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㏆㊥㞳ᚐṌᖐᏯ⪅ 㐲㊥㞳ᚐṌᖐᏯ⪅ ᖐᏯ᩿ᛕ⪅

ϭϯ

◊✲᪉ἲ 䡚䐠ᅾ⪅䛾ᖐᏯពᛮ䛾Ỵᐃ䡚

5HVHDUFKPHWKRGVa

䐠

'HWHUPLQHZKRLQWHQGVWRUHWXUQKRPHa

୰ᚰ㒊䛻ேཱྀ

䛜㞟୰

ϭϰ ᕷෆ䜈䛾ᖐᏯ⪅

䛜ẚ㍑ⓗከ䛔 2䠄㉳Ⅼ䠅䛾ศᕸ

)LJ'LVWULEXWLRQ RI 22ULJLQ

ç䠄⤊Ⅼ䠅䛾ศᕸ )LJ'LVWULEXWLRQ RI ''HVWLQDWLRQ

◊✲᪉ἲ 䡚䐡Ṍ⾜⪅䝅䝭䝳䝺䞊䝅䝵䞁䡚

5HVHDUFKPHWKRGVa

䐡

3HGHVWULDQVLPXODWLRQa

䛂୍ḟඖṌ⾜⪅䝰䝕䝹䜢⏝䛔䛯㧗㏿㑊㞴䝅䝭䝳䝺䞊䝍䛾㛤Ⓨ䛃䠈

๪⏣䜙㻔㻞㻜㻝㻜㻕

㻤䠅

䞉

6RFLDO)RUFH0RGHO

䜢

ϭ

ḟඖ໬䛧䠈䝛䝑䝖䝽䞊䜽✵㛫䛻㐺⏝䛷䛝䜛䜘䛖䛻䛧䛯䝰䝕䝹 䞉Ṍ⾜⪅✵㛫䜢㛗䛥䛸ᖜ䜢ᣢ䛳䛯䜶䝑䝆䛸䝜䞊䝗䛷⾲⌧

эᮏ◊✲䛷䛿䠈㐨㊰䜢䜶䝑䝆䠈஺ᕪⅬ䜢䝜䞊䝗䛷⾲⌧

ϭϱ ᅗṌ⾜⪅䝰䝕䝹䠄ᩥ⊩^㻣䠅䜘䜚ᘬ⏝䠅

3HGHVWULDQPRGHOTXRWHGIURP UHIHUHQFH

ᐇ✵㛫

5HDOVSDFH ϭ

ḟඖ✵㛫

2QHGLPHQVLRQDOVSDFH

◊✲᪉ἲ 䡚䐡Ṍ⾜⪅䝅䝭䝳䝺䞊䝅䝵䞁䡚

5HVHDUFKPHWKRGVa

䐡

3HGHVWULDQVLPXODWLRQa

ϭϲ

䝺䞊䞁

/DQH

䞉ே䚻䛜Ṍ⾜䛾㝿䛻ᙧᡂ䛩䜛ิ䜢 ௬᝿䝺䞊䞁䛷⾲⌧䛩䜛

эṌ⾜⪅䛿䝺䞊䞁ୖ䜢Ṍ䛟

ܮ௠

䠖䜶䝑䝆䠈

݈௡

䠖䝺䞊䞁䠈

ݔ௜

䠖ྠ୍䝺䞊䞁ୖ䛾ඛ㢌䛛䜙

݅

␒┠䛾Ṍ⾜⪅

ᚐṌᖐᏯ㛤ጞ Ṍ⾜⪅䛜 䝜䞊䝗ୖ䛛 Ṍ⾜⪅䜢ධ䜛䛣䛸䛜

ྍ⬟䛺䝺䞊䞁䛻஌䛫䜛

Ṍ⾜㏿ᗘ䜢Ỵᐃ䛩䜛 Ṍ⾜⪅䜢⛣ື䛥䛫䜛

䝺䞊䞁䜢Ώ䜚 ษ䛳䛯䛛 Ṍ⾜⪅䜢䝜䞊䝗䛻

␃䛥䛫䜛

┠ⓗᆅ䛻

╔䛔䛯䛛

