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Сравнительная оценка фильтров в задаче сопровождения воздушных целей.

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2015
ɇȺɍɑɇɕɃ ȼȿɋɌɇɂɄ ɆȽɌɍ ȽȺ
ʋ 220
ɍȾɄ 621.396.969
ɋɊȺȼɇɂɌȿɅɖɇȺə ɈɐȿɇɄȺ ɎɂɅɖɌɊɈȼ ȼ ɁȺȾȺɑȿ ɋɈɉɊɈȼɈɀȾȿɇɂə
ȼɈɁȾɍɒɇɕɏ ɐȿɅȿɃ
ə.ɂ. ɋɌɊȿɄȺɅɈȼɋɄȺə
ɇɚ ɨɫɧɨɜɟ ɢɦɢɬɚɰɢɨɧɧɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɩɨɬɨɤɚ ɪɚɡɧɨɯɚɪɚɤɬɟɪɧɵɯ ɜɨɡɞɭɲɧɵɯ ɰɟɥɟɣ ɩɪɨɜɟɞɟɧɚ ɫɪɚɜɧɢɬɟɥɶɧɚɹ ɨɰɟɧɤɚ ɮɢɥɶɬɪɚ Ʉɚɥɦɚɧɚ ɢ Įȕ, ĮȕȖ ɮɢɥɶɬɪɨɜ ɫɬɚɬɢɫɬɢɱɟɫɤɢɦ ɦɟɬɨɞɨɦ ɨɞɧɨɮɚɤɬɨɪɧɨɝɨ ɞɢɫɩɟɪɫɢɨɧɧɨɝɨ
ɚɧɚɥɢɡɚ. ɇɚ ɦɨɞɟɥɢ ɦɚɥɨɦɚɧɟɜɪɢɪɭɸɳɟɣ ɰɟɥɢ ɪɚɡɥɢɱɢɟ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɡɧɚɱɢɦɵɦ. ȼ ɭɫɥɨɜɢɹɯ ɠɟ ɪɟɡɤɢɯ ɦɚɧɟɜɪɨɜ
ɮɢɥɶɬɪ Ʉɚɥɦɚɧɚ ɞɚɟɬ ɡɧɚɱɢɦɨ ɛɨɥɟɟ ɬɨɱɧɵɟ ɡɧɚɱɟɧɢɹ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɜɬɨɪɢɱɧɚɹ ɨɛɪɚɛɨɬɤɚ, ɮɢɥɶɬɪ Ʉɚɥɦɚɧɚ, Įȕ ɮɢɥɶɬɪ, ĮȕȖ ɮɢɥɶɬɪ, ɨɞɧɨɮɚɤɬɨɪɧɵɣ ɞɢɫɩɟɪɫɢɨɧɧɵɣ ɚɧɚɥɢɡ.
1. ɉɈɋɌȺɇɈȼɄȺ ɁȺȾȺɑɂ
Ɂɚɞɚɱɚ ɫɪɚɜɧɟɧɢɹ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɮɢɥɶɬɪɨɜ ɧɟ ɧɨɜɚ, ɢ ɜ ɤɥɚɫɫɢɱɟɫɤɢɯ ɦɨɧɨɝɪɚɮɢɹɯ,
ɧɚɩɪɢɦɟɪ, [2,4], ɦɨɠɧɨ ɧɚɣɬɢ ɪɚɡɧɨɨɛɪɚɡɧɵɟ ɤɚɱɟɫɬɜɟɧɧɵɟ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɨɰɟɧɤɢ. Ɉɞɧɚɤɨ
ɨɰɟɧɤɢ ɷɬɢ ɜ ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ ɡɚɜɢɫɹɬ ɤɚɤ ɨɬ ɫɜɨɣɫɬɜ ɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɨɛɴɟɤɬɨɜ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɝɨ ɧɚɛɥɸɞɟɧɢɹ (ɬɢɩ ɰɟɥɢ, ɟɺ ɫɤɨɪɨɫɬɶ, ɦɚɧɟɜɪɟɧɧɨɫɬɶ, ɨɬɪɚɠɚɬɟɥɶɧɵɟ ɫɜɨɣɫɬɜɚ ɨɛɨɥɨɱɤɢ), ɬɚɤ ɢ ɨɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɝɨ ɨɛɨɪɭɞɨɜɚɧɢɹ (ɱɚɫɬɨɬɚ ɨɛɡɨɪɚ, ɫɜɨɣɫɬɜɚ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɤɨɦɩɥɟɤɫɚ ɢ ɩɪɨɝɪɚɦɦ ɨɛɪɚɛɨɬɤɢ ɩɟɪɜɢɱɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɢ ɬ.ɩ.). ȼɨ ɜɫɟɯ ɩɟɪɟɱɢɫɥɟɧɧɵɯ ɫɜɨɣɫɬɜɚɯ ɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɡɚ ɩɨɫɥɟɞɧɟɟ ɞɟɫɹɬɢɥɟɬɢɟ ɨɱɟɜɢɞɟɧ ɡɧɚɱɢɬɟɥɶɧɵɣ ɤɚɱɟɫɬɜɟɧɧɵɣ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɵɣ ɩɪɨɝɪɟɫɫ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɨɩɪɨɜɨɠɞɟɧɢɟ ɰɟɥɟɣ ɩɪɨɢɫɯɨɞɢɬ ɜ
ɩɪɢɧɰɢɩɢɚɥɶɧɨ ɧɨɜɵɯ ɭɫɥɨɜɢɹɯ, ɱɬɨ ɧɟ ɩɨɡɜɨɥɹɟɬ ɫɱɢɬɚɬɶ ɩɪɟɠɧɢɟ ɨɰɟɧɤɢ ɫɪɚɜɧɢɬɟɥɶɧɨɣ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɮɢɥɶɬɪɨɜ ɨɤɨɧɱɚɬɟɥɶɧɵɦɢ ɢ/ɢɥɢ ɛɟɡɭɫɥɨɜɧɨ ɩɪɢɦɟɧɢɦɵɦɢ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ.
ȼ ɫɜɟɬɟ ɞɢɧɚɦɢɱɧɨɝɨ ɪɚɡɜɢɬɢɹ ɬɚɤɬɢɱɟɫɤɨɣ ɢ ɨɩɟɪɚɬɢɜɧɨ-ɬɚɤɬɢɱɟɫɤɨɣ ɚɜɢɚɰɢɢ ɛɨɥɶɲɨɟ
ɜɧɢɦɚɧɢɟ ɭɞɟɥɹɟɬɫɹ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɸ ɦɚɧɟɜɪɟɧɧɵɯ ɜɨɡɦɨɠɧɨɫɬɟɣ ɜɨɡɞɭɲɧɵɯ ɰɟɥɟɣ (ȼɐ), ɜ
ɱɚɫɬɧɨɫɬɢ, ɩɪɢɞɚɧɢɹ ɢɦ ɧɨɜɵɯ ɚɷɪɨɞɢɧɚɦɢɱɟɫɤɢɯ ɫɯɟɦ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ ɨɰɟɧɤɟ ɜɨɡɦɨɠɧɨɫɬɟɣ ȼɐ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɢɯ ɜɵɫɨɤɭɸ ɫɤɨɪɨɫɬɶ ɢ ɡɧɚɱɢɬɟɥɶɧɨɟ ɭɥɭɱɲɟɧɢɟ ɦɚɧɟɜɪɟɧɧɵɯ
ɫɜɨɣɫɬɜ, ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɜɟɞɟɧɢɹ ɜɨɡɞɭɲɧɨɝɨ ɛɨɹ ɢ ɭɤɥɨɧɟɧɢɹ ɨɬ ɚɬɚɤ ɤɚɤ ɫ ɜɨɡɞɭɯɚ, ɬɚɤ ɢ ɫ
ɡɟɦɥɢ. Ʉɚɱɟɫɬɜɨ ɨɰɟɧɤɢ ɜɨɡɦɨɠɧɨɫɬɟɣ ȼɐ ɜ ɛɨɥɶɲɟɣ ɫɬɟɩɟɧɢ ɡɚɜɢɫɢɬ ɨɬ ɤɚɱɟɫɬɜɚ ɩɨɫɬɭɩɚɸɳɟɣ
ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ. ɉɨɷɬɨɦɭ ɜɨɩɪɨɫɵ ɩɨɜɵɲɟɧɢɹ ɤɚɱɟɫɬɜɚ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɩɪɢɜɥɟɤɚɸɬ ɛɨɥɶɲɨɟ ɜɧɢɦɚɧɢɟ.
