情報地質 第26巻 第2号 052-053頁 2015年
Geoinformatics, vol.26, no.2, pp.052-053, 2015ྲྀᚓᆅ⌫᝟ሗࡢホ౯㸸࢚ࣥࢺࣟࣆ࣮࡜ศᩓࡢ㐪࠸ṇ㊰ ᚭஓ

Evaluation of Obtained Geoinformation: Comparison of Entropy and Variance
Tetsuya Shoji** ᮾி኱Ꮫ The University of Tokyo, Tokyo 113 -8639, Japan.
E-mail: t-t_shoji@jcom.home.ne.jp࣮࣮࢟࣡ࢻ㸸ᆅ⌫᝟ሗ㸪ㄪᰝ㸪ホ౯㸪࢚ࣥࢺࣟࣆ࣮㸪ศᩓ
Key words: Geoinformation, Survey, Evaluation, Entropy, Variance1㸬ࡣࡌࡵ࡟ᆅ⌫⛉Ꮫⓗㄪᰝࡣ㸪ㄪᰝࡑࡢࡶࡢࡀ┠ⓗ࡛ࡣ࡞ࡃ㸪ࡑࡢ㐣⛬࡛ᚓࡽࢀࡓ᝟ሗ࡟ࡶ࡜࡙࠸࡚㐺ษ࡞⾜ືࢆ㑅ᢥ࣭᥇⏝ࡍࡿࡓࡵ࡟⾜࡞ࢃࢀࡿ㸬౛࠼ࡤ㸪ཎᏊຊⓎ㟁ᡤ(ཎⓎ)࿘㎶࡛᭱㏆⾜ࢃࢀࡓά᩿ᒙࡢㄪᰝࡣ㸪෌✌ാᚋࡢཎⓎࡢᏳ඲ᛶࡢホ౯ࡀ┠ⓗ࡛࠶ࡿ㸬ࡇࡢㄢ㢟࡛㸪➹⪅࡟ࡣ⌮ゎ࡛ࡁ࡞࠸Ⅼࡀ 1 ࡘ࠶ࡿ㸬㏻ᖖ㸪Ᏻ඲ᛶࡀ᏶඲࡟ࡣホ౯࡛ࡁ࡞࠸ࡇ࡜࡟㉳ᅉࡍࡿࣜࢫࢡࡣᦆᐖಖ㝤࡛⿵ൾࡍࡿ㸬ࡇࡢ࡜ࡁ㸪ಖ㝤ᩱࡣࣜࢫࢡࡢⓎ⏕☜⋡࡟ᇶ࡙࠸࡚⟬ᐃࡉࢀ㸪ࡑࡢ⟬ᐃࡣㄪᰝࡢ⢭ᗘࢆ㛵ᩘ࡟⾜ࢃࢀࡿ㸬ࡋࡓࡀࡗ࡚㸪ಖ㝤ᩱ⋡ࡢ⟬ᐃ࡟ࡣㄪᰝࡢ⢭ᗘ࡟㛵ࡍࡿࢹ࣮ࢱࡀᚲせ࡛࠶ࡿ㸬ཎⓎ࡟㝈ࡽࡎఱࡽ࡞ࡢ᧯ᴗ࡟ᑐࡍࡿಖ㝤ࡢ᥃㔠ࡣࢥࢫࢺ࡟⟬ධࡉࢀࡿ㸬୍᪉㸪ㄪᰝ⢭ᗘࢆྥୖࡉࡏࡿࡓࡵ࡟ࡣࡼࡾከࡃࡢࢥࢫࢺࡀᚲせ࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪᭱㐺࡞ㄪᰝࡣ୧ࢥࢫࢺࢆࣂࣛࣥࢫࡉࡏࡿࡇ࡜࡟ࡼࡗ࡚ᐇ⌧ࡉࢀࡿ㸬࡜ࡇࢁࡀ㸪➹⪅ࡀ▱ࡿ㝈ࡾ㸪ཎⓎࡢᏳ඲ᛶ࡟㛵ࡍࡿᢏ⾡ⓗ࣭⤒῭ⓗ㆟ㄽ࡛ࡣ㸪᭱ࡶ⫢ᚰ࡞Ᏻ඲ᛶ㸪ಖ㝤㸪ㄪᰝࡢ㛫ࡀ⤖ࡧࡘࡅ࡚࠸࡞࠸㸬ࡑࡢ⌮⏤ࡢ 1 ࡘ࡜ࡋ࡚㸪ㄪᰝ⢭ᗘࡀ㸪ಖ㝤ᩱ⋡ࡢ⟬ᐃ࡟ᙺ❧ࡘᐃ㔞ⓗᩘ್࡜ࡋ࡚♧ࡉࢀ࡚࠸࡞࠸ࡇ࡜ࡀ⪃࠼ࡽࢀࡿ㸬ᆅ⌫⛉Ꮫⓗㄪᰝࡢ㐣⛬࡛ྲྀᚓࡉࢀࡓᆅ⌫᝟ሗࡣ㸪ḟࡢᘧ(1)࡛ᐃ⩏ࡉࢀࡿ࢚ࣥࢺࣟࣆ࣮s ࠶ࡿ࠸ࡣᘧ(2)࡛ᐃ⩏ࡉࢀࡿศᩓ V ࢆ౑ࡗ࡚ホ౯࡛ࡁࡿ(ṇ㊰㸪2003㸹Shoji, 2006㸹ṇ㊰㸪2014)㸬
msik / K = −∑ p jik / K ln p jik / K

