NMFに基づいた多変量回帰について

2019年度日本分類学会シンポジウムで使用した発表資料です．

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1. NMFに基づいた多変量回帰について 同志社⼤学⼤学院⽂化情報学研究科 加藤秀佳 東京理科⼤学⼯学部 ⼟⽥ 潤 同志社⼤学⽂化情報学部 宿久 洋 2019年度 ⽇本分類学会シンポジウム 2019/12/14
2. 2 /31 発表内容 1. はじめに ・複数の⽬的変数がある状況 ・多変量回帰 ・本研究の⽬的 2. 提案⼿法 ・NMFと多変量回帰の同時分析 ・提案モデルの考え⽅ ・NMF-MRのモデル ・推定⽅法 3. 適⽤例 ・数値例 ・実データ例 -⼈⼝流動データ 4. おわりに ・まとめ ・今後の課題
3. 3 /31 おわりに 適⽤例 提案⼿法 はじめに 3 / 複数の⽬的変数がある状況 はじめに 副教科から主要教科の点数の推定 説明変数⾏列 ⽬的変数⾏列 国語と社会の点数，数学と理科の点数，英語と国語の点数は互いに相関関係がある →この関係を考慮することができず，適切な推定が⾏えない ・主要教科同⼠に構造がある ・⽂系科⽬間，理系科⽬間に相関 この関係を利⽤して回帰の予測精度 が向上が期待できる
4. 4 /31 おわりに 適⽤例 提案⼿法 はじめに 4 / 複数の⽬的変数がある状況 はじめに 複数地点の1時間後の⼈⼝の推定 浅草と上野とスカイツリーには相関関係がある この関係を考慮した⼿法に多変量回帰がある ೔ ෇ ؾ Թ ఱ ؾ ༵ ೔ Πϕ ϯτ 上野 浅草 スカイツリー 隅⽥川花⽕⼤会の⼈⼝のplot 説明変数⾏列 ઙ૲ ͷ ਓޱ ্໺ ͷ ਓޱ εΧΠ πϦͷ ਓޱ Y <latexit sha1_base64="BXEJvnXaCBcmo46G6Vj9SGBn/sA=">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</latexit> … ⽬的変数⾏列 地点数=3000
5. 5 /31 おわりに 適⽤例 提案⼿法 はじめに 5 / 多変量回帰 定式化 はじめに ︓⽬的変数⾏列 ︓説明変数⾏列 ︓回帰係数⾏列 ︓誤差⾏列 X 2 Rn⇥q <latexit sha1_base64="bF4OX0imlSH62d1FhS6oAaVGOro=">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</latexit> Y 2 Rn⇥p <latexit sha1_base64="2fZJ3vNGgJLX1eFEja22b7T+2Qg=">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</latexit> E 2 Rn⇥p <latexit sha1_base64="XAsR357RXnfbWVGJD9hdgPOIp14=">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</latexit> Y = XW + E <latexit sha1_base64="z3L5fMEt9DkovqGpWS1J+sW0TG8=">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</latexit> W 2 Rq⇥p <latexit sha1_base64="1+gWXwxKiX4cTtDWnEmjikgVA9I=">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</latexit> 利点 ・⽬的変数間の関係を考慮 ・複数の⽬的変数を同時に推定 ・予測と解釈を⾏うことができる 課題 ・次元が⼤きいと解釈が困難な状況がある ・データが⾼次元のとき， 計算コストが増⼤し実⾏が困難 ・⾮負データに対して負の予測値を 出すことがある
6. 6 /31 おわりに 適⽤例 提案⼿法 はじめに 6 / 多変量回帰の課題 はじめに ೔ ෇ ؾ Թ ఱ ؾ ༵ ೔ Πϕ ϯτ 説明変数⾏列 ઙ૲ ͷ ਓޱ ্໺ ͷ ਓޱ εΧΠ πϦͷ ਓޱ Y <latexit sha1_base64="BXEJvnXaCBcmo46G6Vj9SGBn/sA=">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</latexit> … ⽬的変数⾏列 地点数=3000 例)複数地点の1時間後の⼈⼝の推定 W <latexit sha1_base64="kKPBi/HosnqSGvmbhPEePKYcruQ=">AAACiXichVG7SgNBFD2u78RH1EawEUPEKsz6QEklprH0lQckQXbXUcfsi91JIIb8gJWdqJWChfgBfoCNP2DhJ4ilgo2FdzcLoqLeYWbOnLnnzhyu7prCl4w9diidXd09vX39sfjA4NBwYmQ07zs1z+A5wzEdr6hrPjeFzXNSSJMXXY9rlm7ygl7NBveFOvd84dhbsuHyiqXt2WJXGJokKl/WrWahtZ1IsjQLY/InUCOQRBRrTuIWZezAgYEaLHDYkIRNaPBplKCCwSWugiZxHiER3nO0ECNtjbI4ZWjEVmndo1MpYm06BzX9UG3QKyZNj5STSLEHds1e2D27YU/s/ddazbBG8JcG7Xpby93t4aPxzbd/VRbtEvufqj8UOmX/7UliF0uhF0He3JAJXBrt+vXDk5fNzEaqOc0u2TP5u2CP7I4c2vVX42qdb5wjRg1Sv7fjJ8jPptW59ML6fHJ5JWpVHyYwhRnqxyKWsYo15OjdAxzjFGdKXFGVJSXTTlU6Is0YvoSS/QDMBZIq</latexit> 係数⾏列 課題 ・次元が⼤きいと解釈が困難な状況がある ・データが⾼次元のとき，計算コストが増⼤し実⾏が困難 ・⾮負データに対して負の予測値を出すことがある
