смод

12. ɄɊɂɌȿɊɂɂ ɋɈȽɅȺɋɂə

12.1.Ɉɛɳɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ

ȼ ɩɪɟɞɵɞɭɳɢɯ ɫɥɭɱɚɹɯ ɩɪɢ ɫɢɧɬɟɡɟ ɚɥɝɨɪɢɬɦɨɜ ɨɰɟɧɨɤ ɩɚɪɚɦɟɬɪɨɜ, ɚɥɝɨɪɢɬɦɨɜ ɨɛɧɚɪɭɠɟɧɢɹ ɢɥɢ ɪɚɡɥɢɱɟɧɢɹ ɫɢɝɧɚɥɨɜ ɩɪɟɞɩɨɥɚɝɚɥɨɫɶ ɧɚɥɢɱɢɟ ɚɩɪɢɨɪɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɢɝɧɚɥɨɜ ɢɥɢ ɩɨɦɟɯ. Ɉɞɧɚɤɨ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɩɪɚɤɬɢɱɟɫɤɢɯ ɫɢɬɭɚɰɢɣ ɩɪɢ ɜɵɩɨɥɧɟɧɢɢ ɩɟɪɟɱɢɫɥɟɧɧɵɯ ɨɩɟɪɚɰɢɣ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɧɟɢɡɜɟɫɬɧɚ. Ɇɨɠɧɨ ɥɢɲɶ ɞɟɥɚɬɶ ɪɚɡɧɨɝɨ ɪɨɞɚ ɩɪɟɞɩɨɥɨɠɟɧɢɹ. ȼ ɬɚɤɨɣ ɫɢɬɭɚɰɢɢ ɧɟɨɛɯɨɞɢɦɨ, ɪɚɫɩɨɥɚɝɚɹ ɜɵɛɨɪɨɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ, ɫɞɟɥɚɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɢɣ ɜɵɜɨɞ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɢɞɚ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɢɡ ɤɨɬɨɪɨɝɨ ɢɡɜɥɟɱɟɧɚ ɜɵɛɨɪɤɚ. ɉɪɟɞɫɬɚɜɥɹɟɬ ɢɧɬɟɪɟɫ ɜɜɟɫɬɢ ɤɨɥɢɱɟɫɬɜɟɧɧɭɸ ɦɟɪɭ ɫɨɨɬɜɟɬɫɬɜɢɹ ɝɢɩɨɬɟɬɢɱɟɫɤɨɝɨ ɢ ɷɦɩɢɪɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɣ, ɢɥɢ, ɤɚɤ ɝɨɜɨɪɹɬ, ɤɪɢɬɟɪɢɣ ɫɨɝɥɚɫɢɹ. ɗɬɨɬ ɤɪɢɬɟɪɢɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɱɢɫɥɨ '(F1*,F1), ɤɨɬɨɪɨɟ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɨɦ ɨɬ ɷɦɩɢɪɢɱɟɫɤɨɝɨ F1* ɢ ɝɢɩɨɬɟɬɢɱɟɫɤɨɝɨ F1 ɪɚɫɩɪɟɞɟɥɟɧɢɣ.

ɋɮɨɪɦɭɥɢɪɨɜɚɧɧɚɹ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ. ɉɨ ɜɵɛɨɪɨɱɧɵɦ ɡɧɚɱɟɧɢɹɦ X ɫɬɪɨɢɬɫɹ ɷɦɩɢɪɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɞɟɥɟɧɢɹ ɩɨ ɮɨɪɦɭɥɟ F1*(x) = ¦ku(x-xk), ɝɞɟ u(x) - ɟɞɢɧɢɱɧɵɣ ɫɤɚɱɨɤ. ɇɚ ɨɫɧɨɜɚɧɢɢ ɷɬɨɝɨ ɷɦɩɢɪɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɧɨ, ɦɨɠɟɬ ɛɵɬɶ, ɢ ɢɡ ɞɪɭɝɢɯ ɫɨɨɛɪɚɠɟɧɢɣ, ɜɵɞɜɢɝɚɟɬɫɹ ɝɢɩɨɬɟɡɚ ɨ ɬɨɦ, ɱɬɨ ɜɵɛɨɪɤɚ ɢɡɜɥɟɱɟɧɚ ɢɡ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F1(x). ɉɨ ɩɪɢɧɹɬɨɦɭ ɡɚɪɚɧɟɟ ɤɪɢɬɟɪɢɸ ɫɨɝɥɚɫɢɹ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ '(F1*,F1) ɢ ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɭɤɚɡɚɧɧɚɹ ɜɟɥɢɱɢɧɚ ɩɪɟɜɨɫɯɨɞɢɬ ɧɟɤɨɬɨɪɵɣ ɩɨɪɨɝ '0. Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɥɢɱɢɧɵ '(F1*,F1). Ɉɩɪɟɞɟɥɢɬɶ ɬɚɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜ ɩɪɨɫɬɨɣ ɮɨɪɦɟ ɭɞɚɟɬɫɹ, ɤɚɤ ɩɪɚɜɢɥɨ, ɥɢɲɶ ɩɪɢ ɛɨɥɶɲɢɯ ɪɚɡɦɟɪɚɯ ɜɵɛɨɪɤɢ (ɬɨɱɧɟɟ, ɨɩɪɟɞɟɥɢɬɶ ɥɢɲɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɪɢ nof). ȿɫɥɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɥɢɱɢɧɵ '(F1*,F1) ɢɡɜɟɫɬɧɨ, ɬɨ, ɡɚɞɚɜɚɹɫɶ ɜɟɪɨɹɬɧɨɫɬɶɸ D (ɭɪɨɜɧɟɦ ɡɧɚɱɢɦɨɫɬɢ ɤɪɢɬɟɪɢɹ) ɬɨɝɨ, ɱɬɨ '>'0, ɧɚɯɨɞɢɦ ɩɨɪɨɝ '0. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɦ D ɩɨɥɭɱɚɟɦ ɩɪɢɟɦɥɟɦɨɟ ɞɥɹ ɩɪɚɤɬɢɤɢ ɩɪɚɜɢɥɨ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ ɨɛ ɢɫɬɢɧɧɨɫɬɢ ɝɢɩɨɬɟɬɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F1(x): ɟɫɥɢ ɞɥɹ ɞɚɧɧɨɣ ɜɵɛɨɪɤɢ ' >'0, ɬɨ ɝɢɩɨɬɟɡɚ ɨɬɜɟɪɝɚɟɬɫɹ, ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɥɭɱɚɟ - ɩɪɢɧɢɦɚɟɬɫɹ. ɉɪɢ ɷɬɨɦ ɜɟɪɨɹɬɧɨɫɬɶ ɨɬɜɟɪɝɧɭɬɶ ɩɪɚɜɢɥɶɧɭɸ ɝɢɩɨɬɟɡɭɛɭɞɟɬ ɪɚɜɧɚ ɭɪɨɜɧɸ ɡɧɚɱɢɦɨɫɬɢ D.