(1)

j =1

Nk / K

S k / K = ∑ sik / Ki =1

Nk / Ks k / K = ∑ s ik / K N k / Ki =1

m −1

Vik / K = ∑ p j 1ik / K
j1=1m∑pj 2 ik / K(2)j 2 = j 1+1

Nk / K

Vk / K = ∑Vik / K N k / Ki =1

ࡇࡇ࡛㸪ῧᏐ୰ࡢ k/K ࡣㄪᰝࡀẁ㝵 K ࡲ࡛㐍ࢇࡔ࡜ࡁࡢ㏵
୰ࡢẁ㝵 k (k = 1,㺃㺃㺃,K)ࢆ⾲ࡋ㸪pjik/K ࡣࢭࣝ ik/K (⥲ᩘ Nk/K)࡟࠾ࡅࡿᆅ㉁య j (⥲✀㢮ᩘ m)ࡢᏑᅾᗘ࡛࠶ࡿ㸬ࡲࡓ㸪Sk/K
࡜ sk / K ࡣࡑࢀࡒࢀᆅᇦ඲యࡢ࢚ࣥࢺࣟࣆ࣮࡜ᖹᆒࢭ࢚ࣝࣥ
ࢺࣟࣆ࣮㸪 Vk / K ࡣᆅᇦ඲యࡢᖹᆒศᩓ࡛࠶ࡿ㸬

ᘧ(1)ࡢ࢚ࣥࢺࣟࣆ࣮࡜ᘧ(2)ࡢศᩓࡣ㸪ῧᏐࡢik/K࡟ࡣ㛵ಀ࡞ࡃᖖ࡟ࡑࢀࡒࢀ∂2s/∂pj2 > 0࡜∂2V/∂pj2 > 0 (j = 1, 2,㺃㺃㺃,m)ࡀᡂࡾ❧ࡘࡢ࡛㸪ࡑࡢᙧ≧ࡣࡇࢀࡽࡢ㍈ࡢṇࡢ᪉ྥ࡟ᑐࡋ࡚ฝ࡛࠶ࡿ㸬ࡇࡢᛶ㉁࡟ࡼࡾ㸪࢚ࣥࢺࣟࣆ࣮࠶ࡿ࠸ࡣศᩓ࡟ࡼࡿ᝟ሗࡢホ౯ࡣ㸪࡯ࡰྠࡌ࡛࠶ࡿ㸬ࡋ࠿ࡋ㸪ཝᐦ࡟⪃࠼ࡿ࡜㐪࠺㸬ࡑࡢ㐪࠸ࢆ௨ୗ᳨࡛ウࡍࡿ㸬