ɋɨɫɬɨɹɧɢɟ ɢ ɪɚɡɜɢɬɢɟ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɢ ɩɨɡɜɨɥɹɸɬ ɩɪɢɦɟɧɢɬɶ ɛɨɥɟɟ ɫɥɨɠɧɵɟ ɢ ɷɮɮɟɤɬɢɜɧɵɟ ɚɥɝɨɪɢɬɦɵ ɨɛɪɚɛɨɬɤɢ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɞɥɹ ɩɨɜɵɲɟɧɢɹ ɬɨɱɧɨɫɬɢ ɨɰɟɧɢɜɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɢ ȼɐ, ɨɩɬɢɦɚɥɶɧɨɫɬɶ ɜɵɛɨɪɚ ɚɥɝɨɪɢɬɦɨɜ ɧɟ ɪɟɲɚɟɬɫɹ ɪɚɡ ɢ ɧɚɜɫɟɝɞɚ.
Ⱥɤɬɭɚɥɶɧɨɫɬɶ ɫɨɫɬɨɢɬ ɜ ɨɰɟɧɤɟ ɦɟɬɨɞɨɜ ɜɬɨɪɢɱɧɨɣ ɨɛɪɚɛɨɬɤɢ ɢɧɮɨɪɦɚɰɢɢ ɜ ɭɫɥɨɜɢɹɯ ɧɨɜɵɯ ɪɟɚɥɢɣ, ɬ.ɟ. ɧɨɜɵɯ ɬɟɯɧɨɥɨɝɢɣ, ɬɟɯɧɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɰɟɥɟɣ, ɧɨɜɵɯ ɜɨɡɦɨɠɧɨɫɬɟɣ ɚɥɝɨɪɢɬɦɨɜ.
ɉɪɢ ɨɛɪɚɛɨɬɤɟ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɫɨɛɟɧɧɨ ɨɫɬɪɨɣ ɡɚɞɚɱɟɣ ɹɜɥɹɟɬɫɹ ɢɫɫɥɟɞɨɜɚɧɢɟ ɚɥɝɨɪɢɬɦɨɜ ɨɛɪɚɛɨɬɤɢ ɧɚ ɭɱɚɫɬɤɚɯ ɦɚɧɟɜɪɚ ȼɐ. ɂɡɜɟɫɬɧɵɟ ɩɨɞɯɨɞɵ ɤ ɩɨɜɵɲɟɧɢɸ
ɬɨɱɧɨɫɬɢ ɫɨɩɪɨɜɨɠɞɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ ȼɐ ɧɚ ɭɱɚɫɬɤɚɯ ɦɚɧɟɜɪɚ, ɜ ɨɫɧɨɜɧɨɦ, ɛɚɡɢɪɭɸɬɫɹ ɧɚ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɨɛɧɚɪɭɠɟɧɢɹ ɧɚɱɚɥɚ ɢ ɨɤɨɧɱɚɧɢɹ ɦɚɧɟɜɪɚ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɢɡɦɟɧɟɧɢɢ ɩɚɪɚɦɟɬɪɨɜ ɮɢɥɶɬɪɚ ɫɨɩɪɨɜɨɠɞɟɧɢɹ. ɗɬɢ ɩɨɞɯɨɞɵ ɩɪɢɜɨɞɹɬ ɤ ɫɯɟɦɟ ɮɢɥɶɬɪɚ Ʉɚɥɦɚɧɚ, ɥɢɛɨ ɤ ɫɯɟɦɚɦ Įȕ ɢ ĮȕȖ ɮɢɥɶɬɪɨɜ.
ɇɚɬɭɪɧɵɟ ɷɤɫɩɟɪɢɦɟɧɬɵ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɚɱɟɫɬɜɚ ɪɚɛɨɬɵ ɮɢɥɶɬɪɨɜ ɧɟɪɟɚɥɶɧɵ ɩɨ ɦɧɨɠɟɫɬɜɭ ɩɪɢɱɢɧ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɞɚɧɧɵɯ ɢɫɩɨɥɶɡɨɜɚɥɚɫɶ ɯɨɪɨɲɨ ɡɚɪɟɤɨɦɟɧɞɨɜɚɜɲɚɹ ɫɟɛɹ ɤɨɦɩɶɸɬɟɪɧɚɹ ɦɨɞɟɥɶ, ɢɦɢɬɢɪɭɸɳɚɹ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɟ ɫɨɩɪɨɜɨɠɞɟɧɢɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɧɨɜɟɣɲɢɯ ɬɟɯɧɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɰɟɥɟɣ. Ȼɨɥɟɟ ɩɨɞɪɨɛɧɨɟ ɨɩɢɫɚɧɢɟ ɫɨɞɟɪɠɢɬɫɹ ɜ
ɧɚɲɢɯ ɩɪɟɞɵɞɭɳɢɯ ɫɬɚɬɶɹɯ [6], [7].
22
ə.ɂ. ɋɬɪɟɤɚɥɨɜɫɤɚɹ
2. ȼɕȻɈɊ ɆȿɌɈȾȺ
ɉɟɪɟɣɞɟɦ ɤ ɢɡɥɨɠɟɧɢɸ ɢɫɩɨɥɶɡɭɟɦɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ. Ⱦɥɹ ɨɞɧɨɡɧɚɱɧɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɬɟɪɦɢɧɨɜ ɜ ɧɢɠɟɢɡɥɨɠɟɧɧɨɦ ɬɟɤɫɬɟ, ɜɜɟɞɟɦ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ
ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡ.
Ɋɟɡɭɥɶɬɚɬɵ ɧɚɛɥɸɞɟɧɢɣ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɩɪɨɜɟɪɤɢ ɩɪɟɞɩɨɥɨɠɟɧɢɣ (ɝɢɩɨɬɟɡ) ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɟɯ ɢɥɢ ɢɧɵɯ ɫɜɨɣɫɬɜ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ. ȼ ɱɚɫɬɧɨɫɬɢ, ɬɚɤɨɝɨ ɪɨɞɚ
ɡɚɞɚɱɢ ɜɨɡɧɢɤɚɸɬ ɩɪɢ ɫɪɚɜɧɟɧɢɢ ɪɚɡɥɢɱɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢɥɢ ɦɟɬɨɞɨɜ ɨɛɪɚɛɨɬɤɢ
ɩɨ ɨɩɪɟɞɟɥɟɧɧɵɦ ɢɡɦɟɪɹɟɦɵɦ ɩɪɢɡɧɚɤɚɦ, ɧɚɩɪɢɦɟɪ, ɩɨ ɬɨɱɧɨɫɬɢ, ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɢ ɢ ɬ.ɞ.