Ɉɬɦɟɬɢɦ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɥɢɱɢɧɵ '(F1*,F1), ɨɩɪɟɞɟɥɹɟɦɨɟ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɜɵɛɨɪɨɱɧɵɟ ɡɧɚɱɟɧɢɹ X, ɩɨ ɤɨɬɨɪɵɦ ɩɨɫɬɪɨɟɧɚ F1*, ɢɡɜɥɟɱɟɧɵ ɢɡ F1, ɡɚɜɢɫɢɬ ɨɬ F1. Ȼɵɥɨ ɛɵ ɠɟɥɚɬɟɥɶɧɨ ɢɦɟɬɶ ɬɚɤɢɟ ɤɪɢɬɟɪɢɢ ɫɨɝɥɚɫɢɹ, ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɤɨɬɨɪɵɯ ɧɟ ɡɚɜɢɫɟɥɢ ɛɵ ɨɬ ɜɢɞɚ F1. Ɍɚɤɢɟ ɤɪɢɬɟɪɢɢ ɧɚɡɵɜɚɸɬ ɧɟɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ. ɇɢɠɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɤɪɢɬɟɪɢɢ, ɤɨɬɨɪɵɟ ɩɪɢ nof ɹɜɥɹɸɬɫɹ ɧɟɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ (ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ).

ɋɥɟɞɭɟɬ ɬɚɤɠɟ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɬɚɤɢɟ ɤɪɢɬɟɪɢɢ ɫɨɝɥɚɫɢɹ ɞɚɸɬ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɦ D ɞɨɫɬɚɬɨɱɧɨ ɷɮɮɟɤɬɢɜɧɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ, ɩɨɡɜɨɥɹɸɳɟɟ ɨɬɜɟɪɝɧɭɬɶ ɧɟɜɟɪɧɭɸ ɝɢɩɨɬɟɡɭ. ɉɪɢ ɷɬɨɦ, ɨɞɧɚɤɨ, ɨɫɬɚɟɬɫɹ ɧɟɢɡɜɟɫɬɧɨɣ ɜɟɪɨɹɬ-

ɧɨɫɬɶ ɨɲɢɛɨɤ ɜɬɨɪɨɝɨ ɪɨɞɚ - ɩɪɢɧɹɬɢɟ ɧɟɜɟɪɧɨɣ ɝɢɩɨɬɟɡɵ. ȿɫɥɢ ɞɥɹ ɞɚɧɧɨɣ ɜɵɛɨɪɤɢ '<'0, ɬɨ, ɯɨɬɹ ɩɪɢɧɹɬɨɟ ɩɪɚɜɢɥɨ ɩɪɟɞɩɢɫɵɜɚɟɬ ɭɬɜɟɪɠɞɚɬɶ ɢɫɬɢɧɧɨɫɬɶ ɜɵɞɜɢɧɭɬɨɣ ɝɢɩɨɬɟɡɵ, ɫɥɟɞɭɟɬ ɤ ɬɚɤɨɦɭ ɭɬɜɟɪɠɞɟɧɢɸ ɨɬɧɟɫɬɢɫɶ ɛɨɥɟɟ ɨɫɬɨɪɨɠɧɨ, ɬɚɤ ɤɚɤ ɫ ɭɱɟɬɨɦ ɡɚɞɚɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɫɨɝɥɚɫɢɹ ɧɟɬ ɨɫɧɨɜɚɧɢɣ ɫɱɢɬɚɬɶ, ɱɬɨ ɜɵɛɨɪɤɚ X ɧɟ ɫɨɝɥɚɫɭɟɬɫɹ ɫ ɝɢɩɨɬɟɬɢɱɟɫɤɢɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ.

ɇɟɞɨɫɬɚɬɤɨɦ ɤɪɢɬɟɪɢɟɜ ɫɨɝɥɚɫɢɹ ɹɜɥɹɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɨɫɬɶ ɜɵɛɨɪɚ ɫɚɦɨɝɨ ɤɪɢɬɟɪɢɹ ɢ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ D, ɚ ɬɚɤɠɟ ɬɪɭɞɧɨɫɬɢ, ɫɜɹɡɚɧɧɵɟ ɫ ɨɩɪɟɞɟɥɟɧɢɟɦ ɢɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɞɥɹ ɦɚɥɵɯ n. Ⱦɪɭɝɨɣ ɩɭɬɶ ɫɨɫɬɨɢɬ ɜ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɦ ɩɨɫɬɪɨɟɧɢɢ ɧɟɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ F1 ɩɨ ɜɵɛɨɪɨɱɧɵɦ ɡɧɚɱɟɧɢɹɦ ɛɟɡ ɜɵɞɜɢɠɟɧɢɹ ɝɢɩɨɬɟɡɵ ɨ ɜɢɞɟ ɷɬɨɣɮɭɧɤɰɢɢ.

12.2. Ʉɪɢɬɟɪɢɣ F2 - ɉɢɪɫɨɧɚ

ɗɬɨɬ ɤɪɢɬɟɪɢɣ ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ ɫɚɦɵɯ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɜ ɩɪɢɥɨɠɟɧɢɹɯ ɤ ɬɟɨɪɢɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɨɰɟɧɨɤ.

Ɋɚɡɨɛɶɟɦ ɨɛɥɚɫɬɶ, ɝɞɟ ɨɩɪɟɞɟɥɟɧɚ ɝɢɩɨɬɟɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F1(x) ɋȼ ɏ, ɧɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɧɟɩɟɪɟɤɪɵɜɚɸɳɢɯɫɹ ɢɧɬɟɪɜɚɥɨɜ 'i, i = 1,..., l. Ɉɛɨɡɧɚɱɢɦ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɟɞɢɧɢɱɧɨɝɨ ɢɡɦɟɪɟɧɢɹ ɜ i-ɣ ɢɧɬɟɪɜɚɥ pi=P(X

l

'i), ¦ pi=1. ɉɭɫɬɶ ɜ ɜɵɛɨɪɤɟ X ɱɢɫɥɨ ɜɵɛɨɪɨɱɧɵɯ ɡɧɚɱɟɧɢɣ, ɩɨɩɚɞɚɸɳɢɯ ɜ

i 1

l

ɢɧɬɟɪɜɚɥ 'i, ɪɚɜɧɨ Qi. əɫɧɨ, ɱɬɨ ¦ Qi=n.

i 1

ɂɫɩɨɥɶɡɭɟɦ ɜ ɤɚɱɟɫɬɜɟ ɤɪɢɬɟɪɢɹ ɫɨɝɥɚɫɢɹ ɜɟɥɢɱɢɧɭ

Ʉɚɤ ɞɨɤɚɡɚɥ ɉɢɪɫɨɧ, ɟɫɥɢ ɩɪɨɜɟɪɹɟɦɚɹ ɝɢɩɨɬɟɡɚ ɨɛ ɢɫɬɢɧɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F1(x) ɜɟɪɧɚ, ɬɨ ɩɪɢ nof ɪɚɫɩɪɟɞɟɥɟɧɢɟ ' ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ

F2 -ɪɚɫɩɪɟɞɟɥɟɧɢɸ ɫ (l 1) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɧɟ ɡɚɜɢɫɹɳɟɦɭ ɨɬ ɜɢɞɚ ɝɢɩɨ-

ɬɟɬɢɱɟɫɤɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ F1(x).

ɂɬɚɤ, ɩɭɫɬɶ F2D - ɩɪɨɰɟɧɬɧɨɟ ɨɬɤɥɨɧɟɧɢɟ ɋȼ, ɢɦɟɸɳɟɣ F2-ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫ (l -1) ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɬ.ɟ. P{F2>F2D}=D. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɨɦ D ɢ ɛɨɥɶɲɨɦ n ɜɟɥɢɱɢɧɚ ', ɜɵɱɢɫɥɟɧɧɚɹ ɫɨɝɥɚɫɧɨ ɤɪɢɬɟɪɢɸ F2-ɉɢɪɫɨɧɚ, ɩɪɚɤɬɢɱɟɫɤɢ ɧɢɤɨɝɞɚ ɧɟ ɛɭɞɟɬ ɩɪɟɜɨɫɯɨɞɢɬɶ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ F2D, ɟɫɥɢ ɬɨɥɶɤɨ ɝɢɩɨɬɟɡɚ ɨ ɜɢɞɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɟɪɧɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɵɛɢɪɚɟɬɫɹ ɫɥɟɞɭɸɳɟɟ ɩɪɚɜɢɥɨ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ: ɝɢɩɨɬɟɡɚ ɨɬɤɥɨɧɹɟɬɫɹ, ɟɫɥɢ '>F2D, ɢ ɩɪɢɧɢɦɚɟɬɫɹ, ɟɫɥɢ '<F2D. ȼɟɪɨɹɬɧɨɫɬɶ ɨɬɤɥɨɧɢɬɶ ɜɟɪɧɭɸ ɝɢɩɨɬɟɡɭɪɚɜɧɚ D.

Ʉɪɨɦɟ ɨɬɦɟɱɟɧɧɵɯ ɭɠɟ ɨɛɳɢɯ ɧɟɞɨɫɬɚɬɤɨɜ ɤɪɢɬɟɪɢɟɜ ɫɨɝɥɚɫɢɹ ɪɚɫɫɦɨɬɪɟɧɧɵɣ ɤɪɢɬɟɪɢɣ F2-ɉɢɪɫɨɧɚ ɨɛɥɚɞɚɟɬ ɟɳɟ ɨɞɧɢɦ ɫɭɳɟɫɬɜɟɧɧɵɦ ɧɟɞɨɫɬɚɬɤɨɦ,

Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɬɚɤɨɣ ɋȼ ɟɫɬɶ 1/(6n), ɜɬɨɪɨɣ ɰɟɧɬɪɚɥɶɧɵɣ ɦɨɦɟɧɬ ɢɦɟɟɬ ɜɢɞ (4n-3)/(180n). ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɬɨɱɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜɟɥɢɱɢɧɵ n

' ɨɱɟɧɶ ɫɥɨɠɧɨɟ, ɧɨ ɩɪɢ n>40 ɨɧɨ ɛɥɢɡɤɨ ɤ ɩɪɟɞɟɥɶɧɨɦɭ. Ʉɚɤ ɢ ɤɪɢɬɟɪɢɣ Ʉɨɥɦɨɝɨɪɨɜɚ, ɤɪɢɬɟɪɢɣ Ɇɢɡɟɫɚ ɜ ɨɬɥɢɱɢɟ ɨɬ F2 ɧɟ ɫɜɹɡɚɧ ɫ ɝɪɭɩɩɢɪɨɜɚɧɢɟɦ ɜɵɛɨɪɨɱɧɵɯ ɞɚɧɧɵɯ, ɢ ɟɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɞɨɫɬɚɬɨɱɧɨ ɛɵɫɬɪɨ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ n ɫɬɪɟɦɢɬɫɹ ɤ ɩɪɟɞɟɥɶɧɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɸ. ɉɪɨɰɟɧɬɧɵɟ ɬɨɱɤɢ ɩɪɟɞɟɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɢɦɟɸɬ ɡɧɚɱɟɧɢɹ, ɩɪɟɞɫɬɚɜɥɟɧɧɵɟ ɜ ɬɚɛɥ.12.1.

12.5.Ɉɰɟɧɤɚ ɩɪɢɧɚɞɥɟɠɧɨɫɬɢ ɞɜɭɯ ɜɵɛɨɪɨɤ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɪɚɫɩɪɟɞɟɥɟɧɢɸ

ɉɭɫɬɶ ɢɦɟɸɬɫɹ ɜɵɛɨɪɤɢ X ɢ Y, ɩɨɥɭɱɟɧɧɵɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɧɟɡɚɜɢɫɢɦɵɯ ɟɞɢɧɢɱɧɵɯ ɢɡɦɟɪɟɧɢɣ, ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɩɪɢɧɚɞɥɟɠɢɬ ɧɟɤɨɬɨɪɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɸ. ɇɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɪɢɬɶ ɝɢɩɨɬɟɡɭ ɨ ɬɨɦ, ɱɬɨ ɨɛɟ ɷɬɢ ɜɵɛɨɪɤɢ ɩɪɢɧɚɞɥɟɠɚɬ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɪɚɫɩɪɟɞɟɥɟɧɢɸ. ɉɭɫɬɶ F1X(z) ɢ F1Y(z) - ɷɦɩɢɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ, ɩɨɫɬɪɨɟɧɧɵɟ ɩɨ ɞɜɭɦ ɜɵɛɨɪɤɚɦ X ɢ Y. (ɪɚɡɦɟɪ ɜɵɛɨɪɤɢ X ɟɫɬɶ n, ɚ ɜɵɛɨɪɤɢ Y - m). ȼ ɤɚɱɟɫɬɜɟ ɤɪɢɬɟɪɢɹ ɫɨɝɥɚɫɢɹ ɩɪɢɦɟɦ ɜɟɥɢɱɢɧɭ '

