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ɱɥɟɧ ɜɨɡɦɨɠɧɨ ɞɥɹ ɜɫɟɯ ɧɟɩɪɟɪɵɜɧɵɯ ɮɭɧɤɰɢɣ. ɇɚ ɨɫɧɨɜɚɧɢɢ ɷɤɫɩɟɪɢɦɟɧ-

ɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɧɭɠɧɨ ɩɨɥɭɱɢɬɶ ɧɟɤɨɬɨɪɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɮɭɧɤɰɢɢ ɨɬɤɥɢ-

ɤɚ f(x1,x2,…xk). ȿɫɥɢ ɚɧɚɥɢɬɢɱɟɫɤɢɣ ɜɢɞ ɷɬɨɣ ɮɭɧɤɰɢɢ ɧɟɢɡɜɟɫɬɟɧ,

ɩɪɢɯɨɞɢɬɫɹ ɨɝɪɚɧɢɱɢɜɚɬɶɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɟɟ ɜ ɜɢɞɟ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ. ɉɨɥɶ-

ɡɭɹɫɶ ɪɟɡɭɥɶɬɚɬɚɦɢ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɥɢɲɶ ɨɰɟɧɤɢ ɩɚɪɚɦɟɬɪɨɜ ɷɬɨɣ ɦɨɞɟɥɢ:

yˆ b0 ¦Ei xi ¦Eij xi x j ¦Eii xi2 ,

ɝɞɟ yˆ – ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ, ɩɪɟɞɫɤɚɡɚɧɧɨɝɨ ɷɬɢɦ ɭɪɚɜɧɟɧɢɟɦ. Ɉɰɟɧɤɢ ɤɨɷɮ-

ɮɢɰɢɟɧɬɨɜ ɨɛɵɱɧɨ ɩɨɥɭɱɚɸɬ ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɚ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ.

ɑɬɨɛɵ ɭɛɟɞɢɬɶɫɹ ɜ ɩɪɢɟɦɥɟɦɨɫɬɢ ɬɚɤɨɣ ɨɰɟɧɤɢ, ɧɭɠɧɨ ɩɪɨɜɟɫɬɢ ɫɬɚɬɢɫɬɢɱɟ-

ɫɤɢɣ ɚɧɚɥɢɡ, ɡɚɞɚɱɚ ɤɨɬɨɪɨɝɨɨɰɟɧɢɬɶ ɩɚɪɚɦɟɬɪɵ ɝɟɧɟɪɚɥɶɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɩɨ ɞɚɧɧɵɦ ɜɵɛɨɪɤɢ, ɭɱɢɬɵɜɚɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ, ɜɧɨɫɢɦɭɸ ɨɝɪɚɧɢɱɟɧɧɨ-

ɫɬɶɸ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɝɨ ɦɚɬɟɪɢɚɥɚ. ɉɪɢ ɫɬɚɬɢɫɬɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ɩɪɨɜɟɪɹ-

ɟɬɫɹ ɡɧɚɱɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ ɢ ɚɞɟɤɜɚɬɧɨɫɬɶ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ

(ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɨɞɟɥɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɞɚɧɧɵɦ ɩɨ ɜɵɛɪɚɧɧɨɦɭ ɤɪɢɬɟ-

ɪɢɸ).

ɉɥɚɧɢɪɨɜɚɧɢɸ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɪɟɞɲɟɫɬɜɭɟɬ ɷɬɚɩ ɜɵɛɨɪɚ ɨɛɥɚɫɬɢ ɷɤɫ-

ɩɟɪɢɦɟɧɬɢɪɨɜɚɧɢɹ, ɰɟɧɬɪɚ ɷɤɫɩɟɪɢɦɟɧɬɚ ɢ ɢɧɬɟɪɜɚɥɨɜ ɜɚɪɶɢɪɨɜɚɧɢɹ ɮɚɤɬɨɪɨɜ

[3]. ɉɪɢ ɷɬɨɦ ɨɰɟɧɢɜɚɸɬɫɹ ɝɪɚɧɢɰɵ ɨɛɥɚɫɬɟɣ ɨɩɪɟɞɟɥɟɧɢɹ ɮɚɤɬɨɪɨɜ, ɡɚɞɚɜɚɟ-

ɦɵɯ ɩɪɢɧɰɢɩɢɚɥɶɧɵɦɢ ɨɝɪɚɧɢɱɟɧɢɹɦɢ ɥɢɛɨ ɬɟɯɧɢɤɨ-ɷɤɨɧɨɦɢɱɟɫɤɢɦɢ ɫɨɨɛ-

ɪɚɠɟɧɢɹɦɢ. ɉɟɪɜɵɣ ɬɢɩ ɨɝɪɚɧɢɱɟɧɢɣ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɧɚɪɭɲɟɧ ɧɢ ɩɪɢ ɤɚɤɢɯ ɨɛɫɬɨɹɬɟɥɶɫɬɜɚɯ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɧɢɠɧɢɦ ɩɪɟɞɟɥɨɦ ɛɭɞɟɬ ɚɛɫɨ-

ɥɸɬɧɵɣ ɧɭɥɶ. ɉɪɢ ɭɫɬɚɧɨɜɥɟɧɢɢ ɜɬɨɪɨɝɨ ɬɢɩɚ ɨɝɪɚɧɢɱɟɧɢɣ ɷɤɫɩɟɪɢɦɟɧɬɚɬɨɪ ɪɭɤɨɜɨɞɫɬɜɭɟɬɫɹ ɤɨɧɤɪɟɬɧɵɦɢ ɨɛɫɬɨɹɬɟɥɶɫɬɜɚɦɢ. ɉɪɢɦɟɪɚɦɢ ɡɞɟɫɶ ɦɨɝɭɬ ɛɵɬɶ: ɫɬɨɢɦɨɫɬɶ ɫɵɪɶɹ, ɞɟɮɢɰɢɬɧɨɫɬɶ ɨɬɞɟɥɶɧɵɯ ɤɨɦɩɨɧɟɧɬɨɜ, ɜɪɟɦɹ ɩɪɨɬɟ-

ɤɚɧɢɹ ɩɪɨɰɟɫɫɚ. ɍɫɬɚɧɨɜɥɟɧɢɟ ɨɛɥɚɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɢɪɨɜɚɧɢɹ ɫɜɹɡɚɧɨ ɫ ɬɳɚ-

