Материал: chap2
Ʉɨɷɮɮɢɰɢɟɧɬ, ɨɩɪɟɞɟɥɹɟɦɵɣ (II,6,6), ɨɛɨɡɧɚɱɚɸɬ ɢɧɨɝɞɚ IJb, ɚ (II,6,1) – IJɚ, ɟɫɥɢ ɧɟɬ ɨɛɴɟɞɢɧɟɧɢɣ ɪɚɧɝɨɜ IJɚ = IJb.
Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɬɪɢ ɤɨɷɮɮɢɰɢɟɧɬɚ r, ȡ, IJ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɫ ɟɞɢɧɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ, ɤɚɤ ɨɛɵɱɧɨ, ɢɦɟɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɢɡ N ɢɧɞɢɜɢɞɨɜ, ɤɚɠɞɵɣɢɡɤɨɬɨɪɵɯɦɨɠɟɬɛɵɬɶɨɯɚɪɚɤɬɟɪɢɡɨɜɚɧɫɩɨɦɨɳɶɸɡɧɚɱɟɧɢɣ ɞɜɭɯ ɩɪɢɡɧɚɤɨɜ X ɢ Y.
ȼɵɛɟɪɟɦ ɩɚɪɭ ɢɧɞɢɜɢɞɨɜ, ɧɚɩɪɢɦɟɪ, i ɢ j ɢ ɫɬɚɧɟɦ ɩɪɢɩɢɫɵɜɚɬɶ ɟɣ ɧɟɤɨɬɨɪɭɸ x – ɨɰɟɧɤɭ ɚij (ɤɨɧɤɪɟɬɢɡɚɰɢɹ ɨɰɟɧɨɤ ɛɭɞɟɬ ɞɚɧɚ ɧɢɠɟ), ɨɛɥɚɞɚɸɳɭɸ ɫɜɨɣɫɬɜɨɦ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɨɫɬɢ: ɚij= - ɚij Ⱥɧɚɥɨɝɢɱɧɨ ɜɜɟɞɟɦ ɭ – ɨɰɟɧɤɭ bij.
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Ɋɚɫɫɦɨɬɪɢɦ ɜɟɥɢɱɢɧɭ
¦¦aijbij
Ƚi j
¦¦aij2 ¦¦bij2
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Ɇɵɭɠɟɜɢɞɟɥɢ (II,6,6), ɱɬɨɞɥɹɜɟɥɢɱɢɧɵ |
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ɟɫɥɢRi |
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aij ® |
1,ɟɫɥɢRi(x) |
% R(jx) |
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(ɝɞɟ R(x) |
– ɪɚɧɝɩɨ X i-ɝɨɷɥɟɦɟɧɬɚ) ɢɚɧɚɥɨɝɢɱɧɨɣ ɜɟɥɢɱɢɧɵ bij: Ƚ = IJ. |
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ɉɭɫɬɶ |
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aij = xj – xi, a, bij = yj – yi |
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ɬɨɝɞɚ |
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¦¦(xj xi )(y j yi ) 2N¦xi yi 2¦¦xi y j |
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¦¦(xj xi )2 |
2N¦xi2 2(¦xi )2 |
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Ɍɟɩɟɪɶ |
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ȿɫɥɢ ɩɨɥɨɠɢɬɶ |
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R(x) |
R(x) , ɚ |
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R( y) |
R( y) , ɬɨ ɦɨɠɧɨ ɚɧɚɥɨɝɢɱɧɨ ɩɪɟɞɵɞɭɳɟɦɭ |
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ɩɨɤɚɡɚɬɶ, ɱɬɨ Ƚ ɨɛɪɚɳɚɟɬɫɹ ɩɪɢ ɷɬɨɦ ɜ ȡ. ɗɬɨ ɪɚɫɫɦɨɬɪɟɧɢɟ ɫɨɫɬɚɜɢɬ ɞɥɹ ɱɢɬɚɬɟɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɟ ɭɩɪɚɠɧɟɧɢɟ 57.
Ɇɵ ɠɟ ɫɨɲɥɟɦɫɹ ɧɚ § 5 ɝɥɚɜɵ II, ɝɞɟ ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ȡ ɹɜɥɹɟɬɫɹ r, ɩɪɢɦɟɧɟɧɧɵɦ ɤ ɪɚɧɝɚɦ, ɚ ɬɚɤ ɤɚɤ ɞɥɹ r ɪɚɫɫɦɨɬɪɟɧɢɟ ɩɪɨɜɟɞɟɧɨ, ɬɨ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɫɬɪɨɝɨɫɬɢ ɢɡɥɨɠɟɧɢɹ, ɜɵɤɥɚɞɤɢ ɞɚɧɧɨɝɨ ɭɩɪɚɠɧɟɧɢɹ ɜ ɬɟɤɫɬɟ ɤɧɢɝɢ ɧɟ ɹɜɥɹɸɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦɢ. ȼ ɡɚɤɥɸɱɟɧɢɟ ɜɵɜɟɞɟɦɧɟɪɚɜɟɧɫɬɜɨ Ʉɨɲɢ.