7. 7 /31 おわりに 適⽤例 提案⼿法 はじめに 7 / 多変量回帰の課題 はじめに ・係数によっては予測値が負の値に ・⼤量に解釈する必要がある ೔ ෇ ؾ Թ ఱ ؾ ༵ ೔ Πϕ ϯτ 説明変数⾏列 ઙ૲ ͷ ਓޱ ্໺ ͷ ਓޱ εΧΠ πϦͷ ਓޱ Y <latexit sha1_base64="BXEJvnXaCBcmo46G6Vj9SGBn/sA=">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</latexit> … ⽬的変数⾏列 地点数=3000 例)複数地点の1時間後の⼈⼝の推定 W <latexit sha1_base64="kKPBi/HosnqSGvmbhPEePKYcruQ=">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</latexit> 係数⾏列 課題 ・次元が⼤きいと解釈が困難な状況がある ・データが⾼次元のとき，計算コストが増⼤し実⾏が困難 ・⾮負データに対して負の予測値を出すことがある
8. 8 /31 おわりに 適⽤例 提案⼿法 はじめに 8 / 多変量回帰の課題 はじめに 計算コストが増⼤し実⾏が困難 ೔ ෇ ؾ Թ ఱ ؾ ༵ ೔ Πϕ ϯτ 説明変数⾏列 ઙ૲ ͷ ਓޱ ্໺ ͷ ਓޱ εΧΠ πϦͷ ਓޱ Y <latexit sha1_base64="BXEJvnXaCBcmo46G6Vj9SGBn/sA=">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</latexit> … ⽬的変数⾏列 地点数=3000 例)複数地点の1時間後の⼈⼝の推定 W <latexit sha1_base64="kKPBi/HosnqSGvmbhPEePKYcruQ=">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</latexit> 係数⾏列 課題 ・次元が⼤きいと解釈が困難な状況がある ・データが⾼次元のとき，計算コストが増⼤し実⾏が困難 ・⾮負データに対して負の予測値を出すことがある
9. 9 /31 おわりに 適⽤例 提案⼿法 はじめに 9 / 多変量回帰の課題 はじめに 課題 ・係数が負の値の場合，解釈が困難な状況がある ・⽬的変数が⾼次元である場合，計算コストが増⼤し実⾏が困難 ・⾮負データに対して負の予測値を出すことがある 解決のためのアイデア 1.⽬的変数に対して次元縮約 2.係数に⾮負制約 〇⽬的変数の共通部分と抽出することができるため予測性能の向上 〇係数⾏列に⾮負制約があるため解釈可能性が向上 ⾮負値⾏列因⼦分解を適⽤
10. 10 /31 おわりに 適⽤例 提案⼿法 はじめに 10 / ⾮負値⾏列因⼦分解 ⾮負値⾏列因⼦分解(Non-negative Matrix Factorization︔NMF)とは， ⾮負値のデータに対する次元縮約法の⼀つ． ⾮負データ⾏列 を得られた時に次で表現できるとする D 2 Rr⇥p + <latexit sha1_base64="oyIPJT36BBRutZHzee5X9Ba4Lhw=">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</latexit> Y <latexit sha1_base64="pbfw6I64CKKFc+ryQJPI16LC34A=">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</latexit> ⾮負データ⾏列 基底⾏列 = <latexit sha1_base64="d7Xl1X83CMeuwvxvso9MGxJlzVY=">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</latexit> 係数⾏列 + <latexit sha1_base64="niLpq4uTD3nJek6jAH71c75wPXQ=">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</latexit> E <latexit sha1_base64="lwLTxlGSLRBjA3EOWkH/hE//Mzo=">AAACiXichVG7SgNBFD2u70RN1EawCYaIVZj1gWIlimCpiYlCEsLuOurovtjdBHTxB6zsRK0ULMQP8ANs/AGLfIJYKthYeHezIBqMd5iZM2fuuTOHq9q6cD3GGh1SZ1d3T29ffyw+MDiUSA6PFF2r5mi8oFm65Wyrist1YfKCJzydb9sOVwxV51vq4Upwv1Xnjissc9M7snnFUPZMsSs0xSOqWFYNf/WkmkyzLAsj1QrkCKQRxbqVfEAZO7CgoQYDHCY8wjoUuDRKkMFgE1eBT5xDSIT3HCeIkbZGWZwyFGIPad2jUyliTToHNd1QrdErOk2HlClk2DO7Y2/sid2zF/b5Zy0/rBH85Yh2tanldjVxOpb/+Fdl0O5h/1vVRqFSdntPHnaxEHoR5M0OmcCl1qxfPz5/yy/mMv4ku2Gv5O+aNdgjOTTr79rtBs9dIUYNkn+3oxUUp7PyTHZuYza9tBy1qg/jmMAU9WMeS1jDOgr07gHOcIFLKS7J0oK02EyVOiLNKH6EtPIFpbOSGA==</latexit> 誤差⾏列 C 2 Rn⇥r + <latexit sha1_base64="E1cCoI3hc/grHFVUlCi8zO04cfI=">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</latexit> 提案⼿法 Y 2 Rn⇥p + <latexit sha1_base64="WfAlcX9LSq+U96dIGu8zgoeHDlQ=">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</latexit> n ⇥ r <latexit sha1_base64="3e5G/kTXVG72eWf4aa02yM0871o=">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</latexit> r ⇥ p <latexit