close

Вход

Забыли?

вход по аккаунту

?

Наилучшие l1-приближения в классах распределений.

код для вставкиСкачать
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 7, ʋ3, 2004 ɝ.
ɫɬɪ.419-424
ɇɚɢɥɭɱɲɢɟ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɜ ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ
ȼ.ɉ. ɉɚɧɬɟɥɟɟɜ
ɗɤɨɧɨɦɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɆȽɌɍ, ɤɚɮɟɞɪɚ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ
ɢ ɟɫɬɟɫɬɜɟɧɧɨɧɚɭɱɧɵɯ ɞɢɫɰɢɩɥɢɧ
Ⱥɧɧɨɬɚɰɢɹ. ɂɡɥɚɝɚɟɬɫɹ ɝɪɚɞɢɟɧɬɧɵɣ ɦɟɬɨɞ ɩɨɢɫɤɚ ɧɚɢɥɭɱɲɟɝɨ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɜ ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ.
Ƚɟɨɦɟɬɪɢɱɟɫɤɚɹ ɮɨɪɦɚ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɰɟɥɟɜɨɣ ɮɭɧɤɰɢɢ ɩɨɡɜɨɥɹɟɬ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶ ɮɭɧɤɰɢɢ,
ɡɚɞɚɜɚɟɦɵɟ, ɜ ɱɚɫɬɧɨɫɬɢ, ɜ ɮɨɪɦɟ ɝɪɚɮɢɤɚ, ɧɚɩɪɢɦɟɪ, ɫɱɢɬɵɜɚɸɳɢɦɢ ɭɫɬɪɨɣɫɬɜɚɦɢ. ɋɯɟɦɚ, ɢɡɥɨɠɟɧɧɚɹ
ɡɞɟɫɶ ɞɥɹ ɤɥɚɫɫɚ ɧɨɪɦɚɥɶɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ, ɥɟɝɤɨ ɩɟɪɟɧɨɫɢɬɫɹ ɧɚ ɞɪɭɝɢɟ ɤɥɚɫɫɵ ɪɚɫɩɪɟɞɟɥɟɧɢɣ.
Abstract. The paper has considered a gradient method of searching the best L1 approximation in classes of
distributions. The method is based on the geometrical differentiation and can be applied for approximation of the
functions, expressed in a graph form, for instance, by reading machines. The method, proposed here for the class
of normal distributions, can readily be extended to other classes of distributions.
ȼɜɟɞɟɧɢɟ
ɉɟɪɟɧɨɫ ɢɡɨɛɪɚɠɟɧɢɣ ɧɚ ɷɤɪɚɧ ɤɨɦɩɶɸɬɟɪɚ ɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɤɨɧɬɭɪɨɜ ɢɡɨɛɪɚɠɟɧɢɣ ɜ ɥɢɧɢɢ,
ɤɚɤ ɢɯ ɩɨɧɢɦɚɟɬ ɦɚɲɢɧɚ, – ɩɪɟɤɪɚɫɧɚɹ ɨɬɩɪɚɜɧɚɹ ɬɨɱɤɚ ɞɥɹ ɦɧɨɝɢɯ ɧɨɜɵɯ ɬɟɯɧɢɱɟɫɤɢɯ ɪɟɲɟɧɢɣ. ɋɪɟɞɢ
ɩɪɨɱɟɝɨ ɫɸɞɚ ɨɬɧɨɫɢɬɫɹ ɩɪɨɛɥɟɦɚ ɧɚɢɥɭɱɲɟɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɤɨɧɬɭɪɧɨɝɨ ɢɡɨɛɪɚɠɟɧɢɹ ɜ ɬɨɦ ɢɥɢ ɢɧɨɦ
ɤɥɚɫɫɟ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɨɛɥɚɫɬɟɣ, ɫɨɫɬɚɜɥɹɸɳɚɹ ɩɪɟɞɦɟɬ ɞɚɧɧɨɣ ɫɬɚɬɶɢ. Ⱦɥɹ ɫɨɞɟɪɠɚɬɟɥɶɧɨɫɬɢ
ɢɡɥɨɠɟɧɢɹ ɦɵ ɨɝɪɚɧɢɱɢɦɫɹ ɱɢɫɥɟɧɧɵɦ ɩɨɢɫɤɨɦ ɧɚɢɥɭɱɲɟɝɨ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɜ ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ,
ɡɚɞɚɜɚɟɦɵɯ ɮɭɧɤɰɢɹɦɢ ɩɥɨɬɧɨɫɬɢ, ɯɨɬɹ ɜɨɡɦɨɠɧɨ ɢ ɩɚɪɚɥɥɟɥɶɧɨɟ ɢɡɥɨɠɟɧɢɟ ɞɥɹ ɢɧɬɟɝɪɚɥɶɧɵɯ ɮɭɧɤɰɢɣ
ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ƚɟɨɦɟɬɪɢɱɟɫɤɢɣ ɩɨɞɯɨɞ ɤ ɩɪɨɛɥɟɦɟ ɨɛɟɫɩɟɱɢɜɚɟɬ ɲɢɪɨɤɭɸ ɫɜɨɛɨɞɭ ɩɪɢ ɜɵɛɨɪɟ ɮɨɪɦɵ
ɡɚɞɚɧɢɹ ɚɩɩɪɨɤɫɢɦɢɪɭɟɦɨɝɨ ɷɥɟɦɟɧɬɚ – ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɮɭɧɤɰɢɢ ɩɥɨɬɧɨɫɬɢ p. ɇɟ ɢɫɤɥɸɱɟɧɨ, ɱɬɨ
ɩɨɫɥɟɞɧɹɹ ɜɵɪɚɠɟɧɚ, ɧɚɩɪɢɦɟɪ, ɝɪɚɮɢɤɨɦ, ɜɵɜɟɞɟɧɧɵɦ ɧɚ ɷɤɪɚɧ ɤɨɦɩɶɸɬɟɪɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɝɪɚɮɢɤ
ɮɭɧɤɰɢɢ p ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɤɚɤ ɥɨɦɚɧɭɸ, ɜɨɡɦɨɠɧɨ, ɫɨɫɬɚɜɥɟɧɧɭɸ ɢɡ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɦɚɥɵɯ
ɡɜɟɧɶɟɜ, ɤɨɬɨɪɚɹ ɨɝɪɚɧɢɱɢɜɚɟɬ ɫɜɟɪɯɭ ɩɨɥɢɝɨɧ, ɥɟɠɚɳɢɣ ɦɟɠɞɭ ɧɟɸ ɢ ɨɫɶɸ ɚɛɫɰɢɫɫ, ɜɨɡɦɨɠɧɨ,
ɧɟɫɜɹɡɧɵɣ. əɫɧɨ, ɱɬɨ ɜɫɟ ɢɧɵɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɮɭɧɤɰɢɢ p ɥɟɝɤɨ ɫɜɨɞɹɬɫɹ ɤ ɝɪɚɮɢɱɟɫɤɨɦɭ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ,
ɤɚɤ ɜɵɹɜɢɬɫɹ ɧɢɠɟ, ɟɫɥɢ ɫɬɚɪɬɨɜɚɬɶ ɨɬ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ p, ɢɧɬɟɪɩɪɟɬɢɪɭɹ ɟɝɨ ɤɚɤ ɥɨɦɚɧɭɸ,
ɩɪɟɞɥɚɝɚɟɦɚɹ ɡɞɟɫɶ ɫɯɟɦɚ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɫɜɨɞɢɬ ɨɲɢɛɤɭ ɦɟɬɨɞɚ ɩɨɬɟɧɰɢɚɥɶɧɨ ɤ ɧɭɥɸ, ɨɫɬɚɜɥɹɹ ɥɢɲɶ
ɜɵɱɢɫɥɢɬɟɥɶɧɭɸ ɨɲɢɛɤɭ.
ɇɟɬɪɢɜɢɚɥɶɧɨɫɬɶ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɜ ɬɨɦ, ɱɬɨ ɷɥɟɦɟɧɬ ɧɚɢɥɭɱɲɟɝɨ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɢɳɟɬɫɹ ɜ
ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ, ɜɨɨɛɳɟ ɧɟ ɹɜɥɹɸɳɢɯɫɹ ɥɢɧɟɣɧɵɦɢ ɦɧɨɝɨɨɛɪɚɡɢɹɦɢ ɢ ɩɨɬɨɦɭ ɧɟ ɞɨɩɭɫɤɚɸɳɢɯ
ɩɪɹɦɨɝɨ ɩɪɢɦɟɧɟɧɢɹ ɥɢɧɟɣɧɵɯ ɦɟɬɨɞɨɜ. Ʉ ɬɨɦɭ ɠɟ, ɦɟɬɪɢɤɚ L1 ɦɟɧɟɟ ɨɛɟɳɚɸɳɚɹ, ɱɟɦ, ɧɚɩɪɢɦɟɪ, L2.
Ⱦɥɹ ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɵɯ ɮɭɧɤɰɢɣ p ɩɪɨɛɥɟɦɚ ɭɫɥɨɠɧɹɟɬɫɹ ɬɪɟɛɨɜɚɧɢɟɦ ɛɨɥɟɟ ɩɨɥɧɨɝɨ ɭɱɺɬɚ
ɫɨɞɟɪɠɚɳɟɣɫɹ ɜ ɧɟɣ ɢɧɮɨɪɦɚɰɢɢ, ɛɟɡ ɭɩɪɨɳɟɧɢɣ, ɬɚɤɢɯ ɤɚɤ ɫɬɹɝɢɜɚɧɢɟ ɜɫɟɣ ɢɧɮɨɪɦɚɰɢɢ ɨ ɮɭɧɤɰɢɢ p ɤ
ɟɺ ɡɧɚɱɟɧɢɹɦ ɜ ɨɬɞɟɥɶɧɵɯ ɬɨɱɤɚɯ ɞɢɫɤɪɟɬɧɨɣ ɦɨɞɟɥɢ. ȼ ɰɟɥɨɦ ɪɟɲɟɧɢɟ ɩɪɨɛɥɟɦɵ ɜɢɞɢɬɫɹ ɡɞɟɫɶ ɜ
ɫɨɫɬɚɜɥɟɧɢɢ ɱɢɫɥɟɧɧɨɝɨ ɚɥɝɨɪɢɬɦɚ ɞɥɹ ɤɨɦɩɶɸɬɟɪɧɨɝɨ ɩɨɢɫɤɚ ɷɥɟɦɟɧɬɚ ɧɚɢɥɭɱɲɟɝɨ L1-ɩɪɢɛɥɢɠɟɧɢɹ
ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɢ p ɜ ɜɵɛɪɚɧɧɨɦ ɤɥɚɫɫɟ ɮɭɧɤɰɢɣ ɩɥɨɬɧɨɫɬɢ K.