Ɉɱɟɜɢɞɧɨɟ ɧɟɪɚɜɟɧɫɬɜɨ (A B )2 |
t 0 ɦɨɠɧɨɩɟɪɟɩɢɫɚɬɶ ɜɜɢɞɟ |
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B2 |
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ɉɨɥɚɝɚɹ |
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Aij |
aij |
ɢ Bij |
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¦¦aij2 |
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[122]
ɢɫɭɦɦɢɪɭɹɜɫɟɜɨɡɦɨɠɧɵɟ ɧɟɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɢɦ:
84
1 |
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¦¦bij2 |
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2 ¦¦aij2 |
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¦¦bij2 |
¦¦aij2 ¦¦bij2 |
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Ɍɚɤ ɤɚɤ ɥɟɜɚɹ ɱɚɫɬɶ ɪɚɜɧɚ 1, ɬɨ ɧɟɪɚɜɟɧɫɬɜɨ Ʉɨɲɢ ɞɨɤɚɡɚɧɨ. ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɨɧɨ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɪɚɜɟɧɫɬɜɨ, ɟɫɥɢ ɜɫɟ aij = Įbij (ɭɛɟɞɢɬɶɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ!), ɱɬɨ ɢ ɛɵɥɨ ɧɚɦɢ ɪɚɧɟɟ ɢɫɩɨɥɶɡɨɜɚɧɨ.
ɇɚɤɨɧɟɰ, ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɨɛɚ ɩɪɢɡɧɚɤɚ ɢɡɦɟɪɟɧɵ ɧɚ ɭɪɨɜɧɟ ɧɚɥɢɱɢɹ – ɨɬɫɭɬɫɬɜɢɹ.
ɉɭɫɬɶ ɢɧɞɟɤɫ 1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɥɢɱɢɸ, ɚ 2 ɨɬɫɭɬɫɬɜɢɸ ɩɪɢɡɧɚɤɚ, ɬɨɝɞɚ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɬɚɛɥɢɰɚɞɥɹɩɪɢɡɧɚɤɨɜ X ɢ Y ɩɪɢɧɢɦɚɟɬɜɢɞ:
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x1 |
N11 |
N12 |
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N21 |
N22 |
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N(yj) |
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N(y2) |
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Ʉɚɠɞɵɣ ɷɥɟɦɟɧɬ ɩɟɪɜɨɣ ɤɥɟɬɤɢ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɞɢɚɝɨɧɚɥɢ ɩɪɢ ɫɨɩɨɫɬɚɜɥɟɧɢɢ ɫ ɷɥɟɦɟɧɬɨɦ ɜɬɨɪɨɣɩɨɪɨɞɢɬ +1, ɜɫɟɝɨɬɚɤɢɯ +1 ɜ S ɜɨɣɞɟɬ N11·N22.
ɋɪɚɜɧɟɧɢɟ ɷɥɟɦɟɧɬɨɜɨɬɪɢɰɚɬɟɥɶɧɨɣɞɢɚɝɨɧɚɥɢɩɨɪɨɞɢɬ N12·N21, ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɟɞɢɧɢɰ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
SN11N22 N12 N21;
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N(x2 )>N(x2 ) 1@; ɚ |
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Ⱥɧɚɥɨɝɢɱɧɨ: |
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[123]
ɬɟɩɟɪɶɤɨɷɮɮɢɰɢɟɧɬɄɟɧɞɷɥɚ, ɨɩɪɟɞɟɥɹɟɦɵɣ (II,6,5):
WN11N22 N12 N21
N(x1)N(x2 )N(y1)N(y2 )
ɬɚɤɢɦɨɛɪɚɡɨɦ, ɫɨɜɩɚɞɚɟɬɫɤɨɷɮɮɢɰɢɟɧɬɨɦ Ɏ (II,3,2).
ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɩɪɨɹɫɧɹɟɬ ɫɦɵɫɥ ɮɨɪɦɚɥɶɧɨ ɜɜɟɞɟɧɧɨɝɨ ɪɚɧɟɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɤɨɧɬɢɧɝɟɧɰɢɢ.
