1-7-7ϑΥϧϚϯτप೾਺‫੻ي‬Λજࡏύϥϝʔλͱͨ͠Ի੠εϖΫτϧੜ੒աఔͷ֬཰Ϟσϧ∗ˑ٢ཬ޾ଠ 1 ɼ๺৚৳ࠀ 1 ɼُԬ߂࿨ 1,2 ɼᜊ౻େี 1 ɼࠧեࢁໜथ 1(1 ౦େӃɾ৘ใཧ޻, 2 NTT CS ‫)ݚ‬1͸͡ΊʹਓؒΒ͍͠Ի੠߹੒Λ࣮‫͢ݱ‬Δʹ͋ͨͬͯ͸ɼԻ੠ͷμΠφϛΫεʹ‫ݱ‬ΕΔඇ‫ޠݴ‬৘ใ΍ύϥ‫ޠݴ‬৘ใΛৄࡉʹϞσϦϯά͢Δ͜ͱ͕ॏཁͰ͋Δɽྫ͑͹ɼԻૉಛ௃ྔͷಈతಛ௃ʹ͸࿩ऀͷ‫ݸ‬ਓੑ͕‫ݱ‬Ε͍ͯ
Δ͜ͱ͕஌ΒΕ͓ͯΓ [1]ɼԻ੠ೝࣝɼ࿩ऀೝࣝɼԻ
੠߹੒ͳͲʹ͓͍ͯॏཁͳಛ௃ྔͷҰͭͱͯ͠ѻΘΕ͍ͯΔ [1, 2, 3, 4]ɽҰํɼԻ੠ͷ‫ج‬ຊप೾਺‫ʹ੻ي‬͸ӆ཯తͳಛ௃͕‫ݱ‬ΘΕ͍ͯΔ͜ͱ͕஌ΒΕ͓ͯΓ[5]ɼӆ཯ղੳ΍Ի੠߹੒ʹ͓͍ͯ‫ج‬ຊप೾਺ͷಈతಛ௃΍ಈతϞσϧ͕ॏཁͳ໾ׂΛՌ͖ͨͯͨ͠ [3, 5]ɽ
·ͨɼۙ೥ར༻ՄೳͳԻ੠σʔλϕʔε͕૿Ճ͖ͯͨ͜͠ͱʹ൐ͬͯɼ౷‫ܭ‬తख๏Λ༻͍ͨԻ੠ॲཧʹؔͯ͠੝Μʹ‫ߦ͕ڀݚ‬ΘΕ͍ͯΔɽͱ͘ʹԻ੠ͷੜ੒աఔΛ֬཰Ϟσϧͱͯ͠ද‫͢ݱ‬ΔΞϓϩʔν͸‫Ͱྗڧ‬ɼύϥϝʔλਪఆʹ༷ʑͳ౷‫ܭ‬తख๏Λ‫͖ͨͰ༻׆‬Γɼઌ‫ݧ‬త৘ใΛύϥϝʔλͷࣄલ෼෍ͱͯ͠Ϟσϧ಺ʹ૊ΈࠐΉ͜ͱ͕Ͱ͖ͨΓɼύϥϝʔλͷ౷‫ܭ‬తͳৼΔ෣͍΍܏޲Λֶश͢Δ͜ͱ͕Ͱ͖ͨΓͱ͍༷ͬͨʑͳར఺͕͋Δɽզʑ͸ɼͦΕΒͷॏཁੑʹ஫໨ͯ͠Ի੠μΠφϛΫεͷ֬཰ϞσϧԽʹऔΓ૊ΜͰ͓ΓɼͦͷҰ‫ͱ؀‬ͯ͜͠Ε·ͰΠϯτωʔγϣϯͷ֬཰ϞσϧԽͷ‫ݕ‬౼ΛਐΊ͖ͯͨ [6, 7]ɽ‫ࡏݱ‬ɼզʑͷ‫Ͱࣨڀݚ‬͸ෳ߹΢ΣʔϒϨοτϞσϧʢComposite Wavelet Model;CWMʣͱ‫Ϳݺ‬εϖΫτϧแབྷϞσϧʹ‫͍͠৽ͮ͘ج‬Ի੠߹੒ํࣜΛ‫ݕ‬౼தͰ͋Δ [8, 9]ɽCWM ͸ LSP ͷΑ͏ʹϑΥϧϚϯτप೾਺ʹରԠ͍ͯ͠ΔͱղऍͰ͖ΔύϥϝʔλΛ༗͓ͯ͠Γɼ౰֘Ի੠߹੒ํࣜ΁কདྷతʹ૊ΈࠐΜͰ͍͘͜ͱΛ‫ݟ‬ਾ͑ɼϑΥϧϚϯτप೾਺‫੻ي‬ͷμΠφϛΫεͷ֬཰ϞσϧԽΛߦ͓͏ͱ͍͏ͷ͕ຊ‫ڀݚ‬ͷಈ‫͋Ͱػ‬ΔɽຊߘͰ͸·ͣɼϑΥϧϚϯτप೾਺ͷ࣌ؒ‫੻ي‬Λજࡏύϥϝʔλͱͯ͠΋ͭԻ੠εϖΫτϧੜ੒աఔͷ֬཰ϞσϧͷఆࣜԽΛߦ͍ɼͦͷύϥϝʔλਪఆΞϧΰϦζϜʹ͍ͭͯड़΂Δɽͦͯ͠ɼ࣮Ի੠ͷεϖΫτϧแབྷ͔ΒϑΥϧϚϯτप೾਺ͷ࣌ؒ‫੻ي‬Λਪఆ͢Δ࣮‫ݧ‬Λ௨ͯ͠ɼఏҊख๏ͷ༗ޮੑΛ֬ೝ͢Δɽ2Ի੠εϖΫτϧੜ੒աఔͷ֬཰ϞσϧԽ2.1 ϑΥϧϚϯτ‫੻ي‬ͷੜ੒աఔʹؔ͢ΔԾઆϑΥϧϚϯτʢຊߘͰ͸εϖΫτϧแབྷͷϐʔΫͱఆٛ͢Δʣ͸ɼԻ੠Λಛ௃͚ͮΔ‫ۃ‬ΊͯॏཁͳཁૉͰ͋Δɽ੠ଳৼಈ͕‫͢ৼڞ‬Δ͜ͱʹΑͬͯੜ͡ΔϑΥϧϚϯτͷप೾਺‫ʹ੻ي‬͸੠ಓͷӡಈʹ൐͏ԿΒ͔ͷ෺ཧతͳ੍໿͕෇ਵ͢Δͱߟ͑ΒΕΔ͕ɼຊ‫Ͱڀݚ‬͸ɼϑΥϧϚϯτप೾਺‫͕੻ي‬౻࡚ͷ F0 ύλʔϯੜ੒աఔϞσϧ [5] ͱಉ༷ͷϝΧχζϜʹΑͬͯੜ͡ΔͱԾఆ͢Δɽ۩ମతʹ͸ɼϑΥϧϚϯτप೾਺ͷର਺ͷ࣌ؒ‫੻ي‬ΛɼFig. 1 ʹࣔ͢Α͏ʹԻૉ۠ؒ͝ͱʹҰఆ஋ΛͱΔ֊ஈঢ়ͷࢦྩؔ਺ʢҎ‫ޙ‬ɼԻૉࢦྩؔ਺ʣʹΠϯύϧεԠ౴ 2 −αt(t ≥ 0)α te(1)
G(t) =0(t < 0)∗