z^ z^

EK

z^

EK

EK

ᖐᏯ᏶஢

◊✲᪉ἲ 䡚䐡Ṍ⾜⪅䝅䝭䝳䝺䞊䝅䝵䞁䡚

5HVHDUFKPHWKRGVa

䐡

3HGHVWULDQVLPXODWLRQa

䝺䞊䞁䛻㛵䛩䜛タᐃ

䞉䝺䞊䞁䜢ᙧᡂ䛷䛝䜛ᩘ䛿㐨㊰䛾ᖜ䛻䜘䛳䛶Ỵ䜎䜛

䞉ᮏ◊✲䛷䛿䠈Ṍ⾜⪅䛿Ṍ㐨䜢⛣ື䛩䜛䜒䛾䛸䛧䠈Ṍ㐨ᖜ䛻䜘䛳䛶䝺䞊䞁ᩘ䜢Ỵᐃ 䞉㐨㊰✀ูẖ䛻Ṍ㐨ᖜ䜢䝃䞁䝥䝸䞁䜾

эṌ㐨ᖜ䜢ィ 䛧䠈ᚓ䜙䜜䛯ᖹᆒ್䜢

㐨㊰✀ูṌ㐨ᖜ䛸䛩䜛

э䝺䞊䞁ᖜ䜢>FP@

䛸䛧䠈䝺䞊䞁ᩘ䜢Ỵᐃ

ϭϳ

㐨㊰䛾✀㢮 Ṍ㐨ᖜ;ŵͿ 䝺䞊䞁ᩘ

୍⯡ᅜ㐨

୍⯡㒔㐨ᗓ┴㐨➼

୺せᆅ᪉㐨䠄ᣦᐃᕷ㐨䠅

୺せᆅ᪉㐨䠄㒔㐨ᗓ┴㐨䠅

ᣦᐃᕷ䛾୍⯡ᕷ㐨

䛭䛾௚䛾㐨㊰

⾲㐨㊰✀ู䛤䛸䛾Ṍ㐨ᖜ䛸䝺䞊䞁ᩘ

7DEOH3HGHVWULDQZLGWKDQGQXPEHURIODQHVE\URDGW\SH

◊✲᪉ἲ 䡚䐡Ṍ⾜⪅䝅䝭䝳䝺䞊䝅䝵䞁䡚

5HVHDUFKPHWKRGVa

䐡

3HGHVWULDQVLPXODWLRQa

䝰䝕䝹䛻䛚䛡䜛Ṍ⾜⪅䛻䛛䛛䜛ຊ

䞉Ṍ⾜⪅䛻

Ϯ

䛴䛾௬᝿ⓗ䛺ຊ䛜స⏝

䐟Ṍ⾜⪅᥎㐍ຊ

͙

࿘ᅖ䛛䜙ᙳ㡪䜢ཷ䛡䛺䛔ሙྜ䛾Ṍ⾜㏿ᗘ䠄⮬⏤Ṍ⾜㏿ᗘ䠅䛻ᡠ䜝䛖䛸䛩䜛ຊ 䐠♫఍ⓗస⏝

͙

࿘ᅖ䛾Ṍ⾜⪅䛸䛾㛫䛻䝟䞊䝋䝘䝹䝇䝨䞊䝇䜢ྲྀ䜝䛖䛸䛩䜛ຊ

Ṍ⾜⪅䝰䝕䝹䛾㏿ᗘᘧ

䞉ྠ䝺䞊䞁ୖ䛾┤๓䛾Ṍ⾜⪅䛸䛾㊥㞳䜢䜒䛸䛻Ṍ⾜㏿ᗘ䜢ィ⟬

ݔሶ௜ݐ ൅ οݐ ൌ ݔሶ௜ݐ ൅ ܽଵݒ௢െ ݔሶ௜ݐ െ ܽଶ‡š’^௥ିο௫_௔

య οݐ

ݐ䠖᫬้;ƐͿ䠈ݔሶ௜ݐ䠖ݐ䛻䛚䛡䜛ྛ䝺䞊䞁ୖ䛾ඛ㢌䛛䜙݅␒┠䛾Ṍ⾜⪅䛾Ṍ⾜㏿ᗘ;ŵͬƐͿ䠈οݔ䠖┤๓䛾Ṍ⾜⪅䛸䛾㊥㞳;ŵͿ䠈οݐ䠖䝅䝭䝳䝺䞊䝅䝵䞁䝇䝔䝑䝥;ƐͿ䠈ݔ௜ሺݐሻ䠖Ṍ⾜⪅݅䛜Ꮡ ᅾ䛩䜛䝺䞊䞁ୖ䛾᫬้ݐ䛻䛚䛡䜛ጞⅬ䝜䞊䝗䛛䜙䛾㊥㞳䠈ݎ䠖Ṍ⾜⪅༙ᚄ;ŵͿ䠈ݒ଴䠖⮬⏤Ṍ⾜㏿ᗘ;ŵͬƐͿ䠈ܽଵǡ ܽଶǡ ܽଷ䠖Ṍ⾜ᐇ㦂➼䛾⤖ᯝ䛛䜙䜻䝱䝸䝤䝺䞊䝅䝵䞁䛩䜛䛣䛸䛷ồ䜑䜙 䜜䜛䝟䝷䝯䞊䝍