ɉɭɫɬɶ X – ɧɚɛɥɸɞɚɟɦɚɹ ɞɢɫɤɪɟɬɧɚɹ ɢɥɢ ɧɟɩɪɟɪɵɜɧɚɹ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ. ɋɬɚɬɢɫɬɢɱɟɫɤɨɣ ɝɢɩɨɬɟɡɨɣ H ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɚɪɚɦɟɬɪɨɜ ɢɥɢ ɜɢɞɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X. ɑɚɫɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢɱɢɧɵ X ɢɡɜɟɫɬɧɨ, ɢ ɩɨ ɜɵɛɨɪɤɟ
ɧɚɛɥɸɞɟɧɢɣ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɪɢɬɶ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɨ ɡɧɚɱɟɧɢɢ ɩɚɪɚɦɟɬɪɨɜ ɷɬɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ɍɚɤɢɟ ɝɢɩɨɬɟɡɵ ɧɚɡɵɜɚɸɬɫɹ ɩɚɪɚɦɟɬɪɢɱɟɫɤɢɦɢ.
ɉɪɨɜɟɪɹɟɦɚɹ ɝɢɩɨɬɟɡɚ ɧɚɡɵɜɚɟɬɫɹ ɧɭɥɟɜɨɣ ɝɢɩɨɬɟɡɨɣ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ H 0 . ɇɚɪɹɞɭ ɫ ɝɢɩɨɬɟɡɨɣ
H 0 ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɨɞɧɭ ɢɡ ɚɥɶɬɟɪɧɚɬɢɜɧɵɯ (ɤɨɧɤɭɪɢɪɭɸɳɢɯ) ɝɢɩɨɬɟɡ H1 . ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɩɪɨɜɟɪɹɟɬɫɹ ɝɢɩɨɬɟɡɚ ɨ ɪɚɜɟɧɫɬɜɟ ɩɚɪɚɦɟɬɪɚ ș ɧɟɤɨɬɨɪɨɦɭ ɡɚɞɚɧɧɨɦɭ ɡɧɚɱɟɧɢɸ θ 0 , ɬɨ ɟɫɬɶ H 0 : θ = θ 0 , ɬɨ
ɜ ɤɚɱɟɫɬɜɟ ɚɥɶɬɟɪɧɚɬɢɜɧɨɣ ɝɢɩɨɬɟɡɵ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɨɞɧɭ ɢɡ ɫɥɟɞɭɸɳɢɯ ɝɢɩɨɬɟɡ:
H1(1) : θ > θ 0 ,
H1(2) : θ < θ 0 ,
H1(3) : θ ≠ θ 0 ,
H1(4) : θ = θ 0 ,
ɝɞɟ θ1 – ɡɚɞɚɧɧɨɟ ɡɧɚɱɟɧɢɟ, θ1 ≠ θ 0 . ȼɵɛɨɪ ɚɥɶɬɟɪɧɚɬɢɜɧɨɣ ɝɢɩɨɬɟɡɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɧɤɪɟɬɧɨɣ
ɮɨɪɦɭɥɢɪɨɜɤɨɣ ɡɚɞɚɱɢ.
ɉɪɚɜɢɥɨ, ɩɨ ɤɨɬɨɪɨɦɭ ɩɪɢɧɢɦɚɟɬɫɹ ɪɟɲɟɧɢɟ ɩɪɢɧɹɬɶ ɢɥɢ ɨɬɤɥɨɧɢɬɶ ɝɢɩɨɬɟɡɭ H 0 , ɧɚɡɵɜɚɟɬɫɹ ɤɪɢɬɟɪɢɟɦ K. Ɍɚɤ ɤɚɤ ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɜɵɛɨɪɤɢ ɧɚɛɥɸɞɟɧɢɣ ɫɥɭɱɚɣɧɨɣ
ɜɟɥɢɱɢɧɵ X, ɧɟɨɛɯɨɞɢɦɨ ɜɵɛɪɚɬɶ ɩɨɞɯɨɞɹɳɭɸ ɫɬɚɬɢɫɬɢɤɭ, ɧɚɡɵɜɚɟɦɭɸ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ Z ɤɪɢɬɟɪɢɹ K. ɉɪɢ ɩɪɨɜɟɪɤɟ ɩɪɨɫɬɨɣ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɝɢɩɨɬɟɡɵ H 0 : θ = θ 0 ɜ ɤɚɱɟɫɬɜɟ ɫɬɚɬɢɫɬɢɤɢ ɤɪɢɬɟɪɢɹ ɜɵɛɢɪɚɸɬ ɬɭ ɠɟ ɫɬɚɬɢɫɬɢɤɭ, ɱɬɨ ɢ ɞɥɹ ɨɰɟɧɤɢ ɩɚɪɚɦɟɬɪɚ θ , ɬ.ɟ. θ .
ɉɪɨɜɟɪɤɚ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɝɢɩɨɬɟɡɵ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɩɪɢɧɰɢɩɟ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦ
ɦɚɥɨɜɟɪɨɹɬɧɵɟ ɫɨɛɵɬɢɹ ɫɱɢɬɚɸɬɫɹ ɧɟɜɨɡɦɨɠɧɵɦɢ, ɚ ɫɨɛɵɬɢɹ, ɢɦɟɸɳɢɟ ɛɨɥɶɲɭɸ ɜɟɪɨɹɬɧɨɫɬɶ,
ɫɱɢɬɚɸɬɫɹ ɞɨɫɬɨɜɟɪɧɵɦɢ. ɗɬɨɬ ɩɪɢɧɰɢɩ ɦɨɠɧɨ ɪɟɚɥɢɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɉɟɪɟɞ ɚɧɚɥɢɡɨɦ
ɜɵɛɨɪɤɢ ɮɢɤɫɢɪɭɟɬɫɹ ɧɟɤɨɬɨɪɚɹ ɦɚɥɚɹ ɜɟɪɨɹɬɧɨɫɬɶ Į , ɧɚɡɵɜɚɟɦɚɹ ɭɪɨɜɧɟɦ ɡɧɚɱɢɦɨɫɬɢ. ɉɭɫɬɶ V –
ɦɧɨɠɟɫɬɜɨ ɡɧɚɱɟɧɢɣ ɫɬɚɬɢɫɬɢɤɢ Z, ɚ Vk ⊆ V – ɬɚɤɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ, ɱɬɨ ɩɪɢ ɭɫɥɨɜɢɢ ɢɫɬɢɧɧɨɫɬɢ
ɝɢɩɨɬɟɡɵ H 0 ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɫɬɚɬɢɫɬɢɤɢ ɤɪɢɬɟɪɢɹ ɜ Vk ɪɚɜɧɚ Į, ɬ.ɟ. P[ Z ∈ Vk / H 0 ] = α .
Ɉɛɨɡɧɚɱɢɦ zɜ ɜɵɛɨɪɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɫɬɚɬɢɫɬɢɤɢ Z, ɜɵɱɢɫɥɟɧɧɨɟ ɩɨ ɜɵɛɨɪɤɟ ɧɚɛɥɸɞɟɧɢɣ.
Ʉɪɢɬɟɪɢɣ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ: ɨɬɤɥɨɧɢɬɶ ɝɢɩɨɬɟɡɭ H 0 , ɟɫɥɢ zɜ ∈ Vk ; ɩɪɢɧɹɬɶ
ɝɢɩɨɬɟɡɭ H 0 , ɟɫɥɢ zɜ ∈ V \ Vk . Ʉɪɢɬɟɪɢɣ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɡɚɪɚɧɟɟ ɡɚɞɚɧɧɨɝɨ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ, ɧɚɡɵɜɚɸɬ ɤɪɢɬɟɪɢɟɦ ɡɧɚɱɢɦɨɫɬɢ. Ɇɧɨɠɟɫɬɜɨ Vk ɜɫɟɯ ɡɧɚɱɟɧɢɣ ɫɬɚɬɢɫɬɢɤɢ ɤɪɢɬɟɪɢɹ Z, ɩɪɢ ɤɨɬɨɪɵɯ ɩɪɢɧɢɦɚɟɬɫɹ ɪɟɲɟɧɢɟ ɨɬɤɥɨɧɢɬɶ ɝɢɩɨɬɟɡɭ H 0 , ɧɚɡɵɜɚɸɬ ɤɪɢɬɢɱɟɫɤɨɣ
ɨɛɥɚɫɬɶɸ; ɨɛɥɚɫɬɶ V \ Vk ɧɚɡɵɜɚɸɬ ɨɛɥɚɫɬɶɸ ɩɪɢɧɹɬɢɹ ɝɢɩɨɬɟɡɵ H 0 .