=max|F1x*(z)-F1y*(z)|. Ʉɚɤ ɞɨɤɚɡɚɥ ɇ.ȼ. ɋɦɢɪɧɨɜ, ɩɪɢ nof ɜɟɪɨɹɬɧɨɫɬɶ P{'(1/n+1/m)-1/2>z} | 1 k(z), ɝɞɟ k(z)-ɮɭɧɤɰɢɹ ɚɧɚɥɨɝɢɱɧɚɹ ɮɭɧɤɰɢɢ k(z), ɢɫ-

ɩɨɥɶɡɭɟɦɨɣ ɜ ɤɪɢɬɟɪɢɢ Ʉɨɥɦɨɝɨɪɨɜɚ. ȿɫɥɢ ɡɚɞɚɧ ɭɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ D, ɬɨ ɦɨɠɧɨ, ɤɚɤ ɢ ɜ ɤɪɢɬɟɪɢɢ Ʉɨɥɦɨɝɨɪɨɜɚ, ɨɩɪɟɞɟɥɢɬɶ ɩɨɪɨɝ 'D, ɫ ɤɨɬɨɪɵɦ ɧɟɨɛɯɨɞɢɦɨ ɫɪɚɜɧɢɜɚɬɶ ɪɚɫɫɱɢɬɚɧɧɨɟ ': 'D= =(1/n+1/m)1/2(0,5ln(2/D))1/2. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɟɫɥɢ '<'D, ɬɨ ɜɵɛɨɪɤɢ X ɢ Y ɩɪɢɧɚɞɥɟɠɚɬ ɨɞɧɨɦɭɢ ɬɨɦɭɠɟ ɪɚɫɩɪɟɞɟɥɟɧɢɸ.

Ⱦɪɭɝɨɣ ɤɪɢɬɟɪɢɣ, ɩɪɟɞɥɨɠɟɧɧɵɣ ȼɢɥɤɨɤɫɨɧɨɦ, ɨɫɧɨɜɚɧ ɧɚ ɩɨɞɫɱɟɬɟ ɱɢɫɥɚ ɢɧɜɟɪɫɢɣ. Ⱦɥɹ ɷɬɨɝɨ ɨɛɟ ɜɵɛɨɪɤɢ ɪɚɫɩɨɥɚɝɚɸɬ ɜ ɜɢɞɟ ɩɨɪɹɞɤɨɜɨɣ ɫɬɚɬɢɫɬɢɤɢ ɩɨ ɜɨɡɪɚɫɬɚɧɢɸ ɡɧɚɱɟɧɢɣ, ɧɚɩɪɢɦɟɪ: y1,y2,x1,y3,x2,... ȿɫɥɢ ɜ ɷɬɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɡɚɞɚɧɧɨɦɭ x ɩɪɟɞɲɟɫɬɜɭɸɬ s ɷɥɟɦɟɧɬɨɜ ɜɵɛɨɪɤɢ Y, ɬɨ ɢɦɟɟɬ ɦɟɫɬɨ s ɢɧɜɟɪɫɢɣ. Ɉɛɳɟɟ ɱɢɫɥɨ ɢɧɜɟɪɫɢɣ U ɪɚɜɧɨ ɫɭɦɦɟ ɢɧɜɟɪɫɢɣ, ɨɛɪɚɡɭɟɦɵɯ ɜɫɟɦɢ ɷɥɟɦɟɧɬɚɦɢ ɩɟɪɜɨɣ ɜɵɛɨɪɤɢ ɫ ɷɥɟɦɟɧɬɚɦɢ ɜɬɨɪɨɣ. ɉɪɚɜɢɥɨ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ ɩɨ ɤɪɢɬɟɪɢɸ ȼɢɥɤɨɤɫɨɧɚ ɫɨɫɬɨɢɬ ɜ ɫɪɚɜɧɟɧɢɢ ɨɛɳɟɝɨ ɱɢɫɥɚ ɢɧɜɟɪɫɢɣ ɫ ɩɨɪɨɝɨɜɵɦ ɱɢɫɥɨɦ, ɨɩɪɟɞɟɥɹɟɦɵɦ ɭɪɨɜɧɟɦ ɡɧɚɱɢɦɨɫɬɢ D.

Ⱦɨɤɚɡɚɧɨ, ɱɬɨ ɩɪɢ m+n > 20 ɢ m > 3 ɫ ɯɨɪɨɲɢɦ ɩɪɢɛɥɢɠɟɧɢɟɦ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɨɛɳɟɝɨ ɱɢɫɥɚ ɢɧɜɟɪɫɢɣ ɧɨɪɦɚɥɶɧɵɦ ɫ ɩɚɪɚɦɟɬɪɚɦɢ m{U} = mn/2; M2{U} = mn(m+n+1)/12. Ɍɨɝɞɚ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ U ɱɢɫɥɚ ɢɧɜɟɪɫɢɣ ɞɥɹ ɡɚɞɚɧɧɨɝɨ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ D ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɨ ɩɨ ɮɨɪɦɭɥɟ UD= mn/2 +

xD(mn(m+n+1)/12)1/2, ɝɞɟ xD- ɩɪɨɰɟɧɬɧɨɟ ɨɬɤɥɨɧɟɧɢɟ ɧɨɪɦɚɥɶɧɨɣ ɋȼ. ȿɫɥɢ ɜɵɱɢɫɥɟɧɧɨɟ ɩɨ ɡɚɞɚɧɧɵɦ ɞɜɭɦ ɜɵɛɨɪɤɚɦ ɡɧɚɱɟɧɢɟ U ɩɪɟɜɨɫɯɨɞɢɬ UD, ɬɨ ɝɢɩɨɬɟɡɚ ɨ ɬɨɦ, ɱɬɨ ɷɬɢ ɜɵɛɨɪɤɢ ɩɪɢɧɚɞɥɟɠɚɬ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɪɚɫɩɪɟɞɟɥɟɧɢɸ, ɨɬɤɥɨɧɹɟɬɫɹ.

