ҼՌϞσϧͱͯ͠ͷճؼੳͷຊ࣭త
Essential problems of regression analysis as a causal model
ழݪਖ਼क∗ ֓ ཁ ճؼੳɼ͋Γ߹Θͤͷσʔλͷͮ͘Γݱʹ͓͚ΔৗཧσʔλΛ༻͍ͯɼݱ ͷഎޙʹ͋ΔҼՌϞσϧΛ୳ࡧ͢Δखஈͱͯ͠༻͍ΒΕΔ͜ͱ͕͋Δɽ͔͠͠ɼճؼੳͷ ݁ՌɼऔΓ্͛ͨઆ໌มؒͷ૬ؔؔଟॏڞઢੑͱݺΕΔɼ͋Δ͍Ϟσϧͷ֎ ʹ͋ΔมΛઆ໌มͱͯ͠औΓ͜Μͩ͜ͱʹΑΔ͜ͱͳͲ͕ݪҼͱͳͬͯҼՌϞσϧͷ୳ࡧ ʹ༻͍Δ͜ͱةݥͰ͋Γɼ༧ଌͷखஈͱͯ͠ͷΈ༻͍Δ͖ͷͰ͋Δ͜ͱΛྫΛத৺ ͱͯٞ͢͠Δɽ
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͡Ίʹ
ճؼੳɼ൧௩ଞ(2019)Ԟଞ(1985)͋Δ͍Ӭాͱ౩ۙʢ2016ʣͰհ͞Ε͍ͯΔΑ͏ ʹɼ࣭ཧͷʹ͓͚Δ͋Γ߹Θͤͷσʔλͷͮ͘Γݱʹ͓͚ΔৗཧͷهͳͲ ൺֱతʹૈѱͳσʔλΛ༻͍ͯɼݱͷഎޙʹ͋ΔҼՌϞσϧΛ୳ࡧ͢Δखஈͱͯ͠༻͍ΒΕΔ ͜ͱ͕͋ΔɽͦΕ࣮ݧܭը๏ʹൺͯσʔλΛ࠾औ͢ΔͨΊͷಛผͳࣝඅ༻͕͔͔Βͳ͍ ͱ͜Ζʹ͋ΔͱࢥΘΕΔɽ ͔͠͠ɼσʔλͷഎޙʹ͋ΔҼՌϞσϧΛ୳ࡧ͢Δͱ͖ʹɼมؒͷଟॏڞઢੑ༨ܭͳม ͷࠞࡏʹΑͬͯಘΒΕͨ݁Ռ͕େ͖͘Өڹ͞ΕΔ͜ͱ͕͋Γɼ҃ΔҙຯͰةݥͳ᠘͕ͪߏ͑ ͍ͯΔ߹͋Δʢͨͱ͑ɼԞଞʢ1985ʣRao(1973))ɽͦͷΑ͏ͳ᠘ʹ͔͔Δͷճؼ ੳΛར༻͍ͯ͠ΔࣄऀͷࣝෆʹݪҼ͕͋Δͱ͍ͬͯ͠·͑ͦΕ·ͰͰ͋Δ͕ɼճؼੳ Λڭ͑Δଆҙשى͚ͩͰͳ͘ɼ͔Γ͍͢ࣄྫΛ௨ͨ͡ڭҭΛߦ͏ͱ͍͏͕͋Δ(Ӭ ా(2019))ɽ ͜͜Ͱɼ۩ମతͳྫΛ༻͍ͯɼճؼੳΛڭ͑Δଆͷਓͼͱʹର͢Δܯͱ͢Δͱͱ ʹɼճؼੳΛ༻͍ͯσʔλͷഎޙʹજΉҼՌؔΛ୳ࡧ͠Α͏ͱ͢Δ࣮ऀʹର͢Δࢀߟͱ͠ ͍ͨɽ2
Ϟσϧ
σʔλͷഎޙʹજΉ๏ଇੑΛϞσϧԽ͠ɼͦͷϞσϧͷԼͰಘΒΕΔσʔλʹճؼੳΛద༻ ͨ͠ͱ͖ɼͲΜͳ݁Ռ͕ಘΒΕΔ͔ΛΔ͜ͱɼೖऀʹݶΒͣηϛϓϩͷ࣮ऀݚڀऀʹ ∗ᴥݩᴦେࡕిؾ௨৴େֶɾใ௨৴ֶ෦ɾใֶՊ 1 大阪電気通信大学 研究論集 （自然科学編） 第 55 号 大阪電気通信大学 研究論集 （自然科学編） 第 55 号ͱ࣮ͬͯײͷ͋ΔͰ͋Ζ͏ɽɹ͜͜ͰɼϞσϧͱͯ͠ɼࡾͭͷઆ໌มx1, x2, x3ͱతม yͷؒʹ yi = β0+ β1xi1+ β1xi2+ β3xi3+ ui, (i = 1, 2, . . . , n) (2.1) ͷҼՌϞσϧཱ͕͢Δͷͱ͠ɼҎԼͰɼҰൠੑΛࣦ͏͜ͱͳ͘ɼ β0 = 0, β1= 1, β2 = 2, β3 = 3 ͱ͢Δɽ ࣜ(2.1)ʹ͓͍͚Δuiɼxi1, xi2, xi3ͱಠཱͳޡࠩΛද͕͢ɼ͜͜Ͱड़Δճؼੳʹ͓͚ Δຊ࣭తͳɼޡ͕ࠩͳ͍߹Ͱى͜ΓಘΔࣄͳͷͰɼui = 0ͷྫΛ༩͑Δɽͨ͠ ͕ͬͯɼҎԼͰ౷ܭॲཧʹΑͬͯআ͔ΕΔҎ֎ͷޡࠩͷଘࡏʹΑΔߟ͑ͳ͍͜ͱͱ ͢Δɽ ͕ͨͬͯ͠ɼҎԼͰ༻͍ΔҼՌϞσϧ y = x1+ 2x2+ 3x3 (2.2) Ͱ͋Δɽ
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͜͜Ͱɼ͋Γ߹ΘͤͷσʔλৗཧσʔλΛ༻͍Δ͜ͱͰൃੜ͢Δઆ໌มؒͷ૬͕ؔ ճؼੳͷ݁ՌΛҼՌϞσϧͱͯ͠׆༻͢Δ͜ͱͷةݥੑΛओͨΔςʔϚͱ͍ͯ͠ΔͨΊɼ ྫ1ʢઆ໌มؒʹڧ͍૬ؔͷ͋Δ߹ʣɼྫ2ʢઆ໌มؒͷ૬͕ؔऑ͍߹ʣɼྫ3 ʢઆ໌มؒʹ૬ؔͷͳ͍߹ʣɼྫ4ʢ૬ؔͷͳ͍ม͕ѱ͞Λ͢Δ߹ʣɼྫ5ʢઆ໌ มؒʹଟॏڞઢੑͷ͋Δ߹ʣɼྫ6ʢઆ໌มؒʹڧ͍૬ؔ͋Δ͕ɼଟॏڞઢੑൃੜ ͍ͯ͠ͳ͍߹ʣͷॱʹ6ݸͷྫΛऔΓ্͛Δɽ 3.1 ྫ 1(આ໌มʹڧ͍૬͕ؔ͋Δ߹) σʔλ͕ද1Ͱ༩͑ΒΕɼ૬ؔߦྻ͕ද2Ͱ༩͑ΒΕΔͱ͖ͷղੳ݁Ռද3Ͱ༩͑ΒΕΔɽ ද1 σʔλ No. x1 x2 x3 y 1 1.00 0.00 0.98 3.94 2 2.00 1.56 0.99 8.09 3 3.00 0.93 0.98 7.80 4 4.00 3.54 1.92 16.84 5 5.00 5.19 2.02 21.44 6 6.00 4.91 1.91 21.55 7 7.00 8.45 3.79 35.27 8 8.00 4.66 2.50 24.82 9 9.00 13.28 4.01 47.59 10 10.00 16.45 3.54 53.52ද2 ૬ؔߦྻ x1 x2 x3 y x1 1.000 0.900 0.900 0.943 x2 0.900 1.000 0.900 0.989 x3 0.900 0.900 1.000 0.946 y 0.943 0.989 0.946 1.000 ͜͜Ͱɼද2͕ࣔ͢Α͏ʹɼઆ໌มؒʹڧ͍૬͕ؔ͋Γɼ୯૬͕ؔͯ͢େ͖ͳʹͳͬ ͍ͯΔɽ ද 3 ੳ݁Ռ ˆ β0 βˆ1 βˆ2 βˆ3 SR φR Se φe Ve F0 P (F0< F ) x1 -4.69 5.23 - - 2258.03 1 279.29 8 34.91 64.68 0.000 x2 5.83 - 3.10 - 2483.36 1 53.96 8 6.75 368.18 0.000 x3 -6.70 - - 13.60 2271.03 1 266.29 8 33.29 68.23 0.000 x1, x2 1.95 1.55 2.31 - 2520.96 2 16.36 7 2.34 539.33 0.000 x1, x3 -7.29 2.68 - 7.34 2384.27 2 153.05 7 21.86 54.52 0.000 x2, x3 1.15 - 2.27 4.22 2525.14 2 12.18 7 1.74 725.61 0.000 x1, x2, x3 0.00 1.00 2.00 3.00 2537.32 3 0.00 6 0.00 - -͜ͷ߹ʹɼ 3.1.1 ୯ճؼੳ ୯ճؼੳʹΑΔճؼҼՌϞσϧͷͱ·ͬͨ͘ҟͳΔʹͳ͍ͬͯΔɽ͜Εɼͦ ΕͧΕͷճؼʹΑΔฏํSRͱࠩʹΑΔฏํSeͷࢄൺ F0 = SSR/φR e/φe = 368.18 ͷ͕େ͖͍͜ͱP (F0< F )ͷখ͍͜͞ͱɼ͋Δ͍د༩ R2x2 = SRS(β2) T = 2483.36 2537.32 = 0.979 ͕ඇৗʹେ͖͍͜ͱ͕ޠΔΑ͏ʹɼͨͱ͑มx2ͷഎޙʹ͋Δઆ໌มx1, x3ͷӨڹΛม x2Ͱઆ໌͠Α͏ͱ͢Δ͜ͱʹىҼ͍ͯ͠Δɽ ݱ࣮ͷ໘ͰɼͲΕ͔Ұͭͷઆ໌มɼͨͱ͑x2ͷΈΛऔΓ্͛ɼx1ͱx2Λແࢹͨ͠ղ ੳΛߦ͏͜ͱ͕ى͜Δ͔͠Εͳ͍ɽͦͷ߹ɼಘΒΕͨճؼࣜΛҼՌϞσϧͱͯ͠ड͚ೖΕΔ ͠·͏ͱɼຊͷҼՌϞσϧͱ·ͬͨ͘ҧͬͨ݁ՌΛड͚ೖΕͨ͜ͱʹͳΔɽ 3.1.2 มબ ճؼੳϞσϧʹ͓͍ͯɼX = (X1(n×(r+1)) : X2(n×(p−r)))ɼβ = (β1((r+1)×1) , β2(p−r)×1) )ͱ͠ɼ y = X1β1+ X2β2+ u (3.1) ͯ
ͱ͢Δɽͨͩ͠ɼ X1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 x11 · · · x1r 1 x21 · · · x2r .. . ... . .. ... 1 xn1 · · · xnr ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, X2= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x1(r+1) · · · x1p x2(r+1) · · · x1p .. . . .. ... xn(r+1) · · · xnp ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, β1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ β0 β1 .. . βr ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, β2= ⎛ ⎜ ⎜ ⎝ βr+1 .. . βp ⎞ ⎟ ⎟ ⎠ ͱ͠ɼβ= (β1, β2)ͷ࠷খ2ਪఆΛβˆ = ( ˆβ 1, ˆβ2)ͱ͢Δɽͳ͓ɼϕΫτϧβʹର͢Δβߦ ྻXʹର͢ΔXɼϕΫτϧߦྻͷసஔΛද͍ͯ͠Δɽ ͜ͷͱ͖ɼ (XX)−1= X 1X1 X1X2 X 2X1 X2X2 −1 = A = A11 A12 A21 A22 (3.2) ͱ͢Εɼ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ X 1X1A11+ X1X2A21= I · · · (1) X 1X1A12+ X1X2A22= 0 · · · (2) X 2X1A11+ X2X2A21= 0 · · · (3) X 2X1A12+ X2X2A22= I · · · (4) Ͱ͋Δ͔Βɽ(2)ͱ(̐)͓Αͼ(1)ͱ(3)ΑΓ A11 = (X1X1) − (X1X2)(X2X2)−1(X2X1) −1 A22 = (X2X2) − (X2X1)(X1X1)−1(X1X2) −1 ɹ A12 = −(X1X1)−1(X1X2)A22 Ͱ͋Δɽ ͕ͨͬͯ͠ɼu ∼ N(0, σ2In)ͱ͢Εɼ ˆ β = ˆ β1 ˆ β2 ∼ Nβ, σ2(XX)−1 ΑΓɼ ˆ β2 ∼ N β2, σ2A22 Ͱ͋Δ͔ΒɼϞσϧ(3.1)ʹ͓͍ͯԾઆ ⎧ ⎨ ⎩ H0 : β2 = 0 ⇔ y = β0+ X1β1+ u H1 : β2 = 0 ⇔ y = X1x1+ X2x2+ u Λݕఆ͢ΔͨΊʹɼؼແԾઆH0ͷԼͰɼ ˆ β2A−122βˆ2 σ2 ∼ χ2(p − r) (3.3) Ͱ͋Δ͜ͱ͕͔Δ͔Βɼݕఆ౷ܭྔ F0 = βˆ2A −1 22βˆ2 ˆ σ2 (3.4)