2㸬࢚ࣥࢺࣟࣆ࣮࡜ศᩓࡢ㛵ಀ
ᘧ(1)ࡢ࢚ࣥࢺࣟࣆ࣮s࡜ᘧ(2)ࡢศᩓVࡣ㸪ᆅ㉁యࡢᏑᅾᗘ
p࡛⤖ࡧࡘࡅࡽࢀ࡚࠸ࡿ㸬ࡇࡢ୧⪅ࡢ㛵ಀࢆ➨1ᅗ࡟♧ࡍ㸬
➨1ᅗ࡛㸪᭱ࡶኴ࠸⥺ࡣᆅ㉁యࡢ✀㢮ᩘmࡀ2ࡢሙྜࢆ♧ࡍ㸬ࡇࡢሙྜ㸪ᆅ㉁య1ࡢᏑᅾᗘࢆp1࡛⾲ࡍ࡜ᆅ㉁య2ࡢᏑ
ᅾᗘp2ࡣ㸪p2 = 1−p1࡛࠶ࡿ㸬ࡋࡓࡀࡗ࡚㸪⊂❧ኚᩘࡢᩘࡀ1ࡍ࡞ࢃࡕ⮬⏤ᗘࡀ1࡞ࡢ࡛㸪s࡜Vࡣ୍ᑐ୍ᑐᛂࡍࡿ㸬ᅗ࡛ࡣ0 ӌ p1 ӌ 1/2ࡢ⠊ᅖࡀ♧ࡋ࡚࠶ࡿ(p2 = 1࡜࡞ࡿⅬࢆA1㸪p2 =1࡜࡞ࡿⅬࢆA2࡜ࡋࡓ࡜ࡁ㸪sࡶVࡶ⥺ศA1A2ࡢ㔜ᚰ≡୰ⅬG12࡟ᑐࡋ࡚ᑐ㇟࡛࠶ࡿ)㸬ࡇࢀ࡟ᑐࡋ࡚㸪m = 3ࡢሙྜࡣ㸪⊂❧ኚᩘࡣp1࡜p2࡛p3 = 1−(p1+p2)࡛࠶ࡿࡓࡵ⮬⏤ᗘࡀ2࡜࡞ࡿ㸬ࡑࡇ࡛㸪➨1ᅗ࡛ࡣ㸪୕ゅᙧA1A2A3 (A3ࡣp3 = 1࡜࡞ࡿⅬ)ࡢ
㔜ᚰࢆⅬG123࡜ࡋ࡚㸪p1㸪p 2㸪p3ࡢ㛵ಀࢆⅬG123࡜ⅬA1࠾ࡼࡧⅬG123࡜ⅬG12ࢆ⤖ࡪ2ᮏࡢ⥺ศ࡟㝈ࡗࡓ࡜ࡁࡢs࡜Vࡢ㛵ಀࢆ2␒┠࡟ኴ࠸⥺࡛⾲ࡋ࡚࠸ࡿ㸬ࡉࡽ࡟㸪➨1ᅗ࡟ࡣ㸪m =4㸪m = 5㸪m = 6ࡢሙྜࢆࡑࢀࡒࢀ3␒┠㸪4␒┠㸪5␒┠࡟ኴ࠸⥺࡛♧ࡋ࡚࠶ࡿ㸬ᅗ♧ࡉࢀࡓྛ⥺࡟࠾࠸࡚㸪p1㸪p2㸪∙∙∙㸪pmࡣm = 3ࡢሙྜࢆᣑᙇࡋࡓ㛵ಀ࡟࠶ࡿ㸬౛࠼ࡤ㸪m = 6ࡢሙྜ㸪5ḟඖࡢ༢యA1A2A3A4A5A6ࡢ㔜ᚰG123456࡜ձⅬA1㸪ղⅬG12㸪ճⅬG123㸪մ3ḟඖࡢ༢యA1A2A3A4(ᅄ㠃య)ࡢ㔜ᚰG1234㸪
յ4ḟඖࡢ༢యA1A2A3A4A5ࡢ㔜ᚰG12345ࢆ⤖ࡪ㛵ಀ࡟࠶ࡿሙྜࡀ♧ࡋ࡚࠶ࡿ㸬➨1ᅗ࡛᫂ࡽ࠿࡞ࡼ࠺࡟㸪ᆅ㉁యࡢ✀㢮ᩘmࡢቑຍ࡜࡜ࡶ࡟㸪࢚ࣥࢺࣟࣆ࣮ࡣ㡰ㄪ࡟ቑຍࡍࡿࡢ࡟ᑐࡋ㸪ศᩓࡣ࡯࡜
ࢇ࡝ቑຍࡋ࡞࠸(ᅗࡢᶓ㍈ࡢ᭱኱್ࡣ2࡛࠶ࡿࡢ࡟ᑐࡋ㸪⦪㍈ࡢࡑࢀࡣ0.5࡛࠶ࡿ)㸬ࡇࢀࡣ㸪ḟࡢᘧ(3)࡜(4)࡛♧ࡍࡼ࠺࡟㸪mࡢቑຍ࡟కࡗ࡚࢚ࣥࢺࣟࣆ࣮ࡢྲྀࡾᚓࡿ᭱኱್smMAX
ࡣⓎᩓࡍࡿࡢ࡟ᑐࡋ㸪ศᩓࡢྲྀࡾᚓࡿ᭱኱್VmMAXࡣ1/2࡟
཰᮰ࡍࡿࡓࡵ࡛࠶ࡿ㸬s mMAX = ln m
(3)
1⎛
1⎞
V mMAX = ⎜1 − ⎟(4)2⎝ m⎠࡞࠾㸪ࡇࢀࡽࡢ᭱኱್ࡣ㸪p1 = p2 = ͐ = pm = 1/m ࡢ࡜ࡁᚓ53ࡽࢀࡿ㸬