13.ɈɐȿɇɄɂ ɉȺɊȺɆȿɌɊɈȼ ɆɇɈȽɈɆȿɊɇɈɃ ɎɍɇɄɐɂɂ ɊȺɋɉɊȿȾȿɅȿɇɂə

13.1. Ɉɛɨɛɳɟɧɢɟ ɨɫɧɨɜɧɵɯ ɨɩɪɟɞɟɥɟɧɢɣ ɧɚ ɦɧɨɝɨɦɟɪɧɵɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ

Ⱦɨ ɫɢɯ ɩɨɪ ɦɵ ɢɡɭɱɚɥɢ ɬɟɨɪɢɸ ɨɰɟɧɨɤ ɩɚɪɚɦɟɬɪɨɜ ɢɥɢ ɫɚɦɢɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɨɞɧɨɣ ɋȼ. ɉɪɚɤɬɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɬɚɤɠɟ ɨɰɟɧɤɢ ɦɧɨɝɨɦɟɪɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɢɥɢ ɢɯ ɩɚɪɚɦɟɬɪɨɜ ɞɥɹ ɫɨɜɨɤɭɩɧɨɫɬɢ ɡɚɜɢɫɢɦɵɯ ɋȼ [1,...,[N. ȼ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɜɢɞ ɦɧɨɝɨɦɟɪɧɨɣ ɮɭɧɤɰɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɷɬɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɢɡɜɟɫɬɟɧ ɢ ɨɰɟɧɢɜɚɸɬɫɹ ɥɢɲɶ ɧɟɢɡɜɟɫɬɧɵɟ ɩɚɪɚɦɟɬɪɵ, ɜ ɧɟɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɨɰɟɧɤɟ ɩɨɞɥɟɠɢɬ ɫɚɦɚ ɮɭɧɤɰɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ȼ ɥɸɛɨɦ ɫɥɭɱɚɟ ɨɫɧɨɜɚɧɢɟɦ ɞɥɹ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɜɵɜɨɞɨɜ ɫɥɭɠɢɬ ɜɵɛɨɪɤɚ ɢɡ ɦɧɨɝɨɦɟɪɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. ȼ ɨɬɥɢɱɢɟ ɨɬ ɨɞɧɨɦɟɪɧɨɝɨ ɫɥɭɱɚɹ, ɷɥɟɦɟɧɬɚɦɢ ɬɚɤɨɣ (ɪɚɡɦɟɪɚ n) ɜɵɛɨɪɤɢ ɫɥɭɠɚɬ ɧɟ ɟɞɢɧɢɱɧɵɟ ɢɡɦɟɪɟɧɢɹ (ɱɢɫɥɚ), ɚ N ɱɢɫɟɥ x1k,...,xNk, (k=1,..., n). Ɍɨɝɞɚ ɪɟɡɭɥɶɬɚɬɵ ɢɡɦɟɪɟɧɢɣ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɹɦɨɭɝɨɥɶɧɭɸ ɦɚɬɪɢɰɭ ɪɚɡɦɟɪɨɦ N ɯ n. (ɩɪɢɦɟɪ: ɨɬɫɱɟɬɵ ɫɢɝɧɚɥɨɜ ɧɚ ɜɵɯɨɞɟ ɚɧɬɟɧɧɨɣ ɪɟɲɟɬɤɢ ɜ ɪɚɡɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ) X =||xik||. Ɏɭɧɤɰɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɜɵɛɨɪɤɢ ɢɡ N-ɦɟɪɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɮɭɧɤɰɢɸ n ɜɟɤɬɨɪɧɵɯ ɚɪɝɭɦɟɧɬɨɜ. ɉɚɪɚɦɟɬɪɵ ɷɬɨɣ ɮɭɧɤɰɢɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɦɚɬɪɢɰɵ (ɢɥɢ ɜɟɤɬɨɪɵ), ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɧɟɢɡɜɟɫɬɧɵɟ ɩɚɪɚɦɟɬɪɵ V = X1,...,Xm ɦɧɨɝɨɦɟɪɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɨɜɨɤɭɩɧɨɫɬɢ ɫɥɭɱɚɣɧɵɯ ɜɟɥɢɱɢɧ. ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɞɥɹ ɧɟɡɚɜɢɫɢɦɵɯ ɷɥɟɦɟɧɬɨɜ

n

ɦɧɨɝɨɦɟɪɧɨɣ ɜɵɛɨɪɤɢ ɮɭɧɤɰɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ LX(V) = = fN(x1k,..., xNk;V),

k 1

ɝɞɟ fN(x1k,...,xNk;V) - ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɜɨɤɭɩɧɨɫɬɢ ɋȼ [1,...,[N.

ɉɨ ɜɵɛɨɪɤɟ X ɨɩɪɟɞɟɥɹɸɬɫɹ s ɜɵɛɨɪɨɱɧɵɯ ɦɚɬɪɢɰ (ɢɥɢ ɜɟɤɬɨɪɨɜ), ɡɚɜɢɫɹɳɢɯ ɨɬ ɜɵɛɨɪɨɱɧɵɯ ɜɟɤɬɨɪɨɜ X1,...,Xn, Mi*=g(X1,...,Xn), i = =1,..., s, ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɩɚɪɚɦɟɬɪɵ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɟɣ fN ɮɢɤɫɢɪɨɜɚɧɵ. ɗɬɢ ɜɵɛɨɪɨɱɧɵɟ ɦɚɬɪɢɰɵ (ɢɥɢ ɜɟɤɬɨɪɵ) ɩɪɟɞɫɬɚɜɥɹɸɬ ɭɫɥɨɜɧɵɟ ɨɰɟɧɤɢ ɦɚɬɪɢɰ Mi (ɢɥɢ ɜɟɤɬɨɪɨɜ), ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɵɯ ɹɜɥɹɸɬɫɹ ɧɟɢɡɜɟɫɬɧɵɟ ɩɚɪɚɦɟɬɪɵ V = X1,...,Xm.