Fig. 1 ઢ‫ʹܥܗ‬ΑΔϑΥϧϚϯτप೾਺‫੻ي‬ͷੜ੒աఔ

Fig. 2 ‫؍‬ଌ৴߸ (Իૉ/e/) ͷεϖΫτϧแབྷʢ఺ઢʣͱ
GMM ʹΑΔͦͷۙࣅʢ࣮ઢʣͷྫʢࠞ߹਺ 10ʣ͕৞Έࠐ·Εʢα ͸‫ݻ‬༗֯प೾਺ʣɼೋ࣍ઢ‫ܥܗ‬ͷग़ྗͱͯ͠ੜͨ͡΋ͷͱߟ͑Δɽͳ͓ɼೋ࣍ઢ‫ܥܗ‬ͷԾఆ͕ஔ͔ΕͨଞͷԻ੠ੜ੒աఔϞσϧͷྫͱͯ͠ɼԻૉೝࣝΛ໨తͱͨ͠ௐԻӡಈͷಈతϞσϧ͕ఏҊ͞Ε͍ͯΔ [10]ɽ࣮Ի੠͔Β௚઀‫؍‬ଌͰ͖Δͷ͸εϖΫτϧแབྷͰ͋ΓɼϑΥϧϚϯτप೾਺͸ཅʹ͸‫؍‬ଌ͞Εͳ͍ɽͦ͜ͰҎԼͰ͸ɼϑΥϧϚϯτप೾਺‫͔੻ي‬ΒͲͷΑ͏ͳϓϩηεΛ௨࣮ͯ͠ࡍʹεϖΫτϧแབྷ͕ੜ੒͞ΕΔ͔ʹ͍ͭͯٞ࿦͠ɼͦͷϓϩηεΛ֬཰Ϟσϧͱͯ͠ఆࣜԽ͢Δɽ

