークな(⽇日の⽬目を⾒見見 ていない)」反応(ここでは失敗した⽔水熱合成)に関する情報で機械学習   • 機械学習モデルは、従来の⼈人間の戦略略よりも優れており、有機鋳型法で新 しい無機⽣生成物を形成する条件を89%の成功率率率で予測
  15. 15. 情報と⽣生命:⽣生命現象は化学の⾔言葉葉で説明可能か? 医薬品化合物   (⽣生体異異物  =  xenobiotic) ⽣生体細胞  =  分⼦子機械 多数の分⼦子と化学反応 から成る巨⼤大な系 多数の因⼦子が絡む現象の解明を⽬目指す⽣生命科学は情報戦に
 →  「情報科学(データ科学)」と「コンピュータ」の出番!
  16. 16. 事例例:⽣生体細胞内の化学反応系  (代謝反応系)
  17. 17. 活性あり(active):  1,737化合物 活性なし(inactive):  26,895化合物 予測モデル 活性(あり  or  なし) 事例例:前⽴立立腺癌細胞株に対する成⻑⾧長阻害アッセイ
  18. 18. ?? ?? ?? 現在までに調べられた物質 可能な新物質の候補 物性 … 1  2        n 物質1 … 1  2        n 物質2 … 1  2        n 物質N … 1  2        n 候補1 … 1  2        n 候補2 … 1  2        n 候補M… 物性 物性 個別に第⼀一   原理理計算 標準的探索索 … < Schrödinger   ⽅方程式を解く 事例例:マテリアルズインフォマティクス
  19. 19. B その利利活⽤用(予測) 法則性A 既知データに潜む   法則性の⾃自動的獲得 ⽬目指すゴール:  データ駆動型の帰納的探索索技術の確⽴立立 Q:  どのようなやり⽅方が筋がよく、⾼高速で⾼高精度度な予測を与えるのか? “演繹的” “帰納的” ?? ?? ?? 現在までに調べられた物質 可能な新物質の候補 物性 … 1  2        n 物質1 … 1  2        n 物質2 … 1  2        n 物質N … 1  2        n 候補1 … 1  2        n 候補2 … 1  2        n 候補M… 物性 物性 個別に第⼀一   原理理計算 標準的探索索 … < Schrödinger   ⽅方程式を解く 事例例:マテリアルズインフォマティクス
  20. 20. 機械学習:データ科学・⼈人⼯工知能の基幹技術の⼀一つ 「経験と勘」≠「偶然」 • データおよび実験から(多⾓角的に?)学習する。 仮説・モデル 実験的な事実(データ) 演繹 機械学習 多様・多量量化 … 既存の知⾒見見・知識識もデータデータに潜む法則性を抽出 ⾃自然(や⼈人間)は無限に複雑…    
 Itʼ’s  impossible  to  model  everything 「論論述的」理理解から   「統計的」理理解へ
  21. 21. 分かりやすい事例例:⼿手書き⽂文字認識識 ⼊入⼒力力した⼿手書き⽂文字の画像を⼊入⼒力力すると、その⽂文字を出⼒力力 してくれるようなコンピュータプログラムをつくりたい コンピュータ   プログラム …しかし、どういうコードを書けばいいのだろう? • ⼈人間にとっては⼦子供でも簡単にできる情報処理理   • が、⾃自分がどうやってこれを処理理しているのか不不明確   • 「なぜか(経験によって?)⾒見見たらすぐ分かる」だけ 「め」
  22. 22. 教師つき学習:お⼿手本からの学習 コンピュータプログラムを作るには⼊入⼒力力から出⼒力力を 得る⼿手順がカンペキにわかってなければいけない。 コンピュータ   プログラム ⼊入⼒力力 出⼒力力 でも、実際には肝⼼心の⼿手順がよく分からないことが多い ?将棋の盤⾯面 勝てる確率率率 ?現在の脳波 ⾒見見ている映像 ?フランス語 ⽇日本語 ?⾳音声信号 テキスト
  23. 23. 教師つき学習  =  新しいプログラミングの形 教師つき学習  =  ⼊入出⼒力力の多量量の⾒見見本から計算部 のロジックを⾃自動的に構築する技術
  24. 24. 教師つき学習のしくみ:統計的法則性の活⽤用 ⼀一つの事例例だけ⾒見見てもわからないが多くの事例例に共通して   浮かび上がってくる統計的な法則性を使う 「論論述的」理理解から   「統計的」理理解へ 「あ」とは◯◯ですと明⽰示的規則   として理理解するのではなく、事例例   から⾮非明⽰示的に定まる統計的法則性
 として理理解する (「⽂文字」は社会的な規則なのでそもそもexplicitな定義はないかも..?)
  25. 25. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る
  26. 26. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  27. 27. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  28. 28. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  29. 29. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  30. 30. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  31. 31. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  32. 32. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 重さ(g) ⾼高さ(cm) ●  りんご   ●  みかん
  33. 33. 統計的に浮かび上がる法則性 重さ(g)と⾼高さ(cm)を測る 5 6.25 7.5 8.75 10 90112.5135157.5180 ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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×