190309_2_2.9-12

Transcript

1. 2019/03/09 ʮਂ૚ֶशʯୈ2ষ 2-9~2-12
2. ٖࣅٯߦྻ ղ͕ෆఆͷํఔࣜΛղ͘
   ɹྫʣ͕ࣜଟ͗ͯ͢ղ͕ଘࡏ͠ͳ͍
   ɹྫʣ͕ࣜগͳͯ͘ղ͕ఆ·Βͳ͍
   ɹྫʣ͕ࣜໃ६ͯ͠ղ͕ଘࡏ͠ͳ͍
   ‣ ͜ͷΑ͏ͳߦྻ͸ٯߦྻΛ࣋ͨͳ͍
   ϜʔΞɾϖϯϩʔζٯߦྻ
   ɹ͜ΕΒ͕੒Γཱͭߦྻɹ
   
   ͸ͨͩͭʹܾ·Δ
   ɹղɹɹɹɹ͸
   ɹɹ͕ॎ௕ͷ৔߹͸
 ɹɹ͕ԣ௕ͷ৔߹͸ɹɹΛ࠷খʹ͢Δ 2 (1)AA+A = A (2)A+AA+ = A+ (3)(AA+)T = AA+ (4)(A+A)T = A+A A+ ∥Ax − b∥2 ∥x∥2 x = A+b A A
3. τϨʔε ߦྻͷର֯੒෼ͷ࿨
   ‣ ߦྻɹͷର֯࿨
   
   
   