2.2 ෳ߹΢ΣʔϒϨοτϞσϧ [11, 12] ͷಋೖεϖΫτϧแབྷʹ͓͚Δ֤ϑΥϧϚϯτΛΨ΢ε෼෍ؔ਺Ͱۙࣅతʹද‫͖Ͱݱ‬Δͱ͢ΔͱɼεϖΫτϧแབྷશମΛΨ΢ε෼෍ؔ਺ͷॏͶ߹Θͤɼ͢ͳΘͪࠞ߹Ψ΢ε෼෍ؔ਺ϞσϧʢGaussian Mixture Model;
GMMʣͰද‫͢ݱ‬Δ͜ͱ͕Ͱ͖ΔɽεϖΫτϧแབྷͷ
GMM ʹΑΔۙࣅͷྫΛ Fig. 2 ʹࣔ͢ɽ͜ͷΑ͏ͳεϖΫτϧแབྷͷද‫ݱ‬Λɼෳ߹΢ΣʔϒϨοτϞσϧʢComposite Wavelet Model; CWMʣ[11, 12] ͱ‫Ϳݺ‬ɽ
CWM ʹ͓͚ΔεϖΫτϧแབྷϞσϧ Fω,t ͸
φω,t =Kψk,ω,tk=1wk,texp
ψk,ω,t = √2πσk,t(2)(ω − μk,t )2−22σk,t(3)Ͱ༩͑ΒΕΔɽͨͩ͠ɼω, t ͸प೾਺ͱ࣌ࠁͷΠϯσοΫεɼk ͸Ψ΢εؔ਺ͷΠϯσοΫεͰ͋ΓɼKProbabilistic model of speech spectral sequences involving formant frequency contours as latent variables.
by YOSHIZATO Kota, HOJO Nobukatsu, KAMEOKA Hirokazu, SAITO Daisuke, SAGAYAMA Shigeki(The University of Tokyo)

日本音響学会講演論文集- 277 -2013年3月

ͨͩ͠ɼGamma(x; a, b) ͸ΨϯϚ෼෍Gamma(x; a, b) = xa−1Fig. 3exp(−x/b)Γ(a) ba(4)

Ͱ͋Δɽ
2.1 અͰٞ࿦ͨ͠Α͏ʹɼఏҊϞσϧͰ͸ࢦྩؔ਺uk,l ʹೋ࣍ઢ‫ܥܗ‬ͷΠϯύϧεԠ౴͕৞Έࠐ·ΕͯΨ΢ε෼෍ؔ਺ͷฏ‫ۉ‬஋ͷ‫ ੻ي‬μk,l ͕ੜ͡Δͱ͢Δɽ۩ମతʹ͸ఏҊ HMM ͷߏ੒l2
P (μk,l |{uk,l }l , sl ) = LN (Gk,l ∗ uk,l , νk,s),l(5)ͱॻ͚Δͱ͢Δɽ͜͜Ͱ LN (x; μ, σ 2 ) ͸ର਺ਖ਼‫ن‬෼෍Ͱ͋Γɼlog x ͕ਖ਼‫ن‬෼෍ N (x; μ, σ 2 ) ʹै͏͜ͱͱ౳ՁͰ͋Δɽ·ͨ Gk,l ͸ࣜ (1) ͷ G(t) ͷ཭ࢄ࣌ؒදlFig. 4