ɬɟɥɶɧɵɦ ɚɧɚɥɢɡɨɦ ɚɩɪɢɨɪɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɛ ɢɡɭɱɚɟɦɨɦ ɨɛɴɟɤɬɟ. Ʌɨɤɚɥɶɧɚɹ ɨɛɥɚɫɬɶ ɷɤɫɩɟɪɢɦɟɧɬɚ ɡɚɞɚɟɬɫɹ ɜɵɛɨɪɨɦ ɤɨɦɛɢɧɚɰɢɢ ɨɫɧɨɜɧɵɯ ɭɪɨɜɧɟɣ ɮɚɤ-

ɬɨɪɨɜ (ɰɟɧɬɪɚ ɷɤɫɩɟɪɢɦɟɧɬɚ) ɢ ɢɧɬɟɪɜɚɥɨɜ ɜɚɪɶɢɪɨɜɚɧɢɹ ɮɚɤɬɨɪɨɜ. ɉɨɫɬɪɨɟ-

ɧɢɟ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɵɯ ɩɥɚɧɨɜ ɫɜɨɞɢɬɫɹ ɤ ɜɵɛɨɪɭ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɬɨɱɟɤ,

ɫɢɦɦɟɬɪɢɱɧɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɷɤɫɩɟɪɢɦɟɧɬɚ. Ɇɵ ɛɭɞɟɦ ɡɞɟɫɶ ɪɚɫɫɦɚɬ-

ɪɢɜɚɬɶ ɢɦɟɧɧɨ ɬɚɤɢɟ ɫɢɦɦɟɬɪɢɱɧɵɟ ɞɜɭɯɭɪɨɜɧɟɜɵɟ ɩɥɚɧɵ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɫɟ k ɮɚɤɬɨɪɨɜ ɢɡɦɟɧɹɸɬɫɹ ɧɚ ɞɜɭɯ ɭɪɨɜɧɹɯ ɢ ɩɥɚɧ ɷɤɫɩɟɪɢɦɟɧɬɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɩɥɚɧɚ ɬɢɩɚ 2k. ɗɬɢ ɭɪɨɜɧɢ ɫɢɦɦɟɬɪɢɱɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɧɨɜɧɨɝɨ ɭɪɨɜɧɹ.

Ɉɞɢɧ ɢɡ ɧɢɯ – ɜɟɪɯɧɢɣ, ɞɪɭɝɨɣ – ɧɢɠɧɢɣ. ɂɧɬɟɪɜɚɥɨɦ ɜɚɪɶɢɪɨɜɚɧɢɹ ɮɚɤɬɨɪɨɜ ɧɚɡɵɜɚɟɬɫɹ ɧɟɤɨɬɨɪɨɟ ɱɢɫɥɨ (ɫɜɨɟ ɞɥɹ ɤɚɠɞɨɝɨ ɮɚɤɬɨɪɚ), ɩɪɢɛɚɜɥɟɧɢɟ ɤɨɬɨɪɨ-

ɝɨ ɤ ɨɫɧɨɜɧɨɦɭ ɭɪɨɜɧɸ ɞɚɟɬ ɜɟɪɯɧɢɣ ɭɪɨɜɟɧɶ, ɚ ɜɵɱɢɬɚɧɢɟ – ɧɢɠɧɢɣ. ɑɬɨɛɵ ɭɩɪɨɫɬɢɬɶ ɢ ɭɧɢɮɢɰɢɪɨɜɚɬɶ ɡɚɩɢɫɶ ɭɫɥɨɜɢɣ ɨɩɵɬɨɜ ɢ ɨɛɥɟɝɱɢɬɶ ɨɛɪɚɛɨɬɤɭ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ, ɦɚɫɲɬɚɛɵ ɩɨɨɫɹɦɡɚɞɚɸɬɫɹ ɜ ɜɢɞɟɤɨɞɢɪɨɜɚɧɧɵɯ ɡɧɚɱɟɧɢɣ +1 ɢ –1. Ⱦɥɹ ɤɨɥɢɱɟɫɬɜɟɧɧɵɯ ɮɚɤɬɨɪɨɜ ɷɬɨ ɜɫɟɝɞɚ ɦɨɠɧɨ ɫɞɟɥɚɬɶ ɫ ɩɨɦɨɳɶɸɩɪɨɫɬɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ:

Xj = (xj – xjɨɫɧ.)/Ij,

ɝɞɟ Xj – ɤɨɞɢɪɨɜɚɧɧɨɟ ɡɧɚɱɟɧɢɟ ɮɚɤɬɨɪɚ, xj – ɧɚɬɭɪɚɥɶɧɨɟ ɟɝɨ ɡɧɚɱɟɧɢɟ, xjɨɫɧ. –

ɧɚɬɭɪɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɨɫɧɨɜɧɨɝɨ ɭɪɨɜɧɹ, Ij – ɢɧɬɟɪɜɚɥ ɜɚɪɶɢɪɨɜɚɧɢɹ.

ɉɭɫɬɶ ɜ ɷɤɫɩɟɪɢɦɟɧɬɟ ɢɡɦɟɧɹɸɬɫɹ ɞɜɚ ɮɚɤɬɨɪɚ ɧɚ ɞɜɭɯ ɭɪɨɜɧɹɯ: x1 –

ɬɟɦɩɟɪɚɬɭɪɚ ɢ x2 – ɜɪɟɦɹ ɪɟɚɤɰɢɢ. Ⱦɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɨɫɧɨɜɧɵɦ ɭɪɨɜɧɟɦ ɹɜɥɹɟɬɫɹ 50°ɋ, ɚ ɢɧɬɟɪɜɚɥ ɜɚɪɶɢɪɨɜɚɧɢɹ ɫɨɫɬɚɜɥɹɟɬ 10°ɋ. Ɍɨɝɞɚ ɞɥɹ x1

50 + 10 = 60°ɋ ɛɭɞɟɬ ɜɟɪɯɧɢɦ ɭɪɨɜɧɟɦ, ɚ 50 – 10 = 40°ɋ – ɧɢɠɧɢɦ. ȼ ɤɨɞɢ-

ɪɨɜɚɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɷɬɨ ɡɚɩɢɲɟɬɫɹ ɬɚɤ: (60 – 50)/10 = 1 ɢ (40 – 50)/10 = –1. ȿɫ-

ɥɢ ɞɥɹ x2 ɜɵɛɪɚɧɵ x2ɨɫɧ. = 30ɦɢɧ ɢ I2 = 5ɦɢɧ, ɬɨ (35 – 30)/5 = 1 ɢ(25 – 30)/5 = –1.