1.
Ⱥɩɩɪɨɤɫɢɦɚɰɢɹ ɤɭɫɨɱɧɨ-ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ
ɋɧɚɱɚɥɚ ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɚɩɩɪɨɤɫɢɦɢɪɭɟɦɚɹ ɮɭɧɤɰɢɹ p ɤɭɫɨɱɧɨ-ɥɢɧɟɣɧɚɹ, ɜɨɡɦɨɠɧɨ, ɫ
ɪɚɡɪɵɜɚɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ ɢ, ɜɨɡɦɨɠɧɨ, ɡɚɞɚɧɧɚɹ ɝɪɚɮɢɤɨɦ, ɜɵɜɟɞɟɧɧɵɦ ɧɚ ɷɤɪɚɧ ɤɨɦɩɶɸɬɟɪɚ ɜ ɯɨɞɟ
ɤɚɤɨɝɨ-ɥɢɛɨ ɷɤɫɩɟɪɢɦɟɧɬɚ. ȼ ɷɬɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɮɭɧɤɰɢɹ p ɢɦɟɟɬ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɥɨɤɚɥɶɧɵɯ
ɷɤɫɬɪɟɦɭɦɨɜ, ɢ ɟɺ ɝɪɚɮɢɤ ɜɦɟɫɬɟ ɫ ɨɫɶɸ ɚɛɫɰɢɫɫ ɫɨɫɬɚɜɥɹɟɬ ɩɨɥɢɝɨɧ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɭɠɟ ɜ ɬɪɟɛɨɜɚɧɢɢ
ɤɭɫɨɱɧɨɣ ɥɢɧɟɣɧɨɫɬɢ p ɧɟɬ ɠɟɫɬɤɨɝɨ ɨɝɪɚɧɢɱɟɧɢɹ, ɩɨɫɤɨɥɶɤɭ, ɧɚɩɪɢɦɟɪ, ɜɫɹɤɚɹ ɤɪɢɜɚɹ ɧɚ ɷɤɪɚɧɟ
ɤɨɦɩɶɸɬɟɪɚ ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɥɨɦɚɧɨɣ. Ʉɪɨɦɟ ɬɨɝɨ, ɜ ɤɨɧɰɟ ɪɚɛɨɬɵ ɛɭɞɟɬ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ
ɭɫɥɨɜɢɟ ɤɭɫɨɱɧɨɣ ɥɢɧɟɣɧɨɫɬɢ ɮɭɧɤɰɢɢ p – ɜɫɟɝɨ ɥɢɲɶ ɜɪɟɦɟɧɧɨɟ ɨɝɪɚɧɢɱɟɧɢɟ ɢ ɱɬɨ ɪɟɡɭɥɶɬɚɬɵ,
ɩɨɥɭɱɟɧɧɵɟ ɞɥɹ ɤɭɫɨɱɧɨ-ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ, ɫɨɯɪɚɧɹɸɬ ɫɢɥɭ ɢ ɞɥɹ ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɵɯ. Ȼɨɥɟɟ ɬɨɝɨ,
ɩɪɟɞɥɨɠɟɧɧɵɣ ɡɞɟɫɶ ɱɢɫɥɟɧɧɵɣ ɚɥɝɨɪɢɬɦ ɢɳɟɬ ɧɚɢɥɭɱɲɢɟ ɩɪɢɛɥɢɠɟɧɢɹ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɮɭɧɤɰɢɣ, ɧɨ, ɤɚɤ
ɭɜɢɞɢɦ ɧɢɠɟ, ɢ ɞɥɹ ɨɛɴɟɤɬɨɜ ɛɨɥɟɟ ɲɢɪɨɤɨɝɨ ɤɥɚɫɫɚ – ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɨɛɥɚɫɬɟɣ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ
ɱɢɫɥɟɧɧɵɣ ɚɥɝɨɪɢɬɦ, ɯɨɬɶ ɢ ɧɟ ɫɭɳɟɫɬɜɟɧɧɨ, ɧɨ ɜɫɺ ɠɟ ɜɚɪɶɢɪɭɟɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɜɵɛɨɪɚ ɤɥɚɫɫɚ K
ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɢɯ ɮɭɧɤɰɢɣ, ɩɨɷɬɨɦɭ ɞɥɹ ɫɨɞɟɪɠɚɬɟɥɶɧɨɝɨ ɢɡɥɨɠɟɧɢɹ ɦɟɬɨɞɚ ɜ ɤɚɱɟɫɬɜɟ K ɜɵɛɢɪɚɟɦ
ɡɞɟɫɶ ɞɜɭɯɩɚɪɚɦɟɬɪɢɱɟɫɤɢɣ ɤɥɚɫɫ ɧɨɪɦɚɥɶɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ:
2.
419
ɉɚɧɬɟɥɟɟɜ ȼ.ɉ.
ɇɚɢɥɭɱɲɢɟ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɜ ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ
f ( x, a , b )
1 exp( ( x a )
2b 2
2ʌ b
2
).
(1)
ɂɧɬɟɝɪɚɥ ɨɬ ɦɨɞɭɥɹ ɪɚɡɧɨɫɬɢ ɮɭɧɤɰɢɣ f ɢ p ɪɚɜɟɧ ɩɥɨɳɚɞɢ S(G''Gt) ɫɢɦɦɟɬɪɢɱɟɫɤɨɣ ɪɚɡɧɨɫɬɢ
G''Gt = (Gt\G')‰(G'\Gt) ɨɛɥɚɫɬɟɣ Gt ɢ G', ɥɟɠɚɳɢɯ ɦɟɠɞɭ ɨɫɶɸ ɚɛɫɰɢɫɫ ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɝɪɚɮɢɤɚɦɢ
ɮɭɧɤɰɢɣ f ɢ p (ɪɢɫ. 1):
f
³|
P
f ( x, a, b) p( x) | dx
S (G'' Gt ) .
(2)
f
Ɉɬ ɨɛɥɚɫɬɟɣ ɧɟ ɬɪɟɛɭɟɬɫɹ ɡɞɟɫɶ ɬɨɩɨɥɨɝɢɱɟɫɤɨɣ
ɫɜɹɡɧɨɫɬɢ, ɩɨɷɬɨɦɭ ɨɛɥɚɫɬɶ ɦɨɠɧɨ ɩɨɧɢɦɚɬɶ ɤɚɤ ɡɚɦɵɤɚɧɢɟ
Q
ɨɬɤɪɵɬɨɝɨ
ɦɧɨɠɟɫɬɜɚ,
ɜɨɡɦɨɠɧɨ,
ɧɟɫɜɹɡɧɨɝɨ
ɢɥɢ
ɦɧɨɝɨɫɜɹɡɧɨɝɨ, ɚ ɩɥɨɳɚɞɶ ɫɱɢɬɚɬɶ ɩɪɨɞɨɥɠɟɧɧɨɣ ɞɨ
a
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɦɟɪɵ Ʌɟɛɟɝɚ. Ɇɢɧɢɦɭɦɭ ɷɬɨɝɨ ɢɧɬɟɝɪɚɥɚ
ɪɢɫ. 1
ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɦɢɧɢɦɭɦɭ ɩɥɨɳɚɞɢ S(G''Gt)
ɫɢɦɦɟɬɪɢɱɟɫɤɨɣ ɪɚɡɧɨɫɬɢ G''Gt ɨɬɜɟɱɚɟɬ ɦɚɤɫɢɦɭɦ ɩɥɨɳɚɞɢ
S(G'ˆGt) ɨɛɳɟɣ ɱɚɫɬɢ G'ˆGt ɨɛɥɚɫɬɟɣ Gt ɢ G', ɱɬɨ ɞɥɹ ɧɨɪɦɢɪɨɜɚɧɧɵɯ ɮɭɧɤɰɢɣ ɩɥɨɬɧɨɫɬɢ ɫɥɟɞɭɟɬ ɢɡ
ɪɚɜɟɧɫɬɜɚ
S(G''Gt) = S(Gt) + S(G') – 2S(G'ˆGt) = 2 – 2S(G'ˆGt),
ɚ ɞɥɹ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɣ p c ɤɨɧɟɱɧɨɣ ɦɟɪɨɣ S(G') – ɢɡ ɪɚɜɟɧɫɬɜɚ
S(G''Gt) = S(Gt)+S(G') – 2S(G'ˆGt) =1+ S(G') – 2S(G'ˆGt).