3㸬㸬࢚ࣥࢺࣟࣆ࣮࡜᝟ሗ㔞᪥ᮏㄒࡢ᝟ሗᏛࡢᩍ⛉᭩࡛ࡣ㸪㺀᝟ሗ࢚ࣥࢺࣟࣆ࣮㺁࡜㺀᝟ሗ㔞㺁ࡀ࡯ࡰྠ⩏ㄒ࡜ࡋ࡚౑ࢃࢀ࡚࠸ࡿ㸬ࡋ࠿ࡋ㸪௨ୗ࡟㏙࡭ࡿࡼ࠺࡟୧⪅ࡣ᫂☜࡟౑࠸ศࡅࡓ᪉ࡀࡼ࠸㸬
Wikipedia (2015/1/25)࡛ࡣ㸪”entropy (information theory)”
ࢆ ”entropy is a measure of unpredictability of informationcontent”(ᩳయࡣཎᩥ㸪ୗ⥺ࡣᘬ⏝⪅)࡜ㄝ᫂ࡋ࡚࠸ࡿ㸬ࡇࡢ㡯ࡢLanguages࡛᪥ᮏㄒ࡟⛣ࡿ࡜㺀᝟ሗ㔞㺁ࡀฟ࡚ࡁ࡚㸪㺀᝟ሗ㔞(ࡌࡻ࠺࡯࠺ࡾࡻ࠺㸪࢚ࣥࢺࣟࣆ࣮࡜ࡶ)ࡣ㸪᝟ሗ⌮ㄽࡢᴫᛕ࡛㸪࠶ࡿ࡛ࡁࡈ࡜(஦㇟)ࡀ㉳ࡁࡓ㝿ࠊࡑࢀࡀ࡝ࢀ࡯࡝㉳ࡇࡾ࡟ࡃ࠸࠿ࢆ⾲ࡍᑻᗘ࡛࠶ࡿ㺁(ୗ⥺㒊ᘬ⏝⪅)࡜ㄝ᫂ࡉࢀ࡚࠸ࡿ㸬➹⪅࡟࡜ࡗ࡚㺀᝟ሗ㔞㺁࡜࠸࠺ㄒࡣ㸪ྲྀᚓࡋࡓ᝟ሗ࡟ྵࡲࢀࡿఱࡽ࠿ࡢ㔞ࢆ⾲ࡋ࡚࠸ࡿ࡜ᛮ࠼ࡿ㸬ࡇࢀ࡟ᑐࡋ㸪࢚ࣥࢺࣟࣆ࣮ࡣ”unpredictability (ண ୙ྍ⬟ᛶ ҷ ୙☜࠿ࡉ)”ࡢᑻᗘ࡞ࡢ࡛㸪ྲྀᚓࡋࡼ࠺ࡋ࡚࠸ࡿ᝟ሗ࡟ྵࡲࢀࡿఱࡽ࠿ࡢ㔞࡛࠶ࡿ㸬ୖグࡢ᝟ሗ࢚ࣥࢺࣟࣆ࣮ࡢᐃ⩏ࢆ㸪ṇ㊰(2014)ࡢㄪᰝ㐍⾜ࡢࣔࢹࣝ࡟㐺⏝ࡍࡿ࡜㸪ㄪᰝẁ㝵(k−1)࡜k࡟࠾ࡅࡿࢭ࢚ࣝࣥ
ࢺࣟࣆ࣮ࢆࡑࢀࡒࢀ s(k−1) / K ࡜ sk / K ࡜᭩࠸ࡓሙྜ㸪ࡑࢀࡽࡢᕪ Δs(k −1)<k = s(k −1) / K − sk / K ࡀẁ㝵kࡢㄪᰝ࡛㝖ཤࡋࡓ࢚ࣥࢺࣟࣆ࣮࠶ࡿ࠸ࡣྲྀᚓࡋࡓ᝟ሗ㔞࡜ᐃ⩏࡛ࡁࡿ㸬ㄪᰝࡀጞࡲࡿ๓ࡢẁ㝵0࡟࠾ࡅࡿ࢚ࣥࢺࣟࣆ࣮ࡣ㸪ྲྀࡾᚓࡿ್ࡢ᭱኱್