͕ࣗ༝(φR, φe) = (p − r, n − p − 1)ͷFʹै͏͜ͱΛ༻͍ͯߦ͏͜ͱ͕Ͱ͖Δɽ Ұํɼ (3.5) y=X1β1+u ʹ͓͚Δ β1ͷਪఆྔΛ βˆ1ͱ͢Δͱ͖ɼճؼࠩ (ResidualSumSquaresɿRSS)Λ RSSr=(y−X1βˆ1)(y−9βˆ1) ͱ͠ɼϞσϧ (3.1)ʹ͓͚ΔճؼࠩΛ RSSp= (Y − X ˆβ)(y − X ˆβ) (3.7) ͱ͢Δͱ͖ɼ RSSr− RSSp = ˆβ2A−122βˆ2 (3.8) ɹͰ͋Δ͜ͱΛར༻͢Εɼ্هͷԾઆ F0= (RSSRSSr− RSSp)/(p − r) p/(n − p − 1) (3.9) Λ༻͍ͯɼ F0≥ F (p − r, n − p − 1; α) (3.10) Λغ٫Ҭͱ͢Δ͜ͱʹΑͬͯɼ༗ҙਫ४ 100α%ͷݕఆΛߦ͏͜ͱ͕Ͱ͖Δ (ٱอ (2019))ɽ ม x1ΛऔΓ্͛ͨͱ͖ͷճؼมಈ SR= 2258.03ͱେ͖ͳͰ͋Δ͕ɼม x2,x3Λऔ Γ্͛ͨճؼϞσϧʹରͯ͠ม x1ΛՃ͢Δҙٛͷ༗ແɼޡࠩΛؚ·ͳ͍ঢ়گʹ͓͍ͯ SR(x1,x2,x3)=STͰ͋Δ͜ͱʹҙ͢Δͱɼ S(β1) ≡ Se(x2, x3) − Se(x1, x2, x3) = SR(x1, x2, x3) − SR(x2, x3)[= Se(x2, x3)] (3.11) ʹΑͬͯධՁͰ͖ͯɼͦͷ SR(β1) = 1.74−0.00 = 1.74Ͱ͋Δ͔ΒɼۃΊͯখ͞ͳͰ͋ Δɽ·ͨɼಉ༷ʹ͢Δͱɼx1,x3ʹΑΔճؼϞσϧʹม x2ΛՃͨ͠Γม x1,x2ʹΑΔճ ؼϞσϧʹม x3ΛՃ͢Δ͜ͱͷҙٛʹର͢ΔධՁ S(β2)=SR(x1,x2,x3)−SR(x1,x3)= Se(x1,x3)=153.05, S(β3)=SR(x1,x2,x3)−SR(x1,x2)=Se(x1,x2)=2.34Ͱ͋ͬͯɼͦΕͧ ΕͷมʹΑΔ୯ճؼੳͷճؼมಈ SR(x2) = 2483.36, SR(x3) = 2271.03ʹൺͯখ͞ͳͰ ͋Δ͜ͱ͕Θ͔Δɽ͜͜ͰɼSe(x1,x2,x3=0.00Ͱ͋Δ͔Β (3.10)ࣜʹ͓͚Δ FݕఆΛߦ͏͜ͱ Ͱ͖ͳ͍͚ΕͲɼޡࠩͷ͋ΓΑ͏ʹΑͬͯɼSR(β1),SR(β2),SR(β3)ͷΛΈΔͱ͖ɼͦΕͧ Εͷ Fݕఆ͕༗ҙʹͳΒͳ͍͜ͱ͕͋ΓಘΔͱ͍͏͜ͱʹҙ͢Δඞཁ͕͋Δɽ 3.1.3 ༧ଌͱҼՌ ճؼੳɼతมͱઆ໌มͷҼՌϞσϧ͕ͲͷΑ͏ʹͳ͍ͬͯΔ͔ෆʹͯ͠ɼతม ͷมಈΛ͍͔ͭ͘ͷઆ໌มͰઆ໌ͨ͠ΓɼతมͷΛ༧ଌͨ͠Γ͢Δ͜ͱΛతͱͯ͠ ͍ΔɽͦͷҙຯͰɼେ͖ͳճؼมಈ͋Δ͍د༩Λ༩͑ΔճؼϞσϧʹΑͬͯతมΛઆ໌ ຢ༧ଌ͢Δ্Ͱճؼੳʹັྗ͕͋Δɽ·ͨɼ͜ͷΑ͏ʹมؒʹۃʹେ͖ͳ૬ؔͷ͋Δ ߹Ͱɼͯ͢ͷมΛऔΓࠐΜͩճؼੳΛߦ͏ͱɼҼՌϞσϧΛਖ਼͘͠ਪఆ͢Δ͜ͱʹ (3.6)
ޭ͢Δ͚ΕͲɼݱ࣮ͷ໘Ͱɼͯ͢ͷઆ໌มΛऔΓ্͍͛ͯΔ͜ͱอূͰ͖ͳ͍ͷͰ͋ Δ͔ΒɼճؼϞσϧΛҼՌϞσϧͱղऍ͢Δ͜ͱʹϦεΫ͕͏ɽͦͷҙຯͰɼճؼੳɼઆ ໌มʹΑΔతมͷઆ໌ຢ༧ଌʹ͍͍ͯΔΑ͏ʹࢥ͑Δ͕ɼҼՌϞσϧͷ୳ڀʹར༻͢ Δ͜ͱਪͰ͖ͳ͍ɽ ·ͨɼճؼੳʹΑΔ݁ՌΛతมʹର͢Δઆ໌༧ଌʹ͏߹ʹɼݱʹ͓͚Δઆ໌ มؒͷ૬͕ؔؔɼखݩʹ͋Δղੳ༻ͷσʔλ͕࣋ͭ૬ؔؔͱಉ͡Α͏ͳʹͳ͍ͬͯΔ͔ Ͳ͏͔Λҙਂ͘ߟ͓ͯ͘͠ඞཁ͕͋Δɽݴ͍͑Δͱɼ͜͜ͰऔΓ্͛ͨΑ͏ʹઆ໌มؒ ʹڧͷ૬͕ؔؔ͋Δ߹ͳΒɼಘΒΕͨճؼϞσϧཱ͕͍ͯ͠Δͷɼʮݱʹ͓͚Δม ؒͷ૬͕ؔؔղੳʹ༻͍ͨσʔλͷ࣋ͭ૬ؔؔʹ͍͜͠ͱ」ཱ͕͢Δ্ͰͷͳͷͰɼ ࣮ࡍͷԠ༻ʹͨͬͯมؒͷ૬ؔؔΛϑΥϩʔ͓ͯ͘͠ඞཁ͕͋Δɽ 3.2 ྫ 2(આ໌มͷ૬͕ؔऑ͍߹) ͜͜Ͱɼ૬ؔߦྻ͕ද 4Ͱ༩͑ΒΕΔมؒͷ૬͕ؔۃʹେ͖͘ͳ͍σʔλͷ߹ Λߟ͑Δ (ද 5ࢀর)ɽ ද 4૬ؔߦྻ x1 x2 x3 y x1 1.000 0.500 0.554 0.771 x2 0.500 1.000 0.477 0.762 x3 0.554 0.477 1.000 0.900 y 0.771 0.762 0.900 1.000 ද5 σʔλ No. x1 x2 x3 y 1 1.00 1.00 0.97 5.91 2 2.00 4.96 0.98 14.86 3 3.00 3.99 0.99 13.95 4 4.00 5.96 3.65 26.87 5 5.00 1.00 0.00 7.00 6 6.00 3.00 1.01 15.03 7 7.00 4.01 6.36 34.10 8 8.00 6.01 1.03 23.11 9 9.00 5.03 4.34 32.08 10 10.00 6.04 4.48 35.52 ͜ͷද 5ʹର͢Δ༷ʑͳճؼϞσϧʹΑΔճؼੳΛߦ͏ͱɼද ͷ݁ՌΛಘΔɽҎԼͰɼ ͦΕΒͷ݁Ռʹ͍ͭͯߟ͢Δɽ
ද6ੳ݁Ռ ˆ β0 βˆ1 βˆ2 βˆ3 SR φR Se φe Ve F0 P (F0 < F ) x1 5.42 2.80 - - 648.34 1 442.28 8 55.29 11.73 0.01 x2 2.85 - 4.39 - 634.01 1 456.61 8 57.08 11.11 0.01 x3 9.79 - - 6.64 882.76 1 207.86 8 25.98 33.98 0.00 x1, x2 -1.41 1.89 2.89 - 855.06 2 235.56 7 33.65 12.70 0.00 x1, x3 4.61 1.43 - 3.52 999.44 2 91.18 7 13.03 38.36 0.00 x2, x3 2.14 - 2.48 3.58 1039.39 2 51.23 7 7.32 71.01 0.00 x1, x2, x3 0.00 1.00 2.00 3.00 1090.62 3 0.00 6 0.00 - -3.2.1 ୯ճؼੳ ୯ճؼੳʹ͓͚ΔճؼͱҼՌϞσϧͷͷဃͷେ͖͞ɼྫ1ʹ͓͚Δဃͱൺ ͯখ͘͞ͳ͍ͬͯΔ͕ɼճؼ͕ࣜҼՌϞσϧͷਪఆͰ͋Δͱ͍͏ʹେ͖ͳဃ͕͋Δɽ·ͨɼ ͜ͷ߹ྫ1ʹ͓͚Δͱಉ༷ʹɼऔΓ্͛ͨઆ໌ม͕എޙʹϞσϧʹऔΓࠐ·Εͳ͔ͬͨม ͷޮՌΛห͠Α͏ͱ͍ͯ͠ΔͨΊɼͦΕΒͷد༩R2(x1) = 0.594, R2(x2) = 0.581, R2(x3) = 0.809͕ޠΔΑ͏ʹɼͦΕͧΕͷճؼϞσϧ౷ܭతʹ༗ҙͱͳ͍ͬͯΔɽ 3.2.2 มબ ͯ͢ͷมΛऔΓೖΕͨͱ͖ͷճؼҼՌϞσϧͷͱҰக͍ͯ͠Δɽޡࠩมಈ͕ Se(x1, x2, x3) = 0.00Ͱ͋ΔͨΊ࣮ߦͰ͖ͳ͍͕ɼͦ͏Ͱͳ͍߹ʹมબΛߦ͏ͱมx2, x3 Λબ͢Δ͔͠Εͳ͍ɽ͢ͳΘͪɼมબʹΑͬͯಘΒΕΔճؼϞσϧΛ࠾͢ΕɼͦΕ ҼՌϞσϧͱҧͬͨϞσϧΛ࠾༻ͨ͜͠ͱʹͳΔ͔͠Εͳ͍ɽ 3.3 ྫ 3(આ໌มʹ૬͕ؔͳ͍߹) ͜͜Ͱɼද7ͷ૬ؔߦྻ͕ࣔ͢Α͏ʹɼઆ໌มؒʹ૬͕ؔͳ͍߹ͷσʔλΛߟ͑Δʢද 8ࢀরʣɽ ද7 ૬ؔߦྻ x1 x2 x3 y x1 1.000 0.000 0.000 0.234 x2 0.000 1.000 0.000 0.355 x3 0.000 0.000 1.000 0.905 y 0.234 0.355 0.905 1.000
ද8 σʔλ No. x1 x2 x3 y 1 1.00 0.95 8.68 28.94 2 2.00 5.62 2.00 19.24 3 3.00 4.43 2.00 17.86 4 4.00 7.64 5.24 35.00 5 5.00 0.99 0.07 7.19 6 6.00 3.03 0.66 14.04 7 7.00 4.24 11.55 50.13 8 8.00 5.69 0.10 19.68 9 9.00 1.03 3.42 21.32 10 10.00 4.73 6.46 38.84 ͜ͷද8ʹର͢Δ༷ʑͳճؼϞσϧʹΑΔճؼੳΛߦ͏ͱɼද9ͷ݁ՌΛಘΔɽҎԼͰɼͦ ΕΒͷ݁Ռʹ͍ͭͯߟ͢Δɽ ද 9 ੳ݁Ռ ˆ β0 βˆ1 βˆ2 βˆ3 SR φR See φe Ve F0 P (F0 < F ) x1 19.72 1.00 - - 82.65 1 1418.75 8 177.34 0.47 0.514 x2 17.56 - 2.00 - 189.22 1 1312.18 8 164.02 1.15 0.314 x3 13.17 - - 3.00 1229.41 1 271.99 8 34.00 36.16 0.000 x1, x2 12.06 1.00 2.00 - 271.66 2 1229.74 7 175.68 0.77 0.497 x1, x3 7.67 1.00 - 3.00 1311.97 2 189.43 7 27.06 24.24 0.001 x2, x3 5.50 - 2.00 3.00 1418.90 2 82.50 7 11.79 60.20 0.000 x1, x2, x3 0.00 1.00 2.00 3.00 1501.40 3 0.00 6 0.00 - -ͯ͢ͷճؼϞσϧʹ͓͚ΔճؼҼՌϞσϧʹҰக͍ͯ͠Δɽ͢ͳΘͪɼઆ໌มؒͷ ૬͕ؔͳ͚ΕɼճؼϞσϧબͨ͠มͱతมͷҼՌϞσϧΛਖ਼͘͠ਪఆ͢Δ͜ͱʹ ޭ͍ͯ͠Δɽಛʹɼͯ͢ͷઆ໌ม͕औΓࠐ·Εͨঢ়ଶͰҼՌϞσϧΛશʹ෮ݩͰ͖͍ͯΔɽ ͔͠͠ɼ࣮ݧܭը๏ͷΑ͏ʹɼઆ໌มͷΛܭըతʹม͑ͳ͍ݶΓɼ͜Ε΄Ͳ͖Ε͍ʹ૬ؔ θϩͷঢ়ଶʢަ͍ͯ͠Δঢ়ଶʣ͕ಘΒΕΔ͜ͱߟ͑ΒΕͳ͍ɽಛʹɼճؼੳ͕ద༻͞ΕΔ Ͱ͋Ζ͏ʮ͋Γ߹ΘͤͷσʔλʯʮৗཧهͷσʔλʯΛ༻͍ΔΑ͏ͳ໘Ͱɼ͜ͷΑ͏ ͳঢ়ଶΛظ͢Δ͜ͱͰ͖ͳ͍Ͱ͋Ζ͏ɽͨͩɼઆ໌มؒͷ૬͕ؔখ͍͜͞ͱΛ֬ೝͰ͖Ε ɼಘΒΕͨճؼࣜΛ༻͍ͯҼՌϞσϧͷ୳ࡧΛߦ͏͜ͱັྗతʹͳΔͱ͍͑Δɽ 3.4 ྫ 4ʢ૬ؔͷͳ͍ม͕ѱ͞Λ͢Δ߹) ͜͜Ͱɼઆ໌มx1, x2, x3ͷ૬ؔྫ3ͱಉ༷ʹθϩͰ͋Δ͕ɼද10ͷ૬ؔߦྻ ͕ࣔ͢Α͏ʹɼͦΕΒͷઆ໌มͱ૬ؔΛ࣋ͪͳ͕ΒҼՌϞσϧʹؚ·Εͳ͍֎෦มx4͕ࠞ ࡏ͍ͯ͠Δ߹Λߟ͑Δ(ද11 ࢀরʣɽ