ɇɚɤɨɧɟɰ, ɫɨɞɟɪɠɚɬɟɥɶɧɵɣ ɫɦɵɫɥ ɪɟɲɚɟɦɨɣ ɡɚɞɚɱɢ ɫɨɯɪɚɧɹɟɬɫɹ ɢ ɜ ɫɥɭɱɚɟ ɛɟɫɤɨɧɟɱɧɨɣ ɦɟɪɵ
S(G'), ɟɫɥɢ ɧɚɢɥɭɱɲɟɟ ɩɪɢɛɥɢɠɟɧɢɟ f ɩɨ-ɩɪɟɠɧɟɦɭ ɜɵɛɢɪɚɬɶ ɤɚɤ ɨɬɜɟɱɚɸɳɟɟ maxS(G'ˆGt). ɂɡɥɚɝɚɟɦɵɣ
ɡɞɟɫɶ ɦɟɬɨɞ ɧɚɰɟɥɟɧ ɧɚ ɦɚɤɫɢɦɢɡɚɰɢɸ ɦɟɪɵ S(G'ˆGt) ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɬɨɝɨ, ɧɨɪɦɢɪɨɜɚɧɚ ɥɢ ɮɭɧɤɰɢɹ p
ɢɥɢ ɧɟ ɧɨɪɦɢɪɨɜɚɧɚ, ɹɜɥɹɟɬɫɹ ɥɢ ɨɧɚ ɮɭɧɤɰɢɟɣ ɩɥɨɬɧɨɫɬɢ ɢɥɢ ɬɚɤɨɜɨɣ ɧɟ ɹɜɥɹɟɬɫɹ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ
ɝɪɚɧɢɰɚ ɨɛɥɚɫɬɢ G' ɫɨɫɬɨɢɬ ɢɡ ɨɞɧɨɣ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɡɚɦɤɧɭɬɵɯ ɥɨɦɚɧɵɯ, ɤɨɬɨɪɵɟ ɛɭɞɟɦ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɢ ɟɞɢɧɨɨɛɪɚɡɧɨ ɨɛɪɚɛɚɬɵɜɚɬɶ. ɉɪɢ ɫɬɨɥɶ ɨɛɳɢɯ ɩɪɟɞɩɨɥɨɠɟɧɢɹɯ ɱɚɫɬɶ ɨɛɥɚɫɬɢ G',
ɥɟɠɚɳɚɹ ɧɢɠɟ ɨɫɢ ɚɛɫɰɢɫɫ, ɤɨɝɞɚ ɬɚɤɚɹ ɱɚɫɬɶ ɢɦɟɟɬɫɹ, ɛɭɞɟɬ ɢɝɧɨɪɢɪɨɜɚɬɶɫɹ, ɢ ɧɢɠɧɟɣ ɝɪɚɧɢɰɟɣ
ɨɛɥɚɫɬɢ G' ɛɭɞɟɬ ɫɱɢɬɚɬɶɫɹ ɱɚɫɬɶ ɨɫɢ ɚɛɫɰɢɫɫ, ɤɨɬɨɪɚɹ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ G'. ȼɵɞɟɥɢɦ ɢɡ ɝɪɚɧɢɰɵ ɞG' ɨɞɧɭ
ɢɡ ɝɪɚɧɢɱɧɵɯ ɥɨɦɚɧɵɯ ȼ1ȼ2…ȼm ɫ ɜɟɪɲɢɧɚɦɢ ȼj(xj, yj), ɡɚɧɭɦɟɪɨɜɚɧɧɵɦɢ ɞɥɹ ɭɞɨɛɫɬɜɚ ɬɚɤ, ɱɬɨ ɫɬɨɪɨɧɚ
ȼmȼ1 ɥɟɠɢɬ ɧɚ ɨɫɢ Ox ɢ ɫɨɜɩɚɞɚɟɬ ɫ ɧɟɣ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ: ȼm(xm, 0), ȼ1(x1, 0), xm < x1. Ⱦɥɹ ɜɫɟɯ ɨɫɬɚɥɶɧɵɯ
ɫɬɨɪɨɧ ȼjȼj+1 ɷɬɨɣ ɥɨɦɚɧɨɣ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɹɬɨɣ ɡɞɟɫɶ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɨɪɢɟɧɬɚɰɢɟɣ, ɢɦɟɟɦ xj +1 d xj,
ȼm+1 { ȼ1. ȿɫɥɢ ɠɟ ɡɚɦɤɧɭɬɚɹ ɝɪɚɧɢɱɧɚɹ ɥɨɦɚɧɚɹ l ɨɛɥɚɫɬɢ G' ɧɟ ɢɦɟɟɬ ɧɚ ɨɫɢ ɚɛɫɰɢɫɫ ɫɬɨɪɨɧɵ, ɬɨ
ɜɟɪɲɢɧɵ ɥɨɦɚɧɨɣ l ɡɚɧɭɦɟɪɨɜɵɜɚɸɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɜɵɛɪɚɧɧɨɣ ɧɚɦɢ ɩɪɚɜɨɣ ɨɪɢɟɧɬɚɰɢɢ, ɧɚɱɢɧɚɹ ɫ
ɥɸɛɨɣ ɟɺ ɜɟɪɲɢɧɵ ȼ1.
ɂɡɦɟɧɟɧɢɸ ɩɚɪɚɦɟɬɪɚ b ɜ ɪɚɜɟɧɫɬɜɟ (1) ɨɬɜɟɱɚɟɬ ɤɨɦɩɨɡɢɰɢɹ ɞɜɭɯ ɫɠɚɬɢɣ ɨɛɥɚɫɬɢ Gt, ɨɞɧɨ
ɜɞɨɥɶ ɨɫɢ ɚɛɫɰɢɫɫ x X, X - a = k(x- a), ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ k ɢ ɞɪɭɝɨɟ ɜɞɨɥɶ ɨɫɢ ɨɪɞɢɧɚɬ, y Y, Y = y/k, c
ɨɛɪɚɬɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ 1/k. ɋɠɚɬɢɟ ɩɨɧɢɦɚɟɬɫɹ ɡɞɟɫɶ ɜ ɲɢɪɨɤɨɦ ɫɦɵɫɥɟ ɫɥɨɜɚ, ɤɨɝɞɚ ɤɨɷɮɮɢɰɢɟɧɬ k
ɞɨɩɭɫɤɚɟɬɫɹ ɥɸɛɨɣ ɩɨɥɨɠɢɬɟɥɶɧɵɣ, 0 < k < +f. ɉɨɫɤɨɥɶɤɭ ɪɟɱɶ ɩɨɣɞɺɬ ɨ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɢ ɩɨ
ɩɚɪɚɦɟɬɪɭ b, ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɠɚɬɢɹ ɛɥɢɡɤɢ ɤ ɬɨɠɞɟɫɬɜɟɧɧɨɦɭ, ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ k ɜɛɥɢɡɢ ɟɞɢɧɢɰɵ.
Ⱦɥɹ ɥɸɛɵɯ ɞɜɭɯ ɧɨɪɦɚɥɶɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ (1), ɨɩɪɟɞɟɥɹɟɦɵɯ ɤɨɧɤɪɟɬɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ a, b ɢ
a', b', ɩɟɪɟɯɨɞ ɨɬ ɨɞɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɤ ɞɪɭɝɨɦɭ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɤɨɦɩɨɡɢɰɢɟɣ ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɫɞɜɢɝɚ ɢ
ɞɜɭɯ ɭɤɚɡɚɧɧɵɯ ɜɵɲɟ ɫɠɚɬɢɣ X - a = k(x- a) ɢ Y = y/k ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ k ɢ 1/k. ɂɡɦɟɧɟɧɢɸ 'a = a'- a
ɩɚɪɚɦɟɬɪɚ a ɨɬɜɟɱɚɟɬ ɩɚɪɚɥɥɟɥɶɧɵɣ ɫɞɜɢɝ X = x+'a ɨɛɥɚɫɬɢ Gt ɜɞɨɥɶ ɨɫɢ ɚɛɫɰɢɫɫ, ɚ ɢɡɦɟɧɟɧɢɸ
ɩɚɪɚɦɟɬɪɚ b ɨɬɜɟɱɚɟɬ ɤɨɦɩɨɡɢɰɢɹ F ɞɜɭɯ ɫɠɚɬɢɣ X- a = k(x- a) ɢ Y = y/k. ɉɪɨɫɥɟɞɢɦ ɫɜɹɡɶ ɦɟɠɞɭ
ɤɨɷɮɮɢɰɢɟɧɬɨɦ k ɤɨɦɩɨɡɢɰɢɢ F ɢ ɩɪɢɪɚɳɟɧɢɟɦ 'b ɩɚɪɚɦɟɬɪɚ b. Ⱦɥɹ ɷɬɨɝɨ ɡɚɦɟɧɢɦ ɜ ɪɚɜɟɧɫɬɜɟ (1)
ɤɨɨɪɞɢɧɚɬɵ x, y =f(x, a, b) ɧɨɜɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ X, Y ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ F.
( X a)2
1 exp( ( X a )2 ), ɨɬɫɸɞɚ Y
1
kY
exp(
).
2ʌb
2(bk ) 2
2 ʌ (bk )
2(bk )2
ȼɢɞɢɦ, ɱɬɨ ɩɪɨɢɡɨɲɥɚ ɡɚɦɟɧɚ ɡɧɚɱɟɧɢɹ b ɩɚɪɚɦɟɬɪɚ ɧɚ ɡɧɚɱɟɧɢɟ bk. Ʉɨɦɩɨɡɢɰɢɢ F ɨɬɜɟɱɚɟɬ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ b bk ɩɚɪɚɦɟɬɪɚ b ɢ ɩɪɢɪɚɳɟɧɢɟ 'b = bk- b = (k-1)b. ɉɪɢ ɷɬɨɦ ɤɨɷɮɮɢɰɢɟɧɬ k ɱɟɪɟɡ
ɩɪɢɪɚɳɟɧɢɟ 'b ɜɵɱɢɫɥɹɟɬɫɹ ɩɨɫɪɟɞɫɬɜɨɦ ɪɚɜɟɧɫɬɜɚ k-1='b/b.
420
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 7, ʋ3, 2004 ɝ.
ɫɬɪ.419-424
ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɨɛɥɚɫɬɢ G' ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ ɨɛɪɚɬɧɨɟ ɤ ɤɨɦɩɨɡɢɰɢɢ F ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ F-1.
ȼɵɱɢɫɥɹɹ ɞɢɮɮɟɪɟɧɰɢɚɥ ɩɥɨɳɚɞɢ S ɜ ɫɥɭɱɚɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ F-1 ɨɛɥɚɫɬɢ G', ɤɨɝɞɚ ɨɛɥɚɫɬɶ Gt
ɧɟɩɨɞɜɢɠɧɚ, ɩɪɟɞɩɨɥɨɠɢɦ ɫɧɚɱɚɥɚ, ɱɬɨ ɝɪɚɧɢɰɚ ɞG' ɨɛɥɚɫɬɢ G' ɤɭɫɨɱɧɨ-ɥɢɧɟɣɧɚɹ.
