смод

ɬɢɧɧɚ. Ɍɨɝɞɚ ɩɪɚɜɢɥɨ ɪɟɲɟɧɢɹ G ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɧɚɛɨɪɨɦ ɪɟɲɟɧɢɣ ɢ ɜɨɡɦɨɠɧɵɦɢ ɪɟɡɭɥɶɬɚɬɚɦɢ ɧɚɛɥɸɞɟɧɢɣ, ɢɥɢ ɩɪɨɫɬɪɚɧɫɬɜɨɦ ɜɵɛɨɪɨɤ. Ɍɨ ɟɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɨ ɜɵɛɨɪɨɤ G ɪɚɡɞɟɥɹɟɬɫɹ ɧɚ m+1 ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɨɛɥɚɫɬɟɣ (ɢɥɢ ɩɨɞɩɪɨɫɬɪɚɧɫɬɜ) G0,..., Gm, ɬɨɝɞɚ ɩɪɚɜɢɥɨ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɪɟɲɟɧɢɹɦɢ Jk ɢ ɨɛɥɚɫɬɹɦɢ Gj(X). ɉɪɢ ɷɬɨɦ ɧɟɨɛɯɨɞɢɦɨ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɩɪɚɜɢɥɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ ɭɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɞɨ ɩɪɨɜɟɞɟɧɢɹ ɧɚɛɥɸɞɟɧɢɣ.

ɉɪɚɜɢɥɚ ɪɟɲɟɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɦɢ (ɢɥɢ ɧɟɪɚɧɞɨɦɢɡɢɪɨɜɚɧɧɵɦɢ). ɉɪɢ ɷɬɨɦ ɞɚɧɧɨɣ ɨɛɥɚɫɬɢ ɜɫɟɝɞɚ ɩɪɢɩɢɫɵɜɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɨɟ ɪɟɲɟɧɢɟ. ɇɨ ɩɪɚɜɢɥɨ ɦɨɠɟɬ ɛɵɬɶ ɢ ɪɚɧɞɨɦɢɡɢɪɨɜɚɧɧɵɦ. ɉɪɢ ɷɬɨɦ ɞɥɹ ɡɚɞɚɧɧɵɯ ɜɵɛɨɪɨɱɧɵɯ ɡɧɚɱɟɧɢɣ ɞɨɩɭɫɤɚɟɬɫɹ ɜɵɛɨɪ ɨɞɧɨɝɨ ɢɡ ɧɟɫɤɨɥɶɤɢɯ ɪɟɲɟɧɢɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɧɟɤɨɬɨɪɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɜɟɪɨɹɬɧɨɫɬɟɣ. ɗɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɟɫɬɶ ɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɪɟɲɟɧɢɣ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɣ ɜɵɛɨɪɤɟ. ɇɚɩɪɢɦɟɪ, ɜ ɡɚɞɚɱɚɯ ɨɛɧɚɪɭɠɟɧɢɹ ɢɧɨɝɞɚ, ɟɫɥɢ ɜɵɛɨɪɤɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɝɪɚɧɢɰɟ ɦɟɠɞɭ ɨɛɥɚɫɬɹɦɢ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɣ, ɩɪɨɢɡɜɨɞɢɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɵɣ ɪɨɡɵɝɪɵɲ, ɬ.ɟ. ɭɫɥɨɜɧɨ ɩɨɥɨɜɢɧɭ ɩɨɩɚɞɚɧɢɣ ɧɚ ɝɪɚɧɢɰɭ ɩɪɢɩɢɫɵɜɚɸɬ ɨɞɧɨɦɭ ɪɟɲɟɧɢɸ, ɚ ɩɨɥɨɜɢɧɭ - ɞɪɭɝɨɦɭ.

Ɏɭɧɤɰɢɢ ɩɨɬɟɪɶ ɢ ɤɪɢɬɟɪɢɢ ɤɚɱɟɫɬɜɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ

Ʌɸɛɨɟ ɩɪɚɜɢɥɨ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ ɩɪɢ ɧɚɥɢɱɢɢ ɷɥɟɦɟɧɬɚ ɫɥɭɱɚɣɧɨɫɬɢ ɧɟɢɡɛɟɠɧɨ ɫɜɹɡɚɧɨ ɫ ɨɲɢɛɨɱɧɵɦɢ ɪɟɲɟɧɢɹɦɢ. ɉɨɫɥɟɞɫɬɜɢɹ ɧɟɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵɦɢ. Ⱥɧɚɥɢɬɢɱɟɫɤɢ ɷɬɨ ɭɫɥɨɜɢɟ ɭɱɢɬɵɜɚɟɬɫɹ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɣ ɮɭɧɤɰɢɟɣ ɩɨɬɟɪɶ, ɤɨɬɨɪɚɹ ɩɪɢɩɢɫɵɜɚɟɬ ɨɩɪɟɞɟɥɟɧɧɵɟ ɩɨɬɟɪɢ ɤɚɠɞɨɦɭ ɨɲɢɛɨɱɧɨɦɭ ɪɟɲɟɧɢɸ. Ɉɞɧɚɤɨ ɦɨɝɭɬ ɛɵɬɶ ɜɜɟɞɟɧɵ ɢ ɜɟɥɢɱɢɧɵ ɜɵɢɝɪɵɲɟɣ ɨɬ ɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ (ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɩɨɬɟɪɢ). Ɍɨɝɞɚ ɭɫɥɨɜɧɵɣ ɪɢɫɤ ɨɬ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ Jj ɟɫɬɶ

m

rj= ¦ ɉjkp{(x1,..., xn) Gk/sj},

k 1

ɝɞɟ p{(x1,..., xn) Gk/sj} - ɭɫɥɨɜɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɩɨɩɚɞɚɧɢɹ ɜɵɛɨɪɤɢ X ɜ ɨɛɥɚɫɬɶ Gk, ɟɫɥɢ ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɫɬɨɹɧɢɟ sj. ɍɫɪɟɞɧɹɹ ɭɫɥɨɜɧɭɸ ɮɭɧɤɰɢɸ ɪɢɫɤɚ ɩɨ ɜɫɟɦ ɜɨɡɦɨɠɧɵɦ ɫɨɫɬɨɹɧɢɹɦ sj, ɩɨɥɭɱɢɦ ɫɪɟɞɧɸɸ ɮɭɧɤɰɢɸ ɪɢɫɤɚ R. ɗɬɚ ɮɭɧɤɰɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɧɹɬɚ ɡɚ ɤɪɢɬɟɪɢɣ ɤɚɱɟɫɬɜɚ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɩɬɢɦɚɥɶɧɵɦ ɩɪɚɜɢɥɨɦ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ ɡɚɞɚɟɬɫɹ ɨɞɢɧ ɫɩɨɫɨɛ ɪɚɡɛɢɟɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɵɛɨɪɨɤ ɧɚ ɧɟɩɟɪɟɫɟɤɚɸɳɢɟɫɹ ɨɛɥɚɫɬɢ, ɤɨɬɨɪɵɣ ɨɛɟɫɩɟɱɢɜɚɟɬ ɩɪɢ ɞɥɢɬɟɥɶɧɨɦ ɟɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɦɢɧɢɦɚɥɶɧɵɟ ɫɪɟɞɧɢɟ ɩɨɬɟɪɢ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɚɤɨɣ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɟɣ ɨɲɢɛɨɱɧɵɟ ɪɟɲɟɧɢɹ ɩɪɢɜɨɞɹɬ ɤ ɩɨɩɚɞɚɧɢɸ ɜɵɛɨɪɤɢ ɜ ɨɛɥɚɫɬɶ, ɧɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɢɫɯɨɞɧɨɦɭ ɫɨɫɬɨɹɧɢɸ ɢɡɭɱɚɟɦɨɝɨ ɹɜɥɟɧɢɹ; ɞɥɹ ɡɚɞɚɱɢ ɨɛɧɚɪɭɠɟɧɢɹ - ɤ ɩɨɩɚɞɚɧɢɸ ɜ ɨɛɥɚɫɬɶ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɨɬɫɭɬɫɬɜɢɸ ɫɢɝɧɚɥɚ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɫɢɝɧɚɥ ɧɚ ɫɚɦɨɦ ɞɟɥɟ ɩɟɪɟɞɚɟɬɫɹ, ɢ ɧɚɨɛɨɪɨɬ.

Ɉɩɬɢɦɚɥɶɧɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ, ɦɢɧɢɦɢɡɢɪɭɸɳɟɟ ɫɪɟɞɧɸɸ ɮɭɧɤɰɢɸ ɪɢɫɤɚ, ɧɚɡɵɜɚɸɬ ɛɚɣɟɫɨɜɫɤɢɦ ɪɟɲɟɧɢɟɦ, ɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɟɦɭ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɪɟɞɧɟɝɨ ɪɢɫɤɚ - ɛɚɣɟɫɨɜɫɤɢɦ ɪɢɫɤɨɦ.

Ɉɞɧɚɤɨ ɛɚɣɟɫɨɜɫɤɨɟ ɩɪɚɜɢɥɨ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ ɨɛɥɚɞɚɟɬ ɫɭɳɟɫɬɜɟɧɧɵɦ ɧɟɞɨɫɬɚɬɤɨɦ - ɬɪɟɛɭɟɬɫɹ ɚɩɪɢɨɪɧɚɹ ɢɧɮɨɪɦɚɰɢɹ ɨɛ ɚɩɪɢɨɪɧɵɯ ɜɟɪɨɹɬɧɨɫɬɹɯ ɢ ɫɬɨɢɦɨɫɬɹɯ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɣ.

ȿɫɥɢ ɚɩɪɢɨɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɨɫɬɨɹɧɢɣ ɧɟ ɢɡɜɟɫɬɧɨ, ɬɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɩɬɢɦɚɥɶɧɨɝɨ ɪɟɲɚɸɳɟɝɨ ɩɪɚɜɢɥɚ ɢɫɩɨɥɶɡɭɸɬ ɭɫɥɨɜɧɵɣ ɪɢɫɤ, ɚ ɢɦɟɧɧɨ ɟɝɨ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ, ɚ ɡɚɬɟɦ ɦɢɧɢɦɢɡɢɪɭɸɬ ɟɝɨ, ɜɵɛɢɪɚɹ ɧɚɢɥɭɱɲɢɦ ɨɛɪɚɡɨɦ ɝɪɚɧɢɰɵ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɣ. Ɍɚɤɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ ɧɚɡɵɜɚɸɬ ɦɢɧɢɦɚɤɫɧɵɦ. ɗɬɨ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ ɞɚɟɬ ɭɜɟɪɟɧɧɨɫɬɶ, ɱɬɨ ɫɪɟɞɧɢɟ ɩɨɬɟɪɢ ɧɟ ɛɭɞɭɬ ɛɨɥɶɲɟ ɧɟɤɨɬɨɪɨɣ ɜɟɥɢɱɢɧɵ. Ɉɞɧɚɤɨ ɛɵɜɚɸɬ ɬɚɤɢɟ ɫɢɬɭɚɰɢɢ, ɤɨɝɞɚ ɬɚɤɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ ɩɪɟɞɥɚɝɚɟɬ ɫɥɢɲɤɨɦ ɨɫɬɨɪɨɠɧɵɣ ɩɨɞɯɨɞ ɤ ɩɪɢɧɹɬɢɸ ɪɟɲɟɧɢɹ.

Ȼɚɣɟɫɨɜɫɤɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ ɩɪɢ ɩɪɨɫɬɨɣ ɝɢɩɨɬɟɡɟ ɢ ɩɪɨɫɬɨɣ ɚɥɶɬɟɪɧɚɬɢɜɟ

Ⱦɚɧɵ: ɝɢɩɨɬɟɡɚ H0 ɢ ɚɥɶɬɟɪɧɚɬɢɜɚ H1. Ɋɚɫɩɪɟɞɟɥɟɧɢɹ ɜɵɛɨɪɨɱɧɵɯ ɡɧɚɱɟɧɢɣ fn(x1,...,xn/s0) ɢ fn(x1,...,xn/s1) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɋɟɲɟɧɢɹ J0 ɢ J1. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɬɨɥɶɤɨ ɨɞɧɚ ɝɢɩɨɬɟɡɚ, ɬɚɤ ɤɚɤ ɨɬɤɥɨɧɟɧɢɟ ɝɢɩɨɬɟɡɵ ɩɪɢɜɨɞɢɬ ɤ ɩɪɢɧɹɬɢɸ ɚɥɶɬɟɪɧɚɬɢɜɵ. Ɉɛɥɚɫɬɶ ɜɵɛɨɪɨɤ ɪɚɡɛɢɜɚɟɬɫɹ ɪɟɲɚɸɳɢɦ ɩɪɚɜɢɥɨɦ ɧɚ ɞɜɟ ɨɛɥɚɫɬɢ G1 ɢ G0. ɉɨɩɚɞɚɧɢɟ ɜ ɨɛɥɚɫɬɶ G1 ɩɪɢɜɨɞɢɬ ɤ ɩɪɢɧɹɬɢɸ ɪɟɲɟɧɢɹ ɜ ɩɨɥɶɡɭ ɝɢɩɨɬɟɡɵ H1, ɚ ɡɧɚɱɢɬ, ɤ ɪɟɲɟɧɢɸ G0. ɉɪɢ ɷɬɨɦ ɨɛɥɚɫɬɶ G0 ɧɚɡɵɜɚɸɬ ɞɨɩɭɫɬɢɦɨɣ, ɚ ɨɛɥɚɫɬɶ G1 - ɤɪɢɬɢɱɟɫɤɨɣ. ɍɪɚɜɧɟɧɢɟ ɩɨɜɟɪɯɧɨɫɬɢ, ɪɚɡɞɟɥɹɸɳɟɣ ɷɬɢ ɨɛɥɚɫɬɢ, ɹɜɥɹɟɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɜɵɪɚɠɟɧɢɟɦ ɩɪɚɜɢɥɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ.

Ɉɲɢɛɤɢ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɪɨɞɚ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɩɨɩɚɞɚɧɢɹɦ ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɛɥɚɫɬɢ. Ɉɲɢɛɤɚ ɩɟɪɜɨɝɨ ɪɨɞɚ - ɩɨɩɚɞɚɧɢɟ ɜ ɨɛɥɚɫɬɶ G1, ɤɨɝɞɚ ɢɡɭɱɚɟɦɨɟ ɹɜɥɟɧɢɟ ɧɚɯɨɞɢɬɫɹ ɜ ɫɨɫɬɨɹɧɢɢ s0. Ɉɲɢɛɤɚ ɜɬɨɪɨɝɨ ɪɨɞɚ - ɧɚɨɛɨɪɨɬ. Ɉɲɢɛɤɚ ɩɟɪɜɨɝɨ ɪɨɞɚ D ɪɚɜɧɚ

ØØ... Øfn(x1,...,xn/s0)dx1...dxn=p{J1/H0}.

