PH24010
MathCAD
More Curve Fitting
Previously on PH24010
• Linear Fitting
• slope(), intercept()
• line()
3
2.7
2.4
2.1
1.8
1.5
1.2
• Pre-process Y-data:
– y = k/x
– y = ekx
0.9
0.6
0.3
0
0
0.5
1
Y data
Y Error
Fit ted
1.5
2
2.5
3
3.5
4
4.5
5
What when can’t pre-process ?
• eg.
• y = a + bx + cx2
10
8
6
• Current through
light bulb
Current
4
2
0
• R changes with
heat
0
0.05
0.1
0.15
Voltage
0.2
0.25
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
 1 


ln
(
x
)


vFunc ( x)   x 
 e 
 3 
 x 
 1 


0

vFun c1
( ) 
 2.7 18


 1 
 1 


0.6
93

vFun c2
( ) 
 7.3 89


 8 
Lightbulb Example
• Quadratic
• A + B x + C x2
1
 
Quadrat( x)    x 
 2
x 
Coeffs   linfit( Voltage Current Quadrat)
• Call linfit() to get
coeffients
 0.118 
Coeffs   3.31 
 131.11 


Create model from linfit() results
• Explicitly put
Coeffs into model
• better to use
vector maths…
(dot product)
2
BulbModel( x)  Coeffs0  xCoeffs1  x Coeffs2
BulbModel ( x)   Quadrat( x) Coeffs
Compare model with data
15
10
Current
BulbModel( v)
5
0
0
0.1
0.2
Voltage v
Peak fitting
• Applications in spectroscopy
• Gaussian peak
• need to find:
– position
– amplitude
– background
– width
Gaussian Peak Fit
10
5
GaussY
0
5
10
5
0
GaussX
5
10
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
2


 x
y( x)  y0  Ae
2
2w
• 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
Function parameters
• Re-write as P0, P1, P2 …
 ( xP2)
y( x)  P 0  P 1e
2
2
2P3
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
d
dP0
d
dP1
2

 ( xP2) 


2

2P3 
P0  P1 e
1
2
2

 ( xP2) 
 1 ( xP2)



2
2
2

2P3 
P3
P0  P1 e
e
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
2


  xP 2


2

2  P 3

P

P

e
0
1




1


2


 1  xP 2



2
2
P


3


e


GaussFit ( x  P)  
2


 1  xP 2



x  P2 2  P 3 2
 P1 

e
2


P3



2

 1  xP 2 



2
2
2
x

P


P
2
 3 

P1 
e


3
P


3


• Function to fit
• dF/dP0
• dF/dP1
• dF/dP2
• dF/dP3
Guess Values for Parameters
• By inspection of graph
The other thing that the f itting routine needs is a v ector of inital gues ses f or the param eters ,
P0 -- P3.
10
5
GaussY
0
5
10
5
0
GaussX
5
10
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
2 
 
6 

Guess P
 3 
 
2 
Call genfit()
GaussCoeffs   genfit( GaussX GaussY  GuessP  GaussFit)
• Form model as before from coeffients & fit
function
GaussModel(x)  GaussFit(x  GaussCoeffs)0
Use vector subscript to extract correct function from vector function
Plot model with data
10

y0A
5
y0
0
5
10
5
Data
Gaussian Fit
0
5
10