ɋɪɚɜɧɢɬɟɥɶɧɚɹ ɨɰɟɧɤɚ ɮɢɥɶɬɪɨɜ…
23
ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ Į ɨɩɪɟɞɟɥɹɟɬ «ɪɚɡɦɟɪ» ɤɪɢɬɢɱɟɫɤɨɣ ɨɛɥɚɫɬɢ Vk . ɉɨɥɨɠɟɧɢɟ ɤɪɢɬɢɱɟɫɤɨɣ ɨɛɥɚɫɬɢ ɧɚ ɦɧɨɠɟɫɬɜɟ ɡɧɚɱɟɧɢɣ ɫɬɚɬɢɫɬɢɤɢ Z ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɭɥɢɪɨɜɤɢ ɚɥɶɬɟɪɧɚɬɢɜɵ
ɝɢɩɨɬɟɡɵ H1 , P[ Z ∈ V \ Vk ] = 1 − α .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɨɜɟɪɤɚ ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɝɢɩɨɬɟɡɵ ɩɪɢ ɩɨɦɨɳɢ ɤɪɢɬɟɪɢɹ ɡɧɚɱɢɦɨɫɬɢ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɛɢɬɚ ɧɚ ɫɥɟɞɭɸɳɢɟ ɷɬɚɩɵ:
1) ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɩɪɨɜɟɪɹɟɦɭɸ ( H 0 ) ɢ ɚɥɶɬɟɪɧɚɬɢɜɧɭɸ ( H1 ) ɝɢɩɨɬɟɡɵ;
2) ɧɚɡɧɚɱɢɬɶ ɭɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ α ;
3) ɜɵɛɪɚɬɶ ɫɬɚɬɢɫɬɢɤɭ Z ɤɪɢɬɟɪɢɹ ɞɥɹ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ H 0 ;
4) ɨɩɪɟɞɟɥɢɬɶ ɜɵɛɨɪɨɱɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɬɚɬɢɫɬɢɤɢ Z ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɜɟɪɧɚ
ɝɢɩɨɬɟɡɚ H 0 ;
5) ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɮɨɪɦɭɥɢɪɨɜɤɢ ɚɥɶɬɟɪɧɚɬɢɜɧɨɣ ɝɢɩɨɬɟɡɵ ɨɩɪɟɞɟɥɢɬɶ ɤɪɢɬɢɱɟɫɤɭɸ ɨɛɥɚɫɬɶ Vk ɨɞɧɢɦ ɢɡ ɧɟɪɚɜɟɧɫɬɜ Z > z1−α , Z < zα ɢɥɢ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɧɟɪɚɜɟɧɫɬɜ Z > z1−α /2 ɢ
Z < zα /2 ;
6) ɩɨɥɭɱɢɬɶ ɜɵɛɨɪɤɭ ɧɚɛɥɸɞɟɧɢɣ ɢ ɜɵɱɢɫɥɢɬɶ ɜɵɛɨɪɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɫɬɚɬɢɫɬɢɤɢ
ɤɪɢɬɟɪɢɹ;
7) ɩɪɢɧɹɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ:
- ɟɫɥɢ zɜ ∈ Vk , ɬɨ ɨɬɤɥɨɧɢɬɶ ɝɢɩɨɬɟɡɭ H 0 ɤɚɤ ɧɟ ɫɨɝɥɚɫɭɸɳɭɸɫɹ ɫ ɪɟɡɭɥɶɬɚɬɚɦɢ ɧɚɛɥɸɞɟɧɢɣ;
- ɟɫɥɢ zɜ ∈ V \ Vk , ɬɨ ɩɪɢɧɹɬɶ ɝɢɩɨɬɟɡɭ H 0 , ɬ.ɟ. ɫɱɢɬɚɬɶ, ɱɬɨ ɝɢɩɨɬɟɡɚ H 0 ɧɟ ɩɪɨɬɢɜɨɪɟɱɢɬ
ɪɟɡɭɥɶɬɚɬɚɦ ɧɚɛɥɸɞɟɧɢɣ.
Ⱦɥɹ ɫɪɚɜɧɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɨɜ, ɜɵɩɨɥɧɟɧɧɵɯ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɤɚɤɨɝɨ-ɥɢɛɨ ɩɚɪɚɦɟɬɪɚ, ɨɛɵɱɧɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɨɞɧɨɮɚɤɬɨɪɧɵɣ ɞɢɫɩɟɪɫɢɨɧɧɵɣ ɚɧɚɥɢɡ.
ɉɭɫɬɶ ɪɟɡɭɥɶɬɚɬɵ ɧɚɛɥɸɞɟɧɢɣ ɫɨɫɬɚɜɥɹɸɬ l ɧɟɡɚɜɢɫɢɦɵɯ ɜɵɛɨɪɨɤ (ɝɪɭɩɩ), ɩɨɥɭɱɟɧɧɵɯ
ɢɡ l ɧɨɪɦɚɥɶɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɝɟɧɟɪɚɥɶɧɵɯ ɫɨɜɨɤɭɩɧɨɫɬɟɣ, ɤɨɬɨɪɵɟ ɢɦɟɸɬ ɪɚɡɥɢɱɧɵɟ ɫɪɟɞɧɢɟ m1 , m2 ,..., ml ɢ ɪɚɜɧɵ ɞɢɫɩɟɪɫɢɢ σ 2 . ɉɪɨɜɟɪɹɟɬɫɹ ɝɢɩɨɬɟɡɚ ɨ ɪɚɜɟɧɫɬɜɟ ɫɪɟɞɧɢɯ
H 0 : m1 = m2 = ... = ml . ɇɚɲɚ ɡɚɞɚɱɚ - ɢɫɫɥɟɞɨɜɚɧɢɟ ɜɥɢɹɧɢɹ, ɤɨɬɨɪɨɟ ɨɤɚɡɵɜɚɟɬ ɢɡɦɟɧɟɧɢɟ ɜɵɛɨɪɚ ɮɢɥɶɬɪɚ ɧɚ ɬɨɱɧɨɫɬɶ ɩɪɨɝɧɨɡɚ. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɧɚɫ ɢɧɬɟɪɟɫɭɟɬ ɜɨɩɪɨɫ, ɢɦɟɸɬ ɥɢ ɪɚɡɥɢɱɧɵɟ
ɮɢɥɶɬɪɵ ɨɞɧɭ ɢ ɬɭ ɠɟ ɫɢɫɬɟɦɚɬɢɱɟɫɤɭɸ ɨɲɢɛɤɭ (ɝɢɩɨɬɟɡɚ H 0 ). ɉɪɢ l = 2 ɞɥɹ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ H 0 ɢɫɩɨɥɶɡɭɸɬɫɹ ɢɡɜɟɫɬɧɵɟ ɤɪɢɬɟɪɢɢ ɡɧɚɱɢɦɨɫɬɢ. ȿɫɥɢ l > 2, ɬɨ ɞɥɹ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ ɨ
ɪɚɜɟɧɫɬɜɟ l ɫɪɟɞɧɢɯ ɩɪɢɦɟɧɹɸɬ ɨɞɧɨɮɚɤɬɨɪɧɵɣ ɞɢɫɩɟɪɫɢɨɧɧɵɣ ɚɧɚɥɢɡ, ɫɭɬɶ ɤɨɬɨɪɨɝɨ ɫɨɫɬɨɢɬ ɜ
ɫɥɟɞɭɸɳɟɦ.