7. ɗɧɬɪɨɩɢɣɧɵɟ ɦɟɪɵ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ
ɉɭɫɬɶ ɧɟɤɨɬɨɪɨɟ ɫɨɛɵɬɢɟ ɦɨɠɟɬ ɢɦɟɬɶ k ɪɚɡɥɢɱɧɵɯ ɢɫɯɨɞɨɜ Ai (i |
1,k) |
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ɤɨɬɨɪɵɯ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ P(Ai). əɫɧɨ, ɱɬɨ |
¦P(Ai ) 1. ɇɚɩɪɢɦɟɪ, |
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ɫɢɦɦɟɬɪɢɱɧɨɣɦɨɧɟɬɵ k = 2, Ⱥ1 — ɜɵɩɚɞɟɧɢɟ ɝɟɪɛɚ, A2 |
— ɪɟɲɤɢ, P(A ) |
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Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɦɵ ɯɨɬɢɦ ɩɪɟɞɫɤɚɡɚɬɶ |
ɢɫɯɨɞ |
ɢɫɩɵɬɚɧɢɹ. ȿɫɥɢ |
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ɩɪɟɞɨɩɪɟɞɟɥɟɧ. ȿɫɥɢ k = 2, ɬɨ ɩɨɹɜɥɹɟɬɫɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ, ɤɨɬɨɪɚɹ ɦɚɤɫɢɦɚɥɶɧɚ ɩɪɢ Ɋ(A1) =
85
= Ɋ(A2). ȿɫɥɢ Ɋ(A1) > Ɋ(A2), ɬɨ ɱɟɦ ɛɨɥɶɲɟ Ɋ(A1), ɬɟɦ ɦɟɧɶɲɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɩɪɟɞɫɤɚɡɚɧɢɹ. ȼ ɩɪɟɞɟɥɟ, ɤɨɝɞɚ Ɋ(Ⱥ1) = 1 (Ɋ(A2) = 0), ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɢɫɱɟɡɚɟɬ: ɜɨ ɜɫɟɯ ɢɫɩɵɬɚɧɢɹɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɢɫɯɨɞ A1.
ɑɟɦ ɛɨɥɶɲɟ k, ɬɟɦ ɦɟɧɟɟ ɨɩɪɟɞɟɥɟɧɧɵ ɩɪɟɞɫɤɚɡɚɧɢɹ, ɬɟɦ ɛɨɥɶɲɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ. ɉɨ
k
Ʉ. ɒɟɧɧɨɧɭ, ɦɟɪɨɣ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ E ¦P(Ai )log P(Ai ) , ɧɚɡɵɜɚɟɦɚɹ
i 1
ɷɧɬɪɨɩɢɟɣ. ȿɫɥɢ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɧɟɬ ɢ, ɫɤɚɠɟɦ, ɪɟɚɥɢɡɭɟɬɫɹ l-ɨɟ ɫɨɫɬɨɹɧɢɟ, ɬ.ɟ. Ɋ (Ⱥl) = 1, ɚ ɜɫɟ ɨɫɬɚɥɶɧɵɟ Ɋ (Ai) = 0, ɬɨ ȿ ɨɱɟɜɢɞɧɨ, ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ. ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɦɚɤɫɢɦɚɥɶɧɚ, ɟɫɥɢ ɜɫɟ ɢɫɯɨɞɵ ɪɚɜɧɨɜɨɡɦɨɠɧɵ, ɬ.ɟ. Ɋ (Ai) = 1/k. ɉɪɢ ɷɬɨɦ ȿmax = log k. ɑɟɦ ɛɨɥɶɲɟ k, ɬɟɦ ɛɨɥɶɲɟ ȿmax. ɂɬɚɤ, 0 ȿ log k.
ɉɭɫɬɶ N ɢɧɞɢɜɢɞɨɜ ɧɟɤɨɬɨɪɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɨɛɥɚɞɚɸɬ ɧɟɤɨɬɨɪɵɦ ɩɪɢɡɧɚɤɨɦ X, ɢ ɫɨɛɵɬɢɟ Ⱥi ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɡɧɚɱɟɧɢɟ ɩɪɢɡɧɚɤɚ ɪɚɜɧɨ xi. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Ni ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ X = xi. ȿɫɥɢ N ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɨ, ɬɨ Ɋi = Ni/N, ɚ ȿ – ɦɟɪɚ «ɪɚɫɩɵɥɟɧɧɨɫɬɢ» ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ⱦɥɹ ɫɨɩɨɫɬɚɜɥɟɧɢɹ ɪɚɡɥɢɱɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɩɟɪɟɣɬɢ ɤ ɧɨɪɦɢɪɨɜɚɧɧɨɦɭ ɤɨɷɮɮɢɰɢɟɧɬɭ İ = ȿ/ȿmax. ȼɟɥɢɱɢɧɚ İ, ɩɪɢɧɢɦɚɸɳɚɹ ɡɧɚɱɟɧɢɹ ɦɟɠɞɭ 0 ɢ 1, ɹɜɥɹɟɬɫɹ ɚɧɚɥɨɝɨɦ ɞɢɫɩɟɪɫɢɢ.
