Non Linear Regression                      

    Y = α + βX Y = a + b X + c X2   a, b , c X .

Simple Non-Linear Regression

    Y = a + b X a, b :

    :

a ( Y = a + b X) Y = aXb () Y = a bx () Y = a + LinX  Y = aXb LinY = Lina + b LinX   LinY = Y , Lina = a , LinX = X Y = a + bX a , b :

Y = a + b/X  Y = a + b(1/x)s s1/X X :

Y = a + b LinX LinX s1/X :

:

    10 (Y) X 250 .

1) Y = aX b        

2)  Y = a + b(1/x)s

3) Y = a + LinX

4) μy/x = C Dx

 

10 9 8 7 6 5 4 3 2 1
300 270 220 200 250 260 255 290 300 250
60 50 40 40 55 50 52 44 42 45

:                            2)                3)                4)

1)  Y = aX b :

LinY = Lin(aX b) = Lin a + Lin(X b) = Lin a + b LinX Then

LinY = Lin a + b LinX

   

LinX LinY LinY LinX LinX2 X2 (Y) (X)
 21.0183  3.8067  5.5215  11.0429 62500 45 250 1
 21.3189   3.7377  5.7038  11.4076 90000 42 300 2
 21.4559  3.7842 5.6699  11.3398 84100 44 290 3
 21.8949  3.9512 5.5413  11.0825 65025 52 255 4
 21.7535  3.9210 5.5607  11.1214 67600 50 260 5
 22.1263  4.0073 5.5215  11.0429 62500 55 250 6
 19.5449  3.6889 5.2983  10.5966 40000 40 200 7
 19.8964  3.6889 5.3936  10.7873 48400 40 220 8
 21.9012  3.9120 5.5984  11.1968 72900 50 270 9
 23.3533  4.0943 5.7038  11.4076 90000 60 300 10
214.2635 38.5832 55.5127 111.0254   478 2595 Total

    b , a

 

 

        10(214.2635) (55.5127)(38.5832)

b =

                10(111.0254)  (55.5127)2

 

 

        2142.635 2141.8576

b =

        1110.254  3081.6600

 

            0.7774

b =

        1971.406

 

b = 0.00039 ≈  0.0004

 

            38.5832 ( 0.0004)55.5127)

Lina =

                                10

 

             38.5832 + 0.0222

Lina =

                       10

 

Lina = 3.8605

 

a = e3.8605

 

a = 47.4891 ≈ 47.5

 

Y = 47.5X 0.0004     Or    LinY = 1.6767 0.0004LinX    (lin(47.5 = 1.6767)

250 :

Y = 47.5X 0.0004    

 

Y = 47.5(250) 0.0004

 

Y = 47.5 0.9978

 

Y = 47.3955  ≈ 47

 

Or

 

LinY = 1.6767 0.0004LinX

 

LinY = 1.6767 0.0004Lin(250)

 

LinY = 1.6767 0.0004 2.3979

 

LinY = 1.6767 0.0010

 

LinY = 1.6757

 

Y = 47.3915 ≈ 47

 


 

2)   :Y = a + b(1/x)s :

 

 

(1/X)2

(1/X)Y

1 / X

(Y) (X)
 0.000016  0.18000 0.00400 45 250 1
 0.000011  0.14000 0.00333 42 300 2
 0.000012  0.15172 0.00345 44 290 3
 0.000015  0.20392 0.00392 52 255 4
 0.000015  0.19231 0.00385 50 260 5
 0.000016  0.22000 0.00400 55 250 6
 0.000025  0.20000 0.00500 40 200 7
 0.000021  0.18182 0.00455 40 220 8
 0.000014  0.18519 0.00370 50 270 9
 0.000011  0.20000 0.00333 60 300 10
 0.000156  1.85496 0.03913 478 2595 Total

 

 

        10(1.85496) (0.03913)(478)

b =

           10(0.000156)  (0.03913)2

 

 

        18.5496 18.70414

b =

        0.00156  0.001531

 

 

        0.15454

b =

           0.00003

 

b = 5151.3 ≈ 5151

 

 

         478 ( 5151.3 0.03913)

a =

                            10

 

        478 + 201.57

a =

                10

 

a = 67.957 ≈ 68

 

250 :

 

Y = 67.957 5151.3(1/x)s

 

Y = 67.957 (5151.3 250)

 

Y = 67.957 20.605

 

Y = 47.352 ≈ 47.4


 

3)   :Y = a + b LinXs :

 

Y LinX

LinX LinX2 X2 (Y) (X)
 248.4657  5.5215  11.0429 62500 45 250 1
 239.5589  5.7038  11.4076 90000 42 300 2
 249.4748 5.6699  11.3398 84100 44 290 3
 288.1457 5.5413  11.0825 65025 52 255 4
 278.0341 5.5607  11.1214 67600 50 260 5
 303.6804 5.5215  11.0429 62500 55 250 6
 211.9327 5.2983  10.5966 40000 40 200 7
 215.7451 5.3936  10.7873 48400 40 220 8
 279.9211 5.5984  11.1968 72900 50 270 9
 342.2269 5.7038  11.4076 90000 60 300 10
2657.1853 55.5127 111.0254   478 2595 Total

 

 

        10(2657.1853) (55.5127)(478)

b =

         10(111.0254)  (55.5127)2

 

 

        26571.853 26535.0706

b =

         1110.254  3081.6599

 

 

           36.7824

b =

       1971.4059

 

b = 0.0187 ≈ 0.02

 

 

         478 ( 0.0187 55.5127)

a =

                            10

 

        478 + 1.0381

a =

                10

 

a = 47.9038 ≈ 47.9

 

250 :

 

Y = a + LinX

 

Y = 47.9 + ( 0.02)(Lin250)

 

Y = 47.9 0.11

 

Y = 47.8

 


4)  

  ( ) μy/x        μy/x = C Dx   C, D c, d μy/x Ŷ   :  Ŷ = c dx 10 :

 

Log y = Log c + X Log d

a = Log c , b = Log d :

Log y = a + b X

    a , b :

 

:

 

X Log y

Log y X2 (Y) (X)
 413.3031   1.6532 62500 45 250 1
 486.9748  1.6233 90000 42 300 2
 476.6013  1.6435 84100 44 290 3
 437.5809  1.7160 65025 52 255 4
 441.7322  1.6990 67600 50 260 5
 435.0907  1.7404 62500 55 250 6
 320.4120  1.6021 40000 40 200 7
 352.4532  1.6021 48400 40 220 8
 458.7219  1.6990 72900 50 270 9
 533.4454  1.7782 90000 60 300 10
4356.3154 16.7565 683025 478 2595 Total

 

:

        10(4356.3154) (2595)(16.7565)

b =

         10(683025)  (2595)2

 

 

        43563.154 43483.1175

b =

        6830250  6734025

 

 

           80.0365

b =

           96225

 

b = 0.000830.001 ,  b = Log d Then d = 1.0019 ≈ 1.002

 

 

         16.7565 (0.0008 2595)

a =

                            10

 

        16.7565 2.1539

a =

                   10

 

a = 1.4681 ≈ 1.5  , a = Log c Then c = 29.3833 ≈ 29.38

 

 Ŷ = c dx

 

 

 Ŷ = 29.3833(1.0019)X

250 :

 

 Ŷ = 29.3833(1.0019)250

 

 

 Ŷ = 47.2275 ≈ 47

250 :

Log y = a + bX

 

Log y = 1.4681 + 0.00083250

 

Log y = 1.4681 + 0.2075

 

Log y = 1.6756

 

Y = 47.3805 ≈ 47