Материал: chap2
[128]
ɱɟɝɨ ɤɥɚɫɫɚ ɋɋɋɊ» (Ɇ., 1970). ɋɨɜɦɟɫɬɧɨ ɫ ɂ.ɇ. Ɍɚɝɚɧɨɜɵɦ Ɉ.ɂ. ɒɤɚɪɚɬɚɧ ɩɪɟɞɩɪɢɧɢɦɚɥ ɩɨɩɵɬɤɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɤɨɥɢɱɟɫɬɜɟɧɧɨɝɨ ɦɟɬɨɞɚ ɞɥɹ ɢɡɭɱɟɧɢɹ ɭɤɚɡɚɧɧɨɣ ɫɬɪɭɤɬɭɪɵ. Ɉɞɧɚ ɢɡ ɧɢɯ, ɫɜɹɡɚɧɧɚɹ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɷɧɬɪɨɩɢɣɧɨɝɨ ɚɧɚɥɢɡɚ, ɛɵɥɚ ɢɡɥɨɠɟɧɚ ɜ ɠɭɪɧɚɥɟ «ȼɨɩɪɨɫɵ ɮɢɥɨɫɨɮɢɢ» (1969, ʋ5) ɢ ɩɪɢɜɥɟɤɥɚ ɜɧɢɦɚɧɢɟ ɫɨɰɢɨɥɨɝɨɜ, ɢɧɬɟɪɟɫɭɸɳɢɯɫɹ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɤɨɥɢɱɟɫɬɜɟɧɧɵɯ ɦɟɬɨɞɨɜ ɜ ɫɨɰɢɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ. Ɋɚɫɫɦɨɬɪɢɦ ɟɟ ɫɭɬɶ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɮɚɤɬɢɱɟɫɤɢ ɪɟɚɥɢɡɨɜɚɧɧɨɣ ɢɫɫɥɟɞɨɜɚɬɟɥɹɦɢ ɩɪɨɝɪɚɦɦɟ, ɧɨ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɨɛɨɡɧɚɱɟɧɢɣ ɩɪɟɞɵɞɭɳɢɯ ɩɚɪɚɝɪɚɮɨɜ.
Ɉɫɧɨɜɧɚɹ ɡɚɞɚɱɚ, ɤɨɬɨɪɚɹ ɪɟɲɚɥɚɫɶ ɚɜɬɨɪɚɦɢ ɫ ɩɨɦɨɳɶɸ ɷɧɬɪɨɩɢɣɧɨɝɨ ɚɧɚɥɢɡɚ, ɫɨɫɬɨɹɥɚ ɜ ɜɵɞɟɥɟɧɢɢ ɫɜɨɣɫɬɜ (ɩɪɢɡɧɚɤɨɜ), ɨɩɪɟɞɟɥɹɸɳɢɯ ɧɟɨɞɧɨɪɨɞɧɨɫɬɶ ɢɡɭɱɚɟɦɨɣ ɫɨɰɢɚɥɶɧɨɣ ɫɬɪɭɤɬɭɪɵ. Ɂɚɞɚɱɚ ɪɚɫɫɦɚɬɪɢɜɚɥɚɫɶ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬ.ɟ. ɢɡ ɝɢɩɨɬɟɬɢɱɟɫɤɨɝɨ ɧɚɛɨɪɚ ɡɧɚɱɢɦɵɯ ɩɪɢɡɧɚɤɨɜ (ɨɧ ɛɵɥ ɫɨɫɬɚɜɥɟɧ ɧɚ ɨɫɧɨɜɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ, ɫɸɞɚ ɜɨɲɥɢ ɬɚɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɤɚɤ ɨɛɪɚɡɨɜɚɧɢɟ, ɤɜɚɥɢɮɢɤɚɰɢɹ, ɩɨɥ, ɩɪɨɮɟɫɫɢɹ ɢ ɬ.ɞ. – ɜɫɟɝɨ 27 ɩɪɢɡɧɚɤɨɜ) ɚɜɬɨɪɵ ɜɵɞɟɥɹɥɢ ɤɚɠɞɵɣ ɪɚɡ ɬɪɨɣɤɭ ɩɪɢɡɧɚɤɨɜ, ɧɚɛɨɪ ɤɨɬɨɪɵɯ ɞɚɜɚɥ ɪɚɡɥɢɱɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɚ. ȼɫɟɝɨ ɬɚɤɢɯ ɩɪɨɫɬɪɚɧɫɬɜ ɦɨɠɧɨ ɜɵɞɟɥɢɬɶ C273 = 2925.
Ʌɨɝɢɤɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɬɚɤɨɜɚ. Ʉɚɠɞɵɣ ɢɧɞɢɜɢɞ ɞɚɧɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɧɨɫɢɬɟɥɟɦ ɪɚɡɥɢɱɧɵɯ ɩɪɢɡɧɚɤɨɜ. ɉɭɫɬɶ ɨɧ ɨɛɥɚɞɚɟɬ i-ɦ ɡɧɚɱɟɧɢɟɦ ɩɪɢɡɧɚɤɚ X, j-ɦ – ɍ, r-ɦ – Z (ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɝɪɚɧɢɱɟɧɢɟɦ, ɩɪɢɧɹɬɵɦ ɚɜɬɨɪɚɦɢ, ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɩɪɨɫɬɪɚɧɫɬɜɨ, ɨɩɪɟɞɟɥɹɟɦɨɟ ɩɪɢɡɧɚɤɚɦɢ X, Y, Z). ɂɧɮɨɪɦɚɰɢɸ ɨɛ ɨɞɧɨɦ ɢɧɞɢɜɢɞɟ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɜɟɤɬɨɪ ɜ ɞɚɧɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. ɋɨɜɨɤɭɩɧɨɫɬɢ ɢɡ N ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɢɧɞɢɜɢɞɨɜ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨɜɨɤɭɩɧɨɫɬɶ N ɜɟɤɬɨɪɨɜ. ɂɡ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɩɪɨɫɬɪɚɧɫɬɜ (ɧɚɛɨɪɨɜ ɩɪɢɡɧɚɤɨɜ) ɧɭɠɧɨ ɜɵɞɟɥɢɬɶ ɬɚɤɨɟ, ɜ ɤɨɬɨɪɨɦ ɜɟɤɬɨɪɵ ɥɟɠɚɬ ɧɚɢɛɨɥɟɟ ɩɥɨɬɧɵɦɢ ɝɪɭɩɩɚɦɢ (ɧɚɛɨɪ ɩɪɢɡɧɚɤɨɜ ɧɚɢɛɨɥɟɟ ɪɟɡɤɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɬ ɫɨɜɨɤɭɩɧɨɫɬɶ ɢɧɞɢɜɢɞɨɜ). Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ ɬɚɤɢɯ ɩɪɨɫɬɪɚɧɫɬɜ ɢ ɛɵɥ ɩɪɢɦɟɧɟɧ ɷɧɬɪɨɩɢɣɧɵɣ ɚɧɚɥɢɡ.
ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɡɚɩɨɥɧɟɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɟɤɬɨɪɚɦɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ
k l m
E¦¦¦Pijr log Pijr , i 1 j 1 r 1
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Nijr |
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ɝɞɟ |
P |
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(ɡɞɟɫɶ Nijr – ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ X = xi, Y = yi, Z = zi); |
i |
1,k; |
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ijr |
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j 1,l ; r |
1,m |
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[129] |
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ȿɫɥɢ ɜɟɤɬɨɪɵ ɪɚɜɧɨɦɟɪɧɨ ɡɚɩɨɥɧɹɸɬ ɩɪɨɫɬɪɚɧɫɬɜɨ, ɬɨ |
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N |
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logklm |
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ijr |
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max |
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klm |
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klm |
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Emax E |
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Ɋɚɫɫɦɨɬɪɢɦ |
ɜɟɥɢɱɢɧɭ |
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. Ɍɚɤ ɤɚɤ 0 E Emax, |
ɬɨ 0 Į 1, ɩɪɢɱɟɦ |
Į |
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Emax |
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ɫɨɨɬɜɟɬɫɬɜɭɟɬ |
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= ȿmɚɯ, |
ɬ.ɟ. |
ɨɬɫɭɬɫɬɜɢɸ ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ |
ɜ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɜɟɤɬɨɪɨɜ |
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(ɨɬɫɭɬɫɬɜɢɸ ɞɢɮɮɟɪɟɧɰɢɚɰɢɢ ɨɛɳɧɨɫɬɢ), ɚ Į = 1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ E = 0, ɬ.ɟ. ɦɚɤɫɢɦɚɥɶɧɨɣ |
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ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ (ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɢɮɮɟɪɟɧɰɢɚɰɢɢ). |
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Ɉɱɟɜɢɞɧɨ, ɪɚɡɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɚɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɚɡɥɢɱɧɵɟ Į ɢ ɮɨɪɦɚɥɶɧɨ ɡɚɞɚɱɚ |
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ɫɜɨɞɢɬɫɹ ɤ ɨɬɵɫɤɚɧɢɸ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɦɚɤɫɢɦɚɥɶɧɵɦ Į. |
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ɍɩɪɚɠɧɟɧɢɟ 66. ɉɨɤɚɡɚɬɶ, ɱɬɨ |
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N log |
klm |
¦¦¦Nijr log Nijr |
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N logklm
89
ɇɚ ɷɦɩɢɪɢɱɟɫɤɨɦ ɦɚɬɟɪɢɚɥɟ ɥɟɧɢɧɝɪɚɞɫɤɢɯ ɫɨɰɢɨɥɨɝɨɜ ɜɟɥɢɱɢɧɚ Į ɨɤɚɡɚɥɚɫɶ ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɥɹ ɧɚɛɨɪɚ ɩɪɢɡɧɚɤɨɜ «ɩɪɨɮɟɫɫɢɹ – ɤɜɚɥɢɮɢɤɚɰɢɹ – ɨɛɪɚɡɨɜɚɧɢɟ». ɂɦɟɧɧɨ ɜ ɷɬɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɟɤɬɨɪɵ ɥɟɠɚɬ ɧɚɢɛɨɥɟɟ ɩɥɨɬɧɵɦɢ ɝɪɭɩɩɚɦɢ, ɞɚɧɧɵɣ ɧɚɛɨɪ ɩɪɢɡɧɚɤɨɜ ɧɚɢɛɨɥɟɟ ɪɟɡɤɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɬ ɢɡɭɱɚɟɦɭɸ ɫɨɰɢɚɥɶɧɭɸ ɨɛɳɧɨɫɬɶ.