ද10 ૬ؔߦྻ x1 x2 x3 x4 y x1 1.000 0.000 0.000 0.347 0.234 x2 0.000 1.000 0.000 0.344 0.355 x3 0.000 0.000 1.000 0.402 0.905 x4 0.347 0.344 0.402 1.000 0.567 y 0.234 0.355 0.905 0.567 1.000 ද11 σʔλ No. x1 x2 x3 x4 y 1 1.000 0.950 8.680 3.29 28.94 2 2.000 5.620 2.000 4.10 19.24 3 3.000 4.430 2.000 7.41 17.86 4 4.000 7.640 5.240 3.02 35.00 5 5.000 0.990 0.070 0.85 7.19 6 6.000 3.030 0.660 1.79 14.04 7 7.000 4.240 11.550 6.56 50.13 8 8.000 5.690 0.100 3.88 19.68 9 9.000 1.030 3.420 1.95 21.32 10 10.000 4.730 6.460 13.41 38.84 ͜ͷද 11ʹର͢Δ༷ʑͳճؼϞσϧʹΑΔճؼੳΛߦ͏ͱɼද ͷ݁ՌΛಘΔɽҎԼͰ ɼͦΕΒͷ݁Ռʹ͍ͭͯߟ͢Δɽ
͜ͷྫʹ͓͚ΔҼՌϞσϧɼ͜Ε·Ͱͷྫͱಉ͡ yi= xi1+ 2xi2+ 3xi3+ ui, (i = 1, 2, . . . , n) Ͱ͋Δ͕ɼҼՌϞσϧࣜʹೖ͍ͬͯͳ͍มx4ΛճؼϞσϧʹࠞೖ్ͤͨ͞ɼ(x1, x4)ɼ(x2, x4)ɼ (x3, x4)ɼ͋Δ͍(x1, x2, x4)ɼ(x1, x3, x4)ɼ(x2, x3, x4)ʹ͓͚Δճؼࣜͷx1, x2, x3ʹର͢Δ ͕ҼՌϞσϧͷ͔ΒมԽ͍ͯ͠Δɽ ͦΕɼมx4͕ҼՌϞσϧͷ্ͰແؔͳมͰ͋ͬͨͱͯ͠ɼͦΕΛઆ໌มͷީิ ʹΈೖΕ్ͨɼͻͱͭͷઆ໌มͱͯࣗ͠ݾओு͠ɼx4ͱ૬ؔͷ͋Δมx1, x2, x3ʹΑΔม ಈͷҰ෦͋Δ͍શ෦Λɼ͔͋ͨx4͕ݪҼͰ͋Δ͔ͷΑ͏ͳৼΔ͍Λ͢ΔͨΊͰ͋Δɽ͜ͷ ͜ͱɼճؼੳ͕తมͷมಈΛ࠷͏·͘આ໌Ͱ͖ΔϞσϧΛ୳ࡧ͍ͯ͠ΔҎ্ɼආ͚Δ ͜ͱͷͰ͖ͳ໋͍॓Ͱ͋Δɽ 3.5 ྫ 5(આ໌มؒʹଟॏڞઢੑͷ͋Δ߹) ͜͜Ͱɼͦͷ૬ؔߦྻ͕ද13Ͱ༩͑ΒΔσʔλʢද14ʣΛߟ͑Δɽ ද13 ૬ؔߦྻ x1 x2 x3 y x1 1.000 0.500 -0.923 -0.973 x2 0.500 1.000 -0.794 -0.686 x3 -0.923 -0.794 1.000 0.987 ͈ -0.973 -0.686 0.987 1.000 ˆ β0 β1ˆ β2ˆ β3ˆ β4ˆ SR φR Se φe Ve F0 P (F0< F ) x1 19.72 1.00 - - - 82.65 1 1418.75 8 177.34 0.47 0.514 x2 17.56 - 2.00 - - 189.22 1 1312.18 8 164.02 1.15 0.314 x3 13.17 - - 3.00 - 1229.41 1 271.99 8 34.00 36.16 0.000 x4 16.06 - 1.98 483.41 1 1017.99 8 127.25 3.80 0.087 x1, x2 12.06 1.00 2.00 - - 271.66 2 1229.74 7 175.68 0.77 0.497 x1, x3 7.67 1.00 - 3.00 - 1311.97 2 189.43 7 27.06 24.24 0.001 x1, x4 15.29 0.18 - - 1.93 485.94 2 1015.46 7 145.07 1.67 0.254 x2, x3 5.5 - 2.00 3.00 - 1418.90 2 82.50 7 11.79 60.20 0.000 x2, x4 13.15 - 1.02 - 1.76 527.13 2 974.27 7 139.18 1.89 0.220 x3, x4 10.56 - - 2.68 0.85 1303.49 2 197.91 7 28.273 23.05 0.001 x1, x2, x3 0.00 1.00 2.00 3.00 - 1501.40 3 0.00 6 0.00 - -x1, x2, x4 11.77 0.3 1.07 - 1.67 4.00 3 1497.40 6 249.57 0.01 0.999 x1, x3, x4 4.59 1.46 - 3.45 0.00 533.25 3 968.15 6 161.36 1.10 0.419 x2, x3, x4 5.12 - 1.78 2.85 39.00 1432.67 3 68.73 6 11.46 41.69 0.000 x1, x2, x3, x4 0.00 1.00 2.00 3.00 0.00 1501.40 4 0.00 5 0.00 - -ද12 ੳ݁Ռ
ද14 σʔλ No. x1 x2 x3 y 1 1.00 1.00 18.00 57.00 2 2.00 4.96 13.04 51.04 3 3.00 3.99 13.01 50.01 4 4.00 5.96 10.04 46.04 5 5.00 1.00 14.00 49.00 6 6.00 3.00 11.00 45.00 7 7.00 4.01 8.99 41.99 8 8.00 6.01 5.99 37.99 9 9.00 5.03 5.97 36.97 10 10.00 6.04 3.96 33.96 ද13ʹ͓͚Δઆ໌มؒͷ૬ؔɼද2ʹ͓͚Δ૬ؔ΄Ͳେ͖͘ͳ͍͕ɼޙड़͢Δ Α͏ʹɼ͜ΕΒͷઆ໌มؒʹxi1+ xi2+ xi3= 20ͱ͍͏ઢܗཱ͕͍ؔͯ͠Δɼ͢ͳΘͪ ଟॏڞઢੑ͕ൃੜ͍ͯ͠Δɽ͜ͷ߹ͷ༷ʑͳճؼϞσϧʹΑΔੳ݁ՌΛද15ʹࣔ͢ɽ ද15ੳ݁Ռ ˆ β0 βˆ1 βˆ2 βˆ3 SR φR Se φe Ve F0 P (F0< F ) x1 57.64 -2.32 - - 442.43 1 24.70 8 3.09 143.30 0.00 x2 55.49 - -2.58 - 219.57 1 247.56 8 30.95 7.10 0.03 x3 27.76 - - 1.65 454.96 1 12.17 8 1.52 299.07 0.00 x1, x2 60.00 -2.00 -1.00 - 467.13 2 0.00 7 0.00 ෆఆ ෆఆ x1, x3 40.00 -1.00 - 1.00 467.13 2 0.00 7 0.00 ෆఆ ෆఆ x2, x3 20.00 - 1.00 2.00 467.13 2 0.00 7 0.00 ෆఆ ෆఆ x1, x2, x3 ෆఆ ෆఆ ෆఆ ෆఆ ෆఆ 3 ෆఆ 6 ෆఆ ෆఆ ෆఆ ද 15ʹ͓͍ͯɼมͷ (x1,x2)ɼ(x1,x3)ɼ(x2,x3)ͷ͍ͣΕతม yͷͯ͢ͷมಈΛ આ໌͍ͯ͠Δɽ͜͜Ͱɼม૿ݮ๏ʹΑΔมબΛߦ͏ͱ (x2,x3)͕બ͞ΕΔ͕ɼ(x1,x2)͋ Δ͍ (x1,x3)ʹΑͬͯɼ্ड़ͷͱ͓Γม yͷมಈΛશʹઆ໌͍ͯ͠Δ͚ͩͰͳ͘ɼม x3ͷΈʹΑΔ୯ճؼੳΛߦͬͯɼͦͷࣗ༝ௐࡁد༩ R∗2=0.971େ͖ͳͰ͋Δ͔ Βɼޡࠩͷ͋ΓΑ͏ʹΑͬͯɼ͜ͷ୯ճؼੳͷ݁ՌͰຬ͢Δ͔͠Εͳ͍ɽ͢ͳΘͪɼ୯ ճؼੳͷ݁Ռɼyi=13.71+3xi3ΛҼՌϞσϧͱͯ͠࠾༻ͯ͠͠·͏͔͠Εͳ͍ɽ ͳ͓ɼม x1,x2,x3ͷภࠩੵɾฏํߦྻͷ࠷খݻ༗ 0Ͱ͋Δ͔ΒɼϥϯΫམ͍ͪͯ͠ ΔͨΊʹٯߦྻଘࡏ͠ͳ͍͚ΕͲɼܭࢉޡ͕ࠩӨڹͯ͠ɼද 15Ͱม x1,x2,x3ʹΑΔճ ؼࣜΛܭࢉͯ͘͠ΔϦεΫഉআͰ͖ͳ͍ɽ ͋Γ߹ΘͤͷσʔλৗཧهσʔλͷΑ͏ͳσʔλΛ׆༻͍ͯ͠Δ߹ɼྫ 4ʹ͓ ͚Δઆ໌มؒͷ૬ؔθϩ͕ى͜Γʹ͍͘ͱಉ༷ʹɼ͜ͷྫ 5ͷΑ͏ʹશͳଟॏڞઢੑͷ ൃੜ͢ΔՄೳੑ͍ɽ͜ͷྫͰޡࠩ ui= 0ͱ͍ͯ͠ΔͨΊɼޡࠩมಈ͕ 0ͱͳ͍ͬͯ Δ͚ΕͲɼޡࠩͷଘࡏ͢Δݱ࣮ͷ໘Ͱɼۃʹେ͖ͳد༩͕ಘΒΕΔ্ʹɼͦΕͧΕͷม