sha1_base64="Y/xrVm8vv18Tg1sgN/B6+IbG7Mc=">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</latexit> n ⇥ p <latexit sha1_base64="6O36WFTf/21/o1krJzInGJ4UJ6o=">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</latexit> n ⇥ p <latexit sha1_base64="6O36WFTf/21/o1krJzInGJ4UJ6o=">AAACjXichVG7SgNBFD1ZXzE+ErURbIJBsQoTHygiIlpoGY2JopGwu446uC92J4EY/AFrwUIUFCzED/ADbPwBi3yCWEawsfBmsyAajHeYmTNn7rkzh6s5hvAkY9WQ0tbe0dkV7o709Pb1R2MDgznPLro6z+q2YbvbmupxQ1g8K4U0+LbjctXUDL6lHa/U77dK3PWEbW3KssP3TPXQEgdCVyVRO1Y8L4XJvbhTiCVYkvkRbwapACQQRNqOPSKPfdjQUYQJDguSsAEVHo1dpMDgELeHCnEuIeHfc5wiQtoiZXHKUIk9pvWQTrsBa9G5XtPz1Tq9YtB0SRnHGHth96zGntkDe2Wff9aq+DXqfynTrjW03ClEz4YzH/+qTNoljr5VLRQaZbf2JHGAOd+LIG+Oz9Rd6o36pZOLWmZ+Y6wyzm7ZG/m7YVX2RA6t0rt+t843LhGhBqV+t6MZ5CaTqankzPp0Ymk5aFUYIxjFBPVjFktYQxpZetfCOa5wrUSVGWVBWWykKqFAM4Qfoax+AdTOk3Q=</latexit> D <latexit sha1_base64="GBSV0o3dhJhWBhxPh5RW1uhs/1c=">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</latexit> C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit>
11. 11 /31 おわりに 適⽤例 提案⼿法 はじめに 11 / 提案モデルの考え⽅ ⾼次元の に対してNMFを適⽤して次元縮約 Y <latexit sha1_base64="pbfw6I64CKKFc+ryQJPI16LC34A=">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</latexit> 提案⼿法 Y <latexit sha1_base64="pbfw6I64CKKFc+ryQJPI16LC34A=">AAACiXichVG7SgNBFD2urxgfidoINmJQrMKsDwypgmksk2iioiK766hj9sXuJhCX/ICVnaiVgoX4AX6AjT9g4SeIZQQbC+9uFkRFvcPMnDlzz505XNXWhesx9tQhdXZ19/TG+uL9A4NDieTwSMW1ao7Gy5qlW866qrhcFyYve8LT+brtcMVQdb6mVvPB/VqdO66wzFWvYfNtQ9k3xZ7QFI+oypZq+BvNnWSKpVkYEz+BHIEUoihYyTtsYRcWNNRggMOER1iHApfGJmQw2MRtwyfOISTCe44m4qStURanDIXYKq37dNqMWJPOQU03VGv0ik7TIeUEptgju2Et9sBu2TN7/7WWH9YI/tKgXW1rub2TOB5beftXZdDu4eBT9YdCpey/PXnYQyb0IsibHTKBS61dv3502lrJlqb8aXbFXsjfJXti9+TQrL9q10VeukCcGiR/b8dPUJlNy3PpheJ8KrcUtSqGcUxihvqxiByWUUCZ3j3ECc5wLvVLspSRsu1UqSPSjOJLSPkP0EeSLA==</latexit> 推定 Y = CD + E <latexit sha1_base64="xVkLKa2PCzqg5ScgV9T4DFiT64A=">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</latexit> 次元縮約した に対して多変量回帰 C = XW + F <latexit sha1_base64="nUA3/QOhDcGkyivrO6uzopYXgR8=">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</latexit> Y = XW D + E⇤ <latexit sha1_base64="WUmUo6ydM2t1NkvJjL0UtyUqQ3c=">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</latexit> 最後に， と の式をまとめる Y <latexit sha1_base64="pbfw6I64CKKFc+ryQJPI16LC34A=">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</latexit> C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> Y 2 Rn⇥p + <latexit sha1_base64="WfAlcX9LSq+U96dIGu8zgoeHDlQ=">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</latexit> X 2 Rn⇥q + <latexit sha1_base64="nHsUsH6tqHf/D2B2daa+VcWGgA8=">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</latexit> ︓⽬的変数⾏列 ︓説明変数⾏列 C 2 Rn⇥r + <latexit sha1_base64="E1cCoI3hc/grHFVUlCi8zO04cfI=">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</latexit> ︓基底⾏列 