   ͷݻ༗஋ͷ࿨
   ‣ ݻ༗஋ͷੵ͕ߦྻࣜͱ౳͘͠ͳΔ 3 Tr(A) = ∑ i Ai,i A A
4. ओ੒෼෼ੳͱݻ༗஋෼ղ ओ੒෼෼ੳ
   ‣ σʔλͷࢄ෍ਤͷதͰ෼ࢄ͕࠷େʹͳΔઢΛҾ͘
 ˠ
   ֤σʔλͷҧ͍ΛΑΓଟ͘ද͢
   ‣ σʔλͷཁ໿ʢଟ͘ͷཁૉΛ·ͱΊΔ͜ͱʣ͕ग़དྷΔ
 ˠ
   ಛ௃͕൑அ͠΍͘͢ͳΔ
 ˠ
   σʔλͷΧςΰϦ෼͚ͳͲʹԠ༻
   4
5. ओ੒෼෼ੳͱݻ༗஋෼ղ ෼ࢄ
   ‣ ͋Δ࣍ݩͷσʔλɹɹɹɹ
   ʹରͯ͠
 ภࠩʢฏۉ஋ɹ͔Βͷࠩʣͷ৐ͷฏۉ
 ɹ
   ڞ෼ࢄ
   ‣ ͋Δଟ࣍ݩͷσʔλɹɹɹɹɹɹɹɹɹɹ
   ʹରͯ͠
 ภࠩʢɹͱɹ͔Βͷࠩʣͷੵͷฏۉ 5 (x1 , x2 , x3 ) ¯ x σxx = (x1 − ¯ x)2 + (x2 − ¯ x)2 + (x3 − ¯ x)2 3 ((x1 , y2 ), (x2 , y2 ), (x3 , y3 )) ¯ x ¯ y σxy = (x1 − ¯ x)(y1 − ¯ y) + (x2 − ¯ x)(y2 − ¯ y) + (x3 − ¯ x)(y3 − ¯ y) 3
6. ओ੒෼෼ੳͱݻ༗஋෼ղ ڞ෼ࢄߦྻ
   ‣ ෼ࢄͱڞ෼ࢄΛཁૉͱͯ͠ฒ΂ͨߦྻ 6 Σ = [ σxx σxy σxy σyy ] = 1 3 [ (x1 − ¯ x)2 + (x2 − ¯ x)2 + (x3 − ¯ x)2 (x1 − ¯ x)(y1 − ¯ y) + (x2 − ¯ x)(y2 − ¯ y) + (x3 − ¯ x)(y3 − ¯ y {(x1 − ¯ x)(y1 − ¯ y) + (x2 − ¯ x)(y2 − ¯ y) + (x3 − ¯ x)(y3 − ¯ y) (y1 − ¯ y)2 + (y2 − ¯ y)2 + (y3 − ¯ y)2 ]
7. ओ੒෼෼ੳͱݻ༗஋෼ղ ڞ෼ࢄߦྻʹ͍ͭͯͷݻ༗஋໰୊
   ‣ ͋Δ఺Λࣔ͢ϕΫτϧΛฒ΂ͨߦྻɹΛఆٛ
 ྫʣͭͷσʔλʹ͍ͭͯͷओ੒෼෼ੳ
   ‣ సஔߦྻɹ
   ͱͷੵΛͱΔ
 ˠ
   දࣜతʹڞ෼ࢄߦྻͱಉ༷ʹͳΔ 7 X = a1x a1y a2x a2y a3x a3y a1 = [a1x a1y], a2 = [a2x a2y], a3 = [a3x a3y] X XT XTX = [ a2 1x + a2 2x + a2 3x a1x a1y + a2x a2y + a3x a3y a1x a1y + a2x a2y + a3x a3y a2 1y + a2 2y + a2 3y ]
8. ओ੒෼෼ੳͱݻ༗஋෼ղ ڞ෼ࢄߦྻʹ͍ͭͯͷݻ༗஋໰୊
   ‣ ͋Δ఺Λࣔ͢ϕΫτϧΛฒ΂ͨߦྻɹΛఆٛ
 ྫʣͭͷσʔλʹ͍ͭͯͷओ੒෼෼ੳ
   ‣ సஔߦྻɹ
   ͱͷੵΛͱΔ
 ˠ
   දࣜతʹڞ෼ࢄߦྻͱಉ༷ʹͳΔ 8 X = a1x a1y a2x a2y a3x a3y a1 = [a1x a1y], a2 = [a2x a2y], a3 = [a3x a3y] X XT XTX = [ a2 1x + a2 2x + a2 3x a1x a1y + a2x a2y + a3x a3y a1x a1y + a2x a2y + a3x a3y a2 1y + a2 2y + a2 3y ] Σ = 1 3 XTX
9. ओ੒෼෼ੳͱݻ༗஋෼ղ σʔλΛࣹӨ͢ΔͨΊͷ୯ҐϕΫτϧɹɹɹ
   Λఆٛ
   σʔλѹॖޙͷ෼ࢄɹͷදࣜΛٻΊΔ
   9 e = [ ex ey] v = (x1 ⋅ e)2 + (x2 ⋅ e)2 + (x3 ⋅ e)2 3 v v = 1 3 [a1 ⋅ e a2 ⋅ e a3 ⋅ e] a1 ⋅ e a2 ⋅ e a3 ⋅ e
10. ओ੒෼෼ੳͱݻ༗஋෼ղ 10 v = (x1 ⋅ e)2 + (x2 ⋅ e)2 + (x3 ⋅ e)2 3 = 1 3 [a1 ⋅ e a2 ⋅ e a3 ⋅ e] a1 ⋅ e a2 ⋅ e a3 ⋅ e = 1 3 (Xe)T(Xe) = 1 3 eTXTXe = eTΣe
11. ओ੒෼෼ੳͱݻ༗஋෼ղ ओ੒෼෼ੳ
    ‣ ɹΛ࠷େԽ͢ΔɹΛٻΊΔඞཁ
 ɹ͸୯ҐϕΫτϧͱ͍͏੍໿෇͖
 ˠ
    ϥάϥϯδΣͷະఆ৐਺๏ͰٻΊΔ 11 Σ [ ex ey] = λ [ ex ey] v e e ڞ෼ࢄߦྻɹʹର͢Δݻ༗ํఔࣜʹͳΔ
    ͜ΕΛղ͍ͯٻΊͨݻ༗஋ͷதͰ
    ࠷େ஋ʹରԠ͢Δݻ༗ϕΫτϧ͕ɹʹͳΔ Σ e