8.ɇɟɤɨɬɨɪɵɟ ɞɪɭɝɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
ȼɞɚɧɧɨɦ ɩɚɪɚɝɪɚɮɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɧɟɫɤɨɥɶɤɨ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɧɟ ɩɨɥɭɱɢɥɢ ɜ ɫɨɰɢɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɬɚɤɨɝɨ ɲɢɪɨɤɨɝɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ, ɤɚɤ, ɫɤɚɠɟɦ, r, ȡ, Ɍ ɢ ɞɚɠɟ Ș). Ɉɞɧɚɤɨ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɥɢɬɟɪɚɬɭɪɟ ɭɠɟ ɜɫɬɪɟɱɚɸɬɫɹ ɭɩɨɦɢɧɚɧɢɹ ɨɛ ɢɯ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɨɬɞɟɥɶɧɵɦɢ ɚɜɬɨɪɚɦɢ.
Ɇɵ ɫɱɢɬɚɟɦ ɰɟɥɟɫɨɨɛɪɚɡɧɵɦ ɪɚɫɫɦɨɬɪɟɬɶ ɨɩɪɟɞɟɥɟɧɢɹ, ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɢɯ ɢ ɩɪɢɜɟɫɬɢ ɩɪɢɦɟɪɵ ɜɵɱɢɫɥɟɧɢɹ ɧɟɤɨɬɨɪɵɯ ɬɚɤɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ27. ɋ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɷɬɨ ɩɨɤɚɠɟɬ ɱɢɬɚɬɟɥɸ, ɱɬɨ ɞɢɚɩɚɡɨɧ ɢɫɩɨɥɶɡɭɟɦɵɯ ɦɟɬɨɞɨɜ ɡɧɚɱɢɬɟɥɶɧɨ ɲɢɪɟ, ɱɟɦ ɦɨɠɟɬ ɩɪɟɞɫɬɚɜɢɬɶɫɹ ɩɨ ɨɫɧɨɜɧɨɣ ɦɚɫɫɟ ɩɭɛɥɢɤɚɰɢɣ, ɫ ɞɪɭɝɨɣ, ɩɨɡɜɨɥɢɬ ɛɨɥɟɟ ɫɜɨɛɨɞɧɨ ɨɪɢɟɧɬɢɪɨɜɚɬɶɫɹ ɜɧɚɭɱɧɵɯ ɫɬɚɬɶɹɯ.
[130]
g – ɤɨɷɮɮɢɰɢɟɧɬ Ƚɭɞɦɚɧɚ (ɞɥɹɧɨɦɢɧɚɥɶɧɵɯɲɤɚɥ)
Ʉɨɷɮɮɢɰɢɟɧɬ Ƚɭɞɦɚɧɚ ɧɟ ɹɜɥɹɟɬɫɹ ɫɢɦɦɟɬɪɢɱɧɵɦ: gyx gxy ȿɫɥɢ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ X ɤɚɤ ɧɟɡɚɜɢɫɢɦɵɣ (ɮɚɤɬɨɪɧɵɣ) ɩɪɢɡɧɚɤ, ɬɨ ɟɝɨ ɜɥɢɹɧɢɟ ɧɚ Y ɨɩɢɫɵɜɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɤɨɷɮɮɢɰɢɟɧɬɚ
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¦max Nij |
max N(yi ) |
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gyx |
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N max N(y j ) |
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ɝɞɟ max N(yj) – ɦɚɤɫɢɦɚɥɶɧɵɣ ɦɚɪɝɢɧɚɥ ɡɚɜɢɫɢɦɨɝɨ ɩɪɢɡɧɚɤɚ, ɚ max Nij – ɦɚɤɫɢɦɚɥɶɧɚɹ ɱɚɫɬɨɬɚ ɜ i-ɨɣ ɫɬɪɨɤɟ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ.