G1

Ɉɲɢɛɤɚ ɜɬɨɪɨɝɨ ɪɨɞɚ E ɟɫɬɶ ØØ... Øfn(x1,...,xn/s1)dx1...dxn=p{J0/H1}. ȼɟɪɨɹɬɧɨɫɬɶ

G0

ɨɲɢɛɤɢ ɩɟɪɜɨɝɨ ɪɨɞɚ ɧɚɡɵɜɚɸɬ ɢɧɨɝɞɚ ɭɪɨɜɧɟɦ ɡɧɚɱɢɦɨɫɬɢ, ɚ (1-E) - ɦɨɳɧɨɫɬɶɸ ɤɪɢɬɟɪɢɹ (ɬɟɪɦɢɧɵ ɢɡ ɦɚɬɟɦɚɬɢɤɢ).

ȼɜɟɞɟɦ ɮɭɧɤɰɢɸ ɩɨɬɟɪɶ, ɤɨɬɨɪɚɹ ɩɪɢɩɢɫɵɜɚɟɬ ɤɚɠɞɨɣ ɤɨɦɛɢɧɚɰɢɢ ɢɫɯɨɞɧɵɯ ɫɨɫɬɨɹɧɢɣ ɢ ɩɪɢɧɹɬɵɯ ɪɟɲɟɧɢɣ ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɩɥɚɬɭ. ɍɞɨɛɧɨ ɮɭɧɤɰɢɸ

ɉ01>ɉ00, ɉ10 > ɉ11. ɋɪɟɞɧɢɣ ɪɢɫɤ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɬɚɤɨɦ ɜɢɞɟ: R=p0r0+p1r1, ɝɞɟ ɭɫɥɨɜɧɵɟ ɪɢɫɤɢ ɩɪɢɧɢɦɚɟɦɵɯ ɪɟɲɟɧɢɣ ɪɚɜɧɵ:

r0 =ɉ00(1-D)+ɉ01D; r1 =ɉ10E+ɉ11(1-E).

ɋɪɟɞɧɢɣ ɪɢɫɤ ɩɨɥɭɱɚɟɦ ɩɨɞɫɬɚɧɨɜɤɨɣ ɭɫɥɨɜɧɵɯ ɪɢɫɤɨɜ ɜ ɮɨɪɦɭɥɭ ɞɥɹ R. ɉɨɞɫɬɚɜɢɦ ɜ ɷɬɭ ɮɨɪɦɭɥɭ ɜɵɪɚɠɟɧɢɹ ɞɥɹ D ɢ E, ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ, ɭɱɢɬɵɜɚɹ, ɱɬɨ ɦɢɧɢɦɭɦ ɫɪɟɞɧɟɝɨ ɪɢɫɤɚ ɛɭɞɟɬ ɩɪɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɨɣ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ:

ɗɬɨ ɢ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ, ɪɚɡɞɟɥɹɸɳɟɟ ɤɪɢɬɢɱɟɫɤɭɸ ɢ ɞɨɩɭɫɬɢɦɭɸ ɨɛɥɚɫɬɢ. Ʌɟɜɭɸ ɱɚɫɬɶ ɧɟɪɚɜɟɧɫɬɜɚ ɧɚɡɵɜɚɸɬ ɨɛɨɛɳɟɧɧɵɦ ɨɬɧɨɲɟɧɢɟɦ ɩɪɚɜɞɨɩɨɞɨ-

ɛɢɹ. ɉɨɪɨɝɨɜɵɣ ɭɪɨɜɟɧɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɚɜɨɣ ɱɚɫɬɶɸ ɷɬɨɝɨ ɠɟ ɧɟɪɚɜɟɧɫɬɜɚ. ɉɨɥɭɱɟɧɧɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ ɢ ɧɚɡɵɜɚɸɬ ɛɚɣɟɫɨɜɫɤɢɦ, ɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɫɪɟɞɧɢɣ ɪɢɫɤ - ɛɚɣɟɫɨɜɫɤɢɦ ɪɢɫɤɨɦ.

Ɉɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɧɟɨɬɪɢɰɚɬɟɥɶɧɭɸ ɜɟɥɢɱɢɧɭ, ɤɨɬɨɪɚɹ ɩɨɥɭɱɚɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɵɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ n ɋȼ ɢ ɨɬɨɛɪɚɠɚɟɬ n- ɦɟɪɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ ɧɚ ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɨɫɶ (ɨɞɧɨɦɟɪɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ). ɋɬɪɭɤɬɭɪɚ ɷɬɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɢ ɹɜɥɹɟɬɫɹ ɫɬɪɭɤɬɭɪɨɣ ɩɪɢɟɦɧɨ-ɪɟɲɚɸɳɟɝɨ ɭɫɬɪɨɣɫɬɜɚ, ɧɚ ɜɵɯɨɞɟ ɤɨɬɨɪɨɝɨ ɜɤɥɸɱɚɟɬɫɹ ɩɨɪɨɝɨɜɨɟ ɭɫɬɪɨɣɫɬɜɨ, ɫɪɚɜɧɢɜɚɸɳɟɟ ɪɟɡɭɥɶɬɚɬ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫ ɩɨɪɨɝɨɜɵɦ ɭɪɨɜɧɟɦ. ȿɫɥɢ ɷɬɨɬ ɭɪɨɜɟɧɶ ɩɪɟɜɵɲɚɟɬɫɹ, ɬɨ ɜɵɧɨɫɢɬɫɹ ɪɟɲɟɧɢɟ ɨ ɧɚɥɢɱɢɢ ɫɢɝɧɚɥɚ ɜɨ ɜɯɨɞɧɨɣ ɫɦɟɫɢ.

ȿɫɥɢ ɫɢɝɧɚɥ ɢ ɫɦɟɫɶ ɫɢɝɧɚɥɚ ɢ ɩɨɦɟɯɢ ɪɚɜɧɨɜɟɪɨɹɬɧɵ, ɬɨ ɩɪɨɰɟɞɭɪɚ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɮɨɪɦɢɪɨɜɚɧɢɢ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɢ ɫɪɚɜɧɟɧɢɢ ɟɝɨ ɫ ɩɨɪɨɝɨɜɵɦ ɭɪɨɜɧɟɦ. Ɋɚɡɥɢɱɧɵɟ ɤɪɢɬɟɪɢɢ ɨɬɥɢɱɚɸɬɫɹ ɥɢɲɶ ɜɵɛɨɪɨɦ ɷɬɨɝɨ ɩɨɪɨɝɨɜɨɝɨ ɭɪɨɜɧɹ.