ͷΈ߹Θͤʹ͓͚Δد༩͕ҟͳͬͯ͘ΔͨΊɼదͳมͷΛબ͢Δ݁ՌʹͳΔɽ͠ ͔͠ɼͦͷ߹Ͱ͋ͬͯɼಘΒΕͨղੳ݁ՌΛతมͷ༧ଌઆ໌ʹ༻͍ΔݶΓɼద༻໘ ʹ͓͚Δઆ໌มؒͷ૬͕ؔղੳʹ༻͍ͨσʔλͷ૬ؔߏͱಉ༷Ͱ͋Δ͜ͱΛ֬ೝͰ͖Δ ͷͰ͋Εɼબ͞Εͨճؼࣜ༗ޮͳͷͰ͋Δɽ 3.6 ྫ 6(આ໌มؒʹڧ͍૬ؔ͋Δ͕ɼଟॏڞઢੑൃੜ͍ͯ͠ͳ͍߹) ͜͜Ͱɼઆ໌มx1, x2, x3ʹؔ͢Δ૬ؔߏྫ1ͱಉ͡Ͱ͋Δ͕ɼͦΕΒͱڧ͍૬ؔ ؔΛ࣋ͪͳ͕ΒҼՌϞσϧͱແؔͳୈ4ͷมx4͕ଘࡏ͢Δ͚ΕͲɼྫ5ͷΑ͏ͳ ଟॏڞઢੑൃੜ͍ͯ͠ͳ͍߹Λߟ͑Δ(ද16ͱද17Λࢀরʣɽ ද16 σʔλ No. x1 x2 x3 x4 y 1 1.000 0.000 0.980 0.94 3.94 2 2.000 1.560 0.990 1.72 8.09 3 3.000 0.930 0.980 1.84 7.80 4 4.000 3.540 1.920 2.6 16.84 5 5.000 5.190 2.020 2.2 21.44 6 6.000 4.910 1.910 2.7 21.55 7 7.000 8.450 3.790 6.52 35.27 8 8.000 4.660 2.500 3.66 24.82 9 9.000 13.280 4.010 6.89 47.59 10 10.000 16.450 3.540 8.99 53.52 ද17 ૬ؔߦྻ x1 x2 x3 x4 y x1 1.000 0.900 0.900 0.885 0.943 x2 0.900 1.000 0.900 0.962 0.989 x3 0.900 0.900 1.000 0.923 0.946 x4 0.885 0.962 0.923 1.000 0.967 y 0.943 0.989 0.946 0.967 1.000 ද16ͷσʔλʹର͢Δ༷ʑͳͳճؼϞσϧʹΑΔੳ݁ՌΛද18ʹࣔ͢ɽ
ˆ β0 β1ˆ β2ˆ β3ˆ β4ˆ SR φR Se φe Ve F0 P (F0< F ) x1 19.72 1.00 - - - 82.65 1 2491.67 8 311.46 0.27 0.620 x2 17.56 - 2.00 - - 189.22 1 2385.10 8 298.14 0.63 0.449 x3 13.17 - - 3.00 - 1229.41 1 1344.91 8 168.11 7.31 0.027 x4 1.16 - 6.02 2374.32 1 200.00 8 25.00 94.97 0.000 x1, x2 12.06 1.00 2.00 - - 271.66 2 2302.66 7 328.95 0.41 0.677 x1, x3 7.67 1.00 - 3.00 - 1311.97 2 1262.35 7 180.34 3.64 0.083 x1, x4 -2.69 2.24 - - 3.80 2463.69 2 110.63 7 15.80 77.94 0.000 x2, x3 5.5 - 2.00 3.00 - 1418.9 2 1155.42 7 165.06 4.30 0.061 x2, x4 4.61 - 2.48 - 1.29 2491.37 2 82.95 7 11.85 105.12 0.000 x3, x4 -2.68 - - 5.17 3.96 2423.14 2 151.18 7 21.60 56.10 0.000 x1, x2, x3 0.00 1.00 2.00 3.00 - 1501.4 3 1072.92 6 178.82 - -x1, x2, x4 1.31 1.47 1.93 - 0.87 2524.53 3 49.79 6 8.30 101.41 0.000 x1, x3, x4 -3.88 1.88 - 2.45 3.18 2472.27 3 102.05 6 17.01 48.45 0.000 , x2, x3, x4 1.18 2.33 - 4.34 -0.17 2525.24 3 49.08 6 8.18 102.90 0.000 x1, x2, x3, x4 0.00 1.00 2.00 3.00 0.00 2574.32 4 0.00 5 0.00 - -ओཁͳͯ͢ͷઆ໌มx1, x2, x3ΛऔΓࠐΜͰ͍Εɼແؔͳઆ໌มx4ΛؚΉઆ໌ม x1, x2, x3, x4ʹΑΔճؼੳΛߦ͏͜ͱ͍Ͱมx4ෆཁͰ͋Δ͜ͱ͕ݕग़͠ɼಘΒΕͨճؼࣜ ʹ͓͚Δมx1, x2, x3ͷճؼҼՌϞσϧͱҰக͍ͯ͠Δɽ͔͠͠ɼ࣮ࡍʹɼ͜ͷྫ ʹ͓͚Δx1, x2, x3, x4ʹର͢Δ݁ՌͷΑ͏ͳͷ͕ಘΒΕͳ͍ݶΓɼओཁͳઆ໌มͷͯ͢Λ औΓೖΕͨͱ͍͏อূಘΒΕͳ͍ͨΊɼճؼ͕ࣜҼՌϞσϧΛਪఆ͍ͯ͠Δ͔Ͳ͏͔͔Β ͳ͍ɽ͕ͨͬͯ͠ɼճؼੳͷ݁ՌɼతมͷมಈΛઆ໌͋Δ͍༧ଌ͢Δ͜ͱ͕Ͱ͖͍ͯ Δͱओு͍ͯ͠Δʹ͗͢ͳ͍ɽ ͜ͷ߹ʹม૿ݮ๏ͰมબΛߦ͏ͱɼมͷऔࣺʹ༻͍ΔS(βi) (i = 1, 2, 3, 4)͕ S(β1) = SR(x1, x2, x3, x4) − SR(x2, x3, x4) = Se(x2, x3, x4) = 11.38 S(β2) = SR(x1, x2, x3, x4) − SR(x1, x3, x4) = Se(x1, x3, x4) = 28.38 S(β3) = SR(x1, x2, x3, x4) − SR(x1, x2, x4) = Se(x1, x2, x4) = 18.87 S(β4) = SR(x1, x2, x3, x4) − SR(x1, x2, x3) = Se(x1, x2, x3) = 0.00 ͱͳ͍ͬͯΔͨΊɼม(x2, x3)ͷબ͞ΕΔՄೳੑ͕͋Δɽ͠ɼͦ͏ͳΕಘΒΕͨճؼࣜ ҼՌϞσϧΛද͢ͷʹͳ͍ͬͯͳ͍͜ͱʹͳΔɽ
4
ྫʹΈΔճؼੳద༻্ͷҙ
͜Ε·ͰͷྫͰड़͖ͯͨΑ͏ʹɼճؼੳΛద༻͢Δͱ͖ɼҎԼͷΑ͏ͳ͜ͱΛҙ ͓ͯ͘͠ඞཁ͕͋Δɽ 13 ද18ੳ݁Ռ(A) આ໌มʹɼओཁͳӨڹྗͷ͋ΔมΛऔΓ͞ͳ͍Α͏ʹɼ·ͨɼແ༻ͳมऔΓೖ Εͳ͍Α͏ʹेͳٕज़తݕ౼Λߦ͏ඞཁ͕͋Δɽͦͷٕज़తݕ౼ͷաఔͰม૿ݮ๏ͳͲ ʹΑΔมબΛߦ͏͜ͱ͋ͬͯɼ͋͘·Ͱٕज़తݕ౼Λ༏ઌ͢Δ͖Ͱ͋ͬͯɼม બٕज़తݕ౼ͷͨΊͷͻͱͭͷखஈʹ͗͢ͳ͍ͱཧղ͓ͯ͘͠ඞཁ͕͋Δɽ (B) આ໌มؒͷ૬͕ؔେ͖͍߹ɼྫ1ͷS(βi)͕খ͘͞ͳͬͨ͜ͱ͕ࣔ͢Α͏ʹɼ ภճؼʹର͢Δݕఆͷݕग़ྗ੬ऑʹͳΔɽ ͦͷ߹ʹɼ࠷ॳͷσʔλn͕େ͖ ࣮ͯ͘ޮతͳσʔλখ͘͞ͳ͍ͯΔ͜ͱʹҙͯ͠ɼ૬ྔͷσʔλΛੵΈ্͓͛ͯ ͘ඞཁ͕͋Δɽ (C) ճؼੳʹΑΔճؼࣜతมʹର͢Δઆ໌༧ଌʹ༻͍Δ͜ͱͰ͖Δ͕ɼۃΊͯಛघ ͳ߹Λྫ֎ͱͯ͠ɼҼՌϞσϧͷਪఆʹ༻͍Δ͜ͱͰ͖ͳ͍ɽҼՌϞσϧͷਪఆΛߦ͏ ͨΊʹɼٕज़తݕ౼ʹج͍ͮͯબ͞ΕͨओཁͳมΛ࣮ݧҼࢠͱͨ͠ަදʹΑΔҰ෦ ࣮ࢪ๏ଟݩஔࢄੳ๏ͳͲͷ࣮ݧܭը๏ʹΑΔղੳΛߦ͏͖Ͱ͋Δɽ Ҏ্ͷΑ͏ͳ͜ͱʹҙ͢ΔͳΒɼ ୯ճؼੳͷ߹ʹɼʹͳΒͳ͍͕ɼॏճؼੳʹ͓͍ͯઆ໌มؒͷଟॏڞઢੑ ʹΑͬͯਪఆࣜͷٻ·Βͳ͍߹͕͋Δʢྫ 5ʣɽͨͩ͠ɼ͜ͷΑ͏ͳ߹ʹɼऔΓೖ ΕΔมΛআ͢Δ͜ͱʹΑͬͯճආͰ͖Δ߹͕͋Δ͕ɼͦͷ݁ՌΛద༻͢Δࡍʹɼղੳ ࣌ͷมؒͷ૬ؔߏ͕อ࣋͞Ε͍ͯΔ͜ͱΛ֬ೝ͢Δඞཁ͕͋Δɽ ࠩͱͯ͠ݟ্͔͚ͷޡ͕ࠩݱΕΔʢྫ 1ͱ 2ʣɽ͜ͷࡍɼ͕ࠩθϩʹͳΔͷɼऔ ΓࠐΜͩมͰతมͷͯ͢ͷมಈΛઆ໌ͭͨ͘͠͠ͱ͖で͋Δ͕ɼ͕ࠩθϩʹͳͬ ͨͱ͖ʹӨڹྗͷ͋Δͯ͢ͷม͕औΓೖΕΒΕ͍ͯΔͱ͍͏อূͳ͍ɽ Өڹྗͷ͋Δม͕औΓࠐ·Εͳ͔ͬͨͱ͖ɼӨڹྗͷͳ͍ม͕ӨڹྗΛ࣋ͭมͱͯ͠ ࣗݾओுͯ͘͠Δʢྫ 4ʣɽͦͷϞσϧʹऔΓࠐ·Εͳ͔ͬͨӨڹͷ͋Δมͱ モσϧʹऔΓࠐΜͩӨڹྗͷͳ͍มͷ૬͕ؔڧ͍΄ͲݦஶʹͳΔɽ ճؼࣜɼݱͷഎޙʹજΉҼՌϞσϧͷਪఆࣜͱͳΔͷɼҎԼʹࣔ͢ۃΊͯݶΒΕͨ ߹Ͱ͋Δɽ B ճؼࣜʹऔΓೖΕͳ͔ͬͨӨڹྗΛ࣋ͭมͱऔΓೖΕͨมؒͷ૬͕ؔθϩͷ ͱ͖ɽ͜͜ͰɼऔΓೖΕͳ͔ͬͨม͕ӨڹྗΛ͔࣋ͭͲ͏͔σʔλ͔ΒஅͰ͖ ͳ͍ʢྫ 3ʣɽ C ӨڹྗΛ࣋ͭมͷ͕ͯ͢ճؼࣜʹऔΓࠐ·Ε͍ͯΔͱ͖ɽͨͩ͠ɼݱ࣮ͷ໘Ͱ ɼͯ͢ͷม͕औΓೖΕΒΕ͔ͨͲ͏͔ɼσʔλ্͔ΒΘ͔Βͳ͍͚ͩͰͳ ͘ɼӨڹྗΛ࣋ͨͳ͍ม͕ओཁมʹͱͬͯସΘΔՄೳੑ͋Δʢྫ 1ͱ ྫ4ʣɽ
͋Δ͍ٯਪఆͳͲʹؔΘΔཧ໘͔ΒͷٞΛ͍ͯ͠ͳ͍ɽճؼੳɼಛʹॏճؼੳ࣭ ཧͷʹ͓͍ͯॏๅ͞ΕΔख๏Ͱ͋Γɼతมͷઆ໌มʹΑΔઆ໌༧ଌʹ༗ޮͳํ๏ Ͱ͋Δ͜ͱ༳Δ͗ͳ͍ͷͰ͋Δ͕ɼͦͷద༻ʹͨͬͯɼ༷ʑͳʹҙΛ͏ඞཁ͕͋ ΔɽಛʹɼճؼਪఆࣜҼՌϞσϧͷਪఆࣜΛఏڙ͍ͯ͠ͳ͍͜ͱʹҙ͢Δඞཁ͕͋Δɽ
ࢀߟจݙ
<> ൧௩ӻޭɼழݪਖ਼कɼؠ࡚ग़ (2019)ɼճؼੳ๏ɼ࣭ཧϕʔγοΫίʔεςΩε τୈ 10ষɼຊՊֶٕज़࿈ໍɽ <> ԞҰɼٱถۉɼ๕լහɼ٢ᖒਖ਼ (1985)ɼɹʢվగ൛ʣଟมྔղੳ๏ɼՊٕ࿈ग़൛ ࣾ. <> ٱอୡ (2019)ɼݱཧ౷ܭֶͷجૅɼڞཱग़൛ɽ <> C.R.Rao(1973),LinearStatisticalInferenceandItsApplication,JoIn8ilFy4POT. <> Ӭా༃ (2019)ɼσϛϯάຊडه೦ߨԋɼຊՊֶٕज़࿈ໍɽ <> Ӭా༃ɼ౩ۙխ (2016)ɼଟมྔղੳ๏ೖɼαΠΤϯεࣾɽ5