smMAX ࡜ᐃ⩏ࡍࡿ࡜㸪Δsk / K = smMAX − sk / K ࢆẁ㝵kࡲ࡛࡟㝖ཤࡉࢀࡓ࢚ࣥࢺࣟࣆ࣮࠶ࡿ࠸ࡣྲྀᚓࡋࡓ᝟ሗ㔞࡜ᐃ⩏࡛ࡁࡿ㸬࡜ࡇࢁ࡛㸪࢚ࣥࢺࣟࣆ࣮ࡣ⇕ຊᏛ࡛ᑟධࡉࢀࡓᴫᛕ࡛㸪ࡑࡢ⤫ィຊᏛⓗㄝ᫂ࡀᘧ(1)࡛࠶ࡿ㸬ᆅ⌫࡛ࡣኴ㝧࠿ࡽࡢ㟁☢Ἴ࢚ࢿࣝࢠ࣮ࡸᆅ⌫ෆ㒊࠿ࡽࡢ⇕࢚ࢿࣝࢠ࣮࡟ࡼࡾ⏕≀άືࢆྵࡴྛ✀ࡢኚ໬ࡀ㉳ࡁ㸪ࡇࡢኚ໬࡟కࡗ࡚࢚ࣥࢺࣟ
ࣆ࣮ࡀⓎ⏕ࡍࡿ㸬⇕ຊᏛ࡟ࡼࡿ࡜㸪ࡇࡢኚ໬ࡀᣢ⥆ࡍࡿࡓࡵ࡟ࡣ㸪ᆅ⌫ࡣⓎ⏕ࡋࡓ࢚ࣥࢺࣟࣆ࣮ࢆእ⏺(㸻Ᏹᐂ✵㛫)࡟ᨺฟࡋ࡞ࡅࢀࡤ࡞ࡽ࡞࠸㸬௒㸪➢ྕࢆྲྀᚓࡣṇ㸪ᨺฟࡣ㈇࡜ᐃ⩏ࡍࡿ࡜㸪ᙜ↛ࡢࡇ࡜࡞ࡀࡽ㸪ᆅ⌫ࡀᨺฟࡍࡿ࢚ࣥࢺࣟࣆ࣮࡜Ᏹᐂ✵㛫ࡀཷࡅྲྀࡿ࢚ࣥࢺࣟࣆ࣮ࡣ➢ྕࡀ㏫࡛㸪⤯ᑐ್ࡣ➼ࡋ࠸㸬ࡇࢀ࡜ྠࡌ㛵ಀࡀ㸪ୖ㏙ࡋࡓ᝟ሗ࢚ࣥࢺ
ࣟࣆ࣮࡜ྲྀᚓ᝟ሗ㔞ࡢ㛫࡛ࡶᡂࡾ❧ࡘ㸬 ࡋࡓࡀࡗ࡚㸪ḟࡢᘧ(3)ࡀᡂࡾ❧ࡘ㸬
Δs k / K = Δs 0 <1 + Δs1< 2 + L + Δs ( K −1) < K(5)ࡍࢀࡤࡼ࠸㸬
ୖ㏙ࡢᩘᏛࡣ㸪ศᩓࡣྲྀᚓࡉࢀࡓ᝟ሗ࡟ྵࡲࢀࡿ࠶࠸ࡲ
࠸ࡉࢆ⾲ࡋ㸪ࡑࡢ㏫ᩘࡣྲྀᚓࡉࢀࡓ᝟ሗࡢ⢭ᗘ࡟ᑐᛂࡍࡿࡇ࡜ࢆ♧ࡋ࡚࠸ࡿ㸬ࡇࢀࢆṇ㊰(2014)ࡢㄪᰝ㐍ᒎࡢࣔࢹࣝ࡟ᙜ࡚ࡣࡵࡿ࡜㸪ㄪᰝẁ㝵(k−1)࡜k࡟࠾ࡅࡿศᩓࢆࡑࢀࡒࢀV(k−1) / K ࡜ Vk / K ࡜ ᭩ ࠸ ࡓ ሙ ྜ 㸪 ࡑ ࢀ ࡽ ࡢ ẚ
r(k−1)<k = Vk / K V(k−1) / K ࡣẁ㝵kࡢㄪᰝ࡟࠾ࡅࡿ࠶࠸ࡲ࠸ࡉࡢῶᑡ⋡ࢆ㸪ࡑࡢ㏫ᩘࡢ1/r(k−1)<kࡣ⢭ᗘࡢྥୖ⋡ࢆ⾲ࡍ㸬ࡲࡓ㸪