ȼɫɩɨɦɨɝɚɬɟɥɶɧɵɣ ɦɚɬɟɪɢɚɥ
ɇɚɦ ɩɨɧɚɞɨɛɹɬɫɹ ɞɜɟ ɩɪɨɫɬɵɟ ɥɟɦɦɵ ɨɬɜɥɟɱɺɧɧɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɢɯ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɪɢɜɢɚɥɶɧɵ, ɧɨ
ɛɟɡ ɜɧɢɦɚɧɢɹ ɤ ɜɵɞɟɥɟɧɧɵɦ ɜ ɧɢɯ ɮɚɤɬɚɦ ɩɨɧɢɦɚɧɢɟ ɩɨɫɥɟɞɭɸɳɟɝɨ ɦɚɬɟɪɢɚɥɚ, ɜɨɡɦɨɠɧɨ, ɛɭɞɟɬ
ɡɚɬɪɭɞɧɟɧɨ.
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ M ɫ ɦɟɪɨɣ m. ɉɪɢɧɹɬɨ ɝɨɜɨɪɢɬɶ, ɱɬɨ ɨɬɨɛɪɚɠɟɧɢɟ H:M M
ɫɨɯɪɚɧɹɟɬ ɦɟɪɭ, ɟɫɥɢ ɜɫɹɤɢɣ ɪɚɡ, ɤɨɝɞɚ ɦɟɪɚ m ɨɩɪɟɞɟɥɟɧɚ ɞɥɹ ɦɧɨɠɟɫɬɜɚ X, ɨɧɚ ɨɩɪɟɞɟɥɟɧɚ ɬɚɤɠɟ ɢ ɞɥɹ
ɦɧɨɠɟɫɬɜɚ H(X) ɢ ɩɪɢɧɢɦɚɟɬ ɧɚ ɦɧɨɠɟɫɬɜɚɯ X ɢ H(X) ɪɚɜɧɵɟ ɡɧɚɱɟɧɢɹ:
3.
m(H(X)) = m(X).
Ʌɟɦɦɚ 1. ɉɭɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ M ɫ ɦɟɪɨɣ m ɜɵɞɟɥɟɧɵ ɞɜɚ ɦɧɨɠɟɫɬɜɚ Gt ɢ G', ɢ ɩɭɫɬɶ ɨɛɪɚɬɢɦɨɟ
ɨɬɨɛɪɚɠɟɧɢɟ H:M M ɫɨɯɪɚɧɹɟɬ ɦɟɪɭ m, ɤɨɬɨɪɚɹ ɩɪɢɧɢɦɚɟɬ ɧɚ ɦɧɨɠɟɫɬɜɟ GtˆH-1(G') ɡɧɚɱɟɧɢɟ
m(GtˆH-1(G')). Ɍɨɝɞɚ ɬɚɤɨɟ ɠɟ ɡɧɚɱɟɧɢɟ ɦɟɪɚ m ɩɪɢɧɢɦɚɟɬ ɢ ɧɚ ɦɧɨɠɟɫɬɜɟ H(Gt)ˆG':
m(H(Gt)ˆG') = m(GtˆH-1(G')).
-1
(3)
-1
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, m(GtˆH (G')) = m(H(GtˆH (G'))) = m(H(Gt)ˆG').
Ʌɟɦɦɚ 2. ɉɭɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ M ɫ ɦɟɪɨɣ m ɜɵɞɟɥɟɧɵ ɞɜɚ ɦɧɨɠɟɫɬɜɚ Gt ɢ G', ɢ ɩɭɫɬɶ ɨɛɪɚɬɢɦɨɟ
~
ɨɬɨɛɪɚɠɟɧɢɟ H:M M ɫɨɯɪɚɧɹɟɬ ɦɟɪɭ m. ȿɫɥɢ ɨɩɪɟɞɟɥɟɧɨ ɩɪɢɪɚɳɟɧɢɟ ' m = m(GtˆH-1(G')) - m(G'ˆGt)
ɦɟɪɵ m ɧɚ ɨɛɳɟɣ ɱɚɫɬɢ G'ˆGt ɞɜɭɯ ɦɧɨɠɟɫɬɜ Gt ɢ G', ɤɨɝɞɚ G' ɩɨɞɜɟɪɝɧɭɬɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ H-1, ɚ Gt
ɧɟɩɨɞɜɢɠɧɨ, ɬɨ ɞɥɹ ɦɟɪɵ m ɨɩɪɟɞɟɥɟɧɨ ɬɚɤɠɟ ɢ ɩɪɢɪɚɳɟɧɢɟ 'm = m(H(Gt)ˆG') - m(G'ˆGt), ɤɨɝɞɚ Gt
~
ɩɨɞɜɟɪɝɧɭɬɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ H, ɚ ɜɬɨɪɨɟ ɦɧɨɠɟɫɬɜɨ G' ɧɟɩɨɞɜɢɠɧɨ, ɢ ɷɬɢ ɞɜɚ ɩɪɢɪɚɳɟɧɢɹ ɪɚɜɧɵ: ' m ='m.
~
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɪɚɜɟɧɫɬɜɨ ɩɪɢɪɚɳɟɧɢɣ 'm= ' m ɦɵ ɩɨɥɭɱɚɟɦ, ɟɫɥɢ ɨɬ ɨɛɟɢɯ ɱɚɫɬɟɣ ɪɚɜɟɧɫɬɜɚ (3)
ɨɬɧɹɬɶ ɱɢɫɥɨ m(G'ˆGt).
Ʉɨɦɩɨɡɢɰɢɹ F-1 ɞɜɭɯ ɫɠɚɬɢɣ xX, X- a = k-1(x- a) ɢ yY, Y = ky ɜɞɨɥɶ ɨɫɟɣ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ
ɤɨɨɪɞɢɧɚɬ ɫɨɯɪɚɧɹɟɬ ɩɥɨɳɚɞɶ S. ɋɨɝɥɚɫɧɨ ɥɟɦɦɟ 2, ɨɬɨɛɪɚɠɟɧɢɹɦ F ɢ F-1 ɨɬɜɟɱɚɸɬ ɪɚɜɧɵɟ ɩɪɢɪɚɳɟɧɢɹ
~
'S = ' S ɩɥɨɳɚɞɢ S, S(F(Gt)ˆG') - S(G'ˆGt) = S(GtˆF-1(G')) – S(G'ˆGt). ɉɨɷɬɨɦɭ, ɩɪɟɨɛɪɚɡɭɹ ɤɨɦɩɨɡɢɰɢɟɣ
F ɨɛɥɚɫɬɶ Gt ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ G' ɧɟɩɨɞɜɢɠɧɚ, ɦɵ ɢɦɟɟɦ ɬɨ ɠɟ ɫɚɦɨɟ ɩɪɢɪɚɳɟɧɢɟ, ɤɚɤ ɟɫɥɢ ɛɵ ɩɨɞɜɟɪɝɥɢ
ɨɛɪɚɬɧɨɦɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ F-1 ɨɛɥɚɫɬɶ G' ɩɪɢ ɧɟɩɨɞɜɢɠɧɨɣ Gt.
Ɉɛɥɚɫɬɢ, ɡɚɦɟɬɚɟɦɨɣ ɧɚɩɪɚɜɥɟɧɧɵɦ ɨɬɪɟɡɤɨɦ (ɢɥɢ ɟɝɨ ɱɚɫɬɶɸ) ɩɪɢ ɦɚɥɨɦ ɫɞɜɢɝɟ ɢɥɢ ɫɠɚɬɢɢ,
ɩɪɢɩɢɫɵɜɚɟɦ ɩɪɚɜɭɸ ɨɪɢɟɧɬɚɰɢɸ, ɟɫɥɢ ɩɪɢ ɞɜɢɠɟɧɢɢ ɨɬɪɟɡɤɚ ɨɧɚ ɩɟɪɟɲɥɚ ɫ ɟɝɨ ɩɪɚɜɨɣ ɫɬɨɪɨɧɵ ɧɚ
ɥɟɜɭɸ. ȿɫɥɢ ɠɟ ɨɧɚ ɩɟɪɟɲɥɚ ɫ ɟɝɨ ɥɟɜɨɣ ɫɬɨɪɨɧɵ ɧɚ ɩɪɚɜɭɸ, ɩɪɢɩɢɲɟɦ ɟɣ ɥɟɜɭɸ ɨɪɢɟɧɬɚɰɢɸ, ɚ ɟɺ
ɩɥɨɳɚɞɢ – ɡɧɚɤ ɦɢɧɭɫ. Ɍɚɤɭɸ ɩɥɨɳɚɞɶ, ɩɪɢɧɢɦɚɸɳɭɸ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ,
ɧɚɡɵɜɚɟɦ ɨɬɧɨɫɢɬɟɥɶɧɨɣ. ɉɪɢ ɫɠɚɬɢɢ ɩɥɨɫɤɨɫɬɢ ɧɚɩɪɚɜɥɟɧɧɵɣ ɨɬɪɟɡɨɤ ɦɨɠɟɬ ɫɞɜɢɧɭɬɶɫɹ ɨɞɧɨɣ ɫɜɨɟɣ
ɱɚɫɬɶɸ ɜɩɪɚɜɨ, ɚ ɞɪɭɝɨɣ – ɜɥɟɜɨ (ɪɢɫ. 2). ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɡɚɦɟɬɚɟɦɚɹ ɢɦ ɨɛɥɚɫɬɶ ɪɚɫɩɚɞɚɟɬɫɹ ɜ ɡɚɦɟɬɚɟɦɭɸ ɜɩɪɚɜɨ
C j'
Cj
ɢ ɜɥɟɜɨ. ɂɯ ɩɥɨɳɚɞɹɦ ɩɪɢɩɢɫɵɜɚɟɬɫɹ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,
ɡɧɚɤ ɩɥɸɫ ɢ ɦɢɧɭɫ. ɉɪɢ ɷɬɨɦ ɨɛɳɚɹ ɨɬɧɨɫɢɬɟɥɶɧɚɹ
ɩɥɨɳɚɞɶ ɷɬɢɯ ɞɜɭɯ ɱɚɫɬɟɣ ɨɛɥɚɫɬɢ, ɡɚɦɟɬɚɟɦɵɯ ɜɩɪɚɜɨ ɢ
+
Dj
ɜɥɟɜɨ,
ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɚɥɝɟɛɪɚɢɱɟɫɤɚɹ ɫɭɦɦɚ ɢɯ
Dj'
ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɩɥɨɳɚɞɟɣ. Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɩɥɨɳɚɞɶ
ɨɛɥɚɫɬɢ, ɤɨɬɨɪɭɸ ɡɚɦɟɬɚɟɬ ɥɨɦɚɧɚɹ, ɬɚɤɠɟ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɨɣ ɨɬɧɨɫɢɬɟɥɶɧɵɯ ɩɥɨɳɚɞɟɣ,
ɪɢɫ. 2
ɫɨɫɬɚɜɥɟɧɧɨɣ ɞɥɹ ɨɛɥɚɫɬɟɣ, ɡɚɦɟɬɚɟɦɵɯ ɜɫɟɦɢ ɟɺ
ɡɜɟɧɶɹɦɢ.
