Texnologik jarayonlar boshqaruvi va kompyuter tizimlari

2. KO'P FAKTORLI REGRESSION TAHLIL


2.1. Korrelyatsion matritsani hisoblash.
Biz quyidagi kirish va chiqish faktorlarni hisobga olgan holda otkazilgan tajriba ma’lumotlariga asosan, korrelyatsion regression tahlil usuludan foydalanib ushbu jarayon uchun matematik model tuzamiz va tuzilgan matematik modelni monandlikka tekshiramiz.



Jarayon uchun quyidagi faktorlar qaralayotgan bo’lsin.
X1 – qurilmaning ishlash vaqti (s);
X2 – qurilmaning quvvati (Vatt);
X3 – olxo'rini quritilshdan oldingi harorati _{ };
Y1 – olxo'rini quritilgandan keyingi harorati _{ };
Laboratoriya sharoitida olingan tajriba ma’lumotlari quyidagi 1.1 – jadvalda keltirilgan.
1.1 – jadval

№	Vaqt	Quvvat	Tem do	Tpos	№	Vaqt	Quvvat	Tem do	T pos
1	258	129	16	68	21	130	183	13	105
2	125	111	13	33	22	163	181	12	113
3	109	121	12	40	23	256	182	11	108
4	213	101	11	65	24	335	183	13	104
5	312	100	13	80	25	197	184	14	107
6	174	100	13	68	26	108	185	14	104
7	189	103	11	66	27	162	186	14	104
8	211	150	13	30	28	311	187	11	106
9	347	150	15	98	29	352	188	11	110
10	256	150	12	40	30	284	189	12	142
11	305	150	14	55	31	309	190	15	123
12	294	120	15	64	32	250	191	12	128
13	316	150	15	77	33	162	192	14	145
14	212	150	11	88	34	334	193	12	114
15	340	180	15	65	35	155	194	12	114
16	110	180	11	56	36	341	195	13	115
17	332	180	15	69	37	215	196	12	114
18	325	181	15	65	38	230	197	15	113
19	256	183	14	79	39	116	188	13	124
20	311	182	15	100	40	117	200	14	128

Ushbu jadvaldan foydalanib jarayon uchun matematik model tuzamiz. Faktorlarning o’zaro bogliqliklarini inobatga olgan holda, ko’plik regressiya tenglamasini quyidagi formulaga asosan tuzamiz _{ }. Matematik modelni tuzishdan oldin kiruvchi va chiquvchi faktorlar orasidagi munosabatni korrelyatsion tahlil orqali tekshiramiz. Buning uchun biz Excel dasturidagi Analiz dannix bo’limidan foydalanib korrelyatsion matritsani hosil qilamiz.

Korrelyatsion matritsaninig umumiy ko’rinishi

2-jadval

Korrelyatsion matritsa
	Y	X1	X2	X3
Y	1
X1	-0,12341	1
X2	0,385415	0,902889	1
X3	-0,57317	0,20946	-0,10297	1

1-rasm. Tajriba natijalarinig koorelyatsion maydoni

Hosil bolgan korrelyatsion matritsaga binoan faktorlar orasidagi bo’gliqliklarni korib chiqishimiz mumkun. Demak, _{ } orasidagi bogliqlik eng yuqori tog’ri boglanishni _{ } esa eng yuqori teskari bog’lanishni ko’rsatmoqda. Faktorlar orasida _{ } ning qolgan faktorlar bilan bog’liqligi judayam kichik miqdorni korsatmoqda.