ɗɤɫɩɟɪɢɦɟɧɬ, ɜ ɤɨɬɨɪɨɦ ɪɟɚɥɢɡɨɜɚɧɵ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɫɨɱɟɬɚɧɢɹ ɭɪɨɜ-

ɧɟɣ ɮɚɤɬɨɪɨɜ, ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɧɵɦ ɮɚɤɬɨɪɧɵɦ (ɉɎɗ). Ⱦɥɹ n ɭɪɨɜɧɟɣ ɷɬɨ ɛɭ-

ɞɟɬ ɉɎɗ ɬɢɩɚ nk. ɍɫɥɨɜɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɪɟɞɫɬɚɜɥɹɸɬɫɹ ɜ ɜɢɞɟ ɬɚɛɥɢɰɵ – ɦɚɬɪɢɰɵ ɩɥɚɧɢɪɨɜɚɧɢɹ. ȼɨɬ ɩɪɢɦɟɪ ɦɚɬɪɢɰɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɫ ɭɱɟɬɨɦ ɫɜɨ-

ɛɨɞɧɨɝɨ ɱɥɟɧɚ bo ɢ ɷɮɮɟɤɬɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ b12:

ɇɨɦɟɪ ɨɩɵɬɚ	xo	x1	x2	x1x2	y
1	+1	–1	–1	+1	y1
2	+1	+1	+1	+1	y2
3	+1	–1	+1	–1	y3
4	+1	+1	–1	–1	y4

ɗɬɨɬ ɩɥɚɧ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɦɨɞɟɥɢ	:	ˆ	+ b1x1 + b2x2	+ b12x1x2.	ɋɬɨɥɛɰɵ
		y b0x0

x1 ɢx2 ɡɚɞɚɸɬ ɩɥɚɧɢɪɨɜɚɧɢɟ – ɩɨ ɧɢɦ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɫɥɨɜɢɹ ɩɪɨɜɟɞɟɧɢɹ ɨɩɵɬɨɜ, ɚ ɫɬɨɥɛɰɵ xo ɢ x1x2 ɫɥɭɠɚɬ ɬɨɥɶɤɨ ɞɥɹ ɪɚɫɱɟɬɨɜ. ɉɨɥɧɨɟ ɱɢɫɥɨ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɷɮɮɟɤɬɨɜ ɪɚɜɧɨ N – ɱɢɫɥɭ ɨɩɵɬɨɜ ɉɎɗ. ɑɬɨɛɵ ɧɚɣɬɢ ɩɨɥɧɨɟ ɱɢɫɥɨ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɧɟɤɨɬɨɪɨɝɨ ɩɨɪɹɞɤɚ, ɦɨɠɧɨ

ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɮɨɪɦɭɥɨɣ ɞɥɹ ɱɢɫɥɚ ɫɨɱɟɬɚɧɢɣ: Cm	k!	,	ɝɞɟ k –

	m!(k m)!
k

ɱɢɫɥɨ ɮɚɤɬɨɪɨɜ, m – ɱɢɫɥɨ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɷɥɟɦɟɧɬɨɜ. Ɍɚɤ ɞɥɹ ɩɥɚɧɚ 24

ɱɢɫɥɨ ɩɚɪɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɪɚɜɧɨ C2	4!	6.
			ɉɪɢ ɩɥɚɧɢɪɨɜɚɧɢɢ

4	2!(4 2)!

ɷɤɫɩɟɪɢɦɟɧɬɚ ɫɭɳɟɫɬɜɟɧɧɵɦɢ ɦɨɝɭɬ ɨɤɚɡɚɬɶɫɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɤɜɚɞɪɚɬɚɯ ɮɚɤɬɨɪɨɜ, ɤɭɛɚɯ ɢ ɬ.ɞ. ɂɡ 2k ɉɎɗ ɧɟɥɶɡɹ ɢɡɜɥɟɱɶ ɢɧɮɨɪɦɚɰɢɸ ɨ ɤɜɚɞɪɚɬɢɱ-

ɧɵɯ ɱɥɟɧɚɯ ɢ ɱɥɟɧɚɯ ɛɨɥɟɟ ɜɵɫɨɤɨɝɨ ɩɨɪɹɞɤɚ. Ⱦɥɹ ɛɨɥɟɟ ɫɥɨɠɧɵɯ ɦɨɞɟɥɟɣ ɜɵɛɢɪɚɸɬɫɹ ɩɥɚɧɵ ɫ ɱɢɫɥɨɦ ɭɪɨɜɧɟɣ, ɛɨɥɶɲɢɦ 2 [23, 24].

Ʉɚɤɢɦɢ ɩɥɚɧɚɦɢ ɧɚɞɨ ɩɨɥɶɡɨɜɚɬɶɫɹ, ɟɫɥɢ ɪɟɱɶ ɢɞɟɬ ɨ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ ɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɱɢɬɚɸɬɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɵɦɢ? ɉɪɢ ɥɢɧɟɣɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ b12o0, ɢ ɜɟɤɬɨɪ-ɫɬɨɥɛɟɰ ɯ1ɯ2 ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɜɜɟɞɟɧɢɹ ɜ ɩɥɚɧ ɧɨɜɨɝɨ ɮɚɤɬɨɪɚ ɯ3. ȼɦɟɫɬɨ ɜɨɫɶɦɢ ɨɩɵɬɨɜ ɞɥɹ 3-ɯ ɮɚɤɬɨɪɨɜ, ɨɤɚɡɵɜɚɟɬɫɹ, ɦɨɠɧɨ ɩɨ-

ɫɬɚɜɢɬɶ ɬɨɥɶɤɨ 4. Ʉɨɥɢɱɟɫɬɜɨ ɨɩɵɬɨɜ ɜ ɉɎɗ 2k ɡɧɚɱɢɬɟɥɶɧɨ ɩɪɟɜɵɲɚɟɬ ɱɢɫɥɨ ɥɢɧɟɣɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢ ɷɤɨɧɨɦɢɹ ɦɨɠɟɬ ɛɵɬɶ ɨɱɟɧɶ ɫɭɳɟɫɬɜɟɧɧɚɹ. ɉɪɢ ɷɬɨɦ ɦɚɬɪɢɰɚ ɩɥɚɧɢɪɨɜɚɧɢɹ ɧɟ ɬɟɪɹɟɬ ɫɜɨɢɯ ɨɩɬɢɦɚɥɶɧɵɯ ɫɜɨɣɫɬɜ ɜ ɪɚɦɤɚɯ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ [2].