4. Ɍɟɨɪɟɦɚ. ɉɥɨɳɚɞɶ S ɨɛɳɟɣ ɱɚɫɬɢ ɩɨɞɜɢɠɧɨɣ ɨɛɥɚɫɬɢ Gt ɢ ɧɟɩɨɞɜɢɠɧɨɣ ɦɧɨɝɨɭɝɨɥɶɧɨɣ ɨɛɥɚɫɬɢ G'
ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɨɣ ɮɭɧɤɰɢɟɣ ɩɟɪɟɦɟɧɧɵɯ a ɢ b ɜɫɸɞɭ ɜ ɩɨɥɭɩɥɨɫɤɨɫɬɢ
- f < a < + f, 0 < b <+ f. ȿɫɥɢ ɩɪɢ ɷɬɨɦ P1Q1, P2Q2,…, PlQl,… – ɩɟɪɟɱɟɧɶ ɜɫɟɯ ɦɚɤɫɢɦɚɥɶɧɵɯ ɩɨ
ɜɤɥɸɱɟɧɢɸ ɞɭɝ ɤɪɢɜɨɣ f ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ (1), ɥɟɠɚɳɢɯ ɜ ɨɛɥɚɫɬɢ G', Pl(xpl, ypl), Ql(xql, yql),
xq" < xpl, ɬɨ ɩɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ S ɩɟɪɟɦɟɧɧɵɯ a ɢ b ɫɭɳɟɫɬɜɭɟɬ ɢ ɪɚɜɟɧ
dS = da6l (yql – ypl) + (db/b)6l{yql(xql – a) – ypl(xpl – a)}.
(4)
421
ɉɚɧɬɟɥɟɟɜ ȼ.ɉ.
ɇɚɢɥɭɱɲɢɟ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɜ ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ
ɋɞɜɢɝ ɚɛɫɰɢɫɫɵ x ɤ ɟɺ ɨɛɪɚɡɭ X ɩɪɢ ɨɛɪɚɬɧɨɦ ɨɬɨɛɪɚɠɟɧɢɢ F-1 ɩɨɫɪɟɞɫɬɜɨɦ ɫɠɚɬɢɣ X- a = k-1(x - a) ɢ Y = ky
ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ X- x = (k-1 -1)(ɯ- a). ɇɚ ɫɬɨɪɨɧɟ BkBk+1 ɨɛɥɚɫɬɢ G' ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɨɞɢɧ ɢɥɢ ɞɜɚ
ɨɬɪɟɡɤɚ, ɰɟɥɢɤɨɦ ɥɟɠɚɳɢɟ ɜ ɨɛɥɚɫɬɢ Gt ɢ ɤɨɬɨɪɵɟ ɧɟɥɶɡɹ ɩɪɨɞɨɥɠɢɬɶ, ɧɟ ɜɵɜɨɞɹ ɢɯ ɡɚ ɩɪɟɞɟɥɵ ɨɛɥɚɫɬɢ
Gt. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ CjDj, Cj(xcj, ycj), Dj(xdj, ydj) ɨɞɢɧ ɢɡ ɬɚɤɢɯ ɨɬɪɟɡɤɨɜ, ɪɚɜɧɨɧɚɩɪɚɜɥɟɧɧɵɣ ɫ BkBk+1.
Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɩɥɨɳɚɞɶ, ɡɚɦɟɬɚɟɦɚɹ ɨɬɪɟɡɤɨɦ CjDj ɩɪɢ ɫɠɚɬɢɢ X - a = k-1(x - a), ɜɵɱɢɫɥɹɟɬɫɹ ɩɨɫɪɟɞɫɬɜɨɦ
ɪɚɜɟɧɫɬɜɚ ǻxjS = 2-1(Xcj – xcj +Xdj – xdj)(ydj – ycj) = 2-1(k-1 – 1)(xcj + xdj – 2a)(ydj – ycj), ɚ ɡɚɦɟɬɚɟɦɚɹ ɜɫɟɦɢ ɨɬɪɟɡɤɚɦɢ
CjDj, ɥɟɠɚɳɢɦɢ ɜ Gt ɢ ɫɨɫɬɚɜɥɟɧɧɵɦɢ ɞɥɹ ɜɫɟɯ ɫɬɨɪɨɧ BkBk+1, ɜ ɤɨɬɨɪɵɯ ɬɚɤɢɟ ɨɬɪɟɡɤɢ CjDj ɢɦɟɸɬɫɹ, –
ɩɨɫɪɟɞɫɬɜɨɦ ɪɚɜɟɧɫɬɜɚ
ǻxS = 6jǻxjS = 2-1(k-1 – 1)6j (xcj + xdj – 2a)(ydj – ycj).
ɉɪɢ ko1 ɩɥɨɳɚɞɶ 'xS c ɬɨɱɧɨɫɬɶɸ ɞɨ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɣ ɜɵɫɲɟɝɨ ɩɨɪɹɞɤɚ ɪɚɜɧɚ ɩɪɢɪɚɳɟɧɢɸ
ɩɥɨɳɚɞɢ S ɨɛɳɟɣ ɱɚɫɬɢ ɨɛɥɚɫɬɟɣ Gt ɢ G', ɤɨɬɨɪɨɟ ɩɨɥɭɱɚɟɬɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɠɚɬɢɹ X- a = k-1(x- a) ɨɛɥɚɫɬɢ G'
ɩɪɢ ɧɟɩɨɞɜɢɠɧɨɣ Gt. ɉɪɢ ɩɨɫɥɟɞɭɸɳɟɦ ɫɠɚɬɢɢ y Y, Y = ky ɫɞɜɢɝ ɨɪɞɢɧɚɬɵ y ɤ ɟɺ ɨɛɪɚɡɭ Y ɪɚɜɟɧ
Y-y = (k-1)y. ɉɨɷɬɨɦɭ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɩɥɨɳɚɞɶ 'yjS, ɤɨɬɨɪɚɹ ɡɚɦɟɬɚɟɬɫɹ ɨɬɪɟɡɤɨɦ CjDj ɩɪɢ ɫɠɚɬɢɢ Y = ky,
ɪɚɜɧɚ
'yjS = 2-1(Ycj – ycj+Ydj – ydj)(xcj – xdj) = 2-1(k – 1)(ycj + ydj)(xcj – xdj),
ɚ ɩɥɨɳɚɞɶ 'yS, ɡɚɦɟɬɚɟɦɚɹ ɜɫɟɦɢ ɨɬɪɟɡɤɚɦɢ CjDj, ɪɚɜɧɚ
'yS = 2-1(k – 1)6j(ycj + ydj)(xcj – xdj).
ɉɪɢ ko1 ɩɥɨɳɚɞɶ 'yS ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɣ ɜɵɫɲɟɝɨ ɩɨɪɹɞɤɚ ɪɚɜɧɚ ɩɪɢɪɚɳɟɧɢɸ
ɩɥɨɳɚɞɢ S, ɤɨɬɨɪɨɟ ɨɧɚ ɩɨɥɭɱɚɟɬ, ɤɨɝɞɚ ɨɛɥɚɫɬɶ G' ɩɨɞɜɟɪɝɚɟɬɫɹ ɫɠɚɬɢɸ Y = ky, ɚ ɨɛɥɚɫɬɶ Gt ɧɟɩɨɞɜɢɠɧɚ.
ɋɭɦɦɚɪɧɨɟ ɩɪɢɪɚɳɟɧɢɟ 'bS ɩɥɨɳɚɞɢ S ɜ ɪɟɡɭɥɶɬɚɬɟ ɤɨɦɩɨɡɢɰɢɢ F-1 ɞɜɭɯ ɫɠɚɬɢɣ ɨɛɥɚɫɬɢ G', X- a =
k-1(ɯ- a) ɢ Y = ky ɩɪɢ ɧɟɩɨɞɜɢɠɧɨɣ Gt ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɣ ɜɵɫɲɟɝɨ ɩɨɪɹɞɤɚ ɪɚɜɧɨ ɫɭɦɦɟ
ɩɪɢɪɚɳɟɧɢɣ 'xS ɢ 'yS.
'bS = 2-1(k-1–1)6j(xcj + xdj – 2a)(ydj – ycj) + 2-1(k –1)6 j(ycj + ydj)(xcj – xdj).