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͜͜ͰɼճؼੳΛద༻͢Δࡍͷຊ࣭తͳΛ໌Β͔ʹ͢ΔͨΊɼҼՌϞσϧʹରͯ͠ ޡࠩΛؚ·ͳ͍σʔλΛ࡞͢Δํ๏Λ࠾༻ͨ͠ɽͦͷͨΊɼճؼʹؔ͢Δݕఆ۠ؒਪఆThe Use of FAB-MS to Study Characters of Silicic Acids
and Silicates in Sodium Silicate Solution
Masa-aki MUROYA
*1, Mihoyo FUJITAKE
*2, Kazuhiko YAGUCHI
*3,
Hiroshi YAMADA
*4and Hitoshi KOSHIMIZU
*4Abstract
Fast atom bombardment mass spectrometry(FAB-MS) has been used to study the structural forms and characters of the silicic acid and metal adduct-silicic acid species in the sodium silicate solution. Many signals were observed in the range of m/z 0-1000, and those signals were analyzed by an electrochemical and colloidal point of views used 28Si. Many characteristic species, which were traced on the presences of about 60 kinds of the silicic acids and metals adduct silicic acids, were observed by this FAB-MS study. In those species, the hydrated silicic acid monomer, dimer, cyclic tetramer, Ca-adduct silicic acid monomer, and Na-adduct silicic acid trimer(Si3(OH)4O6Na4) have the character of the weak electrolyte like feature. On the other hand, the hydrated silicic acid linear-tetramers, linear- and cyclic-pentamers, linear- and cyclic-hexamers, and Na- and Ca-adducts poly-silicic acids are presented to be the oligomers which are the lyophilic colloids forming the polyhedral geometric structures. The hydrated Si3(OH)4O6Na4 silicic acid trimer played an important role as the inhibitor for the polymerization action by the self-condensation between the active orthosilicic acid monomers. The isotope effect was observed in the silicic acid dimer.
1. Introduction
The sodium silicate solution is particularly interesting, for they exist in the mixture system in NaHSiO3 and colloidal silica[1-4]. The silica particles in solution system are in the form of particles from 0.8 to 2.0nm in diameter which corresponded to monomer and oligomers, and the surface of the particles is saturated with adsorbed sodium ions[5]. These particles are in equilibrium with smaller monomeric and oligomeric silicate ions which are also combined with sodium ions[5]. The observation result of 29Si NMR revealed that the building units as polymers for liquid silicates was imagined from that three kinds of broad resonance lines and the singlet sharp resonance lines which are the end-group(E, Q1), middle-group(M, Q2), trifunctional branching(T, Q3), tetrafunctional branching(Q, Q4), and neso-group(N, Q0)[6]. From a different point of view described above, F. Kohlrausch and H. Ukihashi concluded that a sodium silicate solution is constituted of hydrated colloid which has highly viscosity and emulsion properties from the result of electric conductivity measurement[7,8]. In addition, it was reported that four kinds of hydrated Na+, H
3SiO4, H2SiO42, and OH species were contributed to electric conductions to be charge carriers[9,10], and the radii of ion sizes in these
*1Professor Emeritus of Osaka Electro-Communication University, 7-6-8 Higashi-tokiwadai, Toyono-Chyo, Osaka, Japan, 563-0103.
*2Osaka University of Pharmaceutical Sciences, 4-20-1, Nasahara, Takatsuki, Osaka, Japan, 569-1094. *3Team MIRAI in Fuji-Silysia Chemical, Ltd., 1846, kozohji Kasugai, Ichi, Japan, 487-0013.
*4Osaka Labo. Site, Fuji Chemical, Co. Ltd., 3-2-33, Higashi-noda, miyakojima, Osaka, Japan, 534-0024.
大阪電気通信大学 研究論集 （自然科学編） 第 55 号 大阪電気通信大学 研究論集 （自然科学編） 第 55 号
hydrated silicic acid anions are estimated to 3.6-4.6Å[11]. Furthermore, the presence of colloidal silicic acid in a sodium silicate solution is confirmed by method of light scattering by UV range, and then the molecular weights are estimated to be about 150-1500[12-14]. These studies are result in knowledge of primary importance to a sodium silicate solution, but further details are obscure for the properties and structural forms of the silicic acids and metals adduct silicic acid species.