䝅䝭䝳䝺䞊䝅䝵䞁᮲௳

6LPXODWLRQFRQGLWLRQV

䝅䝭䝳䝺䞊䝅䝵䞁䜿䞊䝇

䐟ᾷᕝ䛻ᯫ䛛䜛ᶫᱱ䛾㏻⾜Ṇ䜑䛾᭷↓

䐠ΰ㞧⌧㇟䜢෌⌧䛩䜛䛛䛹䛖䛛

э෌⌧䛧䛺䛔ሙྜ䠈ᖐᏯ⪅඲ဨ䛜᭱▷⤒㊰䜢⮬

⏤Ṍ⾜㏿ᗘ䠄

сϭ͘ϬϮϯ΀ŵͬƐ΁

䠅䛷Ṍ⾜䛩䜛䜿䞊䝇 䠄䜿䞊䝇

ϭĂ͕ϮĂͿ

䛣䜜䜙䛾⤌䜏ྜ䜟䛫䛸䛧䛶䠈ᅄ䛴䛾䜿䞊 䝇䛻ᑐ䛧䛶䝅䝭䝳䝺䞊䝅䝵䞁䜢⾜䛖

䜿䞊䝇 ᾷᕝ䛻ᯫ䛛䜛ᶫᱱ䛾

㏻⾜Ṇ䜑䛾᭷↓

ΰ㞧⌧㇟䛾᭷↓

䜿䞊䝇

ϭĂ

↓ ↓

䜿䞊䝇

ϭď

↓ ᭷

䜿䞊䝇

ϮĂ

᭷ ↓

䜿䞊䝇

Ϯď

᭷ ᭷

⾲ 䝅䝭䝳䝺䞊䝅䝵䞁䜿䞊䝇

ϭϵ

䐟 䐠

䝅䝭䝳䝺䞊䝅䝵䞁᮲௳

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㏻⾜Ṇ䜑䛻㛵䛩䜛᮲௳

䞉ண᝿ᾐỈ⠊ᅖෆ䛾ᾷᕝ䛻ᯫ䛛䜛

ϱ

ᮏ䛾ᶫᱱ 䠄ఏἲ኱ᶫ䠈ᾷᕝ኱ᶫ䠈᪂༑୕኱ᶫ䠈༑୕኱

ᶫ䠈᪂ᾷᕝ኱ᶫ䠅䛾㏻⾜Ṇ䜑䜢᝿ᐃ 䞉ᶫᱱ䛾㏻⾜Ṇ䜑䛿䠈䝅䝭䝳䝺䞊䝅䝵䞁䛾 ᮇ㛫୰䠈⥅⥆䛩䜛䜒䛾䛸䛩䜛

ϮϬ

㏻⾜Ṇ䜑䜢௬ᐃ 䛩䜛ᶫᱱ ᾐỈᇦ

ఏἲ኱ᶫ ᾷᕝ኱ᶫ

᪂༑୕኱ᶫ

༑୕኱ᶫ ᪂ᾷᕝ኱ᶫ

⤖ᯝ 䡚ᖐᏯ⪅䛾ᖐᏯ㊥㞳䡚

5HVXOWa5HWXUQLQJKRPHGLVWDQFHRIUHWXUQLQJKRPHa

ⰱᒇ䞉すᐑᆅᇦ䛾 ᖐᏯ⪅䛾ᖐᏯ㊥㞳 䛜ቑຍ

Ϯϭ

䜿䞊䝇

DE

&DVHDE

䜿䞊䝇

DE

&DVHDE

⤖ᯝ 䡚ᖐᏯ⪅䛾ᖐᏯ᫬㛫䠄ΰ㞧⌧㇟䛺䛧䠅䡚

5HVXOWa5HWXUQLQJKRPHWLPHQRFRQJHVWLRQSKHQRPHQRQa