D 2 Rr⇥p + <latexit sha1_base64="oyIPJT36BBRutZHzee5X9Ba4Lhw=">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</latexit> ︓係数⾏列 ︓誤差⾏列 C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> W 2 Rq⇥r + <latexit sha1_base64="jEs2P7N5566D7S0MsGUP3LSFvLY=">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</latexit> ︓回帰係数⾏列 ← 提案モデル E 2 Rn⇥p <latexit sha1_base64="XAsR357RXnfbWVGJD9hdgPOIp14=">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</latexit>
12. 12 /31 おわりに 適⽤例 提案⼿法 はじめに 12 / 提案⼿法(NMF-Multivariate Regression;NMF-MR) 提案⼿法 Y = XW D + E⇤ <latexit sha1_base64="WUmUo6ydM2t1NkvJjL0UtyUqQ3c=">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</latexit> 提案モデル D 2 Rr⇥p + <latexit sha1_base64="oyIPJT36BBRutZHzee5X9Ba4Lhw=">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</latexit> Y 2 Rn⇥p + <latexit sha1_base64="WfAlcX9LSq+U96dIGu8zgoeHDlQ=">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</latexit> X 2 Rn⇥q + <latexit sha1_base64="nHsUsH6tqHf/D2B2daa+VcWGgA8=">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</latexit> ︓⽬的変数⾏列 ︓説明変数⾏列 ︓係数⾏列 ︓誤差⾏列 W 2 Rq⇥r + <latexit sha1_base64="jEs2P7N5566D7S0MsGUP3LSFvLY=">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</latexit> ︓回帰係数⾏列 R + <latexit sha1_base64="buYS4alX16LCwdPxlI2CbN/GH40=">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</latexit> ︓⾮負の実数値 E⇤ 2 Rn⇥p <latexit sha1_base64="w/gg/YVC5W2kOnmf9jKmQy4klxo=">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</latexit> 𝑾𝑫の rank が r となることから，提案法は Reduced ranked regression に⾮負制約を課したものと ⾒なすことができる
13. 13 /31 おわりに 適⽤例 提案⼿法 はじめに 13 / (C, D, W |X, Y , ↵, , ) = kXW Ck2 + kY CDk2 + ↵kCk2 + kDk2 ! minimize <latexit sha1_base64="C6cqEfpDtRzdc/njXvvAUBh5Hi8=">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</latexit> NMF-MRの⽬的関数 提案⼿法 s.t.C, D, W 0 <latexit sha1_base64="/YZh3hYoj8OVBKv8/mIXT1gNU60=">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</latexit> 本⼿法では誤差関数を2乗関数とする．そのとき⽬的関数は以下のようになる． 多変量回帰 における実測値と予測値の誤差を最⼩にする項 C = XW + F <latexit sha1_base64="nUA3/QOhDcGkyivrO6uzopYXgR8=">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</latexit> NMFで次元縮約した を復元する時の誤差を最⼩にする項 Y = CD + E <latexit sha1_base64="xVkLKa2PCzqg5ScgV9T4DFiT64A=">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</latexit> 正則化項 ︓調整パラメータ ↵, , 0 <latexit sha1_base64="ouXB5esCLKpIg4OBwJuVfGBq4Ps=">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</latexit>