ȿɫɥɢ ɞɚɧɧɨɦɭ X ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɩɪɟɞɟɥɟɧɧɵɣ Y, ɬɨ ɜ ɫɬɪɨɤɟ ɥɢɲɶ ɨɞɧɚ ɱɚɫɬɨɬɚ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɦɚɪɝɢɧɚɥɨɦ, ɢɫɤɨɦɚɹ ɫɭɦɦɚ ɦɚɤɫɢɦɭɦɨɜ ɨɛɪɚɳɚɟɬɫɹ ɜ N, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, gyx = 1
ȿɫɥɢ ɩɪɢɡɧɚɤɢ ɧɟɡɚɜɢɫɢɦɵ, ɬɨ |
Nij |
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N(xi )N(y j ), ɤɚɤ ɦɵ ɜɢɞɟɥɢ, ɢ ɦɚɤɫɢɦɚɥɶɧɚɹ |
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ɱɚɫɬɨɬɚ ɜ i-ɨɣ ɫɬɪɨɤɟ ɬɚɦ, ɝɞɟ ɦɚɤɫɢɦɚɥɟɧ Y-ɦɚɪɝɢɧɚɥ, ɬ.ɟ. |
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¦max Nij |
max N(y j |
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max N(y j ) |
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Ɍɟɩɟɪɶ gyx = 0. ɂɬɚɤ, 0 gyx 1. Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ gxy, ɨɩɢɫɵɜɚɸɳɢɣ ɜɥɢɹɧɢɟ Y
ɧɚ X.
Ʉɨɷɮɮɢɰɢɟɧɬɵ Ƚɭɞɦɚɧɚ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ, ɟɫɥɢ ɢɡ ɫɨɞɟɪɠɚɬɟɥɶɧɵɯ ɫɨɨɛɪɚɠɟɧɢɣ ɹɫɧɨ, ɱɬɨ X ɦɨɠɟɬ ɜɥɢɹɬɶ ɧɚ Y (ɢ ɧɚɨɛɨɪɨɬ) ɢ ɷɬɨ ɜɥɢɹɧɢɟ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɧɟ ɫɢɦɦɟɬɪɢɱɧɨ.
ȼ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ X ɧɟ ɦɨɠɟɬ ɜɥɢɹɬɶ ɧɚ Y (ɧɚɩɪɢɦɟɪ, X – ɤɜɚɥɢɮɢɤɚɰɢɹ, Y – ɜɨɡɪɚɫɬ), ɫɥɟɞɭɟɬ ɜɵɱɢɫɥɹɬɶ ɬɨɥɶɤɨ gxy (ɜɨɡɪɚɫɬɜɥɢɹɟɬ ɧɚ ɤɜɚɥɢɮɢɤɚɰɢɸ).
ɍɩɪɚɠɧɟɧɢɟ 67. Ɋɚɫɫɱɢɬɚɬɶ gyx ɢ gxy ɞɥɹ ɫɥɟɞɭɸɳɟɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ:
X |
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y1 |
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x1 |
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27 Ɉɛɡɨɪ ɪɹɞɚ ɞɪɭɝɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɤɧ.: ȿɥɢɫɟɟɜɚ ɂ.ɂ., Ɋɭɤɚɜɢɲɧɢɤɨɜ ȼ.Ɉ. Ƚɪɭɩɩɢɪɨɜɤɚ, ɤɨɪɪɟɥɹɰɢɹ, ɪɚɫɩɨɡɧɚɜɚɧɢɟ ɨɛɪɚɡɨɜ. Ɇ., 1977, ɝɥ. III, IV.
90
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45 |
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20 |
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30 |
65 |
Ɉɬɜɟɬ: gyx = 0,57; gxy = 1.