In recent years, mass spectrometry(MS) was applied to the speculation of species in a solution, and then the silicate oligomers were detected in an aqueous solution[15]. FAB-MS and ESI-MS analyses revealed that the presence of a number of silicic acid anions and complex species consisting of silicic acid were found in some kind of solutions[16,17].
In this work, FAB-MS analysis was directly applied to the sodium silicate solution. The goal of this investigation was to detect smaller silicic acids, colloidal silicic oligomers as building units[6], various silicic acids, and metal adducts-silicic acid and to determine the characteristics of these species via electrochemical analyses and using a colloidal approach. Moreover, the dependency of the signals on the molar ratios is detected, and the suitable inhibition factor and/or the contribution specie for a polymerization of silicic acid are tried to be found. In addition, the detection of isotope effect in silicic acid species is also one of objects in this study[18,19].
2. Experimental procedure
Materials;
Four kinds of the sodium silicate solutions were used as samples in this experiment. The molar ratios of these samples, respectively, are the values of 2.11, 3.19, 3.40, and 3.75, the chemical compositions and containing impurities of these samples were determined by chemical and ICP analysis, and these values were shown in Table 1. These samples were obtained from Fuji Chemical Co. and the samples were used in this experiment without further purification. Glycerol, which is reagent grade, was used to be the matrix for FAB-MS measurement[20,21].Table 1. Chemical compositions and impurities contained in the sodium silicate solutions used in this experiment.
Composition Impurity Concentration (%) Concentration (ppm)
n*1 SiO2 Na2O H2O Ca K Fe Al Mg Ti Zr
2.11 26.03 12.83 61.24 21.2 444.9 38.68 123.5 0.44 42.42 30.03
3.18 29.25 9.49 61.26 12.64 510.8 42.17 116 0.802 49.18 29.18
3.40 25.29 7.66 67.05 10.19 105.5 36.18 123.55 6.52 39.67 8.78
3.75 25.85 7.11 67.04 0.37 128.3 15.21 72.89 2.498 34.95 19.63
*1molar ratio
Experimental procedures;
In the case of FAB-MS measurement in this experiment, the fluidity of the sample was solidified by using glycerol. Using glycerol, we achieved slight dehydration of the sample, but it was not a problem, as confirmed via infrared spectroscopy. FAB-MS experiment was carried out as follows; the sample mixture was fit on the target, the bulb was sealed off after highly evacuated bulb, and the fast atom Xe was irradiated to the sample. Xe was generated from Xe+ ions which were accelerated to 6keV. Ions evolved from the sample on target were analyzed by the spectrometer. FAB-MS was carried out by using theapparatus of JMS-700(2)(JEOL Ltd., Tokyo, Japan) in this experiment[20,21]. Using in this apparatus, a signal was detected to be 2 times of S/N ratios, and this signal intensity corresponded to trace level. In FAB-MS observation, the signals based on negative and positive charges are observed, and then negative charge was mainly detected in this work. And the intensities of signals have been classified for convenience into six categories which are vs(very strong), s(strong), m(medium), w(weak), vw(very weak), and trace, respectively.
3. Results and Discussion
3.1 FAB-MS Spectrum of the sodium silicate solution
FAB-MS spectrum observed at 6keV for the sodium silicate solution prepared at the molar ratio of 3.40 was shown in Fig.1. Many signals appeared at various intensities in the range of m/z from 0 to 1000. The vs-signal appeared in the position of m/z 205. The signal is not assigned to the silicic acid and metals adduct silicic acid anions and is assigned to the dimerized Mg-adduct glycerol anion(C3(OH)2H5OMgOC3H5(OH)O̼).
Fig.1. Signal pattern observed in the sodium silicate solution prepared by the molar ratio at 3.40.
Four kinds of the m-level signals appeared in m/z 133, 173, 225, and 229, respectively. In these signals, 173 and 133, respectively, originated from the silicic acid anion species[17], but 225 and 229 signals, respectively, are not related to the silicic acid anion species. To interpret and assign the signals, the vertical axis in Fig.1 was magnified 2.1 times to the intensity of the m/z 173 signal, which was defined as 100%. The general treatment of data was the same as that described above.
3.2 Origin and generation mechanism of signal
The sodium silicate solution can be regarded as an electrolyte solution, and then an electrochemical approach to such solution system can be considered that such a treatment is reasonable. However, the electrochemical treatment for this solution resulted in problems because of the strong and thick electrolyte solution. However, it is thought that the electrochemical and colloidal treatments are possibly brought to the interpretation of the signal feature.
In order to speculate the generation mechanism of anion from the sodium silicate solution media, a useful model was stipulated to be for example, and the illustrations for the silicic acid monomer(Si(OH)4) were shown in Fig.2. The orthsilicic acid monomer(Fig.2 (a)) which are hydrated to H2O, and such hydrated silicic acid is surrounded by cloudily ionic atmosphere(ionic atmosphere) which consists of H+, H
3O+, and hydrated Na+ marked at symbols Ɣ.In such a system, an electric attraction and repulsion forces are acted to between the hydrated tetrahedral center ion and ionic atmosphere, and the electrolyte like system is kept to be a stable electrochemical equilibrium state. In this system, a potential, ij, at the any point P is a function of only the distance, r, from central ion, as is illustrated in Fig.2(a). In this case, Poisson-Boltzmann equation[22]can be applied to the system, a potential ij at P-point existed in space at a distance r from i-specific ion was as follows;
1/r2ād/drÂ(r2dij/dr)= ȡ/İ
0İi (1) where, ȡ, is charge density in solution. The potential, ij, at P-point;
ij=(zie/4ʌİ0İir)exp(br)§(zie/4ʌİ0İir)(zieb/4ʌİ0İi) (2)
where, zie is charge of the i-specific ion, İ0 and İi are dielectric constants in vacuum and specific dielectric constants in media, respectively. 1/b is Debye length which corresponded to thickness of the ionic atmosphere surrounding i-ion(tetrahedral hydrated Si(OH)4). In the equation (2), the first term in right-side terms is a potential ij at P-point existed in space at a distance r from i-specific ion which is postulated ion, and the second term is a potential formed by counter ionic atmosphere surrounding postulated center ion.
Fig.2. Schematic illustration of generation mechanism in the silicic acid anions by the fast atom Xe bombardment for the silicic acid monomer(a and a’).
When the fast atom Xe is irradiated on such solution system, the hydrated center ion is set off by removing from the ionic atmosphere inside as a result of the disorder in system zone, and then such hydrated center ion is escaped to be anion(Si(OH)3O̼) to outer side of the solution system. If the center ion is the colloidal silicic acid, the ionic atmosphere corresponding to the electric double layer. On the basis of the view, the facility of anion escape is distinctly reflected to the signal intensity. For example, a postulate center ion is alkali metal such as is Na case, the ion is hydrated in silicate solution inside, and the hydration coordinate number is estimated to be about 4±1 for such a cation[23].No signal corresponded to such a specie was detected in the
m/z 23. Probably the hydrated Na specie is very difficult for the escape from the silicate solution because of the
interactionbetween the hydrated Na cation and ionic atmosphere is very strong.In the other case, the intensity
of m/z 173 signal, which is assigned to dimeric silicic acid anion[17], is strong compared with those of the other anion signals. It is suggested that the escape of dimer silicic acid anion is most easy compared with those of the other anions. If the facility of escape depended upon anion size, the intensity of dimer anion is expected to be weak than that of the monomer, but the intensity of dimer anion(m-level) is stronger than that of the monomer(vw-level). In this case, the anion size of hydrated dimer is probably larger than that of hydrated monomer which is estimate to about 3.6-4.6Å[11]. This is one of questions. It has probably been believed that the silicic acid monomer is completely solvated by hydration to solvent water molecule. Four kinds of dipole-dipole interactions formed between four hydroxyls of tetrahedral silicic acid monomer and four solvent water molecules, as shown in Fig.2(a). Then the ionic atmosphere is not easily allowed for the escape of hydrate monomer. On the other hand, the intensity of m/z 173 signal(dimer silicic acid anion(so called hydrate dimer)) is vs-level, and it intensity is suggested that the escape of hydrate dimer is easy compared with that of the hydrate monomer. This means that the value of second term of right side in equation (2) and the solvation affinity to water for the hydrate dimer, are small compared with these of the case in hydrate monomer.
It has become apparent that the facility of escape is generally decided by three kinds of factors which are a density of ionic atmosphere(1/b), a strength of interaction between center silicic acid and ionic atmosphere, and a solvation affinity to water of a silicic acid. One of these factors, the influence of density of ionic atmosphere on the signal intensity can be extracted from already observed data. It is thought that the ionic valency of dimer silicic acid anion probably reflected to signal intensity. In order to confirm of this matter, the relative intensity against as a function of m/z was plotted, and the result was shown by the symbolic mark ی in Fig.3. In this case, if the anions of different ionic valency are occurrence from dissociations of one by one proton which is released by the fast atom Xe bombardment, these signals corresponded to m/z 173, 172, 171, and 170, and various charging dimer anions are generally formed as follows; Si2(OH)6O ̼ Hn+disso. ĺ Si2(OH)lOmn + Hn+, where the univalent(m/z 173) is of n=1, l=5, and m=2. In the cases of n=2, 3, 4, l=4, 3, 2, and m=3, 4, 5, these anions corresponding to those of bivalent Si2(OH)4O32(m/z 172), trivalent Si2(OH)3O43(m/z 171), and tetravalent Si2(OH)2O54(m/z 170), respectively. The intensity value of 100 at m/z 173 steeply decreased with decreasing in m/z, and such a value reached to about 15 at m/z 172, as seen in Fig.3. This phenomenon is indicated that the charge density of univalent anion(m/z 173) is less than those of bivalent(172) and other anions(171, 170).These results suggested that the interaction between the univalent anion and the ionic
atmosphere is fairly weaker than those of the cases in the bivalent and other polyvalent anions. Therefore the univalent anion is easily produced by the fast atom Xe bombardment, and escape of such a uivalent anion is easily than those of the bivalent or other polyvalent anions. In addition to this fact, it is suggested that the
escape to silicic acid anion is dependent on the degrees of ionic atmosphere density(1/b) and/or electrical charging of anions species. The dependency of the anion charges on the signal intensities was also found in the monomer silicic acid anions. As can be seen from these results, the signal reflected the characteristics of the anion, which was assumed to be dependent on the composition.
Fig.3. Influence of charge density on signal intensity for the silicic acid dimer anions.
In order to see validity and reliability of the positions of m/z in signals and those intensities detected at 6keV, the experiment was carried out by 4keV, and it result was shown in symbol Ŷ of Fig.3. Where, the signal intensities at 4keV were relatively corrected to that of 6keV which was referred. The signal positions and those relative intensities observed at 4keV(Ŷ)are almost in agreement with those of 6keV(ی), and the complementarily distribution of data between 6keV and 4keV were obtained by this experiment.
3.3 Signals related to silicic acid anions
The observed signals and assignments related to the silicic acid anions are shown in Table 2.
These anion species were divided into three anion groups: a single bond
(>Si<͐), a double
bond
(O=Si<͐), and alkaline. Signal appeared at m/z 77 and 95,which were assigned to monomeric
silicic acid, specifically, meta- and ortho-silicic acid, respectively. The 77-signal appeared with a
m-level intensity, which signal originated from the
Si(OH)O2Ѹ (O=Si(OH)O) anion,and the anion was
derived from metasilicic acid.