ɉɭɫɬɶ xik ɨɛɨɡɧɚɱɚɟɬ i-ɣ ɷɥɟɦɟɧɬ k-ɣ ɜɵɛɨɪɤɢ, i = 1, 2, …, nk ; k = 1, 2, …, l; xk - ɜɵɛɨɪɨɱɧɨɟ ɫɪɟɞɧɟɟ k-ɣ ɜɵɛɨɪɤɢ, ɬ.ɟ.
1
xk =
nk
nk
n
1
1 l k
1
.
,
x
=
x
⋅
x
–
ɨɛɳɟɟ
ɜɵɛɨɪɨɱɧɨɟ
ɫɪɟɞɧɟɟ,
ɬ.ɟ.
x
=
xik = x_ ,
¦
¦¦
ik
k
n k =1 i =1
n
nk
i =1
ɝɞɟ n – ɨɛɳɟɟ ɱɢɫɥɨ ɧɚɛɥɸɞɟɧɢɣ, n =
l
nk .
¦
k =1
Ɉɛɳɚɹ ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ ɧɚɛɥɸɞɟɧɢɣ ɨɬ ɨɛɳɟɝɨ ɫɪɟɞɧɟɝɨ
ɩɪɟɞɫɬɚɜɥɟɧɚ ɬɚɤ:
l
nk
l
l
nk
( xik − x ) = ¦ nk ( xk − x ) + ¦¦ ( xik − xk )2 .
¦¦
k =1 i =1
k =1
k =1 i =1
2
2
ɦɨɠɟɬ ɛɵɬɶ
24
ə.ɂ. ɋɬɪɟɤɚɥɨɜɫɤɚɹ
ɗɬɨ – ɨɫɧɨɜɧɨɟ ɬɨɠɞɟɫɬɜɨ ɞɢɫɩɟɪɫɢɨɧɧɨɝɨ ɚɧɚɥɢɡɚ. Ɂɚɩɢɲɟɦ ɟɝɨ ɜ ɜɢɞɟ:
Q = Q1 + Q2 ,
ɝɞɟ Q – ɨɛɳɚɹ ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ ɧɚɛɥɸɞɟɧɢɣ ɨɬ ɨɛɳɟɝɨ ɫɪɟɞɧɟɝɨ;
Q1 – ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ ɜɵɛɨɪɨɱɧɵɯ ɫɪɟɞɧɢɯ xk ɨɬ ɨɛɳɟɝɨ ɫɪɟɞɧɟɝɨ x (ɦɟɠɞɭ
ɝɪɭɩɩɚɦɢ);
Q2 – ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ ɧɚɛɥɸɞɟɧɢɣ ɨɬ ɜɵɛɨɪɨɱɧɵɯ ɫɪɟɞɧɢɯ ɝɪɭɩɩ (ɜɧɭɬɪɢ
ɝɪɭɩɩɵ).
Ɉɩɢɫɚɧɧɨɟ ɜɵɲɟ ɬɨɠɞɟɫɬɜɨ ɥɟɝɤɨ ɩɪɨɜɟɪɹɟɬɫɹ, ɟɫɥɢ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɨɱɟɜɢɞɧɵɦ
ɪɚɜɟɧɫɬɜɨɦ:
( xik − x ) = [( xk − x ) + ( xik − xk )] ,
ɜɨɡɜɟɫɬɢ ɨɛɟ ɟɝɨ ɱɚɫɬɢ ɜ ɤɜɚɞɪɚɬ, ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɩɨ i ɢ k ɢ ɭɱɟɫɬɶ, ɱɬɨ
l
nk
( xik − xk )( xk − x ) = 0
¦¦
k =1 i =1
ɜ ɫɢɥɭ ɨɩɪɟɞɟɥɟɧɢɹ ɫɪɟɞɧɢɯ xk ɢ x .
ȿɫɥɢ ɜɟɪɧɚ ɝɢɩɨɬɟɡɚ H 0 : m1 = m2 = ... = m1 , ɬɨ ɫɬɚɬɢɫɬɢɤɢ Q1 / σ 2 ɢ Q2 / σ 2 ɧɟɡɚɜɢɫɢɦɵ ɢ
ɢɦɟɸɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ χ 2 ɫ l − 1 ɢ n − l ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɬɚɬɢɫɬɢɤɢ
Q
Q
S12 = 1 ɢ S 22 = 2 ɹɜɥɹɸɬɫɹ ɧɟɫɦɟɳɟɧɧɵɦɢ ɨɰɟɧɤɚɦɢ ɧɟɢɡɜɟɫɬɧɨɣ ɞɢɫɩɟɪɫɢɢ σ 2 . Ɉɰɟɧɤɚ
l −1
n−l
2
S1 ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɪɚɫɫɟɹɧɢɟ ɝɪɭɩɩɨɜɵɯ ɫɪɟɞɧɢɯ, ɚ ɨɰɟɧɤɚ S 22 – ɪɚɫɫɟɹɧɢɟ ɜɧɭɬɪɢ ɝɪɭɩɩ, ɤɨɬɨɪɨɟ ɨɛɭɫɥɨɜɥɟɧɨ ɫɥɭɱɚɣɧɵɦɢ ɜɚɪɢɚɰɢɹɦɢ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɛɥɸɞɟɧɢɣ. Ɂɧɚɱɢɬɟɥɶɧɨɟ ɩɪɟɜɵɲɟɧɢɟ
ɜɟɥɢɱɢɧɵ S12 ɧɚɞ ɡɧɚɱɟɧɢɟɦ ɜɟɥɢɱɢɧɵ S 22 ɦɨɠɧɨ ɨɛɴɹɫɧɢɬɶ ɪɚɡɥɢɱɢɟɦ ɫɪɟɞɧɢɯ ɜ ɝɪɭɩɩɚɯ. Ɉɬɧɨɲɟɧɢɟ ɷɬɢɯ ɨɰɟɧɨɤ ɢɦɟɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ Ɏɢɲɟɪɚ ɫ l − 1 ɢ n − l ɫɬɟɩɟɧɹɦɢ ɫɜɨɛɨɞɵ, ɬ.ɟ.
S12 Q1 / (l − 1)
=
= F (l − 1, n − l ).
S22 Q2 / (n − l )
Ⱦɚɧɧɚɹ ɫɬɚɬɢɫɬɢɤɚ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɩɪɨɜɟɪɤɢ ɝɢɩɨɬɟɡɵ H 0 : m1 = m2 = ... = ml = m . Ƚɢɩɨɬɟɡɚ H 0 ɧɟ ɩɪɨɬɢɜɨɪɟɱɢɬ ɪɟɡɭɥɶɬɚɬɚɦ ɧɚɛɥɸɞɟɧɢɣ, ɟɫɥɢ ɜɵɛɨɪɨɱɧɨɟ ɡɧɚɱɟɧɢɟ Fɜ ɫɬɚɬɢɫɬɢɤɢ
Q
ɦɟɧɶɲɟ ɤɜɚɧɬɢɥɢ F1−α (l − 1, n − l ) , ɬ.ɟ. ɟɫɥɢ Fɜ < F1−α (l − 1, n − l ) . ȼ ɷɬɨɦ ɫɥɭɱɚɟ x ɢ 2 ɹɜɥɹɸɬɫɹ
n−l
2
ɧɟɫɦɟɳɟɧɧɵɦɢ ɨɰɟɧɤɚɦɢ ɩɚɪɚɦɟɬɪɨɜ m ɢ σ . ȿɫɥɢ Fɜ ≥ F1−α (l − 1, n − l ) , ɬɨ ɝɢɩɨɬɟɡɚ H 0 ɨɬɤɥɨɧɹɟɬɫɹ ɢ ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɪɟɞɢ ɫɪɟɞɧɢɯ m1 , m2 ,..., ml ɢɦɟɟɬɫɹ ɯɨɬɹ ɛɵ ɞɜɚ ɧɟ ɪɚɜɧɵɯ ɞɪɭɝ ɞɪɭɝɭ.
