Regresie Muiltilineara

7/25/2019 Regresie Muiltilineara

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Regresie multilineara y(x1,x2,x3,x4) = Fm[(Va)T*Vx] = a + bx1 + cx2 + dx3 + hx4

In spatiul cu 5 dimensiuni : Fm este o functie de variabila vectoriala cu valori reale, astfel :

Fm a b c d h x1 x2 x3 x4 a b x1 c x2 d x3 h x4

x1t 1.11 1.25 1.49 1.61 1.82 2.05 2.27 2.44 2.61 2.89( ) Unde s-au notat cu : xit , i = 1,2,3,4 vectorii transpusi ai valorilormasurate.

x2t 1.25 1.39 1.85 1.93 2.35 2.72 3.08 3.74 3.93 4.26 ( )

x3t 2.36 2.64 2.87 2.99 3.65 4.44 4.86 5.69 5.88 6.33( )

a b x1 c x2 d x3 h x4 a b c d h( )

1

x1

x2

x3

x4

= VaT

Vx( )=

x4t 1.51 1.96 2.31 2.45 2.84 3.14 3.59 4.75 5.02 5.66 ( )

Yt 14.2 15.5 18.4 20.4 22.3 22.9 26.7 35.2 44.4 53.9( )

x1t T

0

0

1

2

3

4

5

6

7

8

9

1.11

1.25

1.49

1.61

1.82

2.05

2.27

2.44

2.61

2.89

x2tT

0

0

1

2

3

4

5

67

8

9

1.25

1.39

1.85

1.93

2.35

2.72

3.083.74

3.93

4.26

x3tT

0

0

1

2

3

4

5

67

8

9

2.36

2.64

2.87

2.99

3.65

4.44

4.865.69

5.88

6.33

x4tT

0

0

1

2

3

4

5

67

8

9

1.51

1.96

2.31

2.45

2.84

3.14

3.594.75

5.02

5.66

Se defineste vectorul: U 1 1 1 1 1 1 1 1 1 1( )


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Necunoscute : a, b, c, d, h

S a b c d h( )0

n

i

Fi yi 2

=

n 5 Fi yi diferenta pentru fiecare punct=

Fm xt

T

1 YtT

1 Fm xtT

2 YtT

2. . .

with :a

Sd

d0=

bS

d

d0=

cS

d

d0=

dS

d

d0=

hS

d

d0= a, b, c, d, h

S a b c d h( )

0

9

j

Fm xitT

.j Yt T

j

2

=

|

|

|

|

|

|

|

|

|

|

|

|

|

|

aS

d

d0=

Sistem de 5 ecuatii cu 5 necunoscute :

Derivatele partiale ale Sumei patratelordistantelor de la valorile masurate la

planul in cinci dimensiuni Fm(Va,Vx),

Unde :

Vx = (xt1 xt2 xt3 xt4)

Va = (a b c d h) = Vectorul necunoscut ce

trebuie determinat

bS

d

d0=

cS

d

d0= 2( )

dS

d

d0=

hS

d

d0=

S a b c d h( )

0

9

i

a b x1tT

i c x2t

T

i

d x3tT

i

h x4tT

i

YtT i 2

U 0 1 2 3 4 5 6 7 8 9

0 1 1 1 1 1 1 1 1 1 1


3/6

aS

d

d2

0

9

i

a b x1tT

i c x2t

T

i

d x3tT

i

h x4tT

i

YtT

i

= 0=

0

9

i

YtT

i

U YtT

= v0 U Yt T

bS

d

d2

0

9

i

a b x1tT

i c x2t

T

i

d x3tT

i

h x4tT

i

YtT

i

x1tT

i

= 0=

0

9

i

YtT

ix1t

T

i

Yt x1tT

= v1 Yt x1tT

cS

d

d2

0

9

i

a b x1tT

i c x2t

T

i

d x3tT

i

h x4tT

i

YtT

i

x2tT

i

= 0=

0

9

i

YtT

ix2t

T

i

Yt x2tT

=

v2 Yt x2tT

dS

d

d2

0

9

i

a b x1tT

i c x2t

T

i

d x3tT

i

h x4tT

i

YtT

i

x3tT

i

= 0=

0

9

i

YtT

ix3t

T

i

Yt x3tT

= v3 Yt x3tT

hS

d

d2

0

9

i

a b x1tT

i c x2t

T

i

d x3tT

i

h x4tT

i

YtT

i

x4tT

i

= 0=

0

9

i

YtT

ix4t

T

i

Yt x4tT

= v4 Yt x4tT

0

9

i

1

10

0

9

i

x1tT

i

U xtT

=

0

9

i

x2tT

i

U x2tT

=


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Definesc :Elementele matricei M[5 x 5]