14. 14 /31 おわりに 適⽤例 提案⼿法 はじめに 14 / NMF-MRのアルゴリズム 提案⼿法 Step1 の更新 Step2 の更新 Step3 の更新 W <latexit sha1_base64="kKPBi/HosnqSGvmbhPEePKYcruQ=">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</latexit> D <latexit sha1_base64="GBSV0o3dhJhWBhxPh5RW1uhs/1c=">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</latexit> C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> ←D,Wが与えられたもとでの更新 ←XとCが与えられたもとでNMFと同等 ←Cが与えられたもとでの更新 ⽬的関数を最⼩化するために補助関数法を⽤いる 各更新において，Jensenの不等式により上界を導出し， 上界を最⼩化することで⽬的関数を最⼩化する ⼊⼒︓ Step0 , , の初期化 C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> D <latexit sha1_base64="GBSV0o3dhJhWBhxPh5RW1uhs/1c=">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</latexit> ⽬的関数の値が収束するまで以下を繰り返す W <latexit sha1_base64="kKPBi/HosnqSGvmbhPEePKYcruQ=">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</latexit> ↵, , , X, Y <latexit sha1_base64="THJCVVJmXOs1TmNHMSBgttRN7aE=">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</latexit>
15. 15 /31 おわりに 適⽤例 提案⼿法 はじめに 15 / 推定アルゴリズム 提案⼿法 の推定 → が与えられたもとでの更新 D <latexit sha1_base64="GBSV0o3dhJhWBhxPh5RW1uhs/1c=">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</latexit> C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> dkj Pn i=1 yijcik + Pn i=1 c2 ik Vijk <latexit sha1_base64="/ZDLnaX2GFFffASS5M1QZ5wOUsc=">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</latexit> <latexit sha1_base64="rCcEchuSmnAXo0wAztmou5QKoGc=">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</latexit> n X i=1 p X j=1 ✓ yij r X k=1 cikdkj ◆2  n X i=1 p X j=1 y2 ij 2 n X i=1 p X j=1 yij ✓ r X k=1 cikdkj ◆ + n X i=1 p X j=1 1 Vijk r X k=1 c2 ik d2 kj <latexit sha1_base64="n7COh/ioVsKTD660Rtuel197soc=">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</latexit> ⽬的関数の に関わる項は 第1項は微分が困難なので，Jensenの不等式を⽤いて補助変数 を導⼊ の第1項 は次の不等式で表すことができる これを⽤いて を偏微分して0とおけば 更新式 を得る D <latexit sha1_base64="GBSV0o3dhJhWBhxPh5RW1uhs/1c=">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</latexit> Vijk <latexit sha1_base64="J6p3b+EHfvDxKAQ/z9vZ+o0mqe8=">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</latexit> G(D) = kY CDk2 + kDk2 <latexit sha1_base64="FpaUJbjlbWIH3cP9Rldsp4fLhX4=">AAACwnichVFLL8RgFD3qPR4z2EhsGhNCxOQbjxCJRJCw9BqPKJO2PqP0pe1Mwsz8AX/AwgqxED/AD7BhSyz8BLEksbFw22niFdymvec795779eQqtq65HmMPZUJ5RWVVdU1tpK6+oTEaa2pedK2so/KUaumWs6zILtc1k6c8zdP5su1w2VB0vqTsTPj1pRx3XM0yF7w9m68ZcsbUNjVV9ohKx0anuiTFyE8Wu0dFqSD6eKUo9vp5oliqSIX1PrFHlDKyYcjU9IlNx+IswYIQf4JkCOIIY8aKXULCBiyoyMIAhwmPsA4ZLj2rSILBJm4NeeIcQlpQ5ygiQtosdXHqkIndoW+GTqsha9LZn+kGapVu0el1SCmig92zc/bMrtkFe2Rvv87KBzP8f9mjrJS03E5HD1rnX/9VGZQ9bH2o/lAo1P23Jw+bGA68aOTNDhjfpVqan9s/fJ4fmevId7IT9kT+jtkDuyKHZu5FPZvlc0eI0IKS39fxEyz2JZL9icHZgfjYeLiqGrShHV20jyGMYRozSNG9p7jBLe6ESWFb2BXcUqtQFmpa8CWEwjtH6KcK</latexit> G(D) <latexit sha1_base64="4XL9jQzKxnkWwLgk/Lo3lcMdNww=">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</latexit> G⇤(D) <latexit sha1_base64="KjBWabY3LYDPFdegEHIJFKIxSEg=">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</latexit> G(D) <latexit sha1_base64="4XL9jQzKxnkWwLgk/Lo3lcMdNww=">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</latexit>