Ʉɚɠɞɚɹ ɢɡ ɭɫɥɨɜɧɵɯ ɨɰɟɧɨɤ Mi* ɧɚɡɵɜɚɟɬɫɹ ɫɨɫɬɨɹɬɟɥɶɧɨɣ, ɟɫɥɢ ɩɪɢ nof ɨɧɚ ɫɯɨɞɢɬɫɹ ɤ Mi. Ɉɰɟɧɤɚ Mi* ɧɚɡɵɜɚɟɬɫɹ ɧɟɫɦɟɳɟɧɧɨɣ, ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ n ɟɟ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɩɨ ɫɨɜɨɤɭɩɧɨɫɬɢ ɜɟɤɬɨɪɧɵɯ ɜɵɛɨɪɨɤ ɪɚɜɧɨ M. ɉɪɢ ɷɬɨɦ ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ, ɱɬɨ ɩɨɞ ɫɪɟɞɧɢɦ ɡɧɚɱɟɧɢɟɦ ɫɥɭɱɚɣɧɨɣ ɦɚɬɪɢɰɵ (ɜɟɤɬɨɪɚ) ɩɨɧɢɦɚɸɬ ɦɚɬɪɢɰɭ (ɜɟɤɬɨɪ), ɷɥɟɦɟɧɬɵ ɤɨɬɨɪɨɣ ɪɚɜɧɵ ɫɪɟɞɧɢɦ ɡɧɚɱɟɧɢɹɦ ɷɥɟɦɟɧɬɨɜ ɫɥɭɱɚɣɧɨɣ ɦɚɬɪɢɰɵ (ɜɟɤɬɨɪɚ). Ɍɚɤɢɦ ɠɟ ɨɛɪɚɡɨɦ ɨɛɨɛɳɚɟɬɫɹ ɩɨɧɹɬɢɟ ɢ

ɫɨɜɦɟɫɬɧɨ ɞɨɫɬɚɬɨɱɧɵɯ ɨɰɟɧɨɤ. Ⱦɥɹ ɷɬɨɝɨ ɫɤɚɥɹɪɧɵɟ ɚɪɝɭɦɟɧɬɵ ɡɚɦɟɧɹɸɬɫɹ ɜɟɤɬɨɪɧɵɦɢ.

Ɉɰɟɧɤɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɩɨɥɭɱɚɸɬɫɹ ɢɡ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ w lnLX(M)/wMi = 0, i=1,..., s.

13.2. Ɉɰɟɧɤɢ ɜɟɤɬɨɪɚ ɫɪɟɞɧɢɯ ɢ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ ɦɧɨɝɨɦɟɪɧɨɝɨ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ

ɉɪɨɢɥɥɸɫɬɪɢɪɭɟɦ ɭɤɚɡɚɧɧɵɟ ɨɛɨɛɳɟɧɢɹ ɧɚ ɩɪɢɦɟɪɟ N-ɦɟɪɧɨɝɨ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɂɚɩɢɲɟɦ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɷɬɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜ ɜɟɤɬɨɪɧɨɣ ɮɨɪɦɟ:

fN(X)=(2S)-N/2(detM)-1/2exp[-0,5(x'-a')M-1(x-a)],

ɝɞɟ a - ɜɟɤɬɨɪ ɫɪɟɞɧɢɯ ɡɧɚɱɟɧɢɣ; M - ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɦɚɬɪɢɰɚ.

ɉɭɫɬɶ x1,...,xn - ɧɟɡɚɜɢɫɢɦɵɟ ɜɵɛɨɪɨɱɧɵɟ N-ɦɟɪɧɵɟ ɜɟɤɬɨɪɵ ɢɡ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɏɭɧɤɰɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɜɵɛɨɪɤɢ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ N-

ɦɟɪɧɵɯ ɧɨɪɦɚɥɶɧɵɯ ɩɥɨɬɧɨɫɬɟɣ ɜɟɪɨɹɬɧɨɫɬɟɣ:

n

LX(a,M)= fN(xk;a,M)=

k 1

n

=(2S)-nN/2(detM)-n/2exp[-0,5 ¦ (xk-a)'M-1(xk-a)].

k 1

1 n

ȼɜɟɞɟɦ ɜɟɤɬɨɪ ɜɵɛɨɪɨɱɧɵɯ ɫɪɟɞɧɢɯ ɡɧɚɱɟɧɢɣ m1*= ¦x k ɢ ɜɵɛɨɪɨɱɧɭɸ

n k 1

1 n

ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɦɚɬɪɢɰɭ M*= n ¦k 1 (xk - m1*)(xk - m1*)' . Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ,

ɱɬɨ ɨɰɟɧɤɚ ɜɟɤɬɨɪɚ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɨɠɢɞɚɧɢɣ ɹɜɥɹɟɬɫɹ ɧɟɫɦɟɳɟɧɧɨɣ ɨɰɟɧɤɨɣ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɨɰɟɧɤɚ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ ɫɦɟɳɟɧɧɚɹ, ɬɚɤ ɤɚɤ ɟɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɪɚɜɧɨ (1-1/n)M. ɉɨɷɬɨɦɭ ɧɟɫɦɟɳɟɧɧɨɣ ɨɰɟɧɤɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ M ɹɜɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ M'=nM*/(n-1).

14.ɎɂɅɖɌɊȺɐɂə ɋɅɍɑȺɃɇɕɏ ɉɊɈɐȿɋɋɈȼ

14.1.ɉɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɮɢɥɶɬɪɚɰɢɢ

ȼɩɪɟɞɵɞɭɳɢɯ ɪɚɡɞɟɥɚɯ ɪɚɫɫɦɚɬɪɢɜɚɥɢɫɶ ɨɰɟɧɤɢ ɧɟɢɡɜɟɫɬɧɵɯ ɩɚɪɚɦɟɬɪɨɜ

ɢɪɚɫɩɪɟɞɟɥɟɧɢɣ ɋɉ ɩɨ ɪɟɚɥɢɡɚɰɢɢ ɤɨɧɟɱɧɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ. Ⱦɪɭɝɨɣ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɨɣ ɡɚɞɚɱɟɣ ɹɜɥɹɟɬɫɹ ɨɩɪɟɞɟɥɟɧɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɧɚɛɥɸɞɚɟɦɨɣ ɪɟɚɥɢɡɚɰɢɢ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɧɚɢɥɭɱɲɟɣ ɜ ɫɦɵɫɥɟ ɡɚɞɚɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɨɰɟɧɤɢ ɡɧɚɱɟɧɢɹ ɋɉ ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɜɧɭɬɪɢ ɢɥɢ ɜɧɟ ɢɧɬɟɪɜɚɥɚ ɧɚɛɥɸɞɟɧɢɹ. Ɍɚɤɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɚɡɵɜɚɟɬɫɹ ɮɢɥɶɬɪɚɰɢɟɣ ɋɉ.

ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɮɢɥɶɬɪɚɰɢɹ ɦɨɠɟɬ ɛɵɬɶ ɥɢɧɟɣɧɨɣ. Ʉɪɢɬɟɪɢɟɦ ɤɚɱɟɫɬɜɚ ɨɰɟɧɤɢ ɡɧɚɱɟɧɢɹ ɋɉ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɢɥɢ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɋɉ ɩɨ ɦɧɨɠɟɫɬɜɭ ɪɟɚɥɢɡɚɰɢɣ ɤɜɚɞɪɚɬɚ ɨɲɢɛɤɢ, ɬ.ɟ. ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɚ ɨɬɤɥɨɧɟɧɢɹ ɨɰɟɧɤɢ ɨɬ ɨɰɟɧɢɜɚɟɦɨɝɨ ɡɧɚɱɟɧɢɹ, ɢɥɢ ɨɬɧɨɲɟɧɢɟ ɦɨɳɧɨɫɬɢ ɩɨɥɟɡɧɨɝɨ ɫɢɝɧɚɥɚ ɤ ɦɨɳɧɨɫɬɢ ɩɨɦɟɯɢ. ɑɚɳɟ ɜɫɟɝɨ ɬɚɤɭɸ ɡɚɞɚɱɭ ɧɚɡɵɜɚɸɬ ɡɚɞɚɱɟɣ ɜɵɞɟɥɟɧɢɹ ɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ [(t), ɧɚɡɵɜɚɟɦɨɝɨ ɫɢɝɧɚɥɨɦ, ɢɡ ɚɞɞɢɬɢɜɧɨɣ ɫɦɟɫɢ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ ɫ ɞɪɭɝɢɦ ɩɪɨɰɟɫɫɨɦ K(t) - ɩɨɦɟɯɨɣ. ɇɚɛɥɸɞɚɟɬɫɹ ɪɟɚɥɢɡɚɰɢɹ x(t) ɫɭɦɦɵ ɭɤɚɡɚɧɧɵɯ ɩɪɨɰɟɫɫɨɜ, ɨɰɟɧɢɜɚɟɬɫɹ ɡɧɚɱɟɧɢɹ ɋɉ [(t), ɤɨɪɪɟɥɢɪɨɜɚɧɧɨɝɨ ɫ ɩɪɨɰɟɫɫɨɦ K(t). ɂɧɨɝɞɚ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɨɰɟɧɤɭ ɧɟ ɫɚɦɨɝɨ ɩɪɨɰɟɫɫɚ [(t), ɚ ɟɝɨ ɥɢɧɟɣɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ. ɉɪɢ ɷɬɨɦ ɚɧɚɥɢɡɢɪɭɸɬ ɫɥɟɞɭɸɳɢɟ ɜɢɞɵ ɥɢɧɟɣɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ: ɫɦɟɳɟɧɢɟ ɦɨɦɟɧɬɚ ɨɰɟɧɤɢ ɩɨ ɨɫɢ ɜɪɟɦɟɧɢ (ɨɰɟɧɤɚ ɫ ɡɚɞɟɪɠɤɨɣ), ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ.

ȼɨ ɜɬɨɪɨɣ ɱɚɫɬɢ ɡɚɞɚɱɢ ɮɢɥɶɬɪɚɰɢɢ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɜɨɩɪɨɫɵ ɧɟɥɢɧɟɣɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɩɨ ɤɪɢɬɟɪɢɸ ɦɢɧɢɦɭɦɚ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɚ ɨɲɢɛɤɢ.

14.2. Ʌɢɧɟɣɧɚɹ ɮɢɥɶɬɪɚɰɢɹ ɩɨ ɤɪɢɬɟɪɢɸ ɦɢɧɢɦɭɦɚ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɚ ɨɲɢɛɤɢ

Ɂɚɞɚɧɚ ɪɟɚɥɢɡɚɰɢɹ x(t) ɫɭɦɦɵ ɋɉ [(t) + K(t). ȼ ɤɚɱɟɫɬɜɟ ɨɰɟɧɤɢ ɋɉ [(t) ɩɪɢɧɢɦɚɟɬɫɹ ɩɪɨɮɢɥɶɬɪɨɜɚɧɧɨɟ ɡɧɚɱɟɧɢɟ ɪɟɚɥɢɡɚɰɢɢ x(t), ɥɢɧɟɣɧɵɣ ɮɢɥɶɬɪ ɜɵɪɚɠɚɟɬɫɹ ɢɦɩɭɥɶɫɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ h(t), ɧɨ ɨɝɪɚɧɢɱɟɧɢɹ, ɫɜɹɡɚɧɧɵɟ ɫ ɮɢɡɢɱɟɫɤɨɣ ɨɫɭɳɟɫɬɜɢɦɨɫɬɶɸ ɮɢɥɶɬɪɚ, ɫɧɢɦɚɸɬɫɹ. ȿɫɬɟɫɬɜɟɧɧɨ, ɨɰɟɧɤɚ ɋɉ [(t) ɹɜɥɹɟɬɫɹ ɨɬɤɥɢɤɨɦ ɥɢɧɟɣɧɨɝɨ ɮɢɥɶɬɪɚ ɧɚ ɪɟɚɥɢɡɚɰɢɸ x(t) ɢ ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɚ ɢɧɬɟɝɪɚɥɨɦ ɫɜɟɪɬɤɢ:

f

Ɉɰɟɧɤɚ (14.1) ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɹɜɥɹɟɬɫɹ ɫɦɟɳɟɧɧɨɣ.

ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɤɜɚɞɪɚɬɚ ɨɲɢɛɤɢ [(t) - [*(t) ɩɪɢ ɭɫɪɟɞɧɟɧɢɢ ɩɨ ɦɧɨɠɟɫɬɜɭ ɪɟɚɥɢɡɚɰɢɣ ɪɚɜɧɨ H2(t)=¢[[(t) - [*(t)]2². ɇɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɢɦɩɭɥɶɫɧɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ h(t) ɥɢɧɟɣɧɨɝɨ ɮɢɥɶɬɪɚ, ɞɥɹ ɤɨɬɨɪɨɝɨ H2 ɢɦɟɟɬ ɦɢɧɢɦɚɥɶɧɭɸ ɜɟɥɢɱɢɧɭɫɪɟɞɢ ɞɪɭɝɢɯ ɥɢɧɟɣɧɵɯ ɫɢɫɬɟɦ.