Ƚɢɩɨɬɟɡɚ ɨ ɪɚɡɥɢɱɟɧɢɢ ɤɚɱɟɫɬɜɚ ɮɢɥɶɬɪɨɜ ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɨɫɥɟ ɧɚɡɧɚɱɟɧɢɹ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ Į, ɤɨɬɨɪɵɣ ɢɦɟɟɬ ɫɦɵɫɥ ɜ ɞɨɩɭɫɬɢɦɨɣ ɨɲɢɛɤɟ ɩɪɢ ɩɪɢɧɹɬɢɢ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ.
ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ ɜɜɢɞɭ ɜɵɫɨɤɨɣ ɰɟɧɵ ɨɲɢɛɨɤ ɭɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ ɧɟ ɦɨɠɟɬ
ɛɵɬɶ ɛɨɥɶɲɟ 0,05 ɢ ɜ ɧɚɲɢɯ ɩɨɫɥɟɞɭɸɳɢɯ ɪɚɫɱɟɬɚɯ ɦɵ ɩɪɢɧɢɦɚɟɦ Į = 0,01.
ɋɪɚɜɧɢɬɟɥɶɧɚɹ ɨɰɟɧɤɚ ɮɢɥɶɬɪɨɜ…
25
ȿɫɥɢ ɝɢɩɨɬɟɡɚ H 0 ɨ ɪɚɜɟɧɫɬɜɟ ɫɪɟɞɧɢɯ ɨɬɤɥɨɧɹɟɬɫɹ, ɬɨ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ, ɤɚɤɢɟ
ɢɦɟɧɧɨ ɝɪɭɩɩɵ ɢɦɟɸɬ ɡɧɚɱɢɦɨɟ ɪɚɡɥɢɱɢɟ ɫɪɟɞɧɢɯ. Ⱦɥɹ ɷɬɢɯ ɰɟɥɟɣ ɢɫɩɨɥɶɡɭɟɬɫɹ ɦɟɬɨɞ ɥɢɧɟɣɧɵɯ ɤɨɧɬɪɚɫɬɨɜ. Ʌɢɧɟɣɧɵɣ ɤɨɧɬɪɚɫɬ Lk ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ:
l
Lk = ¦ ck mk ,
k =1
ɝɞɟ ck , k = 1, 2, …, l, – ɤɨɧɫɬɚɧɬɵ, ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɦɵɟ ɢɡ ɮɨɪɦɭɥɢɪɨɜɤɢ ɩɪɨɜɟɪɹɟɦɵɯ
Lk = Σl c x , ɚ ɨɰɟɧɤɚ ɞɢɫɩɟɪɫɢɢ Lk ɪɚɜɧɚ
ɝɢɩɨɬɟɡ, ɩɪɢɱɟɦ Σl c = 0 . Ɉɰɟɧɤɚ Lk ɪɚɜɧɚ k
k =1 k k
k =1 k
ck2 Q2 l ck2
=
.
n −l¦
k =1 nk
k =1 nk
l
2
= D[ Lk ] = σ 2 ¦
sLk
Ƚɪɚɧɢɰɵ ɞɨɜɟɪɢɬɟɥɶɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɞɥɹ Lk ɢɦɟɸɬ ɜɢɞ:
k
Lk ± s
Lk
(l − 1) ⋅ F1−α (l − 1, n − l ),
3. ɈɉɂɋȺɇɂȿ ɂɋɏɈȾɇɈɃ ɂɇɎɈɊɆȺɐɂɂ ɂ ɉɊɂɆȿɊ ɊȺɋɑȬɌɈȼ
ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ ɬɪɢ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɬɪɟɦ ɮɢɥɶɬɪɚɦ: Įȕ,
ĮȕȖ ɢ ɮɢɥɶɬɪɭ Ʉɚɥɦɚɧɚ; ɬ.ɨ. l = 3 . Ɂɚɞɚɱɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɩɪɨɜɟɪɤɟ ɝɢɩɨɬɟɡɵ H 0 : m1 = m2 = m3 ,
ɝɞɟ mk – ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɫɭɦɦɵ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɹ ɷɤɫɬɪɚɩɨɥɹɰɢɢ. Ɉɛɳɟɟ ɱɢɫɥɨ
ɧɚɛɥɸɞɟɧɢɣ n = 72 , ɚ nk = 24 . ɍɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ Į = 0,01.
ɇɚɦɢ ɩɪɨɜɨɞɢɥɨɫɶ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɩɪɢ ɩɨɦɨɳɢ ɜɵɲɟɭɩɨɦɹɧɭɬɨɣ ɤɨɦɩɶɸɬɟɪɧɨɣ ɦɨɞɟɥɢ,
ɢɦɢɬɢɪɭɸɳɟɣ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɟ ɫɨɩɪɨɜɨɠɞɟɧɢɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɧɨɜɟɣɲɢɯ ɬɟɯɧɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɰɟɥɟɣ. Ⱦɥɹ ɤɚɠɞɨɝɨ ɢɡ ɬɪɟɯ ɬɢɩɨɜ ȼɐ (ɪɚɤɟɬɵ, ɫɚɦɨɥɟɬɵ, ɜɟɪɬɨɥɟɬɵ), ɡɚɞɚɜɚɥɢɫɶ
ɫɥɟɞɭɸɳɢɟ ɩɚɪɚɦɟɬɪɵ ɦɨɞɟɥɢɪɨɜɚɧɢɹ:
move_1 – ȼɐ ɥɟɬɢɬ ɩɨ ɩɪɹɦɨɣ, ɬ.ɟ. ɧɟɬ ɦɚɧɟɜɪɚ ɧɢ ɩɨ ɭɝɥɭ ɦɟɫɬɚ, ɧɢ ɩɨ ɫɤɨɪɨɫɬɢ, ɧɢ ɩɨ
ɜɵɫɨɬɟ;
move_2 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɬɨɥɶɤɨ ɩɨ ɤɭɪɫɭ;
move_3 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɬɨɥɶɤɨ ɩɨ ɫɤɨɪɨɫɬɢ;
move_4 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɬɨɥɶɤɨ ɩɨ ɜɵɫɨɬɟ;
move_5 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɩɨ ɤɭɪɫɭ ɢ ɫɤɨɪɨɫɬɢ;
move_6 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɩɨ ɤɭɪɫɭ ɢ ɜɵɫɨɬɟ;
move_7 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɩɨ ɫɤɨɪɨɫɬɢ ɢ ɜɵɫɨɬɟ;
move_8 – ȼɐ ɦɚɧɟɜɪɢɪɭɟɬ ɩɨ ɤɭɪɫɭ, ɫɤɨɪɨɫɬɢ ɢ ɜɵɫɨɬɟ.