[124]
ɉɟɪɟɣɞɟɦ ɤ ɞɜɭɯɦɟɪɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹɦ ɞɥɹ ɩɪɢɡɧɚɤɨɜ X ɢ Y ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɷɦɩɢɪɢɱɟɫɤɢɣɦɚɬɟɪɢɚɥɫɜɟɞɟɧ ɜɤɨɪɪɟɥɹɰɢɨɧɧɭɸɬɚɛɥɢɰɭ {Nij}.
Ɍɟɩɟɪɶ
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E¦¦Pij log Pij , i 1 j 1
ɝɞɟ Ɋij = Nij/N ɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɜɟɞɟɬɫɹ ɩɨ ɜɫɟɦ ɤɥɟɬɤɚɦ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ. Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɦɵ ɧɟ ɭɤɚɡɵɜɚɟɦ ɨɫɧɨɜɚɧɢɟ ɥɨɝɚɪɢɮɦɚ, ɬɚɤ ɤɚɤ ɨɛɫɭɠɞɚɟɦɵɟ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɩɨɤɚɡɚɬɟɥɢ İɢȜ, ɨɬɧɟɝɨɧɟɡɚɜɢɫɹɬ.
ɍɩɪɚɠɧɟɧɢɟ 58. ɉɨɤɚɡɚɬɶ, ɱɬɨ ȿmax = log kl
ɍɩɪɚɠɧɟɧɢɟ 59. ɉɨɤɚɡɚɬɶ, ɱɬɨɬɟɩɟɪɶ
kl
N log N ¦¦Nij log Nij
Hi 1 j 1
N logkl
ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɢɫɩɨɥɶɡɭɟɬɫɹɞɥɹɪɚɫɱɟɬɚ ɷɧɬɪɨɩɢɣɧɨɣɦɟɪɵɞɢɫɩɟɪɫɢɢ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɬɚɤ ɧɚɡɵɜɚɟɦɭɸ ɷɧɬɪɨɩɢɣɧɭɸ ɦɟɪɭ ɫɜɹɡɢ. ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ Y- ɪɚɫɩɪɟɞɟɥɟɧɢɹ
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ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ Y-ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɭ ɢɧɞɢɜɢɞɨɜ ɫ X = ɯi, ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɭɫɥɨɜɧɚɹ |
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ȼ ɢɬɨɝɨɜɭɸ ɭɫɥɨɜɧɭɸ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɤɚɠɞɚɹ ɫɬɪɨɱɤɚ ɬɚɛɥɢɰɵ ɞɚɟɬ ɜɤɥɚɞ c ɭɞɟɥɶɧɵɦ ɜɟɫɨɦ N(xi)/N, ɬ.ɟ. ɩɨɥɧɚɹɭɫɥɨɜɧɚɹɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ Y-ɪɚɫɩɪɟɞɟɥɟɧɢɹ:
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Ɇɟɪɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ X ɢ Y ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɜɟɥɢɱɢɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ
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[125]
ɍɩɪɚɠɧɟɧɢɟ 60. Ɋɚɫɫɦɨɬɪɟɬɶ ɞɥɹ ɩɪɨɫɬɟɣɲɢɯ ɬɚɛɥɢɰ 2x2 ɫɥɭɱɚɣ ɨɬɫɭɬɫɬɜɢɹ ɫɜɹɡɢ ɢ ɩɨɤɚɡɚɬɶ, ɱɬɨȜ = 0. ɍɤɚɡɚɧɢɟ: ɢɫɩɨɥɶɡɨɜɚɬɶ, ɱɬɨ Nij = N(xi)N(yj)/N.
ɍɩɪɚɠɧɟɧɢɟ 61. Ɋɚɫɫɦɨɬɪɟɬɶ ɫɥɭɱɚɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɫɜɹɡɢ ɢ ɩɨɤɚɡɚɬɶ, ɱɬɨ Ȝ = 1. ɍɤɚɡɚɧɢɟ: ɭɱɟɫɬɶ, ɱɬɨɬɚɛɥɢɰɚ ɩɪɢɧɢɦɚɟɬɞɢɚɝɨɧɚɥɶɧɵɣ ɜɢɞ. ɂɬɚɤ, 0 Ȝ 1. ɑɟɦɛɨɥɶɲɟȜ, ɬɟɦ ɛɨɥɶɲɟɫɜɹɡɶɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ.