Ko’lpik regressiya tenglamasini aniqlash uchun _{ } faktorlarning o’za’ro munosabatlarini inobatga olgan holda quyidagi modelni quyida funksiyalar yordamida quramiz va uni tahlil qilib ko’ramiz umuiy korinishga keltiramiz (2.1).
_{ 
 }





2.2. Regressiya tenglamasining koeffitsientlarini kichik kvadratlar usuli yordamida hisoblash.
Matematik model tuzish ikki hil usulda amalga oshiriladi birinchi teglama ko’rinishi chiziqli ikkinchisiniki esa ko’rsatkichli funksiya yordamida aniqlanadi (2.1). Demak, biz har ikkalla tenglamani tuzishni ko’rib chiqamiz.
Hosil bolgan tenlamani noma’lum koeffitsientlarni topish uchun kichik kvadratlar usuli yoki Excel dasturidagi mazsus bo’limlaridan foydalanamiz. Yuqoridagi shartga ko’ra olxo'rinining zararsizlantirilgandan keyingi harorati uchun matematik model (2.2) korinishga keltiramiz va noma’lum koeffitsientlarni topib birinchi chiziqli funksiyani hosil qilamiz. Quyidagi jadvalga muofiq regressiya tenglamasininig koeffitsientlari mos ravishda aniqlanadi. (2.2)

b3	b2	b1	b0
-2,88301	0,625777	-0,11304	56,71201
1,585472	0,185804	0,069966	46,10185
0,449478	26,37325	#Н/Д	#Н/Д
9,797486	36	#Н/Д	#Н/Д
20443,87	25039,73	#Н/Д	#Н/Д

Chiziqli tenglama hosil qilingandan so’ng jarayon uchun ko’rsatkichli tenglama tuzib ikkalasidan jarayonga mosini ajratib olamiz. Buning uchun quyidagi matematik amalni bajaramiz. Tenlamani har ikki tomonini logorifmlab regressiya tenglamasini koeffitsientlarini aniqlaymiz (2.3).

_{ } (2.3)

N					ln(y)	ln(x1)	ln(x2)	ln(x3)
1	77	247	139	17	4,343805422	5,509388	4,934474	2,833213
2	45	253	139	17	3,80666249	5,533389	4,934474	2,833213
3	41	241	139	17	3,713572067	5,484797	4,934474	2,833213
4	52	211	104	19	3,951243719	5,351858	4,644391	2,944439
5	63	213	108	19	4,143134726	5,361292	4,682131	2,944439
6	58	178	108	19	4,060443011	5,181784	4,682131	2,944439
7	60	198	108	16	4,094344562	5,288267	4,682131	2,772589
8	25	211	160	15	3,218875825	5,351858	5,075174	2,70805
9	30	350	160	25	3,401197382	5,857933	5,075174	3,218876
10	89	260	160	12	4,48863637	5,560682	5,075174	2,484907
11	91	249	160	12	4,510859507	5,517453	5,075174	2,484907
12	64	294	179	15	4,158883083	5,68358	5,187386	2,70805
13	78	316	179	15	4,356708827	5,755742	5,187386	2,70805
14	68	260	179	11	4,219507705	5,560682	5,187386	2,397895
15	65	260	171	13	4,17438727	5,560682	5,141664	2,564949
16	75	260	183	13	4,317488114	5,560682	5,209486	2,564949
17	69	332	183	13	4,234106505	5,805135	5,209486	2,564949
18	56	325	183	17	4,025351691	5,783825	5,209486	2,833213
19	88	277	183	14	4,477336814	5,624018	5,209486	2,639057
20	99	307	180	12	4,59511985	5,726848	5,192957	2,484907
21	100	125	180	12	4,605170186	4,828314	5,192957	2,484907
22	121	166	181	13	4,795790546	5,111988	5,198497	2,564949
23	108	288	191	11	4,682131227	5,66296	5,252273	2,397895
24	106	288	191	13	4,663439094	5,66296	5,252273	2,564949
25	107	288	191	14	4,672828834	5,66296	5,252273	2,639057
26	125	288	191	14	4,828313737	5,66296	5,252273	2,639057
27	114	162	186	11	4,736198448	5,087596	5,225747	2,397895
28	116	311	187	11	4,753590191	5,739793	5,231109	2,397895
29	120	360	188	15	4,787491743	5,886104	5,236442	2,70805
30	146	284	189	13	4,983606622	5,648974	5,241747	2,564949
31	150	309	190	15	5,010635294	5,733341	5,247024	2,70805
32	128	250	163	12	4,852030264	5,521461	5,09375	2,484907
33	160	165	163	14	5,075173815	5,105945	5,09375	2,639057
34	118	334	163	12	4,770684624	5,811141	5,09375	2,484907
35	114	231	163	11	4,736198448	5,442418	5,09375	2,397895
36	117	231	195	13	4,762173935	5,442418	5,273	2,564949
37	114	231	198	21	4,736198448	5,442418	5,288267	3,044522
38	125	231	198	12	4,828313737	5,442418	5,288267	2,484907
39	129	113	198	15	4,859812404	4,727388	5,288267	2,70805
40	133	121	201	11	4,890349128	4,795791	5,303305	2,397895