ঢ়ଶ/k/ͷ 4 ͭͷখঢ়ଶ΁ͷ෼ׂ2͸ GMM ͷࠞ߹਺Ͱ͋Δɽ·ͨɼμk,t , σk,t, wk,t ͸ɼͦΕͧΕΨ΢ε෼෍ؔ਺Λ౷‫ܭ‬෼෍ͱ‫ࡍͨ͠ͳݟ‬ͷฏ‫ۉ‬ɾ෼ࢄɾॏΈʹରԠ͠ɼϑΥϧϚϯτप೾਺ɾϑΥϧϚϯτϐʔΫͷӶ͞ɾϑΥϧϚϯτͷ‫ʹ౓ڧ‬ରԠ͢ΔύϥϝʔλͰ͋Δɽ

2.3 ֬཰ϞσϧͷఆࣜԽຊઅͰ͸ɼҎ্ͷ४උΛ΋ͱʹɼϑΥϧϚϯτप೾਺ͷ࣌ؒ‫੻ي‬Λજࡏύϥϝʔλͱͯ͠΋ͭԻ੠εϖΫτϧੜ੒աఔͷ֬཰ϞσϧΛఆࣜԽ͢Δɽ
[6] ͷΞΠσΟΞͱಉ༷ɼ֊ஈঢ়ͷؔ਺͕ӅΕϚϧίϑϞσϧʢHidden Markov Model; HMMʣʹΑΓද‫͖Ͱݱ‬Δ͜ͱʹண໨͢ΔͱɼԻૉ‫ڥ‬քʹΑΓஈ͕੾ΓସΘΔ֊ஈঢ়ͷԻૉࢦྩؔ਺ΛɼFig. 3 ͷΑ͏ͳԻૉʹରԠͨ͠ঢ়ଶ͔ΒͳΔ HMM ʹΑΓ֬཰ϞσϧԽ͢Δ͜ͱ͕Ͱ͖Δɽ͜Ε͸ɼকདྷతʹఏҊϞσϧΛ HMM Ի੠߹੒ͷ࿮૊Έʹ૊ΈࠐΜͰ͍͘͜ͱΛߟྀͯ͠΋޷౎߹Ͱ͋Δɽ͢ͳΘͪɼ͜ͷ HMM ͸ɼ֤཭ࢄ࣌ࠁ l ʹ͓͍ͯ CWM ύϥϝʔλͰ͋Δ wk,l ɼσk,lʢศٓతʹҎ‫ ޙ‬σk,l ͷ‫਺ٯ‬Λ ρk,l ͱஔ͖ɼσk,l ͷ୅ΘΓʹ ρk,l Λύϥϝʔλͱ‫͢ͳݟ‬ɻʣɼ͓ΑͼɼԻૉࢦྩؔ਺ uk,l Λग़ྗ͢Δ֬཰తδΣωϨʔλͱ‫ݟ‬ͳͤΔɽՃ͑ͯɼࣗ‫ݾ‬ભҠͷ࣋ଓ௕ΛύϥϝʔλԽ͢ΔͨΊʹɼӅΕηϛϚϧίϑϞσϧ (Hidden Semi-MarkovModel; HSMMʣ[13] Λಋೖ͢ΔɽHSMM ͸ɼ֤ঢ়ଶΛे෼େ͖ͳ਺ͷখঢ়ଶʹ෼ׂ͢Δ͜ͱͱ౳ՁͰ
͋Δɽ͜͜Ͱɼ෼ׂ‫ޙ‬ͷ֤খঢ়ଶ͸͢΂ͯಉ͡ग़ྗ෼෍Λ࣋ͭɽFig. 4 ʹঢ়ଶ/k/Λ෼ׂͨ͠ྫΛࣔͨ͠ɽ͜ͷΑ͏ͳ෼ׂʹΑΓɼ͋Δঢ়ଶʹ͋Δ཭ࢄ͚࣌ؒͩཹ·Δ֬཰Λ‫ݸ‬ผʹύϥϝʔλԽ͢Δ͜ͱ͕ՄೳʹͳΔɽҎ্ΑΓɼఏҊ HMM ͷߏ੒͸ҎԼͱͳΔɽग़ྗ஋‫ྻܥ‬: {wk,l , ρk,l , uk,l }k,lঢ়ଶू߹: {/a/i , /k/i , /o/i , . . . }iঢ়ଶ‫ྻܥ‬: s = {sl }lঢ়ଶग़ྗ෼෍:(w) (w)P (wk,l |sl ) = Gamma(wk,l ; ak,sl , bk,sl )
(ρ)