Ɇɵ ɫɨɫɬɚɜɢɥɢ ɩɪɢɪɚɳɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɩɥɨɳɚɞɢ S ɨɛɳɟɣ ɱɚɫɬɢ ɞɜɭɯ ɨɛɥɚɫɬɟɣ Gt ɢ G', ɤɨɝɞɚ
ɨɛɥɚɫɬɶ G' ɩɨɞɜɟɪɝɚɟɬɫɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ F-1, ɚ ɨɛɥɚɫɬɶ Gt ɧɟɩɨɞɜɢɠɧɚ. ɋɨɝɥɚɫɧɨ ɥɟɦɦɟ 2, ɷɬɨ ɠɟ
ɩɪɢɪɚɳɟɧɢɟ 'bS ɩɨɥɭɱɚɟɬ ɩɥɨɳɚɞɶ S, ɤɨɝɞɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ F, X - a = k(ɯ- a), Y = k-1y ɩɨɞɜɟɪɝɚɟɬɫɹ ɨɛɥɚɫɬɶ
Gt ɩɪɢ ɧɟɩɨɞɜɢɠɧɨɣ G'. ɉɨɫɪɟɞɫɬɜɨɦ F ɡɧɚɱɟɧɢɟ ɩɚɪɚɦɟɬɪɚ b ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ kb, ɱɬɨ ɨɬɜɟɱɚɟɬ ɩɪɢɪɚɳɟɧɢɸ
'b = kb - b = (k-1)b ɩɚɪɚɦɟɬɪɚ b. Ɉɬɫɸɞɚ, k-1 = 'b/b. ɉɪɢ 'b o 0 ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɩɪɢ ko1 ɛɟɫɤɨɧɟɱɧɨ
ɦɚɥɚɹ (1-k)/k ɷɤɜɢɜɚɥɟɧɬɧɚ -'b/b. ɍɱɢɬɵɜɚɹ ɷɬɨ, ɫɨɫɬɚɜɥɹɟɦ ɱɚɫɬɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ S ɩɨ
ɩɟɪɟɦɟɧɧɨɣ b ɞɥɹ ɫɥɭɱɚɹ, ɤɨɝɞɚ ɩɨɫɪɟɞɫɬɜɨɦ F ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɨɛɥɚɫɬɶ Gt, ɚ G' ɨɫɬɚɺɬɫɹ ɧɟɩɨɞɜɢɠɧɨɣ.
dbS = 2-1(db/b)6 j{xcj + xdj – 2a)(ycj – ydj) + (ycj + ydj)(xcj – xdj)}.
ɉɨɫɥɟ ɩɪɢɜɟɞɟɧɢɹ ɩɨɞɨɛɧɵɯ ɱɥɟɧɨɜ ɢɦɟɟɦ
dbS = (db/b)6 j{xcj(xcj – a) – ydj(ydj – a)}.
(5)
Ɉɛɪɚɬɢɦ ɬɟɩɟɪɶ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɫɭɦɦɢɪɨɜɚɧɢɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (5) ɩɨɞɱɢɧɟɧɨ
ɩɪɚɜɢɥɭ ɫɥɨɠɟɧɢɹ ɜɟɤɬɨɪɨɜ. ȿɫɥɢ ɧɚɩɪɚɜɥɟɧɧɵɟ ɨɬɪɟɡɤɢ CjDj, Cj+1Dj+1,…, CkDk, ɥɟɠɚɳɢɟ ɜ ɨɛɥɚɫɬɢ Gt,
ɫɨɫɬɚɜɥɹɸɬ ɥɨɦɚɧɭɸ l ɞG', ɦɚɤɫɢɦɚɥɶɧɭɸ ɩɨ ɜɤɥɸɱɟɧɢɸ ɢ ɧɚɩɪɚɜɥɟɧɧɭɸ ɫɨɝɥɚɫɧɨ ɨɪɢɟɧɬɚɰɢɢ ɝɪɚɧɢɰɵ
ɞG', ɬɨ ɜ ɪɚɜɟɧɫɬɜɟ (5) ɩɪɢ ɫɭɦɦɢɪɨɜɚɧɢɢ ɜɡɚɢɦɧɨ ɭɧɢɱɬɨɠɚɸɬɫɹ ɜɫɟ ɫɥɚɝɚɟɦɵɟ, ɨɬɜɟɱɚɸɳɢɟ ɜɧɭɬɪɟɧɧɢɦ
ɜɟɪɲɢɧɚɦ ɥɨɦɚɧɨɣ l, ɨɫɬɚɸɬɫɹ ɥɢɲɶ ɫɥɚɝɚɟɦɵɟ ydk(xdk - a) - ycj(xcj- a), ɤɨɬɨɪɵɟ ɨɬɜɟɱɚɸɬ ɤɨɧɰɚɦ
ɡɚɦɵɤɚɸɳɟɝɨ ɨɬɪɟɡɤɚ CjDk ɥɨɦɚɧɨɣ l – ɬɨɱɤɚɦ ɜɯɨɞɚ ɝɪɚɧɢɰɵ ɞG' ɜ ɨɛɥɚɫɬɶ Gt ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜɵɯɨɞɚ ɢɡ
ɧɟɺ. ȼ ɪɚɜɟɧɫɬɜɟ (4) ɷɬɢɦ ɬɨɱɤɚɦ ɨɬɜɟɱɚɸɬ ɨɛɨɡɧɚɱɟɧɢɹ Pl-1 ɢ Ql. ɉɨɷɬɨɦɭ ɩɨ ɡɚɜɟɪɲɟɧɢɸ ɝɪɭɩɩɢɪɨɜɤɢ
ɫɥɚɝɚɟɦɵɯ ɜ ɩɪɚɜɵɯ ɱɚɫɬɹɯ ɪɚɜɟɧɫɬɜɚ (5) ɨɫɬɚɧɭɬɫɹ ɥɢɲɶ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ Ql(xql, yql) ɜɯɨɞɚ ɝɪɚɧɢɰɵ ɞG' ɜ
ɨɛɥɚɫɬɶ Gt ɢ ɬɨɱɟɤ Pl(xpl, ypl) ɜɵɯɨɞɚ ɝɪɚɧɢɰɵ ɞG' ɢɡ ɨɛɥɚɫɬɢ Gt. ɉɪɢ ɷɬɨɦ ɬɨɱɤɚɦ ɜɵɯɨɞɚ Pl ɜ ɪɚɜɟɧɫɬɜɟ (5)
ɨɬɜɟɱɚɸɬ ɫɥɚɝɚɟɦɵɟ ypl(xpl - a), ɚ ɬɨɱɤɚɦ ɜɯɨɞɚ Ql – ɫɥɚɝɚɟɦɵɟ -yql(xql - a), ɧɚɞɟɥɟɧɧɵɟ ɡɧɚɤɨɦ ɦɢɧɭɫ. ȼɜɢɞɭ
ɢɦɟɸɳɟɣɫɹ ɡɞɟɫɶ ɞɜɨɣɫɬɜɟɧɧɨɫɬɢ ɩɪɨɢɫɯɨɞɢɬ ɡɚɦɟɳɟɧɢɟ ɪɨɥɟɣ ɬɨɱɟɤ ɜɯɨɞɚ ɢ ɜɵɯɨɞɚ: ɬɨɱɤɢ Ql ɜɯɨɞɚ
ɝɪɚɧɢɰɵ ɞG' ɜ ɨɛɥɚɫɬɶ Gt – ɷɬɨ ɜ ɬɨ ɠɟ ɜɪɟɦɹ ɬɨɱɤɢ ɜɵɯɨɞɚ ɝɪɚɧɢɰɵ ɞGt ɢɡ ɨɛɥɚɫɬɢ G', ɚ ɬɨɱɤɢ Pl ɜɵɯɨɞɚ
ɝɪɚɧɢɰɵ ɞG' ɢɡ Gt ɫɨɜɩɚɞɚɸɬ ɫ ɬɨɱɤɚɦɢ ɜɯɨɞɚ ɝɪɚɧɢɰɵ ɞGt ɜ ɨɛɥɚɫɬɶ G'. ȼ ɢɬɨɝɟ ɪɚɜɟɧɫɬɜɨ (5) ɩɪɢɧɢɦɚɟɬ
ɫɥɟɞɭɸɳɢɣ ɜɢɞ
dbS = (db/b) 6l{xql(xql – a) – ypl(ypl – a)},
(6)
ɝɞɟ Pl(xpl, ypl) ɢ Ql(xql, yql), ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɬɨɱɤɢ ɜɯɨɞɚ ɝɪɚɧɢɰɵ ɞGt ɜ ɨɛɥɚɫɬɶ G' ɢ ɜɵɯɨɞɚ ɷɬɨɣ ɝɪɚɧɢɰɵ ɢɡ
G'. ɑɚɫɬɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ dbS ɪɚɜɟɧɫɬɜɚ (6), ɤɚɤ ɜɢɞɢɦ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɚɤɨɦɭ ɠɟ ɜ ɪɚɜɟɧɫɬɜɟ (4). Ⱦɥɹ
ɡɚɜɟɪɲɟɧɢɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɟɨɪɟɦɵ ɨɫɬɚɺɬɫɹ ɜɵɹɜɢɬɶ ɮɨɪɦɭ ɱɚɫɬɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ daS ɩɨ ɩɟɪɟɦɟɧɧɨɣ a.
422
ȼɟɫɬɧɢɤ ɆȽɌɍ, ɬɨɦ 7, ʋ3, 2004 ɝ.
ɫɬɪ.419-424
ɉɪɢɪɚɳɟɧɢɸ 'a ɩɚɪɚɦɟɬɪɚ a ɨɬɜɟɱɚɟɬ ɩɚɪɚɥɥɟɥɶɧɵɣ ɫɞɜɢɝ ɨɛɥɚɫɬɢ Gt ɜɞɨɥɶ ɨɫɢ ɚɛɫɰɢɫɫ.
ɉɪɨɰɟɞɭɪɚ ɫɨɫɬɚɜɥɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɚ daS ɩɥɨɳɚɞɢ S ɩɨ ɩɟɪɟɦɟɧɧɨɣ a ɡɞɟɫɶ ɞɨɜɨɥɶɧɨ ɩɪɨɫɬɚ, ɩɨɷɬɨɦɭ
ɫɪɚɡɭ ɜɵɩɢɫɵɜɚɟɦ ɤɨɧɟɱɧɵɣ ɪɟɡɭɥɶɬɚɬ. ɑɚɫɬɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ daS ɩɪɢ ɫɞɜɢɝɟ ɨɛɥɚɫɬɢ Gt ɨɬɧɨɫɢɬɟɥɶɧɨ
ɧɟɩɨɞɜɢɠɧɨɣ G' ɪɚɜɟɧ
daS = da6l(yql – ypl),
ɝɞɟ Pl(xpl, ypl) ɢ Ql(xql, yql) – ɬɨɱɤɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜɯɨɞɚ ɜɟɪɯɧɟɣ ɝɪɚɧɢɰɵ ɞGt ɜ ɨɛɥɚɫɬɶ G' ɢ ɜɵɯɨɞɚ ɟɺ ɢɡ
ɧɟɺ, ɚ da – ɩɪɢɪɚɳɟɧɢɟ ɩɚɪɚɦɟɬɪɚ a ɮɭɧɤɰɢɢ f. ɉɨɥɧɵɣ ɞɢɮɮɟɪɟɧɰɢɚɥ ɮɭɧɤɰɢɢ S ɩɨ ɩɟɪɟɦɟɧɧɵɦ a ɢ b
ɞɥɹ ɩɨɞɜɢɠɧɨɣ ɨɛɥɚɫɬɢ Gt ɢ ɧɟɩɨɞɜɢɠɧɨɣ G' ɪɚɜɟɧ ɫɭɦɦɟ ɱɚɫɬɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɨɜ dS = daS+dbS.