ɍɩɪɚɠɧɟɧɢɟ 62. ȼɵɱɢɫɥɢɬɶİɢȜy/x ɞɥɹɫɥɟɞɭɸɳɟɣ ɬɚɛɥɢɰɵ:
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Ɉɬɜɟɬ: İ = 0,872, ȿɭ/ɯ1 = 0,664, ȿɭ/ɯ2= 0,660, ȿɭ/ɯ3 = 0,582, Ȝy/x= 0,086.
ɍɩɪɚɠɧɟɧɢɟ 63. Ɉɛɪɚɬɢɦɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɢɸ ɫɜɹɡɢ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɪɚɛɨɬɨɣ ɢ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɨɣ. Ⱦɥɹ ɬɚɛɥɢɰɵ 18 (ɪɚɛɨɬɧɢɤɢ ɜ ɜɨɡɪɚɫɬɟ ɞɨ 30 ɥɟɬ) ɧɚɣɬɢȜ.
Ɉɬɜɟɬ: ȿɭ/ɯ1 = 0,289, ȿɭ/ɯ2 = 0,396, ȿɭ/ɯ3 = 0,383, ȿɭ/ɯ = 0,348, Ȝ = 0,030.
ɍɩɪɚɠɧɟɧɢɟ 64. Ⱦɥɹɬɚɛɥɢɰɵ 19 (ɪɚɛɨɬɧɢɤɢɫɬɚɪɲɟ 30 ɥɟɬ) ɧɚɣɬɢȜ. Ɉɬɜɟɬ: Ȝ = 0,014. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɜɹɡɶ ɦɟɠɞɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ ɩɨɤɚɡɚɬɟɥɹɦɢ ɛɨɥɟɟ ɬɟɫɧɚɹ ɞɥɹ ɦɨɥɨɞɵɯ
ɪɚɛɨɬɧɢɤɨɜ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɜɟɪɧɟɦɫɹ ɤ ɷɬɨɦɭ ɜɨɩɪɨɫɭ ɟɳɟ ɪɚɡ, ɢɫɩɨɥɶɡɭɹ ɞɪɭɝɢɟ ɦɟɬɨɞɵ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨɢɡɭɱɟɧɢɹɫɜɹɡɟɣ (§ 8 ɝɥɚɜɵ II).
ɉɪɢɦɟɪ 24. ɉɪɟɞɫɬɚɜɥɹɟɬ ɧɟɫɨɦɧɟɧɧɵɣ ɢɧɬɟɪɟɫ ɡɚɞɚɱɚ ɨ ɫɜɹɡɢ ɢɧɬɟɝɪɚɥɶɧɨɣ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ ɫ ɱɚɫɬɧɵɦɢ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɹɦɢ (ɨɬɞɟɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɪɚɛɨɱɟɣ ɫɢɬɭɚɰɢɢ).
ȼ ɤɚɱɟɫɬɜɟ ɷɥɟɦɟɧɬɨɜ ɨɛɵɱɧɨ ɜɵɞɟɥɹɸɬ: 1) ɫɨɞɟɪɠɚɧɢɟ ɬɪɭɞɚ (ɫɨɜɨɤɭɩɧɨɫɬɶ ɬɪɭɞɨɜɵɯ ɮɭɧɤɰɢɣ, ɜɵɩɨɥɧɹɟɦɵɯ ɜ ɩɪɨɰɟɫɫɟ ɫɨɡɞɚɧɢɹ ɩɨɬɪɟɛɢɬɟɥɶɧɵɯ ɫɬɨɢɦɨɫɬɟɣ ɜ ɩɪɨɰɟɫɫɟ ɬɪɭɞɚ), 2) ɭɫɥɨɜɢɹ (ɮɚɤɬɨɪɵ, ɩɨɞ ɜɨɡɞɟɣɫɬɜɢɟɦ ɤɨɬɨɪɵɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɪɭɞɨɜɚɹ ɞɟɹɬɟɥɶɧɨɫɬɶ: ɫɦɟɧɧɨɫɬɶ, ɮɢɡɢɱɟɫɤɚɹ ɧɚɝɪɭɡɤɚ, ɫɨɫɬɨɹɧɢɟ ɨɤɪɭɠɚɸɳɟɣɫɪɟɞɵɢɬ.ɞ.);
[126]
3) ɨɪɝɚɧɢɡɚɰɢɹ (ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɟɪɨɩɪɢɹɬɢɣ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ ɪɚɰɢɨɧɚɥɶɧɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɪɚɛɨɱɟɣ ɫɢɥɵ); 4) ɨɩɥɚɬɚ; 5) ɦɟɠɥɢɱɧɨɫɬɧɵɟ ɨɬɧɨɲɟɧɢɹ ɢ ɬ.ɞ.