ɉɪɢ ɦɨɞɟɥɢɪɨɜɚɧɢɢ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɬɪɚɟɤɬɨɪɢɢ, ɤɨɬɨɪɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɬɟɯɧɢɱɟɫɤɢɦ ɜɨɡɦɨɠɧɨɫɬɹɦ ɫɨɜɪɟɦɟɧɧɵɯ ɫɪɟɞɫɬɜ ɧɚɩɚɞɟɧɢɹ; ɬɨɱɧɟɟ:
– ɩɨɞɥɟɬ ɧɚ ɩɪɟɞɟɥɶɧɨ ɧɢɡɤɢɯ ɜɵɫɨɬɚɯ ɩɪɨɬɢɜɨɤɨɪɚɛɟɥɶɧɵɯ ɪɚɤɟɬ;
– ɦɚɧɟɜɪɢɪɨɜɚɧɢɟ «ɩɢɤɢɪɨɜɚɧɢɟ», «ɝɨɪɤɚ» ɫɜɟɪɯɡɜɭɤɨɜɵɯ ɫɚɦɨɥɟɬɨɜ;
– ɦɚɧɟɜɪɵ «ɜɢɪɚɠ», «ɫɩɢɪɚɥɶ» ɢ «ɛɨɟɜɨɣ ɪɚɡɜɨɪɨɬ» ɜɟɪɬɨɥɟɬɨɜ.
ȼ ɩɪɢɜɟɞɟɧɧɨɦ ɧɢɠɟ ɜɚɪɢɚɧɬɟ ɪɚɫɱɟɬɚ ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ ɩɨɤɚ ɧɟ ɜɫɟ ɷɬɢ ɜɨɡɦɨɠɧɨɫɬɢ.
Ⱦɚɧɧɵɟ ɤɨɧɤɪɟɬɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɢɦɟɸɬ ɫɥɟɞɭɸɳɭɸ ɫɬɪɭɤɬɭɪɭ. ɂɡɦɟɪɹɟɦɨɣ ɜɟɥɢɱɢɧɨɣ
ɹɜɥɹɟɬɫɹ ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ, ɩɪɟɞɫɤɚɡɚɧɧɵɯ ɮɢɥɶɬɪɚɦɢ ɩɨɥɨɠɟɧɢɣ ɰɟɥɢ ɨɬ ɮɚɤɬɢɱɟɫɤɢɯ
ɩɨɥɨɠɟɧɢɣ ɦɨɞɟɥɢ ɞɜɢɠɟɧɢɹ ɡɚ ɨɞɢɧ ɫɟɚɧɫ ɦɨɞɟɥɢɪɨɜɚɧɢɹ. Ɋɟɡɭɥɶɬɚɬɵ ɤɚɠɞɨɝɨ ɫɟɚɧɫɚ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɫɨɫɬɚɜɥɹɸɬ ɫɬɪɨɤɭ ɬɚɛɥɢɰɵ; ɩɪɢɜɟɞɟɧɵ ɮɚɤɬɢɱɟɫɤɢɟ ɞɚɧɧɵɟ ɧɟɫɤɨɥɶɤɢɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ.
26
ə.ɂ. ɋɬɪɟɤɚɥɨɜɫɤɚɹ
Įȕ ɮɢɥɶɬɪ
ĮȕȖ ɮɢɥɶɬɪ
ɮɢɥɶɬɪ Ʉɚɥɦɚɧɚ
ɇɨɦɟɪ ɮɚɤɬɨɪɚ, k
1
2
3
move_1
56382663,6300
7872007,2640
1587014,2730
move_2
60198448,1254
8342543,5941
1846487,1435
move_3
63598740,9743
8764831,5148
2015496,0156
…
…
…
…
move_8
140695471,1640
19354865,1573
5846529,6812
move_1
51756849,7145
7156834,1582
1284599,3324
move_2
54268311,2184
7294693,1462
1176481,1427
move_3
57154930,1540
7712943,9411
1469810,1610
…
…
…
…
move_8
133847021,3084
18481023,3855
4827691,0211
move_1
49871247,3415
6891275,3694
1108495,1496
move_2
51486928,1560
7024569,2313
1194783,0115
move_3
52147801,1109
7214568,1694
1247168,1978
…
…
…
…
move_8
135198731,5972
18972004,9836
5815722,4878
1981337497,9005
300024555,5715
79684790,9806
ɋɚɦɨɥɟɬɵ
Ɋɚɤɟɬɵ
Ɂɧɚɱɟɧɢɟ ɮɚɤɬɨɪɚ
ȼɟɪɬɨɥɟɬɵ
ɋɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɨɬɤɥɨɧɟɧɢɣ xik (ɤɜ.ɦ)
Ɍɚɛɥɢɰɚ
ɋɭɦɦɚ x.k
Ⱦɚɧɧɵɟ ɩɪɢɜɟɞɟɧɵ ɱɚɫɬɢɱɧɨ ɫ ɬɟɦ, ɱɬɨɛɵ ɧɟ ɡɚɝɪɨɦɨɠɞɚɬɶ ɫɬɚɬɶɸ.
ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ ɬɪɢ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ l = 3 , ɨɛɳɟɟ ɱɢɫɥɨ ɧɚɛɥɸɞɟɧɢɣ
n = 72 , nk = 24 , ɭɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ Į = 0,01.
ȼɵɱɢɫɥɹɟɦ ɜɵɛɨɪɨɱɧɨɟ ɡɧɚɱɟɧɢɟ ɫɬɚɬɢɫɬɢɤɢ ɩɨ ɜɵɲɟɩɪɢɜɟɞɟɧɧɨɦɭ ɚɥɝɨɪɢɬɦɭ:
Fɜ =
Q1 / (l − 1)
Q /2
= 1
≈ 0,807004.
Q2 / (n − l ) Q2 / 69
F0,99 (2, 69) ≈ F0,99 (2, 60) = 4,98. Ɍɚɤ ɤɚɤ Fɜ = 0,81 < 4,98 , ɬɨ ɝɢɩɨɬɟɡɚ H 0 ɨ ɧɟɪɚɡɥɢɱɟɧɢɢ
ɮɢɥɶɬɪɨɜ ɧɟ ɩɪɨɬɢɜɨɪɟɱɢɬ ɪɟɡɭɥɶɬɚɬɚɦ ɧɚɛɥɸɞɟɧɢɣ.
Ɋɚɡɭɦɟɟɬɫɹ, ɮɨɪɦɚɬ ɫɬɚɬɶɢ ɧɟ ɩɨɡɜɨɥɹɟɬ ɧɚɦ ɩɪɢɜɟɫɬɢ ɜ ɩɨɥɧɨɦ ɨɛɴɟɦɟ ɱɢɫɥɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ, ɩɨɥɭɱɟɧɧɵɟ ɩɪɢ ɦɨɞɟɥɢɪɨɜɚɧɢɢ ɩɨɥɟɬɚ ɫ ɮɢɤɫɚɰɢɟɣ ɧɟɫɤɨɥɶɤɢɯ ɫɨɬɟɧ ɬɨɱɟɤ. ȼ ɪɟɚɥɶɧɨɣ
ɫɢɬɭɚɰɢɢ: ɞɚɥɶɧɨɫɬɶ ɨɛɧɚɪɭɠɟɧɢɹ ɰɟɥɢ ɪɚɞɢɨɥɨɤɚɬɨɪɨɦ – 50 ɤɦ, ɰɢɤɥ ɨɛɡɨɪɚ ɪɚɞɢɨɥɨɤɚɬɨɪɚ – 5 ɫ.