ɋɤɥɚɞɵɜɚɹ ɢɯ, ɩɨɥɭɱɚɟɦ ɬɪɟɛɭɟɦɨɟ ɪɚɜɟɧɫɬɜɨ (4). ɇɚ ɷɬɨɦ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ ɡɚɜɟɪɲɚɟɬɫɹ.
5. ɑɢɫɥɟɧɧɚɹ ɨɩɬɢɦɢɡɚɰɢɹ ɮɭɧɤɰɢɢ S
ȼɵɩɢɫɵɜɚɹ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɢ ɝɪɚɞɢɟɧɬ
ɞS/ɞa = 6l (yql – ypl), ɞS/ɞb = b-16l{yql(xql – a) – ypl(xpl – a)},
(7)
g = ’S = i6l(yql – ypl) + jb-16l{yql(xql – a) – ypl(xpl – a)} =
= 6l{i(yql – ypl) + jb-1[yql(xql – a) – ypl(xpl – a)]},
ɡɚɦɟɱɚɟɦ, ɱɬɨ ɜɫɟ ɷɬɢ ɮɭɧɤɰɢɢ ɨɩɪɟɞɟɥɟɧɵ ɢ ɧɟɩɪɟɪɵɜɧɵ ɜɫɸɞɭ ɜ ɩɨɥɭɩɥɨɫɤɨɫɬɢ -f < a < +f, 0 < b<+f.
ɉɪɢ ɷɬɨɦ ɞɥɹ ɢɯ ɫɨɫɬɚɜɥɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ ɡɧɚɬɶ ɥɢɲɶ ɬɨɱɤɢ ɩɟɪɟɫɟɱɟɧɢɹ ɜɟɪɯɧɢɯ ɝɪɚɧɢɰ ɨɛɥɚɫɬɟɣ Gt ɢ G' ɢ
ɪɚɡɛɢɟɧɢɟ ɦɧɨɠɟɫɬɜɚ M ɷɬɢɯ ɬɨɱɟɤ ɧɚ ɞɜɚ ɩɨɞɦɧɨɠɟɫɬɜɚ Mi ɢ Mo, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɬɨɱɟɤ ɜɯɨɞɚ ɝɪɚɧɢɰɵ ɞGt
ɜ ɨɛɥɚɫɬɶ G' ɢ ɬɨɱɟɤ ɟɺ ɜɵɯɨɞɚ ɢɡ G'. ȼ ɪɚɜɟɧɫɬɜɚɯ (4) ɢ (7) ɧɟɬɪɭɞɧɨ ɡɚɦɟɬɢɬɶ ɚɥɶɬɟɪɧɢɪɨɜɚɧɧɵɟ ɫɭɦɦɵ.
ɉɭɫɬɶ R1R2…R2t-1R2t – ɤɨɪɬɟɠ ɜɫɟɯ ɬɨɱɟɤ Rl(xrl, yrl) ɩɟɪɟɫɟɱɟɧɢɹ ɜɟɪɯɧɢɯ ɝɪɚɧɢɰ ɨɛɥɚɫɬɟɣ Gt ɢ G', ɜ ɤɨɬɨɪɨɦ
ɧɟɱɺɬɧɵɟ ɦɟɫɬɚ ɡɚɧɢɦɚɸɬ ɬɨɱɤɢ R2k-1 ɜɯɨɞɚ ɝɪɚɧɢɰɵ ɞGt ɜ ɨɛɥɚɫɬɶ G', ɚ ɱɺɬɧɵɟ ɦɟɫɬɚ – ɬɨɱɤɢ R2k ɜɵɯɨɞɚ
ɝɪɚɧɢɰɵ ɞGt ɢɡ G', x1 > x2 >…>x2t. Ɍɨɝɞɚ ɞɢɮɮɟɪɟɧɰɢɚɥ dS ɢ ɪɚɜɟɧɫɬɜɚ (7) ɩɪɢɧɢɦɚɸɬ ɜɢɞ ɚɥɶɬɟɪɧɢɪɨɜɚɧɧɵɯ
ɫɭɦɦ
dS = {6l(–1)lyrl}da + {b-16l(-1)lyrl(xrl – a)}db =
= 6l(-1)lyrlda + b-1yrl(xrl – a)db},
ɞS/ɞa = 6l(-1)lyrl, ɞS/ɞb = b-16l(-1)lyrl(xrl – a),
g = i6l(-1)lyrl + jb-16l(-1)lyrl(xrl – a) =
= 6l(-1)l{yrli + b-1yrl(xrl – a)j}.
ȿɫɥɢ ɝɪɚɞɢɟɧɬ g = ’S ɧɟ ɪɚɜɟɧ ɧɭɥɸ, ɬɨ ɜɫɹɤɢɣ ɦɚɥɵɣ ɲɚɝ 'r = ('a, 'b),
'r = hg =h{i6 l(-1)lyrl + jb-16l(-1)lyrl(xrl – a)}
ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɝɪɚɞɢɟɧɬɚ, ɜɟɞɺɬ ɧɚɫ ɤ ɛóɥɶɲɟɦɭ ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ S. ɉɪɨɰɟɫɫ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ
ɩɟɪɟɯɨɞɨɜ ɤ ɛóɥɶɲɟɦɭ ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ S, ɧɚɩɪɚɜɥɟɧɧɵɣ ɧɚ ɨɬɵɫɤɚɧɢɟ ɦɚɤɫɢɦɭɦɚ ɮɭɧɤɰɢɢ S,
ɨɪɝɚɧɢɡɭɟɬɫɹ ɩɨɫɪɟɞɫɬɜɨɦ ɢɬɟɪɚɰɢɣ rn+1= rn+hngn ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɫɟɯ ɫɨɫɬɨɹɧɢɣ r(a, b), ɨɩɪɟɞɟɥɹɸɳɢɯ
ɜɡɚɢɦɧɨɟ ɩɨɥɨɠɟɧɢɟ ɨɛɥɚɫɬɟɣ Gt ɢ G'. ȼɟɤɬɨɪɨɦ r(a, b) ɮɢɤɫɢɪɭɸɬɫɹ, ɧɚɩɪɢɦɟɪ, ɩɨɥɨɠɟɧɢɹ ɩɨɞɜɢɠɧɨɣ
ɨɛɥɚɫɬɢ Gt ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɞɪɭɝɚɹ ɨɛɥɚɫɬɶ G' ɧɟɩɨɞɜɢɠɧɚ. Ɂɞɟɫɶ gn – ɝɪɚɞɢɟɧɬ, ɜɵɱɢɫɥɟɧɧɵɣ ɞɥɹ n-ɝɨ
ɫɨɫɬɨɹɧɢɹ, ɢ hn – ɪɟɝɭɥɹɬɨɪ ɲɚɝɚ, ɜ ɤɚɱɟɫɬɜɟ ɤɨɬɨɪɨɝɨ ɜ ɩɪɨɫɬɟɣɲɟɦ ɫɥɭɱɚɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɩɨɥɨɠɢɬɟɥɶɧɚɹ
ɤɨɧɫɬɚɧɬɚ h. ɂɬɟɪɚɰɢɹ ɫɨɫɬɨɢɬ ɜ ɩɪɟɨɛɪɚɡɨɜɚɧɢɢ ɩɚɪɚɦɟɬɪɨɜ a ɢ b ɨɛɥɚɫɬɢ Gt ɩɨ ɫɥɟɞɭɸɳɟɦɭ ɩɪɚɜɢɥɭ
an+1 = an + hnɞS/ɞa, bn+1 = bn + hnɞS/ɞb,
ɝɞɟ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɮɭɧɤɰɢɢ S ɜɵɱɢɫɥɹɸɬɫɹ ɫɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ (7). ȼɨ ɢɡɛɟɠɚɧɢɟ ɫɛɨɟɜ ɜ ɧɚɱɚɥɟ
ɜɵɩɨɥɧɟɧɢɹ ɩɪɨɝɪɚɦɦɵ, ɪɟɝɭɥɹɬɨɪ hn ɨɝɪɚɧɢɱɢɜɚɟɬɫɹ ɫɜɟɪɯɭ, ɧɚɩɪɢɦɟɪ, ɧɟɪɚɜɟɧɫɬɜɨɦ hn|ɞS/ɞb| < b2-4
ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɱɢɫɥɨɦ
b2-4|ɞS/ɞb|-1 = b22-4|6l(-1)lyrl(xrl – a)|-1 > hn .
6. Ɂɚɤɥɸɱɟɧɢɟ
ɉɨɫɤɨɥɶɤɭ ɞɢɮɮɟɪɟɧɰɢɚɥ dS ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɨɝɨ, ɤɚɤ ɜɟɞɺɬ ɫɟɛɹ ɝɪɚɧɢɰɚ ɞG' ɜ ɩɪɨɦɟɠɭɬɤɚɯ
ɦɟɠɞɭ ɬɨɱɤɚɦɢ ɟɺ ɩɟɪɟɫɟɱɟɧɢɹ ɫ ɝɪɚɧɢɰɟɣ ɞGt, ɧɢɱɬɨ ɧɟ ɦɟɲɚɟɬ ɩɪɢɦɟɧɹɬɶ ɪɚɫɫɦɨɬɪɟɧɧɵɣ ɜɵɲɟ ɦɟɬɨɞ
ɨɩɬɢɦɢɡɚɰɢɢ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɤɭɫɨɱɧɨ-ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɩɥɨɬɧɨɫɬɢ p, ɧɨ ɢ ɞɥɹ ɤɭɫɨɱɧɨ-ɧɟɩɪɟɪɵɜɧɵɯ.