rk / K = Vk / K VmMAX ࡜1/rk/Kࡣࡑࢀࡒࢀẁ㝵kࡲ࡛࡟࠾ࡅࡿ࠶࠸ࡲ࠸ࡉࡢῶᑡ⋡࡜⢭ᗘࡢྥୖ⋡ࢆ⾲ࡋ࡚࠸ࡿ㸬ࡇࡢࡼ࠺࡟ᐃ⩏ࡍࡿ࡜㸪ศᩓ࡟ᑐࡋ࡚ࡣḟࡢᘧ(6)ࡀᡂࡾ❧ࡘ㸬
rk / K = r0 <1 ⋅ r1< 2 L r( K −1)< K(6)5㸬ࡲ࡜ࡵᆅ⌫⛉Ꮫⓗㄪᰝࡢ㐣⛬࡛ྲྀᚓࡉࢀࡓ᝟ሗ࡟ࡶ࡜࡙ࡃྛᆅ㉁యࡢᏑᅾᗘ࠿ࡽồࡵࡽࢀࡿ࢚ࣥࢺࣟࣆ࣮࠶ࡿ࠸ࡣศᩓࡣ㸪࡜ࡶ࡟ṇࡢ್࡛㸪ㄪᰝࡢ㐍⾜࡟క࠺⤯ᑐ್ࡢῶᑡࡣ୙☜ᐇᛶࡢῶᑡ࡟ᑐᛂࡋ࡚࠸ࡿ㸬୧⪅ࢆẚ㍑ࡍࡿ࡜㸪ࡑࢀࡽࡢ᭱኱್ࡣ㸪๓⪅ࡀᆅ㉁యࡢ✀㢮ࡢቑຍ࡟ᑐࡋᩄឤ࡛↓㝈኱࡟Ⓨᩓࡍࡿࡢ࡟ᑐࡋ㸪ᚋ⪅ࡣ㕌ឤ࡛1/2࡟཰᮰ࡍࡿ㸬୙☜ᐇᛶ
ࡢ㏫㸪ࡍ࡞ࢃࡕ☜ᐇᛶ࠶ࡿ࠸ࡣ⢭ᗘࢆ♧ࡍᅉᏊ࡜ࡋ࡚㸪๓⪅ࡣྲྀᚓ᝟ሗ㔞(㸻᭱኱࢚ࣥࢺࣟࣆ࣮࠿ࡽࡢᕪ)㸪ᚋ⪅ࡣศᩓ
ࡢ㏫ᩘ(㸻᭱኱ศᩓ࡟ᑐࡍࡿẚ)ࡀᐃ⩏࡛ࡁࡿ㸬ᘬ⏝ᩥ⊩ṇ㊰ᚭஓ (1972) 㹖⥺⢊ᮎἲ࡟ࡼࡿ࢔ࣝ࢝ࣜࢳࣙ࢘▼ࡢ⤌ᡂ࠾ࡼࡧᵓ㐀(Al/Si ⛛ᗎ↓⛛ᗎ)ࡢỴᐃ㸬㖔≀Ꮫ㞧ㄅ,vol. 10, no. 6, pp.413-425㸬ṇ㊰ᚭஓ (2003) ᆅ⌫⛉Ꮫ᝟ሗࡢホ౯㸬᝟ሗᆅ㉁㸪vol. 14, no.
4, pp.285-299.
Shoji, T. (2006) Optimal systems of geoscience surveying – apreliminary discussion. Computers & Geosci., vol. 32, no. 1,pp.1128-1138.
ṇ㊰ᚭஓ (2014) ᥈ᰝࡢ㐍⾜࡟క࠺ᆅ⌫᝟ሗ࢚ࣥࢺࣟࣆ࣮
ࡢኚ໬㸬᝟ሗᆅ㉁㸪vol. 25, no. 2, pp.68-71.