Ʉɪɢɬɟɪɢɣ ɇɟɣɦɚɧɚ-ɉɢɪɫɨɧɚ

ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɢɧɮɨɪɦɚɰɢɢ ɨ ɩɨɬɟɪɹɯ ɢ ɜɟɪɨɹɬɧɨɫɬɹɯ ɫɨɫɬɨɹɧɢɹ ɢɡɭɱɚɟɦɨɝɨ ɨɛɴɟɤɬɚ ɩɪɢɦɟɧɹɸɬ ɷɬɨɬ ɤɪɢɬɟɪɢɣ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɧɢɦ ɜɵɛɢɪɚɟɬɫɹ ɬɚɤɨɟ ɩɪɚɜɢɥɨ, ɤɨɬɨɪɨɟ ɨɛɟɫɩɟɱɢɜɚɟɬ ɦɢɧɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɭɸ ɜɟɥɢɱɢɧɭ E ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɤ ɜɬɨɪɨɝɨ ɪɨɞɚ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɨɲɢɛɨɤ ɩɟɪɜɨɝɨ ɪɨɞɚ ɧɟ ɩɪɟɜɵɲɚɟɬ ɧɚɩɟɪɟɞ ɡɚɞɚɧɧɭɸ ɜɟɥɢɱɢɧɭ, ɬ.ɟ. ɤɪɢɬɟɪɢɣ ɇɟɣɦɚɧɚ - ɉɢɪɫɨɧɚ ɫɪɟɞɢ ɜɫɟɯ ɪɟɲɚɸɳɢɯ ɩɪɚɜɢɥ ɨɛɟɫɩɟɱɢɜɚɟɬ ɧɚɢɛɨɥɶɲɭɸ ɦɨɳɧɨɫɬɶ 1-E, ɟɫɥɢ ɭɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ ɧɟ ɩɪɟɜɨɫɯɨɞɢɬ D.

ɉɪɢ ɷɬɨɦ ɩɨɪɨɝɨɜɨɟ ɡɧɚɱɟɧɢɟ C ɜɵɛɢɪɚɟɬɫɹ ɢɡ ɫɥɟɞɭɸɳɟɝɨ ɭɫɥɨɜɢɹ:

p{l(X)tC/s0}=D,

ɝɞɟ l(X) = fn(X/s1)/fn(X/s0). ȿɫɥɢ ɢɡɜɟɫɬɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ l(X), ɬɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɚɜɢɥɶɧɵɯ ɢ ɧɟɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɤɚɤ ɩɥɨɳɚɞɢ ɨɝɪɚɧɢɱɟɧɧɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɩɥɨɬɧɨɫɬɹɦɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɩɨɪɨɝɨɜɵɦ ɭɪɨɜɧɟɦ ɋ. ɗɬɨɬ ɤɪɢɬɟɪɢɣ ɧɚɲɟɥ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɜ ɥɨɤɚɰɢɢ ɢ ɹɜɥɹɟɬɫɹ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɤɪɢɬɟɪɢɹ Ȼɚɣɟɫɚ.

ȼɵɱɢɫɥɟɧɢɟ ɭɫɥɨɜɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɲɢɛɨɤ

Ɉɬɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ ɨɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɩɨɞɜɟɪɝɚɟɬɫɹ ɦɨɧɨɬɨɧɧɨɦɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ Ɏ(l), ɬɨ ɫɪɚɜɧɟɧɢɟ l(X) ɫ ɩɨɪɨɝɨɜɵɦ ɡɧɚɱɟɧɢɟɦ ɋ ɷɤɜɢɜɚɥɟɧɬɧɨ ɫɪɚɜɧɟɧɢɸ Ɏ(l) = O(X) ɫ ɩɨɪɨɝɨɜɵɦ ɭɪɨɜɧɟɦ Ɏ(ɋ). Ɍɨɝɞɚ ɩɪɟɞɟɥɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɭɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ Ɏ(ɋ), ɚ ɜ ɩɨɞɵɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɨɞɫɬɚɜɥɹɟɬɫɹ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ, ɭɱɢɬɵɜɚɸɳɚɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ Ɏ(l).

ȿɫɥɢ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɜɯɨɞɧɨɣ ɜɵɛɨɪɤɢ ɹɜɥɹɸɬɫɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɦɢ, ɜ ɱɚɫɬɧɨɫɬɢ ɧɨɪɦɚɥɶɧɵɦɢ, ɬɨ ɢɫɩɨɥɶɡɭɸɬ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɪɟɲɚɸɳɟɝɨ ɩɪɚɜɢɥɚ ɥɨɝɚɪɢɮɦɢɪɨɜɚɧɢɟ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ln(l).

8.2. ɋɢɧɬɟɡ ɨɩɬɢɦɚɥɶɧɨɝɨ ɨɛɧɚɪɭɠɢɬɟɥɹ ɫɢɝɧɚɥɨɜ

Ɉɛɧɚɪɭɠɟɧɢɟ ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɨɝɨ ɫɢɝɧɚɥɚ ɧɚ ɮɨɧɟ ɧɨɪɦɚɥɶɧɨɣ ɩɨɦɟɯɢ (ɲɭɦɚ)

ɉɭɫɬɶ ɩɪɢɧɢɦɚɟɬɫɹ ɢɡɜɟɫɬɧɨɟ ɩɨ ɮɨɪɦɟ ɤɨɥɟɛɚɧɢɟ ɧɚ ɮɨɧɟ ɧɨɪɦɚɥɶɧɨɝɨ ɛɟɥɨɝɨ ɲɭɦɚ. Ɍɨɝɞɚ ɩɪɢɧɹɬɨɟ ɤɨɥɟɛɚɧɢɟ x(t) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ [(t)= P s(t) + n(t), 0dtdT, ɝɞɟ n(t) - ɧɨɪɦɚɥɶɧɵɣ ɛɟɥɵɣ ɲɭɦ; s(t) - ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɵɣ ɨɛɧɚɪɭɠɢɜɚɟɦɵɣ ɫɢɝɧɚɥ; ɩɚɪɚɦɟɬɪ P ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ "1" ɢ "0". ɉɪɢ ɷɬɨɦ ɚɩɪɢɨɪɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɫɢɝɧɚɥɚ ɦɨɝɭɬ ɛɵɬɶ ɢɡɜɟɫɬɧɵ (ɪɚɞɢɨɫɜɹɡɶ) ɢ ɧɟɢɡɜɟɫɬɧɵ (ɥɨɤɚɰɢɹ). ɇɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ, ɚ ɡɧɚɱɢɬ, ɢ ɫɬɪɭɤɬɭɪɭ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɪɢɟɦɧɨ-ɪɟɲɚɸɳɟɝɨ ɭɫɬɪɨɣɫɬɜɚ.

Ɋɚɫɫɦɨɬɪɢɦ ɬɪɢ ɨɫɧɨɜɧɵɯ ɤɪɢɬɟɪɢɹ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɬɟɯɧɢɤɟ ɨɛɧɚɪɭɠɟɧɢɹ ɫɢɝɧɚɥɨɜ: ɤɪɢɬɟɪɢɣ ɢɞɟɚɥɶɧɨɝɨ ɧɚɛɥɸɞɚɬɟɥɹ (ɤɪɢɬɟɪɢɣ Ʉɨɬɟɥɶɧɢɤɨɜɚ - Ɂɢɝɟɪɬɚ); ɤɪɢɬɟɪɢɣ ɇɟɣɦɚɧɚ - ɉɢɪɫɨɧɚ ɢ ɤɪɢɬɟɪɢɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ. Ʉɚɠɞɵɣ ɢɡ ɷɬɢɯ ɤɪɢɬɟɪɢɟɜ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɤɚɤɚɹ ɚɩɪɢɨɪɧɚɹ ɢɧɮɨɪɦɚɰɢɹ ɨ ɩɪɢɧɢɦɚɟɦɵɯ ɫɢɝɧɚɥɚɯ ɢ ɲɭɦɚɯ ɢɦɟɟɬɫɹ.