(ρ)

P (ρk,l |sl ) = Gamma(ρk,l ; ak,sl , bk,sl )2
P (uk,l |sl ) = N (uk,l ; mk,sl , ηk,s)lঢ়ଶભҠ֬཰: Φi ,i = P (sl = i|sl−1 = i )ॳ‫ظ‬ঢ়ଶ֬཰: Φi = P (s1 = i)‫͋Ͱݱ‬Γʢ‫ݻ‬༗प೾਺͸ αk ͱ͓͘ʣɼ∗ ͸཭ࢄ࣌ࠁʹؔ͢Δ৞ΈࠐΈΛද͢ɽҎ߱ɼύϥϝʔλΛ·ͱΊͯ ρ = {ρk,l }k,l , w = {wk,l }k,l , u = {uk,l }k,l , μ ={μk,l }k,l ͱॻ͘ɽ͢΂ͯͷ CWM ύϥϝʔλͱঢ়ଶ‫ ྻܥ‬s ͕༩͑ΒΕͨͱ͖ʹ‫؍‬ଌεϖΫτϧแབྷ yω,l ͕ੜ͡Δ֬཰ΛP (yω,l |ρ, w, μ, s) = Poisson(yω,l ; φω,l )ͱ͢Δɽ͜͜Ͱ Poisson(x; λ) ͸ϙΞιϯ෼෍Ͱ͋Δɽ
ͳ͓ɼ͜ͷԾఆͷԼͰͷ λ ͷ࠷໬ਪఆ໰୊͸ɼεϖΫτϧؒͷۙ͞ΛଌΔई౓ͷҰͭͱͯۙ͠೥Ի‫ڹ‬৴߸ॲཧ෼໺Ͱଟ༻͞ΕΔ I μΠόʔδΣϯεͱ‫Ϳݺ‬࿪Έई౓Λ‫ن‬४ͱͨ͠ x ͱ λ ͷ࠷దϑΟοςΟϯά໰୊ͱ౳ՁͱͳΔ͜ͱ͕஌ΒΕ͍ͯΔ [14]ɻҎ্ͷఏҊϞσϧͷਪఆ͢΂͖ύϥϝʔλΛ·ͱΊͯ Θ = {ρ, w, μ, u, s, θ} ʢ ͨ ͩ ͠ ɼθ =(w) (ρ){bk,i , bk,i , mk,i }k,i ʣͱ͠ɼ࣍ষͰ͸ɼ‫؍‬ଌεϖΫτϧ‫ ྻܥ‬Y = {yω,l }ω,l ͕༩͑ΒΕͨԼͰͷࣄ‫཰֬ޙ‬P (Θ|Y ) Λ࠷େԽ͢ΔύϥϝʔλਪఆΞϧΰϦζϜʹ͍ͭͯड़΂Δɽ3ύϥϝʔλਪఆΞϧΰϦζϜ؆ ୯ ͷ ͨ Ί ɼຊ ߘ Ͱ ͸ ύ ϥ ϝ ʔ λ(w) (ρ) 2
2{ak,i , ak,i , ηk,i, νk,i}k,i , {αk }k Λ ͢ ΂ ͯ ఆ ਺ ͱ͢Δɽ·ͨɼθ ͷࣄલ෼෍͸Ұ༷෼෍ͱ͢Δɽ
P (Θ|y) Λ࠷େԽ͢Δ Θ Λղੳతʹ‫ٻ‬ΊΔ͜ͱ͸೉͍͕͠ɼҎԼʹड़΂ΔΑ͏ʹิॿؔ਺๏ʹ‫͖ͮج‬‫࠷ॴہ‬దԽΞϧΰϦζϜΛಋ͘͜ͱ͕Ͱ͖Δɽ
log P (Θ|y) ͸ɼclog P (Θ|y) = log P (y|Θ)P (Θ)(7)

clog P (Θ) = log P (s)P (ρ|s, θ)P (w|s, θ)P (u|s, θ)P (μ|s, u, θ)(8)cͱද͞ΕΔɽͨͩ͠ɼ= ͸ఆ਺෦෼Λআ͍ͨ৔߹ͷ౳߸Λҙຯ͢Δɽઌड़ͷΑ͏ʹɼ− log P (y|Θ) ͸ఆ਺߲Λআ͚͹‫؍‬ଌεϖΫτϧแབྷ yω,l ͱεϖΫτϧ
แབྷϞσϧ φω,l ͱͷؒͷ I μΠόʔδΣϯεͱ౳͘͠ [14]ɼ͞Βʹɼ ∞
wk,l(ω − μk,l )2√dω
ψk,ω,l 
exp −22σk,l2πσk,l−∞ω= wk,l日本音響学会講演論文集(6)- 278 -(9)2013年3月