Ɉɫɨɡɧɚɜɚɹ, ɱɬɨ ɱɟɥɨɜɟɤ ɧɟ ɦɨɠɟɬ ɬɨɱɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɤɥɚɞ, ɤɨɬɨɪɵɣ ɜɧɨɫɢɬ ɜ ɨɛɳɟɟ ɫɨɫɬɨɹɧɢɟ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ ɭɞɨɜɥɟɬɜɨɪɟɧɢɟ ɨɬɞɟɥɶɧɵɯ ɩɨɬɪɟɛɧɨɫɬɟɣ, ɦɵ ɨɬɤɚɡɚɥɢɫɶ ɨɬ ɦɟɬɨɞɚ ɪɚɧɠɢɪɨɜɚɧɢɹ ɪɚɡɥɢɱɧɵɯ ɮɚɤɬɨɪɨɜ. Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɨɛɫɭɠɞɚɟɦɨɣ ɫɜɹɡɢ ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ ɪɚɡɥɢɱɧɵɟ ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɩɨɤɚɡɚɬɟɥɢ, ɤɨɬɨɪɵɟ ɜɵɱɢɫɥɹɥɢɫɶ ɞɥɹ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɞɧɢɦ ɢɡ ɩɪɢɡɧɚɤɨɜ ɹɜɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɚɹ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɢ ɞɪɭɝɢɦ – ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ-ɱɚɫɬɧɵɟ.
Ⱦɥɹ T ɢ Ȝ ɷɥɟɦɟɧɬɵ ɪɚɫɩɨɥɨɠɢɥɢɫɶ ɬɚɤ: ɫɨɞɟɪɠɚɧɢɟ ɬɪɭɞɚ, ɨɪɝɚɧɢɡɚɰɢɹ, ɨɩɥɚɬɚ, ɨɬɧɨɲɟɧɢɹ ɫ ɚɞɦɢɧɢɫɬɪɚɰɢɟɣ ɢ ɬ.ɞ. (ɫɦ. ɬɚɤɠɟ § 8 ɝɥ. II). Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɫɥɟɞɭɟɬ ɭɱɢɬɵɜɚɬɶ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɷɥɟɦɟɧɬɵ ɧɟ ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ: ɫɨɞɟɪɠɚɧɢɟ ɬɪɭɞɚ, ɧɚɩɪɢɦɟɪ, ɧɟɥɶɡɹ ɫɱɢɬɚɬɶ «ɨɱɢɳɟɧɧɵɦ» ɨɬ ɜɥɢɹɧɢɹ ɡɚɪɩɥɚɬɵ, ɢɛɨ ɜ ɫɪɟɞɧɟɦ ɛɨɥɟɟ ɫɨɞɟɪɠɚɬɟɥɶɧɚɹ ɪɚɛɨɬɚ ɜɵɲɟ ɨɩɥɚɱɢɜɚɟɬɫɹ ɢ ɬ.ɞ. ɋɥɟɞɭɟɬ ɬɚɤɠɟ ɭɱɢɬɵɜɚɬɶ, ɱɬɨ ɪɟɱɶ ɢɞɟɬ ɨɛ ɨɰɟɧɤɚɯ ɷɥɟɦɟɧɬɨɜ, ɚ ɫɜɹɡɶ ɦɟɠɞɭ ɷɥɟɦɟɧɬɨɦ ɢ ɨɰɟɧɤɨɣ ɧɨɫɢɬ ɫɥɨɠɧɵɣ, ɨɩɨɫɪɟɞɫɬɜɨɜɚɧɧɵɣ
87
ɯɚɪɚɤɬɟɪ. ɇɚɩɪɢɦɟɪ, ɧɟɬ ɩɪɹɦɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɡɚɪɩɥɚɬɨɣ ɢ ɟɟ ɜɟɥɢɱɢɧɨɣ (ɜ ɧɚɲɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɛɵɥɨ ɭɫɬɚɧɨɜɥɟɧɨ ɧɚɥɢɱɢɟ U-ɨɛɪɚɡɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ25). Ɂɚɜɢɫɢɦɨɫɬɢ ɨɩɨɫɪɟɞɫɬɜɭɸɬɫɹ ɩɨɬɪɟɛɧɨɫɬɹɦɢ, ɩɪɢɬɹɡɚɧɢɹɦɢ. Ɍɚɤ, ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɡɚɪɩɥɚɬɨɣ ɡɚɜɢɫɢɬ ɧɟ ɫɬɨɥɶɤɨ ɨɬ ɟɟ «ɚɛɫɨɥɸɬɧɨɣ» ɜɟɥɢɱɢɧɵ, ɫɤɨɥɶɤɨ ɨɬ ɞɨɫɬɢɠɟɧɢɹ «ɧɨɪɦɵ», ɜ ɤɚɱɟɫɬɜɟ ɤɨɬɨɪɨɣ, ɤɚɤ ɭɞɚɥɨɫɶ ɭɫɬɚɧɨɜɢɬɶ, ɜɵɫɬɭɩɚɟɬ ɫɪɟɞɧɹɹ ɩɪɨɝɪɟɫɫɢɜɧɚɹ ɪɟɮɟɪɟɧɬɧɨɣ ɝɪɭɩɩɵ (ɞɥɹ ɪɚɛɨɬɧɢɤɨɜ ɩɪɨɦɵɲɥɟɧɧɵɯ ɩɪɟɞɩɪɢɹɬɢɣ ɟɸ ɨɤɚɡɚɥɚɫɶ ɢɯ ɫɨɰɢɚɥɶɧɨɩɪɨɮɟɫɫɢɨɧɚɥɶɧɚɹ ɝɪɭɩɩɚ). ȼɨ ɜɫɹɤɨɦ ɫɥɭɱɚɟ ɧɚɦɢ ɭɫɬɚɧɨɜɥɟɧɚ ɬɟɫɧɚɹ ɤɨɪɪɟɥɹɰɢɹ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɡɚɪɩɥɚɬɨɣ ɪɚɛɨɱɢɯ ɞɚɧɧɨɣ ɝɪɭɩɩɵ ɢ ɱɢɫɥɨɦ ɪɚɛɨɬɧɢɤɨɜ, ɩɨɥɭɱɚɸɳɢɯ ɡɚɪɩɥɚɬɭɧɟɧɢɠɟɫɪɟɞɧɟɩɪɨɝɪɟɫɫɢɜɧɨɣ26.
ɉɪɢɦɟɪ 25. Ʉɨɷɮɮɢɰɢɟɧɬ Ȝ, ɨɩɪɟɞɟɥɟɧɧɵɣ ɜɵɲɟ, ɨɩɢɫɵɜɚɟɬ ɜɥɢɹɧɢɟ X ɧɚ Y. Ɇɵ ɨɛɨɡɧɚɱɢɦ ɟɝɨ Ȝy/x. Ⱥɧɚɥɨɝɢɱɧɨ
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ɦɨɠɧɨ ɜɜɟɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬ O |
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Ex Ex / y |
, ɤɨɬɨɪɵɣ ɨɩɢɫɵɜɚɟɬ ɜɥɢɹɧɢɟ Y ɧɚ X. Ȝ |
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ɧɟɫɢɦɦɟɬɪɢɱɟɧ: ɜɨɨɛɳɟ ɝɨɜɨɪɹ Ȝy/x Ȝx/y ȿɫɥɢ ɢɡ ɫɨɞɟɪɠɚɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ ɹɫɧɨ, ɱɬɨ X ɦɨɠɟɬ ɜɥɢɹɬɶ ɧɚ Y ɢ Y ɧɚ X, ɬɨ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɜɵɱɢɫɥɢɬɶ ɨɛɚ ɤɨɷɮɮɢɰɢɟɧɬɚ. ɇɚɩɪɢɦɟɪ, ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɪɚɛɨɬɨɣ (Y), ɜɥɢɹɟɬ ɧɚ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ (X) ɢ ɧɚɨɛɨɪɨɬ. ɉɨɷɬɨɦɭɦɵɜɵɱɢɫɥɹɟɦɨɛɚɤɨɷɮɮɢɰɢɟɧɬɚ, ɢɫɩɨɥɶɡɭɹ ɢɯ ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɭɤɚɡɚɧɧɵɯ ɜɥɢɹɧɢɣ. Ɍɚɤ, ɜ ɤɨɧɤɪɟɬɧɨɦ ɢɫɫɥɟɞɨɜɚɧɢɢ ɪɚɛɨɱɢɯ ɂɥɶɢɱɟɜɫɤɨɝɨ ɫɭɞɨɪɟɦɨɧɬɧɨɝɨ ɡɚɜɨɞɚ (1974ɝ.) ɧɚɦɢ ɛɵɥɨ ɩɨɥɭɱɟɧɨɬɚɤɨɟɞɜɭɦɟɪɧɨɟɪɚɫɩɪɟɞɟɥɟɧɢɟɨɛɫɭɠɞɚɟɦɵɯɩɪɢɡɧɚɤɨɜ:
Ɍɚɛɥɢɰɚ31
ɋɜɹɡɶɦɟɠɞɭɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸɪɚɛɨɬɨɣɢɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸɫɩɟɰɢɚɥɶɧɨɫɬɶɸ
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y1 |
y2 |
y3 |
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x1 |
1105 |
30 |
110 |
1245 |
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x2 |
