PH24010

MathCADMore Curve Fitting

Previously on PH24010

• Linear Fitting• slope(), intercept()• line()

• Pre-process Y-data:– y = k/x– y = ekx

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

Y dataY Error

Fitted

What when can’t pre-process ?

• eg.

• y = a + bx + cx2

• Current through light bulb

• R changes with heat

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

Current

Voltage

linfit()

• Fitting routine for ‘linear’ combinations of functions.

• eg: y = A + B ln(x) + C ex + D x3

Find A,B,C & D to give best fit to data.

• Needs ‘vector of functions’

A vector of functions

• Define function of variable x

• Create vector with n rows & 1 column

• Fill placeholders with expressions involving x

vFunc x( )

1

ln x( )

ex

x3

vFunc 1( )

1

0

2.718

1

vFunc 2( )

1

0.693

7.389

8

Lightbulb Example

• Quadratic

• A + B x + C x2

• Call linfit() to get coeffients

Quadrat x( )

1

x

x2

Coeffs linfitVoltage Current Quadrat( )

Coeffs

0.118

3.31

131.11

Create model from linfit() results

• Explicitly put Coeffs into model

• better to use vector maths…(dot product)

BulbModel x( ) Coeffs0 x Coeffs1 x2

Coeffs2

BulbModel x( ) Quadrat x( ) Coeffs

Compare model with data

0 0.1 0.20

5

10

15

Current

BulbModelv( )

Voltage v

Peak fitting

• Applications in spectroscopy

• Gaussian peak

• need to find:– position– amplitude– background– width

Gaussian Peak Fit

10 5 0 5 105

0

5

10

GaussY

GaussX

genfit()

• Generalised fit of any function

• Need:– model function– partial derivatives of model wrt parameters– vector of initial guesses for each parameter

Gaussian Peak Function

• Where:– y0 is the background level– m is the x value of the peak centre– w is the width of the peak at half amplitude– A is the amplitude of the peak maximum

y x( ) y0 A e

x 2

2w2

Function parameters

• Re-write as P0, P1, P2 …

y x( ) P0 P1 e

x P2( )2

2P32

P0 is the background level (y0)P1 is the amplitude of the peak maximum (A) P2 is the x value of the peak centre (m)P3 is the width of the peak at half amplitude (w)

Form Partial Derivatives

• Use symbolic differentiation

P0P0 P1 e

x P2( )2

2P32

d

d1

P1P0 P1 e

x P2( )2

2P32

d

de

12

x P2( )2

P32

Create Function Vector for genfit()

• Function takes 2 parameters:– Independent variable, x– Parameter Vector, P

• Re-write P0,P1,P2 etc to use vector subscripts P0, P1, P2

Vector function for Gaussian fit

• Function to fit

• dF/dP0

• dF/dP1

• dF/dP2

• dF/dP3

GaussFit x P( )

P0 P1 e

x P2 2

2 P3 2

1

e

12

x P2 2

P3 2

P1x P2

P3 2e

12

x P2 2

P3 2

P1x P2 2

P3 3e

12

x P2 2

P3 2

Guess Values for Parameters

• By inspection of graphThe other thing that the fitting routine needs is a vector of inital guesses for the parameters,P0 -- P3.

10 5 0 5 105

0

5

10

GaussY

GaussX

P0 is the background level (y0) = 2 P1 is the amplitude of the peak maximum (A) = 6 P2 is the x value of the peak centre () = -3 P3 is the width of the peak at half amplitude (w) = 2

GuessP

2

6

3

2

Call genfit()

• Form model as before from coeffients & fit function

GaussCoeffs genfit GaussX GaussY GuessP GaussFit( )

GaussModel x( ) GaussFit x GaussCoeffs( )0

Use vector subscript to extract correct function from vector function

Plot model with data

10 5 0 5 105

0

5

10

DataGaussian Fit

y0

y0 A