PENCOCOKAN KURVA/DATA

(CURVE/DATA FITTING)

Three attempts to fit a “best” curve through five data points. (a) Least

squares regression, (b) linear interpolation, and (c) curvilinear

interpolation

Least-Squares RegressionCriteria for a “Best” Fit

Examples of some criteria for “best fit” that are inadequate for regression:

(a) minimizes the sum of the residuals, (b) minimizes the sum of the

absolute values of the residuals, and (c) minimizes the maximum error of

any individual point.

Pencocokan Data (Data Fitting)

Pencocokan Data ke Garis Lurus

Pencocokan Data ke Polinomial

y a bx

n

nxaxaxaxaay 3

3

2

210

m

l

ll xfaxy0

)()(

Contoh Pencocokan Data ke Garis Lurus

i xi yi si

1 1.0 1,5 0,3

2 2.0 1,7 0,2

3 3.0 3,6 0,2

4 4.0 4,3 0,1

5 5.0 5,8 0,2

6 6.0 6,1 0,3

7 7.0 6,4 0,1

8 8.0 7,4 0,1

9 9.0 9,8 0,3

akan dicari persamaan

garis lurus yang cocok

untuk pasangan data ini ii bxay ˆ

yang berarti mencari

koefisian a, sa ,b dansb

berdasarkan pasangan data

eksperimental

Data Eksperimental

Dengan asumsi bahwa pasangan data merupakan

sampel dari populasi yang berditribusi Normal (Gauss)

dengan kebolehjadian total n pasangan data adalah:

n

i

yy

i

i i

ii

eyP1

ˆ

2

12

2

1),(

s

ss

dan menggunakan asas kuadrat terkecil (least squares)

maka koefisien a, sa ,b dan sb ditentukan dengan persamaan:

i i

ii yy2

2 ˆ

s

minimum

02

a

0

2

b

ax y x y xi

i

i

i

i

i

i i

i

12

2 2 2 2 s s s s

bx y x y

i

i i

i

i

i

i

i

1 12 2 2 2 s s s s

2

21

i

ia

x

ss

2

11

i

bs

s

Persamaan Regresi Linear untuk ralat pengukuran tidak sama

12

2

2 2

2

s s si

i

i

i

i

x x

Contoh Pencocokan Data ke Garis Lurus

(ralat pengukuran sama)

i xi yi si

1 1.0 1,5 0,2

2 2.0 1,7 0,2

3 3.0 3,6 0,2

4 4.0 4,3 0,2

5 5.0 5,8 0,2

6 6.0 6,1 0,2

7 7.0 6,4 0,2

8 8.0 7,4 0,2

9 9.0 9,8 0,2

akan dicari persamaan

garis lurusyang cocok

untuk pasangan data ini ii bxay ˆ

yang berarti mencari

koefisian a, sa ,b dansb

berdasarkan pasangan data

eksperimental

Data Eksperimental

iiiii yxxyxa 21

iiii yxyxNb1

22 ii xxN

Persamaan Regresi Linear untuk ralat pengukuran sama

2

2

ia xs

s

2s

sN

b

2222 )(2

1)(

2

1iiii yy

Nbxay

Ns

s

GOODNESS OF FIT

2

2

2 )(1

ii

i

xyys

22

If the fitting function is a good approximation

to the parent function, the value of the

reduced chi-square should be approximately

unity

12

Estimasi Ralat untuk Fungsi Termodifikasi

Jika kita memodifikasi fungsi dalam pencocokan data, yaitu yi

dengan koefisien a, b dimodifiksai menjadi yi’=f(yi) dengan

koefisien a’, b’, maka :

a’=fa(a) b’=fb(b)

aa

aa

afss

d

)(d' bb

bb

bfss

d

)(d'

Linearization of Nonlinear Relationships

(a)The exponential equation, (b) the power equation, and (c) the saturation-

growth-rate equation. Parts (d), (e), and (f) are linearized versions of these

equations that result from simple transformations.

POLYNOMIAL REGRESSIONThe second-order polynomial or quadratic

The squares of the residuals is

take the derivative of Eq. respect to each of the unknown coefficients of the

polynomial, as in

POLYNOMIAL REGRESSION

These equations can be set equal to zero and rearranged to develop the

following set of normal equations:

MULTIPLE LINEAR REGRESSIONFor example, y might be a linear function of x1 and x2, as in

Graphical depiction of multiple linear regression where y is a linear

function of x1 and x2.

MULTIPLE LINEAR REGRESSION

Quantification of Error of Linear Regression

Regression data showing (a) the spread of the data around

the mean of the dependent variable and (b) the spread of the

data around the best-fit line. The reduction in the spread in

going from (a) to (b), as indicated by the bell-shaped curves

at the right, represents the improvement due to linear

regression.

Quantification of Error of Linear Regression

Quantification of Error of Linear Regression

r2 is called the coefficient of determination and r is the

correlation coefficient

Quantification of Error of Linear Regression

Quantification of Error of Linear Regression

Quantification of Error of Linear Regression

Quantification of Error of Linear Regression

Quantification of Error of Linear Regression

Quantification of Error of Linear Regression

(a) Results using linear regression to comparepredictions computed with the theoretical model [Eq.(1.10)] versus measured values. (b) Results usinglinear regression to compare predictions computed withthe empirical model [Eq. (E17.3.1)] versus measuredvalues

Latihan Curve Fitting Impedansi rangkaian RL dinyatakan oleh persamaan :

Suatu percobaan untuk mengukur R dan L telah dilakukan

menggunakan rangkaian RL. Frekuensi f divariasi kemudian Z

diukur, dan didapatkan data sebagai barikut:

Dengan menggunakan analisis regresi linear hitunglah L ±L dan

R ± R.

2222 4 LfRZ

No. f (Hz) Z (ohm) Z

1 120 7,4 0,2

2 160 8,4 0,1

3 190 9,1 0,2

4 200 9,6 0,2

5 230 10,3 0,1

6 240 10,5 0,2

7 270 11,4 0,1

8 290 11,9 0,1

9 300 12,2 0,1

LL s LL s

Latihan Curve Fitting Gunakan analisis regresi linear untuk mencocokkan

model eksponensial :

terhadap data eksperimen di bawah ini:

Hitunglah A ± sA dan B ± sB

BXAeY

X 0,05 0,4 0,8 1,2 1,6 2,0 2,4

Y 550 750 1000 1400 2000 2700 3750

Latihan Curve Fitting

PENCOCOKAN DATA KE POLINOMIAL

LEGENDRE

Data Distribusi Cacah Emisi Sinar Gamma

terhadap Sudut