ͱͳΔ͜ͱΛ༻͍Ε͹ɼyω,lcyω,l log− log P (y|Θ) =− yω,l + φω,lφω,lω,lc=(φω,l − yω,l log φω,l )ω,l

wk,l −k,lyω,l log φω,l(10)ω,l͕‫͑ݴ‬Δɽ͜ͷࣜͷ −yω,l log φω,l ͷ߲ʹର͠ɼෛͷର਺ؔ਺ͷತੑΛར༻͠ɺJensen ͷෆ౳ࣜΛ༻͍Δ͜ͱͰ−yω,l log φω,l ≤ −yω,lk

ψk,ω,lγk,ω,l logγk,ω,l(11)Fig. 6 No.436 ͷԻૉ/a/ͷεϖΫτϧแབྷʢ఺ઢʣ
ͷ GMM ۙࣅʢ࣮ઢʣͷΑ͏ʹ্քؔ਺Λઃ‫͢ܭ‬Δ͜ͱ͕Ͱ͖Δɽ·ͨɼ
2− log P (μ|s, u) ͷͷ߲ʹର͠ɼೋτ Gk,l−τ uk,τ࣍ؔ਺ͷತੑΛར༻͠ɼಉ༷ʹ Jensen ͷෆ౳ࣜΛ༻͍Δ͜ͱͰ

2Gk,l−τ uk,τ≤τ (Gk,l−τ uk,τ )2τλτ,k,l(12)1+μk,l2 log ξk,l1+ 2ξk,lξk,l+ | log ξk,l |2 − 2 log ξk,l −2ξk,lGk,l−τ uk,ττ  Gk,l−τ uk,τψk,ω,lφω,lξk,l = μk,lͱͳΔɽ͜͜Ͱ Ti = {l|sl = i} Ͱ͋Δɽμ ʹ͍ͭͯ͸ɼp3 μ3k,l + p2 μ2k,l + p1 μk,l + p0 = 0yω,l γk,ω,l ρk,lp3 =1(13)

日本音響学会講演論文集

(23)
(24)
(25)

p0 = −

122νk,sl(26)(14)ͱ͓͍ͯɼࣜ (22) ͷਖ਼ͷղͷ͏ͪ − log P (Θ|y) Λ࠷΋খ͘͢͞Δ μk,l Λબ΂͹Α͍ɽҎ্ͷߋ৽ଇΛे෼ͳճ਺൓෮͢Δ͜ͱͰɼP (Θ|y)Λ‫࠷ॴہ‬େԽ͢Δύϥϝʔλ Θ Λਪఆ͢Δ͜ͱ͕Ͱ͖Δɽ

(15)

4(17)
ρk,lwk,l(ρ)(w)
bk,i =  l∈Ti (ρ) , bk,i =  l∈Ti (w)l∈Ti ak,sll∈Ti ak,sl
mk,slGk,τ −l log μk,τ+ τ ≥l22ηk,sνk,sτ
luk,l =2
Gk,τ −l1+ τ ≥l ν 2 λl,k,τ
2
ηk,sk,sτl(22)τ ≤t(16)l2l∈Ti ηk,sl2ξk,l log ξk,l + 1 −yω,l γk,ω,l ρk,l ω
2 ξ22νk,sl k,lω
1 Gk,l−τ uk,τp1 = 1 − 2νk,slp2 =Ͱ༩͑ΒΕΔɽϞσϧύϥϝʔλ Θ ͷߋ৽ࣜ͸ɼิॿؔ਺Λ࠷খԽ͢Δղɼ͢ͳΘͪภඍ෼͕ 0 ͱͳΔ
ղͱͳΔɽμ Ҏ֎ͷύϥϝʔλͷߋ৽ࣜ͸ɼ
uk,l2l∈Ti ηk,s(21)llμk,lγk,ω,l =mk,i = (w)
ak,sl − 1 + ω yω,l γk,ω,lwk,l =1(w) + 1bk,sωͷΑ͏ʹ্քؔ਺Λઃ‫͢ܭ‬Δ͜ͱ͕Ͱ͖Δ [16]ɽ͜͜
Ͱ γk,ω,l , λτ,k,l , ξk,l ͸ิॿม਺Ͱ͋Δɽࣜ (11)ʙ(13) ΑΓɼ− log p(Θ|Y ) ͷ্քؔ਺Λઃ‫͢ܭ‬Δ͜ͱ͕Ͱ͖ɼ͜ΕΛิॿؔ਺ͱͯ͠ิॿؔ਺๏Λద༻͢Δ͜ͱ͕Ͱ͖Δɽ·ͣɼิॿม਺ͷߋ৽ࣜ͸ɼ্ड़ͷෆ౳ࣜͷ౳߸੒ཱ৚݅ɼ͢ͳΘͪɼ