In addition, the 77-signal also appeared by that this signal occurred via dehydration process in Si(OH)3O(m/z 95)H2O ĺ O=Si(OH)Oí(m/z 77)[24], moreover, the 77-signal is also agreement with that of the mass of Al(OH)2Oí anion. Al in such anion contain as impurity in the starting material shown in Table 1. It is not clear that the 77-signal corresponded to either O=Si(OH)Oanion or Al(OH)2Oí anion. If the m/z 77-signal originated to two kinds of anions, the matter is suggested the escape facilities of these anions are analogous to be same order. When 77-signal corresponded to the Al(OH)2O anion, the detection of such anion suggested that the Al(OH)3(gibbsite) is contained in the sample solution. Furthermore, single signal corresponded to two or above anions species will be reported together with results on the other signals which are the m/z 311, 467, 155, 233, and 389, as seen in Table 2.Table 2. Signals(m/z, intensities) corresponded silicic acid anions,and the assignments of signals.
Signal Intensity Anion specie Anion Silicic acid Overlap anion
m/z 6keV
>Si< 䈈, Si䇴O single bond only in siloxan chain
95 13.5 w Si(OH)3O䇴 monome Si(OH)4 㻌㻌㻌㻌㻌㻌㻌㻌䇷
173 100 vs Si2(OH)5O2䇴 dimer Si2(OH)6O 㻌㻌㻌㻌㻌㻌㻌㻌䇷
251 4.2 vw Si3(OH)7O3䇴 trimer Si3(OH)8O2 㻌㻌㻌㻌㻌㻌㻌㻌䇷
329 1.7 vw Si4(OH)9O4䇴 linear tetramer Si4(OH)10O3 㻌㻌㻌㻌㻌㻌㻌㻌䇷
311 17.3 w Si4(OH)7O5䇴 cyclic tetramer Si4(OH)8O4 Si3(OH)7Al(OH)O4ʷ
407 2.0 vw Si5(OH)11O5䇴 linear pentamer Si5(OH)12O4 㻌㻌㻌㻌㻌㻌㻌㻌䇷
389 5.7 vw Si5(OH)9O6䇴 cyclic pentamer Si5(OH)10O5 㻌㻌㻌㻌㻌㻌㻌㻌䇷
485 3.3 vw Si6(OH)13O6䇴 linear hexamer Si6(OH)14O5 㻌㻌㻌㻌㻌㻌㻌㻌䇷
467 2.4 vw Si6(OH)11O7䇴 cyclic hexamer Si 3+2+1 Si6(OH)12O6 Si5(OH)11Al(OH)O6䇴
O=Si< 䈈, Si=O contained double bond in siloxane chain
77 36.3 m Si(OH)O2䌦 monomer Si(OH)2O Al(OH)2Oʷ
155 12.3 w Si2(OH)3O3ʷ dimer Si2(OH)4O2 Si(OH)3Al(OH)O2ʷ
137 4.5 vw Si2(OH)O4ʷ dimer Si2(OH)2O3 䇷
233 9.3 vw Si3(OH)5O4ʷ trimer Si3(OH)6O3 Si2(OH)5Al(OH)O3ʷ
215 21.0 w Si3(OH)3O5ʷ trimer Si3(OH)4O4 䇷
197 9.4 vw Si3(OH)O6ʷ trimer Si3(OH)2O5 䇷
311 17.3 w Si4(OH)7O5ʷ linear tetramer Si4(OH)8O4 Si3(OH)7Al(OH)O4䇴
293 6.3 vw Si4(OH)5O6ʷ linear tetramer Si4(OH)6O5 䇷
275 trace Si4(OH)3O7ʷ linear tetramer Si4(OH)4O6 䇷
257 trace Si4(OH)O8ʷ linear tetramer Si4(OH)2O7 䇷
389 5.7 vw Si5(OH)9O6ʷ linear pentamer Si5(OH)10O5 Si4(OH)9Al(OH)O5ʷ
371 3.2 vw Si5(OH)7O7ʷ linear pentamer Si5(OH)8O6 䇷
353 6.8 vw Si5(OH)5O8ʷ linear pentamer Si5(OH)6O7 䇷
335 5.2 vw Si5(OH)3O9ʷ linear pentamer Si5(OH)4O8 䇷
317 7.0 vw Si5(OH)O10ʷ linear pentamer Si5(OH)2O9 䇷
449 3.4 vw Si6(OH)9O8䇴 cyclic hexamer 3x2 Si6(OH)10O7 䇷
Alkaline
30 0.9 trac Li2O䇴 ʊ ʊ 䇷
62 1.8 vw Na2Oʷ ʊ ʊ 䇷
94 2.9 vw K2O䇴 ʊ ʊ 䇷
In many signals, the interesting signals related to silicic acid anions appeared in the m/z 95(vw), 311(vw), and 329(trace), and these signals are assigned to the orthosilicic acid monomer(95,Si(OH)3Oí), cyclic tetramer(311,Si4(OH)7O5í), and linear tetramer(329,Si4(OH)9O4) anions, respectively. In these anions, the
mass of cyclic tetramer Si4(OH)7O5í anion is also in agreement with that of the linear tetramer O=Si4(OH)7O5 anion. Further, 311-signal also corresponded to Al-adduct silicic acid anion as showing in Table 2. The exact assignment for the 311-signal is undecided yet in any event. However, the detection of 311-signal suggested that these three kinds of anions are produced by the fast atom Xe bombardment. In addition to these tetramer, the m/z 329 signal also corresponded to the mass of linear silicic acid tetramer anion, Si4(OH)9O4.
Compare the two linear anions, the intensity of linear tetramer anion, Si4(OH)9O4(329 trace) is less than that of the linear tetramer anion of O=Si4(OH)7O5(311 vw). It is suggested that the escape of the O=Si4(OH)7O5 anion(311) from the sodium silicate solution is 10 times easy compared with that of the Si4(OH)9O4 anion(m/z 329), by the two kinds of reasons, the O=Si< double bond is existent in the O=Si4(OH)7O5 anion and the solvation affinity to water of O=Si4(OH)8O4 silicic acid is less than that of Si4(OH)10O3 silicic acid. In other example, the intensity of orthosilicic acid anion((Si(OH)3O-), m/z 95) is less than that of metasilicic acid anion((O=Si(OH)O-), m/z 77), therefore, the presence of O=Si< double bond will result in molecular ion and it promotes the anion escape. Fig.4 shows the comparison of anions having double bond with cyclic(symbols Ŷ) and without double bond(䕺). The intensities of cyclic anions(symbols Ŷ㻕 except for the cyclic-hexamer(467-signal), which is single bond only, were generally strong compared with those of the linear anions(symbols 䕺). On the other hand, the intensities of linear anions are smaller values, the escape of such linear anions are not easy by the reason that of the linear anions are governed by strong electrical atmospheres which are electric double layer in colloidal system. If the linear anions may have been the character of oligomers as colloidal mode, the problem will now be discussed from a slightly different point of view, that is, this difficulty makes it necessary to consideration of a colloidal stability in the linear silicic acids based on the applications of DLVO theory and of Schultze-Hardy rule to such a system[25-28]. If the linear silicic acids are present as colloidal substances, the electrical double layer probably formed on surrounding such hydrate silicic acids. Therefore, it is thought that the escape of the linear tetramer sillicic acid anions is fairly difficult than that of cyclic tetramer silicic acid anions. It is presumed that an anion surrounding electric atmosphere is the ionic atmosphere for the cyclic species and is the electric double layer as colloidal mode for the linear species. From this point of view, it was recognized that the silicic acid species, which are the monomer, dimer, and cyclic tetramer, are fitted to the characters as weak electrolyte like substances, but the linear silicic acid species, which are tetramer, pentamer, and hexamer, are probably fitted to the oligomer as colloidal mode.It is thought that these poly-siloxane structures of oligomer silicic acids probably corresponded to the building units observed 29NMR spectrum[6].
The other linear- and cyclic-silicic acid anions described above were detected to be the signals corresponded to the tetramers, pentamers, and hexamers, as seen in Table 2. And those anion signals appeared at vw-level intensities, and it is assumed that those anions are probably colloid like oligomers.
Fig.4.Intensities of cyclic and linear anions species vs m/z.
3.4 Signals of the metal adduct silicic acid anions
Many signals corresponded to Na- and Ca-adduct silicic acid anions appeared in the range of m/z 0-1000, and m/z of those signals and intensities were listed in Tables 3 and 4. The relation between the intensities of those signals and m/z was plotted, and the result was shown in Fig.5. The way in which the values of the intensity distributions for m/z in signals for the Na-adduct silicic acid anions(symbols Ŷ) differed compared with those of the cases of Ca-adduct anions(Ƈ). Very weak(vw) intensities are found on the Na-adduct silicic acid anions except for m/z 339 signal. Compare the intensities of Na-adduct monomer silicic acid anion with that of Ca-adduct monomer, the interactions between the univalent Na adduct anions(m/z 117, 139) and these ionic atmospheres are fairly strong than that of bivalent Ca adduct anion(133). In addition, the interaction between Na-adduct silicic acid and solvent water, on the basis of dipole-dipole interaction between the hydroxyl dipole in Na-adduct silicic acid and the dipole moment of solvent water molecule, are also fairly strong than that of the Ca-adduct anions, and then the escapes of both these Na-adduct anions(m/z 117, 139) are resulted in very lower level. Then further mode of the interaction will be discussed to the arranging hydroxyl number in anions. The strength of this interaction reflected to the signal intensity, the intensity of m/z 117 signal(3.4) corresponded to the Si(OH)2O2Na anions(plural hydroxyls) is less than that of 139 signal(8.1) corresponded to the Si(OH)O3Na2 anion(singular hydroxyl), therefore, the facility of anion escape is dependent on the strength of interaction between anion and solvent water. In other various Na-adduct silicic acid anions except for m/z 117 and 139, the signal intensities of the Na-adduct silicic acids, which are the tetramers, pentamers, and hexamers for linear and cyclic anions, are uniformly vw-levels. It is suggested that the Na-adduct silicic acids are probably composed of form of the poly-silicic acid or oligomers as colloidal mode. From their general results, it may be concluded that the poly-siloxane also corresponded to the building unit observed on 29NMR spectrum[6].
Table 3. Signals(m/z, intensities), and the assignments.