4.2. Ɋɟɝɪɟɫɫɢɨɧɧɵɣ ɚɧɚɥɢɡ ɞɥɹ ɨɪɬɨɝɨɧɚɥɶɧɵɯ ɞɜɭɯɭɪɨɜɧɟɜɵɯ ɩɥɚɧɨɜ

Ɋɚɫɫɦɨɬɪɢɦ 3-ɯ ɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ. ɇɚɫ ɛɭɞɭɬ ɢɧɬɟɪɟɫɨɜɚɬɶ ɥɢ-

ɧɟɣɧɵɟ ɷɮɮɟɤɬɵ ɢ ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ. Ɇɨɞɟɥɶ ɨɛɴɟɤɬɚ ɢɦɟɟɬ ɜɢɞ:

yˆ b0x0 + b1x1 + b2x2 + b3x3 + b12x1x2 + b13x1x3 + b23x2x3. ɇɚɢɛɨɥɟɟ ɩɪɨɫɬɨɣ ɩɥɚɧ,

ɞɨɩɭɫɤɚɸɳɢɣ ɨɰɟɧɤɭ ɜɫɟɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɷɬɨɣ ɦɨɞɟɥɢ – ɩɨɥɧɵɣ ɮɚɤɬɨɪɧɵɣ

ɷɤɫɩɟɪɢɦɟɧɬ 23. Ⱦɥɹ ɨɰɟɧɤɢ ɨɲɢɛɤɢ ɜɨɫɩɪɨɢɡɜɨɞɢɦɨɫɬɢ ɜɫɟ ɨɩɵɬɵ ɞɭɛɥɢɪɨ-

ɜɚɥɢɫɶ.

Ɉɰɟɧɤɚ ɞɢɫɩɟɪɫɢɣ ɫɪɟɞɧɟɝɨ ɚɪɢɮɦɟɬɢɱɟɫɤɨɝɨ.

	¦( yiq				i )2
	¦( yiq			y	i )2
	q 1
Si2	q 1					,


	n (n 1)
ɝɞɟ yiq – ɪɟɡɭɥɶɬɚɬ ɨɬɞɟɥɶɧɨɝɨ ɨɩɵɬɚ,			i – ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɩɨ ɩɨ-
ɝɞɟ yiq – ɪɟɡɭɥɶɬɚɬ ɨɬɞɟɥɶɧɨɝɨ ɨɩɵɬɚ,		y	i – ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɩɨ ɩɨ-

ɜɬɨɪɧɵɦ ɨɩɵɬɚɦ, ɩ – ɤɨɥɢɱɟɫɬɜɨ ɩɚɪɚɥɥɟɥɶɧɵɯ ɨɩɵɬɨɜ, q – ɧɨɦɟɪ ɩɚɪɚɥɥɟɥɶ-

ɧɨɝɨ ɨɩɵɬɚ, i – ɧɨɦɟɪ ɫɬɪɨɤɢ ɦɚɬɪɢɰɵ ɩɥɚɧɚ, i = 1… N.

ɉɪɨɜɟɪɤɚ ɨɞɧɨɪɨɞɧɨɫɬɢ ɞɢɫɩɟɪɫɢɣ ɫ ɩɨɦɨɳɶɸ ɤɪɢɬɟɪɢɹ Ʉɨɯɪɟɧɚ. Ʉɪɢ-

ɬɟɪɢɣ Ʉɨɯɪɟɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɢɫɩɟɪɫɢɢ ɤ ɫɭɦɦɟ ɜɫɟɯ ɞɢɫɩɟɪɫɢɣ: SNi2max . ɋ ɷɬɢɦ ɤɪɢɬɟɪɢɟɦ ɫɜɹɡɚɧɵ ɱɢɫɥɚ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ

¦Si2

i 1

n–1 ɢ N. Ƚɢɩɨɬɟɡɚ ɨɛ ɨɞɧɨɪɨɞɧɨɫɬɢ ɞɢɫɩɟɪɫɢɣ ɧɟ ɨɬɜɟɪɝɚɟɬɫɹ, ɟɫɥɢ ɷɤɫɩɟɪɢ-

ɦɟɧɬɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ Ʉɨɯɪɟɧɚ ɧɟ ɩɪɟɜɵɫɢɬ ɬɚɛɥɢɱɧɨɝɨ. ȼ ɪɟɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɝɢɩɨɬɟɡɚ ɨɛ ɨɞɧɨɪɨɞɧɨɫɬɢ ɞɢɫɩɟɪɫɢɣ ɩɨɞɬɜɟɪɠɞɚɟɬɫɹ ɞɚɥɟɤɨ ɧɟ ɜɫɟ-

ɝɞɚ. Ɍɨɝɞɚ ɦɨɠɧɨ ɢɞɬɢ ɪɚɡɥɢɱɧɵɦɢ ɩɭɬɹɦɢ. ɇɚɩɪɢɦɟɪ, ɧɚɣɬɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɡɚɜɢɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɨɬɵɫɤɚɬɶ ɢɧɨɣ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɥɭɱɚɣɧɨɣ ɜɟɥɢ-

ɱɢɧɵɢɥɢ ɨɛɪɚɬɢɬɶɫɹɤɤɚɤɨɦɭ-ɧɢɛɭɞɶɪɨɛɚɫɬɧɨɦɭ ɫɬɚɬɢɫɬɢɱɟɫɤɨɦɭ ɦɟɬɨɞɭ.

Ɉɰɟɧɤɚ ɭɫɪɟɞɧɟɧɧɨɣ ɞɢɫɩɟɪɫɢɢ ɜɨɫɩɪɨɢɡɜɨɞɢɦɨɫɬɢ. ɗɬɚ ɨɰɟɧɤɚ ɪɚɫɫɱɢ-

ɬɵɜɚɟɬɫɹ, ɟɫɥɢ ɞɢɫɩɟɪɫɢɢ ɨɞɧɨɪɨɞɧɵ:

N n

¦¦( yiq

i )2

Si2

S^2y

i 1 q

i 1

n (n 1)

ȼɵɱɢɫɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɪɚɜɧɟɧɢɹ. ɉɪɢɜɟɞɟɦ ɮɨɪɦɭɥɵ ɞɥɹ ɪɚɫɱɟɬɚ

ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɪɚɜɧɟɧɢɹ ɪɟɝɪɟɫɫɢɢ:

¦x ji

¦xui x ji

i 1

buj

i 1

ɫɢɟɣ: S^2 `

ɝɞɟ xji – ɡɧɚɱɟɧɢɟ j-ɝɨ ɮɚɤɬɨɪɚ			ɜ	i-ɨɦ ɨɩɵɬɟ; u,	j – ɧɨɦɟɪɚ ɮɚɤɬɨɪɨɜ;
j, u = 0, 1,… k, j z u.
ɉɪɨɜɟɪɤɚ ɝɢɩɨɬɟɡɵ ɨɛ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ. ɉɪɨɜɟɪɤɚ ɨɫɧɨɜɚɧɚ ɧɚ
ɪɚɫɱɟɬɚɯ ɞɢɫɩɟɪɫɢɢ ɚɞɟɤɜɚɬɧɨɫɬɢ Sɚɞ2				ɢ ɤɪɢɬɟɪɢɹ Ɏɢɲɟɪɚ (F-ɤɪɢɬɟɪɢɹ).
N					2
Sɚɞ2 ¦		i yˆi	2 / f ; f N p; F		Sɚɞ		,
	y

i 1					S^2y	`

ɝɞɟ yˆi – ɪɚɫɫɱɢɬɚɧɧɨɟ ɩɨ ɭɪɚɜɧɟɧɢɸ ɪɟɝɪɟɫɫɢɢ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ, f – ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ, ɫɜɹɡɚɧɧɨɟ ɫ ɞɢɫɩɟɪɫɢɟɣ ɚɞɟɤɜɚɬɧɨɫɬɢ, ɪ – ɱɢɫɥɨ ɨɰɟɧɢ-

ɜɚɟɦɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ. Ɋɚɫɫɱɢɬɚɧɧɨɟ ɡɧɚɱɟɧɢɟ F-ɤɪɢɬɟɪɢɹ ɫɪɚɜ-

ɧɢɜɚɟɬɫɹ ɫ ɬɚɛɥɢɱɧɵɦ ɡɧɚɱɟɧɢɟɦ, ɨɩɪɟɞɟɥɹɟɦɵɦ ɱɢɫɥɚɦɢ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ f

ɢ N (ɩ–1). ȿɫɥɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɚɹ ɜɟɥɢɱɢɧɚ F-ɤɪɢɬɟɪɢɹ ɧɟ ɩɪɟɜɵɲɚɟɬ ɬɚɛ-

ɥɢɱɧɨɝɨ ɡɧɚɱɟɧɢɹ, ɬɨ ɝɢɩɨɬɟɡɚ ɨɛ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ ɧɟ ɨɬɜɟɪɝɚɟɬɫɹ.

ɉɪɨɜɟɪɤɚ ɡɧɚɱɢɦɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ. ɉɨɫɤɨɥɶɤɭ ɩɥɚɧ ɨɪ-

ɬɨɝɨɧɚɥɟɧ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɟɝɪɟɫɫɢɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɞɢɫɩɟɪ-

S^2y` . Ⱦɚɥɟɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ

ɞɨɜɟɪɢɬɟɥɶɧɵɣ ɢɧɬɟɪɜɚɥ ɫ ɧɟɤɨɬɨɪɨɣ ɞɨɜɟɪɢɬɟɥɶɧɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ. ȼ ɷɬɨɦ ɜɵɪɚɠɟɧɢɢ t-ɤɪɢɬɟɪɢɣ (ɤɪɢɬɟɪɢɣ ɋɬɶɸɞɟɧɬɚ) ɢɦɟɟɬ ɬɨ ɠɟ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ

ɫɜɨɛɨɞɵ, ɱɬɨ ɢ ɞɢɫɩɟɪɫɢɹ ɜɨɫɩɪɨɢɡɜɨɞɢɦɨɫɬɢ 'bj rt S^2b`. Ʉɨɷɮɮɢɰɢɟɧɬ

ɡɧɚɱɢɦ, ɟɫɥɢ ɟɝɨ ɚɛɫɨɥɸɬɧɚɹ ɜɟɥɢɱɢɧɚ ɛɨɥɶɲɟ ɞɨɜɟɪɢɬɟɥɶɧɨɝɨ ɢɧɬɟɪɜɚɥɚ.

ɇɟɡɧɚɱɢɦɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɟɝɪɟɫɫɢɢ ɢɫɤɥɸɱɚɸɬɫɹ ɢ ɜɧɨɜɶ ɩɪɨɜɨɞɢɬ-

ɫɹ ɩɪɨɜɟɪɤɚ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ ɫɨ ɡɧɚɱɢɦɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ.

ɋɬɚɬɢɫɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɡɚɜɟɪɲɚɟɬɫɹ ɢɧɬɟɪɩɪɟɬɚɰɢɟɣ ɦɨɞɟɥɢ ɜ ɬɟɪɦɢ-

ɧɚɯ ɨɛɴɟɤɬɚ ɢɫɫɥɟɞɨɜɚɧɢɹ.

4.3. Ɉɰɟɧɤɚ ɤɨɦɩɥɟɤɫɧɨɝɨ ɜɥɢɹɧɢɹ ɧɟɫɤɨɥɶɤɢɯ ɮɚɤɬɨɪɨɜ ɧɚ ɄɉȾ ɯɢɦɢɱɟɫɤɨɝɨ ɪɟɚɤɬɨɪɚ

ɐɟɥɶ ɪɚɛɨɬɵ: ɇɚɭɱɢɬɶɫɹ ɩɥɚɧɢɪɨɜɚɬɶ ɷɤɫɩɟɪɢɦɟɧɬ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɪɟɡɭɥɶɬɚɬɨɜ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ, ɩɪɨɜɨɞɢɬɶ ɚɧɚɥɢɡ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɢ ɞɟɥɚɬɶ ɜɵɜɨɞɵ ɨ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɚɯ.

Ɉɛɴɟɤɬ ɢɫɫɥɟɞɨɜɚɧɢɹ: ɩɪɨɬɟɤɚɧɢɟ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ.

ɗɬɚɩɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ:

1.ȼɵɛɨɪ ɜɯɨɞɧɵɯ ɩɟɪɟɦɟɧɧɵɯ (ɮɚɤɬɨɪɨɜ): ɯ1, ɯ2, …, ɯn.

2.Ɉɩɪɟɞɟɥɟɧɢɟ ɨɛɥɚɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɢɪɨɜɚɧɢɹ.

3.ȼɵɛɨɪ ɜɵɯɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ (ɨɬɤɥɢɤɚ): y.

