Parabolik empirik bog’liqlik qurish (parabolik regressiya funksiyasi koeffisiyentlarini aniqlash)

3.1.2. Parabolik empirik bog’liqlik qurish (parabolik regressiya funksiyasi koeffisiyentlarini aniqlash)
Parabolik regressiya funksiyasi quyidagi ko’rinishga ega, ya’ni
Bu yerda - regressiya koeffisiyenlari.
Regressiya funksiyasining - koeffisiyenlarini aniqlaymiz. Ularni eng kichik kvadratlar usulidan foydalanib, - koeffisiyenlarni hisoblash uchun olingan tenglamalar sistemasining ildizlari sifatida topamiz. U quyidagi tenglamalar sistemasidir:
Yuqoridagilarga qo’shimcha ravishda yozamiz:
U holda quyidagi ko’rinishdagi tenglamalr sistemaga ega bo’lamiz:
Bu sistemani yechib quyidagi ildizlarga ega bo’lamiz:
B0=- 43,773; b1= 3,2395; b2= -0,0336
Demak, X1 va Y1 o’rtasidagi parabolik regression bo’gliqlik funksiyasi quyidagi ko’rinishda bo’ladi:
y = - 43,773+ 3,2395xi -0,0336xi2
3.2. X1 – kirish omili va Y2 – chiqish o’rtasidagi empirik bog’liqlik ifodasini topish
3.2.1. Chiziqli empirik bog’liqlik qurish (chiziqli regressiya funksiyasi koeffisiyentlarini aniqlash)
Hisoblashlarni yuqoridagi X1 va Y1 uchun bajarilgani kabi olib boramiz.
X1 va Y2 uchun quyidagi jadvalni tuzamiz (2.3-jadval):
2.3-jadval.
№
|
x1
|
y2
|
x1^2
|
x1y2
|
x1^3
|
x1^4
|
x1^2 Y2
|
1
|
44,50723
|
10,64869
|
1980,893
|
473,9437
|
88164,07392
|
3923938,6
|
21093,91904
|
2
|
43,61856
|
10,94456
|
1902,579
|
477,3859
|
82987,76071
|
3619806,8
|
20822,88586
|
3
|
43,41233
|
11,37678
|
1884,63
|
493,8924
|
81816,19144
|
3551831,4
|
21441,01752
|
4
|
45,74424
|
11,51029
|
2092,536
|
526,5294
|
95721,45243
|
4378705,2
|
24085,68735
|
5
|
44,3006
|
10,65859
|
1962,543
|
472,182
|
86941,83613
|
3851575,5
|
20917,94632
|
6
|
45,58919
|
11,27352
|
2078,374
|
513,9508
|
94751,37571
|
4319638,1
|
23430,59653
|
7
|
46,06464
|
10,96483
|
2121,951
|
505,0909
|
97746,88296
|
4502674,5
|
23266,82725
|
8
|
44,19028
|
10,88966
|
1952,781
|
481,2173
|
86293,94997
|
3813354,1
|
21265,12907
|
9
|
45,53594
|
11,9723
|
2073,522
|
545,17
|
94419,77764
|
4299493,5
|
24824,82729
|
10
|
45,8312
|
12,05893
|
2100,499
|
552,6752
|
96268,41633
|
4412097,5
|
25329,77039
|
11
|
45,05244
|
10,81117
|
2029,723
|
487,0698
|
91443,9569
|
4119773,6
|
21943,68349
|
12
|
47,64148
|
13,06317
|
2269,711
|
622,3488
|
108132,3778
|
5151586,6
|
29649,62016
|
summa
|
541,4881
|
136,1725
|
24449,74
|
6151,456
|
1104688,052
|
49944475
|
278071,9103
|
o'rtacha
|
45,12401
|
11,34771
|
2037,478
|
512,6213
|
92057,33767
|
4162039,6
|
23172,65919
|
Olingan natijalar asosida sistema quyidagi ko’rinishda bo’ladi:
Bu sistemani yechib quyidagi ildizlarga ega bo’lamiz:
B0=- 8,3102; b1=0,4356
Demak, X1 va Y2 o’rtasidagi chiziqli regression bo’gliqlik funksiyasi quyidagi ko’rinishda bo’ladi:
y = - 8,3102+ 0,4356xi
3.2.2. Parabolik empirik bog’liqlik qurish (parabolik regressiya funksiyasi koeffisiyentlarini aniqlash)
Olingan natijalar asosida sistema quyidagi ko’rinishda bo’ladi:
Bu sistemani yechib quyidagi ildizlarga ega bo’lamiz:
B0=358,4; b1= - 15,742; b2= 0,1783
Demak, X1 va Y2 uchun parabolik regression bo’gliqlik funksiyasi quyidagi ko’rinishda bo’ladi:
y i= 358,4 - 15,742xi+0,1783xi2
3.3. X2 – kirish omili va Y1 – chiqish o’rtasidagi empirik bog’liqlik ifodasini topish
3.3.1. Chiziqli empirik bog’liqlik qurish (chiziqli regressiya funksiyasi koeffisiyentlarini aniqlash)
Hisoblashlarni yuqoridagi X1 va Y1 uchun bajarilgani kabi olib boramiz.
X2 va Y1 uchun quyidagi jadvalni tuzamiz (2.4-jadval):
2.4-jadval.