ɉɨɞɫɬɚɜɢɜ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ H ɮɨɪɦɭɥɭ (14.1) ɢ ɩɨɦɟɧɹɜ ɦɟɫɬɚɦɢ ɢɧɬɟɝɪɚɥɵ ɢ ɨɩɟɪɚɰɢɢ ɭɫɪɟɞɧɟɧɢɹ, ɩɨɥɭɱɢɦ

H2(t)=¢[2(t)² -2 ³h(t,W) ¢x(W)[(t)²dW + ³ ³h(t, u)h(t, v)¢x(u)x(v)²dudv .

ȼ ɷɬɨɦ ɜɵɪɚɠɟɧɢɢ ɭɫɪɟɞɧɟɧɢɸ ɩɨɞɥɟɠɚɬ ɪɟɚɥɢɡɚɰɢɹ x(t) ɜ ɪɚɡɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɢ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɟɚɥɢɡɚɰɢɢ ɧɚ ɋɉ [(t). Ɋɟɡɭɥɶɬɚɬɵ ɬɚɤɨɝɨ ɭɫɪɟɞɧɟɧɢɹ ɟɫɬɶ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɮɭɧɤɰɢɹ ɜɯɨɞɧɨɣ ɫɦɟɫɢ ɢ ɜɡɚɢɦɧɚɹ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɮɭɧɤɰɢɹ ɜɯɨɞɧɨɣ ɫɦɟɫɢ ɢ ɨɰɟɧɢɜɚɟɦɨɝɨ ɋɉ. ɋɱɢɬɚɟɦ, ɱɬɨ ɤɨɪɪɟɥɹɰɢɨɧ-

ɧɵɟ ɮɭɧɤɰɢɢ B[(t1,t2), BK(t1,t2) ɢ B[K(t1,t2) ɢɡɜɟɫɬɧɵ ɢ ɨɛɨɡɧɚɱɢɦ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɮɭɧɤɰɢɸ ɜɯɨɞɧɨɣ ɫɦɟɫɢ ɢ ɜɡɚɢɦɧɭɸ ɤɨɪɪɟɥɹɰɢɨɧɧɭɸ ɮɭɧɤɰɢɸ ɜɯɨɞɧɨɣ ɫɦɟɫɢ ɢ ɲɭɦɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ Bx(t1,t2) ɢ Bx[(t1,t2). Ɍɚɤ ɤɚɤ <x(u)x(v)> = Bx(u,v) =

B[(u,v) + B[K(u,v) + BK[(u,v) + BK(u,v); <x(W)[(t)> = Bx(u,v) = = B[(u,v) + B[K(u,v) + BK[(u,v) + BK(u,v); <x(W)[(t)> = Bx[(W,t) = B[(W,t) + B[K(W,t), ɬɨ, ɩɨɞɫɬɚɜɢɜ ɷɬɢ ɜɵ-

ɪɚɠɟɧɢɹ ɜ ɮɨɪɦɭɥɭɞɥɹ H2(t), ɧɚɯɨɞɢɦ

ɂɡ ɷɬɨɣ ɮɨɪɦɭɥɵ ɫɥɟɞɭɟɬ, ɱɬɨ ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɟɫɤɚɹ ɨɲɢɛɤɚ ɥɢɧɟɣɧɨɣ ɨɰɟɧɤɢ [(t) ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɢ ɜɡɚɢɦɧɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɮɭɧɤɰɢɣ ɩɪɨɰɟɫɫɨɜ [(t) ɢ K(t) ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɛɨɥɟɟ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ.

Ⱦɨɤɚɠɟɦ, ɱɬɨ ɩɪɢ ɡɚɞɚɧɧɵɯ B[, BK, B[K ɧɚɢɥɭɱɲɭɸ (ɜ ɫɦɵɫɥɟ ɩɪɢɧɹɬɨɝɨ ɤɪɢɬɟɪɢɹ) ɥɢɧɟɣɧɭɸ ɮɢɥɶɬɪɚɰɢɸ ɩɪɨɰɟɫɫɚ [(t) ɢɡ ɟɝɨ ɚɞɞɢɬɢɜɧɨɣ ɫɦɟɫɢ ɫ K(t) ɛɭɞɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶ ɥɢɧɟɣɧɚɹ ɫɢɫɬɟɦɚ, ɢɦɩɭɥɶɫɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɤɨɬɨɪɨɣ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɢɧɬɟɝɪɚɥɶɧɨɦɭɭɪɚɜɧɟɧɢɸ

f

ɉɨɞɫɬɚɜɢɦ ɜɵɪɚɠɟɧɢɟ (14.2) ɜ ɮɨɪɦɭɥɭ ɞɥɹ H2(t) ɢ ɩɨɤɚɠɟɦ, ɱɬɨ ɨɲɢɛɤɚ H 2(t) ɦɢɧɢɦɚɥɶɧɚ ɩɪɢ ɭɫɥɨɜɢɢ h(t,W) = h*(t,W). ɇɚ ɫɚɦɨɦ ɞɟɥɟ, ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɤɚɡɚɧɧɨɣ ɜɵɲɟ ɩɨɞɫɬɚɧɨɜɤɢ ɩɨɥɭɱɢɦ

ff

H2(t) = B[(t,t) -2ҏ³ ³ h(t,W)h*(t,y)Bx(W,y)dWdy +

f f

-hҏ*(t,u)][h(t,v) -hҏ*(t,v)]dudv.

Ɍɚɤ ɤɚɤ ɬɨɥɶɤɨ ɩɨɫɥɟɞɧɟɟ ɫɥɚɝɚɟɦɨɟ ɜ ɷɬɨɦ ɜɵɪɚɠɟɧɢɢ ɫɨɞɟɪɠɢɬ ɧɟɢɡɜɟɫɬɧɭɸ ɮɭɧɤɰɢɸ h(t,u) ɢ ɬɚɤ ɤɚɤ ɷɬɨ ɫɥɚɝɚɟɦɨɟ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨ (ɜ ɫɢɥɭ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɩɨɥɭɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɨɪɦɵ ɜ ɩɨɤɚɡɚɬɟɥɟ ɷɤɫɩɨɧɟɧɬɵ ɧɨɪɦɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ), ɬɨ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ H2(t) ɛɭɞɟɬ ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ ɬɚɤɨɣ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɟ, ɩɟɪɟɞɚɬɨɱɧɚɹ ɮɭɧɤɰɢɹ ɤɨɬɨɪɨɣ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ. Ⱥ ɷɬɨ ɛɭɞɟɬ ɢɦɟɬɶ ɦɟɫɬɨ ɩɪɢ ɭɫɥɨɜɢɢ h(t,W) = h*(t,W), ɱɬɨ ɢ ɬɪɟɛɨɜɚɥɨɫɶ ɞɨɤɚɡɚɬɶ.