4.ȼɵɛɨɪ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɣ ɛɭɞɭɬ ɩɪɟɞɫɬɚɜɥɹɬɶɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɞɚɧɧɵɟ – ɮɭɧɤɰɢɢ ɜɡɚɢɦɨɫɜɹɡɢ y ɢ x1, x2, …, xn.

5.ȼɵɛɨɪ ɩɥɚɧɚ ɷɤɫɩɟɪɢɦɟɧɬɚ.

6.Ʉɨɞɢɪɨɜɚɧɢɟ ɮɚɤɬɨɪɨɜ, ɩɨɫɬɪɨɟɧɢɟ ɦɚɬɪɢɰɵ ɩɥɚɧɢɪɨɜɚɧɢɹ.

7.ɉɪɨɜɟɞɟɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ.

8.ɋɬɚɬɢɫɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɪɟɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚ.

9.ɂɧɬɟɪɩɪɟɬɚɰɢɹ ɢ ɪɟɤɨɦɟɧɞɚɰɢɢ.

ɉɨɪɹɞɨɤ ɜɵɩɨɥɧɟɧɢɹ ɪɚɛɨɬɵ:

1)Ɏɚɤɬɨɪɵ:	ɯ1	– ɬɟɦɩɟɪɚɬɭɪɚ;
	ɯ2	– ɜɪɟɦɹ ɪɟɚɤɰɢɢ;
	ɯ3	– ɞɨɡɚ ɨɛɥɭɱɟɧɢɹ ɢɫɯɨɞɧɨɝɨ ɫɵɪɶɹ.

2)Ɉɛɥɚɫɬɶ ɷɤɫɩɟɪɢɦɟɧɬɢɪɨɜɚɧɢɹ.

Ɏɚɤ-	Ɉɛɨɡɧɚ-	ɋɨɞɟɪɠɚ-	ȿɞɢɧɢɰɵ	Ɉɫɧɨɜɧɨɣ	ɂɧɬɟɪɜɚɥ	ɇɢɠ-	ȼɟɪɯɧɢɣ
ɬɨɪ	ɱɟɧɢɟ	ɧɢɟ	ɢɡɦɟɪɟɧɢɹ	ɭɪɨɜɟɧɶ	ɜɚɪɶɢɪɨ-	ɧɢɣ	ɭɪɨɜɟɧɶ
				ɯiɨɫɧ.	ɜɚɧɢɹ Ii	ɭɪɨɜɟɧɶ
1	ɯ1	Ɍɟɦɩɟɪɚɬɭ-	qɋ	45	15	30	60
1	ɯ1		qɋ	45	15	30	60
		ɪɚ

2	ɯ2	ȼɪɟɦɹ	ɫɟɤ	55	25	30	80
3	ɯ3	Ⱦɨɡɚ	ɦɢɥɥɢɪɟɧɬ-	0,7	0,3	0,4	1
3	ɯ3			0,7	0,3	0,4	1
		ɨɛɥɭɱɟɧɢɹ	ɝɟɧ

Ʉɨɥɢɱɟɫɬɜɨ ɮɚɤɬɨɪɨɜ q = 3 (i = 1…q).

Ʉɨɥɢɱɟɫɬɜɨ ɭɪɨɜɧɟɣ k = 2 (j = 1…k).

3)Ɉɬɤɥɢɤ: y – ɜɵɯɨɞ ɨɫɧɨɜɧɨɝɨ ɜɟɳɟɫɬɜɚ.

4)Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɩɪɨɰɟɫɫɚ:

yb0 x0 b1 x1 b2 x2 b3 x3 b12 x1x2 b13 x1x3 b23 x2 x3

(ɥɢɧɟɣɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ)

5)ɉɥɚɧ ɷɤɫɩɟɪɢɦɟɧɬɚ – ɩɨɥɧɵɣ ɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ. Ʉɨɥɢɱɟɫɬɜɨ

ɨɩɵɬɨɜ: N		kq 23
6)Ʉɨɞɢɪɨɜɚɧɢɟ ɮɚɤɬɨɪɨɜ: xi = (xi – xiɨɫɧ.)/Ii,
				Ɋɚɫɱɟɬɧɚɹ ɬɚɛɥɢɰɚ

ɇɨɦɟɪ	Ⱥɞɞɢɬɢɜɧɚɹ		Ɇɚɬɪɢɰɚ			ȼɟɤɬɨɪɵ-ɫɬɨɥɛɰɵ			ɗɤɫɩɟɪɢɦɟɧ-
ɨɩɵɬɚ	ɩɨɫɬɨɹɧɧɚɹ		ɩɥɚɧɢɪɨɜɚɧɢɹ			ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ			ɬɚɥɶɧɵɣ ɨɬɤɥɢɤ
N		x0	x1	x2	x3	x1x2	x1x3	x2x3	y
1		+1	–1	–1	–1	1	1	1	y1
2		+1	–1	+1	+1	–1	–1	1	y2
3		+1	–1	+1	–1	–1	1	–1	y3
4		+1	–1	–1	+1	1	–1	–1	y4
5		+1	+1	–1	–1	–1	–1	1	y5
6		+1	+1	–1	+1	–1	1	–1	y6
7		+1	+1	+1	–1	1	–1	–1	y7
8		+1	+1	+1	+1	1	1	1	y8

7)ɉɪɨɜɟɞɟɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ.

Ʉɚɠɞɵɣ ɨɩɵɬ ɛɵɥ ɩɪɨɜɟɞɟɧ ɞɜɚ ɪɚɡɚ (ɞɥɹ ɨɰɟɧɤɢ ɨɲɢɛɤɢ ɜɨɫɩɪɨɢɡɜɨ-

ɞɢɦɨɫɬɢ). ɉɨɥɭɱɟɧɵ ɫɥɟɞɭɸɳɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɞɚɧɧɵɟ:

ɇɨɦɟɪ ɨɩɵɬɚ		Ɏɚɤɬɨɪɵ				Ɉɬɤɥɢɤɢ
N	x1	x2	x3		y1		y2
1	–1	–1		–1		3,40	4,10
2	–1	+1		+1		2,35	3,15
3	–1	+1		–1		–0,40	–0,60
4	–1	–1		+1		2,70	1,80
5	+1	–1		–1		2,20	3,30
6	+1	–1		+1		0,60	0,90
7	+1	+1		–1		–0,84	–1,16
8	+1	+1		+1		0,60	0,40

8)ɋɬɚɬɢɫɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɪɟɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚ.