࠸ࡲኚ㔞 x ࡟㛵ࡋ࡚ n ಶࡢ᥎ᐃ್ x1 㸪 x2 㸪͐㸪 xn ࡜ࡑࢀࡽࡢศᩓࡑࢀࡒࢀ V ( x1 ) 㸪V ( x 2 ) 㸪͐㸪V ( x n ) ࡀ࠶ࡿ࡜ࡁ㸪ࡇࢀࡽ࠿ࡽࡼࡾⰋ࠸᥎ᐃ್ x ࡣ㸪ḟᘧࡢࡼ࠺࡟㔜ࡳ௜ࡁᖹᆒ࡜ࡋ࡚ồࡵࡿࡇ࡜ࡀ࡛ࡁࡿ㸬n⎛ n⎞⎜⎜ ∑ a i = 1⎟⎟
x = ∑ ai xii =1⎝ i =1⎠ศᩓࡢຍἲᛶ࡟ࡼࡾ㸪 x ࡢศᩓ V (x ) ࡣ㸪
⎛ n
⎞ nV ( x ) = V ⎜⎜ ∑ ai xi ⎟⎟ = ∑ ai2V ( xi )⎝ i =1⎠ i =1࡛࠶ࡿ㸬ࡇࡢ V (x ) ࡣ λ ࢆ Lagrange ࡢᮍᐃ஌ᩘ࡜ࡍࡿ࡜㸪a1V ( x1 ) = a 2V ( x2 ) = L = anV ( xn ) = λ 2ࡢ࡜ࡁ᭱ᑠ࡟࡞ࡿ(౛࠼ࡤ㸪ṇ㊰㸪1972)㸬ࡋࡓࡀࡗ࡚㸪ࡼࡾⰋ࠸᥎ᐃ್ࢆồࡵࡿ࡟ࡣ㸪ྛ᥎ᐃ್ࡢศᩓࡢ㏫ᩘࢆ㔜ࡳ࡜㼂㼍㼞㼕㼍㼚㼏㼑

4㸬ศᩓࡢ㏫ᩘ࡜᝟ሗࡢ⢭ᗘ㻜㻜㻚㻡㻝㻱㼚㼠㼞㼛㼜㼥㻝㻚㻡㻞➨ 1 ᅗ㸬࢚ࣥࢺࣟࣆ࣮s ࡜ศᩓ V ࡢ㛵ಀ㸬ྛ⥺ࡣ㸪m ᡂศ࠿ࡽ࡞ࡿ⣔࡛㸪(m−1)ḟඖࡢ༢యࡢ㡬Ⅼ࡜㔜ᚰࢆ⤖ࡪ⥺࡟ἢࡗ࡚Ꮡᅾᗘࡀኚ໬ࡋࡓሙྜ࡟ᑐᛂࡋ㸪ኴ࠸᪉࠿ࡽ⣽࠸᪉࡬ m ࡀ 2㸪3㸪4㸪5㸪6 ࡢሙྜ㸬