313 |
55 |
62 |
430 |
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x3 |
35 |
4 |
36 |
75 |
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N(yj) |
1453 |
89 |
208 |
1750 |
Ⱦɥɹ ɷɬɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ, ɨɤɚɡɵɜɚɟɬɫɹ, Ȝɭ/ɯ = 0,073, ɚ Ȝx/y = 0,057. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ ɜ ɛɨɥɶɲɟɣ ɦɟɪɟ ɜɥɢɹɟɬ ɧɚ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɪɚɛɨɬɨɣ (ɩɪɟɞɩɪɢɹɬɢɟɦ), ɱɟɦ ɧɚɨɛɨɪɨɬ. ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɨɬɧɨɫɢɬɫɹ ɤ ɥɨɤɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ɨɩɪɟɞɟɥɟɧɧɨɝɨ, ɜɟɫɶɦɚ ɫɩɟɰɢɮɢɱɟɫɤɨɝɨ ɩɪɟɞɩɪɢɹɬɢɹ. Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɝɨ ɜɨɩɪɨɫɚ ɜ ɰɟɥɨɦ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɫɬɢ ɞɚɥɶɧɟɣɲɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ. ȼ ɧɚɲɭ ɡɚɞɚɱɭ ɡɞɟɫɶ ɜɯɨɞɢɥɨ ɨɡɧɚɤɨɦɥɟɧɢɟ ɫ ɢɞɟɟɣ ɦɟɬɨɞɚ ɢ ɬɟɯɧɢɤɨɣ ɜɵɱɢɫɥɟɧɢɹ.
ɍɩɪɚɠɧɟɧɢɟ 65. ȼɵɱɢɫɥɢɬɶ Ȝɭ/ɯ ɢ Ȝx/y ɞɥɹ ɬɚɛɥɢɰɵ ɢɡ ɭɩɪɚɠɧɟɧɢɹ 62 ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.
ɉɪɢɦɟɪ 26. ɗɧɬɪɨɩɢɣɧɵɣ ɚɧɚɥɢɡ ɫɨɰɢɚɥɶɧɵɯ ɫɬɪɭɤɬɭɪ.
ȼ ɲɟɫɬɢɞɟɫɹɬɵɟ ɝɨɞɵ Ɉ.ɂ. ɒɤɚɪɚɬɚɧ ɫ ɝɪɭɩɩɨɣ ɫɨɬɪɭɞɧɢɤɨɜ ɢɡɭɱɚɥ ɫɨɰɢɚɥɶɧɭɸ ɫɬɪɭɤɬɭɪɭ ɫɨɜɪɟɦɟɧɧɨɝɨ ɩɪɨɦɵɲɥɟɧɧɨɝɨ ɩɪɟɞɩɪɢɹɬɢɹ. Ɋɟɡɭɥɶɬɚɬɵ ɬɟɨɪɟɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ, ɛɚɡɢɪɭɸɳɟɝɨɫɹ ɧɚ ɡɧɚɱɢɬɟɥɶɧɨɦ ɷɦɩɢɪɢɱɟɫɤɨɦ ɦɚɬɟɪɢɚɥɟ, ɢɡɥɨɠɟɧɵ ɜ ɤɧɢɝɟ «ɉɪɨɛɥɟɦɵ ɫɨɰɢɚɥɶɧɨɣ ɫɬɪɭɤɬɭɪɵ ɪɚɛɨ-
25Ⱥɧɚɥɨɝɢɱɧɵɣ ɯɚɪɚɤɬɟɪ ɢɦɟɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɨɛɪɚɡɨɜɚɧɢɟɦ ɢ ɮɚɤɬɢɱɟɫɤɢɦ ɨɛɪɚɡɨɜɚɧɢɟɦ (ɨɛɫɥɟɞɨɜɚɥɢɫɶɪɚɛɨɬɧɢɤɢɩɪɨɦɵɲɥɟɧɧɵɯɩɪɟɞɩɪɢɹɬɢɣɝ. Ɉɞɟɫɫɵ).
26Ɇɚɤɫɢɦɟɧɤɨ ȼ. ɋ., ɉɨɩɨɜɚ ɂ. Ɇ. Ɂɚɪɚɛɨɬɧɚɹ ɩɥɚɬɚ ɤɚɤ ɮɚɤɬɨɪ ɫɬɢɦɭɥɢɪɨɜɚɧɢɹ ɬɪɭɞɨɜɨɣɞɟɹɬɟɥɶɧɨɫɬɢ.— ȼɤɧ.: ɉɪɨɛɥɟɦɵɷɤɨɧɨɦɢɤɢɦɨɪɹɢɦɢɪɨɜɨɝɨɨɤɟɚɧɚ. Ɉɞɟɫɫɚ, 1973, ʋ 2
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