Quyidagi amallarni bajargandan so’ng yuqoridagi operatsiyani takrorlaymiz va ko’rsatkichli funksiyaning koeffitsientlarini topamiz.

b3	b2	b1	bo'	b0
-0,78307	0,930262	-0,26263	3,203241	24,61216
0,326342	0,361566	0,196275	2,41489
0,433481	0,342175	#Н/Д	#Н/Д
9,181985	36	#Н/Д	#Н/Д
3,225187	4,215019	#Н/Д	#Н/Д

Quyidagi amallarni bajarish orqali biz ushbu jarayon uchun ikkinchi tenglamani hosil qildik va u quyidagicha ko’rinishga ega

Keyingi bisqichda ko’p faktorli tajriba ma’lumotlari asosida olingan tenglamalarga asosan biz tajriba natijasi va matematik model o’rtasidsagi bog’liqliklarni aniqlashimiz mumkun bo’ladi.

N	T pos y				Ychiziqli	Ykorsat
1	77	247	139	17	66,76318	62,05919
2	45	253	139	17	66,08494	61,66924
3	41	241	139	17	67,44141	62,46129
4	52	211	104	19	43,16435	45,26389
5	63	213	108	19	45,44138	46,76525
6	58	178	108	19	49,39775	49,02272
7	60	198	108	16	55,78601	54,53753
8	25	211	160	15	89,73992	81,31733
9	30	350	160	25	45,19737	47,72425
10	89	260	160	12	92,85005	91,67534
11	91	249	160	12	94,09348	92,72205
12	64	294	179	15	92,24745	82,73371
13	78	316	179	15	89,7606	81,18053
14	68	260	179	11	107,6228	108,9377
15	65	260	171	13	96,85059	91,59964
16	75	260	183	13	104,3599	97,56513
17	69	332	183	13	96,22111	91,4983
18	56	325	183	17	85,48033	74,57818
19	88	277	183	14	99,55524	90,54572
20	99	307	180	12	100,0528	97,92315
21	100	125	180	12	120,6259	123,9851
22	121	166	181	13	113,734	108,6505
23	108	288	191	11	111,9671	112,6491
24	106	288	191	13	106,201	98,83606
25	107	288	191	14	103,318	93,26369
26	125	288	191	14	103,318	93,26369
27	114	162	186	11	123,0811	127,8306
28	116	311	187	11	106,8641	108,2464
29	120	360	188	15	90,41888	82,11118
30	146	284	189	13	105,4016	98,23311
31	150	309	190	15	97,43542	86,31796
32	128	250	163	12	95,85778	94,23905
33	160	165	163	14	99,70006	93,15355
34	118	334	163	12	86,3625	87,33552
35	114	231	163	11	100,8885	103
36	117	231	195	13	115,1474	106,7686
37	114	231	198	21	93,9606	74,39046
38	125	231	198	12	119,9077	115,3009
39	129	113	198	15	124,5973	116,8153
40	133	121	201	11	137,1023	148,3376

2-rasm. Tajriba natijasi va matematik model o’rtasidagi boglanish grafigi