№
|
x2
|
y1
|
x2^2
|
x2y1
|
x2^3
|
x2^4
|
x2^2 y1
|
1
|
2,731342
|
33,90532
|
7,460227
|
92,60701
|
20,37643
|
55,65498
|
252,9414
|
2
|
2,885333
|
33,58244
|
8,325145
|
96,8965
|
24,02081
|
69,30804
|
279,5787
|
3
|
2,746667
|
33,51405
|
7,54418
|
92,05194
|
20,72135
|
56,91465
|
252,836
|
4
|
2,795006
|
34,11232
|
7,812058
|
95,34413
|
21,83475
|
61,02825
|
266,4874
|
5
|
2,76048
|
33,66283
|
7,620249
|
92,92558
|
21,03554
|
58,06819
|
256,5192
|
6
|
2,617546
|
33,98334
|
6,851548
|
88,95296
|
17,93424
|
46,94371
|
232,8385
|
7
|
2,674556
|
34,26062
|
7,153248
|
91,63194
|
19,13176
|
51,16896
|
245,0747
|
8
|
2,973181
|
33,65661
|
8,839807
|
100,0672
|
26,28235
|
78,14219
|
297,518
|
9
|
2,886661
|
34,06488
|
8,332811
|
98,33375
|
24,054
|
69,43574
|
283,8562
|
10
|
2,686996
|
33,92456
|
7,219947
|
91,15515
|
19,39997
|
52,12764
|
244,9335
|
11
|
2,957826
|
33,84605
|
8,748736
|
100,1107
|
25,87724
|
76,54039
|
296,1101
|
12
|
2,89884
|
34,21423
|
8,403271
|
99,18156
|
24,35974
|
70,61497
|
287,5114
|
summa
|
33,61443
|
406,7272
|
94,31123
|
1139,258
|
265,0282
|
745,9477
|
3196,205
|
o'rtacha
|
2,801203
|
33,89394
|
7,859269
|
94,9382
|
22,08568
|
62,16231
|
266,3504
|
Olingan natijalar asosida sistema quyidagi ko’rinishda bo’ladi:
Bu sistemani yechib quyidagi ildizlarga ega bo’lamiz:
B0= -0,4456; b1=35,142;
Demak, X2 va Y1 o’rtasidagi chiziqli regression bo’gliqlik funksiyasi quyidagi ko’rinishda bo’ladi:
y = 35,142-0,4456xi
3.3.2. Parabolik empirik bog’liqlik qurish (parabolik regressiya funksiyasi koeffisiyentlarini aniqlash)
Olingan natijalar asosida sistema quyidagi ko’rinishda bo’ladi:
Bu sistemani yechib quyidagi ildizlarga ega bo’lamiz:
B0=47,083; b1= -8,9768; b2=1,5213
Demak, X2 va Y1 uchun parabolik regression bo’gliqlik funksiyasi quyidagi ko’rinishda bo’ladi:
y = 47,083-8,9768xi+1,5213xi2
3.4. X2 – kirish omili va Y2 – chiqish o’rtasidagi empirik bog’liqlik ifodasini topish
3.4.1. Chiziqli empirik bog’liqlik qurish (chiziqli regressiya funksiyasi koeffisiyentlarini aniqlash)
Hisoblashlarni yuqoridagi X1 va Y1 uchun bajarilgani kabi olib boramiz.
X2 va Y2 uchun quyidagi jadvalni tuzamiz (2.5-jadval):
2.5-jadval.
№
|
x2
|
y2
|
x2^2
|
x2y2
|
x2^3
|
x2^4
|
x2^2y2
|
1
|
2,731342
|
10,64869
|
7,460227
|
29,08521
|
20,37643
|
55,65498
|
79,44164
|
2
|
2,885333
|
10,94456
|
8,325145
|
31,57869
|
24,02081
|
69,30804
|
91,11503
|
3
|
2,746667
|
11,37678
|
7,54418
|
31,24822
|
20,72135
|
56,91465
|
85,82845
|
4
|
2,795006
|
11,51029
|
7,812058
|
32,17132
|
21,83475
|
61,02825
|
89,91904
|
5
|
2,76048
|
10,65859
|
7,620249
|
29,42283
|
21,03554
|
58,06819
|
81,22112
|
6
|
2,617546
|
11,27352
|
6,851548
|
29,50897
|
17,93424
|
46,94371
|
77,24109
|
7
|
2,674556
|
10,96483
|
7,153248
|
29,32605
|
19,13176
|
51,16896
|
78,43415
|
8
|
2,973181
|
10,88966
|
8,839807
|
32,37694
|
26,28235
|
78,14219
|
96,26252
|
9
|
2,886661
|
11,9723
|
8,332811
|
34,55997
|
24,054
|
69,43574
|
99,76291
|
10
|
2,686996
|
12,05893
|
7,219947
|
32,40229
|
19,39997
|
52,12764
|
87,06482
|
11
|
2,957826
|
10,81117
|
8,748736
|
31,97757
|
25,87724
|
76,54039
|
94,58411
|
12
|
2,89884
|
13,06317
|
8,403271
|
37,86804
|
24,35974
|
70,61497
|
109,7734
|
summa
|
33,61443
|
136,1725
|
94,31123
|
381,5261
|
265,0282
|
745,9477
|
1070,648
|
urtacha
|
2,801203
|
11,34771
|
7,859269
|
31,79384
|
22,08568
|
62,16231
|
89,22069
|
Olingan natijalar asosida sistema quyidagi ko’rinishda bo’ladi:
Bu sistemani yechib quyidagi ildizlarga ega bo’lamiz:
B0=9,8699; b1= 0,5276
Demak, X2 va Y2 o’rtasidagi chiziqli regression bo’gliqlik funksiyasi quyidagi ko’rinishda bo’ladi:
y = 9,8699+0,5276xi
|