[131]
ɂɧɬɟɪɩɪɟɬɢɪɭɟɦ ɪɟɡɭɥɶɬɚɬ:
Ɂɚɞɚɧɢɟ Y ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ X (ɫɦ. ɬɚɛɥɢɰɭ). ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ gyx = 1; ɧɨ ɡɚɞɚɧɢɟ X ɧɟ ɨɩɪɟɞɟɥɹɟɬ ɟɳɟ Y (ɧɚɩɪɢɦɟɪ, ɟɫɥɢ X = x2, ɬɨ Y ɦɨɠɟɬ ɛɵɬɶ ɢ ɭ2, ɢ y3), ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
gyx < 1
ɍɩɪɚɠɧɟɧɢɟ 68. 1. Ɂɚɩɢɫɚɬɶ ɥɸɛɭɸɞɢɚɝɨɧɚɥɶɧɭɸ ɬɚɛɥɢɰɭ ɢɭɛɟɞɢɬɶɫɹ, ɱɬɨ gyx = gxy = 1. 2. ɋɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɬɚɛɥɢɰɭ, ɞɥɹ ɤɨɬɨɪɨɣ gxy < 1, ɚ gyx = 1, ɂɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ
ɪɚɫɱɟɬɚ ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɩɪɟɞɵɞɭɳɢɦ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɵɩɨɥɧɟɧɢɟ ɷɬɢɯ ɧɟɫɥɨɠɧɵɯ ɭɩɪɚɠɧɟɧɢɣ ɩɨɦɨɝɚɟɬ ɭɹɫɧɢɬɶ ɫɦɵɫɥ ɢ ɪɚɡɥɢɱɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ gyx ɢ gxy. Ɉɬɦɟɬɢɦ ɬɚɤɠɟ ɩɪɟɞɥɚɝɚɟɦɵɣ Ȼ. Ɇɢɪɤɢɧɵɦ ɩɨɞɯɨɞ ɤ ɨɛɪɚɛɨɬɤɟ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɢɧɮɨɪɦɚɰɢɢ28, ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧ ɞɚɠɟ ɞɥɹ ɫɥɭɱɚɹ ɧɨɦɢɧɚɥɶɧɵɯ ɲɤɚɥ. ȼ ɤɚɱɟɫɬɜɟ ɦɟɪɵ ɛɥɢɡɨɫɬɢ ɩɪɢɡɧɚɤɨɜ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɦɟɪɚ ɛɥɢɡɨɫɬɢ ɪɚɡɛɢɟɧɢɣ ɨɛɳɧɨɫɬɢ, ɨɫɭɳɟɫɬɜɥɹɟɦɵɯ ɷɬɢɦɢ ɩɪɢɡɧɚɤɚɦɢ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɛɥɢɡɨɫɬɢ ɪɚɡɛɢɟɧɢɣ
Ɉɛɨɛɳɢɦ ɮɨɪɦɭɥɭ ɞɥɹ ɦɟɪɵ ɛɥɢɡɨɫɬɢ ɦɟɠɞɭ ɞɜɭɦɹ ɪɚɡɛɢɟɧɢɹɦɢ ɧɚ ɫɥɭɱɚɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ k × l. ȼ ɤɚɱɟɫɬɜɟ ɢɫɯɨɞɧɨɣ ɜɨɡɶɦɟɦ ɮɨɪɦɭɥɭ, ɩɪɢɜɨɞɢɦɭɸ Ȼ.Ƚ. Ɇɢɪɤɢɧɵɦ ɢ Ʌ.Ȼ. ɑɟɪɧɵɦ ɜ ɫɬɚɬɶɟ «Ɉɛ ɢɡɦɟɪɟɧɢɢ ɦɟɪɵ ɛɥɢɡɨɫɬɢ ɦɟɠɞɭ ɪɚɡɥɢɱɧɵɦɢ ɪɚɡɛɢɟɧɢɹɦɢ ɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɨɛɴɟɤɬɨɜ»29.
ȿɫɥɢ R ɢ S ɞɜɚ ɪɚɡɛɢɟɧɢɹ ɦɧɨɠɟɫɬɜɚ ɢɡ N ɷɥɟɦɟɧɬɨɜ ɢ R ɪɚɡɛɢɜɚɟɬ ɟɝɨ ɧɚ m, ɚ S ɧɚ n ɤɥɚɫɫɨɜ, ɩɪɢɱɟɦ ɜ i-ɨɦ ɤɥɚɫɫɟ |Ri| ɷɥɟɦɟɧɬɨɜ, ɚ ɜ j-ɨɦ |Si| ɷɥɟɦɟɧɬɨɜ, ɬɨ ɦɟɪɚ ɛɥɢɡɨɫɬɢ ɪɚɡɛɢɟɧɢɣ
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(Ɂɞɟɫɶ R S – ɩɟɪɟɫɟɱɟɧɢɟ ɤɥɚɫɫɨɜ R ɢ S). |
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Ɍɚɤ ɤɚɤ d |
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N(N 1) , ɧɨɪɦɢɪɨɜɚɧɧɚɹ ɦɟɪɚ G |
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, ɩɪɢɱɟɦ 0 į 1, ɝɞɟ į = 0 |
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ɫɨɨɬɜɟɬɫɬɜɭɟɬ
[132]
ɦɚɤɫɢɦɚɥɶɧɨɣ ɫɜɹɡɢ, į = 1 ɦɢɧɢɦɚɥɶɧɨɣ (ɨɬɫɭɬɫɬɜɢɟ ɫɜɹɡɢ).
Ⱦɥɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ {Nij}: ɩɪɢɡɧɚɤ X ɨɫɭɳɟɫɬɜɥɹɟɬ ɪɚɡɛɢɟɧɢɟ ɨɛɳɧɨɫɬɢ N ɧɚ k ɤɥɚɫɫɨɜ ɯi, ɜ ɤɚɠɞɨɦ ɢɡ ɤɨɬɨɪɵɯ N(xi) ɷɥɟɦɟɧɬɨɜ: ɩɪɢɡɧɚɤ Y ɧɚ l ɤɥɚɫɫɨɜ yi, ɜ ɤɚɠɞɨɦ ɢɡ
ɤɨɬɨɪɵɯ N(yj) ɷɥɟɦɟɧɬɨɜ. |
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Ɍɚɤ ɤɚɤ Nij |
N(xi ) N(y j ) , ɬɨ ɦɟɪɚ ɛɥɢɡɨɫɬɢ ɞɜɭɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɪɚɡɛɢɟɧɢɣ |
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ª k |
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«¦N 2 (xi ) ¦N 2 (y j ) 2¦¦Nij2 |
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N(N 1) |
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i 1 j 1 |
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Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɦɵ ɢɫɫɥɟɞɭɟɦ ɧɟɤɨɬɨɪɨɟ ɪɚɡɛɢɟɧɢɟ, ɨɫɭɳɟɫɬɜɥɹɟɦɨɟ Y, ɢ ɯɨɬɢɦ ɜɵɹɫɧɢɬɶ ɡɧɚɱɢɦɨɫɬɶ ɪɹɞɚ ɩɪɢɡɧɚɤɨɜ X(ɪ) (ɪ = 1, 2, ...) ɞɥɹ ɜɵɹɜɥɟɧɢɹ ɞɚɧɧɨɝɨ ɪɚɡɛɢɟɧɢɹ.
28Ɇɢɪɤɢɧ Ȼ.Ƚ. ɇɨɜɵɣ ɩɨɞɯɨɞ ɤ ɨɛɪɚɛɨɬɤɟ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɢɧɮɨɪɦɚɰɢɢ. – ȼ ɤɧ.: ɂɡɦɟɪɟɧɢɟ ɢ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɜ ɫɨɰɢɨɥɨɝɢɢ. ɇɨɜɨɫɢɛɢɪɫɤ, 1969.
29Ⱥɜɬɨɦɚɬɢɤɚ ɢɬɟɥɟɦɟɯɚɧɢɤɚ, 1970, ʋ5.
91
Ɂɧɚɱɢɦɨɫɬɶ ɏ(ɪ) ɛɭɞɟɬ ɬɟɦ ɛɨɥɶɲɟɣ, ɱɟɦ ɛɥɢɠɟ ɪɚɡɛɢɟɧɢɹ, ɬ.ɟ. ɱɟɦ ɦɟɧɶɲɟ į (ɏ(ɪ), Y) Łįp. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɧɚɱɢɦɨɫɬɶ ɩɪɢɡɧɚɤɚ ɏ(ɪ) ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɪɚɡɛɢɟɧɢɸ Y ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɪɚɫɫɬɨɹɧɢɸ įp. ɗɬɭ ɡɧɚɱɢɦɨɫɬɶ («ɫɢɥɭ ɜɥɢɹɧɢɹ») ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ, ɫɥɟɞɭɹ Ȼ.Ƚ. Ɇɢɪɤɢɧɭ, ɤɚɤ ɦɟɪɭ ɫɜɹɡɢ ɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ. ɉɭɫɬɶ, ɧɚɩɪɢɦɟɪ, ɪɚɡɛɢɟɧɢɟ ɍ – ɷɬɨ ɫɨɰɢɚɥɶɧɨ-ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɵɟ ɝɪɭɩɩɵ, ɚ ɏ(ɪ) – ɪɚɡɥɢɱɧɵɟ ɫɨɰɢɚɥɶɧɨɞɟɦɨɝɪɚɮɢɱɟɫɤɢɟ ɩɪɢɡɧɚɤɢ (ɩɪɨɮɟɫɫɢɹ, ɤɜɚɥɢɮɢɤɚɰɢɹ, ɨɛɪɚɡɨɜɚɧɢɟ, ɞɨɯɨɞ, ɦɟɫɬɨ ɠɢɬɟɥɶɫɬɜɚ ɢ ɬ.ɞ.), ɜɵɱɢɫɥɹɹ įp, ɦɵ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ ɡɧɚɱɢɦɨɫɬɶ (ɜɥɢɹɧɢɟ) ɪɚɡɥɢɱɧɵɯ ɏ(ɪ) ɞɥɹ ɜɵɹɜɥɟɧɢɹ Y-ɪɚɡɛɢɟɧɢɹ, ɜɵɞɟɥɢɬɶ ɧɚɢɛɨɥɟɟ ɢɧɮɨɪɦɚɬɢɜɧɵɟ ɩɪɢɡɧɚɤɢ.