¾ɉɪɨɜɟɪɤɚ ɜɨɫɩɪɨɢɡɜɨɞɢɦɨɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨ ɤɪɢɬɟɪɢɸ Ʉɨɯɪɟɧɚ.

ɉɟɪɟɞ ɩɪɨɜɟɞɟɧɢɟɦ ɞɢɫɩɟɪɫɢɨɧɧɨɝɨ ɚɧɚɥɢɡɚ ɧɟɨɛɯɨɞɢɦɨ ɭɛɟɞɢɬɶɫɹ ɜ ɨɞɧɨɪɨɞɧɨɫɬɢ ɞɢɫɩɟɪɫɢɣ. ȼ ɫɥɭɱɚɟ ɨɞɢɧɚɤɨɜɨɝɨ ɱɢɫɥɚ ɩɨɜɬɨɪɧɵɯ

ɨɩɵɬɨɜ ɞɥɹ ɨɰɟɧɤɢ ɨɞɧɨɪɨɞɧɨɫɬɢ ɞɢɫɩɟɪɫɢɢ ɩɪɢɦɟɧɹɟɬɫɹ ɤɪɢɬɟɪɢɣ

Ʉɨɯɪɟɧɚ (ɍ):

ɍɷɤɫɩ k max ,

¦Si2

i 1

ɝɞɟ ɍɷɤɫɩ. – ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ Ʉɨɯɪɟɧɚ;

Smax2 – ɦɚɤɫɢɦɚɥɶɧɚɹ ɞɢɫɩɟɪɫɢɹ;

¦Si2 – ɫɭɦɦɚ ɜɫɟɯ ɞɢɫɩɟɪɫɢɣ.

i 1

Ɇɚɤɫɢɦɚɥɶɧɚɹ ɞɢɫɩɟɪɫɢɹ Smax2 = 0,3025

Ʉɪɢɬɟɪɢɣ Ʉɨɯɪɟɧɚ: ɍɷɤɫɩ.= 0,3025/0,8556 = 0,35355

Ɉɩɪɟɞɟɥɢɬɶ ɍɬɚɛɥ. (ɬɚɛɥɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ Ʉɨɯɪɟɧɚ). Ɍɚɛɥɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɛɟɪɟɬɫɹ ɞɥɹ r = 2 (ɱɢɫɥɨ ɢɡɦɟɪɟɧɢɣ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ), N = 8 (ɱɢɫɥɨ ɨɩɵɬɨɜ) ɢ ɜɵɛɪɚɧɧɨɝɨ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ D = 0,05 (5%).

Ʉɪɢɬɟɪɢɣ Ʉɨɯɪɟɧɚ ɍɬɚɛɥ			ɍ(D;r 1; N)		ɍ(0,05;1;8) =0,679 (ɫɦ. ɉɪɢ-
ɥɨɠɟɧɢɟ 1).

	N	y1	y2	yɫɪ	Ⱦɢɫɩɟɪɫɢɹ S2
	1	3,40	4,10	3,75	0,1225
	2	2,35	3,15	2,75	0,16
	3	–0,40	–0,60	–0,50	0,01
	4	2,70	1,80	2,25	0,2025
	5	2,20	3,30	2,75	0,3025
	6	0,60	0,90	0,75	0,0225
	7	–0,84	–1,16	–1,00	0,0256
	8	0,60	0,40	0,50	0,01
	ɋɭɦɦɚ				0,8556

ɋɪɚɜɧɢɬɶ ɍɬɚɛɥ. ɢ ɍɷɤɫɩ. ɢ ɫɞɟɥɚɬɶ ɜɵɜɨɞ. ȿɫɥɢ ɍɷɤɫɩ. ɍɬɚɛɥ. , ɬɨ ɷɤɫɩɟɪɢ-

ɦɟɧɬ ɜɨɫɩɪɨɢɡɜɨɞɢɦ (ɝɢɩɨɬɟɡɚ ɨɛ ɨɞɧɨɪɨɞɧɨɫɬɢ ɞɢɫɩɟɪɫɢɣ ɩɨɞɬɜɟɪɠɞɚ-

ɟɬɫɹ) ɢ ɦɨɠɧɨ ɩɪɢɫɬɭɩɚɬɶ ɤ ɞɢɫɩɟɪɫɢɨɧɧɨɦɭ ɚɧɚɥɢɡɭ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯɞɚɧɧɵɯ.

Ⱦɢɫɩɟɪɫɢɹ ɜɨɫɩɪɨɢɡɜɨɞɢɦɨɫɬɢ ɪɚɜɧɚ ɫɪɟɞɧɟɦɭ ɡɧɚɱɟɧɢɸ ɞɢɫɩɟɪɫɢɣ:

S2V = ɋɊɁɇȺɑ(S2i) = 0,107

¾ȼɵɱɢɫɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɪɚɜɧɟɧɢɹ ɥɢɧɟɣɧɨɣ ɪɟɝɪɟɫɫɢɢ (ɦɚɬɟɦɚ-

ɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ).

					N						N
					¦xi yicp						¦xi xk yicp
		bi			i 1			ɢ bik			i 1		:
		bi			N			ɢ bik					:
					N							N

N	yɫɪ	x0 yɫɪ		x1 yɫɪ			x2 yɫɪ			x3 yɫɪ		x1 x2 yɫɪ		x1 x3 yɫɪ	x2 x3 yɫɪ
1	3,75	3,75			–3,75			–3,75		–3,75		3,75		3,75	3,75
2	2,75	2,75			–2,75		2,75			2,75		–2,75		–2,75	2,75
3	–0,50	–0,50		0,50				–0,50		0,50		0,50		–0,50	0,50
4	2,25	2,25			–2,25			–2,25		2,25		2,25		–2,25	–2,25
5	2,75	2,75		2,75				–2,75		–2,75		–2,75		–2,75	2,75
6	0,75	0,75		0,75				–0,75		0,75		–0,75		0,75	–0,75
7	–1,00	–1,00			–1,00			–1,00		1,00		–1,00		1,00	1,00
8	0,50	0,50		0,50			0,50			0,50		0,50		0,50	0,50
		b0 = 1,406	(ɫɪɟɞɧɟɟ			ɡɧɚɱɟɧɢɟ			x0 yɫɪ –			ɮɭɧɤɰɢɹ		ɋɊɁɇȺɑ)

b1 = –0,656

b2 = –0,969

b3 = 0,156

b12 = –0,031

b13 = –0,281

b23 =1,031

¾ɉɪɨɜɟɪɤɚ ɡɧɚɱɢɦɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨ ɤɪɢɬɟɪɢɸ ɋɬɴɸɞɟɧɬɚ.

