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An introduction of statistical learning
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Nakamura, Ryotaro
June 28, 2017
Science
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An introduction of statistical learning
Nakamura, Ryotaro
June 28, 2017
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Transcript
ػցֶशษڧձ ୈ 1 ճ தଜ ྒྷଠ June 13, 2017
Table of contents Supervised Learning 1. Classification 2. Perceptron 3.
Regression Unsupervised Learning 4. Clustering 1
ࠓͷඪ ࣍ճҎ߱ʹֶͿΞϧΰϦζϜͷ֓ཁΛΔ ΞϧΰϦζϜͱద༻ྫ ΞϧΰϦζϜ ద༻ྫ ྨ εύϜϝʔϧఆ ճؼੳ ച্༧ଌ ΫϥελϦϯά
ը૾ͷݮ৭ॲཧ 2
ύϥϝτϦοΫ๏ ϞσϧʢࣜʣΛԾఆ͠ɼϞσϧͷ࠷దͳύϥϝλΛֶश͢Δ ύϥϝτϦοΫ๏ͷखॱ 1. σʔλͷ༧ଌϞσϧΛԾఆ 2. Ϟσϧͷύϥϝλͷ ධՁج४ΛܾΊΔ 3. ύϥϝλΛܾΊΔ
0.0 0.2 0.4 0.6 0.8 1.0 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 Ұ࣍ؔͷϞσϧͷύϥϝλௐ 3
Classification
ྨ Ϋϥεʹྨ͞ΕͨطଘσʔλΛݩʹ৽نσʔλΛྨ͢Δ ΞϧΰϦζϜ • ύʔηϓτϩϯ • ϩδεςΟοΫճؼ 4
Perceptron
ύʔηϓτϩϯ, Ϟσϧ ઢܗͳϞσϧ f Λઃఆ͢Δ f (x, y) = w0
+ w1x + w2y f (x, y) > 0 ⇒ t = +1 f (x, y) < 0 ⇒ t = −1 −20 −10 0 10 20 30 x −30 −20 −10 0 10 20 y t = +1 t = -1 ଐੑ t = ±1 Λͭσʔλ܈ ઢ্ͷ (x′, y′) f (x′, y′) = 0 ΛΈͨ͢ 5
ύʔηϓτϩϯ, ධՁج४ʢޡࠩؔʣ ޡࠩؔ E ͕࠷খʹͳΔ wi ΛٻΊΔ E = N
∑ i=1 {− (w0 + w1x + w2y) ti } = N ∑ i=1 (−f (xi , yi )ti ) • N σʔλ • ޡྨͩͱ −f (xi , yi )ti > 0 −20 −10 0 10 20 30 x −30 −20 −10 0 10 20 y t = +1 t = -1 ଐੑ t = ±1 Λͭσʔλ܈ 6
ϩδεςΟοΫճؼ, Ϟσϧ ύʔηϓτϩϯͱಉ͘͡ઢܗϞσϧ f Λઃఆ͢Δ f (x, y) = w0
+ w1x + w2y f (x, y) > 0 ⇒ t = +1 f (x, y) < 0 ⇒ t = −1 −30 −20 −10 0 10 20 30 x −20 −15 −10 −5 0 5 10 15 20 y t = +1 t = -1 f (x, y) ͕૿Ճ͢Δ͖ 7
ϩδεςΟοΫճؼ, Ϟσϧ ͨͩ͠ɼ|f | ͕େ͖͍΄Ͳ t Ͱ͋Δ͕֬ߴ͍ͱ͢Δ ϩδεςΟοΫؔ σ (α)
= 1 1 + e−α Λಋೖ͠ɼ (x′, y′) ͕ t = 1 Ͱ͋Δ֬Λ 0 < σ ( f ( x′, y′ )) < 1 ͱ͢Δ −4 −3 −2 −1 0 1 2 3 4 α 0.0 0.2 0.4 0.6 0.8 1.0 σ (α) ϩδεςΟοΫؔͷάϥϑ 8
ϩδεςΟοΫճؼ, ධՁج४ʢ࠷ਪఆʣ ܇࿅σʔλ͕ಘΒΕΔ֬ P Λ࠷େʹ͢Δ wi ΛٻΊΔ p(x, y) =
σ(x0 + w1x + w2y) P = N ∏ i p (xi , yi )tn {1 − p (xi , yi )}1−tn ܇࿅σʔλ࠷ൃੜ͕֬ߴ͍σʔλͰ͋ΔͱԾఆ͍ͯ͠Δ 9
Regression
ճؼੳ, ϞσϧͱධՁج४ʢ࠷খೋʣ σʔλ͕ M ࣍ଟ߲ࣜ f ʹै͏ͱͯ͠ɼೋޡࠩ ED Λ࠷খʹ͢Δ ύϥϝλ
wi ΛબͿ f (x) = M ∑ m=0 wmxm ED = 1 2 N ∑ n=1 {f (xn) − tn}2 0 2 4 6 8 10 −15 −10 −5 0 5 ground truth degree 3 degree 4 degree 5 training points M ∈ {3, 4, 5} ͷଟ߲ࣜۙࣅྫ 10
Clustering
k ฏۉ๏ σʔλؒͷڑΛٻΊɼσʔλΛ k ݸͷΫϥελʹ͚Δ −2 −1 0 1 2
3 0 1 2 3 4 5 σʔλू߹ −2 −1 0 1 2 3 0 1 2 3 4 5 cluster 1 cluster 2 cluster 3 centroids k = 3 ͷΫϥελ Ϋϥελ͝ͱʹදσʔλΛܾΊɼදͷۙ͘ͷσʔλू߹Ͱ ΫϥελΛ࡞Δ 11
k ฏۉ๏ͷΞϧΰϦζϜ ೖྗ: σʔλू߹ D = { x(1), x(2), ·
· · , x(|D|) } : Ϋϥελ k ແ࡞ҝʹ m1, m2 · · · , mk ΛܾΊΔ until ऩଋ foreach x(i) ∈ D cmax = arg max c sim ( x(i), mc ) σʔλू߹ͷׂ insert x(i)into cmax end foreach ∀c, mc = 1 |c| ∑ x(i)∈c x(i) දϕΫτϧΛ࠶ܭࢉ end until 12