ᖐᏯ㊥㞳䛾ศᕸ䛻ᑐ ᛂ䛧䛶䠈䛣䛱䜙䜒ቑຍ

ϮϮ

䜿䞊䝇

ϭĂ

&DVHD

䜿䞊䝇

ϮĂ

&DVHD

඲య䛾ᖹᆒᖐᏯ᫬㛫䠖ϰϳϴϳ΀Ɛ΁ ඲య䛾ᖹᆒᖐᏯ᫬㛫䠖ϰϵϮϱ΀Ɛ΁

⤖ᯝ 䡚ᖐᏯ⪅䛾ᖐᏯ᫬㛫䠄ΰ㞧⌧㇟䛒䜚䠅䡚

5HVXOWa5HWXUQLQJKRPHWLPHRIWKHUHWXUQLQJKRPHZLWKFRQJHVWLRQ SKHQRPHQRQa

ᾷᕝ௨໭䛾ᖐᏯ⪅䛾 ᖐᏯ᫬㛫䛜ቑຍ

Ϯϯ

䜿䞊䝇

ϭď

&DVHE

䜿䞊䝇

Ϯď

&DVHE

඲య䛾ᖹᆒᖐᏯ᫬㛫䠖ϲϴϱϱ΀Ɛ΁ ඲య䛾ᖹᆒᖐᏯ᫬㛫䠖ϵϮϳϲ΀Ɛ΁

⤖ᯝ 䡚㐨㊰ୖ䛷䛾ΰ㞧≧ἣ䡚

5HVXOWa&URZGHGQHVVRQWKHURDGa

ΰ㞧≧ἣ䛾ᢕᥱ

͙

஧ḟ⅏ᐖ䛾༴㝤ᛶ䛺䛹䛾ᢕᥱ䛻ᚲせ ΰ㞧䛾ྍど໬᮲௳

䞉ΰ㞧䜢⾲䛩ᣦᶆ

͙

Ṍ⾜⪅ᐦᗘ䠄ே

ͬ

䟝䠅

䞉Ṍ⾜⪅ᐦᗘ

Ϭ͘ϳ

䠄ே

ͬ

䟝䠅௨ୖ䛷ΰ㞧䛜Ⓨ⏕䛧䛶䛔䜛䛸ᐃ⩏

э䜶䝑䝆ୖ䛾኱㒊ศ䛾Ṍ⾜⪅䛜䠈Ṍ⾜䛻㛵䛧䛶ΰ㞧䛻䜘䜛ไ⣙䜢ཷ䛡䜛್

:͘:͘&ƌƵŝŶ䛾Ṍ⾜㊰䝃䞊䝡䝇Ỉ‽^ϵͿ䜘䜚

䞉ᾐỈ䛜ᠱᛕ䛥䜜䜛ᱵ⏣௜㏆䛾ᾷᕝ䛾ᶫᱱ䛻╔┠

䞉䛭䜜䛮䜜䛾䜿䞊䝇䛻㛵䛧䛶ΰ㞧䛜⥅⥆䛩䜛᫬㛫䜢ྍど໬

⤖ᯝ 䡚㐨㊰ୖ䛷䛾ΰ㞧≧ἣ䡚

5HVXOWa&URZGHGQHVVRQWKHURDGa

䜿䞊䝇

ϭď

&DVHE

:Z኱㜰㥐

Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ Ϯ Ϯ͘ϱ ϯ