Ʉɪɢɬɟɪɢɣ ɢɞɟɚɥɶɧɨɝɨ ɧɚɛɥɸɞɚɬɟɥɹ

Ⱥɩɪɢɨɪɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɧɚɥɢɱɢɹ ɢɥɢ ɨɬɫɭɬɫɬɜɢɹ ɫɢɝɧɚɥɚ ɢɡɜɟɫɬɧɵ. ȼɯɨɞɧɚɹ ɫɦɟɫɶ ɫɢɝɧɚɥɚ ɢ ɲɭɦɚ ɢɥɢ ɨɞɧɨɝɨ ɲɭɦɚ ɩɨɞɜɟɪɝɚɟɬɫɹ ɞɢɫɤɪɟɬɢɡɰɢɢ, ɩɨɫɥɟ ɱɟɝɨ ɩɨɥɭɱɚɟɬɫɹ ɜɵɛɨɪɤɚ X ɪɚɡɦɟɪɚ n. Ⱦɢɫɤɪɟɬɧɵɟ ɡɧɚɱɟɧɢɹ ɫɢɝɧɚɥɚ, ɩɨɦɟɯɢ ɢ ɢɯ ɫɦɟɫɢ ɨɛɨɡɧɚɱɟɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ si, [i ɢ xi. ɋɱɢɬɚɟɦ ɞɥɹ ɩɪɨɫɬɨɬɵ, ɱɬɨ ɢɧɬɟɪɜɚɥ ɞɢɫɤɪɟɬɢɡɚɰɢɢ ɩɪɟɜɵɲɚɟɬ ɢɧɬɟɪɜɚɥ ɤɨɪɪɟɥɹɰɢɢ, ɚ ɡɧɚɱɢɬ, ɟɞɢɧɢɱɧɵɟ ɢɡɦɟɪɟɧɢɹ ɜɵɛɨɪɨɤ x(t) ɧɟ ɤɨɪɪɟɥɢɪɨɜɚɧɵ, ɚ ɪɚɡ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɨɦɟɯɢ ɧɨɪɦɚɥɶɧɨɟ, ɬɨ ɨɧɢ ɢ ɧɟɡɚɜɢɫɢɦɵ. ɉɨɦɟɯɚ ɹɜɥɹɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ ɫɥɭɱɚɣɧɵɯ ɩɪɨɰɟɫɫɨɦ ɫ ɧɭɥɟɜɵɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɠɢɞɚɧɢɟɦ ɢ ɢɡɜɟɫɬɧɨɣ ɞɢɫɩɟɪɫɢɟɣ, ɧɟ ɢɡɦɟɧɹɸɳɟɣɫɹ ɜɨ ɜɪɟɦɟɧɢ. Ⱦɨɛɚɜɥɟɧɢɟ ɫɢɝɧɚɥɚ ɧɟ ɢɡɦɟɧɹɟɬ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ, ɢɡɦɟɧɹɟɬɫɹ ɬɨɥɶɤɨ ɟɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɧɚ ɜɟɥɢɱɢɧɭ si. Ɂɚɩɢɲɟɦ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɨɞɧɨɦɟɪɧɵɯ ɩɥɨɬɧɨɫɬɟɣ ɜɟɪɨɹɬɧɨɫɬɟɣ ɟɞɢɧɢɱɧɵɯ ɢɡɦɟɪɟɧɢɣ ɩɪɢ

ɧɚɥɢɱɢɢ ɢ ɨɬɫɭɬɫɬɜɢɢ ɩɨɥɟɡɧɨɝɨ ɫɢɝɧɚɥɚ:

f([i/s0) = (V)-1(2S)-1/2exp{-([i-Psi)2/(2V2)}

ɩɪɢ P = 0, ɚ f([i/s1) ɩɪɢ P = 1. ɍɱɢɬɵɜɚɹ, ɱɬɨ ɞɥɹ ɧɨɪɦɚɥɶɧɵɯ ɩɪɨɰɟɫɫɨɜ ɧɟɤɨɪɪɟɥɢɪɨɜɚɧɧɨɫɬɶ ɢ ɧɟɡɚɜɢɫɢɦɨɫɬɶ ɨɞɧɨ ɢ ɬɨ ɠɟ, n-ɦɟɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ n ɨɞɧɨɦɟɪɧɵɯ ɩɥɨɬɧɨɫɬɟɣ ɜɟɪɨɹɬɧɨɫɬɟɣ. ɉɟɪɟɦɟɫɬɢɦ ɷɬɨ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜ ɫɭɦɦɭɜ ɩɨɤɚɡɚɬɟɥɟ ɷɤɫɩɨɧɟɧɬɵ, ɬɨɝɞɚ ɩɨɥɭɱɢɦ

n

f(X/P) = V-n(2S)-1/2exp{-0,5V-2 ¦ (xi-Psi)2}.

i 1

ɉɨɞɫɬɚɜɢɦ ɷɬɭ ɮɨɪɦɭɥɭ ɜ ɨɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɢ ɩɪɨɥɨɝɚɪɢɮɦɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ:

n n

-0,5V-2{ ¦ [xi-P1si]2+ ¦ [xi-P0si]2}tln(p0/p1).

i 1 i 1

ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɨɛɧɚɪɭɠɟɧɢɹ P1 =1, ɚ P0 =0. ȿɫɥɢ ɭɫɬɪɟɦɢɬɶ ɢɧɬɟɪɜɚɥ ɦɟɠɞɭ ɟɞɢɧɢɱɧɵɦɢ ɢɡɦɟɪɟɧɢɹɦɢ ɤ ɧɭɥɸ ɩɪɢ ɧɟɢɡɦɟɧɧɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ ɜɵɛɨɪɤɢ Ɍ, ɫɭɦɦɚ ɩɟɪɟɣɞɟɬ ɜ ɢɧɬɟɝɪɚɥ ɜ ɩɪɟɞɟɥɚɯ ɢɧɬɟɪɜɚɥɚ Ɍ ɢ ɦɵ ɩɨɥɭɱɢɦ ɚɥɝɨɪɢɬɦ ɨɛɧɚɪɭɠɟɧɢɹ ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɨɝɨ ɫɢɝɧɚɥɚ ɨɝɪɚɧɢɱɟɧɧɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ

T

³x(t)s(t)dt > E/2,

0

T

ɝɞɟ ȿ = ³s2(t)dt - ɷɧɟɪɝɢɹ ɨɛɧɚɪɭɠɢɜɚɟɦɨɝɨ ɫɢɝɧɚɥɚ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɨɥɭɱɟɧ-

0

ɧɨɣ ɮɨɪɦɭɥɨɣ ɩɨɥɭɱɚɟɦ ɤɨɪɪɟɥɹɰɢɨɧɧɵɣ ɢɧɬɟɝɪɚɥ, ɤɨɬɨɪɵɣ ɹɜɥɹɟɬɫɹ ɚɥɝɨɪɢɬɦɨɦ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɜɡɚɢɦɨɤɨɪɪɟɥɹɰɢɨɧɧɨɝɨ ɩɪɢɟɦɧɢɤɚ. ȼɪɟɦɟɧɧɵɟ ɞɢɚɝɪɚɦɦɵ ɧɚɩɪɹɠɟɧɢɣ ɧɚ ɜɵɯɨɞɚɯ ɭɡɥɨɜ ɩɪɢɟɦɧɢɤɚ ɩɪɢ ɨɛɧɚɪɭɠɟɧɢɢ ɪɚɞɢɨɢɦɩɭɥɶɫɚ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɢɥɥɸɫɬɪɚɰɢɟɣ ɪɚɛɨɬɵ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɪɢɟɦɧɢɤɚ.

Ʉɪɢɬɟɪɢɣ ɇɟɣɦɚɧɚ -ɉɢɪɫɨɧɚҏ

Ɉɬɥɢɱɚɟɬɫɹ ɨɬ ɩɪɟɞɵɞɭɳɟɝɨ ɬɨɥɶɤɨ ɫɩɨɫɨɛɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɪɨɝɨɜɨɝɨ ɭɪɨɜɧɹ ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ. ɗɬɨɬ ɩɨɪɨɝ ɜɵɛɢɪɚɟɬɫɹ ɢɡ ɪɟ-

f

ɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ³ f(l/s0)dl=D, ɜ ɤɨɬɨɪɨɦ f(l/s0) - ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɨɬɧɨ-

C

ɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ. ȿɫɥɢ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɢɫɯɨɞɧɨɣ ɜɵɛɨɪɤɢ ɹɜɥɹɟɬɫɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɣ, ɬɨ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɩɪɢɦɟɧɹɸɬ ɥɨɝɚɪɢɮɦɢɪɨɜɚɧɢɟ ɤɚɤ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ, ɬɚɤ ɢ ɭɪɨɜɧɹ ɋ. ȿɫɥɢ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜɵɛɨɪɤɚ ɛɨɥɶɲɨɝɨ ɪɚɡɦɟɪɚ (n ɜɟɥɢɤɨ), ɬɨ ɩɨ ɰɟɧɬɪɚɥɶɧɨɣ ɩɪɟɞɟɥɶɧɨɣ ɬɟɨɪɟɦɟ Ʌɹɩɭɧɨɜɚ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɟɣ n ɢɡɦɟɪɟɧɢɣ ɫɬɪɟɦɢɬɶɫɹ ɤ ɧɨɪɦɚɥɶɧɨɣ ɢ ɬɨɝɞɚ ɩɥɨɬɧɨɫɬɶ ɜɟɪɨɹɬ-

ɧɨɫɬɢ l ɟɫɬɶ

f1(l/s0)|(2Snm20)-1/2exp[-(l-nm10)2/(2nm20)];

f1(l/s1)|(2Snm21)-1/2exp[-(l-nm11)2/(2nm21)],

ɝɞɟ m10=m1[lnl(x)/s0], m20=m2[lnl(x)/s0]; m11=m1[lnl(x)/s1], m21= =m2[lnl(x)/s1]. ȼ

ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɸɬ ɦɟɫɬɨ ɫɥɟɞɭɸɳɢɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ:

D=1-F [(lnC-nm10)/(2Snm20)1/2] ɢ E=1-F [(lnC-nm11)/(2Snm20)1/2],

x

ɝɞɟ F(x)=(2S)-1/2 ³ exp(-t2/2)dt - ɬɚɛɥɢɱɧɵɣ ɢɧɬɟɝɪɚɥ Ʌɚɩɥɚɫɚ.

f

ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɵɣ ɚɧɚɥɢɡ (ɤɪɢɬɟɪɢɣ Ⱥ.ȼɚɥɶɞɚ)

Ɉɬɥɢɱɢɬɟɥɶɧɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ ɜɫɟɯ ɢɡɭɱɟɧɧɵɯ ɤɪɢɬɟɪɢɟɜ ɩɪɢ ɩɪɨɜɟɪɤɟ ɩɪɨɫɬɨɣ ɝɢɩɨɬɟɡɵ ɹɜɥɹɟɬɫɹ ɧɟɢɡɦɟɧɧɨɫɬɶ ɪɚɡɦɟɪɚ n ɜɵɛɨɪɤɢ X. ɂɦɟɟɬɫɹ ɞɪɭɝɨɣ ɩɨɞɯɨɞ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɩɪɚɜɢɥɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ. ɉɪɢ ɷɬɨɦ ɪɚɡɦɟɪ ɜɵɛɨɪɤɢ ɨɝɪɚɧɢɱɢɜɚɸɬ ɜ ɩɪɨɰɟɫɫɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɛɥɸɞɟɧɢɣ. ȼɧɚɱɚɥɟ ɧɚɛɥɸɞɚɸɬ ɨɞɧɨ ɟɞɢɧɫɬɜɟɧɧɨɟ ɢɡɦɟɪɟɧɢɟ x1 (ɪɚɡɦɟɪ ɜɵɛɨɪɤɢ n=1) ɢ ɩɪɢɧɢɦɚɸɬ ɨɞɧɨ ɢɡ ɬɪɟɯ ɜɨɡɦɨɠɧɵɯ ɪɟɲɟɧɢɣ: ɩɪɢɧɹɬɶ ɝɢɩɨɬɟɡɭ H0; ɨɬɜɟɪɝɧɭɬɶ ɝɢɩɨɬɟɡɭ H0 ɢɥɢ ɩɪɢɧɹɬɶ ɚɥɶɬɟɪɧɚɬɢɜɭ H1; ɩɪɨɞɨɥɠɢɬɶ ɧɚɛɥɸɞɟɧɢɹ, ɬ.ɟ. ɧɟ ɩɪɢɧɢɦɚɬɶ ɧɢ ɨɞɧɨɝɨ ɢɡ ɞɜɭɯ ɩɪɟɞɵɞɭɳɢɯ ɪɟɲɟɧɢɣ. ȿɫɥɢ ɜ ɩɪɨɰɟɫɫɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɪɢɧɢɦɚɸɬ ɨɞɧɨ ɢɡ ɞɜɭɯ ɩɟɪɜɵɯ ɜ ɫɩɢɫɤɟ ɪɟɲɟɧɢɣ, ɬɨ ɷɤɫɩɟɪɢɦɟɧɬ ɡɚɤɚɧɱɢɜɚɟɬɫɹ. ȿɫɥɢ ɩɪɢɧɢɦɚɟɬɫɹ ɬɪɟɬɶɟ ɪɟɲɟɧɢɟ, ɬɨ ɩɨɥɭɱɚɸɬ ɫɥɟɞɭɸɳɟɟ ɟɞɢɧɢɱɧɨɟ ɢɡɦɟɪɟɧɢɟ, n ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɵɦ ɞɜɭɦ ɢ ɫɧɨɜɚ ɩɪɨɰɟɞɭɪɚ ɩɨɜɬɨɪɹɟɬɫɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ n ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɋȼ. Ɉɛɥɚɫɬɶ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɣ (ɨɛɥɚɫɬɶ ɜɵɛɨɪɨɤ ɪɚɡɛɢɜɚɟɬɫɹ ɭɠɟ ɧɚ ɬɪɢ ɨɛɥɚɫɬɢ: ɞɨɩɭɫɬɢɦɭɸ, ɤɪɢɬɢɱɟɫɤɭɸ ɢ ɩɪɨɦɟɠɭɬɨɱɧɭɸ). ȼ ɤɚɱɟɫɬɜɟ ɤɪɢɬɟɪɢɹ ɩɪɢɧɢɦɚɸɬ ɦɢɧɢɦɭɦ ɫɪɟɞɧɟɝɨ ɪɚɡɦɟɪɚ ɜɵɛɨɪɤɢ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɭɪɨɜɟɧɶ ɡɧɚɱɢɦɨɫɬɢ ɧɟ ɩɪɟɜɵɲɚɟɬ D, ɚ ɦɨɳɧɨɫɬɶ ɤɪɢɬɟɪɢɹ ɧɟ ɦɟɧɶɲɟ 1-E. Ʉɚɤ ɩɨɤɚɡɚɥ Ⱥ.ȼɚɥɶɞ, ɧɚɢɥɭɱɲɢɦ ɪɟɲɚɸɳɢɦ ɩɪɚɜɢɥɨɦ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɹɜɥɹɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɟ ɩɪɚɜɢɥɨ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ. Ɉɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɫɪɚɜɧɢɜɚɟɬɫɹ ɫ ɩɨɪɨɝɨɜɵɦɢ ɭɪɨɜɧɹɦɢ C0 ɢ C1.