M1 10 M2 U x1tT

M3 U x2tT

M4 U x3tT

M5 U x4tT

M2 U x1t

T N1 x

1tx

1t

T

N2 x1t x2tT

N3 x1t x3tT

N4 x1t x4tT

M3 U x2tT

N2 x1tx2tT

P1 x2t x2tT

P2 x3tx2tT

P3 x4t x2tT

M4 U x3tT

P2 x2t x3tT

Q1 x3t x3tT

Q2 x4t x3tT

N3 x1t x3t

T

Q2 x3t x4tT

R1 x4t x4tT

M5 U x4tT

N4 x1t x4tT

P3 x2t x4tT

M

M1

M2

M3

M4

M5

M2

N1

N2

N3

N4

M3

N2

P1

P2

P3

M4

N3

P2

Q1

Q2

M5

N4

P3

Q2

R1

M

10

19.54

26.5

41.71

33.23

19.54

41.392

57.528

89.311

72.317

26.5

57.528

80.629

124.678

101.491

41.71

89.311

124.678

193.391

156.821

33.23

72.317

101.491

156.821

128.072

V1

v0

v1

v2

v3

v4

V1

273.9

601.08

845.39

1.304 103

1.071 103

Determinantul Matricei M este notat : |M| M 0.192 M1

10.205

15.201

15.671

3.796

1.834

15.201

30.073

22.887

1.39

3.399

15.671

22.887

36.151

9.657

7.966

3.796

1.39

9.657

5.647

0.939

1.834

3.399

7.966

0.939

3.727


5/6

Fm a b c d h x1 x2 x3 x4 a b x1 c x2 d x3 h x4

M1

V1

6.594

13.995

19.901

2.283

20.733

a 6.594 b 13.995 c 19.901 d 2.283 h 20.733

Rezulta Fs :

Fs x1tx2t x3t x4t 6.594 13.995x1t 19.901 x2t 2.283 x3t 20.733 x4t

Fs x1tT

x2tT

x3tT

x4tT

Reprezinta valorile functiei calculate in punctele initiale masurate

Fs x1tT

x2tT

x3tT

x4tT

YtT

Reprezinta valorile erorilor, ca diferenta dintre valorile calculate si cele masurate

Fs x1tT

x2tT

x3tT

x4tT

0

0

1

2

3

4

5

6

7

8

9

9.983

17.847

18.783

21.499

22.658

22.93

27.216

38.616

42.378

51.971

Yt( )T

0

0

1

2

3

4

5

6

7

8

9

14.2

15.5

18.4

20.4

22.3

22.9

26.7

35.2

44.4

53.9

Fs x1tT

x2tT

x3tT

x4tT

Yt( )T

0

0

1

2

3

4

5

6

7

8

9

-4.217

2.347

0.383

1.099

0.358

0.03

0.516

3.416

-2.022

-1.929


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Calculul si graficul erorilor : epsilon = |Fs[(Vx)T] - (Yt)T|

0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9

5

10

15

20

25

5

10

15

20

25

Fs x1tT

x2tT x3t

T x4tT i Yt

T i

i i

m

0

9

i

Fs x1tT

x2tT

x3tT

x4tT

i YtT

i

10

m 1.632

p0

9

i

Fs x1tT

x2tT

x3tT

x4tT

i YtT

i

2

10 9

p 2.165 103

p 0.68 1.472 103

1 m 1 1.633

2 m 2 1.63

1.630 1.633

Fm1 x y a b c( ) a bx

cy

Fm3 x y a b c d( )a

xb ln x( )

c

y d ln y( )

Fm2 x y a b c( ) a xb cy Fm4 x y a b c( ) a sin x( )

b cos y( )c

Fm02 x y z a b c d( ) a xb

yc

zd

Fm5 x y a b c( ) a sin x( )b

sin y( )c

Fm03 x y z u a b c d e( ) a xb

yc

zd

ue

Fm6x y a b c d( ) a b x c y d x y

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