Ɋɚɫɫɦɚɬɪɢɜɚɥɚɫɶ ɢ ɬɚɤɚɹ ɡɚɞɚɱɚ: ɩɭɫɬɶ Y – ɷɬɨ ɪɚɫɫɟɥɟɧɢɟ ɪɚɛɨɬɧɢɤɨɜ ɩɨ «ɡɨɧɚɦ ɞɨɫɬɭɩɧɨɫɬɢ ɩɪɟɞɩɪɢɹɬɢɹ»30, ɚ ɏ(ɪ) – ɧɟɤɨɬɨɪɵɟ ɫɨɰɢɚɥɶɧɨ-ɞɟɦɨɝɪɚɮɢɱɟɫɤɢɟ ɩɪɢɡɧɚɤɢ, ɡɧɚɱɢɦɵɟ ɞɥɹ ɪɚɫɫɟɥɟɧɢɹ. ɇɚɢɛɨɥɟɟ ɡɧɚɱɢɦɵɦ ɩɪɢɡɧɚɤɨɦ ɨɤɚɡɚɥɚɫɶ ɩɪɢɧɚɞɥɟɠɧɨɫɬɶ ɤ ɫɨɰɢɚɥɶɧɨ-ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɨɣ ɝɪɭɩɩɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɟɳɟ ɪɚɡ ɜɨɩɪɨɫ ɨ ɫɜɹɡɢ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɹɦɢ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɨɣ ɢ ɪɚɛɨɬɨɣ, ɢɫɩɨɥɶɡɭɹ ɞɥɹ ɟɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɨɛɫɭɠɞɚɟɦɭɸ ɦɟɪɭ (ɫɦ. ɩɪɢɦɟɪ ʋ 14 § 1 ɷɬɨɣ ɝɥɚɜɵ).
[133] |
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D = N(N–1)=1015056, į – 0,490.
ɍɩɪɚɠɧɟɧɢɟ 69. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɬɚɛɥɢɰɵ 19 § 1 ɷɬɨɣ ɝɥɚɜɵ į = 0,528.
ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ į ɦɟɧɶɲɟ, ɧɨ ɬɚɤ ɤɚɤ ɫɜɹɡɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ 1/ į ɬɨ ɨɧɚ ɛɨɥɶɲɟ, ɱɟɦ ɜɨ ɜɬɨɪɨɦ; ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɞɟɥɚɧɧɵɣ ɪɚɧɟɟ ɜɵɜɨɞ (§ 1) ɩɨɞɬɜɟɪɠɞɚɟɬɫɹ.
ǻ-ɤɨɷɮɮɢɰɢɟɧɬ (ɧɨɦɢɧɚɥɶɧɵɟ ɲɤɚɥɵ)
ȼ ɪɚɛɨɬɟ ɂ.Ⱥ. ɒɤɪɚɛɤɢɧɨɣ ɢ Ƚ.ɂ. ɋɦɢɪɧɨɜɨɣ «ɉɪɨɝɪɚɦɦɚ ɢɡɦɟɪɟɧɢɹ ɬɟɫɧɨɬɵ ɫɜɹɡɢ ɦɟɠɞɭ ɞɜɭɦɹ ɩɪɢɡɧɚɤɚɦɢ»31 ɩɪɟɞɥɚɝɚɟɬɫɹ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɫɜɹɡɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɨɞɭɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ǻ.
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ɜɜɟɞɟɦ ɜɟɥɢɱɢɧɭ |
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Ɇɟɪɨɣ ɫɜɹɡɢ, ɬɨɱɧɟɟ, ɜɥɢɹɧɢɹ X ɧɚ Y, ɦɨɠɟɬ ɫɥɭɠɢɬɶ |
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ɉɨɤɚɠɟɦ ɷɬɨ. ȿɫɥɢ ɩɪɢɡɧɚɤɢ ɧɟɡɚɜɢɫɢɦɵ, ɬɨ nij |
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ɫɭɦɦɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ.
Ʌɨɝɢɤɚ ɢɡɦɟɪɟɧɢɹ ɫɜɹɡɢ ɬɚɤɨɜɚ: ɟɫɥɢ ɩɪɢɡɧɚɤɢɧɟɡɚɜɢ-
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30«Ɂɨɧɚ ɞɨɫɬɭɩɧɨɫɬɢ ɩɪɟɞɩɪɢɹɬɢɹ» ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɪɟɦɟɧɟɦ, ɡɚɬɪɚɱɢɜɚɟɦɵɦ ɪɚɛɨɬɧɢɤɨɦ ɧɚ ɩɟɪɟɞɜɢɠɟɧɢɟ ɨɬ ɦɟɫɬɚ ɠɢɬɟɥɶɫɬɜɚ ɞɨ ɦɟɫɬɚ ɪɚɛɨɬɵ. ɉɨ ɧɨɪɦɚɦ ɝɪɚɞɨɫɬɪɨɢɬɟɥɶɫɬɜɚ ɜɵɞɟɥɹɸɬɫɹ ɱɟɬɵɪɟ ɡɨɧɵ: Ⱥ (ɞɨ 30 ɦɢɧ.), Ȼ (ɨɬ 30 ɞɨ 45 ɦɢɧ.), ȼ (ɨɬ 45 ɦɢɧ. ɞɨ ɱɚɫɚ), Ƚ (ɫɜɵɲɟ ɱɚɫɚ).