ɇɚɢɥɭɱɲɟɟ ɪɚɡɛɢɟɧɢɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɵɛɨɪɨɤ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦɢ ɧɟɪɚɜɟɧɫɬɜɚɦɢ:

- ɞɨɩɭɫɬɢɦɚɹ ɨɛɥɚɫɬɶ C0 < l(x1,..., xk) < C1, k=1,..., n-1;

l(x1,..., xk) < C0;

- ɤɪɢɬɢɱɟɫɤɚɹ ɨɛɥɚɫɬɶ C0 < l(x1,..., xk) < C1, k=1,..., n-1;

l(x1,..., xk) > C0;

- ɤɪɢɬɢɱɟɫɤɚɹ ɨɛɥɚɫɬɶ C0 < l(x1,..., xk) < C1, k=1,..., n.

ȼɵɱɢɫɥɟɧɢɟ ɩɨɪɨɝɨɜɵɯ ɭɪɨɜɧɟɣ C0 ɢ C1 ɜɵɡɵɜɚɟɬ ɡɧɚɱɢɬɟɥɶɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ, ɨɞɧɚɤɨ ɟɫɥɢ ɭɫɥɨɜɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɲɢɛɨɤ ɧɟ ɩɪɟɜɵɲɚɸɬ ɡɧɚɱɟɧɢɹ 0,5, ɬɨ ɩɨɪɨɝɨɜɵɟ ɡɧɚɱɟɧɢɹ ɨɩɪɟɞɟɥɹɸɬ ɢɡ ɫɥɟɞɭɸɳɢɯ ɧɟɪɚɜɟɧɫɬɜ: C1d

(1-E)/D ɢ C0tE/(1-D).

ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɤɪɢɬɟɪɢɹ ɞɨɫɬɢɝɚɟɬɫɹ ɷɤɨɧɨɦɢɹ ɪɚɡɦɟɪɚ ɜɵɛɨɪɤɢ ɜ ɫɪɟɞɧɟɦ, ɤɨɝɞɚ ɟɟ ɪɚɡɦɟɪ ɢɦɟɟɬ ɫɭɳɟɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ, ɧɚɩɪɢɦɟɪ, ɜ ɥɨɤɚɰɢɢ ɩɪɢ ɨɛɡɨɪɟ ɩɪɨɫɬɪɚɧɫɬɜɚ ɦɨɠɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɜɵɢɝɪɚɬɶ ɜɪɟɦɹ ɨɛɡɨɪɚ,

ɷɤɨɧɨɦɹ ɟɝɨ ɧɚ ɫɢɝɧɚɥɚɯ, ɤɨɬɨɪɵɟ ɨɛɧɚɪɭɠɢɜɚɸɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɛɵɫɬɪɨ ɢɡ-ɡɚ ɩɪɟɜɵɲɟɧɢɹ ɢɦɢ ɲɭɦɨɜ ɩɨ ɢɧɬɟɧɫɢɜɧɨɫɬɢ. Ɉɞɧɚɤɨ ɟɫɥɢ ɫɢɝɧɚɥ ɫɪɚɜɧɢɦ ɫ ɲɭɦɨɦ ɩɨ ɢɧɬɟɧɫɢɜɧɨɫɬɢ, ɬɨ ɬɚɤɨɟ ɧɚɛɥɸɞɟɧɢɟ ɩɪɢɜɨɞɢɬ ɤ ɧɟɭɜɟɪɟɧɧɨɦɭ ɪɟɲɟɧɢɸ, ɞɥɢɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɞɨɥɝɨ ɢ ɧɢɤɚɤɨɝɨ ɜɵɢɝɪɵɲɚ ɜ ɪɚɡɦɟɪɟ ɜɵɛɨɪɤɢ ɧɟ ɩɨɥɭɱɚɟɬɫɹ. Ɍɨɝɞɚ ɩɪɢɦɟɧɹɸɬ ɭɫɟɱɟɧɧɨɟ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ: ɪɚɡɦɟɪ ɜɵɛɨɪɤɢ ɨɝɪɚɧɢɱɢɜɚɸɬ ɢ ɨɫɬɚɧɚɜɥɢɜɚɸɬ ɩɪɢɧɹɬɢɟ ɪɟɲɟɧɢɹ, ɟɫɥɢ n ɨɤɚɠɟɬɫɹ ɛɨɥɶɲɟ mɦɚɤɫ.

Ɇɧɨɝɨɚɥɶɬɟɪɧɚɬɢɜɧɵɟ ɩɪɚɜɢɥɚ ɜɵɛɨɪɚ ɪɟɲɟɧɢɣ

ȿɫɥɢ ɜɵɞɜɢɝɚɟɬɫɹ (m+1) ɩɪɨɫɬɵɯ ɝɢɩɨɬɟɡ, ɬɨ ɞɥɹ ɜɵɛɨɪɚ ɪɟɲɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɝɨ, ɤɚɤɚɹ ɢɡ ɧɢɯ ɢɫɬɢɧɧɚ, ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɪɢɬɟɪɢɣ ɦɢɧɢɦɭɦɚ ɫɪɟɞɧɟɝɨ ɪɢɫɤɚ. ɉɪɢ ɷɬɨɦ ɨɛɥɚɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɵɛɨɪɨɤ ɪɚɡɛɢɜɚɟɬɫɹ ɧɚ (m+1) ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ ɨɛɥɚɫɬɟɣ ɢ ɩɪɢɩɢɫɵɜɚɟɬɫɹ ɤɚɠɞɨɣ ɢɡ ɨɛɥɚɫɬɟɣ Gk ɨɞɧɨɝɨ ɪɟɲɟɧɢɹ Jk ɨ ɬɨɦ, ɱɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɫɨɫɬɨɹɧɢɟ sk ɢɫɫɥɟɞɭɟɦɨɝɨ ɹɜɥɟɧɢɹ. ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬɫɹ ɫɢɫɬɟɦɚ m ɧɟɪɚɜɟɧɫɬɜ.

Ɋɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɥɨɝɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɣ, ɨɛɴɟɞɢɧɹɸɳɢɯ ɪɟɡɭɥɶɬɚɬɵ ɫɪɚɜɧɟɧɢɹ ɨɬɧɨɲɟɧɢɣ ɩɪɚɜɞɨɩɨɞɨɛɢɹ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ ɨɛɧɚɪɭɠɟɧɢɟ ɤɚɠɞɨɝɨ ɫɢɝɧɚɥɚ ɧɚ ɮɨɧɟ ɲɭɦɚ, ɚ ɡɚɬɟɦ ɫɪɟɞɢ ɨɛɧɚɪɭɠɟɧɧɵɯ ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬɫɹ ɜ ɩɨɥɶɡɭ ɬɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɞɥɹ ɤɨɬɨɪɨɝɨ ɛɨɥɶɲɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɨɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ. ɋɬɪɭɤɬɭɪɚ ɫɨɞɟɪɠɢɬ m ɛɥɨɤɨɜ ɮɨɪɦɢɪɨɜɚɧɢɹ ɨɬɧɨɲɟɧɢɣ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɢ ɛɥɨɤ ɩɨɢɫɤɚ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ (ɪɢɫ.8.1).