16. 16 /31 おわりに 適⽤例 提案⼿法 はじめに 16 / 推定アルゴリズム 提案⼿法 の推定 → が与えられたもとで， を更新する C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> D, W <latexit sha1_base64="/mDu6LMQlAWC7Ycs/pVSLISz+Eo=">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</latexit> C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> @H(C) @cik = 2 p X j=1 yijdkj + 2 p X j=1 d2 kj Vijk cik + 2↵cik 2 q X `=1 xi`w`k + 2cik = 0 <latexit sha1_base64="NSlfb+Y6ngAiDAM9cOIH9rRl4YQ=">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</latexit> ⽬的関数で に関わる項は も の更新と同様にJensenの不等式を⽤いて補助変数 を導⼊ 偏微分して0とおけば， の更新式 を得る C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> H(C) <latexit sha1_base64="SLQiDE7jpr97Sa5fPW7cfiSMYIM=">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</latexit> D <latexit sha1_base64="GBSV0o3dhJhWBhxPh5RW1uhs/1c=">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</latexit> C <latexit sha1_base64="CQsCECCNfU3U/q3nYsgFXFjo87k=">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</latexit> Vijk <latexit sha1_base64="J6p3b+EHfvDxKAQ/z9vZ+o0mqe8=">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</latexit> H(C) = kY CDk2 + ↵kCk2 + kXW Ck2 <latexit sha1_base64="Y8eFIAdzUwYoO9mxneS9qufMFXg=">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</latexit> cik Pp j=1 yijdkj + Pq `=1 xi`w`k ↵ + 1 + Pp j=1 d2 kj Vijk <latexit sha1_base64="fuI77/hXJ5GoEcOcQRyPXuXOrzU=">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</latexit>
17. 17 /31 おわりに 適⽤例 提案⼿法 はじめに 17 / 数値例 適⽤例 ⽬的 予測精度が担保されているかの検証 ⽐較⼿法 通常の多変量回帰(MR) 部分的最⼩2乗(PLS)回帰 評価指標 テストデータに対して， 相関係数(Correlation Coefficient; CC) 平均2乗誤差(Mean-Squared Error︔MSE) を⽤いる 予測値 Ypred = Xtest ˆ W ˆ D <latexit sha1_base64="WL4HSPgLsai58DBoenBbXJLYEQk=">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</latexit>
18. 18 /31 おわりに 適⽤例 提案⼿法 はじめに 18 / 数値例 適⽤例 [1] p < q の時 (p,q) = (5,10) [2] p = q の時 (p,q) = (10,10) [3] p > q の時 (p,q) = (20,10) N=30 数値例の状況設定 次の9つの状況を考える． N=50 N=100 [4] p < q の時 (p,q) = (5,10) [5] p = q の時 (p,q) = (10,10) [6] p > q の時 (p,q) = (20,10) [7] p < q の時 (p,q) = (5,10) [8] p = q の時 (p,q) = (10,10) [9] p > q の時 (p,q) = (20,10) p = ⽬的変数の次元，q = 説明変数の次元 ⽬的変数間に 関係がある
19. 19 /31 おわりに 適⽤例 提案⼿法 はじめに 19 / 数値例 C = XW + F <latexit sha1_base64="nUA3/QOhDcGkyivrO6uzopYXgR8=">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</latexit> Y = CD + E <latexit sha1_base64="xVkLKa2PCzqg5ScgV9T4DFiT64A=">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</latexit> (X)i` ⇠ N(0, 102)0  (X)i`  10 <latexit sha1_base64="U2nSIHJoHoDrdah9lR2vfPazpw4=">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</latexit> (D)kj ⇠ N(2, 12)0  (D)kj  10 <latexit sha1_base64="OyXyQGbIgq4IDCvaGZiwz70/yP0=">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</latexit> (E)np ⇠ N(0, 102) 50  (E)np  50 <latexit sha1_base64="b3Hub/+lCNIJsXU8bUt/2P/EbLk=">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</latexit> (F )nr ⇠ N(2, 102) 50  (F )nr  50 <latexit sha1_base64="8OHvZqGYz7NQOcsrncHDowSnm7w=">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</latexit> (W )qr ⇠ N(1, 12)0  (W )qr  10 <latexit sha1_base64="eAE5n14YgAod550zDUPggVRiqg0=">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</latexit> n = 30, 50, 100 <latexit sha1_base64="/naCTMQ6pNFpPSU1tyqL3KBLzZA=">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</latexit> r = 3 <latexit sha1_base64="sz/q3bDKL1crNNq/Kpk5F+dzUAE=">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</latexit>
20. 20 /31 おわりに 適⽤例 提案⼿法 はじめに 20 / 数値例の結果(30回平均) 適⽤例 n = 30 n = 50 n = 100 相関係数(CC) p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 既存1︓通常のMR 0.8672 0.9101 0.9223 0.9284 0.9299 0.8672 0.8714 0.9395 0.9330 既存2︓PLS回帰 0.8694 0.9057 0.9084 0.9269 0.9293 0.8694 0.8704 0.9391 0.9354 提案︓NMF-MR 0.8835 0.8844 0.8540 0.9213 0.9067 0.8835 0.8689 0.9357 0.9355 n = 30 n = 50 n = 100 平均2乗誤差(MSE) p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 既存1︓通常のMR 4259.1 3244.0 2927.6 1982.6 2288.7 4259.1 1682.9 1993.4 1754.2 既存2︓PLS回帰 3522.1 2958.2 3502.3 1899.5 2379.3 3522.1 1730.8 2011.3 1774.3 提案︓NMF-MR 3368.4 4215.3 6537.7 2066.0 3366.4 3368.4 1715.4 2173.3 1775.3
21. 21 /31 おわりに 適⽤例 提案⼿法 はじめに 21 / 数値例の結果(中央値⽐較) 適⽤例 n = 30 n = 50 n = 100 CC p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 既存1︓通常のMR 0.8953 0.9197 0.9337 0.9364 0.9368 0.8953 0.9419 0.9451 0.9419 既存2︓PLS回帰 0.8958 0.9210 0.9209 0.9395 0.9265 0.8958 0.9382 0.9458 0.9347 提案︓NMF-MR 0.9034 0.8970 0.8827 0.9320 0.9048 0.9034 0.9347 0.9398 0.9382 n = 30 n = 50 n = 100 MSE p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 p = 5 p = 10 p = 20 既存1︓通常のMR 3335.0 2651.1 2631.6 1982.6 2180.2 3335.0 1682.9 1953.3 1682.9 既存2︓PLS回帰 3050.8 2644.0 3225.2 2066.0 2321.7 3050.8 1715.4 1919.1 1715.4 提案︓NMF-MR 3116.4 3023.9 4479.1 1899.5 2512.5 3116.4 1730.8 1893.2 1730.8
22. 22 /31 おわりに 適⽤例 提案⼿法 はじめに 22 / 実データ例 適⽤例 東京都墨⽥区・台東区の⼈⼝流動データ 36地点(1kmメッシュ)の⼈⼝が1時間間隔に3ヶ⽉分記録 説明変数⾏列(200×16) ・平⽇，休⽇フラグ ・時間帯フラグ(朝，昼，⼣⽅，夜，深夜) ・浅草の1,2,3時間前の⼈⼝ ・東京スカイツリーの1,2,3時間前の⼈⼝ ・上野の1,2,3時間前の⼈⼝ ⽬的変数⾏列(200×16) ・36地点200時間分の⼈⼝流動 Y <latexit sha1_base64="pbfw6I64CKKFc+ryQJPI16LC34A=">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</latexit> データ分割割合 train︓test = 1:1 隅⽥川花⽕⼤会19:00の⼈⼝のplot
23. 23 /31 おわりに 適⽤例 提案⼿法 はじめに 23 / 実データ例の解釈 適⽤例 推定結果 = XW + F 説明変数⾏列(100×16) ・平⽇，休⽇フラグ ・時間帯フラグ(朝，昼，⼣⽅，夜，深夜) ・浅草の1,2,3時間前の⼈⼝ ・東京スカイツリーの1,2,3時間前の⼈⼝ ・上野の1,2,3時間前の⼈⼝ 基底1 : 平⽇・昼・深夜の時間帯 基底2 : 休⽇・夜の時間帯 基底3 : 平⽇・朝・⼣⽅・夜の時間帯 基底1 基底2 基底3