Ϭ ϭ Ϯ ϯ ϰ ϱ ϲ ϳ ϴ ϵ ϭϬ ϭϭ ϭϮ ϭϯ

Ṍ⾜⪅ᐦᗘ΀ேͬ䟝΁

⤒㐣᫬㛫΀᫬΁

ᶫᱱ䛾ΰ㞧≧ἣ

&RQJHVWLRQVWDWXVRIEULGJHV

༑୕኱ᶫ ᪂ᾷᕝ኱ᶫ 㛗᯶ᶫ

༑୕኱ᶫ ᪂ᾷᕝ኱ᶫ 㛗᯶ᶫ

⤖ᯝ 䡚㐨㊰ୖ䛷䛾ΰ㞧≧ἣ䡚

5HVXOWa&URZGHGQHVVRQWKHURDGa

Ϯϲ

䜿䞊䝇

Ϯď

&DVHE

Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ Ϯ Ϯ͘ϱ ϯ

Ϭ ϭ Ϯ ϯ ϰ ϱ ϲ ϳ ϴ ϵ ϭϬ ϭϭ ϭϮ ϭϯ ϭϰ ϭϱ ϭϲ ϭϳ ϭϴ ϭϵ ϮϬ Ϯϭ ϮϮ Ϯϯ Ϯϰ

Ṍ⾜⪅ᐦᗘ΀ேͬ䟝΁

⤒㐣᫬㛫΀᫬΁

ᶫᱱ䛾ΰ㞧≧ἣ

&RQJHVWLRQVWDWXVRIEULGJHV

㛗᯶ᶫ :Z኱㜰㥐

㛗᯶ᶫ

䜎䛸䜑

6XPPDU\

䞉ᮏ◊✲䛷䛿䠈኱㜰ᕷ䜢ᑐ㇟䛻኱つᶍ䛺ᚐṌᖐᏯ䝅䝭䝳䝺䞊䝅䝵䞁䜢ᐇ᪋

э኱㜰ᕷ䛷䛣䜜䜎䛷༑ศ⾜䜟䜜䛶䛣䛺䛛䛳䛯኱ᆅ㟈᫬䛻䛚䛡䜛ᖐᏯ⪅䛾ᖐᏯ⾜ື䛻

㛵䛩䜛ᇶ♏ⓗ▱ぢ䜢ᢕᥱ

䞉ᾷᕝ௨໭䜈ᖐᏯ䛩䜛ᖐᏯ⪅䛻኱䛝䛺ᙳ㡪

䞉ᶫᱱୖ䛷䛾Ṍ⾜⪅䛾␃эᾐỈ䛛䜙䛾㏨䛢㐜䜜䛾ၥ㢟

эᅾ⪅䠄≉䛻ᾷᕝ䛾ᶫ䜢Ώ䛳䛶ᖐᏯ䛩䜛ᚲせ䛾䛒䜛ᅾ⪅䠅䜢䛩䛠䛻䛿ᖐᏯ䛥䛫䛪

䛻䠈㏆㎶䛾㑊㞴ሙᡤ➼䛻㑊㞴䛥䛫䜛䛣䛸䜢᳨ウ䛩䜛䛺䛹䛜ᚲせ

Ϯϳ

䜎䛸䜑

6XPPDU\

ドキュメント内 Vol.78 MI Lecture Notes|九州大学 マス・フォア・インダストリ研究所 math 5acd7940c3b8c (ページ 130-135)