ȼ ɧɚɛɨɪɟ ɧɟɩɪɚɜɢɥɶɧɵɯ ɪɟɲɟɧɢɣ ɢɦɟɟɬɫɹ m ɥɨɠɧɵɯ ɬɪɟɜɨɝ ɢ ɨɲɢɛɤɢ ɧɟɩɪɚɜɢɥɶɧɨɣ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɫɢɝɧɚɥɨɜ (ɫɨɫɬɨɹɧɢɣ).

ȿɫɥɢ ɩɥɚɬɵ ɡɚ ɩɪɚɜɢɥɶɧɵɟ ɪɟɲɟɧɢɹ ɪɚɜɧɵ ɧɭɥɸ, ɚ ɡɚ ɧɟɩɪɚɜɢɥɶɧɵɟ ɪɟɲɟɧɢɹ - ɨɞɢɧɚɤɨɜɵ, ɬɨ ɩɨɪɨɝɨɜɵɣ ɭɪɨɜɟɧɶ ɪɚɜɟɧ ɟɞɢɧɢɰɟ ɢ ɨɛɥɚɫɬɶ ɩɪɢɧɹɬɢɹ ɪɟɲɟɧɢɹ ɪɚɡɛɢɜɚɟɬɫɹ ɧɚ ɬɪɢ ɨɛɥɚɫɬɢ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ.8.2 (l1 ɢ l2 - ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɬɧɨɲɟɧɢɹ ɩɪɚɜɞɨɩɨɞɨɛɢɹ).

f

pD= ³f(z/s1)dz = 1-Ɏ((z0-Q2)/Q).

z0

Ɋɢɫ.8.3

ɉɊɂɆȿɊ. ɇɚ ɜɯɨɞ ɩɪɢɟɦɧɢɤɚ ɜɨɡɞɟɣɫɬɜɭɟɬ ɩɨɦɟɯɚ ɫ ɩɨɥɨɫɨɣ 'F = 1ɆȽɰ, V =100ɦɤȼ. ɇɟɨɛɯɨɞɢɦɨ ɨɛɧɚɪɭɠɢɬɶ ɫɢɝɧɚɥ ɫ ɚɦɩɥɢɬɭɞɨɣ Ⱥ=10ɦɤȼ ɩɪɢ pF = 10-9, pD = 0,9. Ʉɚɤɨɣ ɞɨɥɠɧɚ ɛɵɬɶ ɞɥɢɬɟɥɶɧɨɫɬɶ Ɍ ɫɢɝɧɚɥɚ, ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɡɚɞɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ (ɩɨ ɤɪɢɜɵɦ ɨɛɧɚɪɭɠɟɧɢɹ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɦ ɡɚɜɢɫɢɦɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɩɪɚɜɢɥɶɧɨɝɨ ɨɛɧɚɪɭɠɟɧɢɹ ɨɬ ɨɬɧɨɲɟɧɢɹ Q ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɜɟɪɨɹɬɧɨɫɬɹɯ ɥɨɠɧɵɯ ɬɪɟɜɨɝ)? Ⱦɥɹ ɡɚɞɚɧɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɨɩɪɟɞɟɥɹɟɦ ɬɪɟɛɭɟɦɨɟ ɨɬɧɨɲɟɧɢɟ Q. Ɉɧɨ ɪɚɜɧɨ 4,5, ɨɬɤɭɞɚ Ɍ=1 ɦɫ.

ɉɪɢ ɨɛɧɚɪɭɠɟɧɢɢ ɩɨɥɧɨɫɬɶɸ ɢɡɜɟɫɬɧɨɝɨ ɫɢɝɧɚɥɚ ɧɚ ɮɨɧɟ ɧɨɪɦɚɥɶɧɨɝɨ ɲɭɦɚ ɩɨɦɟɯɨɭɫɬɨɣɱɢɜɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɨɬɧɨɲɟɧɢɟɦ ɫɢɝɧɚɥ/ɲɭɦ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɮɨɪɦɵ ɫɢɝɧɚɥɚ. Ʉ ɧɟɞɨɫɬɚɬɤɚɦ ɤɨɪɪɟɥɹɰɢɨɧɧɨɝɨ ɩɪɢɟɦɧɢɤɚ ɫɥɟɞɭɟɬ ɨɬɧɟɫɬɢ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɡɧɚɬɶ "ɤɨɩɢɸ" ɫɢɝɧɚɥɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɟɫɥɢ ɜɪɟɦɹ ɩɪɢɯɨɞɚ ɫɢɝɧɚɥɚ ɧɟɢɡɜɟɫɬɧɨ, ɬɨ ɧɟɢɡɜɟɫɬɧɨ, ɤɨɝɞɚ ɩɨɞɚɜɚɬɶ "ɤɨɩɢɸ" ɫɢɝɧɚɥɚ ɧɚ ɨɞɢɧ ɢɡ ɜɯɨɞɨɜ ɩɟɪɟɦɧɨɠɢɬɟɥɹ.

8.4.Ɉɬɧɨɲɟɧɢɟ ɩɪɚɜɞɨɩɨɞɨɛɢɹ ɩɪɢ ɧɚɥɢɱɢɢ ɫɥɭɱɚɣɧɵɯ ɧɟɢɡɦɟɪɹɟɦɵɯ ɩɚɪɚɦɟɬɪɨɜ ɫɢɝɧɚɥɚ ɢɥɢ ɲɭɦɚ

ɋɢɝɧɚɥ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧ ɜ ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɟ: s(t,a1,...,al; b1,...,bm) = s(t, A, B), ɝɞɟ A - ɜɟɤɬɨɪ ɩɨɥɟɡɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɪɚɡɦɟɪɚ l; B - ɜɟɤɬɨɪ ɧɟɢɡɜɟɫɬɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɪɚɡɦɟɪɚ m.

ɉɨɥɟɡɧɵɦ ɩɚɪɚɦɟɬɪɨɦ ɦɨɠɟɬ ɛɵɬɶ a = 1 ɩɪɢ ɧɚɥɢɱɢɢ ɫɢɝɧɚɥɚ (ɜ ɡɚɞɚɱɟ ɨɛɧɚɪɭɠɟɧɢɹ) ɢ a = 0 ɩɪɢ ɟɝɨ ɨɬɫɭɬɫɬɜɢɢ. ɇɟɢɡɜɟɫɬɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ ɦɨɝɭɬ ɛɵɬɶ: ɚɦɩɥɢɬɭɞɚ ɫɢɝɧɚɥɚ Um, ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ M ɢ ɜɪɟɦɹ ɩɪɢɯɨɞɚ ɫɢɝɧɚɥɚ t0 (ɡɚɞɟɪɠɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɡɨɧɞɢɪɭɸɳɟɝɨ ɫɢɝɧɚɥɚ ɜ ɪɚɞɢɨɥɨɤɚɰɢɢ).