Ɇɟɠɞɭ ɬɟɦ, ɫɧɢɦɚɹ ɬɪɟɛɨɜɚɧɢɟ ɤɭɫɨɱɧɨɣ ɥɢɧɟɣɧɨɫɬɢ, ɧɟɨɛɯɨɞɢɦɨ ɫɨɯɪɚɧɢɬɶ ɫɨɞɟɪɠɚɳɟɟɫɹ ɜ ɧɺɦ ɞɥɹ
ɞɚɧɧɨɝɨ ɤɥɚɫɫɚ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɬɪɟɛɨɜɚɧɢɟ ɪɟɝɭɥɹɪɧɨɫɬɢ, ɤɨɬɨɪɨɟ ɞɨ ɫɢɯ ɩɨɪ ɜɵɩɨɥɧɹɥɨɫɶ ɚɜɬɨɦɚɬɢɱɟɫɤɢ
ɢ ɤɨɬɨɪɨɟ ɩɪɟɞɩɨɥɚɝɚɟɬ, ɱɬɨ ɦɧɨɠɟɫɬɜɨ ɞG'ˆɞGt ɬɨɱɟɤ ɩɟɪɟɫɟɱɟɧɢɹ ɝɪɚɧɢɰ ɧɚ ɤɚɤɨɣ-ɥɢɛɨ ɢɡ ɷɬɢɯ ɝɪɚɧɢɰ
ɢɦɟɟɬ ɦɟɪɭ ɧɭɥɶ. ȿɫɥɢ ɜ ɤɚɱɟɫɬɜɟ ɪɟɝɭɥɹɬɨɪɚ ɜɵɛɢɪɚɟɬɫɹ ɤɨɧɫɬɚɧɬɚ hn = h, ɟɺ ɧɚɞɨ ɜɡɹɬɶ ɧɟɫɤɨɥɶɤɨ
423
ɉɚɧɬɟɥɟɟɜ ȼ.ɉ.
ɇɚɢɥɭɱɲɢɟ L1-ɩɪɢɛɥɢɠɟɧɢɹ ɜ ɤɥɚɫɫɚɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ
ɦɟɧɶɲɟ, ɧɨ ɬɨɝɨ ɠɟ ɩɨɪɹɞɤɚ, ɱɬɨ ɢ ɭɤɚɡɚɧɧɚɹ ɜɵɲɟ ɜɟɪɯɧɹɹ ɝɪɚɧɢɰɚ, ɩɨɫɤɨɥɶɤɭ ɡɚɧɢɠɟɧɢɟ ɟɺ ɦɨɠɟɬ
ɡɚɦɟɞɥɢɬɶ ɪɚɛɨɬɭ ɤɨɦɩɶɸɬɟɪɧɨɣ ɩɪɨɝɪɚɦɦɵ.
Ⱦɥɹ ɧɟɩɪɟɪɵɜɧɵɯ ɤɭɫɤɨɜ ɮɭɧɤɰɢɢ ɩɥɨɬɧɨɫɬɢ p, ɪɟɝɭɥɹɪɧɨ ɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɫ ɤɪɢɜɵɦɢ
ɫɟɦɟɣɫɬɜɚ (1), ɜɟɪɧɵ ɬɟ ɠɟ ɜɵɜɨɞɵ, ɱɬɨ ɢ ɞɥɹ ɤɭɫɨɱɧɨ-ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ p, ɢ ɞɥɹ ɨɬɵɫɤɚɧɢɹ
ɧɚɢɥɭɱɲɟɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɩɪɢɝɨɞɟɧ ɨɩɢɫɚɧɧɵɣ ɜɵɲɟ ɚɥɝɨɪɢɬɦ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɥɢɲɶ ɜ ɫɥɭɱɚɟ
ɪɟɝɭɥɹɪɧɨɫɬɢ ɮɭɧɤɰɢɹ S ɡɚɜɟɞɨɦɨ ɢɦɟɟɬ ɧɟɩɪɟɪɵɜɧɵɣ ɝɪɚɞɢɟɧɬ, ɥɟɠɚɳɢɣ ɜ ɨɫɧɨɜɟ ɚɥɝɨɪɢɬɦɚ.
Ɍɟɫɬɢɪɨɜɚɧɢɟ ɤɨɦɩɶɸɬɟɪɧɨɣ ɩɪɨɝɪɚɦɦɵ ɦɨɠɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶɫɹ ɩɨɫɪɟɞɫɬɜɨɦ ɜɵɜɨɞɚ ɧɚ ɷɤɪɚɧ
ɞɢɫɩɥɟɹ ɤɨɧɟɱɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɨɛɥɚɫɬɟɣ Gt ɢ G' ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɝɪɚɮɢɤɨɜ ɮɭɧɤɰɢɣ f ɢ p, ɞɥɹ
ɜɢɡɭɚɥɶɧɨɣ ɨɰɟɧɤɢ ɤɨɧɟɱɧɨɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɨɛɥɚɫɬɟɣ Gt ɢ G' ɧɚ ɩɪɟɞɦɟɬ ɨɩɬɢɦɚɥɶɧɨɫɬɢ,
ɪɚɜɧɨ ɤɚɤ ɢ ɩɨɫɪɟɞɫɬɜɨɦ ɜɵɜɨɞɚ ɢ ɧɚɛɥɸɞɟɧɢɹ ɜ ɩɨɲɚɝɨɜɨɦ ɪɟɠɢɦɟ ɩɚɪɚɥɥɟɥɶɧɵɯ ɫɞɜɢɝɨɜ da ɢ ɫɠɚɬɢɣ
X- a = k(x- a), Y = y/k ɜɞɨɥɶ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ, ɩɪɨɢɡɜɨɞɢɦɵɯ ɧɚɞ ɩɨɞɜɢɠɧɨɣ ɨɛɥɚɫɬɶɸ. ȼ ɤɚɱɟɫɬɜɟ
ɩɨɫɥɟɞɧɟɣ ɧɢɱɬɨ ɧɟ ɦɟɲɚɟɬ ɜɵɛɪɚɬɶ ɥɸɛɭɸ ɢɡ ɨɛɥɚɫɬɟɣ Gt, ɢ G', ɧɨ ɜ ɜɵɱɢɫɥɢɬɟɥɶɧɨɦ ɨɬɧɨɲɟɧɢɢ
ɩɪɟɞɩɨɱɬɢɬɟɥɶɧɟɟ ɬɚ, ɞɥɹ ɤɨɬɨɪɨɣ ɛɨɥɟɟ ɫɤɨɪɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɟɺ ɧɨɜɨɟ ɩɨɥɨɠɟɧɢɟ ɩɨɫɥɟ ɨɱɟɪɟɞɧɨɣ
ɢɬɟɪɚɰɢɢ.
Ʉɨɷɮɮɢɰɢɟɧɬ k ɥɟɝɤɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɫɢɥɭ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɧɢɦ ɢ ɩɚɪɚɦɟɬɪɨɦ b, k = 1+'b/b, ɱɬɨ
ɩɨɥɟɡɧɨ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɜɵɜɨɞɚ ɧɚ ɷɤɪɚɧ ɨɛɧɨɜɥɺɧɧɨɝɨ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ f ɢ ɱɬɨ ɟɫɬɟɫɬɜɟɧɧɨ ɨɫɭɳɟɫɬɜɥɹɬɶ
ɩɨ ɡɚɜɟɪɲɟɧɢɸ ɨɞɧɨɣ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɢɬɟɪɚɰɢɣ. ɇɚɤɚɩɥɢɜɚɸɳɢɟɫɹ ɜ ɯɨɞɟ ɜɵɱɢɫɥɟɧɢɣ ɨɲɢɛɤɢ
ɭɫɬɪɚɧɹɸɬɫɹ ɛɨɥɟɟ ɬɨɱɧɵɦ ɩɨɫɬɪɨɟɧɢɟɦ ɝɪɚɮɢɤɚ ɮɭɧɤɰɢɢ f ɩɨ ɡɚɜɟɪɲɟɧɢɸ ɧɟɤɨɟɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɱɢɫɥɚ
ɢɬɟɪɚɰɢɣ. ɉɨɲɚɝɨɜɚɹ ɨɰɟɧɤɚ ɩɪɚɜɢɥɶɧɨɫɬɢ ɜɵɩɨɥɧɟɧɧɵɯ ɨɩɟɪɚɰɢɣ ɦɨɠɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶɫɹ ɬɚɤɠɟ
ɩɨɫɪɟɞɫɬɜɨɦ ɜɵɜɨɞɚ ɧɚ ɷɤɪɚɧ ɢɡɦɟɧɟɧɢɹ ɮɭɧɤɰɢɢ p ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɠɚɬɢɹ ɢɥɢ ɫɞɜɢɝɚ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ
ɨɱɟɪɟɞɧɨɣ ɢɬɟɪɚɰɢɢ.
Ⱦɚɧɧɚɹ ɫɬɚɬɶɹ ɧɚɩɢɫɚɧɚ ɜ ɪɚɡɜɢɬɢɟ ɫɨɨɛɳɟɧɢɹ (Panteleev, 2001), ɫɞɟɥɚɧɧɨɝɨ ɚɜɬɨɪɨɦ ɧɚ Ɍɪɟɬɶɟɣ
Ɇɨɫɤɨɜɫɤɨɣ ɦɟɠɞɭɧɚɪɨɞɧɨɣ ɤɨɧɮɟɪɟɧɰɢɢ ɩɨ ɢɫɫɥɟɞɨɜɚɧɢɸ ɨɩɟɪɚɰɢɣ (ORM2001).
Ʌɢɬɟɪɚɬɭɪɚ
Panteleev V. P. A method of computing the best L1 approximations in classes of distributions. The Third Moscow
International Conference on Operations Research (ORM2001). Abstracts. Moscow, p.92-93, 2001.
424
Документ
Категория
Без категории
Просмотров
4
Размер файла
95 Кб
Теги
приближение, наилучший, распределение, класса
1/--страниц
Пожаловаться на содержимое документа