ɋɪɚɜɧɢɬɟɥɶɧɚɹ ɨɰɟɧɤɚ ɮɢɥɶɬɪɨɜ…
27
4. ȼɕȼɈȾɕ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɟ ɪɚɫɱɟɬɵ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɩɪɢ ɜɨɫɶɦɢɤɪɚɬɧɨɦ ɫɤɚɧɢɪɨɜɚɧɢɢ ɫɞɟɥɚɬɶ ɧɚɞɟɠɧɵɣ ɜɵɜɨɞ ɨ ɪɚɡɥɢɱɟɧɢɢ ɮɢɥɶɬɪɨɜ ɧɟ ɭɞɚɟɬɫɹ. ɗɬɨ ɢ ɧɟ ɭɞɢɜɢɬɟɥɶɧɨ, ɩɨɬɨɦɭ ɱɬɨ ɧɚɞɟɠɧɵɟ ɜɵɜɨɞɵ ɨ ɩɪɨɜɟɪɤɟ ɝɢɩɨɬɟɡ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɥɢɲɶ ɩɪɢ ɡɧɚɱɢɬɟɥɶɧɨɦ ɨɛɴɺɦɟ
ɫɬɚɬɢɫɬɢɤɢ. ȼɚɠɧɨ, ɱɬɨ ɩɪɢɜɟɞɺɧɧɵɟ ɪɚɫɱɟɬɵ ɫɞɟɥɚɧɵ ɧɚ ɦɨɞɟɥɢ ɦɚɥɨɦɚɧɟɜɪɢɪɭɸɳɟɣ ɰɟɥɢ.
ȼ ɭɫɥɨɜɢɹɯ ɠɟ ɪɟɡɤɢɯ ɦɚɧɟɜɪɨɜ ɮɢɥɶɬɪ Ʉɚɥɦɚɧɚ ɨɛɵɱɧɨ ɞɚɟɬ ɡɧɚɱɢɦɨ ɛɨɥɟɟ ɬɨɱɧɵɟ ɡɧɚɱɟɧɢɹ.
ɉɨɞɪɨɛɧɵɟ ɤɚɱɟɫɬɜɟɧɧɵɟ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɜɵɜɨɞɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɮɢɥɶɬɪɨɜ ɜ ɪɚɦɤɚɯ ɪɟɚɥɶɧɨɝɨ ɜɪɟɦɟɧɢ ɛɭɞɭɬ ɨɩɭɛɥɢɤɨɜɚɧɵ ɜ ɫɥɟɞɭɸɳɢɯ ɪɚɛɨɬɚɯ.
ɅɂɌȿɊȺɌɍɊȺ
1. Ʉɭɡɶɦɢɧ ɋ.Ɂ. Ɉɫɧɨɜɵ ɬɟɨɪɢɢ ɰɢɮɪɨɜɨɣ ɨɛɪɚɛɨɬɤɢ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ. – Ɇ.: ɋɨɜɟɬɫɤɨɟ ɪɚɞɢɨ, 1974.
2. Ʉɭɡɶɦɢɧ ɋ.Ɂ. Ɉɫɧɨɜɵ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦ ɰɢɮɪɨɜɨɣ ɨɛɪɚɛɨɬɤɢ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ. – Ɇ.:
Ɋɚɞɢɨ ɢ ɫɜɹɡɶ, 1986.
3. Ɏɚɪɢɧɚ Ⱥ., ɋɬɭɞɟɪ Ɏ. ɐɢɮɪɨɜɚɹ ɨɛɪɚɛɨɬɤɚ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ: ɩɟɪ. ɫ ɚɧɝɥ. – Ɇ.: Ɋɚɞɢɨ ɢ
ɫɜɹɡɶ, 1993.
4. Ȼɚɤɭɥɟɜ ɉ.Ⱥ. Ɋɚɞɢɨɥɨɤɚɰɢɨɧɧɵɟ ɫɢɫɬɟɦɵ: ɭɱɟɛ. ɞɥɹ ɜɭɡɨɜ. – Ɇ.: Ɋɚɞɢɨɬɟɯɧɢɤɚ, 2004.
5. ȿɥɢɫɟɟɜɚ ɂ.ɂ. Ɉɛɳɚɹ ɬɟɨɪɢɹ ɫɬɚɬɢɫɬɢɤɢ: ɭɱɟɛ. ɞɥɹ ɜɭɡɨɜ. – Ɇ.: Ɏɢɧɚɧɫɵ ɢ ɫɬɚɬɢɫɬɢɤɚ, 2009.
6. ɋɬɪɟɤɚɥɨɜɫɤɚɹ ə.ɂ. ɂɦɢɬɚɰɢɨɧɧɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɩɨɬɨɤɚ ɪɚɡɧɨɯɚɪɚɤɬɟɪɧɵɯ ɜɨɡɞɭɲɧɵɯ ɰɟɥɟɣ ɜ ɭɫɥɨɜɢɹɯ ɨɞɧɨɜɪɟɦɟɧɧɨɝɨ ɫɨɩɪɨɜɨɠɞɟɧɢɹ. ɇɚɭɱɧɵɣ ɜɟɫɬɧɢɤ ɆȽɌɍ ȽȺ, 2013. ʋ 194. ɋ. 79-84.
7. ɋɬɪɟɤɚɥɨɜɫɤɚɹ ə.ɂ. ȼɵɛɨɪ ɨɩɬɢɦɚɥɶɧɨɝɨ ɮɢɥɶɬɪɚ ɜ ɫɥɭɱɚɟ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɝɨ ɫɨɩɪɨɜɨɠɞɟɧɢɹ ɜɨɡɞɭɲɧɨɣ ɰɟɥɢ ɫ ɞɢɜɟɪɫɢɮɢɰɢɪɨɜɚɧɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ. ɇɚɭɱɧɵɣ ɜɟɫɬɧɢɤ ɆȽɌɍ ȽȺ, 2013. ʋ 194. ɋ. 85-89.
COMPARATIVE EVALUATION OF FILTERS USED IN TRACKING AIR TARGETS
Strekalovskaya Y.I.
Using an imitation model for a flow of heterogeneous air targets the comparative assessment of the Įȕ, ĮȕȖ and the
Kalman filters efficiency is evaluated. In the case of slightly maneuvering target the difference in filters’ efficiency is statistically insignificant; in the case of sharp maneuvering the Kalman filter is significantly more precise.
Keywords: secondary data processing, Kalman filter, Įȕ filter, ĮȕȖ filter, single-factor dispersion analysis.
REFERENCES
1. Kuz'min S.Z. Osnovy teorii cifrovoj obrabotki radiolokacionnoj informacii (Foundations of the Theory for Numeric Processing of Radio-locating Data), Moscow, Sovetskoe radio, 1974.
2. Kuz'min S.Z. Osnovy proektirovanija sistem cifrovoj obrabotki radiolokacionnoj informacii (Foundations for Design of Numeric Processing of Radio-locating Data), M.: Radio i svjaz', 1986.
3. Farina A., Studer F. Cifrovaja obrabotka radiolokacionnoj informacii. Per. s angl. (Numerical processing of of
Radio-locating Data), Moscow, Radio i svjaz', 1993.
4. Bakulev P.A. Radiolokacionnye sistemy: Uchebnik dlja vuzov (Radio-locating Systems. University textbook).
Moscow, Radiotehnika, 2004.
5. Eliseeva I.I. Obshhaja teorija statistiki: Uchebnik dlja vuzov (General Theory of Statistics. University textbook),
Moscow, Finansy i statistika, 2009, 657 p.
6. Strekalovskaya Y.I. Nauchnyj Vestnik MGTUCA, 2013, no 194, pp. 79 - 84.
7. Strekalovskaya Y.I. Nauchnyj Vestnik MGTUCA, 2013, no 194, pp. 85- 89.
ɋɜɟɞɟɧɢɹ ɨɛ ɚɜɬɨɪɟ
ɋɬɪɟɤɚɥɨɜɫɤɚɹ əɧɚ ɂɧɧɨɤɟɧɬɶɟɜɧɚ, ɨɤɨɧɱɢɥɚ ɆȽɌɍ ȽȺ (2011), ɚɜɬɨɪ 2 ɧɚɭɱɧɵɯ ɪɚɛɨɬ, ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɜɬɨɪɢɱɧɚɹ ɨɛɪɚɛɨɬɤɚ ɪɚɞɢɨɥɨɤɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ.
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