Ʉɪɢɬɟɪɢɣ ɋɬɶɸɞɟɧɬɚ ɞɥɹ ɭɪɨɜɧɹ ɡɧɚɱɢɦɨɫɬɢ D = 0,05 ɢ ɱɢɫɥɚ ɨɩɵɬɨɜ

N = 8 ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ ɋɌɖɘȾɊȺɋɉɈȻɊ. Ʉɨɷɮ-

ɮɢɰɢɟɧɬ ɋɬɴɸɞɟɧɬɚ t(D,N) = 2,306006.

ɂɧɬɟɪɜɚɥ ɡɧɚɱɢɦɨɫɬɢ ' t(D; N ) (S2V / N )1/ 2 = 0,26663.

ȼɚɠɧɨ! ȼɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɛɨɥɶɲɢɟ ɩɨ ɦɨɞɭɥɸ, ɱɟɦ ', ɡɧɚɱɢɦɵ. Ɍɚ-

ɤɢɦ ɨɛɪɚɡɨɦ, ɤɨɷɮɮɢɰɢɟɧɬɵ b0, b1, b2, b13, b23 – ɡɧɚɱɢɦɵ; ɨɬɛɪɚɫɵɜɚɟɦ ɤɨɷɮɮɢɰɢɟɧɬɵ: b3, b12.

¾ɍɪɚɜɧɟɧɢɟ ɪɟɝɪɟɫɫɢɢ:

yb0 x0 b1 x1 b2 x2 b13 x1x3 b23 x2 x3

¾ɉɪɨɜɟɪɤɚ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ ɩɨ ɤɪɢɬɟɪɢɸ Ɏɢɲɟɪɚ.

Ⱦɢɫɩɟɪɫɢɸ ɚɞɟɤɜɚɬɧɨɫɬɢ S2A ɧɚɯɨɞɢɦ ɩɨ ɮɨɪɦɭɥɟ:

S	2	A	¦( yicp yit )2
			N m	,

ɝɞɟ yiɫɪ – ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɭ ɜ i-ɬɨɦ ɨɩɵɬɟ; 99

yit – ɬɟɨɪɟɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ ɨɬɤɥɢɤɚ ɜ i-ɬɨɦ ɨɩɵɬɟ, ɜɵɱɢɫ-

ɥɟɧɧɨɟ ɩɨ ɭɪɚɜɧɟɧɢɸ ɪɟɝɪɟɫɫɢɢ;

N – ɱɢɫɥɨ ɨɩɵɬɨɜ (N = 8);

m – ɱɢɫɥɨ ɡɧɚɱɢɦɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ ɭɪɚɜɧɟɧɢɢ ɪɟɝɪɟɫɫɢɢ (m = 5).

	N	yiɫɪ	yit	(yiɫɪ – yit)2
	1	3,750	3,781	0,001
	2	2,750	2,406	0,118
	3	–0,500	–0,219	0,079
	4	2,250	2,281	0,001
	5	2,750	3,031	0,079
	6	0,750	0,406	0,118
	7	–1,000	–0,969	0,001
	8	0,500	0,531	0,001
	ɋɭɦɦɚ			0,398
Ⱦɢɫɩɟɪɫɢɹ ɚɞɟɤɜɚɬɧɨɫɬɢ S2A = 0,133
Ʉɪɢɬɟɪɢɣ Ɏɢɲɟɪɚ Fɷɤɫɩ		S2 A S2V = 1,241819

Ɍɚɛɥɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ Ɏɢɲɟɪɚ ɨɩɪɟɞɟɥɹɟɦ ɞɥɹ ɭɪɨɜɧɹ ɡɧɚɱɢ-

ɦɨɫɬɢ D = 0,05 ɫ ɩɨɦɨɳɶɸ ɮɭɧɤɰɢɢ FɊȺɋɉɈȻɊ:

Fɬɚɛɥ F(D; N m; N ) F(0,05;3;8) = 4,06618.

ȼɚɠɧɨ! ȿɫɥɢ Fɷɤɫɩ Fɬɚɛɥ, ɬɨ ɝɢɩɨɬɟɡɚ ɨɛ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ ɧɟ ɨɬ-

ɜɟɪɝɚɟɬɫɹ.

9)ɂɧɬɟɪɩɪɟɬɚɰɢɹ ɢ ɪɟɤɨɦɟɧɞɚɰɢɢ.

ɇɟɨɛɯɨɞɢɦɨ ɭɫɬɚɧɨɜɢɬɶ, ɜ ɤɚɤɨɣ ɦɟɪɟ ɤɚɠɞɵɣ ɢɡ ɮɚɤɬɨɪɨɜ ɜɥɢɹɟɬ ɧɚ ɩɚ-

ɪɚɦɟɬɪ ɨɩɬɢɦɢɡɚɰɢɢ. ȼɟɥɢɱɢɧɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɟɝɪɟɫɫɢɢ – ɤɨɥɢɱɟɫɬɜɟɧ-

ɧɚɹ ɦɟɪɚ ɷɬɨɝɨ ɜɥɢɹɧɢɹ. ɑɟɦ ɛɨɥɶɲɟ ɤɨɷɮɮɢɰɢɟɧɬ, ɬɟɦ ɫɢɥɶɧɟɟ ɜɥɢɹɟɬ ɮɚɤɬɨɪ. Ɉ ɯɚɪɚɤɬɟɪɟ ɜɥɢɹɧɢɹ ɮɚɤɬɨɪɨɜ ɝɨɜɨɪɹɬ ɡɧɚɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ.

Ɂɧɚɱɢɦɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ ɩɹɬɶ:		b0 = 1,406
b1	= –0,656	b13	= –0,281
b2	= –0,969	b23	= 1,031

¾ABS(b2) > ABS(b1) Ɇɨɞɭɥɶ ɤɨɷɮɮɢɰɢɟɧɬɚ b2 ɛɨɥɶɲɟ ɦɨɞɭɥɹ ɤɨɷɮ-

ɮɢɰɢɟɧɬɚ b1. ȼɪɟɦɹ ɩɪɨɬɟɤɚɧɢɹ ɩɪɨɰɟɫɫɚ ɜɥɢɹɟɬ ɧɚ ɜɵɯɨɞ ɩɪɨɞɭɤɬɚ ɫɭɳɟɫɬɜɟɧɧɟɟ, ɱɟɦ ɬɟɦɩɟɪɚɬɭɪɚ.

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