Signal / m/z Intensity Anion specie
Anion
S
ilicic acid Na-adduct silicic acid anion117 3.4 vw Si(OH)2O2Na䇴 monomer Si(OH)3ONa 139 8.1 vw Si(OH)O3Na2䇴 monomer Si(OH)2O2Na2 161 trace SiO4Na3䇴 monomer Si(OH)O3Na3 195 7.6 vw Si2(OH)4O3Na䇴 dimer Si2(OH)5O2Na 217
11.4 w Si2(OH)3O4Na2䇴 dimer Si2(OH)4O3Na2 239 trace Si2(OH)2O5Na3䇴 dimer Si2(OH)3O4Na3 261 6.4 vw Si2(OH)O6Na4䇶 dimer Si2(OH)2O5Na4 273 trace Si3(OH)6O4Na䇴 trimer Si3(OH)7O3Na 317 7.0 vw Si3(OH)4O6Na3䇴 trimer Si3(OH)5O5Na3 339 41.2 m Si3(OH)3O7Na4䇴
trimer
Si3(OH)4O6Na4 361 6.8 vw Si3(OH)2O8Na5䇴 trimer Si3(OH)3O7Na5 295 9.2 䡒w Si3(OH)5O5Na2䇴trimer
Si3(OH)6O4Na2 333 5.6 vw Si4(OH)6O6Na䇴 cyclic tetramer, Si2x2 Si4(OH)7O5Na 351 4.5 vw Si4(OH)8O5Na䇴 linear tetramer Si4(OH)9O4Na 355 6.8 vw Si4(OH)5O7Na2䇴 cyclic tetramer, Si2x2 Si4(OH)6O6Na2 373 3.1 vw Si4(OH)7O6Na2䇴 linear tetramer Si4(OH)8O5Na2 377 4.0 vw Si4(OH)4O8Na3䇴 cyclic tetramer, Si2x2 Si4(OH)5O7Na3 395 4.5 vw Si4(OH)6O7Na3䇴 linear tetramer Si4(OH)7O6Na3 399 5.6 vw Si4(OH)3O9Na4䇴 cyclic tetramer, Si 2x2 Si4(OH)4O8Na4 417 6.7 vw Si4(OH)5O8Na4䇴 linear tetramer Si4(OH)6O7Na4 411 6.1 vw Si5(OH)8O7Na䇴 cyclic pentamer, Si 3+2 Si5(OH)9O6Na 429 3.6 vw Si5(OH)10O6Na䇴 linear pentamer Si5(OH)11O5Na 433 6.4 vw Si5(OH)7O8Na2䇴 cyclic pentamer, Si 3+2 S䡅5(OH)8O7Na2 451 2.6 vw Si5(OH)9O7Na2䇴 linear pentamer Si5(OH)10O6Na2 455 4.2 vw Si5(OH)6O9Na3䇴 cyclic pentamer, Si 3+2 Si5(OH)7O8Na3 471 2.1 vw Si6(OH)8O9Na䇴 cyclic hexamer(1), Si 3x2 Si6(OH)9O8Na 489 2.6 vw Si6(OH)10O8Na䇴 cyclic hexamer(2), Si 3+2+1 Si6(OH)11O7Na 507 2.8 vw Si6(OH)12O7Na䇴 linear hexamer Si6(OH)13O6NaTable 4. Signals(m/z, intensities), and the assignments.
Signal / m/z Intensity Anion specie Anion Silicic acid Overlap anion Ca-adduct silicic acid
133 92.7 vs Si(OH)O3Ca䇴 monomer Si(OH)2O2Ca 㻌㻌㻌㻌㻌㻌㻌㻌䇷
211 㻌㻌㻌㻌trace Si2(OH)3O4Caʷ dimer Si2(OH)4O3Ca ʊ
249 12.5 w Si2(OH)O6Ca2䇴 dimer Si2(OH)2O5Ca2 㻌㻌㻌㻌㻌㻌㻌㻌䇷
327 2.0 vw Si3(OH)3O7Ca2䇴 trimer Si3(OH)4O6Ca2 ʊ
289 12.8 w Si3(OH)5O5Ca䇴 trimer Si3(OH)6O4Ca, 㻌㻌㻌㻌㻌㻌㻌㻌䇷
361 6.8 vw Si3(OH)O9Ca3ʷ trimer Si3(OH)2O8Ca3 Si3(OH)2O8Na5䇴
349 trace Si4(OH)5O7Caʷ cyclic tetramer, Si 2x2 Si4(OH)6O6Ca ʊ
405 3.4 vw Si4(OH)5O8Ca2䇴 linear tetramer Si4(OH)6O7Ca2 ʊ
425 5.7 vw Si4(OH)O11Ca3䇴 cyclic tetramer, Si 2x2 Si4(OH)2O10Ca3 ʊ
443 4.0 vw Si4(OH)3O10Ca3䇴 linear tetramer Si4(OH)4O9Ca3 ʊ
481 12.0 w Si4(OH)O12Ca4䇴 linear tetramer Si4(OH)2O11Ca4 ʊ
427 2.3 vw Si5(OH)7O8Ca䇴 cyclic pentamer, Si 3+2 Si5(OH)8O7Ca ʊ
445 2.7 vw Si5(OH)9O7Ca䇴 linear pentamer Si5(OH)10O6Ca ʊ
465 3.0 vw Si5(OH)5O10Ca2䇴 cyclic pentamer, Si 3+2 Si5(OH)6O9Ca2 ʊ
483 3.2 vw Si5(OH)7O9Ca2䇴 linear pentamer Si5(OH)8O8Ca2 ʊ
503 4.9 vw Si5(OH)3O12Ca3䇴 cyclic pentamer, Si 3+2 Si5(OH)4O11Ca3 ʊ
521 5.2 vw Si5(OH)5O11Ca3䇴 linear pentamer Si5(OH)6O10Ca3 ʊ
487 2.0 vw Si6(OH)7O10Ca䇴 cyclic hexamer(1), Si 3x2 Si6(OH)8O9Ca ʊ
505 2.8 vw Si6(OH)9O9Ca䇴 cyclic hexamer(2), Si 3+2+1 Si6(OH)10O8Ca ʊ
523 7.4 vw Si6(OH)11O8Caʷ linear hexamer Si6(OH)12O7Ca ʊ
525 4.6 vw Si6(OH)5O12Ca2䇴 cyclic hexamer(1), Si 3x2 Si6(OH)6O11Ca2 ʊ
543 4.0 vw Si6(OH)7O11Ca2䇴 cyclic hexamer(2), Si 3+2+1 Si6(OH)8O10Ca2 ʊ
561 10.7 w Si6(OH)9O10Ca2䇴 linear hexamer Si6(OH)10O9Ca2 ʊ
Other metal-adduct and overlap anions
117 3.4 vw SiO4K3ʷ monomer Si(OH)O3K Si(OH)2O2Na䇴
133 92.7 vs Si(OH)2O2K䇴 monomer Si(OH)3OK Si(OH)O3Ca䇴
117 3.4 vw Si(OH)O3Mgʷ monomer Si(OH)2O2Mg Si(OH)2O2Na䇴
217 11.4 w Si2(OH)O6Mg2ʷ dimer Si2(OH)2O5Mg2 Si2(OH)3O4Na2䇴
449 3.4 vw Si3(OH)5O5Mgʷ trimer Si3(OH)6O4Mg Si6(OH)9O8䇴
417 6.7 vw Si4(OH)O12Mg4ʷ linear tetramer Si4(OH)2O11Mg4 Si4(OH)5O8Na4䇴
517 1.8 vw Si5(OH)O15Mg5䇴 㻌linear pentamer Si5(OH)2O14Mg5
77 36.3 m Al(OH)2O䇴 monomer Al(OH)3 Si(OH)O2䇴
311 17.3 w Si3(OH)8O4Alʷ trimer Si3(OH)9O3Al Si4(OH)7O5䇴
193 㻌㻌㻌㻌㻌 2.1 vw Si(OH)3OTi(OH)2O䇴 monomer Si(OH)3OTi(OH)3 ʊ
236 㻌 1.1 vw Si(OH)3OZr(OH)2O䇴 monomer Si(OH)3OZr(OH)3 ʊ
Fig.5. Comparison of the relative intensities for the Ca- and Na-adduct silicic acid anions.
The outstanding signal appears in m/z 339 which is m-level intensity, the intensity of signal is most strong within many Na-adduct anions, and the signal was assigned to the trimeric Na-adduct silicic acid anion which corresponded to Si3(OH)3O7Na4̼. Besides this trimer anion, four kinds of other trimer anions were observed in
m/z 273(trace), 295(vw), 317(vw), and 361(vw), respectively, and the assignments of these signals were shown
in Table 3. In order to see the character of m/z 339-signal, the intensity of 339-signals is plotted as a function of Na2O concentration, and it result was shown in marked at symbol 䕺 of Fig.6 together with the signal intensity of m/z 95(Ŷ) which is Si(OH)3O monomer anion. The value of signal intensity of m/z 339 is about 44 at 7.1% Na2O, it value proportionally decreased with increasing of Na2O concentration until about 9.5%, and then such a intensity decreased till 12.8%. The intensity dependency of m/z 339 on the Na2O concentration indicated that the affinity of the hydrated Si3(OH)4O6Na4 silicic acid to the solvent water increased; in addition, the restriction power of the ion atmosphere to the center Si3(OH)4O6Na4 species also increased with increase in the Na2O concentration.
In contrast to the 339-signal, the intensity of m/z 95-signal corresponded monomer silicic acid anion is very weak(vw) with the same concentration of 7.1% Na2O described above. The intensity of 95-signal proportionally increased with increasing of Na2O concentration until about 9.5% Na2O, and the intensity of 95-signal slightly increased in the region of about from 9.5 to 12.8% Na2O. In the region of Na2O concentration between about 7.1 and 9.5%, the intensity dependency of the 95-signal on the Na2O concentration are proportionally opposite direction compared with that of 339-signal intensity, and the facilities of escape in these anions are also inversely proportional to the Na2O concentration. These results suggested that the freedom of hydrated Si3(OH)4O6Na4 silicic acid trimer specie is relatively high when Na2O concentration is relatively lower state which is about 7.1-9.5 Na2O%, but the freedom was depressed by the increasing Na2O concentration at about above 9.5% Na2O. In the case of the hydrated Si(OH)4 silicic acid monomer, the freedom of this monomer was depressed by under the condition of about 7.1-9.5% Na2O, but such freedom is enhanced as a result of the freedom of hydrated Si3(OH)4O6Na4 silicic acid trimer specie depressed in the region of about 9.5 and above Na2O%. When the freedom of monomer is enhanced, the frequency of polymerization between monomers is also enhanced, and the intensity enhancement of monomer is discontinued as a result of probability occurrences in the number of monomer anions are reduced. These phenomena are commonly analogous to that of the behavior of specific conductivity investigated by H. Ukihashi[8,11], and it is thought that one of some kind of charge carriers corresponded to the hydrated silicic acid monomer. Perhaps it is thought that the Si3(OH)4O6Na4 specie detected in this experiment corresponded to the contribution specie for a control of the electrical conduction in a sodium silicate solution. From these results, the hydrated silicic acid monomer has a strongly active character, such activity is reduction by the behavior of high freedom in the Si3(OH)4O6Na4 specie, but the activity of silicic acid monomer is heightened by emancipation from reduction state by inactivation of the active Si3(OH)4O6Na4. In such Na2O higher concentration region, it is thought that the active silicic acid monomers are rapidly polymerized by the condensation reaction between active silicic acid monomers. Therefore, the hydrated Si3(OH)4O6Na4 silicic acid trimer specie appears to play an important role in the polymerization action as a inhibitor for self-condensations of the active silicic acid monomers in the sodium silicate solution inside.
The interesting signal was observed in m/z 62 position. This signal corresponded to the mass of Na2O, the intensity of the signal is trace-level in spite of high concentration of Na2O, which is one of important components in the sodium silicate solution, and such 62-signal may be assigned to Na2O̼ anion. On the other hand, the molar refractive index of a sodium silicate solution is measured by the sodium D-line(589nm), and the observation value is slightly larger than that of the theoretical value[29]. Generally, the free or isolated Na2O is not presence in a sodium silicate solution[1,29]. If such a deviation arise from the existence of the free or isolated Na2O, the 62-signal is possibly generated from Na2O specie. It seems reasonable to assume that the concentration of the free or isolated Na2O specie in the sodium silicate solution is well not defined. Furthermore, the concentration of the free or isolated Na2O cannot be estimated by FAB-MS analysis. On the species of K2O̼ and Li2O̼, these signals detected in m/z 94 and 30 as shown in Table 2, and the presences of K2O and Li2O species can be presumed in the sodium silicate solution.
On the other hand, the signal appeared in m/z 133 with vs intensity, this signal was assigned to the Ca-adduct silicic acid monomer anion corresponded to Si(OH)O3Ca[17], but the m/z of it signal is also correspondence
![Figure 5. Flowchart of our posituion/orientation regiastration method based on digital neigobors (Source: Noborio [10] (2014))](https://thumb-ap.123doks.com/thumbv2/123deta/7853122.1235063/38.892.285.595.126.332/figure-flowchart-posituion-orientation-regiastration-digital-neigobors-noborio.webp)