λτ,k,l = (20)bk,sͷΑ͏ʹ্քؔ਺Λઃ‫͢ܭ‬Δ͜ͱ͕Ͱ͖Δ [15]ɽ͞Βʹ − log P (μ|s, u) ͷதͷ (log μk,l )2 ͷ߲ʹؔͯ͠͸ɼ(log μk,l )2 ≤(ρ)
2(ak,sl − 1) + ω yω,l γk,ω,lρk,l = 2
2(ρ) +ω yω,l γk,ω,l (ω − μk,l )

(18)

(19)

࣮‫ݧ‬

ఏҊϞσϧ͕࣮Ի੠ͷϑΥϧϚϯτप೾਺ͷ࣌ؒ‫੻ي‬ΛΑ͘ද‫͍͖ͯͰݱ‬Δ͜ͱΛ֬ೝ͢ΔͨΊʹ࣮‫ݧ‬Λߦͬͨɽ࣮‫ݧ‬͸େ͖͘෼ֶ͚ͯशϑΣΠζͱਪఆϑΣΠζͷ 2 ஈ֊͔ΒͳΔɽֶशϑΣΠζͰ͸ɼATR೔ຊ‫ޠ‬Ի੠σʔλϕʔεͷ B ηοτ [17] ͔Βஉੑ࿩
ऀ 1 ਓΛબ୒͠ɼNo.1ʙNo.400 ·Ͱͷ 400 จΛର৅ͱͯ͠ɼԻૉ͝ͱʹఆ·Δύϥϝʔλ θ ͷֶशΛߦͬͨɽଓ͘ਪఆϑΣΠζͰ͸ɼֶशʹ࢖͍ͬͯͳ͍ൃ࿩จΛର৅ʹ CWM ύϥϝʔλͷਪఆΛߦͬͨɽ͜͜Ͱ θ ͸ֶशϑΣΠζͰͷਪఆ஋Λ༻͍ͯఆ਺ͱΈͳͨ͠ɽͳ͓ɼຊ࣮‫͍͓ͯʹݧ‬εϖΫτϧแབྷͷநग़ʹ͸ STRAIGHT[18] Λ༻͍ɼ·ͨԻૉϥϕϧͷσʔλΛ༩͑Δ͜ͱͰঢ়ଶ‫ ྻܥ‬s ͸ఆ਺ͱͨ͠ɽຊ࣮‫Ͱݧ‬͸ɼGMM ͷࠞ߹਺͸ 10ɼύϥϝʔλਪఆΞϧΰϦζϜͷ൓෮ճ਺͸ 10ɼαk = 50 ͱ͠ɼͦͷଞͷ CWM ύϥϝʔλͷॳ‫ظ‬஋͸ [8] ͷ Chain Λಋೖ͠ͳ͍ਪఆΞϧΰϦζϜΛ༻͍ܾͯఆͨ͠ɽ
No.436 ͷʮԖචͩͱɼྗΛೖΕͯॻ͘ͷͰɼεϐʔυ͕ʹͿΔͷͰ͋ΔʯͷಡΈ্͛Ի੠Λର৅ʹਪఆΛߦͬͨ݁ՌΛ Fig. 5 ͱ Fig. 6 ʹࣔͨ͠ɽFig. 5 ͸ɼ