31Ⱥɧɚɥɢɡ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɢɧɮɨɪɦɚɰɢɢ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɗȼɆ, ɱ.1. Ɇ., 1973, ɫ.143-157
92
[134] |
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ɫɢɦɵ, ɬɨ ɩɪɢ ɢɡɦɟɧɟɧɢɢ X ɡɧɚɱɟɧɢɟ Y ɧɟ ɞɨɥɠɧɨ ɦɟɧɹɬɶɫɹ, ɬ.ɟ. ɱɢɫɥɚ ɢɧɞɢɜɢɞɨɜ |
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ɪɚɡɧɵɦɢ X ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ Y ɞɨɥɠɧɵ ɛɵɬɶ ɩɪɢɦɟɪɧɨ ɨɞɢɧɚɤɨɜɵ, ɬ.ɟ. ɪɚɜɧɵɦɢ |
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ɠɟ X ɜɥɢɹɟɬ ɧɚ Y, ɬɨ nij ɞɨɥɠɧɵ ɨɬɥɢɱɚɬɶɫɹ ɨɬɫɪɟɞɧɟɝɨ nj . |
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Ɉɛɫɭɠɞɚɟɦɚɹ ɫɭɦɦɚ ɧɟ ɹɜɥɹɟɬɫɹ ɧɨɪɦɢɪɨɜɚɧɧɨɣ. ȼ ɪɚɛɨɬɟ ɒɤɪɚɛɤɢɧɨɣ ɢ ɋɦɢɪɧɨɜɨɣ ɜ
ɤɚɱɟɫɬɜɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɪɢ ɫɭɦɦɟ ɩɪɟɞɥɚɝɚɟɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜɟɥɢɱɢɧɭ 1 . Ɉɞɧɚɤɨ ɤɚɤ ɥɟɝɤɨ k
ɜɢɞɟɬɶ, 'c S ɧɟ ɹɜɥɹɟɬɫɹ ɧɨɪɦɢɪɨɜɚɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. k
Ⱦɥɹ ɧɨɪɦɢɪɨɜɤɢ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɭɦɦɵ S. Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ ɜ ɫɥɭɱɚɟ ɩɨɥɧɨɣ ɫɜɹɡɢ (ɫɜɹɡɶ ɦɵ ɧɚɡɵɜɚɟɦ ɩɨɥɧɨɣ, ɟɫɥɢ ɤɚɠɞɨɦɭ X
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɞɧɨ ɡɧɚɱɟɧɢɟ Y) ɢ ɪɚɜɧɨ 2 (m 1)(2k m) , ɝɞɟ m = min(k,l) – ɦɟɧɶɲɟɟ ɢɡ ɱɢɫɟɥ k
k ɢ l.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɨɪɦɢɪɨɜɚɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ, ɨɩɢɫɵɜɚɸɳɢɣ ɜɥɢɹɧɢɟ X ɧɚ Y:
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ȼɫɟ ɪɚɧɟɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɡɞɟɫɶ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢɦɟɧɢɦɵ ɞɚɠɟ ɞɥɹ ɧɨɦɢɧɚɥɶɧɵɯ ɲɤɚɥ. ɉɟɪɟɣɞɟɦ ɤ ɤɨɷɮɮɢɰɢɟɧɬɚɦ, ɤɨɬɨɪɵɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɧɚɥɢɱɢɢ ɭɩɨɪɹɞɨɱɟɧɢɹ ɡɧɚɱɟɧɢɣ ɩɪɢɡɧɚɤɨɜ.
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Ȗ-ɤɨɷɮɮɢɰɢɟɧɬ Ƚɭɞɦɚɧɚ |
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[135] |
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ɝɞɟ Ɋ – ɱɢɫɥɨ ɩɚɪ ɨɛɴɟɤɬɨɜ, ɭ ɤɨɬɨɪɵɯ ɨɛɚ ɩɪɢɡɧɚɤɚ ɭɩɨɪɹɞɨɱɟɧɵ ɜ ɨɞɢɧɚɤɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ, ɚ Q – ɬɨ ɠɟ, ɧɨ ɜɨɛɪɚɬɧɨɣ.
ɉɭɫɬɶ ɡɧɚɱɟɧɢɹ X ɢ Y ɜ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɟ ɜɵɩɢɫɚɧɵ ɜ ɨɞɢɧɚɤɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ. ȼɟɥɢɱɢɧɭ Ɋ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɤɚɤ ɫɭɦɦɭ ɪɟɡɭɥɶɬɚɬɨɜ ɭɦɧɨɠɟɧɢɹ ɱɚɫɬɨɬ ɤɚɠɞɨɣ
Ɍɚɛɥɢɰɚ 32
ɉɪɢɦɟɪ ɪɚɫɱɟɬɚ Ȗ-ɤɨɷɮɮɢɰɢɟɧɬɚ Ƚɭɞɦɚɧɚ
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93