An introduction of statistical learning

Transcript

1. ػցֶशษڧձ ୈ 1 ճ தଜ ྒྷଠ࿠ June 13, 2017
2. Table of contents Supervised Learning 1. Classification 2. Perceptron 3. Regression Unsupervised Learning 4. Clustering 1
3. ࠓ೔ͷ໨ඪ ࣍ճҎ߱ʹֶͿΞϧΰϦζϜͷ֓ཁΛ஌Δ ΞϧΰϦζϜͱద༻ྫ ΞϧΰϦζϜ ద༻ྫ ෼ྨ εύϜϝʔϧ൑ఆ ճؼ෼ੳ ച্༧ଌ ΫϥελϦϯά ը૾ͷݮ৭ॲཧ 2
4. ύϥϝτϦοΫ๏ Ϟσϧʢ਺ࣜʣΛԾఆ͠ɼϞσϧͷ࠷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 Ұ࣍ؔ਺ͷϞσϧͷύϥϝλௐ੔ 3
5. Classification
6. ෼ྨ Ϋϥεʹ෼ྨ͞ΕͨطଘσʔλΛݩʹ৽نσʔλΛ෼ྨ͢Δ ΞϧΰϦζϜ • ύʔηϓτϩϯ • ϩδεςΟοΫճؼ 4
7. Perceptron
8. 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 Λ΋ͭσʔλ܈ ௚ઢ্ͷ఺ (x′, y′) ͸ f (x′, y′) = 0 ΛΈͨ͢ 5
9. ύʔηϓτϩϯ, ධՁج४ʢޡࠩؔ਺ʣ ޡࠩؔ਺ E ͕࠷খʹͳΔ wi ΛٻΊΔ E = N ∑ i=1 {− (w0 + w1x + w2y) ti } = N ∑ i=1 (−f (xi , yi )ti ) • N ͸σʔλ਺ • ޡ෼ྨͩͱ −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 Λ΋ͭσʔλ܈ 6
10. 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) ͕૿Ճ͢Δ޲͖ 7
11. ϩδεςΟοΫճؼ, Ϟσϧ ͨͩ͠ɼ|f | ͕େ͖͍΄Ͳ t Ͱ͋Δ֬཰͕ߴ͍ͱ͢Δ ϩδεςΟοΫؔ਺ σ (α) = 1 1 + e−α Λಋೖ͠ɼ (x′, y′) ͕ t = 1 Ͱ͋Δ֬཰Λ 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) ϩδεςΟοΫؔ਺ͷάϥϑ 8
12. 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͸࠷΋ൃੜ֬཰͕ߴ͍σʔλͰ͋ΔͱԾఆ͍ͯ͠Δ 9
13. Regression
14. ճؼ෼ੳ, ϞσϧͱධՁج४ʢ࠷খೋ৐ʣ σʔλ͕ M ࣍ଟ߲ࣜ f ʹै͏ͱͯ͠ɼೋ৐ޡࠩ 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
15. Clustering
16. k ฏۉ๏ σʔλؒͷڑ཭ΛٻΊɼσʔλΛ k ݸͷΫϥελʹ෼͚Δ −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 Ϋϥελ͝ͱʹ୅දσʔλΛܾΊɼ୅දͷۙ͘ͷσʔλू߹Ͱ ΫϥελΛ࡞Δ 11
17. 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߹ͷ෼ׂ insert x(i)into cmax end foreach ∀c, mc = 1 |c| ∑ x(i)∈c x(i) ୅දϕΫτϧΛ࠶ܭࢉ end until 12