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Fig. 5 No.436 ͷεϖΫτϧแབྷ‫( ྻܥ‬ΧϥʔϚοϓදࣔ) ͱఏҊ๏ʹΑΔਪఆϑΥϧϚϯτप೾਺‫࣮( ੻ي‬ઢ)(఺ઢ͸Իૉ‫ڥ‬ք)
No.436 ͷεϖΫτϧแབྷʹఏҊख๏Ͱਪఆͨ͠ϑΥϧϚϯτप೾਺ͷ࣌ؒ‫੻ي‬ΛॏͶͯඳ͍ͨਤͰ͋Δɽ
·ͨ Fig. 6 ͸ɼಉ͘͡ No.436 ͷԻૉ/a/ͷεϖΫτϧแབྷͱਪఆύϥϝʔλʹΑΔ GMM ۙࣅΛࣔͨ͠ਤͰ͋Δɽ͜ΕΒͷ݁Ռ͔Βɼਪఆͨ͠εϖΫτϧϐʔΫ͕࣮Ի੠ͷϑΥϧϚϯτप೾਺‫੻ي‬Λ͏·͘ද‫͍͖ͯͰݱ‬Δ͜ͱ͕֬ೝͰ͖Δɽ5͓ΘΓʹຊ‫Ͱڀݚ‬͸ɼϑΥϧϚϯτप೾਺ͷ࣌ؒ‫੻ي‬Λજࡏύϥϝʔλͱͯ͠΋ͭԻ੠εϖΫτϧੜ੒աఔͷ֬཰ϞσϧͷఆࣜԽΛߦ͍ɼͦͷύϥϝʔλਪఆΞϧΰϦζϜΛಋग़ͨ͠ɽ࣮ͦͯ͠‫ݧ‬Λ௨ͯ͠ɼఏҊख๏͕࣮Ի੠ͷεϖΫτϧϐʔΫΛΑ͘ද‫͖Ͱݱ‬Δ͜ͱΛ֬ೝͨ͠ɽࠓ‫ޙ‬͸ɼϑΥϧϚϯτप೾਺‫੻ي‬Λೋ࣍ઢ‫ܥܗ‬ͷग़ྗͱΈͳͨ͠ͱ͖ɼ‫ݻ‬༗֯प೾਺ α ʹ‫ݸ‬ਓੑ͕දΕΔͷͰ͸ͳ͍͔ͱ͍ͬͨɼΑΓৄࡉͳϞσϧͷ‫ݕ‬౼Λߦͳ͍͖͍ͬͯͨɽ·ͨɼఏҊϞσϧΛ‫ج‬ຊप೾਺ύλʔϯͷੜ੒աఔͷ֬཰Ϟσϧ [7] ͱ౷߹͢Δ͜ͱͰɼΑΓߴ඼࣭ͳςΩετԻ੠߹੒Λ࣮‫͢ݱ‬Δ‫ڀݚ‬ΛਐΊ͍ͯ͘͜ͱΛߟ͍͑ͯΔɽࢀߟจ‫ݙ‬[1] ࠧեࢁ, ൘૔, “Ի੠ͷಈతई౓ʹ‫·ؚ‬ΕΔ‫ݸ‬ਓੑ৘ใ,” Իߨ࿦ (य़), No. 3-2-7, pp. 589–590,
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[7] ٢ཬଞ, “F0 ύλʔϯੜ੒աఔͷ֬཰ϞσϧʹΑΔ౻࡚Ϟσϧύϥϝʔλͷਪఆ,” ৘ॲ‫ݚ‬ใ,Vol. 2012-SLP-92, No. 9, pp. 1–6, 2012.

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[14] H. Kameoka, “Statistical Approach to Multipitch Analysis,” Ph.D. Thesis, The University ofTokyo, 2007.
[15] ُԬଞ, “Ի੠ͷεύʔεੑͱඇෛ੍໿͖ͭ৞ΈࠐΈϞσϧʹ‫ͮ͘ج‬ύϫʔεϖΫτϧྖҬ࢒‫ڹ‬আ‫ڈ‬,” Իߨ࿦ (ळ), No. 3-8-10, pp. 705–708, 2008.
[16] H. Kameoka et al., “Constrained and regularized variants of non-negative matrix factorization incorporating music-speciﬁc constraints,”
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