24. 24 /31 おわりに 適⽤例 提案⼿法 はじめに 24 / 実データ例 適⽤例 推定結果 = XW + F 基底1 : 平⽇・昼・深夜の時間帯 Y = CD + E <latexit sha1_base64="xVkLKa2PCzqg5ScgV9T4DFiT64A=">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</latexit> 基底2 : 休⽇・夜の時間帯 基底3 : 平⽇・朝・⼣⽅・夜の時間帯 基底1 基底2 基底3 基底1 基底2 基底3
25. 25 /31 おわりに 適⽤例 提案⼿法 はじめに 25 / 実データ例 適⽤例 推定結果 Y = CD + E <latexit sha1_base64="xVkLKa2PCzqg5ScgV9T4DFiT64A=">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</latexit> = XW + F 基底1 : 平⽇・昼・深夜の時間帯 基底2 : 休⽇・夜の時間帯 基底3 : 平⽇・朝・⼣⽅・夜の時間帯 基底1 基底2 基底3 基底1 基底2 基底3
26. 26 /31 おわりに 適⽤例 提案⼿法 はじめに 26 / 実データ例 適⽤例 0.1301 -0.5111 -1.7746 最⼩値 NMF_MRの推定値は負の値を取らない NMF_MR 通常の多変量回帰(MR) PLS回帰
27. 27 /31 おわりに 適⽤例 提案⼿法 はじめに 27 / まとめ おわりに 成果 提案⼿法 ⽬的変数に対して⾮負値⾏列因⼦分解を適⽤した多変量回帰を提案 適⽤例 数値例→PLSと相関係数を⽐較したとき，同等orそれ以上 東京都の⼈⼝予測→負の予測値が出る問題に対処，解釈可能性が向上 多変量回帰 の課題 ・係数が負の値の場合，解釈が困難な状況がある ・⽬的変数が⾼次元である場合，計算コストが増⼤し実⾏が困難 ・⾮負データに対して負の予測値を出すことがある
28. 28 /31 おわりに 適⽤例 提案⼿法 はじめに 28 / 今後の展望 おわりに ・零過剰に対応した⾮負値⾏列因⼦分解に基づく多変量回帰 への拡張 ・異なる誤差関数を⽤いた場合と．の結果の⽐較 ・その他の複数⽬的変数を推定するデータに対して実装 ・分析者が決めるパラメータの検討 ・地点数を増やした実データ例を実装 ・バイナリデータへの拡張も可能
29. 34 /31 おわりに 適⽤例 提案⼿法 はじめに 34 / 参考⽂献 おわりに [1] Borchani, H., Varando, G., Bielza, C., Larrañaga, P. (2015). A survey on multi- output regression. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 5, 216–233. [2] Chen, Y. N., Lin, H. T. (2012). Feature-aware label space dimension reduction for multi- label classification. In Advances in Neural Information Processing Systems (pp. 1529-1537). [3] Hsu, D. J., Kakade, S. M., Langford, J., Zhang, T. (2009). Multi-label prediction via compressed sensing. In Advances in Neural Information Processing Systems (pp. 772-780). [4] Spyromitros-Xioufis, E., Tsoumakas, G., Groves, W., Vlahavas, I. (2016). Multi-target regression via input space expansion: treating targets as inputs. Machine Learning, 104, (pp.55–98). [5] Tai, F., Lin, H. T. (2012). Multilabel classification with principal label space transformation. Neural Computation, 24, (pp.2508–2542). [6] Xu, D., Shi, Y., Tsang, I. W., Ong, Y. S., Gong, C., Shen, X. (2019). A survey on multi-output learning. arXiv preprint arXiv:1901.00248.
30. 35 /31 おわりに 適⽤例 提案⼿法 はじめに 35 / なぜ多変量回帰なのか︖ おわりに 想定しているデータが多変量回帰を⽤いなければ解くことができない．また は⽤いた⽅がいい状況がある． 個別に推定してはいけないから． ⽬的変数間の関係を考慮できない． ⾼次元のデータ
31. 36 /31 おわりに 適⽤例 提案⼿法 はじめに 36 / なぜNMFを⽤いるのか︖ おわりに 想定しているデータが，⽬的変数と説明変数が共に⾮負であるから． 係数に負の値がある場合，解釈ができない場合がある． 今回，Wが⾮負の制約を⼊れているため，解釈可能性が向上する．
32. 37 /31 おわりに 適⽤例 提案⼿法 はじめに 37 / 数値例 適⽤例 組み合わせは9通り (n, p, q, r) = (300, 5, 10, 3)，(300, 10, 10, 3) ，(300, 20, 10, 3) ， (500, 5, 10, 3)， (500, 10, 10, 3)， (500, 20, 10, 3)， (1000, 5, 10, 3) ， (1000, 10, 10, 3)， (1000, 20, 10, 3) = (300, 5, 10, 6)，(300, 10, 10, 6) ，(300, 20, 10, 6) ， (500, 5, 10, 6)， (500, 10, 10, 6)， (500, 20, 10, 6)， (1000, 5, 10, 6) ， (1000, 10, 10, 6)， (1000, 20, 10, 6) 時間があれば基底数r = 6のパターンもやる．合計18通り