PH24010

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# PH24010 - PowerPoint PPT Presentation

PH24010. MathCAD More Curve Fitting. Previously on PH24010. Linear Fitting slope(), intercept() line() Pre-process Y-data: y = k/x y = e kx. What when can’t pre-process ?. eg. y = a + bx + cx 2 Current through light bulb R changes with heat. linfit().

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Presentation Transcript

### PH24010

More Curve Fitting

Previously on PH24010
• Linear Fitting
• slope(), intercept()
• line()
• Pre-process Y-data:
• y = k/x
• y = ekx
What when can’t pre-process ?
• eg.
• y = a + bx + cx2
• Current through light bulb
• R changes with heat
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
Lightbulb Example
• A + B x + C x2
• Call linfit() to get coeffients
Create model from linfit() results
• Explicitly put Coeffs into model
• better to use vector maths…(dot product)
Peak fitting
• Applications in spectroscopy
• Gaussian peak
• need to find:
• position
• amplitude
• background
• width
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
Function parameters
• Re-write as P0, P1, P2 …

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
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
Guess Values for Parameters
• By inspection of graph

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 (m) = -3

P3 is the width of the peak at half amplitude (w) = 2

Call genfit()
• Form model as before from coeffients & fit function

Use vector subscript to extract correct function from vector function